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[
"BOUNDARIES OF GRAPHS OF RELATIVELY HYPERBOLIC GROUPS WITH CYCLIC EDGE GROUPS",
"BOUNDARIES OF GRAPHS OF RELATIVELY HYPERBOLIC GROUPS WITH CYCLIC EDGE GROUPS"
]
| [
"Ravi Tomar "
]
| []
| []
| We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups with dynamically quasi-convex edge groups (Theorem 1.3). To prove these results, we use a modification of Dahmani's technique[Dah03]. Then we show that the fundamental group of a graph of relatively hyperbolic groups with edge groups either parabolic or infinite cyclic is relatively hyperbolic and construct Bowditch boundary. Finally, we show that the homeomorphism type of Bowditch boundary of the fundamental group of a graph of relatively hyperbolic groups with parabolic edge groups is determined by the homeomorphism type of the Bowditch boundaries of vertex groups (under some additional hypotheses)(Theorem 7.1). In the last section of the paper, we give some applications and examples.2010 Mathematics Subject Classification. Primary 20F65,20F67. | 10.1007/s12044-022-00694-3 | [
"https://arxiv.org/pdf/2104.08843v2.pdf"
]
| 233,296,397 | 2104.08843 | 5d9b238adfa2b864c5002909cebf7142add2614a |
BOUNDARIES OF GRAPHS OF RELATIVELY HYPERBOLIC GROUPS WITH CYCLIC EDGE GROUPS
5 Feb 2022
Ravi Tomar
BOUNDARIES OF GRAPHS OF RELATIVELY HYPERBOLIC GROUPS WITH CYCLIC EDGE GROUPS
5 Feb 2022arXiv:2104.08843v2 [math.GR]
We prove that the fundamental group of a finite graph of convergence groups with parabolic edge groups is a convergence group. Using this result, under some mild assumptions, we prove a combination theorem for a graph of convergence groups with dynamically quasi-convex edge groups (Theorem 1.3). To prove these results, we use a modification of Dahmani's technique[Dah03]. Then we show that the fundamental group of a graph of relatively hyperbolic groups with edge groups either parabolic or infinite cyclic is relatively hyperbolic and construct Bowditch boundary. Finally, we show that the homeomorphism type of Bowditch boundary of the fundamental group of a graph of relatively hyperbolic groups with parabolic edge groups is determined by the homeomorphism type of the Bowditch boundaries of vertex groups (under some additional hypotheses)(Theorem 7.1). In the last section of the paper, we give some applications and examples.2010 Mathematics Subject Classification. Primary 20F65,20F67.
Introduction
In [BF92], Bestvina and Feighn proved a combination theorem for hyperbolic groups. Motivated by this, Mj and Reeves proved a combination theorem for relatively hyperbolic groups [MR08]. Using the characterization of relative hyperbolicity given by Yaman (see [Yam04]), Dahmani proved a combination theorem for acylindrical finite graph of relatively hyperbolic groups with fully quasi-convex edge groups [Dah03].
In Dahmani's paper [Dah03], Theorem 0.1(1) produces convergence groups only in the case of an acylindrical finite graph of relatively hyperbolic groups with fully quasi-convex edge groups (see the main Theorem of [Dah03]). From first part of this theorem, if we remove the hypothesis that vertex groups are relatively hyperbolic or edge groups are fully quasi-convex, it is unclear whether the fundamental group of a graph of groups is a convergence group. This motivates the following natural question:
Question 1.1. Let Γ be the fundamental group of a finite graph of convergence groups with dynamically quasi-convex edge groups. Under which condition(s) is Γ a convergence group?
We answer the above question in the cases: (a) when the edge groups are parabolic (b) when the edge groups are cyclic (c) when the collection of edge groups forms a dynamical malnormal family of dynamically quasi-convex subgroups in adjacent vertex groups (Definition 2.4(3),(4)). We have the following result for parabolic edge groups:
Theorem 1.2. Let Γ be the fundamental group of a finite graph of countable convergence groups with parabolic edge groups. Then Γ is a convergence group.
Note: Proof of parts (2),(3),(4) of Theorem 0.1 in [Dah03] also go through when convergence groups replace relatively hyperbolic groups, and rest of the hypotheses remain same. Therefore, with a little effort, proof of the above theorem also follows by combining parts (2),(3),(4) of Dahmani's theorem. In part (2),(3),(4) of Dahmani's theorem, the domain of edge boundary point (see Defintion 3.1) is infinite, but it is of star-like form. On the other hand, we allow domains to be infinite subtrees of the Bass-Serre tree. For example, let Γ = G 1 * P G 2 , where G 1 , G 2 are convergence groups and P is isomorphic to parabolic subgroups and not maximal parabolic in G 1 and G 2 , respectively. Suppose P 1 , P 2 are maximal parabolic containing isomorphic copy of P in G 1 , G 2 respectively. Then domain of edge parabolic point is the Bass-Serre tree of P 1 * P P 2 . Thus, we give a more direct proof by generalizing the technique of Dahmani [Dah03]. In fact, we explicitly construct a compact metrizable space (see Section 3) on which Γ acts as a convergence group. Also, we use this construction for producing Bowditch boundary in Theorem 1.6.
We state a combination theorem for convergence groups that also answers Question 1.1. Theorem 1.3. Let Γ be the fundamental group of a finite graph of countable convergence groups such that stabilizers of the limit sets of edge groups form a dynamically malnormal family of dynamically quasi-convex subgroups (Definition 2.4 (3),(4)) in adjacent vertex groups. Then Γ is a convergence group.
In the above theorem, by the definition of a dynamically malnormal family (Definition 2.4 (4)), we see that action of Γ on its Bass-Serre tree is 2-acylindrical. Thus Γ is the fundamental group of an acylindrical graph of convergence groups satisfying the hypotheses of the above theorem. When vertex groups are convergence groups and edge groups are fully dynamically quasi-convex in Theorem 0.1(1) of Dahmani [Dah03], it is different from the above theorem. In Theorem 0.1(1) of Dahmani, the limit sets of edge groups in adjacent vertex groups are homeomorphic, which is not the case here. There was no identification involved inside Bowditch boundaries of vertex groups. On the other hand, to prove the above theorem, we identify the translates of the limit set of edge groups in adjacent vertex groups with points, see Section 5. We have the following proposition that gives an answer to Question 1.1 when edge groups are infinite cyclic.
Proof. Groups with non-trivial Floyd boundary act as a convergence group on their Floyd boundary (see [Kar03]). Thus, the corollary follows from Proposition 5.3.
In the above corollary, if we take vertex groups to be relatively hyperbolic then Γ is relatively hyperbolic by Theorem 1.6, and hence, by [Ger12], it has non-trivial Floyd boundary. However, if vertex groups are not relatively hyperbolic, it is unclear whether Γ has a non-trivial Floyd boundary; see Questions 8.10 and 8.11. Now, we state a combination theorem for a graph of relatively hyperbolic groups with parabolic edge groups.
Theorem 1.6. Let Γ be the fundamental group of a finite graph of relatively hyperbolic groups whose edge groups are parabolic subgroups of the adjacent vertex groups. Then Γ is relatively hyperbolic.
We explain parabolic structure of Γ after proof of the above theorem in section 6. In the above theorem, if edge group is not maximal parabolic in adjacent vertex groups, it does not satisfy the hypotheses of [Dah03, Theorem 0.1]. However, proof of the above theorem still follows from parts (2),(3),(4) of Theorem 0.1 in [Dah03]. Here, we give a different proof by constructing a compact metrizable space on which Γ acts geometrically finitely. Therefore, we have an explicit construction of Bowditch boundary of Γ. Note that relatively hyperbolicity in Theorem 1.6 also follows from the work of Bigdely and Wise [BW13]. Since in our situation, parabolic edge groups can be infinitely generated, the first condition of the theorem of Mj-Reeves [MR08] is not satisfied. Thus, relative hyperbolicity does not follow from the theorem of Mj-Reeves. The following is a combination theorem for a graph of relatively hyperbolic groups with cyclic edge groups:
Theorem 1.7. Let Γ be the fundamental group of a finite graph of relatively hyperbolic groups with infinite cyclic edge groups. Then Γ is relatively hyperbolic.
To prove the above theorem, we use Theorem 1.6 (see Section 6 ). In particular, by extending the parabolic structure on vertex groups, we convert a graph of reltively hyperbolic groups with infinite cyclic edge groups into a graph of relative hyperbolic groups with parabolic edge groups. Thus, we can explicitly construct Bowditch boundary for the group Γ. Now, we state a theorem for the homeomorphism type of Bowditch boundary of the fundamental group of a graph of relatively hyperbolic groups with parabolic edge groups.
Theorem 1.8 (Theorem 7.1). Let Y be a finite connected graph and let G(Y ), G ′ (Y ) be two graph of groups satisfying the following:
(1) For each vertex v ∈ V (Y ), let (G v , P v ), (G ′ v , P ′ v ) be relatively hyperbolic groups.
(2) Let e ∈ E(Y ) be an edge with vertices v, w and let P e , P ′ e are parabolic edge groups in G(Y ), G ′ (Y ), respectively. Then either P e , P ′ e have infinite index in corresponding maximal parabolic subgroups in G v , G ′ v , respectively or P e , P ′ e have the same finite index in maximal parabolic subgroups in G v , G ′ v , respectively. Similarly, either P e , P ′ e have infinite index in maximal parabolic subgroups in G w , G ′ w , respectively or P e , P ′ e have the same finite index in corresponding maximal parabolic subgroups in G w , G ′ w , respectively.
(3) Let B v , B ′
v be the set of translate of parabolic points corresponding to adjacent edge groups under the action of G v , G ′ v on their Bowditch boundaries
respectively. For each vertex v ∈ V (Y ), suppose we have a homeomorphism from ∂G v → ∂G ′ v that maps B v onto B ′ v . Let Γ = π 1 (G(Y )), Γ ′ = π 1 (G ′ (Y )
). (By Theorem 1.6, the groups (Γ, P), (Γ ′ , P ′ ) are relatively hyperbolic) Then there exists a homeomorphism from ∂Γ to ∂Γ ′ preserving edge parabolic points, i.e. taking parabolic points corresponding to edge groups of G(Y ) to parabolic points corresponding to edge groups of G ′ (Y ).
Remark 1.9. The proof of Theorem 0.1 in [Dah03] works only for infinite edge groups. In particular, if the edge groups are finite then the space M constructed in [Dah03] need not be compact. In this paper, in all of the above theorems, we are also taking infinite edge groups. In [MJ15], Martin and Świątkowski constructed Gromov boundary for a graph of hyperbolic groups with finite edge groups, see also [Mar14]. If we take a graph of convergence groups with finite edge groups then by [Bow12], the fundamental group of graph of groups is relatively hyperbolic with respect to infinite vertex groups and hence it a convergence group.
A few words on the proofs: To prove Theorem 1.2, we generalize Dahmani's technique. Here, we explicitly construct a space on which the fundamental group of the graph of groups acts as a convergence group. In [Dah03], the main role of fully quasi-convex edge groups and acylindrical action was to get uniformly bounded domains (see Definition 3.1 ). In our situation, domains for points in edge spaces may be infinite (when edge groups are not maximal parabolic), therefore the space constructed by Dahmani does not work, See 3.6. Thus, we modify the space. For that, we look at the domain of a point ξ in an edge space and identify all the boundary points (in the visual boundary of Bass-Serre tree) of the domain of ξ with ξ itself. In our situation, domains for points other than edge spaces are singletons. We get a new set by going modulo the equivalence relation generated by this. Then we define a topology on this set and see that this is our candidate space. In Section 4, we prove Theorem 1.2 as Corollary 4.3. For proving Theorem 1.3, we use a result of Manning from [Man20]. Using this result, proof of this theorem boils down to the parabolic edge group case, and this is done by Theorem 1.2. Proof of Theorem 1.2 also gives proof of Theorem 1.6. In the proof of Theorem 1.6, for the construction of candidate space (see Section 3 ), we take Bowditch boundary over which vertex groups act geometrically finitely and the rest of the things remain the same.
For proving these theorems, it is sufficient to consider only the amalgam and the HNN extension case. For a general graph of groups, we may take a maximal tree in the graph over which we are taking a graph of groups. By proving the theorems for the amalgam, we are done for a graph of groups over a maximal tree. By adding the remaining edges one by one, we are in the HNN extension case. By proving the theorems in the HNN extension case, we are done.
Acknowledgement: I would like to thank my supervisor, Pranab Sardar, for many helpful discussions and comments on the exposition of the paper.
preliminaries
We collect here some necessary definitions from [Bow99], [Tuk94b], and [BM74] A group Γ is said to be a convergence group if there exists a compact metrizable space M and Γ acts on M such that: given any sequence (γ n ) n∈N in Γ, there exists a subsequence (γ σ(n) ) n∈N and two points ξ, η in M such that for all compact subset K of M \ {η}, γ σ(n) K converges uniformly to ξ.
Let Γ be a convergence group on a compact metrizable space M . An infinite order element γ ∈ Γ is said to be loxodromic if it has exactly two distinct fixed points in M . A point ξ ∈ M is said to be a conical limit point if there exists a sequence (γ n ) n∈N in Γ and two distinct points ζ, η such that γ n ξ converges to ζ and for all ξ ′ ∈ M \ {ξ}, γ n ξ ′ converges to η. A subgroup G of Γ is said to be parabolic if it is infinite, it does not have any loxodromic element and it fixes a point ξ. Such a point is unique and called a parabolic point. The stabilizer of a unique parabolic point is called maximal parabolic subgroup. A parabolic point ξ is called a bounded parabolic point if Stab Γ (ξ) acts co-compactly on M \ {ξ}. The group Γ is said to be geometrically finite if every point of M is either a conical limit point or a bounded parabolic point. Now, we define relatively hyperbolic groups. There are several equivalent definitions of relative hyperbolicity (see [Hru10]), but we use the definition given by Bowditch [Bow12, Definition 1].
We say that a group Γ is hyperbolic relative to a family G of finitely generated subgroups if Γ admits a properly discontinuous isometric action on a proper hyperbolic space X such that induced action of Γ on ∂X is convergence, every point of ∂X is either conical or bounded parabolic, and the subgroups in the family G are precisely the maximal parabolic subgroups.
In short, we say that the pair (Γ, G) is a relatively hyperbolic group. In this case, the boundary of X is canonical and called the Bowditch boundary of (Γ, G). We shall denote it by ∂Γ. We have the following topological characterization for relatively hyperbolic group given by Yaman [Yam04].
Theorem 2.2. [Yam04] Let Γ be a geometrically finite group acting on a non-empty perfect metrizable compactum. Assume that the quotient of bounded parabolic points is finite under the action of Γ and the corresponding maximal parabolic subgroups are finitely generated. Let G be the family of maximal parabolic subgroups. Then (Γ, G) is a relatively hyperbolic group and M is equivariantly homeomorphic to Bowditch boundary of Γ.
Remark 2.3. The assumption that maximal parabolic subgroups are finitely generated does not play any role in the proof of the above theorem, but it is there merely to satisfy the hypothesis in Bowditch's definition of a relatively hyperbolic group. Also by a result of Tukia [Tuk98, Theorem 1B], one can remove the assumption of finiteness of the set of orbits of bounded parabolic points.
Definition 2.4. We collect the following definitions:
(1) (Limit set of a subgroup [Dah03]) Let Γ be a convergence group on M . The limit set Λ(H) of an infinite subgroup H is the unique minimal non-empty closed H-invariant subset of M . The limit set of a finite set is empty.
i ) ∩ Λ(H j ) = ∅ unless i = j and g ∈ H i .
Since the limit set of a parabolic subgroup in a convergence group is a singleton, parabolic subgroups are dynamically quasi-convex. Also, by [GP16], relatively quasi-convex subgroups of a relatively hyperbolic group are dynamically quasiconvex. Now, we prove the following:
Lemma 2.5. Let M be a compact metrizable space. Suppose a group G acts on M as a convergence group. Then infinite cyclic subgroups of G are dynamically quasi-convex.
First of all, we record the following proposition:
Proposition 2.6. Let G be a group having a convergence action on a compact metrizable space M . Let H be a subgroup of G and ΛH be its limit set. Then H is dynamically quasi-convex if and only if for any sequence (g n ) in distinct left cosests of H in G there exists a subsequence (g σ(n) ) of (g n ) such that g σ(n) ΛH uniformly converges to a point.
For relatively hyperbolic groups, the above proposition also appears in [Dah03, Proposition 1.8]. We skip proof of the above proposition as it follows directly from the definition of dynamically quasi-convex subgroup. For the definition of hyperbolic space and Gromov boundary, one is referred to [BH99]. Before proving Lemma 2.5, we prove the following:
Lemma 2.7. Let X be a proper geodesic hyperbolic space and let G be a group acting by isometries on X. Suppose G acts as a convergence group on ∂X(∂X denotes Gromov boundary of X). Then, infinite cyclic subgroups of G are dynamically quasi-convex.
Proof. Let g ∈ G be an infinite order element. If g is a parabolic element for G ∂X. Then, clearly g is dynamically quasi-convex. Suppose Λ( g ) = {x, y}. Suppose g is not dynamically quasi-convex. Then, by the above proposition, there exists a sequence (g n ) ⊂ G \ g such that g n Λ( g ) converges to two distinct points ξ, η (say). WLOG, assume that g n x → ξ and g n y → η. Since the action of G on ∂X is convergence, there exists a subsequence (g n k ) of (g n ) and two points a, b ⊂ ∂X such that for all z ∈ ∂X \{b}, g n k z converges to a. Note that G also acts on X ∪∂X as a convergence group with the same attracting and repelling points. If x = b and y = b then both g n k x, g n k y converge to a. This implies ξ = η, a contradiction. Now, suppose x = b and y = b. Then g n k x → a and g n k y → η. Thus, a = ξ. Now, all the points, except y, on bi-infinite geodesic ray joining x and y converge to a under the action of g n k . Now, choose a point p on bi-infinite geodesic ray joining ξ and η close enough to η and consider a ball around p of radius R for some R > 0. Then this ball does not intersect the bi-infinite geodesic rays joining g n k x to g n k y for sufficiently large k. This is a contradiction as g n x and g n y converging to two different points in the boundary of a proper hyperbolic space. We get a similar contradiction when x = b and y = b. Hence g is dynamically quasi-convex for G ∂X.
Proof of Lemma 2.5: If G acts on M as an elementary convergence group, then every infinite cyclic subgroup is dynamically quasi-convex. Now, suppose G acts on M as a non-elementary convergence group. Let Q be the set of all distinct triples. Then, by [Bow98],[Sun19, Proposition 6.4], there is a G-invariant hyperbolic path quasi-metric on Q. Thus, we can define the boundary, ∂Q, of Q as in [Bow98]. By [Bow98,Proposition 4.7], ∂Q is G-equivariantly homeomorphic to M . Now ∂Q is compact as M is compact. Then, by [Bow98, Proposition 4.8], there is a locally finite hyperbolic path quasi-metric space Q ′ (quasi-isometric to Q) and M is G-equivariantly homeomorphic to ∂Q ′ . As Q ′ is locally finite path quasi-metric space, by [Bow98, Section 3], Q ′ is G-equivariantly quasi-isometric to a locally finite graph G r (Q ′ ) for some r ≥ 0. As Q ′ is a locally finite hyperbolic path quasimetric space, G r (Q ′ ) is a proper hyperbolic geodesic metric space. Also, ∂Q ′ is G-equivariantly homeomorphic to ∂G r (Q ′ ). Finally, by the above discussion, M is G-equivariantly homeomorphic to ∂G r (Q ′ ). Let φ be the homeomorphism induced by quasi-isometry from M to G r (Q ′ ). Since G acts on M as a convergence group, G also acts on ∂G r (Q ′ ) as a convergence group [Bow99]. Also, G acts as a convergence group on G r (Q ′ ) ∪ ∂G r (Q ′ ). Suppose g ∈ G such that order of g is infinite. If g is a parabolic element. Then, clearly g is dynamically quasi-convex for G M . Now, suppose g loxodromic for the action of G on M . Then, g is loxodromic for the action of G on ∂G r (Q ′ ). By the lemma 2.7, g is dynamically quasi-convex subgroup of G for G ∂G r (Q ′ ). Now, let if possible g is not dynamically quasiconvex for G M . Then there exist disjoint closed subsets K, L of M such that the set {g ∈ G \ g |gΛ( g ) ∩ K = ∅, gΛ( g ) ∩ L = ∅} is infinite. Since φ is a homeomorphism, φ(K), φ(L) are disjoint closed subsets of ∂G r (Q ′ ). Then, by G-equivariance of φ, g is not dynamically quasi-convex for G ∂G r (Q ′ ). This gives us a contradiction.
Construction of boundary of graphs of groups
As we mentioned in the introduction, it is sufficient to consider the amalgam and the HNN extension case for proving our theorems. This will be our standing assumption for the next two sections. Let Γ be an amalgamated free product or an HNN extension of convergence groups along with parabolic edge group. Let T be the Bass-Serre tree of this splitting and let τ be a subtree of T , an edge in case of amalgam, and a vertex in case of HNN extension. Here, we construct our candidate space M on which Γ acts as a convergence group.
We fix some notation: If v is a vertex of T , we write Γ v for its stabilizer in Γ. Similarly, for an edge e, we write Γ e for its stabilizer. For each vertex v and edge e incident on v, Γ v is a convergence group, and Γ e is a parabolic subgroup in Γ v . We denote X v and X e as compact metrizable spaces on which the vertex group Γ v and the edge group Γ e act as convergence groups. In our situation, X e is singleton.
Definition of M as a set.
Contribution of the vertices of T Let V(τ ) be the set of vertices of τ . For a vertex v ∈ V(τ ), the group Γ v is by assumption a convergence group. So, we have compactum (compact metrizable
space) X v for Γ v . Set Ω to be Γ × ( v∈V(τ ) X v ) divided by the natural relation (γ 1 , x 1 ) = (γ 2 , x 2 ) if ∃v ∈ V(τ ), x i ∈ X v , γ −1 2 γ 1 ∈ Γ v , (γ −1 2 γ 1 )x 1 = x 2 for i = 1, 2
In this way, Ω is the disjoint union of compactums corresponding to the stabilizers of the vertices of T . Also, for each v ∈ V(τ ), the space X v naturally embeds in Ω as the image of {1} × X v . We identify it with its image. The group Γ naturally acts on the left on Ω. For γ ∈ Γ, γX v is the compactum for the vertex stabilizer Γ γv .
Contribution of the edges of T Each edge allows us to glue together compactums corresponding to the vertex stabilizers along with the limit set of the stabilizer of the edge. Each edge group embeds as a parabolic subgroup in adjacent vertex groups, so its limit set is a singleton. Let e = (v 1 , v 2 ) be the edge in τ , there exist equivariant maps Λ e,vi : X e ֒→ X vi for i = 1, 2. Similar maps are defined by translation for edges in T \ τ .
The equivalence relation ∼ on Ω is the transitive closure of the following: Let v and v ′ be vertices of T . The points x ∈ X v and x ′ ∈ X v ′ are equivalent in Ω if there is an edge e between v and v ′ and a point x e ∈ X e satisfying x = Λ e,v (x e ) and x ′ = Λ e,v ′ (x e ) simultaneously. Let Ω/ ∼ be the quotient under this relation and let π ′ : Ω → Ω/ ∼ be the corresponding projection. An equivalence class [x] of an element x ∈ Ω is denoted by x itself. Let ∂T be the (visual) boundary of the tree. We define M ′ as a set:
M ′ = ∂T ⊔ (Ω/ ∼ ). Definition 3.1. (Domains) For all x ∈ Ω/ ∼ , we define the domain of x to be D(x) = {v ∈ V(T )|x ∈ π ′ (X v )}.
We also say that domain of a point ξ ∈ ∂T is {ξ} itself.
3.2.
Final construction of M . It turns out that M ′ with the topology defined in [Dah03] is not a Hausdorff space (see 3.6 ). Therefore we need to modify M ′ further. For getting the desired space, we further define an equivalence relation on the set M ′ . Firstly observe that the domain of each element in Ω/ ∼ is either singleton or an infinite subtree of T . So, we also identify the boundary points of infinite domain D(x) in ∂T to x itself. By considering the equivalence relation generated by these relations, we denote the quotient of M ′ by M . Again, M can be written as a disjoint union of two sets of equivalence classes :
M = Ω ′ ⊔ (∂T ) ′
where Ω ′ (as a set it is same as Ω/ ∼ ) is the set of equivalence classes of elements in Ω/ ∼ , and, (∂T ) ′ is the equivalence classes of the remaining elements in ∂T as some elements of ∂T are identified with parabolic points of edge groups. The equivalence class of each remaining element in ∂T is a singleton. We write p for an element of M , if p ∈ Ω ′ then p = x, y, z, ... and if p ∈ (∂T ) ′ then p = ξ, η, ζ, ....
Remark 3.2.
In M , we define domain of each element as previously defined. Additionally for those points η of ∂T which are identified with parabolic points x of edge groups, we define domain of η same as that of x.
It is clear that for each v in T , the restriction of projection map π ′ from X v to Ω/ ∼ is injective. Let π ′′ be the projection map from M ′ to M . Let π be the composition of the restriction of π ′′ to Ω/ ∼ and π ′ . Again it is clear that restriction of π to X v is injective for all v in T .
Definition of neighborhoods in M.
We define a family (W n (p)) n∈N,p∈M of subsets of M , which generates a topology on M . For a vertex v and an open subset
U of X v , we define the subtree T v,U of T as {w ∈ V(T ) : X e ∩ U = ∅}, where e is the first edge of [v, w]. For each vertex v in T , let us choose U(v),U(v) contains X v . Let x be in Ω ′ and D(x) = {v 1 , ..., v n , ...} = (v i ) i∈I . Here I is a subset of N. For each i ∈ I, let U i ⊂ X vi be an element of U(v i ) containing x such that for all but finitely many indices i ∈ I, U i = X vi . The set W (Ui)i∈I (x) is the disjoint union of three subsets W (Ui)i∈I (x) = A ∪ B ∪ C,
where A is nothing but the collection of all boundary points of subtrees T vi,Ui which are not identified with some parabolic point corresponding to edge group, B is collection of all points y outside i∈I X vi in Ω ′ whose domains lie inside i∈I T vi,Ui , C is simply the union of all neighborhood U i around x in each vertex of D(x). In notations A,B,C are defined as follows:
A = ( i∈I ∂T vi,Ui ) ∩ (∂T ) ′ B = {y ∈ Ω ′ \ ( i∈I X vi )|D(y) ⊂ i∈I T vi,Ui } C = {y ∈ j∈I X vj |y ∈ m∈I|ζ∈Xv m U m } As A ⊂ (∂T ) ′ , the remaining elements in i∈I ∂T vi,Ui are in B. In this way, A, B, C are disjoint subsets of M .
Remark 3.3. The set W (Ui)i∈I (x) is completely defined by the data of the domain of x, the data of a finite subset J of I, and the data of an element of U(v j ) for each index j ∈ J. Therefore there are only countably many different sets
W (Ui)i∈I (x), for x ∈ Ω ′ , and U i ∈ U(v i ), v i ∈ D(x).
For each x, we choose an arbitrary order and denote them by W m (x). Proof. If p ∈ (∂T ) ′ , then both of the vertices of e are not in D(p). The lemma follows by taking one of the vertices as base and m ≥ 1. If p ∈ Ω ′ then a unique segment exists from D(p) to the edge e. Let v be the vertex from where this segment starts and e 0 be the first edge. Then X e0 does not contain p. So to find a W m (p) such that it does not intersect X e , it is sufficient to find a neighborhood around p in X v which does not intersect X e0 . But it is evident as X e0 is just a point.
Topology of M.
Consider the smallest topology T on M such that the family of sets {W n (p) :
n ∈ N, p ∈ M } are open subsets of M . Lemma 3.5. (M, T ) is a Hausdorff space Proof. Let p and q be two distinct points in M . If D(p) ∩ D(q) = ∅.
Then there is a unique geodesic segment from a vertex of D(p) to a vertex of D(q) in T having an edge e on this segment such that both vertices of e are neither contained in D(p) nor in D(q). Then using the above lemma, we can find disjoint neighborhoods around p and q. Now, suppose D(p) ∩ D(q) = ∅. Then there is only one vertex in this intersection, let D(p) ∩ D(q) = {v}. Since X v is Hausdorff, we can find disjoint open subsets of X v around p and q, respectively. Using these neighborhoods in X v , we have found disjoint neighborhoods around p and q in M .
Remark 3.6. The reason why we need to define a further equivalence relation on the set M ′ is the following:
In M ′ , if we consider an edge group parabolic point p, then D(p) is infinite subtree of T , and there are uncountably many boundary points of D(p). If we take one boundary point η of D(p), then on the geodesic ray [v 0 , η) there is no edge with at least one vertex not belonging to D(p), where v 0 is some vertex in D(p). Thus we can not find a disjoint neighborhood around η and p. Same kind of situation will arise if we prove that M ′ is a regular space.
Lemma 3.7. (Filtration) For every p ∈ M , every integer n, and every q ∈ W n (ξ), there exists m such that W m (q) ⊂ W n (p).
Proof. Suppose p ∈ (∂T ) ′ and W n (p) is a neighborhood around p. Let q ∈ W n (p). If q ∈ (∂T ) ′ then choose m = n and W m (q) = W n (p). Let q be some point in Ω ′ . Suppose the subtree T n (p) starts at the vertex v and e be the last edge on the segment from a base vertex to v then except X e , all the points in M corresponding to subtree T n (p) is in W n (p). By this observation, it is clear that there exists a neighborhood W m (q) such that W m (q) ⊂ W n (p) for sufficiently large m. Now, suppose that p ∈ Ω ′ and q ∈ W n (p). If D(p) ∩ D(q) = ∅, then there exists an edge e on a unique geodesic segment from D(p) to D(q). Then by Lemma 3.4, one can find a neighborhood W m (q) sitting inside W n (p). Finally, if D(p) ∩ D(q) = ∅, then it is a singleton and let this intersection be {v}. As q ∈ W n (p), q is in some U i , where U i is in neighborhood basis of X vi and D(p) = (v i ) i∈I . Then find a neighborhood V i around q sitting inside U i and then using this V i we see that we have found W m (q) such that W m (q) ⊂ W n (p).
Lemma 3.8. The family of sets {W n (p) : n ∈ N, p ∈ M } forms a basis for the topology T .
Proof. Using the previous lemma, it remains to prove that if W n1 (p 1 ) and W n2 (p 2 ) are two neighborhoods and q ∈ W n1 (p 1 )∩W n2 (p 2 ) then there exists a neighborhood around q, namely W m (q) such that W m (q) ⊂ W n1 (p 1 ) ∩ W n2 (p 2 ). Again by the previous lemma there exists m 1 and m 2 such that
W m1 (q) ⊂ W n1 (p 1 ) and W m2 (q) ⊂ W n1 (p 2 ). Now W k (q) ⊂ W m1 (q) ∩ W m2 (q) for some k, hence the lemma.
Lemma 3.9. For each v ∈ V(T ), the restriction of π to X v to M is a continuous map.
Proof. Let x be an element of X v , and π(x) be its image in M . We denote this image by x. Consider the neighborhood W n (x) around x. Then by definition of neighborhoods, it is clear that the inverse image of this neighborhood under π is an open subset of X v . Hence restriction of π to X v is continuous.
Lemma 3.10. (Regularity) The topology T is regular, that is, for all p ∈ M and for all W m (p) there exists n such that W n (p) ⊂ W m (p).
Proof. Case(1) Let p ∈ (∂T ) ′ and W m (p) be a neighborhood around p. Let v be a vertex from where the subtree T m (ξ) starts. Let e be the last edge of the geodesic segment from v 0 to v. Observe that closure of W m (p) contains only one extra point, namely X e . By taking n to be sufficiently large, we see W n (p) ⊂ W m (p).
Case(2) Let p ∈ Ω ′ and let W m (p) be a neighborhood around p. Let D(p) = (v i ) i∈I . Again observe that in the closure of W m (p) only extra points are the points in the closure of each U i in X vi .(If some point is not in the closure of U i then one can easily find a neighborhood around that point disjoint from W m (p)). Also since each vertex boundary is regular so choose a neighborhood V i of ξ in X vi such that
V i ⊂ U i . Then W {Vi}i∈I (p) ⊂ W m (p).
Note: Now by previous lemmas, we see that the topology on M is second countable, Hausdorff and regular. Then by Urysohn's metrization theorem, we see that M is metrizable space. Also, M is a perfect space as every point of M is a limit point. Finally, we have the following convergence criterion:
A sequence (p n ) n∈N in M converges to a point p if and only if ∀n, ∃m 0 ∈ N such that ∀m > m 0 , p m ∈ W n (p).
Theorem 3.11. The metrizable space M is compact.
Proof. It is sufficient to prove that M is sequentially compact. Let (p n ) n∈N be a sequence in M . Let us choose a vertex v in T and for each n, choose a vertex v n (if p n ∈ (∂T ) ′ then choose v n = p n ) in D(p n ). We see that up to extraction of a subsequence, either the Gromov inner products (v n , v m ) v remain bounded or they go to infinity. In the latter case (v n ) converges to a point q in ∂T . If q is the point which is identified with some edge boundary point then there is a ray in the domain of that edge boundary point converging to q, then by definition of the neighborhood around q, we see that p n converges to q. If q ∈ (∂T ) ′ then again there is a ray converges to q, and by convergence criterion, p n converges to q. Now, in the first case up to extracting a subsequence, we assume that Gromov inner products is equal to some constant N . Let g n be the geodesic from v to v n then there exist geodesic g = [v, v ′ ] of length N such that g lies in each g n . For different n and m, g n and g m do not have a common prefix whose length is longer than g. We have the following two cases:
Case(1): There exist a subsequence (g n k ) such that g n k = g. Then in this case as X ′ v is compact; we get a subsequence of (p n ) n∈N which converges to a point of X ′ v . Case(2): There exists a subsequence (g n k ) of (g n ) such that each g n k strictly longer than g. Let e n k be the edges just after v ′ ; they all are distinct. As for each edge e, X e is a singleton, so (X en k ) forms a sequence of points in X v ′ . Since X v ′ is compact, there exists a subsequence which converges to a point in X v ′ . Then, by convergence criterion, we see that there exists a subsequence of (p n ) converging to this point of X v ′ . (It may be possible that all the X n k are equal to p for some p in X v ′ , sequence converges to p in this situation also.)
Dynamics of Γ on M
In this section, we prove that Γ acts on M as a convergence group. For that we need the following two lemmas.
Lemma 4.1. (Large Translation) Let (γ n ) n∈N be a sequence in Γ. Assume that, for some (hence any) vertex v 0 ∈ T , dist(v 0 , γ n v 0 ) → ∞.
Then, there is a subsequence (γ σ(n) ) n∈N , two points p ∈ M and ζ ∈ (∂T ) ′ such that for all compact K ⊂ (M \ {ζ}), γ σ(n) K converges uniformly to p.
Proof. Let p 0 be in X v0 . Since M is sequentially compact, there exists a subsequence (γ σ(n) ) n∈N such that (γ σ(n) p 0 ) n∈N converges to a point p ∈ M . Also, we still have dist(v 0 , γ σ(n) v 0 ) → ∞. Let v 1 be another vertex in T . The lengths of the segments [γ n v 0 , γ n v 1 ] equal to the length of [v 0 , v 1 ]. As the dist(v 0 , γ σ(n) v 0 ) goes to ∞, for all m there is n m such that for all n > n m , the segments [v 0 , γ σ(n) v 0 ] and [v 0 , γ σ(n) v 1 ] have prefix of length more than m. Then, by convergence criterion 3.4, for all v ∈ T , γ σ(n) X v converges uniformly to p. Let ζ 1 , ζ 2 ∈ (∂T ) ′ . As triangles in T are degenerate, so the triangle with vertices v 0 , ζ 1 , ζ 2 have center some vertex v in T . Therefore for all m ≥ 0, the segments [v 0 , γ σ(n) v 0 ] and [v 0 , γ σ(n) v] coincide on a subsegment of length more than m for sufficiently large n. Then at least for one ζ i , the sequence of rays [v 0 , γ σ(n) ζ i ] have a common prefix of length at least m. Then by convergence criterion γ σ(n) ζ i converges to p. Therefore, there exists at most one (since for any two points in (∂T ) ′ at least one of them converges to p under the action of γ n ) ζ in (∂T ) ′ such that for all ζ ′ ∈ ((∂T ) ′ \ {ζ}), we have γ σ(n) ζ ′ → p.
Let K be a compact subspace of M \ {ζ}. Then there exists a vertex v, x ∈ Ω ′ , and a neighborhood W n (x) of x containing K such that ζ / ∈ W n (x). Consider the segment [v 0 , v]. As by the discussion at the beginning of proof, the sequence (γ σ(n) X v ) n∈N uniformly converges to p, the sequence (γ σ(n) W m (x)) n∈N uniformly converges to p. Hence the convergence is uniform on K.
Lemma 4.2. (Small Translation) Let (γ n ) n∈N be a sequence of distinct elements in Γ, and assume that for some (hence any) vertex v 0 , the sequence (γ n v 0 ) n is bounded in T . Then there exists a subsequence (γ σ(n) ) n∈N , a vertex v, a point p ∈ X v , and another point p ′ ∈ Ω ′ , such that for all compact subspaces K of M \ {p ′ }, one has (γ σ(n) ) n∈N K → p uniformly.
Proof. We prove the lemma in two cases.
Case(1): Assume that for some vertex v and some element γ ∈ Γ there exists a subsequence (h n ) n∈N in Γ v such that γ n = h n γ for all n. Since Γ v acts as a convergence group on X v . Then one can further extract a subsequence of (γ n ) (we shall denote it again by γ n ) and a point p ′ in X γ −1 v such that for all compact subsets K of X γ −1 v \ {p ′ }, γ n K → p uniformly, for some p ∈ X v . Suppose p ′ is a conical limit point. then it will not be in any edge boundary contained in X γ −1 v . Let e n be the possible edges starting from the vertex γ −1 v. For any q ∈ M \ {X γ −1 v }, we see that unique edge path from γ −1 v to w contains e n , for some n and q ∈ X w . Since γ n r → p for all r in X γ −1 v \ {p ′ } then by convergence criterion, we see that γ n q converges to p, and the same is true for the points in (∂T ) ′ . Hence for all compact K ⊂ M \ {p ′ } we have γ n K → p uniformly. Suppose p ′ is a parabolic point, γp ′ is also a parabolic point in X v . Then γ n p ′ also converges to p. Otherwise, γp ′ is a conical limit point. Thus again, by the same argument as above we see that for all q ∈ M , γ n q → p.
Case (2): Suppose such a sequence (h n ) n∈N and a vertex v does not exist. After possible extraction we can assume that the distance dist(v 0 , γ n (v 0 )) is constant. Let us choose a vertex v such that there exists a subsequence (γ σ(n) ) n∈N such that the segments [v 0 , γ σ(n) v 0 ] share a common segment [v 0 , v] and the edges e σ(n) located just after v are all distinct. Then one can extract a subsequence (e σ ′ (n) ) n∈N such that spaces corresponding to these edges converges to a point p in X v . Then by convergence criterion, γ σ ′ (n) (X v0 ) converge uniformly to p. Let ξ ∈ (∂T ) ′ . Then v is not on the ray [γ σ ′ (n) v 0 , γ σ ′ (n) ξ] for sufficiently large n for if v is there then, for infinitely many n, γ −1 σ ′ (n) v = w for some fixed vertex w on [v 0 , ξ), we see that we are in the first case, which is a contradiction. Thus for all ξ ∈ (∂T ) ′ , we see that the unique rays from v to γ σ ′ (n) ξ contains the edge e ′ σ(n) for some n. Hence by convergence criterion γ σ ′ (n) ξ → p for all ξ ∈ (∂T ) ′ . Now, let x be in the space except X v0 . Suppose x ∈ X v ′ . Again by the same reasoning, we see that v / ∈ [γ σ ′ (n) v 0 , γ σ ′ (n) v ′ ] for sufficiently large n. Then the unique geodesic segment from v to γ σ ′ (n) v ′ contains e ′ (n) for sufficiently large n. Hence by definition of neighborhoods γ σ ′ (n) x → p. Thus, for all compact subsets K ⊂ M , we see that γ σ ′ (n) K → p uniformly.
Using the previous two lemmas, we get the following: Proof. Fix a vertex v 0 in T . Let (γ n ) n∈N be a sequence in Γ. Then up to extraction of a subsequence, there are two cases: either the distance from v 0 to γ n v 0 goes to infinity or the distance from v 0 to γ n v 0 is bounded. In either case, previous lemmas imply the corollary, which proves Theorem 1.2.
Combination of convergence groups
In this section, we prove Theorem 1.3. First of all, we recall a construction from [Man20] that is used in the proof of Theorem 1.3. Let Γ be a hyperbolic group and let (Γ, G) be a relatively hyperbolic group. In [Man20], Manning constructed a space which is the quotient of Gromov's boundary of Γ and showed that Γ acts geometrically finitely on the quotient. The quotient is obtained by collapsing all the translates of the limit set of subgroups in G. Using Yaman's characterization (Theorem 2.2) of relative hyperbolicity, this quotient is Bowditch boundary of (Γ, G). There, to prove that the action of Γ on the quotient is convergence, we do not require that Γ is hyperbolic and G is a malnormal family of quasi-convex subgroups.
Suppose Γ acts on a compact metrizable space X as a convergence group. Let G is a dynamically malnormal family of dynamically quasi-convex subgroups. Then form a quotient space X/ ∼ as in [Man20] by collapsing translates of limit sets of subgroups in G. Also, assume that |Λ(H)| ≥ 2, where H ∈ G. To prove Proposition 2.2 in [Man20], we require that the collection of limit sets of the cosets of the elements in G form a null sequence which follows from Proposition 2.6. We immediately have the following lemma:
Lemma 5.1.
(1) X/ ∼ is a compact metrizable space.
(2) The group Γ acts on X/ ∼ as a convergence group.
Also, note that each subgroup in G become a parabolic subgroup for the action of Γ on X/ ∼. Now, we give proof of Theorem 1.3.
Proof of Theorem 1.3. Let Γ be as in the statement of the theorem. Let Γ v be a vertex group that acts as a convergence group on X v . Take the collection of edges incident to vertex v and take a collection of those edge groups which are not parabolic in the vertex group Γ v . Consider the stabilizers of the limit sets of these edge groups in the vertex group Γ v . By assumption, they form a dynamical malnormal family of dynamically quasi-convex subgroups. Therefore, by Lemma 5.1, we obtain a quotient of X v , namely X v / ∼ on which Γ v acts as a convergence group and all edge groups incident to v become parabolic subgroups of Γ v . Hence by following the same process at each vertex group, we are in the situation where we have a graph of convergence groups with edge groups parabolic in adjacent vertex groups. This completes the proof using Theorem 1.2.
Proof of Proposition 1.4. It is sufficient to consider the amalgam and HNN extension case.
Case(1): Let Γ = Γ 1 * γ1 ≃ γ2 Γ 2 . Suppose Γ 1 is acting on X 1 as a convergence group and γ 1 is a loxodromic element in Γ 1 . Then, by [Yan12, Lemma 2.6], stabilizer of the limit set of γ 1 in Γ 1 is a dynamically quasi-convex subgroup of Γ 1 . Also, by assumption, the stabilizer of the limit set of γ 1 in Γ 1 is dynamically malnormal. Similarly, the stabilizer of the limit set of γ 2 in Γ 2 is dynamically quasi-convex and dynamically malnormal. Hence, by Theorem 1.3, Γ is a convergence group.
Case(2): Let Γ = Γ 1 * γ1 ≃ γ2 . Suppose Γ 1 acts as a convergence group X 1 and γ 1 is loxodromic for this action. Then, by Lemma 5.1, Γ 1 acts as a convergence group on the quotient X 1 / ∼ of X 1 , γ 1 is a parabolic element for the action of Γ 1 on X 1 / ∼ . Now, if γ 2 is parabolic for the action of Γ 1 on X 1 / ∼ then we have HNN extension of convergence group with parabolic edge group, and hence by Theorem 1.2, Γ is a convergence group. If γ 2 is loxodromic for the action of Γ 1 on X 1 / ∼ then again, by Lemma 5.1, we have the quotient of X 1 / ∼ such that Γ 1 acts as a convergence group. γ 2 is a parabolic element for the action of Γ 1 on the quotient of X 1 / ∼. Hence, we have the HNN extension of convergence group with parabolic edge group, and, by Theorem 1.2, Γ is a convergence group.
Note: The amalgam case in the above corollary is exactly a corollary of Theorem 1.3. However, the HNN extension is not exactly a corollary of Theorem 1.3 as stabilizers of limit sets of γ 1 , γ 2 in Γ 1 respectively need not form a dynamical malnormal family.
When the vertex groups in Corollary 1.4 are torsion-free, then, by the following lemma, we do not need to assume dynamical malnormality of edge group in adjacent vertex groups.
Lemma 5.2. Let Γ be a torsion-free group that acts on X as a convergence group. Let γ ∈ Γ be a loxodromic element and let H = Stab Γ (Λ( γ )) Then H is a dynamically quasi-convex and dynamically malnormal subgroup of Γ.
Proof. The dynamical quasi-convexity of H follows from Lemma 2.6 of [Yan12]. Let γ 1 ∈ Γ \ H and assume that γ 1 Λ(H) ∩ Λ(H) = ∅. Let Λ(H) = {x 1 , x 2 } and let γ 1 fixes, either x 1 or x 2 . Since Γ is torsion-free, γ 1 has infinite order. Thus γ 1 is either a parabolic or a loxodromic element. By [Bow99, Proposition 3.2], a parabolic point can not be a fixed point of a loxodromic element so γ 1 can not be parabolic. By [Tuk94a,Theorem 2G], if γ 1 is loxodromic, then γ 1 must fix the other point. But this implies that γ 1 is in H, which is a contradiction. Now, suppose that γ 1 does not fix any of x i and γ 1 x 1 = x 2 . Consider the element γ ′ 1 = γ −1 1 γγ 1 which fixes x 1 . Again γ ′ 1 can not be parabolic so it has to be loxodromic, but this implies that γ ′ 1 x 2 = x 2 . Thus γ 1 x 2 = x 1 and this implies that γ 1 is in H, which is again a contradiction.
Hence for torsion-free groups, we have the following:
Proposition 5.3. Let Γ be the fundamental group of a finite graph of torsionfree countable convergence groups with infinite cyclic edge groups. Then Γ is a convergence group.
Remark 5.4. Although we have answered Question 1.1 in the special cases but the general case is still not answered. For the general case, if we try to work with Dahmani's construction, we need to identify more points in the space constructed in [Dah03] but it is unclear which points are needed to be identified.
6. Proof of Theorems 1.6, 1.7
So far, we have proved combination theorems for convergence groups. Let Γ be as in the Theorem 1.2. First of all, if each vertex group Γ v acts geometrically finitely on compact metrizable space X v , then the group Γ acts geometrically finitely on the space M constructed in Section 3 . Using this, we give the proof of Theorems 1.6,1.7. To prove that Γ acts geometrically finitely , we demonstrate that every point of M is either a conical limit point or a bounded parabolic point. So, we start proving the following lemmata:
Lemma 6.1. Every point in (∂T ) ′ ⊂ M is a conical limit point for Γ in M .
Proof. Let η ∈ (∂T ) ′ ⊂ M and v 0 be a vertex of T . Then there exists a sequence (γ n ) n∈N in Γ such that γ n v 0 lies on the unique geodesic ray [v 0 , η) for all n. Then by lemma 4.1, there exists a subsequence denoted by γ n and a point p ∈ M such that for all q in M except possibly a point in (∂T ) ′ , we have γ −1 n q converges to p. Now after multiplying each γ n on the the right by an element of Γ v0 , we can assume that p does not belong to X v0 . To prove that η is a conical limit point of Γ in M , it is sufficient to prove that γ n η does not converges to p. Observe that the ray [γ −1 n v 0 , γ −1 n η) always have v 0 on this ray for all n. If the sequence γ −1 n η converges to p then as the sequence γ n x also converges to p for any x ∈ X v0 , we see that p belongs to Γ v0 , which is a contradiction to our choice of p.
Now, we prove that each conical limit point for the action of the vertex group Γ v on X v is conical for the action of Γ on M .
Lemma 6.2. Every point in Ω ′ which is the image of a conical limit point in the vertex stabilizer's boundary is a conical limit point for Γ in M .
Proof. Let x ∈ X v be a conical limit point for Γ v in X v . There exists a sequence (γ n ) n∈N and two distinct points y and z in X v such that γ n (x) → y and γ n (x ′ ) → z for all x ′ = x. Now, we show that π(x) is a conical limit point for Γ in M . Since the restriction of π to X v is continuous from X v to M . Therefore π(γ n x) = γ n π(x) → π(y) and π(γ n x ′ ) = γ n π(x ′ ) → π(z). Since restriction of π is injective, so π(y) and π(z) are distinct. Hence π(x) is a conical limit point for Γ v in M , hence for Γ in M .
The following lemma proves that the image of each bounded parabolic point in vertex space is bounded parabolic for Γ in M . Proof. (Amalgam Case) We prove the lemma in two cases:
Case(1) Let p be a bounded parabolic point for a vertex group Γ v in X v , which is not in any edge space attached to X v . We denote π(p) by p. Let D(p) = {v}. Let P be the stabilizer of p in Γ. Since P fixes the vertex v, P ≤ Γ v . In fact, P is the stabilizer of p in Γ v . As p is bounded parabolic point for Γ v in X v , P acts co-compactly on X v \ {p}. Let K be a compact subset of X v \ {p} such that P K = X v \ {p}. Consider E the set of edges whose boundaries intersect K. Let e be the edge with one vertex v, then there exists h ∈ P such that X e ∩ hK = ∅. Therefore the set of edges ∪ h∈P hE contains every edge with one and only one vertex v. Let V be the set of vertices w of the tree T such that the first edge of [v, w] is in E. Let K ′ be the subset of M consisting of points whose domains are in V. Define K ′′ = K ∪ K ′ . As the sequence of points in K ′ has limit in K, K ′′ is a compact subset of M . Now, it is clear that P K ′′ = M \ {p}.
Case(2) Suppose p is an element of edge boundary. In this case, D(p) is infinite. Suppose that v 1 and v 2 are vertices of an edge e. Let X e = {p} and P 1 , P 2 be maximal parabolic subgroups in vertex groups Γ v1 , Γ v2 respectively, and P is the parabolic edge group. Then D(p) is nothing but the Bass-Serre tree of the amalgam Q = P 1 * P P 2 , which is the stabilizer of p in Γ. Under the action of P 1 * P P 2 the quotient of D(p) is the edge e. Since P 1 , P 2 acts co-compactly on X v1 \{p}, X v2 \{p} respectively, there exists a compact subset K i of X vi \ {p} such that P i K i = X vi for i = 1, 2. Consider E i the set of edges starting at v i whose boundary intersects K i but does not contain p. Let e be an edge with only one vertex in D(p) and v i be this vertex. Then there exists h ∈ P i such that X e ∩ hK i = ∅ for i = 1, 2. Therefore the set of edges ∪ i=1,2 QE i contains every edge with one and only one vertex in D(p). Let V i be the set of vertices w such that first edge of [v i , w] is in E i . Let K ′ i be the subset of M consisting of the points whose domain is in V i for i = 1, 2. Since a sequence of points in the spaces corresponding to the stabilizers of distinct edges in E i have only accumulation points in K i , the set K ′′ i = K i ∪ K ′ i is a compact for i = 1, 2. Hence K = ∪ i=1,2 K ′′ i is a compact set of M not containing p and QK = M \ {p}. Therefore p is bounded parabolic point for Γ in M .
(HNN extension case) Case(1) Let Γ v be vertex group in HNN extension and P is a parabolic subgroup sitting inside a maximal parabolic subgroup P 1 and isomorphic to a subgroup P ′ of P 1 . In this case, the proof remains the same as in the amalgam case except that the maximal parabolic subgroup corresponding to edge boundary point is P 1 * P ≃P ′ , and maximal parabolic subgroups corresponding to parabolic points which are not in any edge spaces are maximal parabolic for Γ in M .
Case(2) Let Γ v be same as in case(1) and suppose P is sitting inside P 1 and is isomorphic to a subgroup P ′ of maximal parabolic subgroup P 2 , which is not conjugate to P 1 in Γ v . Then, in this case, we can write Γ v * P ≃P ′ = (Γ v * P P ′ ) * P ′ , and we apply the amalgam and case(1) of HNN extension respectively to get the result.
From the above lemmata, it is clear that if each Γ v acts on X v geometrically finitely then Γ acts on M geometrically finitely. Now, we are in the position of proving the following proposition:
Proposition 6.4. Let Γ be a finitely generated group that splits as a finite graph of convergence groups with finitely generated parabolic edge groups. Then Γ acts geometrically finitely on M (constructed in Section 3) if and only if each vertex group Γ v acts geometrically finitely on compact metrizable space X v .
Proof. Note that, by [BW13, Lemma 2.5], each vertex group Γ v is finitely generated. Suppose each vertex group Γ v acts geometrically finitely on X v . Then, by Lemma 6.1, 6.2, 6.3, Γ acts geometrically finitely on M . Conversely, suppose that Γ acts geometrically finitely on M . Since each edge group is parabolic for the action of Γ on M , each edge group is a relatively quasi-convex subgroup of Γ. By [HH21, Proposition 5.2], each vertex group Γ v is a relatively quasi-convex subgroup of Γ.
In particular, each Γ v acts geometrically finitely on X v as X v is the limit set of Γ v for Γ acting on M .
Proof of Theorem 1.6. Let Γ be either amalgam or HNN extension of relatively hyperbolic groups with parabolic edge groups. Since each vertex group Γ v is relatively hyperbolic, Γ acts geometrically finitely on its Bowditch boundary. To prove Γ is relatively hyperbolic, we use Yaman's characterization Theorem 2.2. For constructing a space on which Γ acts geometrically finitely, we follow the same construction as in Section 3 by taking compactum X v , X e for Γ v , Γ e as Bowditch boundaries of these groups, respectively. Let Γ be as in the theorem. From the above discussion, we have a space M which is compact metrizable as proved in Section 3. Now, the proof of Theorem 1.2 gives that Γ acts on M as a convergence group. Since each vertex group acts geometrically on its Bowditch boundary, from the above lemmata, the groups Γ acts geometrically finitely on M . Hence by Yaman's characterization 2.2, Γ is relatively hyperbolic, and M is Bowditch boundary for Γ.
Let Γ be as in the proof of the above theorem. Consider the collection G containing the two type of subgroups of Γ: (1) stabilizers of bounded parabolic points in Bowditch boundary of vertex groups which are not identified with edge parabolic point.
(2) stabilizers of edge parabolic points in Γ. Then Γ is hyperbolic relative to G.
Remark 6.5. The limit set of each vertex group, Γ v for the action of Γ on M , is homeomorphic to its Bowditch boundary ∂Γ v . It is clear that Γ v acts geometrically finitely on its limit set, and therefore, Γ v is a relatively quasi-convex subgroup of Γ. Now, we prove Theorem 1.7. For that, let us recall some basic definitions and results from [Osi06].
Let G be a group hyperbolic relative to a collection of subgroups {H α , α ∈ Λ}. A subgroup Q of G is said to be hyperbolically embedded in G if G is hyperbolic relative to {H α , α ∈ Λ} ∪ {Q}. We say that an element g of G is parabolic if it is conjugate to an element of H α for some α ∈ Λ. Otherwise, it is called hyperbolic. For any hyperbolic element g of infinite order, we set E(g) = {f ∈ G : f −1 g n f = g ± n}. For any hyperbolic element of infinite order, Osin proved the following theorem:
Theorem 6.6. [Osi06, Theorem 4.3] Every hyperbolic element g of infinite order in G is contained in a unique maximal elementary subgroup E(g).
From proof of the above theorem, it follows that [E(g) : g ] < ∞ and therefore E(g) is elementary. Also, for infinite order hyperbolic element, the unique maximal elementary subgroup is hyperbolically embedded in the relatively hyperbolic group G, see [Osi06, Corollary 1.7]. To prove Theorem 1.7, as we mentioned in the introduction, it is sufficient to consider the amalgam and the HNN extension case.
Proof of Theorem 1.7. Case(1): Let Γ = Γ 1 * Z1≃Z2 Γ 2 . Suppose Z 1 = γ 1 and Z = γ 2 . If both γ 1 , γ 2 are parabolic elements in Γ 1 , Γ 2 respectively, then we are in amalgam case of Theorem 1.6, and hence Γ is relatively hyperbolic. Suppose at least one of them is a hyperbolic element, then by Theorem 6.6, we have a maximal elementary subgroup containing cyclic subgroup generated by a hyperbolic element which is hyperbolically embedded. Thus, we are in the amalgam case of the Theorem 1.6 and therefore Γ is relatively hyperbolic. Case(2): Let Γ = Γ 1 * Z≃Z ′ and let Z = γ , Z ′ = γ ′ are isomorphic subgroups of Γ 1 . Again if both γ, γ ′ are parabolic elements, then we are in the HNN extension case of Theorem 1.6, and hence Γ is relatively hyperbolic. If at least one of them is a hyperbolic element, then by applying the Theorem 6.6, we get maximal elementary subgroups containing that cyclic subgroup that is hyperbolically embedded. Thus, we are in the HNN extension case of Theorem 1.6, and hence Γ is relatively hyperbolic.
In either case, we have proved that Γ is relatively hyperbolic and applying Theorem 1.6, we also have a description of Bowditch boundary.
Homeomorphism type of Bowditch boundary
In [MJ15], authors proved that the homeomorphism type of Gromov boundary of the fundamental group of a graph of hyperbolic groups with finite edge groups depends only on the set of homeomorphism type of Gromov boundary of nonelementary hyperbolic vertex groups. It is not clear that the same result can be extended in the case of a graph of relatively hyperbolic groups with finite edge groups. However, under some assumptions, we prove a similar result for a graph of relatively hyperbolic groups with parabolic edge groups. For the convenience of the reader, we are again stating the following theorem:
Theorem 7.1. Let Y be a finite connected graph and let G(Y ), G ′ (Y ) be two graph of groups satisfying the following:
(1) For each vertex v ∈ V (Y ), let (G v , P v ), (G ′ v , P ′ v )
be relatively hyperbolic groups.
(2) Let e ∈ E(Y ) be an edge with vertices v, w and let P e , P ′ e are parabolic edge groups in G(Y ), G ′ (Y ), respectively. Then either P e , P ′ e have infinite index in corresponding maximal parabolic subgroups in G v , G ′ v , respectively or P e , P ′ e have the same finite index in maximal parabolic subgroups in G v , G ′ v , respectively. Similarly, either P e , P ′ e have infinite index in maximal parabolic subgroups in G w , G ′ w , respectively or P e , P ′ e have the same finite index in corresponding maximal parabolic subgroups in G w , G ′ w , respectively.
v ∈ V (Y ), suppose we have a homeomorphism from ∂G v → ∂G ′ v that maps B v onto B ′ v . Let Γ = π 1 (G(Y )), Γ ′ = π 1 (G ′ (Y )
). (By Theorem 1.6, the groups (Γ, P), (Γ ′ , P ′ ) are relatively hyperbolic) Then there exists a homeomorphism from ∂Γ to ∂Γ ′ preserving edge parabolic points, i.e. taking parabolic points corresponding to edge groups of G(Y ) to parabolic points corresponding to edge groups of G ′ (Y ).
Remark 7.2. In the above theorem, it is not possible that for G(Y ), edge groups are maximal parabolic, and for G ′ (Y ), edge groups are parabolic (Not maximal parabolic). Thus in both graphs of groups, either edge groups are maximal parabolic or edge groups are parabolic. The example below justifies this situation. respectively. Here, we just take identity map between Bowditch boundaries of vertex groups. From the construction of Bowditch boundaries, it is clear that if we remove parabolic points from Bowditch boundaries ∂Γ, ∂Γ ′ respectively, we have two connected components, infinitely many connected components, respectively. Therefore there is no homeomorphism from ∂Γ to ∂Γ ′ preserving edge parabolic points.
Also, the above theorem does not deal with the case when parabolic edge groups have different finite indexes in corresponding maximal parabolic subgroups. Here, we give a specific example in this direction. F (a, b) is relatively hyperbolic with respect to [a, b] . Both the groups Γ, Γ ′ are relatively hyperbolic with Bowditch boundary ∂Γ, ∂Γ ′ respectively. Again, from the construction of Bowditch boundary, removing a parabolic point from ∂Γ gives two connected components but removing a parabolic point from ∂Γ ′ gives three connected components. Thus there is no homeomorphism from ∂Γ to ∂Γ ′ preserving edge parabolic points. Similarly, if we take
Example 7.4. Consider the two groups Γ = F (a, b) * [a,b] ≃ [ā,b] F (ā,b) and Γ ′ = F (a, b) * [a,b] ≃ [a,b] 2 F (ā,b). Assume thatΓ = F (a, b) * [a,b] ≃ [a,b] 3 F (ā,b) and Γ ′ = F (a, b) * [a,b] ≃ [a,b] 2 F (ā,b)
then there is no homeomorphism from ∂Γ to ∂Γ ′ preserving edge parabolic points.
To prove Theorem 7.1, it is sufficient to consider amalgam and HNN extension case. 7.1. Proof of Theorem 7.1 in amalgam case: Let Y be an edge. Then G(Y ) and G ′ (Y ) are amalgams of two relatively hyperbolic groups with parabolic edge groups. Let Γ = π 1 (G(Y )) and let Γ ′ = π 1 (G ′ (Y )). Let T, T ′ be the Bass-Serre trees for G(Y ), G ′ (Y ), respectively. For each edge e ∈ T and e ′ ∈ T ′ , let P e , P ′ e be parabolic edge groups in Γ, Γ ′ respectively. Also, in adjacent vertices of e and e ′ , let P v , P w and let P ′ v , P ′ w be maximal parabolic subgroups corresponding to P e , P ′ e respectively. Let ∂Γ, ∂Γ ′ denotes Bowditch boundaries of Γ, Γ ′ respectively. Keeping the construction of Bowditch boundaries in mind, we define a map f from ∂Γ to ∂Γ ′ .
Let e, e ′ be two edges of T, T ′ with vertices v, w and v ′ , w ′ respectively. Suppose we have homeomorphisms ∂G v → ∂G ′ v and ∂G w → ∂G ′ w as in Theorem 7.1(3). By definition of these homeomorphisms, we have bijections between cosets of P v in G v and cosets of P ′ v in G ′ v . Also, there is a bijection between cosets of P e in P v and cosets of P ′ e in P ′ v . Combining these two, we get a bijection between cosets of P e in G v and cosets of P ′ e in G ′ v . Similarly, we have a bijection between cosets of P e in G w and cosets of P ′ e in G ′ w . By following this process inductively, we have
an isomorphism φ from T to T ′ . Let ξ ∈ ∂G v for some vertex v ∈ V (T ). Define f (ξ) := f v (ξ), where f v is a homeomorphism from ∂G v to ∂G φ(v)
. Note that if ξ is a parabolic point in ∂G v and let D(ξ) be its domain then φ| D(ξ) = D(f (ξ)). Since φ is an isomorphism, we have a homeomorphism ∂φ from ∂T to ∂T ′ . Observe that if some point of ∂T is identified with some parabolic point, then its image under ∂φ is also identified with some parabolic point. If η ∈ ∂T such that it is not identified with some edge parabolic point, define f (η) := φ(η). Clearly, f is a bijection. Thus, we have a map f from ∂Γ to ∂Γ ′ . To prove that f is a homeomorphism, it is sufficient to prove that f is continuous as Bowditch boundaries ∂Γ, ∂Γ ′ are compact Hausdorff. Let ξ ∈ ∂G v for some v ∈ V (T ) and let U be a neighborhood of f (ξ) in ∂Γ ′ . Note that φ(D(ξ)) = D(f (ξ)). For each vertex u ∈ D(ξ), we can choose a neighborhood V u such that f u (V u ) ⊂ U φ(u) as f u is a homeomorphism, where U φ(u) is a neighborhood around f (ξ) in ∂G φ(u) . Now, it is clear from the definition of neighborhoods in ∂Γ and definition of maps f, φ that we can find a neighborhood V of ξ in ∂Γ such that f (V ) ⊂ U . Now, let η ∈ ∂T such that it is not identified with a parabolic point. Let U be a neighborhood of f (η) in ∂Γ ′ . From the construction of map φ, it is clear that φ takes subtree W m (η) (see section 3 ) onto the subtree W m (f (η)). Then again, by definition of neighborhoods, we can find a neighborhood V of η in ∂Γ such that f (V ) ⊂ U . The map f is continuous, and hence f is a homeomorphism. 7.2. Proof of Theorem 7.1 in HNN extension case: Here, we prove it in the following two subcases:
Subcase(1) Let Γ = G * P1≃P2 , Γ ′ = G ′ * P ′ 1 ≃P ′ 2 ,
where (G, P), (G ′ , P ′ ) are relatively hyperbolic groups. Assume that P 1 , P 2 , and P ′ 1 , P ′ 2 are both sitting inside the same maximal parabolic subgroups in G, G ′ , respectively. Let T, T ′ be the Bass-Serre trees of Γ, Γ ′ respectively. Also, we have a homeomorphism between Bowditch boundaries ∂G, ∂G ′ satisfying (3) in Theorem 7.1. We get an isomorphism φ from T to T ′ in a similar manner as in the amalgam case. Also, we can define a map from ∂Γ to ∂Γ ′ in the same way as we define in the amalgam case. Note that f is a bijection. To prove that f is a homeomorphism, it is sufficient to prove that f is continuous as ∂Γ, ∂Γ ′ are compact Hausdorff. Again continuity is clear from the definition of map f and definition of neighborhoods in ∂Γ, ∂Γ ′ respectively.
Subcase(2) Let Γ = G * P1≃P2 , Γ ′ = G ′ * P ′ 1 ≃P ′ 2 ,
where (G, P), (G ′ , P ′ ) are relatively hyperbolic groups. In this case, P 1 , P 2 , and P ′ 1 , P ′ 2 are sitting inside in different(not conjugate) maximal parabolic subgroups in G, G ′ , respectively. Now, we can write Γ = (G * P1 P 2 ) * P2 and Γ ′ = (G ′ * P ′ 1 P ′ 2 ) * P ′ 2 , respectively. By applying amalgam and Subcase(1) of the HNN extension respectively, we get the desired homeomorphism from ∂Γ to ∂Γ ′ .
Applications and examples
8.1. Example of a subgroup of a relatively hyperbolic group with exotic limit set. In this subsection, following the construction of space given in Section 3, we give an example of a relatively hyperbolic group having a non-relatively quasi-convex subgroup whose limit set is not equal to the limit of any relatively quasi-convex subgroup. This is motivated by the work of I.Kapovich [Kap95a], where he gave such an example in the case of hyperbolic group.
Consider the torus with one puncture and let ψ be a pseudo-Anosov homeomorphism fixing the puncture. Suppose M ψ is the mapping torus for the homeomorphism ψ. Let G, F be the fundamental groups of M ψ , puncture torus respectively. Then it is well known that G is relatively hyperbolic with respect to a subgroup isomorphic to Z ⊕ Z, and subgroup F is not relatively quasi-convex in G. Let Γ = G * z G and let H = F * z F , where z ∈ F , be doubles of G and H respectively along cyclic subgroup z . The groups Γ, H are relatively hyperbolic, by Theorem 1.7. We have the following: Proof. Suppose H is relatively quasi-convex in Γ. Since F is relatively quasi-convex in H and H is relatively quasi-convex in Γ, F is relatively quasi-convex in Γ by [BW13, Lemma 2.3]. Since F is a normal subgroup of G, the limit set of F in G is the same as Bowditch boundary of G that is homeomorphic to the limit set of G in Γ. Also the limit set of F in Γ is the same as Bowditch boundary of G. Hence F acts geometrically finitely on its limit in G. Thus, F is relatively quasi-convex in G, which is a contradiction.
As we observe in the proof of Theorem 1.7, the edge group in Γ is parabolic, or we change the parabolic structure in vertex group G so that edge group become parabolic. Therefore, following Section 3, we can give a construction of Bowditch boundary of relatively hyperbolic group Γ. Now, we prove the following:
Lemma 8.2. Stab Γ (Λ(H)) = H, i.e. H is maximal in its limit set.
Proof. Let T, T ′ be the Bass-Serre trees of the groups Γ, H respectively. Note that the tree T ′ embeds in T , and each vertex group in H is normal in the corresponding vertex group of Γ. Also, there is topological embedding between Gromov boundaries of T ′ and T . Let M be Bowditch boundary of Γ. Here we explicitly know the construction of M (see Section 3). Let p be a map from M → T ∪ ∂T defined as follows: for ξ ∈ ∂G v , define p(ξ) = v and for η ∈ ∂T , define p(η) = η. Consider N , a subset of M , the inverse image of T ′ ∪ ∂T ′ under the map p. It is clear from the definition of topology on M that N is the minimal closed H-invariant set. Thus Λ(H) = N . Now, the lemma follows immediately from the construction of Bowditch boundary M . Now, we prove that the limit set of the subgroup H is exotic, i.e. there is no relatively quasi-convex subgroup of Γ whose limit set is equal to the limit set of H.
Recall that a subgroup of a relatively hyperbolic group is dynamically quasi-convex if and only if it is relatively quasi-convex (see [GP16]).
Lemma 8.3. There does not exist a relatively quasi-convex subgroup of Γ whose limit set is equal to the limit set of H.
Proof. If possible, there is a relatively quasi-convex subgroup Q of Γ such that Λ(Q) = Λ (H). Since Stab Γ (Λ(H)) = Stab Γ (Λ(Q)) = H, we see that Q ⊂ H. Thus, by [Yan12, Lemma 2.6], we see that H is a dynamically quasi-convex subgroup of Γ; it is relatively quasi-convex, which gives a contradiction as H is not relatively quasi-convex.
Remark 8.4. In [Dah03], Dahmani gave a construction of Bowditch boundary for the fundamental group of an acylindrical graph of relatively hyperbolic groups with fully quasi-convex edge groups. In particular, we can construct Gromov boundary of the fundamental group of an acylindrical graph of hyperbolic groups with quasiconvex edge groups. Let S g , g ≥ 2 be a closed orientable surface of genus g and let φ be a pseudo-Anosov homeomorphism of S g . Let M φ be the mapping torus corresponding to φ and let G be the fundamental group of M φ . Then G is a hyperbolic group and F = π 1 (S g ) is a non-quasiconvex subgroup of G. Let z ∈ F be such that z is not a proper power in F and hence it is not a proper power in G. Consider the double Γ of group G along z , i.e. Γ = G * z G. Note that the group Γ is hyperbolic by [BF92] and the subgroups G, G of Γ are quasi-convex. Consider the group H = F * z F . Again, by [BF92], H is hyperbolic. Now, using the construction of Gromov boundary of Γ from [Dah03], we can explicitly construct the limit set of subgroup H (as we did in Lemma 8.2). Then, we have Stab Γ Λ(H) = H. Then, using the same idea as above, H is not quasi-convex, and there is no quasiconvex subgroup of Γ whose limit set equal Λ(H). Thus, we have a different proof of I.Kapovich's result from [Kap95b].
Above, we have given an example of a graph of relatively hyperbolic groups for which we have a subgraph of groups such that the fundamental group of subgraph of groups is not relatively quasi-convex in the fundamental group of the graph of groups. Contrary to that, we prove that when the subgraph of groups is obtained by restricting graph of groups (as in Theorem 1.6) to a subgraph, then the fundamental group of the subgraph of groups is relatively quasi-convex in the fundamental group of graph of groups.
Let G(Y) be a graph of groups as in Theorem 1.6 and G(Y 1 ) be a subgraph of groups obtained by restricting graph of groups G(Y) to a subgraph Y 1 ⊂ Y . Let Γ, Γ 1 be the fundamental groups of G(Y), G(Y 1 ), respectively. Let T 1 be the Bass-Serre tree for G(Y 1 ). As an application of Theorem 1.6, we have the following:
Proposition 8.5. Γ 1 is a relatively quasi-convex subgroup of Γ.
Proof. G(Y 1 ) is also a graph of relatively hyperbolic groups with parabolic edge groups. Using the construction of Section 3, we have a set M 1 ⊂ M for Γ 1 . By definition of neighborhoods on M , we see that M 1 is a closed subset of M . It is also minimal Γ 1 -invariant subset of M . Thus, M 1 is the limit set of Γ 1 . By the proof of Theorem 1.6, it is clear that every point of M 1 coming from vertex boundaries is either conical or bounded parabolic. Now, let η be a point in (∂T 1 ) ′ . Since the action of Γ 1 on T 1 is co-compact, there exists a sequence (γ 1n ) n∈N ⊂ Γ 1 and a vertex v ∈ T 1 such that γ 1n v meet the geodesic ray [v 0 , η) for each n. Now by applying the proof of Lemma 6.1, we see that up to extraction, a subsequence of (γ −1 1n ), (γ −1 1n ) is a conical sequence for the action of Γ on M . Since M 1 is a closed subset of M and (γ 1n ) n∈N ⊂ Γ 1 , (γ −1 1n ) is conical for the action of Γ 1 on M 1 . Thus the proposition follows.
8.2. Example of a family of non-convergence groups. Here, we give an example of a family of groups that do not act on a compact metrizable space as a non-elementary convergence group.
Proposition 8.6. Let G be a torsion-free group and let H be a subgroup of G satisfying the following:
• H is malnormally closed in G, i.e. there is no proper subgroup of G containing H, which is malnormal in G First of all, we collect some basic facts about subgroups of a convergence group.
Lemma 8.7. Suppose G acts on a compact metrizable space as a convergence group. Then a subgroup P of G isomorphic to Z ⊕ Z is parabolic.
Proof. Since P is abelian, |Λ(P )| ≤ 2. Let if possible |Λ(P )| = 2. Then P contains a loxodromic element p (say). The fixed point set of p, Fix(p) = Λ(P ) and Stab G (Λ(P )) contains P . By [Tuk94a, Theorem 2I], p has finite index in Stab G (Λ(P )). In particular, p has finite index in P , which is impossible. Hence |Λ(P )| = 1 and P is a parabolic subgroup.
A subgroup K of G is said to be weakly malnormal if for all g ∈ G \ H, |H ∩ gHg −1 | < ∞. We observe the following:
Lemma 8.8. Let G be a group that acts on a compact metrizable space as a convergence group. Then maximal parabolic subgroups are weakly malnormal.
Proof. Let P be a maximal parabolic subgroup with Λ(P ) = {p}. Suppose for some g ∈ G, |P ∩ gP g −1 | = ∞. As P ∩ gP g −1 ⊂ P , Λ(P ∩ gP g −1 ) = Λ(P ) = {p}. Similarly, Λ(P ∩ gP g −1 ) = Λ(gP g −1 ) = gΛ(P ). Hence g fixes the parabolic point p and therefore g ∈ P .
Note that if G is torsion-free in the above lemma, then maximal parabolic subgroups are malnormal. Now, we obtain the following: Lemma 8.9. Let Γ be the double of G along H as above. If H ∩ gHg −1 = {1} then H ∩ gHg −1 is a subgroup of a parabolic subgroup.
Proof. Let w ∈ H ∩ gHg −1 . Then w = ghg −1 for some h ∈ H. This implies h = g −1 wg. Since Γ is double of group G along H, g −1 wg =ḡ −1 wḡ. Thusḡg −1 commute with w. As Γ is torsion-free, ḡg −1 , w ≃ Z ⊕ Z. By the above lemma, ḡg −1 , w is a parabolic subgroup. Since w is an arbitrary element of H ∩ gHg −1 , H ∩ gHg −1 is a subgroup of a parabolic subgroup.
Now, we define neighborhoods around the points of (∂T ) ′ . Let ξ ∈ (∂T ) ′ and choose a base point v 0 in T . Firstly we define the subtree T m (ξ): it consists of the vertices w such that [v 0 , w] ∩ [v 0 , ξ) has length bigger than m. We set W m (ξ) = {ζ ∈ M |D(ζ) ⊂ T m (ξ)}. This definition does not depend on the choice of base point v 0 , up to shifting the indices.
an edge) Let p be a point in M and e an edge in T with at least one vertex not in D(p). Then there exists an integer m such that W n (p)∩X e = ∅.
Corollary 4. 3 .
3The group Γ acts on M as a convergence group.
Lemma 6 . 3 .
63Every point in Ω ′ which is image by π of a bounded parabolic point in some vertex stabilizer's boundary is a bounded parabolic point for Γ in M .
( 3 )
3Let B v , B ′ v be the set of translates of parabolic points corresponding to adjacent edge groups under the action of G v , G ′ v on their Bowditch boundaries respectively. For each vertex
Example 7 . 3 .
73Let Y be an edge and let F (a, b) be a free group of rank 2. LetG(Y ) be a double of free group F (a, b) along [a, b] and G ′ (Y ) be a double of free group F (a, b) along [a, b] 2 . Thus Γ = F (a, b) * [a,b] ≃ [ā,b] F (ā,b) and Γ ′ = F (a, b) * [a,b] 2 ≃ [ā,b] 2 F (ā,b). Assume thatF (a, b)is relatively hyperbolic with respect to [a, b] . By Theorem 1.6, Γ and Γ ′ are relatively hyperbolic with respect to[a, b] and [a, b] * [a,b] 2 ≃ [ā,b] 2 [ā,b]
Lemma 8.1. H is not a relatively quasi-convex subgroup of Γ.
• [Comm G (H) : H] > 1, where Comm G (H) denotes the commensurator of H in G.Consider the double Γ of the group G along H, i.e. Γ = G * H≃H G. Then, Γ does not act on a compact metrizable space as a non-elementary convergence group.
Remark 2.1. If H is a subgroup of Γ and Γ acts as a convergence group on a compact space M , then every conical limit point for H action on ΛH ⊂ M (see Definition 2.4(1) below) is a conical limit point for H acting in M and hence for Γ in M . Each parabolic point for H in ΛH is a parabolic point for Γ in M and its maximal parabolic subgroup in H is exactly the intersection of maximal parabolic subgroup in Γ with H.
a countable basis of open neighborhoods of X v . Without loss of generality, we can assume that for all v, the collection of open subsets
Answer to the above question does not give a counterexample to Olshanskii-Osin-Sapir conjecture because of the following:Question 8.11. Give an example of a non-elementary convergence group with a trivial Floyd boundary?Answer to Question 8.11 will also answer Question 8.10. Given a finitely presented group Q, Rips has constructed a hyperbolic group G and a surjection from G to Q. Let N be the kernel of this surjection. Then it is not known whether N is relatively hyperbolic with respect to a collection of proper subgroups of N or if it has trivial Floyd boundary.
H ∩ gHg −1 is a subgroup of a parabolic subgroup of Γ. Since [H : H ∩ gHg −1 ] < ∞, H and H ∩ gHg −1 sit inside the same maximal parabolic subgroup P (say). Now, by Lemma 8.8, the subgroup P is malnormal in Γ. As H is malnormally closed in G, G ⊂ P . Similarly, we can show that G ⊂ P . Hence Γ ⊂ P and therefore Γ is not a non-elementary convergence group. of Propostion 8.6: Since [Comm G (H) : H] > 1, let g ∈ Comm G (H) such that g / ∈ H. By Lemma 8.9Note: Let H, K be two subgroups of G such that H ⊂ K ⊂ G and 1 < [K :of Propostion 8.6: Since [Comm G (H) : H] > 1, let g ∈ Comm G (H) such that g / ∈ H. By Lemma 8.9, H ∩ gHg −1 is a subgroup of a parabolic subgroup of Γ. Since [H : H ∩ gHg −1 ] < ∞, H and H ∩ gHg −1 sit inside the same maximal parabolic subgroup P (say). Now, by Lemma 8.8, the subgroup P is malnormal in Γ. As H is malnormally closed in G, G ⊂ P . Similarly, we can show that G ⊂ P . Hence Γ ⊂ P and therefore Γ is not a non-elementary convergence group. Note: Let H, K be two subgroups of G such that H ⊂ K ⊂ G and 1 < [K :
Then one can check that K ⊂ Comm G (H). Also, assume that H is malnormally closed in G. Then the double Γ = G * H≃H G is not a non-elementary convergence group. < ∞ , < ∞. Then one can check that K ⊂ Comm G (H). Also, assume that H is malnormally closed in G. Then the double Γ = G * H≃H G is not a non-elementary convergence group.
By Floyd mapping theorem in [Ger12], we see that relatively hyperbolic groups have non-trivial Floyd boundary. The converse of this fact is a question of Olshanskii-Osin-Sapir [OOS09, Problem 7.11], i.e. if a finitely generated group has non-trivial Floyd boundary then it is hyperbolic relative to a collection of proper subgroups. We have the following question: Question 8.10. Give an example of a non. elementary convergence group that is not relatively hyperbolic? ReferencesBy Floyd mapping theorem in [Ger12], we see that relatively hyperbolic groups have non-trivial Floyd boundary. The converse of this fact is a question of Olshanskii- Osin-Sapir [OOS09, Problem 7.11], i.e. if a finitely generated group has non-trivial Floyd boundary then it is hyperbolic relative to a collection of proper subgroups. We have the following question: Question 8.10. Give an example of a non-elementary convergence group that is not relatively hyperbolic? References
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Non-positively curved complexes of groups and boundries. A Martin, Geom. Topol. 181A. Martin, Non-positively curved complexes of groups and boundries, Geom. Topol. 18(1) (2014), 31-102.
Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries. Alexandre Martin, Jacek , J. Group Theory. 182Alexandre Martin and Jacek, Infinitely-ended hyperbolic groups with homeomorphic Gromov boundaries, J. Group Theory 18 (2015), no. 2, 273-289.
A combination theorem for strong relative hyperbolicity. Mahan Mj, Lawrence Reeves, Geom. Topol. 123Mahan Mj and Lawrence Reeves, A combination theorem for strong relative hyperbol- icity, Geom. Topol. 12 (2008), no. 3, 1777-1798.
Lacunary hyperbolic groups. A Olshanskii, Denis V Osin, Mark V Sapir, Geom. Topol. Michael Kapovich and Bruce Kleiner134A. Olshanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051-2140, With an appendix by Michael Kapovich and Bruce Kleiner.
Elementary subgroups of relatively hyperbolic groups and bounded generation. Denis V Osin, Internat. J. Algebra Comput. 161Denis V. Osin, Elementary subgroups of relatively hyperbolic groups and bounded gen- eration, Internat. J. Algebra Comput. 16 (2006), no. 1, 99-118.
Bin Sun, A dynamical characterization of acylindrically hyperbolic groups. Algebr. Geom. Topol. 194Bin Sun, A dynamical characterization of acylindrically hyperbolic groups, Algebr. Geom. Topol. 19 (2019), no. 4, 1711-1745.
Convergence groups and Gromov's metric hyperbolic spaces. Pekka Tukia, New Zealand J. Math. 232Pekka Tukia, Convergence groups and Gromov's metric hyperbolic spaces, New Zealand J. Math. 23 (1994), no. 2, 157-187.
Generalizations of Fuchsian and Kleinian groups. Paris; Birkhäuser, BaselIIProgr. Math., Generalizations of Fuchsian and Kleinian groups, First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 447-461.
Conical limit points and uniform convergence groups. J. Reine Angew. Math. 501, Conical limit points and uniform convergence groups, J. Reine Angew. Math. 501 (1998), 71-98.
A topological characterisation of relatively hyperbolic groups. Asli Yaman, J. Reine Angew. Math. 566Asli Yaman, A topological characterisation of relatively hyperbolic groups, J. Reine Angew. Math. 566 (2004), 41-89.
Limit sets of relatively hyperbolic groups. Wen-Yuan Yang, Geom. Dedicata. 156Wen-yuan Yang, Limit sets of relatively hyperbolic groups, Geom. Dedicata 156 (2012), 1-12.
| []
|
[
"MULTI-MODALITY IMAGE SUPER-RESOLUTION USING GENERATIVE ADVERSARIAL NETWORKS",
"MULTI-MODALITY IMAGE SUPER-RESOLUTION USING GENERATIVE ADVERSARIAL NETWORKS"
]
| [
"Aref Abedjooy [email protected] \nFaculty of Science\nFaculty of Science\nOntario Tech University\nOshawaOntarioCanada\n",
"Mehran Ebrahimi [email protected] \nOntario Tech University\nOshawaOntarioCanada\n"
]
| [
"Faculty of Science\nFaculty of Science\nOntario Tech University\nOshawaOntarioCanada",
"Ontario Tech University\nOshawaOntarioCanada"
]
| []
| Over the past few years deep learning-based techniques such as Generative Adversarial Networks (GANs) have significantly improved solutions to image super-resolution and image-to-image translation problems. In this paper, we propose a solution to the joint problem of image super-resolution and multi-modality image-to-image translation. The problem can be stated as the recovery of a high-resolution image in a modality, given a low-resolution observation of the same image in an alternative modality. Our paper offers two models to address this problem and will be evaluated on the recovery of high-resolution day images given low-resolution night images of the same scene. Promising qualitative and quantitative results will be presented for each model. | 10.33965/mccsis2022_202206l013 | [
"https://arxiv.org/pdf/2206.09193v2.pdf"
]
| 249,889,271 | 2206.09193 | 1a30e479d2223a9ea025414c811136d09d25f44d |
MULTI-MODALITY IMAGE SUPER-RESOLUTION USING GENERATIVE ADVERSARIAL NETWORKS
Aref Abedjooy [email protected]
Faculty of Science
Faculty of Science
Ontario Tech University
OshawaOntarioCanada
Mehran Ebrahimi [email protected]
Ontario Tech University
OshawaOntarioCanada
MULTI-MODALITY IMAGE SUPER-RESOLUTION USING GENERATIVE ADVERSARIAL NETWORKS
Image super-resolutionImage-to-image translationGenerative adversarial networksDeep learning
Over the past few years deep learning-based techniques such as Generative Adversarial Networks (GANs) have significantly improved solutions to image super-resolution and image-to-image translation problems. In this paper, we propose a solution to the joint problem of image super-resolution and multi-modality image-to-image translation. The problem can be stated as the recovery of a high-resolution image in a modality, given a low-resolution observation of the same image in an alternative modality. Our paper offers two models to address this problem and will be evaluated on the recovery of high-resolution day images given low-resolution night images of the same scene. Promising qualitative and quantitative results will be presented for each model.
RELATED WORK
To best of our knowledge, no studies have been conducted on the use of GANs to combine image superresolution with image-to-image translation.
The idea of image-to-image translation dates back to Hertzmann et al.'s image analogies (Hertzmann, A. et al, 2001), a non-parametric model that uses a pair of images to generate image transformations . There are many problems relating to computer vision and computer graphics applications that are instances of the image-to-image translation problem. Image-to-image translation involves mapping of an image given in a domain to a copy of the image in a different target domain. An example of the translation problem could be the mapping of grayscale images to RGB. In order to learn how to map one visual representation to another, it is necessary to understand the underlying features that are shared among these representations, such features can be either domain independent or domain-specific (Pang, Y. et al, 2021).
During translation, domain-independent features represent the underlying spatial structure and should be preserved (e.g., the content should be preserved when translating a natural image to Van Gogh's style), whereas domain-specific features relate to the rendering of the structure and may need to be changed during translation (e.g., if the style needs to be changed when translating the image to Van Gogh's style) (Alotaibi, A., 2020).
It is challenging to learn the mapping between two or more domains. It is sometimes difficult to collect a pair of images or the relevant images may not exist. Another challenge is where one input image can be mapped to multiple outputs. The development and use of GANs and their variants in image-to-image translation have provided state-of-the-art solutions in recent years (Alotaibi, A., 2020), (Park, T. et al, 2020), (Emami, H. et al, 2020).
Over the past few decades, researchers have been exploring image super-resolution, a technique for reconstructing images from low-resolution observations into higher resolution images. In particular, deep learning-based SR approaches have attracted much attention and have significantly improved reconstruction accuracy on synthetic data (Bashir, S.M.A. et al, 2021). GANs can be used to enhance lower quality images by reducing noise and enhancing sharpness and contrast along with the resolution (Gupta, R. et al, 2020), (Nazeri, K. et al, 2019). A super-resolution GAN combines a deep network with an adversary network to create higher resolution images (Wang, X. et al, 2021).
METHODOLOGY
The problem description and a possible methodology are presented in this Section.
Problem description
It is possible to formalize the task of combining super-resolution and image-to-image translation in the following way. While this paper presents the problem in the context of daytime and nighttime images, our approach can be generalized to any arbitrary multi-modality super-resolution scenario.
A LR input image in nighttime mode of a scene is provided to the model. The model will generate a corresponding HR daytime image of the same scene such that A. Its contents are meaningful and coherent with so that human eye would accept it as a HR daytime image of . B. It is similar to ground truth image , when it is available. C. Based on a daytime image, it has a realistic-looking texture and color spectrum.
System overview
To address this problem, we propose and investigate two different two-stage models based on GANs.
M1: SR-first model
First stage in this model would be to generate using Real-ESRGAN (Wang, X. et al, 2021). is the generated HR version of . Then, CUT (Park, T. et al, 2020), which is a newer version of CycleGAN , is fed by NHR as an input image.
would be translated to HR daytime image in the second stage. Both Real-ESRGAN and CUT are GAN-based models and have to be trained and tuned. This approach is depicted in Figure 1.
M2: Translation-first model
This model examines the opposite order of the operations of the SR-first model. It means that the CUT (Park, T. et al, 2020) is used in the first stage to translate into LR daytime image followed by the second stage which is using the Real-ESRGAN (Wang, X. et al, 2021) to generate HR version of daytime image . Dimensions of input images and dimensions of output images in the CUT (Park, T. et al, 2020) are the same. Therefore, the zooming factor could be set in Real-ESRGAN (Wang, X. et al, 2021). Figure 2 illustrates how this approach works.
EXPERIMENTS AND RESULTS
In this Section, the proposed models are assessed quantitatively and qualitatively. We will also explain the dataset and setup.
Dataset
The Transient Attributes dataset (Laffont, P.Y. et al, 2014) was used to create the night2day dataset. Each original sample is of size 256 × 512 containing two 256 × 256 images, i.e., one day image along with its corresponding night image. These images have been divided into 20,110 night images and 20,110 corresponding day images. The dataset was then randomly split into training, validation, and test subsets for each night and day categories. More precisely, 2,011 or 10% for testing, 2,011 or 10% for validation, and 16,088 or 80% for training phases.
In SR, down-sampling is the basic operation for synthesizing LR images. In general, we consider both down-sampling and up-sampling as resizing. In addition to the nearest neighbor interpolation algorithm, there are also bilinear interpolation and bicubic interpolation algorithms. A resize operation can produce a variety of effects; some produce blurry results, whereas others may produce overly-sharp images.
In this study, the bilinear interpolation is used to convert dataset into LR images. To resize an image, OpenCV's "cv2.resize()" function can be used. This function which uses the bilinear interpolation (INTER LINEAR) as its default interpolation method, is used to produce low-resolution version (64×64) of each sample data.
Considering zooming factor of 4 in each direction, after creating a low-resolution version of each image, we have 80% or 16,088 LR night images (64×64) and 80% or 16,088 LR day images (64 × 64) for training, 10% or 2,011 LR night images (64×64) and 10% or 2,011 LR day (64×64) images for validation, and 10% or 2,011 LR night images (64×64) and 10% or 2,011 LR day images (64×64) for testing. The nearest-neighbor interpolation was used to up-sample the LR images when comparing it to the HR image.
Training
There are two stages of training for each model. To start the training for the first stage of the M1 model, a pretrained Real-ESRGAN (Wang, X. et al, 2021) was used. Real-ESRGAN uses only HR images (256×256) for training and validation. Thus, there is no need to use the resized training data and the resized validation data at this stage. The pre-trained model is trained for 300 epochs on the night2day dataset. It is optimized using the Adam optimizer using an initial learning rate of 0.0002.
In the second stage of the M1 model, we follow the setting of the CUT (Park, T. et al, 2020). For training parameters, the number of epochs is 200. The Adam optimizer (Kingma, D.P. and Ba, J., 2014) with β1 = 0.5 and β2 =0.999 is used to optimize the model. The Initial learning rate for Adam optimizer is 0.0002. The type of GAN objective is LSGAN (Mao, X. et al, 2017). Size of image buffer that stores previously generated images is 50.
In the M2 model, first stage is translation of the LR night image of size (64×64) into the LR day image (64×64). The CUT (Park, T. et al, 2020) input image and output image sizes are changed from 256×256 to 64×64 by adjusting the scaling factor. Therefore, LR night images and LR day images are used to train the translation module. In order to keep consistency, the CUT's settings from the previous approach were used. In the second stage of the M2 model, a trained Real-ESRGAN (Wang, X. et al, 2021) model was used with the same settings as the first stage of the M2 model.
We conducted all experiments using the Linux Ubuntu operating system and the NVIDIA GeForce GTX TITAN X GPU.
SR-first model (M1) Evaluation
A model evaluation is performed using 2,011 LR nighttime images (64×64). Three evaluation phases have been defined for a better quantitative evaluation of the M1 model. Figure 3. Figure 3 shows that the RMSE (top-left) and the MAE (top-right) are shifted to the left from the "M1-Pre" state to the "M1-Post" state. Lower value of these measures for final results in "M1-Post" indicates better performance. Similarly, the "M1-Post" phase provides higher values for SSIM (bottom-left), and the NCC (bottom-right) in comparison with the "M1-Intermediate" and the "M1-Pre" states. Table I In addition, M2-Intermediate phase could have also been defined as comparing the generated LR day image (64×64) (after translation) VS Real day image i.e. day ground truth image. However, the output size of the translation module in the M2 model would be different from that of the first stage if it had been up-sampled. As a result, this evaluation phase was not considered and the total performance of the M2 model is evaluated only by comparing M2-Pre with M2-Post.
Translation-first model (M2) Evaluation
For RMSE, MAE, SSIM, NCC, and FID, we compare the results of these evaluation phases. Figure 5 illustrates these comparisons.
In Figure 5, it can be seen that the RMSE (top-left) and the MAE (top-right) are shifted to the left significantly from the "Pre" to the "Post". Therefore, lower values of these measures for final results in M2-Post indicate better performance. The SSIM (bottom-left) and NCC (bottom-right) are shifted to the right in the "Post", which means that the generated images are more similar to the desired output.
In Table 2, these measures are listed along with their means and standard deviations for M2-Pre, and M2-Post. Furthermore, FID metrics for each phase are calculated. There is a ↓ sign next to each measure indicates that a lower value is "better", and similarly, a ↑ sign means a higher value is "better". Some examples of images for different stages of the M2 model is provided in Figure 6 for qualitative evaluation. From left to right, each column shows the following: a) Input image (LR night image), b) The generated LR day image (after translation), c) Output (generated HR day), and d) Real day image (day ground truth image).
SR-first model (M1) VS Translation-first model (M2)
Comparing the M1-Post and M2-Post results will enable us to compare the M1 and M2 models' performance. In both models, the Post phase is comparing the generated HR day image (Output) VS Real day image or day ground truth image. This comparison using the RMSE (top-left), the MAE (top-right), the SSIM (bottom-left), and the NCC (bottom-right) is depicted in Figure 7. Table 3 summarizes these comparisons. The models can be assessed qualitatively using figures 4 and 6. The quantitative and qualitative results indicate that the two models perform similarly and that there is no significant difference between them.
CONCLUSION
The problem of combining the image-to-image translation with the image super-resolution was proposed and addressed by concatenating two existing modules and rigorously evaluating the performance of both models after training. This paper describes the steps involved in preparing the selected dataset for the defined problem. In addition, a variety of evaluation phases are used to evaluate M1 and M2 models. Qualitative and quantitative analysis indicate that both models perform similarly.
By performing both tasks simultaneously, an end-to-end generative network may provide more accurate results. Our study focuses on the night-to-day image super-resolution and provides promising results.
In many image enhancement tasks including medical imaging, a similar approach can be used to generates realistic high-resolution image of a desired modality based on a given low resolution image in a different modality.
Figure 1 .
1SR-first Model (M1); Super-resolution module followed by an image-to-image translation module.
Figure 2 .
2Translation-first Model (M2); An image-to-image translation module followed by a super-resolution module.
Figure 3 .
3Comparison of final results (Pre vs Intermediate vs Post) of the SR-first model (M1) using different measures. Some examples of images for different stages of the M1 model is provided inFigure 4for qualitative evaluation. From left to right, each column shows the following: a) Input image (LR night image), b) The generated HR night image (after super-resolution), c) Real night image (night ground truth image), d) Output (generated HR day), and e) Real day image (day ground truth image).
Figure 4 .
4Qualitative evaluation of M1 model; from left to right; (a) Input image, (b) Generated HR night image, (c) Night ground truth, (d) Output image, (e) Day ground truth.
Figure 5 .
5Comparison of final results (Pre vs Post) of the Translation-first model (M2) using different measures.
Figure 6 .
6Qualitative evaluation of the translation-first model (M2) model; from left to right; (a) Input (LR night image), (b) The generated LR day (after translation), (c) Output (generated HR day), and (d) Real day image (day ground truth)
Figure 7 .
7Comparison of final results of the M1: SR-first with M2: Translation-first models using different metrics.
• M1 -
M1Pre: Comparing the LR night image (64 × 64) (Input) VS Real day image i.e. Day Ground truth image. • M1-Intermediate: Comparing the generated HR night image with 256×256 resolution (After SR) VS Real day image i.e. Day ground truth image. • M1-Post: Comparing the generated HR day image with 256×256 resolution (Output) VS Real day image i.e. Day ground truth image. By comparing M1-Pre with M1-Intermediate and M1Post, the overall performance of the M1 model can be assessed. Root Mean Square Error (RMSE), Mean Absolute Error (MAE), Structural Similarity Index Measure (SSIM), Normalized Cross-Correlation (NCC), and Fréchet inception distance (FID) are calculated for each phase and histograms of these similarity measures over the test data are provided. The comparisons are shown in
presents the mean and the standard deviation of these measures for Pre, Intermediate, and Post. The FID metric for each phase is computed as well. Based on all these measures,Table 1suggests that final results from the model are more consistent with ground truth.Table 1. Comparing final results M1-Pre vs M1-Intermediate vs M1-Post of the SR-first model (M1) using different measures.M1-Pre
M1-Intermediate
M1-Post
RMSE (↓)
0.48±0.14
0.36±0.10
0.23±0.08
MAE (↓)
0.43±0.14
0.30±0.09
0.06±0.04
SSIM (↑)
0.34±0.14
0.36±0.14
0.43±0.14
NCC (↑)
0.81±0.14
0.83±0.13
0.91±0.08
FID (↓)
173.95
146.76
90.47
Table 2 .
2Comparing final results M2-Pre vs M2-Post of the Translation-first model (M2) using different measures.M2-Pre
M2-Post
RMSE (↓)
0.48±0.14
0.24±0.08
MAE (↓)
0.43±0.14
0.07±0.07
SSIM (↑)
0.34±0.14
0.44±0.14
NCC (↑)
0.81±0.14
0.90±0.08
FID (↓)
173.95
96.56
Table 3 .
3Comparing final results of the M1: SR-first with M2: Translation-first models using different measures.M2
M1
RMSE (↓)
0.24±0.08
0.23±0.08
MAE (↓)
0.07±0.07
0.06±0.04
SSIM (↑)
0.44±0.14
0.43±0.14
NCC (↑)
0.90±0.08
0.91±0.08
FID (↓)
96.56
90.47
ACKNOWLDGEMENTSThis work was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC).
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| []
|
[
"Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes",
"Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes",
"Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes",
"Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes"
]
| [
"Lotte Paulissen \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Alex Alvarado [email protected] \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Kaiquan Wu \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Alexios Balatsoukas-Stimming \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Lotte Paulissen \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Alex Alvarado [email protected] \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Kaiquan Wu \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n",
"Alexios Balatsoukas-Stimming \nEindhoven University of Technology\nThe Netherlands5600MBEindhoven\n"
]
| [
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven",
"Eindhoven University of Technology\nThe Netherlands5600MBEindhoven"
]
| []
| Some low-complexity LDPC decoders suffer from error floors. We apply iteration-dependent weights to the degree-3 variable nodes to solve this problem. When the 802.3ca EPON LDPC code is considered, an error floor decrease of more than 3 orders of magnitude is achieved. | 10.48550/arxiv.2206.11532 | [
"https://export.arxiv.org/pdf/2206.11532v1.pdf"
]
| 249,953,732 | 2206.11532 | 8fda6438dedc3818022ddedf6eba1965c489d1eb |
Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes
Lotte Paulissen
Eindhoven University of Technology
The Netherlands5600MBEindhoven
Alex Alvarado [email protected]
Eindhoven University of Technology
The Netherlands5600MBEindhoven
Kaiquan Wu
Eindhoven University of Technology
The Netherlands5600MBEindhoven
Alexios Balatsoukas-Stimming
Eindhoven University of Technology
The Netherlands5600MBEindhoven
Reducing the Error Floor of the Sign-Preserving Min-Sum LDPC Decoder via Message Weighting of Low-Degree Variable Nodes
Some low-complexity LDPC decoders suffer from error floors. We apply iteration-dependent weights to the degree-3 variable nodes to solve this problem. When the 802.3ca EPON LDPC code is considered, an error floor decrease of more than 3 orders of magnitude is achieved.
Introduction
Increasing data rates in fiber optical communication systems require powerful forward error correction schemes to provide reliable transmission. High-throughput systems also have stringent power consumption requirements, therefore, hardware-friendly decoders are key importance. The 802.3ca EPON standard [1] has adopted lowdensity parity check (LDPC) codes due to their near-capacity performance, but also because LDPC decoders are highly parallelizable.
The belief-propagation (BP) [2] algorithm for LDPC decoding offers excellent performance [3] but is complex to implement due to the hyperbolic tangent update rule at the check nodes (CNs). The complexity can be reduced by simplifying the update rule at the CNs, leading for example to the popular min-sum (MS) decoder [4] . Recently, the sign-preserving min-sum (SP-MS) decoder was introduced [5] as a low-complexity alternative for the MS decoder. The SP-MS decoder is designed for quantized messages taken from a finite alphabet (with typical alphabet sizes of 4, 8, or 16) and it was shown to perform well for the regular LPDC code with degree-6 variable nodes (VNs) [5] defined in the IEEE 802.3 10G Ethernet standard [6] . Using the SP-MS decoder with small alphabet sizes is potentially very interesting for high-speed optical fiber applications.
In this paper, we study the applicability of very low complexity versions of the SP-MS decoder (alphabet sizes 4 and 8) for the 802.3ca EPON LDPC code. The first contribution of this paper is to show that such decoders result in an error floor for the EPON LDPC code. We also show that the reason for this floor is the degree-3 VNs present in the 802.3ca EPON LDPC code and the difficulties very low complexity versions of the SP-MS decoder has dealing with such low-degree VNs [7] . 978-1-6654-3868-1/21/$31.00 ©2022 IEEE The second contribution of this paper is to propose a method to reduce the error floor for the SP-MS decoder. The main idea is to weight up the CN-to-VN messages to allow the decoder to "escape" situations where the channel messages dominate the strongly quantized messages in the SP-MS decoder. This idea also means that weights are only required for low-degree VNs, while for high-degree VNs, the incoming messages can make the decoder escape problematic situations.
Numerical results for the 802.3ca EPON LDPC code show that the bit error rate (BER) error floor can be reduced by more than three orders of magnitude by applying weights to degree-3 VNs only. These results are based on a hardwarefriendly implementation of our algorithm, where all weights are represented using three bits only.
Quantized Decoding
As shown in Fig. 1, LDPC coded bits x n (n = 1, 2, . . . , N ) are modulated using binary-phase shift keying (BPSK) and transmitted over the additive white Gaussian noise (AWGN) channel. At the receiver, log-likelihood ratios (LLRs) L n are calculated. In this paper, we use 4 bits for the quantization of the LLRs. Using more bits for the LLR quantization does not affect the conclusions in this paper. In the quantization process, a constant scaling factor α is used to map the channel LLRs to their corresponding quantized LLRs I n . When required, the scaling α was numerically optimized in the numerical simulations. These quantized LLRs are used as input of the LDPC decoder.
In the LDPC decoder, messages are iteratively exchanged between VNs and CNs in order for the BPSK Enc.
AWGN LLR
Quan. Dec.
inf. bits x n y n L n I nxn Fig. 1
m ( +1) vn→cm = L n + c ext m m ( ) c→vn .(1)
We target low-complexity decoders where the messages in the LDPC decoder are strongly quantized. We consider q = {2, 3, 4} precision bits, with alphabet sizes 4, 8 and 16 resp. When the messages in the decoder are quantized to 2 or 3 precision bits, there is a mismatch between the channel LLRs (quantized using 4 bits) and incoming messages in the VN update rule (see (1)). As we will show below, this causes an error floor.
In this paper, we propose to modify the SP-MS decoder [5] by adding an iteration dependent weight w ( ) n . The modified SP-MS update rule is
m ( +1) vn→cm = Ψ I n + w ( ) n µ ( ) vn→cm 2 + c ext m m ( ) c→vn ,(2)
which coincides with the original SP-MS decoder when w ( ) n = 1 for n = 1, 2, .., N and ∀ . In (2), µ ( ) vn→cm is the sign-preserving factor to ensure the VN update rule never generates an erased message, which is a message that carries no information, and therefore, does not contribute to the convergence of the decoder. Additionally, quantized decoders require a function Ψ(m s ) = sign(m s ), S max ( |m s | − ϕ v , 0) , 2 q−1 − 1 , to saturate the outgoing messages back to the specified message alphabet. S(·) is a saturation function clipping the message m s to 2 q−1 − 1 when its magnitude is larger than 2 q−1 − 1, and ϕ v is an offset factor dependent on the magnitude of the message [5] .
In this paper, we consider the IEEE 802.3ca EPON irregular LDPC code [1] with N = 17664, R = 0.826, and with VN degree distribution polynomial λ(X) = 12800 17664 X 2 + 4352 17664 X 5 + 256 17664
X 10 + 256 17664
X 11 .(3)
Throughout this paper, we always consider a maximum of 12 decoding iterations. For all simulation results, a minimum of 500 frames are sent
SNR [dB]
Frame error rate (FER) with at least 30 frame errors occurring for each signal-to-noise ratio (SNR) point. Fig. 2 shows the BER and frame error rate (FER) performance of the SP-MS decoder (using (2) with w ( ) n = 1) with q = {2, 3, 4} (dashed lines). These results were obtained using α = 0.75, 0.95, 1.15 for q = 2, 3, 4, resp. As reference, we also show the pre-FEC BER and the performance of the BP decoder using (1) (solid lines). These results indicate that SP-MS with q = 4 (dashed green) offers performance close to that of the BP decoder, but they also show an unacceptably high error floor for the cases q = 2 and q = 3 (BER ≈ 10 −3 and ≈ 10 −6 , resp). Similar conclusions can be drawn from the FER results. We will show in the next section that by adjusting w ( ) n in (2), the error floor can be significantly lowered.
2-bit SP-MS 3-bit SP-MS 4-bit SP-MS 2-bit weighted SP-MS 3-bit weighted SP-MS
Weighted SP-MS Decoder
In this section, we propose to modify the SP-MS decoder by introducing the coefficient w ( ) n in (2). The rationale behind this is as follows. For the BP decoder, by performing the update rules and exchanging messages iteratively, more bits are likely to be correct at each iteration. Therefore the importance of the incoming messages increases compared to the channel LLRs as the iteration number grows. The introduced weighting factor is used to mimic this evolution and to adjust the importance of the magnitude of the incoming messages compared to the channel LLRs in the VN and tentative update rule. This is realized by applying an iteration-dependent weighting factor to the incoming messages. As mentioned before, for low values of q, there is a mismatch between the alphabets of the channel LLRs and the messages inside the decoder. We conjecture that the decoder is in general able to overcome this mismatch for VNs with a large number of incoming messages. However, the decoder might be unable to recover from bad channel LLRs for low-degree VNs. In what follows we will show that this is indeed the case and that the error floors in Fig. 2 are due to the degree-3 VNs.
Based on the observation that the LDPC code under consideration has a large number of degree-3 VNs (≈ 72 % of the total, as shown in (3)), we propose to apply weights w ( ) n = 1 to degree-3 VNs so that the mismatched alphabets can be taken care of. This approach does not only allow us to prove our conjecture but also to reduce complexity as not all VNs need to be weighted. Furthermore, using this approach, the LLR quantization coefficient α does not need to be optimized. For simplicity, in our results we use the same α values used for the SP-MS decoder.
The weights need to be chosen carefully, because suboptimal weights can significantly decrease the performance of the decoder. The weight vectors are determined by selecting the one resulting in the lowest BER/FER performance from a set of random candidate weight vectors. Furthermore, to simplify parameter optimization, avoid overestimation, and to follow the rationale explained above, the weights are constrained such that 0 < w ( )
n ≤ w ( +1) n .
The modified SP-MS update rule in (2) requires a multiplication operation. The hardware implementation of a multiplication has a high complexity when compared to the remaining operations required by the SP-MS decoder (simple additions and comparisons). Moreover, supporting opti-mum weights requires a large number of quantization bits, which further complicates the multiplication operation. We thus propose a hardwarefriendly implementation of the modified SP-MS decoder, where the weights w BER and FER results using the weights in Tab. 1 are shown in Fig. 2 (solid lines with squares). By applying iteration-dependent weights to degree-3 VNs quantized to only 3 bits, the error floor can be reduced by more than three orders of magnitude. The weights reported in Tab. 1 were optimized for an SNR of 2.8 dB and 3.1 dB for q = 3 and q = 2, resp. For the first four iterations, the obtained weights w ( ) n are equal to one. Therefore, we only require shift-and-add operations for the remaining 8 iterations.
Conclusions
In this paper, we have shown that an error floor occurs when we have mismatched alphabets between channel LLRs and incoming messages for the SP-MS decoder. The error floor occurs due to degree-3 VNs, which are the majority of the VNs present in the 802.3ca EPON LDPC code. We proposed a method to reduce this error floor by applying iteration-dependent weights to the incoming messages of degree-3 VNs. Our hardware friendly implementation of the modified SP-MS decoder is able to reduce the error floor by more than three orders of magnitude. This method is a general method that next to the SP-MS decoder, can be used for all low-complexity decoders that suffer an error floor due to mismatched alphabets. Further research should be conducted to identify why we observe an error floor at lower BER in order to design a lowcomplexity LDPC decoder that can achieve error floor-free performance close to the BP decoder.
Fig. 2 :
2BER (top) and FER (bottom) performance of the q-bit SP-MS decoder (dashed lines) and the proposed q-bit degree-3 weighted SP-MS decoder (solid lines with squares).
limited to a sum of up to three powers of two. The obtained results are reported in Tab. 1 for q = {2, 3}.
: System model. cm→vn denotes a CN-to-VN message at iteration . All CNs that are connected to VN n except for CN m are denoted by c ext m . At the VNs of the BP decoder, we add the likelihoods from all neighbouring CNs to the channel message. For the unquantized BP decoder, this can be defined asarXiv:2206.11532v1 [cs.IT] 23 Jun 2022
decoder to converge towards a valid codeword.
A VN-to-CN message at iteration is denoted as
m
( )
vn→cm , and m
( )
Acknowledgements:The work of K. Wu and A. Alvarado has received funding from the Netherlands Organisation for Scientific Research via the VIDI Grant ICONIC (project number 15685). The work of A. Alvarado has also received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (project 963945).
Standard for Ethernet Amendment 9: physical layer specifications and management parameters for 25 Gb/s and 50 Gb/s passive optical networks. IEEE Std. 802"Standard for Ethernet Amendment 9: physical layer specifications and management parameters for 25 Gb/s and 50 Gb/s passive optical networks", IEEE Std P802.3ca-2020 (2020).
The capacity of lowdensity parity-check codes under message-passing decoding. T J Richardson, R L Urbanke, 10.1109/18.910577IEEE Trans. Inf. Theory. 472T. J. Richardson and R. L. Urbanke, "The capacity of low- density parity-check codes under message-passing de- coding", IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 599- 618, Feb. 2001. DOI: 10.1109/18.910577.
On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit. S.-Y Chung, G Forney, T J Richardson, R Urbanke, 10.1109/4234.905935DOI: 10.1109/ 4234.905935IEEE Commun. Lett. 52S.-Y. Chung, G. Forney, T. J. Richardson, and R. Ur- banke, "On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit", IEEE Commun. Lett., vol. 5, no. 2, pp. 58-60, Feb. 2012. DOI: 10.1109/ 4234.905935.
Reduced complexity iterative decoding of low-density parity check codes based on belief propagation. M P C Fossorier, M Mihaljević, H Imai, 10.1109/26.768759DOI: 10 . 1109/26.768759IEEE Trans. Commun. 475M. P. C. Fossorier, M. Mihaljević, and H. Imai, "Reduced complexity iterative decoding of low-density parity check codes based on belief propagation", IEEE Trans. Com- mun., vol. 47, no. 5, pp. 673-680, May 1999. DOI: 10 . 1109/26.768759.
Signpreserving min-sum decoders. F Cochachin, E Boutillon, D Declercq, 10.1109/TCOMM.2021.3099173DOI: 10.1109/ TCOMM.2021.3099173IEEE Trans. Commun. 6910F. Cochachin, E. Boutillon, and D. Declercq, "Sign- preserving min-sum decoders", IEEE Trans. Commun., vol. 69, no. 10, pp. 6439-6454, Oct. 2021. DOI: 10.1109/ TCOMM.2021.3099173.
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| []
|
[
"On propagation of positive and negative streamers in air in uniform electric fields",
"On propagation of positive and negative streamers in air in uniform electric fields"
]
| [
"G V Naidis [email protected] \nJoint Institute for High Temperatures\nRussian Academy of Sciences\n125412MoscowRussia\n",
"Yu Babaeva \nJoint Institute for High Temperatures\nRussian Academy of Sciences\n125412MoscowRussia\n"
]
| [
"Joint Institute for High Temperatures\nRussian Academy of Sciences\n125412MoscowRussia",
"Joint Institute for High Temperatures\nRussian Academy of Sciences\n125412MoscowRussia"
]
| []
| Recently published results of numerical simulations of positive and negative streamers propagating in uniform electric fields in air are analyzed here in the framework of an analytical approach. Obtained approximate relations between the streamer radius, velocity and length, depending on the value of applied electric field, are in reasonable agreement with the results of numerical simulations. Ionization wavesstreamers formed in dielectric media at application of sufficiently strong electric fields are actively studied as key elements in pre-breakdown phenomena [1,2]. Recently a number of papers have been published [3-10] on numerical modeling of streamers propagating in air in uniform electric fields. Various aspects of the dynamics of accelerating and decelerating positive and negative streamers have been studied in a wide range of external conditions: gas pressure, gap length, applied voltage, etc. In this work, results obtained in [3-10] and in earlier publications on this topic [11,12] are analyzed in the framework of approximate analytical approach [13,14].The dynamics of streamers developing in short regions of strong electric field, e.g.near needle protrusions at plate electrodes, and later propagating in much longer regions of relatively weak uniform electric field, e.g. between plate electrodes, is considered. The character of streamer motion in uniform fields depends on the relation between the values of the applied field E and the field in the streamer channel Ec. Below the conditions are considered when the losses of electrons in the channel, due to attachment to gas molecules and electron-ion recombination, during streamer propagation are small. In this case, variation of the field Ec along the channel is rather weak. The fall Uc of electric field potential along the head and channel of a streamer having the length L can be evaluated as Uc = Uh + LEc, | null | [
"https://export.arxiv.org/pdf/2305.09560v1.pdf"
]
| 258,714,652 | 2305.09560 | e91d26f9be1ea57bb15ecb2746a9490834a7e1b5 |
On propagation of positive and negative streamers in air in uniform electric fields
G V Naidis [email protected]
Joint Institute for High Temperatures
Russian Academy of Sciences
125412MoscowRussia
Yu Babaeva
Joint Institute for High Temperatures
Russian Academy of Sciences
125412MoscowRussia
On propagation of positive and negative streamers in air in uniform electric fields
Recently published results of numerical simulations of positive and negative streamers propagating in uniform electric fields in air are analyzed here in the framework of an analytical approach. Obtained approximate relations between the streamer radius, velocity and length, depending on the value of applied electric field, are in reasonable agreement with the results of numerical simulations. Ionization wavesstreamers formed in dielectric media at application of sufficiently strong electric fields are actively studied as key elements in pre-breakdown phenomena [1,2]. Recently a number of papers have been published [3-10] on numerical modeling of streamers propagating in air in uniform electric fields. Various aspects of the dynamics of accelerating and decelerating positive and negative streamers have been studied in a wide range of external conditions: gas pressure, gap length, applied voltage, etc. In this work, results obtained in [3-10] and in earlier publications on this topic [11,12] are analyzed in the framework of approximate analytical approach [13,14].The dynamics of streamers developing in short regions of strong electric field, e.g.near needle protrusions at plate electrodes, and later propagating in much longer regions of relatively weak uniform electric field, e.g. between plate electrodes, is considered. The character of streamer motion in uniform fields depends on the relation between the values of the applied field E and the field in the streamer channel Ec. Below the conditions are considered when the losses of electrons in the channel, due to attachment to gas molecules and electron-ion recombination, during streamer propagation are small. In this case, variation of the field Ec along the channel is rather weak. The fall Uc of electric field potential along the head and channel of a streamer having the length L can be evaluated as Uc = Uh + LEc,
where Uh is the potential of streamer head.
On the other hand, the potential of applied field at the distance L from the point of streamer initiation is equal to U0 + LE, where U0 is the fall of potential in the small region near initiating electrode. According to these relations, the head potential Uh = U0 + L(EEc) varies during streamer propagation proportionally to the streamer length, increasing at E > Ec and decreasing at E < Ec. Stable propagation of streamers, with the velocity and radius independent of the streamer length, takes place at the applied field value Es equal to the field in the channel Ec.
The streamer head potential Uh is nearly proportional to the product of streamer radius R and electric field Eh in the streamer head. Results of numerous experimental and computational studies show that in conditions of streamer propagation in the applied fields E not much stronger than Es the values of Eh vary in a rather narrow interval. In particular, for streamers in air the reduced electric field in the head Eh/, where is the ratio of gas density to its normal value (at room temperature and atmospheric pressure), is typically within the intervals 120-160 kV/cm and 100-120 kV/cm for positive and negative streamers, respectively [14]. Therefore, the streamer radius R varies with the length L, similarly to Uh, nearly proportionally to L(EEs), according to the equation [13] dR/dL = a(E/Es-1).
(1)
The reduced streamer radius R is related with the streamer velocity V nearly as V = bR, where the coefficient b is governed by the value Eh/ of reduced electric field in the streamer head [13,14]. The use of this relation, together with equation (1), gives, in assumption that variation of Eh/ and hence of the coefficient b during streamer propagation is weak (note that this assumption is not valid for decelerating streamers approaching stopping points, see below), the equation
dV/d(L) = bdR/dL = ab(E/Es-1).(2)
The coefficient a in equation (1) is roughly estimated, using the relations presented above, as a ~ Es/Eh. Note that accurate values of the parameters Es, a and b in equations (1) and (2) cannot be obtained in the framework of approximate analytical approach. Below they are estimated using the results of computations of positive [3][4][5][6][7][8][9]11] and negative [10,12] air streamers propagating in uniform applied fields. The profiles of streamer radius and velocity versus the length obtained in these computations are rather close to linear (at least as some rather long intervals of the streamer length variation), with slopes depending on the value of applied field.
Note that the data on streamer radius presented in the mentioned papers refer to the optical radius [4,7,10] or to the streamer head radius [3,6,8], the latter corresponding to the radial coordinate, just behind the streamer head, corresponding to the maximum of radial electric field. Though generally these radii do not coincide, in the case when the streamer head has a hemispherical shape the head radius is close to the optical radius [15].
In figure 1 the values dR/dL estimated for positive streamers using the plots of the streamer radius versus the length, presented in [3,4,6,8] for pressure P = 1 bar and in [7] for P = 0.1 bar, are shown versus the reduced applied electric field E/. Linear dependence (1), with parameters a = 0.011 and Es/ = 4.7 kV/cm, is also given. It is seen that all the data are close to the same linear dependence. Note that at given E/ the slopes dR/dL evaluated using the data obtained in [3] for the head radius and, for similar conditions, in [4] for the optical radius are close to each other, hence it can be expected that in these conditions the relative difference between the head and optical radii is rather small. Computational data obtained for negative streamers in air can be also analyzed using equations (1) and (2), with parameters Es, a and b differing from those for positive streamers.
In figure 3 the slope dR/dL evaluated using the plot of the streamer radius versus the length presented in [10] is shown, together with linear dependence (1) with parameters a = 0.08 and Es/ = 11.7 kV/cm. Thus estimated parameter a for negative streamers is about 7 times larger than that obtained above for positive streamers. This difference is partly caused by the difference, of about 3 times, between the values of Es/Eh for positive and negative streamers.
Unfortunately, there is no available computational data allowing to check if the slope dR/dL for negative streamers is, like that shown in figure 1 for positive streamers, rather insensitive to variation of gas pressure, discharge gap, etc. Figure 3. The rate of variation of negative streamer radius with length versus reduced electric field. Pointsestimates obtained using calculated data [10] for pressure 1 bar. Dashed lineequation (1) with parameters a = 0.08 and Es/ = 11.7 kV/cm. Figure 4 shows the values of dV/d(L) versus E/ for negative streamers, estimated using the plots of streamer velocity versus the length presented in [10,13]. The linear dependence (2) with parameters Es/ = 11.7 kV/cm, a = 0.08 and b = 1.8 ns -1 is also given.
This estimate of the parameter b is close to the ratio V/(R) evaluated in [14] for negative streamers at Eh/ = 110 kV/cm. Figure 4. The rate of variation of negative streamer velocity with reduced length versus reduced electric field. Pointsestimates obtained using calculated data [10,12] for pressure 1 bar. Dashed lineequation (2) with parameters Es/ = 11.7 kV/cm, a = 0.08 and b = 1.8 ns -1 .
Results presented above show that equations (1) and (2) describe reasonably well the behavior of velocity and radius of accelerating streamers. For decelerating streamers, the behavior of streamer characteristics is more complex. When streamers approach stopping points, the reduced field in the streamer head Eh/ changes substantially, decreasing in negative streamers [10,16] and increasing in positive streamers [5,16,17].
It should be noted that obtained estimates for the parameters in equations (1) and (2) are rather approximate. Depending on external conditions, these parameters can vary in some ranges. E.g., the values of Es/ corresponding to steady propagation of positive streamers obtained in computations [5] vary inside the interval 4.0-5.4 kV/cm. Nevertheless equations (1) and (2), although imprecise, could be useful to easily evaluate approximately the rates of variation of the streamer velocity and radius with the length depending on the value of applied uniform field.
Figure 1 .
1The rate of variation of positive streamer radius with length versus reduced electric field. Pointsestimates obtained using calculated data for pressure 1 bar[3,4,6,8] and 0.1 bar[7]. Dashed lineequation (1) with parameters a = 0.011 and Es/ = 4.7 kV/cm. In figure 2 the slopes dV/d(L), evaluated for positive streamers using the profiles of streamer velocity versus the length presented in[3,4,6,7,9,11], are shown versus the reduced applied electric field E/. The linear dependencies (2) are also given, corresponding to three values of the parameter b = V/(R): 4.2, 2.6 and 1.5 ns -1 , estimated in[14] for positive streamers at Eh/ = 160, 140 and 120 kV/cm, respectively. It is seen that all the data are within the region between lines 1 and 3, corresponding to the upper and lower boundaries of the interval of Eh/ typical for positive streamers in air. The values of Eh/ obtained in simulations of weakly accelerating streamers, about 160 kV/cm in[11] and 120 kV/cm in[4], correspond to the maximal and minimal slopes. Note that though these slopes differ substantially, the values Es/ of reduced electric field corresponding to zero slopes are close to each other.
Figure 2 .
2The rate of variation of positive streamer velocity with reduced length versus reduced electric field. Pointsestimates obtained using calculated data for pressure 1 bar[3,4,6,9,11] and 0.1 bar[7]. Dashed linesequation(2) with parameters Es/ = 4.7 kV/cm, a = 0.011 and b = 4.2 ns -1 (line 1), 2.6 ns -1 (line 2), 1.5 ns -1 (line 3).
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| []
|
[
"Exotic Hadrons from Scattering in the Diabatic Dynamical Diquark Model",
"Exotic Hadrons from Scattering in the Diabatic Dynamical Diquark Model"
]
| [
"Richard F Lebed \nDepartment of Physics\nArizona State University\n85287TempeAZUSA\n",
"Steven R Martinez \nDepartment of Physics\nArizona State University\n85287TempeAZUSA\n"
]
| [
"Department of Physics\nArizona State University\n85287TempeAZUSA",
"Department of Physics\nArizona State University\n85287TempeAZUSA"
]
| []
| The diabatic framework generalizes the adiabatic approximation built into the Born-Oppenheimer (BO) formalism, and is devised to rigorously incorporate the mixing of BO-approximation eigenstates with two-particle thresholds. We recently applied this framework in a bound-state approximation to the mixing of hidden-charm dynamical-diquark tetraquark states with open-charm di-meson thresholds. Since almost all of these states are observed as above-threshold resonances, we here implement the corresponding scattering formalism to allow for a study of exotic tetraquark resonances within the diabatic framework. We calculate elastic open-charm di-meson cross sections (in channels with zero, open, and hidden strangeness) as functions of center-of-mass energy, and observe the development of true resonances, near resonances, and various threshold cusp effects. As an example, χc1(3872) can originate in the 1 ++ channel as a diquark-antidiquark state enhanced by the D 0 D * 0 threshold, with or without an additional contribution from the conventional charmonium χc1(2P ) state. | null | [
"https://export.arxiv.org/pdf/2305.09146v1.pdf"
]
| 258,714,835 | 2305.09146 | 8bf1e5817dd7da11a3d8975eab71cbd7508ad360 |
Exotic Hadrons from Scattering in the Diabatic Dynamical Diquark Model
Richard F Lebed
Department of Physics
Arizona State University
85287TempeAZUSA
Steven R Martinez
Department of Physics
Arizona State University
85287TempeAZUSA
Exotic Hadrons from Scattering in the Diabatic Dynamical Diquark Model
(Dated: May, 2023)Exotic hadronsdiquarksscattering
The diabatic framework generalizes the adiabatic approximation built into the Born-Oppenheimer (BO) formalism, and is devised to rigorously incorporate the mixing of BO-approximation eigenstates with two-particle thresholds. We recently applied this framework in a bound-state approximation to the mixing of hidden-charm dynamical-diquark tetraquark states with open-charm di-meson thresholds. Since almost all of these states are observed as above-threshold resonances, we here implement the corresponding scattering formalism to allow for a study of exotic tetraquark resonances within the diabatic framework. We calculate elastic open-charm di-meson cross sections (in channels with zero, open, and hidden strangeness) as functions of center-of-mass energy, and observe the development of true resonances, near resonances, and various threshold cusp effects. As an example, χc1(3872) can originate in the 1 ++ channel as a diquark-antidiquark state enhanced by the D 0 D * 0 threshold, with or without an additional contribution from the conventional charmonium χc1(2P ) state.
I. INTRODUCTION
Reaching the 20-year anniversary of the first clear experimental evidence for the existence of heavy-quark exotic hadrons-the observation of the charmoniumlike state now called χ c1 (3872) [1]-the field of hadron spectroscopy now faces the same scientific challenges shared by many other areas of study. Some definitive answers on the nature of these states have been obtained; but many of the original questions remain, and many new questions have arisen. More than 60 heavy-quark exotic candidates have been observed to date, notably some of which that were first seen shortly after the 2003 discovery of χ c1 (3872) by Belle [1] (e.g., Y (4260) in 2005 by BaBar [2], which has subsequently been determined by BESIII to consist of more than one state [3]). Despite a longstanding need for a theoretical paradigm to describe the structure, production, and decays of these states, no universally predictive model has emerged capable of accommodating all of them [4][5][6][7][8][9][10][11][12][13][14][15][16]. A number of these exotic candidates (some of which are listed in Table I) lie remarkably close to some particular di-hadron threshold, the most notable example being χ c1 (3872): (1) using the averaged mass value for each particle provided by the Particle Data Group (PDG) [17]. Clearly, it can be no coincidence that so many of these states appear near a threshold. Some of them, such as χ c1 (3872), lie close below the corresponding threshold, suggesting a possible description via a di-hadron molecular picture, with the hadron pair (in this case, D 0D * 0 plus its chargeconjugate) being bound in part via π 0 exchange. In fact, this interpretation has a rich history, in some cases long predating the χ c1 (3872) discovery [18][19][20]. Others, such as the Z states of Table I, lie close above a threshold, discouraging the naive meson-exchange molecular description. A complete, self-consisent model must be able to describe the relation between these exotic states and their nearby thresholds, as well as states that lie relatively far from any di-hadron threshold, such as Z c (4430) or many of the J P C = 1 −− Y states.
m χc1(3872) − m D 0 − m D * 0 = −0.04 ± 0.09 MeV,
Adding to the puzzle, χ c1 (3872) exhibits some behaviors that seem to imply the importance of short-distance components of its wavefunction, such as in its appreciable decays to J/ψ, χ c1 , and γψ(2S), with the radiative decays being especially significant. However, given the tiny binding energy [Eq. (1)] available to a molecular χ c1 (3872), one would expect its observables to be utterly dominated by long-distance interactions. This contradiction, in part, has led to the long-standing view that χ c1 (3872) contains at least some component of the fundamental charmonium state χ c1 (2P ) [21]. However, an alternate short-range, color-attractive configuration is available to the χ c1 (3872), in the form of a diquarkantidiquark pair: (cu)3(cū) 3 .
In fact, one approach using this paradigm, the dynamical diquark model [22,23], has made strides in successfully representing the χ c1 (3872) as an exotic diquarkantidiquark state, as well generating the full accompanying spectra of both tetraquark and pentaquark exotic multiplets in multiple flavor sectors [23][24][25][26][27][28][29]. These advances include the incorporation of effects such as spinand isospin-dependent interactions, SU(3) flavor mixing, and most recently, mixing between diquark-antidiquark states and nearby di-hadron thresholds [30].
While the original dynamical diquark model calculations were performed assuming that di-hadron thresholds close in mass to those of the diquark-antidiquark states can be neglected-which imposes the framework of the Born-Oppenheimer (BO) approximation-the incorpora-
ΣcD * Z b (10610) BB * Z b (10650) B * B *
tion of di-hadron threshold mixing can be accomplished through its rigorous generalization; this so-called diabatic formalism was originally developed for, and has long been used in, molecular physics [31]. First introduced into hadronic physics by Ref. [32] to analyze exotic states produced by the mixing of heavy quarkonium QQ with dihadron thresholds, the diabatic framework also provides a method through which diquark-antidiquark states mixing with di-hadron thresholds can be analyzed [30]. Almost all exotic states lie above the energy threshold of the lowest possible open-heavy-flavor hadron pair with the same J P C and flavor quantum numbers. While these states may, in some cases, be approximated as bound states (which is the assumption of Refs. [30,32,33]), the more accurate treatment is to view these states as resonant poles within scattering processes. The unification of the diabatic formalism with scattering theory, again using QQ/di-hadron mixing, was pioneered in Ref. [34]. Here, we expand upon the work of Ref. [30] by developing the same techniques for δ-δ/di-hadron mixing.
This paper is organized as follows. In Sec. II we define the features of the dynamical diquark model, which generates the spectrum of heavy-quark exotic hadrons studied here. Section III describes the diabatic formalism that generalizes the adiabatic formalism inherent in the BO approximation used by the original dynamical diquark model. The diabatic formalism is incorporated in Sec. IV into scattering theory, particularly in order to study open-flavor heavy-meson elastic scattering processes, in which exotic resonances (ultimately originating as dynamical-diquark states) may occur. In Sec. V, we first reprise our previous bound-state calculations, and then present numerical results for hidden-charm scattering cross sections and discuss the diverse interesting features that arise. Section VI summarizes our conclusions and indicates the next directions for research.
II. THE DYNAMICAL DIQUARK MODEL
The dynamical diquark picture [22] provides key context for the construction of the full scattering model developed in this paper. In the original picture, quark pairs (qQ) and (qQ) in (attractive) color-triplet configurations (Q being heavy) are produced within relative proximity of each other, and with a high relative momentum with respect to the opposite pair; such a scenario occurs in an appreciable fraction of QQ production processes. Thus, the diquarks δ ≡ (qQ)3 andδ ≡ (qQ) 3 can naturally form as compact objects, especially since heavy Q have less Fermi motion. Due to confinement, they remain bound to each other via a color flux tube. The kinetic energy associated with the high relative momentum is then converted into the potential energy of the flux tube as the distance between the diquarks increases, the δ-δ separation eventually reaching a maximum as the relative momentum between the compact diquarks drops toward zero. With an appreciable distance now separating the quark-antiquark pairs that can form color singlets, this configuration has difficulty hadronizing, allowing it to persist long enough to be observed as an exotic tetraquark resonance. The analogous process for the pentaquark case [35] can also be described using this mechanism by substitutingδ →θ, where the color-triplet
triquark is defined byθ ≡ [Q(q 1 q 2 )3] 3 .
The dynamical diquark model is then constructed from this picture by implementing the BO approximation for QCD, as described in detail in Sec. III. This approximation, which has been extensively used to study heavy hybrid mesons, provides the most natural formalism for describing such a quasi-static system. The end result of applying the BO approximation is the generation of a set of effective static potentials, which in turn are used to produce a full spectrum of state multiplets. These BO potentials may be explicitly calculated on the lattice (see, e.g., Refs. [36][37][38]). The multiplets of states within these potentials are denoted by a set of five quantum numbers: Λ η (nL), where Λ η define the BO potentials through the symmetries of the light degrees of freedom (d.o.f.), and n, L indicate the familiar radial and orbital quantum numbers defining the orbitals of each BO potential. Explicitly, the labels Λ η designate irreducible representations of the group D ∞h , which describes the symmetries inherent to a cylinder whose axis coincides with the characteristic radial separation vectorr of the heavy quasiparticle pair.
A more detailed discussion of these potentials, as well as their application to δδ and δθ systems, may be found in Refs. [23,24]. For the purpose of this analysis, Refs. [24][25][26] are especially important by providing clear numerical indications that these potentials correctly describe multiplet mass averages for heavy-quark exotic states in each light-flavor sector (e.g., ccqq in 1S and 1P states, ccqqq, ccss, bbqq ), and these multiplets are shown to accommodate the J P C quantum numbers of all known exotics. The multiplet mass averages may then be resolved into a fine-structure spectrum by introducing Hamiltonian spin-and isospin-dependent operators that are expected to be the ones most relevant for describing fine-structure effects. In general, the number of free parameters in the model is then (n + 1), consisting of n fine-structure operators included in the analysis, plus the diquark (triquark) mass m δ (m θ ). A phenomenological fixing of these parameters, where one fits to the numerical value of each so that the best-understood exotic states emerge naturally, is the approach of Refs. [24][25][26][27][28][29]39]; mass predictions for every member of the complete spectrum of states then immediately follows.
III. THE DIABATIC APPROACH
The incorporation of the diabatic approach into the dynamical diquark model [30] signifies a departure from the strict framework of the BO approximation to its rigorous generalization [31], and we reprise its development for hadronic systems here. To describe a (nonrelativistic) system consisting of two heavy color sources interacting through light (quark and gluon) fields, one begins with the Hamiltonian
H = K heavy + H light = p 2 2µ heavy + H light ,(2)
where H light contains the light-field static energy, as well as the heavy-light interaction. Under the BO framework, one writes the solutions to the corresponding Schrödinger equation as
|ψ = i drψ i (r) |r |ξ i (r) ,(3)
where |r are defined as states of heavy source pairs with separation vector r, and |ξ i (r) are eigenstates of H light . Note that the heavy and light states here reference the same value of r; Eq. (3) is called the adiabatic expansion, although the expression at this point remains general. The set {|ξ i (r) } forms a complete, orthonormal basis for the light d.o.f. at any given r, but in general, configuration mixing occurs at different values of r: ξ j (r )|ξ i (r) = 0 even for j = i. Inserting Eq. (3) into the Schrödinger equation and taking inner products with ξ j (r)|, after some manipulations one arrives at
i − 2 2µ QQ [∇ + τ (r)] 2 ji + [V j (r) − E] δ ji ψ i (r) = 0,(4)
where the functions τ (r) ji , known as Non-Adiabatic Coupling Terms (NACTs), are defined as
τ ji (r) ≡ ξ j (r)|∇ξ i (r) .(5)
If, in addition, the heavy d.o.f.'s are sufficiently heavy compared to the light d.o.f.'s, then one may approximate the light d.o.f.'s as instantaneously (adiabatically) adapting to changes in the heavy-source separation, which in this notation reads ξ i (r )|ξ i (r) ≈ 1 for small separation r = r, the adiabatic approximation. Additionally, at values of r , r where the light-field eigenstates do not appreciably mix, one has ξ j (r )|ξ i (r) ≈ 0 for j = i, which is called the single-channel approximation. These two approximations define the full BO approximation, and are conveniently summarized by the single condition on the NACTs:
τ ji (r) = ξ j (r)|∇ξ i (r) ≈ 0.(6)
For systems containing a heavy (hence static) QQ pair, unquenched lattice-QCD calculations have long found that this approximation works well in regions far from energy thresholds for on-shell di-meson production. Close to these thresholds, the static light-field energies experience an avoided level-crossing, thus demonstrating the explicit breaking of the single-channel approximation [40,41]. In order to discuss more general mixed states that may have such energies, one may adopt the rigorous generalization of the BO approximation known as the diabatic formalism [31]. This method rewrites the expansion of the solution Eq. (3) as
|ψ = i dr ψ i (r , r 0 ) |r |ξ i (r 0 ) ,(7)
where r 0 is a free parameter. Here again, the completeness of the basis {|ξ i (r) }, regardless of the choice of r, is crucial. In analogy to the previous procedure, one inserts the expansion Eq. (7) into the Schrödinger equation and takes inner products with ξ j (r 0 )|, thus producing:
i − 2 2µ i δ ij ∇ 2 + V ji (r, r 0 ) − Eδ ji ψ i (r, r 0 ) = 0. (8)
Now the object of interest is V ji , which is known as the diabatic potential matrix; it is defined as
V ji (r, r 0 ) ≡ ξ j (r 0 )|H light |ξ i (r 0 ) .(9)
The NACT method and the diabatic-potential method are rigorously equivalent, as shown in Refs. [31,32], but the latter is more convenient for our numerical simulations. As discussed in Ref. [32], one may choose r 0 far from potential-energy level crossings, such that the states |ξ i (r 0 ) may be unambiguously identified with pure, unmixed configurations. For the specific application to dynamical-diquark states with a fixed value of r 0 , we identify the diagonal elements of this matrix as the static light-field energies V δδ associated with a pure δδ state and its corresponding di-meson thresholds V
(i) M1M 2 , i = 1, 2, . . . , N . Explicitly, V ji may then be written as V = V δδ (r) V (1) mix (r) · · · V (N ) mix (r) V (1) mix (r) V (1) M1M 2 (r) . . . . . . V (N ) mix (r) V (N ) M1M 2 (r) ,(10)
where we ignore direct mixing terms between any two dimeson configurations (i.e., the suppressed elements are zero). For the purposes of this work, we set each pure di-meson energy to be the free energy of the state, i.e.,
V (i) M1M 2 (r) → T M1M 2 = M 1 + M 2 .(11)
One could of course instead replace V
(i) M1M 2
(r) with a mildly attractive potential (e.g., pion-exchange interactions or the effects of triangle singularities), as suggested in Ref. [30].
IV. SCATTERING THEORY
As noted in the Introduction, the diabatic formalism provides a method to study mixed but still formally bound states. In contrast, nearly all of the exotic candidates have been observed solely through their strong-interaction decays, and therefore should properly be treated as resonances in scattering theory, i.e., as poles in a scattering S-matrix.
Here we review the construction of the K-matrix formalism as a method of retrieving the S-matrix for coupled-channel eigenstates of the Schrödinger equation, specifically using the method of Ref. [42]. The K-matrix has several advantages over the S-matrix, in particular that it can be chosen to be real and symmetric (assuming time-reversal symmetry), and that pole terms induced by distinct resonances, even heavily overlapping ones, may be simply added together in the K-matrix (unlike for the S-matrix). In this work, we consider only elastic scattering of asymptotically pure di-meson configurations. As discussed in Ref. [33], this type of scattering, mediated by the short-range mixing of di-meson and δδ states, is the natural physical process in which to study the asymptotic behavior of solutions to Eq. (8). Collecting the set of linearly independent solutions to the Schrödinger equation into a matrix Ψ, one may write the asymptotic behavior as
Ψ(r) = J(r) − N(r)K,(12)
where K denotes the K-(or reaction) matrix, and J and N are the (diagonal) solutions to the Schrödinger equation in the r → ∞ limit, at which only the centrifugal part of the potential remains significant. Following Ref. [42], we choose the closed-channel elements (channels with thresholds above the total energy E) of both matrices to be proportional to their corresponding modified spherical Bessel functions (x i ≡ rk i , where k i is the wave number for the i th channel):
J ij = x i ·i j (x i ) δ ij , N ij = x i ·k j (x i ) δ ij ,(13)
while the open-channel elements (channels with thresholds below the total energy E) are set to be the Riccati-Bessel functions,
J ij = x i ·j j (x i ) δ ij , N ij = x i ·n j (x i ) δ ij .(14)
Formally, one may then write K as a function of J, N, and the log-derivative y of the matrix solution Ψ, y ≡ Ψ Ψ −1 :
K = (yN − N ) −1 (yJ − J ) .(15)
In the sign convention for K imposed by Eq. (12) (see Ref. [43] for alternate sign conventions for all of these quantities), the S-matrix is obtained as:
S = (I − iK oo ) −1 (I + iK oo ),(16)
where K oo denotes the sub-matrix of K containing only elements that connect open channels to other open channels. That Eq. (16) can be expressed solely in terms of K oo relies directly upon the specific forms of Eqs. (13)- (14), as is thoroughly explained in Ref. [44]. Reference [42] also provides a method for numerically calculating Eq. (15) using the reduced Numerov method, which has already been employed extensively for solving dynamical-diquark Schrödinger equations (starting with Ref. [24]). We now briefly comment on the form of the solutions contained in Ψ. Since this analysis is concerned only with the elastic scattering of asymptotically pure di-meson states, we restrict this discussion to the elements of Ψ associated with those states. With V M1M 2 (r) = T M1M 2 , the well-known unmixed solutions are: , spherical harmonics Y m (r), and spinors ξ ms s . In addition, k and (i) denote the k th partial wave of the i th di-meson threshold with quantum numbers J P C , while j is the th spherical Bessel function of the first kind, p (i) = 2µ (i) (E − T (i) ) is the relative momentum (or wave number) of the di-meson pair, and 2 π µ (i) p (i) is a factor introduced in Ref. [34] to normalize the full solution in terms of energy E:
ψ (i) J P C ,m J (r) = 2 π µ (i) p (i) i (i) k j (i) k (p (i) r)Y J,m J (i) k s (i) k (r),(17)Ψ E |Ψ E = δ(E − E).(19)
One may go further by using the large-argument asymptotic expression for spherical Bessel functions,
j (pr) → 1 pr sin pr − π 2 .(20)
This form allows for mixed solutions to be clearly expressed using well-known elastic scattering theory (e.g., Eq. (11.17) in Ref. [45]): The effect of mixing with a short-range attractive state, in this case δδ, enters as a channel-and momentum-dependent phase shift δ in the unmixed asymptotic wavefunctions of the di-meson configurations. Explicitly,
1 p (i) r sin p (i) r − π 2 −→ e iδ (i) 1 p (i) r sin p (i) r − π 2 + δ (i) .(21)
Summing over all partial waves k (and adopting the notation of Ref. [34] as closely as possible), we have
ψ (i) J P C ,m J (r) = 1 r 2µ (i) πp (i) k i (i) k a (i) J P C ;k × 1 p (i) r sin p (i) r − k π 2 + δ (i) J P C ;k Y J,m J (i) k s (i) k (r),(22)
where the usual scattering coefficients a (i) J P C ;k keep track of the weighted amplitude that each partial wave contributes to the overall J P C state. Finally, one may now write the asymptotic wavefunction of a di-meson to dimeson scattering state (i ← i ), in specific partial waves
k ≡ ( (i ) , s (i ) ) → k ≡ ( (i) , s (i) ), as ψ i←i J P C ,m J ;k (r)= 1 r 2µ (i) πp (i) k i (i) k × δ ii δ kk sin p (i) r − (i) k π 2 + p (i) f i←i J P C ;k,k e i(p (i) r− (i) k π 2 ) × Y J,m J k (r),(23)
with f i←i J P C ;k,k being the partial-wave scattering amplitude. In the present analysis, f i←i J P C ;k,k are the objects of interest, since one may extract the elastic-scattering cross sections directly from these scattering amplitudes.
We do so, again following the work of Ref. [34], and thus provide a proof-of-concept calculation of elasticscattering cross sections for the di-meson configurations (mediated by coupling to δδ states) as discussed in Sec. III. This may be done using the S-matrix by calculating the scattering amplitude
f i←i J P C = (S − I) ii 2ip (i) ,(24)
with which one may calculate the J P C -specific partial cross section
σ i←i J P C = 4π(2J + 1) (2s M (i ) 1 + 1)(2s M (i ) 2 + 1) k,k |f i←i J P C ;k,k | 2 .(25)
For the purposes of this calculation, we instead calculate a normalized cross sectionσ [34],
σ i←i J P C = k,k |p (i) f i←i J P C ;k,k | 2 ,(26)
which allows for a clearer investigation of the behavior near threshold (where phase space, and hence σ, vanishes), as well as providing more unequivocal indications of resonant behavior, in which fully saturated resonances are expected to reach the maximum allowed value of unity forσ.
V. RESULTS
In this analysis, we assume the mixing elements of the diabatic-potential matrix in Eq. (10) to have the simple Gaussian form [32]:
|V (i) mix (r)| = ∆ 2 exp − 1 2 V δδ (r) − T (i) M1M 2 2 Λ 2 ,(27)
where ∆ is the strength of the mixing and Λ is a width parameter, both with units of energy. To produce meaningful results, ∆ must be large enough to induce sufficient mixing with δδ states that clearly indicates the importance of nearby di-meson thresholds, while Λ must be small enough not to induce excess mixing with thresholds far from the original δδ state; until lattice-QCD simulations are able to provide specific values for these parameters, their magnitudes remain constrained only by these qualitative constraints. One may rewrite Λ as
Λ = ρσ,(28)
where ρ may be identified as the radial scale of the mixing, while σ is the string tension of the δδ configuration. As discussed in Ref. [32], this particular form of the mixing potential, which is motivated by results of lattice QCD [41], acts as a phenomenological placeholder, in anticipation of future precision lattice simulations. We then reproduce the results of the bound-state formalism in Ref. [30], with a slight variation of model parameters:
ρ = 0.165 fm,(29)
and, for the ground-state BO potential Σ + g ,
V δδ (r) = − α r + σr + V 0 + m δ + mδ ,(33)
where α, σ, and V 0 are 0.053 GeV·fm, 1.097 GeV/fm, and −0.380 GeV, respectively [46]. Hence, Λ in Eq. (27) is 0.181 GeV. We note that applying the hybrid QQ potential inputs obtained from lattice simulations for the δδ case is reasonable, since both are color 3-3 potentials between two heavy sources. For BO potentials above Σ + g , which generally tend to mix with each other, the extension of this formalism is straightforward. However, in this work we focus solely on the Σ + g potential, since all exotics found to date appear to be accommodated within its orbitals [24]. These results are presented in Tables II, III, and IV. As in Ref. [30], the mixing parameters are retrieved by fitting to the χ c1 (3872) mass central value 3871.65 MeV reported by the PDG [17], while keeping the same diquark mass m cq [Eq. (31)] found in Ref. [29]. Additionally, the mixing parameters are moderately constrained to reproduce certain behaviors of the mixing angle between the δδ and DD * components: i.e., the mixing angle θ(r) must smoothly and quickly vary between 0 and π/2 as r decreases/increases away from the critical radius r c , which is defined as the separation for which V δδ (r c ) equals the DD * threshold mass (see Refs. [30,32]). Again, we note that these mixing parameters ρ, ∆ are not uniquely defined by this fit, and thus only serve as working values for the present analysis. With these inputs, the diquark mass m cs is then fixed [Eq. (32)] by requiring the 0 ++ ccss state to have mass equal to the central value 3921.7 MeV for X(3915) given by the PDG [17].
Once these parameters are fixed, the diabatic dynamical-diquark model Hamiltonian (not yet including fine structure) for each tetraquark flavor sector, ccqq , ccqs, and ccss, is completely specified. This assertion, of course, assumes that the mixing parameters are universal, and not unique to each threshold or flavor sector. Some work towards this end, specifically to include heavy-quark spin-symmetry breaking effects, has been carried out in Ref. [47], where the author calculates transition rates between the elementary state (in that case, QQ) and its corresponding thresholds. Using the formalism described in Sec. IV, we may then directly produce flavor-and J P C -specific cross sections as functions of center-of-mass energy. In aggregate, these results are presented in Figs. 1-11. Some universal characteristics include the stability of all major functional features inσ [Eq. (26)] upon minor variations of the phenomenologically determined parameters ρ, ∆, and m δ . Additionally, we find resonant behavior to occur in all but one of the cross sections, which is consistent with the calculations performed under the bound-state framework of Refs. [30,32]. That is, we find resonances in the near proximity of all predicted bound states.
A. ccqq
Unique amongst the ccqq sector are the 1 ++ results presented in Fig. 1. Here, in addition to δδ, we incorporate a cc channel [representing the fundamental χ c1 (2P ) state] into the diabatic-potential matrix, with a mixing potential connecting this channel to the (same) corresponding di-meson thresholds. This particular simulation, unlike others in the ccqq category, necessarily produces only isosinglet amplitudes. The mixing, for which we adopt the same form as that for δδ-M 1 M 2 , is parameterized using the results of Ref. [32]: specifically, ρ cc = 0.3 fm and ∆ cc = 0.130 GeV. A direct comparison with Fig. 2, in which the cc channel is removed, reveals that its inclusion in Fig. 1 can result in the formation of a secondary peak containing significant overlap with the peak appearing at the mass of χ c1 (3872). Additional contributions from the processes D * D * and D sD * s are also induced by the inclusion of the χ c1 (2P ) component in Fig. 1. We also note the appearance of threshold effects at the D sD * s mass (∼4081 MeV) in Figs. 1 and 2, since this model makes no attempt to address the Okubo-Zweig-Iizuka (OZI) suppression required for transitions between states containing qq and ss. Lastly, we find (but do not exhibit here) that the usual elastic scattering phase δ , defined by
S ii ≡ e 2iδ i←i ,(34)
exhibits resonant behavior (sharp transitions in δ reaching above π/2) at both major peaks in Fig. 1. However, this conclusion is only true for the mixed S-D partial wave, i.e., DD * ( = 0) ↔ DD * ( = 2). The pure S-wave process, DD * ( = 0) ↔ DD * ( = 0), nearly reaches π/2 at the χ c1 (3872) mass, and then smoothly trails off at higher energies; but if one varies the parameters ρ or ∆, then it is possible to induce a value of δ that rises above π/2 in the pure S-wave process at the same mass. Under this variation, the same resonant behavior as in the other partial-wave channels is still observed. This result implies that the current framework can easily accommodate a pair of resonant states δδ and χ c1 (2P ), either fully overlapping or clearly discernable, each mixing with nearby di-hadron thresholds. Our 0 ++ results in Fig. 3 show a wide, fully saturated peak at 3900 MeV in the process DD → DD, with nontrivial modifications from both D sDs and D * D * thresholds as well. This result is consistent with expectations inferred from Table II, where the bound-state approximation produces significant contributions from the corresponding thresholds to a state with matching energy, 3903.83 MeV. In Fig. 3, the impacts of threshold effects in the lineshapes are clearly visible.
Conversely, for 2 ++ scattering (Fig. 4), we observe a sharp peak near 3910 MeV, which can be unambiguously assigned to the corresponding state (3917.44 MeV) of Table II. Outside of this peak in the 2 ++ cross section, there are relatively small contributions in all but the D * D * channel. We also observe the same preferential 2 ++ coupling to D * D * in Table II, despite its threshold (∼4014 MeV) being significantly higher in mass than the D sDs threshold (∼3937 MeV). In Ref. [30], this enhancement is attributed to the fact that the D * D * threshold coupling to 2 ++ allows an S-wave coupling, which is naturally expected to dominate over > 0 configurations (D-wave for D (s)D(s) in 2 ++ ) in scattering processes.
While the sharpness of the peak in Fig. 4 suggests the existence of a clear δδ resonance with J P C = 2 ++ that should be immediately detectable by experiment, it is important to point out that the width of peaks in this analysis arises solely from the diabatic mixing of di-meson threshold components, rather than through direct decay couplings. In particular, if an elementary δδ (or, for that matter, cc) state lies far from all diabatic-coupled thresholds, then its width inferred from these figures may turn out to be much narrower than the physical width one would obtain when direct transition couplings are properly included. In the present case, the isoscalar 2 ++ channel is already known to feature the cc candidate χ c2 (2P ) at 3922.5±1.0 MeV [17] (which could certainly have been included in this analysis, in the same manner as done in Fig. 1), and this state has a substantial width of about 35 MeV, likely largely due to its observed (D-wave) DD decay mode. The calculation of these widths through conventional methods (i.e., as performed in Ref. [33] for cc states in the diabatic formalism) will appear in future work.
In this work we now include bound-state results (Table II) for the 1 −− ccqq channel (which did not appear in the results of Ref. [30]), and also present the corresponding cross section (Fig. 5). The energy interval (4.15-4.50 GeV) exhibited for the analysis in this channel is restricted to impose stringent requirements upon which thresholds to include, in order to admit only those expected to generate the most physically significant effects. Thus, we include only thresholds for meson pairs with relatively small individual widths (< 50 MeV), and (with the exception of D * sD * s ) that couple to 1 −− in an S-wave. This calculation produces a resonant peak with an extraordinarily small width (only 4.2 MeV), but again we caution the reader that the widths of states appearing in these plots are based upon incomplete physical input. At a mass of about 4240 MeV, this peak is clearly sensitive to the D * sD * s threshold, which again requires an OZI-suppressed amplitude to couple to ccqq. It is natural to identify this peak with ψ(4230), even though this state's open-charm decay modes are poorly known (only π + D 0 D * − has thus far been seen [17]). We also note a nearly 30-MeV shift of the resonant peak from the bound-state energy predicted by the corresponding state in Table II Table II. Beyond this peak, the 1 −− channel as displayed in Fig. 5 exhibits an abundance of threshold behaviors in all presented cross sections.
B. ccss and ccqs
The full suite of ccss results is presented in Figs. 6-8, while the ccqs results appear in Figs. 9-11. Beginning with our 0 ++ findings for the ccss sector (Fig. 6), we observe further agreement with our bound-state predictions (Table III) in the appearance of a fully saturated peak at 3920 MeV in DD → DD. One may note the similarity of this lineshape with the analogous one in the ccqq sector (Fig. 3). Such results are a direct result of the fact that this formalism is currently "blind" to any effects due to strangeness, other than through explicit differences in diquark and meson masses. We expect this effect to diminish as additional SU(3) flavor symmetry breaking is incorporated.
In the 1 ++ results for this sector (Fig. 7), we find a relatively wide peak centered at 3925 MeV in the DD * → DD * cross section. While this result may appear to discourage assignment to the 1 ++ state in Table III (3968.47 MeV), we note the relatively long tail present in this peak structure, and also recall the up-to-30 MeV downwards shift that may be caused by the introduction of open thresholds. These two facts argue that an assignment of the peak in Fig. 7 to the 1 ++ bound state in Table III is not unreasonable, and indeed, show how strong threshold effects can be in certain channels. As threshold structures are abundant throughout the full results of this analysis, we draw attention to their absence in both hidden-flavor 1 ++ resonances [Figs. 2 and 7] at the D * D * threshold. As symmetry forbids an S-wave 1 − 1 − → 1 ++ coupling, this threshold has only a D-wave coupling to 1 ++ . Thus, these results provide further evidence for the dominance of S-wave couplings in scattering processes.
Lastly, we find the 2 ++ -channel scattering (Fig. 8) to yield a sharp (but not fully saturated) peak around 3925 MeV, which falls within the aforementioned 30-MeV interval for reasonable identification with the corresponding bound state of Table III. In Fig. 8, we observe a uniquely interesting case, in which the state appears to be dragged below the previously open threshold of D sDs [although Table III disallows admixture to this state because the bound state (3949.33 MeV) was found to lie above the D sDs threshold (∼3937 MeV)]. In the scattering context, D sDs only couples to 2 ++ through a D-wave, and therefore is still expected to be suppressed compared to S-waves.
The ccqs sector provides another opportunity to examine the nearly unbroken SU(3) flavor symmetry present in this calculation. A near-perfect overlap is observed for 1 + elastic D * D s and DD * s scattering processes (Fig. 9). We find no resonant behavior in these results, consistent with the 1 + prediction of Table IV, which indicates an eigenstate (3912.73 MeV) below the lowest available di-meson threshold (∼3975 MeV). Additionally, we find fully saturated peaks in both the 0 + (Fig. 10) and 2 + (Fig. 11) results, centered just above and just below 3950 MeV, respectively. Of the two, the peak found in 2 + DD s → DD s notably has the smallest apparent width of any appearing in this analysis (but with the same caveats discussed above). In both cases, the location of the peak differs only slightly from the predictions of Table IV, which, interestingly, our calculations show can be attributed to the introduction of the DD s threshold (∼3833 MeV), which lies well below the predicted eigenvalues. One may also contrast the contributions of the D * D s and DD * s processes in Figs. 9 and 11. We see that a bound-state calculation in which over 90% of the content is δδ (i.e., Fig. 11 but not Fig. 9) produces no obvious structure inσ for scattering processes with thresholds far above the resonance. This conclusion is corroborated by the results of Figs. 7 and 10.
In addition, the inputs in this sector are completely fixed by the phenomenological fits to the other flavor sectors, and thus provide useful benchmarks for comparison against experiment. The 1 + state of Table IV (3912.73 MeV) in particular, which is generally unaffected by the changes introduced in the present calculation, may ultimately be associated with the observed Z cs (3985) [17], once multiplet fine-structure effects are included, especially the mixing of strange states in distinct 1 + SU(3) flavor multiplets [29]. This assignment works especially well when one compares the admixtures of the Table IV state with the fact that Z cs (3985) has been observed as a D sD * + D * sD resonance [48]. The difference between these two masses (∼70 MeV) is well within the largest fine-structure mass-splitting effect predicted for diquark-antidiquark states in this sector [29].
An additional comparison is available from the 1 ++ state of Table III: Although the mass difference is much larger [∼170 MeV, corresponding to the ccss candidate χ c1 (4140)], it is not yet known how the fine structure of diabatic dynamical diquark states differs from those that are blind to threshold effects, particularly once effects sensitive to the larger strange-quark mass are properly included.
VI. CONCLUSIONS
We have reviewed the incorporation of the diabatic formalism, a rigorous extension of the well-known Born-Oppenheimer approximation that is designed to include effects due to the presence of two-particle thresholds, into the dynamical diquark model. While our previous work addresses states formed in the immediate vicinity of these thresholds (the bound-state approximation), this paper develops a scattering framework capable of describing not only exotic states lying close to such thresholds, but also those that lie quite far from them (and thus have no obvious interpretation as a di-hadron molecular state).
Using the bound-state approximation, we first reproduce our previous flavor-and J P C -specific calculations of energy eigenvalues and fractions of both diquarkantidiquark and di-meson components within the corresponding eigenstates. We then summarize the construction of the K-matrix formalism as a method to retrieve the S-matrix, in order to calculate asymptotic scattering amplitudes of coupled-channel, elastic meson-meson collision processes (the most natural ones in which to study resonance and threshold behaviors). We validate the physical expectation that asymptotically free mesonmeson pairs develop resonance structures through their short-range interaction with diquark-antidiquark channels. These scattering amplitudes are calculated numerically for the hidden-charm system (with zero, hidden, and open strangeness), and then are directly used to produce all corresponding CM energy-dependent cross sections, which comprise the main results of this work.
We confirm the expected resonant behavior in all flavor-and J P C -specific cross sections, and also observe several instances of threshold-induced structures such as cusp effects. In addition, the peak of every resonance is calculated to occur not far from the energy of its corresponding bound-state eigenvalue. We observe shifts of these resonances down from the bound-state energies once the couplings to open thresholds are included, in agreement with expectations that thresholds are generally "attractive." While nearly all of these resonant behaviors reach the maximum value allowed by unitarity, some prominent examples reach as low as ∼ 75% of this value.
Although this analysis is mostly limited to mesonmeson scattering coupled to diquark-antidiquark channels described by the dynamical diquark model, we find evidence that the conventional cc state χ c1 (2P ) may be incorporated separately into the ccqq 1 ++ channel, producing two resonant components that may overlap to form χ c1 (3872). In general, a complete calculation would include all diquark-antidiquark and cc states in every allowed J P C channel.
While these results are quite promising, they do not yet distinguish explicit spin-and isospin-multiplet members. The incorporation of such fine-structure analysis has been accomplished for multiple flavor sectors in the original (adiabatic) dynamical diquark model, and thus will be straightforward to include in its diabatic form; this extension will be one major thrust of future work. In addition, this analysis does not incorporate SU(3) flavor symmetry-breaking effects beyond explicit differences in the diquark masses m cq and m cs , and in meson masses m D ( * ) , m D ( * ) s , etc. Such additional effects, not to mention OZI suppression, are expected to have substantial impact on the scattering processes discussed here. Lastly, the widths of the resonances implied by these cross-section plots should not be directly compared to experiment, as they are produced solely through diabatic mixing. Thus, future work will also use well-known techniques to calculate physical strong-decay widths and shifts of energy eigenvalues due to open-threshold di-meson pairs that lie well below the diabatically mixed eigenstates studied here-i.e., the pairs that represent their physical decay channels.
≡
r| , s, J, m J = m ,ms C m ,ms,m J ,s,J Y m (r) ξ ms s ,
built with the conventional Clebsch-Gordan coefficients C m ,ms,m J ,s,J
m cq = m qc = 1927.1 MeV, (31) m cs = m sc = 1944.6 MeV,
. While we have yet to explicitly calculate the expected bound-state mass shifts that arise from the perturbative introduction of couplings to open thresholds, Ref. [33] provides a rough estimate of what might be expected through their analogous calculation in cc-D ( * ) (s)D ( * ) (s) mixing. A comparison to the largest shift noted in that work, roughly 28 MeV, allows for the reasonable identification of the peak in Fig. 5 with the 1 −− bound state of
FIG. 1 .
1The dimensionless cross sectionsσ of Eq. (26) (solid lines) for elastic open-charm D scattering processes as functions of center-of-momentum frame energy E CM . Theσ curves are presented in the same order as that of increasing mass for their corresponding thresholds T (dashed lines). This figure presents results for the flavor content ccqq in the isosinglet channel with J P C = 1 ++ , and includes a contribution from the conventional cc state χc1(2P ).
FIG. 2 .
2The same as in Fig. 1, except suppressing the χc1(2P ) contribution, and (if the small DsD * s contributions are also suppressed) not necessarily limited to the isoscalar combination of ccqq . FIG. 3. The same as in Fig. 2, for elastic open-charm ccqq D
TABLE I .
IExamples of heavy-quark exotic candidates lying particularly close in mass (< 15 MeV) to a di-hadron threshold.Exotic candidate Di-hadron threshold
χc1(3872)
D 0D * 0
Zc(3900)
DD *
Zc(4020)
D * D *
Pc(4312)
ΣcD
Pc(4450)/Pc(4457)
TABLE II .
IICalculated eigenvalues and component-state admixtures for the ccqq sector obtained from solving Eq. (8) for specific J P C numbers. Suppressed entries indicate contributions that are individually finite but < 1% or give no contribution.J P C
E (MeV)
δδ
DD *
DsDs
D * D *
D *
sD
*
s
0 ++
3903.83
69.8%
22.7%
6.9%
1 ++
3871.65
9.1%
90.9%
2 ++
3917.44
86.0%
1.5%
10.4%
1.5%
DD1
DD *
2
D * D
1
1 −−
4269.58
44.0%
51.2%
2.4%
1.5%
TABLE III .
IIIThe same as inTable II, for the ccss sector.J P C E (MeV) δδ DsDs D * D * DsD * s D * sD *s 0 ++ 3921.69 55.7% 35.4% 7.1% 1.2% 1 ++ 3968.47 90.4% 1.2% 7.8% 2 ++ 3949.33 82.1% 15.5% 2.1% TABLE IV. The same as in Tables II & III, for the ccqs sector. J P E (MeV) δδ D * D s DD * s D * D * s 0 + 3969.04 95.2% 4.5% 1 + 3912.73 71.9% 13.7% 13.4% 2 + 3951.42 92.5% 1.5% 1.5% 4.5%
( * ) (s)D ( * ) (s) scattering processes with J P C = 0 ++ . FIG. 4. The same as in Fig. 2, for elastic open-charm ccqq D ( * ) (s)D ( * ) (s) scattering processes with J P C = 2 ++ .
ACKNOWLEDGMENTSThis work was supported by the National Science Foundation (NSF) under Grants No. PHY-1803912 and PHY-2110278.
The same as in Fig. 2, for elastic open-charm ccqq scattering processes with J P C = 1 −− . The energy range has been adjusted to 4.15-4.50 GeV in order to capture the relevant behavior of the cross section. The thresholds are all S-wave. except for D * sD * s , which is P -waveFIG. 5. The same as in Fig. 2, for elastic open-charm ccqq scattering processes with J P C = 1 −− . The energy range has been adjusted to 4.15-4.50 GeV in order to capture the relevant behavior of the cross section. The thresholds are all S-wave, except for D * sD * s , which is P -wave.
The same as in Fig. 3, but now for the ccss channel J P C = 0 ++. FIG. 6. The same as in Fig. 3, but now for the ccss channel J P C = 0 ++ .
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[
"Quasinormal modes and grey-body factors of regular black holes with a scalar hair from the Effective Field Theory",
"Quasinormal modes and grey-body factors of regular black holes with a scalar hair from the Effective Field Theory"
]
| [
"R A Konoplya *[email protected] \nInstitute of Physics and Research Centre of Theoretical Physics and Astrophysics\nFaculty of Philosophy and Science\nSilesian University\nCZ-746 01Opava, OpavaCzech Republic\n"
]
| [
"Institute of Physics and Research Centre of Theoretical Physics and Astrophysics\nFaculty of Philosophy and Science\nSilesian University\nCZ-746 01Opava, OpavaCzech Republic"
]
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| The Effective Field Theory (EFT) of perturbations on an arbitrary background geometry with a timelike scalar profile has been recently constructed in the context of scalar-tensor theories. Unlike General Relativity, the regular Hayward metric is realized as an exact background metric in the Effective Field Theory with timelike scalar profile without resorting to special matter field, such as nonlinear electrodynamics. The fundamental quasinormal mode for axial graviational perturbations of this black hole has been considered recently with the help of various methods. Here we make a further step in this direction and find that, unlike the fundamental mode, a few first overtones deviate from their Schwarzschild limit at a much higher rate. This outburst of overtones occurs because the overtones are extremely sensitive to the least change of the near-horizon geometry. The analytical formula for quasinormal modes is obtained in the eikonal regime. In addition, we calculated greybody factors and showed that regular Hayward black hole with a scalar hair has smaller grey-body factor than the Schwarzschild one. Integration of the wave-like equation in time-domain shows that the power-law tails following the ring-down phase at late times are indistinguishable from the Schwarzschild ones. | null | [
"https://export.arxiv.org/pdf/2305.09187v1.pdf"
]
| 258,714,879 | 2305.09187 | b074df98dffcb64b7d94e36daaea0b0cbc08fe8c |
Quasinormal modes and grey-body factors of regular black holes with a scalar hair from the Effective Field Theory
16 May 2023
R A Konoplya *[email protected]
Institute of Physics and Research Centre of Theoretical Physics and Astrophysics
Faculty of Philosophy and Science
Silesian University
CZ-746 01Opava, OpavaCzech Republic
Quasinormal modes and grey-body factors of regular black holes with a scalar hair from the Effective Field Theory
16 May 2023PACS numbers: 04.50.Kd,04.70.-s 2
The Effective Field Theory (EFT) of perturbations on an arbitrary background geometry with a timelike scalar profile has been recently constructed in the context of scalar-tensor theories. Unlike General Relativity, the regular Hayward metric is realized as an exact background metric in the Effective Field Theory with timelike scalar profile without resorting to special matter field, such as nonlinear electrodynamics. The fundamental quasinormal mode for axial graviational perturbations of this black hole has been considered recently with the help of various methods. Here we make a further step in this direction and find that, unlike the fundamental mode, a few first overtones deviate from their Schwarzschild limit at a much higher rate. This outburst of overtones occurs because the overtones are extremely sensitive to the least change of the near-horizon geometry. The analytical formula for quasinormal modes is obtained in the eikonal regime. In addition, we calculated greybody factors and showed that regular Hayward black hole with a scalar hair has smaller grey-body factor than the Schwarzschild one. Integration of the wave-like equation in time-domain shows that the power-law tails following the ring-down phase at late times are indistinguishable from the Schwarzschild ones.
I. INTRODUCTION
The Effective Field Theory (EFT) offers a model-independent framework for studying the dynamics of perturbations on a particular spacetime. The fundamental components of the EFT include the configuration of matter fields, the symmetry breaking mechanism, and the background spacetime. The EFT of scalar-tensor theories on Minkowski and de Sitter spacetimes, known as ghost condensation, was formulated long time ago in [1,2], under the assumption that a scalar field's timelike gradient spontaneously breaks time diffeomorphism and that the EFT remains invariant under spatial diffeomorphism, time translation, and time reflection, subject to the shift and reflection of the scalar field. Recent work [3] generalized the EFT action with a timelike scalar profile to arbitrary background geometries. Then, a dictionary was developed to relate this EFT to concrete covariant theories such as the Horndeski's theory [3] and shift-symmetric quadratic higher-order scalar-tensor (HOST) theories [6]. As a result of the scalar field's timelike nature, the EFT of [3] can describe both cosmological and black hole scales in a single framework, making it possible to extract some information about scalar-field dark energy from astrophysical black hole observations. This feature induces additional interest to the EFT, making it appealing to consider the EFT of black hole perturbations with a timelike scalar profile. It is worth noting that an EFT of black hole perturbations with a spacelike scalar profile on a static and spherically symmetric background was developed in [4] (see also [8] for the formulation of the EFT on a slowly rotating black hole).
Thus the above approach allows one to study perturbations and proper oscillation frequencies, quasinormal modes, of black holes in the above theories, which could hardly be a treatable problem otherwise. The first step in this direction was made in [9], where the formalism for the EFT of perturbations on a static and spherically symmetric black hole with a timelike scalar profile was constructed and the two examples of background solutions were considered: the stealthy Schwarzschild black hole and Hayward spacetime with a non-trivial scalar hair. While the first metric has rather trivial spectrum, which can be easily found via re-scaling of the well-known Schwarzschild spacetime, the second solution is remarkable. First of all, in some range of parameters it represents a regular black hole without resorting to any specific types of matter such as the non-linear electrodynamics with a particular source of nonlinearity. Secondly, it has a different quasinormal spectrum from either Schwarzschild metric or Hayward solution in the non-linear electrodynamics. The fundamental mode (for the lowest overtone and multipole numbers) for this case were calculated in [9] with the help of the Leaver, direct integration and 6th order WKB methods. The corresponding master wave-like equation describing axial gravitational perturbations was earlier deduced in [6].
In this work, we will take a further step and investigate not only the fundamental mode but also several first overtones of the EFT of gravitational perturbations of the Hayward spacetime. Although it is commonly believed that the principal contribution to the signal is due to the fundamental mode, recent research [10] (and subsequently examined in [11][12][13]) has demonstrated that in order to simulate the ringdown phase accurately, which is obtained within the precise numerical relativity simulations at the start of the quasinormal ringing and not just at the final stage, the first few overtones must be considered. This discovery also suggested that the actual QN ringing begins earlier than expected. Once the overtones are taken into account, the modelling and linear profile of the ringdown are in full agreement, allowing for the extraction of the black hole's angular mass and momentum. Despite the current LIGO/VIRGO observational data not allowing the detection of the overtones in the gravitational-wave signals [14,15], and the higher-overtone contributions requiring nonlinear corrections [16][17][18] when analyzing them, there are indications that the overtones are significantly excited in some events and may be detectable by LISA during the early ringdown phase [19].
An important aspect related to the overtones' behavior is related to the near horizon geometry of a black hole. If modifications to Einstein's theory induce noticeable deformations of the black-hole geometry only near the event horizon, the fundamental mode remains largely unaffected. However, even a small change near the event horizon can significantly impact the first few overtones [20], providing a means to probe the geometry of the event horizon. As was shown in [20] overtones are stable against small deformations of spacetime at a distance from the black hole, allowing the event horizon to be distinguished from the surrounding environment. In contrast to echoes, overtones make a much larger energy contribution. This finding gives us further motivation to study overtones of various black holes [21][22][23].
Taking into consideration the above motivations we will study here several first overtones for the axial gravitational perturbations following from the Effective Field Theory with timelike scalar profile. In addition we will consider timedomain integration of the above wave-like equation which allow us to see the late time tails following the ring-down phase. We will also study grey-body factors of the gravitational perturbations.
The paper is organized as follows. In Sec. II we will briefly review perturbations of the Haywrad black hole in the Effective Field Theory. Sec. III is devoted to numerical and semi-analytic methods used for calculations of quasinormal modes. Sec. IV discusses quasinormal modes, both fundamental one and outburst of overtones, as well as time-domain evolution of the perturbation. In Sec. V the grey-body factors are considered. Finally, we summarize the obtained results and mention some open question.
II. BLACK HOLE PERTURBATIONS IN THE EFFECTIVE FIELD THEORY
Here we will provide a brief overview of the Effective Field Theory (EFT) for perturbations on an arbitrary background with a timelike scalar profile [3,6]. For a similar EFT construction for shift-symmetric scalar-tensor theories, see [24]. The main concept behind our EFT, which is similar to the EFT of ghost condensation [1] and the EFT of inflation/dark energy [25,26], is that the scalar field's time-dependent background,Φ, breaks time diffeomorphism spontaneously, establishing a preferred (Φ = const.) time-slicing. This time-slicing can be defined by the unit normal vector, n µ , as shown in,
n µ ≡ − ∂ µ Φ √ −X → − δ τ µ √ −g τ τ ,(1)
where X ≡ g µν ∂ µ Φ∂ ν Φ is the scalar field's kinetic term, and n µ n µ = −1. Here, we use τ as the time coordinate such thatΦ =Φ(τ ) and δΦ ≡ Φ −Φ = 0 (unitary gauge). The expression on the right-hand side of (1) refers to the one in the unitary gauge. The EFT's residual symmetry, therefore, is 3d diffeomorphism invariance. Consequently, the EFT we will write down in the unitary gauge may contain a scalar function of, for example, the 4d and 3d curvatures, the extrinsic curvature, the (τ τ )-component of the inverse metric tensor, and the time coordinate τ . We will use the Arnowitt-Deser-Misner (ADM) 3 + 1 decomposition, in which the metric can be expressed as shown in,
ds 2 = −N 2 dτ 2 + h ij (dx i + N i dτ )(dx j + N j dτ ).(2)
Here, N is the lapse function, N i is the shift vector, and h µν ≡ g µν + n µ n ν is the induced metric on a spacelike hypersurface of constant τ . The indices i and j refer to the spatial components, and the spatial indices are raised and lowered by the induced metric h ij . The extrinsic curvature is defined using the induced metric h µν :
K µν ≡ h ρ µ ∇ ρ n ν .(3)
Here ∇ µ is the 4d covariant derivative. Then, one can find the spatial components of K µν and its trace in terms of the ADM variables in the following form:
K ij = 1 2N ḣ ij − D i N j − D j N i , K = h ij K ij ,(4)
with a dot being the derivative with respect to τ and D i the 3d covariant derivative constructed from the induced metric h ij . In addition, the 3d curvature (3) R can be found via the induced metric h ij . The unitary-gauge EFT action we are formulating is invariant, that is, not affected by 3d diffeomorphism. Thus, in addition to the 4d covariant terms such as the 4d Ricci scalarR, the action can rely on any geometrical quantities that are covariant under the 3d diffeomorphism, such as g τ τ (= −1/N 2 ), K µν , and (3) !R µν . It's worth noting that the symbol R (without the tilde) denotes the 4d Ricci scalar with the divergence term subtracted. Moreover, the action can depend on τ explicitly. With all the possibilities above, the unitary-gauge action takes the form
S = d 4 x √ −g F (R µναβ , K µν , g τ τ , ∇ µ , τ ) ,(5)
where F is a scalar function of those 4d and 3d diffeomorphism covariant quantities, andR µναβ is the 4d Riemann tensor of the metric g µν . The 3d curvature tensor can be written in terms of the 4d curvature and the extrinsic curvature by use of the Gauss relation, so that one does not include it explicitly in the action. The action (5) could be applied to any background geometries without assuming a particular symmetry of the background. Following [3], the perturbations are defined as follows:
δg τ τ ≡ g τ τ −ḡ τ τ (τ, x) , δK µ ν ≡ K µ ν −K µ ν (τ, x) , δ (3) R µ ν ≡ (3) R µ ν − (3)Rµ ν (τ, x) ,(6)
where a bar denotes the background value. Practically, the EFT action is written as a polynomial of these perturbation variables as well as their derivatives. Each EFT coefficient can have an explicit dependence on both τ and x due to the spacetime dependence of the background quantitiesḡ τ τ ,K µ ν , and (3)Rµ ν . Note that such a dependence on x is not compatible with the 3d diffeomorphism invariance in general. Therefore, in order for the EFT action to respect the 3d diffeomorphism invariance, a set of consistency relations on the EFT coefficients was imposed. The technically difficult formalism for writing-down the linearized perturbation equations and reduction them to the wave-like form was developed in [9] and we refer a reader to this work for details. The Hayward metric [27] which is the background solution in the above Effective Field Theory has the form:
g tt = −g −1 rr = 1 − µr 2 r 3 + σ 3 .(7)
For σ > 0, the Hayward metric corresponds to a regular black hole, but not at σ < 0, as there exists a curvature singularity at r = −σ. One could regard σ as just a phenomenological parameter that controls the deviation from the Schwarzschild metric. The perturbation equations, however, depend not upon the event horizon of the above metric function, but on a new function, corresponding to axial gravitational perturbations built from EFT. From here and on, we will use the resultant wave like equation for the axial gravitational perturbations in the EFT deduced in [9]:
d 2 dr 2 * Ψ(r * ) + (ω 2 − V eff )Ψ(r * ) = 0 .(8)
The function F ≡ dr/dr * is
F (r) = r 4 − µ(r 3 + σ 3 ) r(r 3 + σ 3 ) .(9)
The position of the odd-mode horizon r g (> 0) is given by F (r g ) = 0, or equivalently,
r 4 g − µ(r 3 g + σ 3 ) = 0 ,(10)
which has a single positive solution so long as µ and σ are positive. The effective potential is
V eff (r) = 1 − µ(r 3 + σ 3 ) r 4 ℓ(ℓ + 1)r 4 (r 3 + σ 3 ) 2 − 3 4µr 9 + 2σ 3 r 6 (8r − µ) + σ 6 r 3 (r − 7µ) − µσ 9 4(r 3 + σ 3 ) 4 .(11)
Instead of choosing r g and σ as independent parameters and fixing µ we fix the radius of the event horizon r g = 1 and change σ, so that
µ = 1 1 + σ 3 .(12)
The above effective potential is positive definite and has single maximum (see fig. 3).
III. METHODS USED FOR CALCULATION OF QUASINORMAL MODES
Here we will briefly review the three methods used for the spectral analysis of black hole perturbations: WKB method, Frobenius method and time-domain integration.
A. WKB method
In the frequency domain we will use the semi-analytic WKB approach applied by Will and Schutz [30] for finding quasinormal modes. The Will-Schutz formula was extended to higher orders in [31][32][33] and made even more accurate when using the Padé approximants [33,34]. The general WKB formula has the form [35],
ω 2 = V 0 + A 2 (K 2 ) + A 4 (K 2 ) + A 6 (K 2 ) + . . . (13) − iK −2V 2 1 + A 3 (K 2 ) + A 5 (K 2 ) + A 7 (K 2 ) + . . . ,
where K = n + 1/2 is half-integer. The corrections A k (K 2 ) of the order k to the eikonal formula are polynomials of K 2 with rational coefficients and depend on the values of higher derivatives of the potential V (r) in its maximum. In order to increase the accuracy of the WKB formula, we will follow the procedure of Matyjasek and Opala [33] and use the Padé approximants. Here we will use the sixth order WKB method withm = 4, wherem is defined in [33,35], because this choice provides the best accuracy in the Schwarzschild limit and there is hope that this will be the case for more general metrics.
B. Frobenius method
In order to find accurate values of quasinormal modes we use the method proposed by Leaver [28]. The wave-like equation (8) always has a regular singularity at the horizon r = r g and the irregular singularity at spatial infinity r = ∞. We introduce the new function,
Ψ(r) = P (r, ω) 1 − r g r −iω/F ′ (rg ) y(r),(14)
where the factor P is chosen in such a way that that y(r) is regular for r 0 ≤ r < ∞, once Ψ(r) corresponds to the purely outgoing wave at spatial infinity and the purely ingoing wave at the event horizon. Therefore, we are able to represent y(r) in terms of the Frobenius series:
y(r) = ∞ k=0 a k 1 − r g r k .(15)
Then, using Guassian eliminations in the recurrence relation for the coefficients of the expansion we reduce the problem to solution of an algebraic equation. In addition for quicker convergence we use the Nollert improvement [29].
C. Time-domain integration
In order to find quasinormal modes and, foremost, analyze possible echo-like phenomena we will use the timedomain integration method. We will integrate the wavelike equation in terms of the light-cone variables u = t − r * and v = t + r * via applying the discretization scheme of Gundlach-Price-Pullin [36],
Ψ (N ) = Ψ (W ) + Ψ (E) − Ψ (S) − ∆ 2 V (S) Ψ (W ) + Ψ (E) 4 + O ∆ 4 ,(16)
where the following notation for the points was used: N ≡ (u + ∆, v + ∆), W ≡ (u + ∆, v), E ≡ (u, v + ∆), and S ≡ (u, v). The Gaussian initial data are imposed on the two null surfaces, u = u 0 and v = v 0 . The dominant quasinormal frequencies can be extracted from the time-domain profiles with the help of the Prony method see, e.g., [37].
IV. QUASINORMAL MODES: OUTBURST OF OVERTONES AND TIME-DOMAIN EVOLUTION
As can be seen from the wave-like equation and the form of the function F (r) relating the Schwarzschild-like radial coordinate and the tortoise one, quasinormal modes for each type (channel) of perturbations is described by a different horizon. Therefore quasinormal modes of the Hayward metric studied, for example, in [21], will be completely different from those studied within the Effective Field Theory. Here, for convenience, we used the units r g = 1. If fixing µ instead of r g , which was the units of [9], then we reproduce the results of table II in [9] both via the Leaver and 6th order WKB methods. Here, however, we used 7th order WKB with the Pade approximants which is in better concordance with the accurate Frobenius method (see table I). Notice, that our main method here is the Frobenius method, because it is based on the convergent procedure, while WKB formula converges only asymptotically and does not guarantee convergence in each order.
One can see that while the fundamental mode is changed relatively softly, the first few overtones are changing at a much higher rate. A similar phenomenon takes place for perturbations of test fields in the Hayward metric [21], because there the metric function deviates from the Schwarzschild limit especially strongly near the event horizon. Using the first order WKB formula and expanding in terms of small σ and 1/ℓ we can find quasinormal modes in analytic form in the eikonal regime ℓ ≫ 1. For this the location of the maximum of the effective potential can be found as follows:
r max ≈ 3 2 + 1 2ℓ 2 − σ 3 6 + 7σ 3 18ℓ 2 .(17)
Using the above expression in the first order WKB formula we find
ω n = ℓ 2 3 √ 3 + 22σ 3 81 √ 3 + O σ 6 + 2in + (1 + i) 3 √ 3 + (70in + (11 + 35i))σ 3 81 √ 3 + O σ 6 + O 1 ℓ 1(18)
When σ = 0, this formula goes over into the one for the Schwarzschild black hole (see, for example, eq. (4.5) in [37] or [38] and references therein). It is worth noticing that the above case of eikonal regime of quasinormal modes, as well as other examples [39][40][41], breaks the correspondence between the eikonal quasinormal modes and null geodesics claimed in [42].
Using the time-domain integration, we can see the evolution of Ψ at a given spacial point as a function of time (see fig. II). The power-law tails at late times are indistinguishable from the Schwarzschild ones:
|Ψ| ∼ t −(2ℓ+3) , t → ∞.(19)
V. SCATTERING PROBLEM AND GREY-BODY FACTORS
The computation of grey-body factors is crucial for determining the proportion of the initial quantum radiation that is reflected back to the event horizon by the potential barrier. Despite the temperature is usually a dominant factor for the intensity and amount of Hawking radiation, in some cases grey-body factors can be more influential than temperature [45].
We will examine the wave equation (8) under boundary conditions that allow for incoming waves from infinity. Due to the symmetry of scattering properties, this is equivalent to the scattering of a wave originating from the near-horizon zone. The boundary conditions for scattering in (8) are:
Ψ = e −iωr * + Re iωr * , r * → +∞, Ψ = T e −iωr * , r * → −∞,(20)
where R and T are the reflection and transmission coefficients. The effective potential has the form of a potential barrier that decreases monotonically towards both infinities, allowing, thereby, for the application of the WKB approach [30][31][32] to determine R and T . As ω 2 is real in the scattering problem, the first-order WKB values for R and T will be real [30][31][32], and
|T | 2 + |R| 2 = 1.(21)
Once the reflection coefficient is obtained, one can find the transmission coefficient for each ℓ
|A ℓ | 2 = 1 − |R ℓ | 2 = |T ℓ | 2 .(22)
In order to study the reflection and transmission coefficients we will use the higher order WKB formula [32]. However, this formula is not suitable for very small values of ω, which correspond to almost complete wave reflection and have negligible contributions to the overall energy emission rate. For this regime, we employed extrapolation of the WKB results at a given order to smaller ω. According to [30,31], the reflection coefficient can be expressed as follows,
R = (1 + e −2iπK ) − 1 2 ,(23)
where K is determined by solving the equation
K − i (ω 2 − V max ) −2V ′′ max − i=6 i=2 Λ i (K) = 0,(24)
involving the maximum effective potential V max , the second derivative V ′′ max with respect to the tortoise coordinate, and higher order WKB corrections Λ i .
The WKB formula for finding grey-body factors is known to provide reasonable accuracy for further estimation of the intensity of Hawking radiation and was, therefore, used in a number of papers (see, for example, [43,44]). From fig. 3 one can see that the larger is σ, the smaller is the grey-body factors, that is the bigger portion of particles is reflected by the effective potential. This can easily be understood from the behavior of the effective potential which becomes higher (i.e. more difficult to penetrate) for larger σ.
VI. CONCLUSIONS
Here we considered quasinormal modes and grey-body factors of gravitational perturbations of the Hayward black hole with a scalar hair built within the Effective Field Theory. The initial study of ℓ = 2, n = 0 frequency has been recently suggested in [9]. Here we studied quasinormal frequencies for various n and ℓ in more details. Our main finding here is that the first few overtones deviate from their Schwarzschild limit at a much higher rate than the fundamental mode and this happens because of the deformation of the spacetime near the horizon. The integration of the wave-like equation in time-domain confirms the dominant quasinormal frequencies found by other methods and shows that the power-law tails remain the same as in the Einstein theory. In the eikonal regime of large multipole numbers ℓ the analytical formula for quasinormal modes has been obtained. In addition we have found grey-body factors for the above gravitational perturbations and showed that the bigger is σ (at a fixed radius of the horizon r g ), the smaller is the grey-body factors, because the effective potential becomes higher in this case.
Our work could be extended in a number of ways. Once the corresponding wave-like equations were obtained for the test fields, the corresponding quasinormal modes and grey-boday factors could be obtained in a similar way. If, in addition, the temperature is properly defined [46,47], one could use the above constituents to estimate the intensity of Hawking radiation.
74915 − 0.17957i 0.749208 − 0.179624i 0.69267 − 0.55235i 0.693900 − 0.553250i 0.59500 − 0.96519i 0.600195 − 0.966697i 0.3 0.75316 − 0.18340i 0.753224 − 0.183484i 0.69362 − 0.56417i 0.694606 − 0.565556i 0.59000 − 0.98734i 0.595137 − 0.989760i 0.4 0.75956 − 0.19007i 0.759561 − 0.190294i 0.69447 − 0.58455i 0.694590 − 0.587283i 0.57925 − 1.02328i 0.583917 − 1.030721i 0.5 0.76674 − 0.19910i 0.766538 − 0.199606i 0.69231 − 0.60992i 0.691925 − 0.617129i 0.55996 − 1.05669i 0.564135 − 1.088089i 0.6 0.77158 − 0.20896i 0.771162 − 0.209718i 0.68398 − 0.63594i 0.684964 − 0.649982i 0.51059 − 1.06490i 0.537282 − 1.154489i 0.7 0.77061 − 0.21740i 0.770339 − 0.218185i 0.67149 − 0.65870i 0.673597 − 0.678287i 0.46298 − 1.06576i 0.511096 − 1.218028i 0.8 0.76151 − 0.22231i 0.762131 − 0.222897i 0.65316 − 0.67322i 0.659015 − 0.695039i 0.45630 − 1.06699i 0.494865 − 1.264176i 0.9 0.74425 − 0.22337i 0.746259 − 0.222918i 0.62929 − 0.67911i 0.642316 − 0.696476i 0.45957 − 1.09357i 0.489861 − 1.280763i 1.0 0.72009 − 0.22301i 0.723789 − 0.218567i 0.60258 − 0.68871i 0.623701 − 0.682879i 0.44887 − 1.16460i 0.489066 − 1.265066i
FIG. 1 .
1Reω (left) and Imω (right) as a function of σ (in units rg = 1) for the fundamental mode and first six overtones from top to bottom.
FIG. 2 .
2Semi-logarithmic time domain profile for ℓ = 2, σ = 0 (blue, middle), σ = −0.4 (green, top) and σ = 1 (red, bottom).
FIG. 3 .
3Left panel: Effective potential for ℓ = 2 perturbations; σ = 0 (red), σ = −0.4 (blue) and σ = 1 (green). Right panel: grey-body factors for ℓ = 2, 3, 4 (from left to right on the plot); σ = 0 (red), σ = −0.4 (blue) and σ = 0.75 (green).
TABLE I .
IFirst three quasinormal modes for ℓ = 2 and various values of σ calculated by the 7th order WKB method with Pade apprximants and accurate Leaver method. Notice, that the units used here, rg = 1, is different from those of[9], which are µ = 1.
ACKNOWLEDGMENTSThe author acknowledges A. Zhidenko and S. Mukohyama for useful discussions.
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| []
|
[
"Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit",
"Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit",
"Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit",
"Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit"
]
| [
"Wataru Izumida ",
"Osamu Sakai ",
"Yukihiro Shimizu \nDepartment of Applied Physics\nTohoku University\n980-77SendaiJapan\n",
"\nDepartment of Physics\nTohoku University\n980-77SendaiJapan\n",
"Wataru Izumida ",
"Osamu Sakai ",
"Yukihiro Shimizu \nDepartment of Applied Physics\nTohoku University\n980-77SendaiJapan\n",
"\nDepartment of Physics\nTohoku University\n980-77SendaiJapan\n"
]
| [
"Department of Applied Physics\nTohoku University\n980-77SendaiJapan",
"Department of Physics\nTohoku University\n980-77SendaiJapan",
"Department of Applied Physics\nTohoku University\n980-77SendaiJapan",
"Department of Physics\nTohoku University\n980-77SendaiJapan"
]
| []
| Tunneling conductance of an Aharonov-Bohm circuit including two quantum dots is calculated based on the general expression of the conductance in the linear response regime of the bias voltage. The calculation is performed in a wide temperature range by using numerical renormalization group method. Various types of AB oscillations appear depending on the temperature and the potential depth of the dots. Especially, AB oscillations have strong higher harmonics components as a function of the magnetic flux when the potential of the dots is deep. This is related to the crossover of the spin state due to the Kondo effect on quantum dots. When the temperature rises up, the amplitude of the AB oscillations becomes smaller reflecting the breaking of the coherency. | 10.1143/jpsj.66.717 | [
"https://export.arxiv.org/pdf/cond-mat/9707144v1.pdf"
]
| 119,446,422 | cond-mat/9707144 | 0b920b168aac4b4f0bdd03d916f087a0dc626362 |
Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit
3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Wataru Izumida
Osamu Sakai
Yukihiro Shimizu
Department of Applied Physics
Tohoku University
980-77SendaiJapan
Department of Physics
Tohoku University
980-77SendaiJapan
Many Body Effects on Electron Tunneling through Quantum Dots in an AB Circuit Many Body Effects on Electron Tunneling through Quantum Dots in an Aharonov-Bohm Circuit
23, 4, 5, 6, 7, 8, 9, 10, 11, 12(Received October 15, 1996)typeset using JPSJ.sty <ver.0.8>
Tunneling conductance of an Aharonov-Bohm circuit including two quantum dots is calculated based on the general expression of the conductance in the linear response regime of the bias voltage. The calculation is performed in a wide temperature range by using numerical renormalization group method. Various types of AB oscillations appear depending on the temperature and the potential depth of the dots. Especially, AB oscillations have strong higher harmonics components as a function of the magnetic flux when the potential of the dots is deep. This is related to the crossover of the spin state due to the Kondo effect on quantum dots. When the temperature rises up, the amplitude of the AB oscillations becomes smaller reflecting the breaking of the coherency.
§1. Introduction
There are many systems that electron-electron interactions play important roles in the physical properties of them. A system of metal with dilute magnetic impurities is a good example. Various anomalous phenomena are observed at low temperatures, related to the Kondo effect, which is caused by competition between quantum coherence effects and electron-electron interactions. 1) In the phenomenon of the electron tunneling through a quantum dot between leads, the effects of the electron-electron interactions in the dot have been investigated extensively. 2,3,4,5,6,7,8,9,10,11,12) The importance of the Kondo effect in a quantum dot has been pointed out by some theoreticians. 2,3,4,5,6,7,8,9) The 'Kondo effect in a quantum dot' will give the following features. In low temperatures than the Kondo temperature of the system, T K , the conductance will show the coherent resonant tunneling caused by the Kondo resonance peak in the quantum dot. When the temperature rises up than T K , the Kondo resonance peak vanishes and the local spin freedom appears, and so the electron tunneling with inelastic spin scattering process will dominate. Therefore the coherent resonant tunneling process will be broken.
The interference phenomenon is the good indication for the coherency of the electronic state. The Aharonov-Bohm (AB) circuits with quantum dots have been investigated to study the coherency on quantum dots. 13,14,15,16,17,18) Akera had studied the system in which two quantum dots are included in AB circuit, and shown that the amplitude of the AB oscillations depends on the spin state of the dots. 13) However, his investigation was restricted only to the high temperature region than T K . Main purpose of the present work is to study how the Kondo effect affects the AB oscillations. We calculate the temperature dependence of the AB oscillations in a wide temperature range including the crossover temperature T K .
In this paper, we consider the system shown in Fig. 1, which is the same system that Akera studied. The essential physics of this system can be given by the two impurity Anderson Hamiltonian. Because our interests are related to the change of the coherency of the electronic state, we have to calculate the conductance without assumptions on tunneling processes. In other words, the assumption, such as the restriction to the 'coherent resonant tunneling process' or to the 'sequential tunneling process', should not be used. Numerical Renormalization Group (NRG) method has been known as a useful method to study the many body systems written by the Anderson Hamiltonian. 19,20,21) This method has reliability in the temperature range near and below T K . In this paper, we develop this method to calculate the conductance without assumptions on tunneling processes.
Main results of this paper are as follows: When the temperature rises up, the amplitude of the AB oscillations becomes smaller reflecting breaking of the coherency by spin inelastic scattering;
Various types of AB oscillations appear due to many body effects when the depth of dot's potential is changed; AB oscillations have strong higher harmonics components as a function of the magnetic flux reflecting the crossover of the spin state.
In section 2, the model Hamiltonian of the system is given and is transformed for numerical calculation. The conductance formula is derived based on the linear response theory. At the same time, for comparisons, the conductance formula based on the coherent resonant tunneling process is derived by using the Friedel sum rule. In section 3, numerical results of the conductance are shown. In section 4, summary and discussion of this study are given. §2. Formulation
Model and transformation
For simplicity, we consider following situation for our system in Fig. 1. Energy separations between states of the single-electron orbits in a dot are much larger than the other energy scales:
the Coulomb repulsion energy U , temperature T , and the level width ∆. This situation corresponds to the assumption that only one single-electron orbit in the dot contributes to the current. Two electrons can occupy in this orbit because of the spin degeneracy, but the second electron need excess energy U due to the Coulomb interaction between two electrons. The orbit in the dot couples to those in leads, and this causes the electron tunneling. Tunneling from the one lead to the other lead can occur only via quantum dots. The essential properties of the system can be written by the Anderson model as follows;
H = H l + H d + H l−d , (2.1) H l = kσ ε a kσ a † kσ a kσ + pσ ε b pσ b † pσ b pσ ,(2.
2)
H d = i=1,2 σ ε d,i d † iσ d iσ + i=1,2 U i d † i↑ d i↑ d † i↓ d i↓ ,(2.
3)
H l−d = kσ (v a k,1 e i ϕ 4 d † 1σ a kσ + v a k,2 e −i ϕ 4 d † 2σ a kσ + h.c.) + pσ (v b p,1 e −i ϕ 4 d † 1σ b pσ + v b p,2 e i ϕ 4 d † 2σ b pσ + h.c.), (2.4)
where a kσ is the annihilation operator of the electron with spin σ and the state k in the left lead A, b pσ is that of in the right lead B, and d i,σ is that of on the orbit in the i-th dot, respectively. The quantity ε d,i is the single-electron level of orbit in the i-th dot, and is the effective depth of dot's potential. The quantity U i is the Coulomb repulsion energy in the i-th dot. The terms H l and H d denote the leads and the dots, respectively, and H l−d is the electron tunneling term between leads and dots. The factor e ±i ϕ 4 is due to the magnetic flux. (ϕ = 2πΦ/Φ 0 , where Φ is the magnetic flux enclosed by the circuit, and Φ 0 is the magnetic flux quantum, hc/e.)
We transform the model Hamiltonian to fit numerical calculation. We consider only the situation that ε d,1 = ε d,2 ≡ ε d and U 1 = U 2 ≡ U for simplicity. Furthermore, we assume that the geometry of the leads has symmetry with respect to the interchange of dot 1 and dot 2. Moreover we also assume the symmetry with respect to the interchange of lead A and lead B. Electron motion in the leads are restricted only to the one-dimensional motion along the lead. (The state index k is reduced to k.) Neglecting the k dependence of the tunneling matrix, we put the relations, v ak,1 = v bp,1 = v ak,2 = v bp,2 ≡ V . Finally, each term is rewritten as
H l = kσ ε k (α † s,kσ α s,kσ + α † a,− U 4 σ 1 σ 2 σ 3 σ 4 ( σ) σ 1 σ 2 · ( σ) σ 3 σ 4 d † e,σ 1 d e,σ 2 d † o,σ 3 d o,σ 4 − U 2 (d † e,↑ d † e,↓ d o,↓ d o,↑ + d † o,↑ d † o,↓ d e,↓ d e,↑ ), (2.6) H l−d = 2 kσ (V cos ϕ 4 d † e,σ α s,kσ + V sin ϕ 4 d † o,σ α a,kσ +h.c.),(2.7)
where α s,kσ ≡ (a kσ + b kσ )/ √ 2 is the symmetric combination of the lead orbits, and α a,kσ ≡ (a kσ − b kσ )/ √ 2 is the anti-symmetric combination of those, and d e,σ ≡ (d 1σ
+ d 2σ )/ √ 2 , d o,σ ≡ −i(d 1σ − d 2σ )/ √
2 are those of the dot orbits, respectively. The quantity σ is the Pauli matrix. The problem is reduced to solve the two-channel Anderson model as seen in eqs. (2.5), (2.6) and (2.7). Hereafter, we call the orbit denoted by d e,σ as 'even-orbit' and the channel of it as 'even-channel', and those of d o,σ as 'odd-orbit' and 'odd-channel'. We denote them by l as l = e or l = o.
Conductance formula
In this paper, we restrict ourselves to the linear response conductance for the applied bias voltage.
We derive the conductance formula without assumptions on electron tunneling processes. We also derive the conductance formula with assumption of the coherent resonant tunneling process.
Calculated results are compared at very low temperature.
Conductance formula at finite temperature
Now, we must calculate the conductance without using approximations for tunneling processes such as the sequential tunneling process picture or the coherent tunneling process picture. We define an electric current from lead A to lead B as follows;
I ≡ −e − Ṅ A + Ṅ B 2 , (2.8)
where −e is the charge of an electron, and Ṅ A is the expectation value of the time differentiation of N A = kσ a † kσ a kσ , the electron number operator in the lead A. The quantity Ṅ B is that of the lead B. We consider the situation applying the voltage 2V between lead A and lead B. This situation is given by adding the term H ′ = N A eV − N B eV to our Hamiltonian H, eq. (2.1). Assuming that the voltage difference 2V is small, and using linear response theory derived by Kubo,22) we obtain the following expression for the conductance formula (see Appendix);
G ≡ I 2V = 2e 2 h lim ω→0 P ′′ (ω) ω , (2.9) with P ′′ (ω) = π 2h2 4 1 Z n,m e −βEm − e −βEn × n Ṅ A −Ṅ B m 2 ×δ (ω − (E n − E m )) ,(2.10)
where Z = n e −βEn is the partition function, β = 1/T , T is the temperature of the system.
The quantities Ṅ A and Ṅ B can be expressed by localized operators near the dots as shown in Appendix. So, this expression is suitable for the numerical calculation by the NRG method, though it needs the delicate limiting process ω → 0.
Conductance formula at zero temperature based on the coherent resonant tunneling process
Here we derive the conductance formula at zero temperature for comparison with numerical results from eqs. (2.9) and (2.10). The conductance at zero temperature is given by
G F = 2πe 2 h σ,S int σ ′ ,S fin |T fin,int (ε F )| 2 ×ρ 2 (ε F )W S int , (2.11)
where initial and final states of transition are denoted as int = (kσ, S int ) and fin = (pσ ′ , S fin ). The quantity k denotes the electron in the lead A, and p denotes that in the lead B. By S int and S fin we represent initial and final states on the quantum dots, and W S int is the probability of state in S int .
The quantity T fin,int is the transition matrix, and ρ(ε F ) is the density of states on the Fermi energy in the leads. The transition matrix is written by using Green's function G as
T = V + V GV , where V = H l−d in this case.
At zero temperature, the system written in eq. (2.1) is expected to be in the local Fermi liquid state. Electron tunneling processes through the quantum dots will be given by only the coherent resonant tunneling process, i.e., S int = S fin . The conductance G F is written as follows;
G F = 2e 2 h |∆ e G e,σ (ε F + i0) − ∆ o G o,σ (ε F + i0)| 2 , (2.12) with ∆ e = 4π|V | 2 ρ(ε F ) cos 2 ϕ 4 , (2.13) ∆ o = 4π|V | 2 ρ(ε F ) sin 2 ϕ 4 .
(2.14)
In the local Fermi liquid state, Green's function G l,σ (ε F + i0) in the l-th channel (l = e, o) satisfies the Friedel sum rule as follows 23,24) ;
G l,σ (ε F + i0) = 1 −∆ l 1 tan δ l + i∆ l , (2.15)
where δ l is the phase shift for the l-th channel. Using the relation between the phase shift δ l and the ground-state occupation number n 0,l on the l-th orbit,
n 0,l = 2 δ l π , (2.16)
we get the conductance as follows;
G F = 2e 2 h sin 2 π 2 ( n 0,e − n 0,o ) . (2.17)
Although the expression (2.17) seems to be different from eq. (2.9), it can be derived from eq. (2.9)
when the system is in the local Fermi liquid state and T = 0.
The expression (2.17) is contrast with the single dot case. In such a case, the conductance is given by (2e 2 /h) sin 2 (π n 0 /2), and it has almost the maximum value 2e 2 /h when there is the Kondo resonance peak on the Fermi energy, i.e., n 0 ∼ 1. 2, 3, 4) On the other hand, the conductance has very small value in the present case as seen from (2.17) if the even and odd components have simultaneously the Kondo resonance on the Fermi energy, i.e., n 0,e ∼ n 0,o ∼ 1. This is caused by the interference cancellation between processes through the even and odd orbit states.
Excitation spectra
We will calculate excitation spectra other than P ′′ (ω) to get insights of the electronic states of the dots system. The single particle excitation spectrum for the l-th orbit (l = e, o) is defined as follows;
ρ l (ω) ≡ 1 Z n,m e −βEm − e −βEn × n|d † l |m 2 δ (ω − (E n − E m )) + | n|d l |m | 2 δ (ω + (E n − E m )) . (2.18)
The magnetic excitation spectrum at T = 0 is calculated as follows;
χ ′′ m,e (ω) ≡ n Gr | n|(S e,z + S o,z )|Gr | 2 ×δ (ω − (E n − E Gr )) , (2.19) where S l,z = (d † l,↑ d l,↑ − d † l,↓ d l,↓ )/2
is the spin operator on the l-th orbit and Gr denotes the ground state of the system. The energy of the peak position will reflect the characteristic energy of the spin fluctuation. In this paper, the energy of the peak position of χ In this paper, we fix the Coulomb repulsion energy to be U = 0.10 and the tunneling intensity ∆ = 0.03π. It is enough to study the only ε d ≥ −0.5U cases because the conductance as a function of ε d is symmetric with respect to ε d = −0.5U . Then the energy level ε d is varied from −0.05 = −0.50U , corresponding to the electron-hole symmetric case, to 0.1 = 1.0U . The conductance is the periodic function of the magnetic flux ϕ and moreover it is even function of ϕ for the present model, so we calculate the conductance only in a half period, 0 ≤ ϕ ≤ π.
Temperature T is varied from 5.8 × 10 −7 to 3.1 × 10 −1 . (See Table I and II.) Before going to detailed discussions of the numerical results, we stress the following fact. The Kondo effects in different channels seem to occur as if they are independent though the original Hamiltonian includes interaction terms between electrons in even and odd orbits. (See eq. (2.6).)
As shown later in §3.2, the Kondo resonance peaks of the single particle excitation in even-channel do not have structure reflecting the energy scale of the odd-channel Kondo effect, and vice versa.
Table I. Parameters ε d , U , ∆ ≡ 4π|V | 2 ρ(εF) in units of the band width D. ε d U ∆/π −0.050 = −0.50U −0.030 = −0.30U −0.025 = −0.25U −0.020 = −0.20U 0.10 0.03 +0.000 = +0.00U
+0.050 = +0.50U
+0.100 = +1.00U Table II. Temperatures denoted as Ti in this paper. These are given in units of the band width D.
T1 T2 T3 T4 T5 T6 T7 5.8 × 10 −7 1.7 × 10 −6 5.2 × 10 −6 1.6 × 10 −5 4.7 × 10 −5 1.4 × 10 −4 4.2 × 10 −4 T8 T9 T10 T11 T12 T13 1.3 × 10 −3 3.8 × 10 −3 1.1 × 10 −2 3.4 × 10 −2 1.1 × 10 −1 3.1 × 10 −1 3.1 Conductance at zero temperature G F (ϕ)
The conductance at zero temperature is written by using the difference of the occupation numbers,
Conductance at finite temperature G(ϕ)
In this subsection we show the numerical results of the conductance G(ϕ) at various temperatures calculated by eqs. (2.9) and (2.10).
In Fig. 4, we show examples of the spectra, P ′′ (ω)/ω, calculated by the NRG method. The calculated raw data are plotted by symbols, and the lines give the values averaged by the spline interpolation method. The data points with energies ω/T < α ∼ 1 are expected to be erroneous because they are calculated from transitions with excitation energy (T ∼ |t L |) larger than ω, where |t L | is the hopping matrix characterizing the low energy scale of the NRG calculation. 20,21) In the interpolation, these low energy data are discarded. The value at ω min = 4.0T is substituted for the limiting value of ω → 0. As shown from Fig. 4, this process does not lead the ambiguity of the limiting value of P ′′ (ω)/ω for the low temperature cases T ≪ T K . But it causes ambiguity for the T > α ′ ∼ T K cases. The quantitative accuracy of the numerical value of the present work should not be so trusted for higher temperature cases. However, it seems not bad for the qualitative discussions.
In the following subsections, we show two typical cases for the numerical results of the conductance at finite temperature.
Deep ε d case (ε d = −0.025)
Numerical results of the conductance G(ϕ) at ε d = −0.025 for various temperatures are shown in Fig. 5. First, we discuss the results in the temperature range T 1 ≤ T ≤ T 7 . The conductance G(ϕ) changes rapidly as ϕ changes between 0.1 < ϕ/π < 0.3. In addition, it has value near 2e 2 /h in ϕ/π < 0.1 region contrasted to the result of G F (ϕ) for ε d = −0.025 in Fig. 3. In the region of ϕ/π > 0.3, the curves of the conductance G(ϕ) overlap to each other, and also coincide with the result from G F (ϕ).
To get insights the origin of this behavior of the conductance, we investigate the electronic state of the system. The one-particle excitation spectra π∆ e ρ e (ω) and π∆ o ρ o (ω), and also the current spectra P ′′ (ω) for various flux cases at T = T 1 are shown in Fig. 6. The same quantities at T = T 3 are shown in Fig. 7. At T = T 1 , we have the Kondo resonance peaks in the even-channel for all flux ϕ cases. (Notice that the abscissa have a logarithmic scale.) On the other hand in ϕ/π < 0.1 cases for the odd-channel, the widths of the Kondo resonance peaks are small compared to the lowest energy calculated in the Fig. 6. The width gradually increases as flux ϕ increases from ϕ/π ∼ 0.10, and the complete Kondo resonance peak appears in the ϕ/π > 0.15 cases. The evolution of the Kondo resonance peak in a certain channel means that the electronic state in the channel is in the spin singlet state. In this case, the excitation properties will be described as the Fermi liquid state. Therefore, the electron tunneling processes through that channel will be given by only the coherent resonant tunneling process.
The even-channel state is always in the Fermi liquid state at T = T 1 for whole range of ϕ. The electronic state of the odd-channel falls into the Fermi liquid state for ϕ/π > 0.15 at T = T 1 . On the other hand for the cases ϕ/π < 0.1, the spin state of the odd-channel at T = T 1 is expected to be magnetic. Fig. 7 shows the spectra at T = T 3 . The intensity of the Kondo resonance of odd-channel with ϕ/π = 0.141 decreases remarkably from the lines in Fig. 6. At the same time the conductance increases.
It may be advisable to consider the Kondo temperature, T K , for given ϕ case. Now, we define The judgments of the boarder between the magnetic state (broken lines) to non-magnetic state (solid lines) in Fig. 5 were given by using T K (ϕ) in Fig. 8. Tunneling processes are expected to occur by the coherent resonant tunneling process in the solid line region. On the broken lines, the contribution from the processes with the inelastic spin excitation will be not small. The conductance show relatively large difference from G F (ϕ) in this region. proportional to sin 2 (ϕ/4) and is very small for ϕ ∼ 0. TK(ϕ) increases as ϕ increases from ϕ = 0 to π.
When the temperature T rises beyond T 7 , the conductance near ϕ/π ∼ 1 gradually increases and the AB oscillations become relatively smaller. When the temperature increases further through T 10 = 1.1 × 10 −2 , the conductance near ϕ/π ∼ 0 gradually decreases and the AB oscillations become very small. 26) The spin states of both even and odd channels gradually change from nonmagnetic to magnetic state for ϕ/π ∼ 1 as temperature rises up.
We can conclude that the behavior of the conductance strongly reflects the spin state in the quantum dots. At this place we comments on the difference of G(ϕ) at very low temperature T = T 1 from the curve of G F (ϕ) at exactly zero temperature. The conductance G F (ϕ = 0) is less than 0.1 × 2e 2 /h, but G(ϕ = 0) is almost 2e 2 /h even at very low temperature T = T 1 . In our calculation, it is very difficult to get the results which agree with G F (ϕ) in the region ϕ/π < 0.15, because we must continue the numerical computation to extremely low energy region. We have checked that calculations up to low energy region ω ∼ 10 −7 , T ∼ 10 −8 gives consistent result with G F (ϕ) for ϕ/π = 0.117. (See Fig. 9). But it is impossible to continue such calculation to ϕ/π < 0.117 cases. In the ϕ/π < 0.15 cases at T = T 1 , the contribution from the coherent resonant tunneling process of the odd-channel will be small because T K (ϕ) is less than T 1 . In such cases, the conductance will have the value near 2e 2 /h as seen from Fig. 3. We think that abrupt increase of G(ϕ ∼ 0) at T = T 1 from G F (ϕ ∼ 0) really reflects the crossover of states due to the increase of the temperature. Here, we note the meaning of the spectrum P ′′ (ω)/ω. This quantity is related to the fluctuation of the bias voltage. It shows peak structure at energy comparable to the width of the Kondo resonance at the Fermi energy. It shows two peaks structure when ϕ is small, as clearly seen in Fig. 9. These peaks seem to relate to the Kondo resonance in each channel.
Shallow
ε d case (ε d = 0.000)
Numerical results of the conductance G(ϕ) at ε d = 0.000 for various T are shown in Fig. 10.
The conductance G(ϕ) does not show rapid changes in this case, contrasted to the results shown in Fig. 5. We show the Kondo temperature T K (ϕ) in the ε d = 0.000 case in Fig. 11. It is much higher than that of ε d = −0.025 case in Fig. 8 at ϕ ∼ 0. In the n 0,o ∼ 0.0 case, spin freedom on the odd-orbit will disappear and so the spin excitation mainly comes from the electrons in the even-orbit. Then the Kondo temperature T K (ϕ) is larger in this case. Even when the occupation number on odd-orbit increases due to the increase of ϕ, the characteristic magnetic excitation energy is not so small because the electrons in both channels are in the mixed valence regime. So the variation of T K (ϕ) for change of ϕ is much smaller than that in the ε d = −0.025 case. Therefore the conductance in Fig. 10 does not show the rapid change caused by the crossover of the spin state.
In the temperature range T < T 9 , the conductance shows simple cos ϕ like behavior contrasted to the complicated ϕ dependence of ε d = −0.025 case. When the temperature T rises up than T 10 , the AB oscillations become smaller because the temperature becomes comparable to T K (ϕ). 26) Fig. 10. The conductance G(ϕ) for various temperatures at ε d = 0.000. AB oscillations become smaller as temperature rises up than T ∼ T9. The phenomenon that G(ϕ) rapidly changes with flux at low temperatures can not be seen in this case. §4. Summary and Discussion
In this paper, we have investigated the temperature and the magnetic flux dependences of the conductance G for the AB circuit including two quantum dots. This quantity is expected to reflect the coherency of the electronic state, and is closely related to the 'Kondo effect in quantum dots'.
The model Hamiltonian had the form of the two-channel Anderson model with even and odd orbits. The conductance was calculated by using the NRG method based on the formula without assumptions on tunneling processes.
There were interaction terms between electrons in even and odd orbits, but the Kondo effects in the even-channel and the odd-channel occurred as if they were independent. When the dots system were in the Kondo regime near ϕ ∼ 0, i.e., the occupation number in the odd-orbit satisfies in this situation. 14, 15) Extension of our calculation to the asymmetric cases will be given in near future. We stress that the present work treated the case that most drastic changes of the tunneling conductance are expected.
Anyway, the conductance will show sharp ϕ-dependence reflecting the crossover of the spin state of dots, when the Kondo effect is caused by the suitable variation of the dot potential energy.
Appendix: Formulation of the conductance in the linear response theory
We define an electric current as follows;
I ≡ −e − Ṅ A + Ṅ B 2 , (A . 1)
where −e is the charge of an electron. The quantityṄ A ≡ ī h [H, N A ] is the time derivative of the electron number operator in the lead A, N A = kσ a † kσ a kσ , andṄ B is that in the lead B. We consider the current when the external perturbation term
H ′ = −N A V A − N B V B isṄ A = σ AA V A + σ AB V B , (A . 2) Ṅ B = σ BA V A + σ BB V B , (A . 3) with σ µν = lim ω→0 σ µν (ω) (A . 4) ≡ lim ω→0 ī h ∞ 0 dt ′ e −δt ′ +iωt ′ ×Tr e −i H h t ′ [N ν , ρ eq ]e i H h t ′Ṅ µ , (A . 5)
where δ = 0+. Using properties of the trace operator and integration by parts, we get the following expression;
σ µν (ω) = 1 iω (K µν (ω) − K µν (0)) , (A . 6) with K µν (ω) ≡ − ī h ∞ 0 e −δt+iωt [Ṅ ν ,Ṅ µ (t)] dt. (A . 7)
The quantity K µν (ω) can be expressed by the following form;
K µν (ω) = K ′ µν (ω) + iK ′′ µν (ω), (A . 8)
where K ′ µν (ω) and K ′′ µν (ω) are given as follows by using the eigenstates of H (i.e., H|m = E m |m ),
K ′ µν (ω) = 1 Z n,m e −βEn − e −βEm × E m − E n +hω (E m − E n +hω) 2 + (hδ) 2 × n|Ṅ ν |m m|Ṅ µ |n (A . 9) = K ′ µν (−ω), (A . 10) K ′′ µν (ω) = 1 Z n,m e −βEn − e −βEm × −hδ (E m − E n +hω) 2 + (hδ) 2 × n|Ṅ ν |m m|Ṅ µ |n (A . 11) = −K ′′ µν (−ω). (A . 12)
The quantity σ µν is given by
σ µν = lim ω→0 K ′′ µν (ω) ω . (A . 13)
We give the value −V A = −eV and −V B = eV for external field, and obtain the electric current as follows;
I = e 2 4 (σ AA + σ BB − σ AB − σ BA ) · 2V. (A . 14)
The conductance G is given by
G ≡ I 2V (A . 15) = 2e 2 h lim ω→0 P ′′ (ω) hω , (A . 16) with P ′′ (ω) ≡ π 2h2 4 1 Z n,m e −βEm − e −βEn × n|Ṅ A −Ṅ B |m 2 ×δ (hω − (E n − E m )) . (A . 17)
We note that the operatorṄ µ (µ = A, B) can be expressed by the localized operators near the dots. For example,Ṅ A −Ṅ B is given as follows; In this place the operator kσ α l,kσ (l = s, a), which is given by linear combination of all k-state, is proportional to the localized orbit near the dots. Therefore the expression of conductance by using the quantity P ′′ (ω) is suitable for the calculation based on the NRG method, in which the approximate eigenstates are obtained successively starting from the localized orbits. (ω)/ω ω/D ϕ/π=0.00(raw data) (spline data) ϕ/π=0.50(raw data) (spline data) ϕ/π=1.00(raw data) (spline data) (ω) ω/D ϕ/π=0.125 ϕ/π=0.188 ϕ/π=0.250 ϕ/π=0.500 ϕ/π=0.750 ϕ/π=1.000
N A −Ṅ B = 2ī h kσ V cos ϕ 4 d † e,
Fig. 1 .
1The geometry of the two quantum dots and leads. Electron tunneling from the one lead to the other lead can occur only via quantum dots. The magnetic flux Φ is applied between leads and dots to cause AB interference effect.
e,↑ n e,↓ + n o,↑ n o,↓ ) + U 4 (n e,↑ + n e,↓ )(n o,↑ + n o,↓ )
′′ m,e (ω) is used as the definition of the Kondo temperature, T K . For the four fold degenerate model, this definition gives rather well approximate value of T K defined by the usual definition based on the susceptibility. 20) §3. Numerical Results We have several parameters for the model. The band width of the leads, D, is chosen to be energy units, D = 1. The parameters ε d , U, ∆ ≡ 4π|V | 2 ρ(ε F ) and T are given in units of D.
n 0,e − n 0,o , as shown in eq.(2.17). The expectation value of the occupation number n 0,l is calculated from the wave function obtained by the NRG method, and is shown inFig. 2. In the electron-hole symmetric case, ε d = −0.050, the expectation values satisfy the relation n 0,e = n 0,o = 1.0 for any ϕ. When ε d moves away from ε d = −0.050, both n 0,e and n 0,o deviate from 1.0 and they depend on the flux ϕ. We note that the hybridization strength for the oddorbit, ∆ o = 4π|V | 2 ρ(ε d ) sin 2 (ϕ/4), is very small in the ϕ ∼ 0 region. Therefore the small change
Fig. 2 .Fig. 3 .
23The expectation values of the occupation number of the even-orbit, n0,e , and the odd-orbit, n0,o , in the ground state as a function of ϕ for various cases of ε d . The flux ϕ is given in units of Φ0 = hc/e. The value n0,o at ϕ ∼ 0 decreases rapidly as ε d increases between ε d = −0.025 to −0.020. of the parameters in a critical region causes drastic change of the ground state properties of the odd-channel. Reflecting these, n 0,o at ϕ ∼ 0 decreases rapidly as ε d increases from −0.025 to −0.020. At the same time the Kondo resonance peak in the odd-channel rapidly moves away from the Fermi energy.We have two regimes for the ϕ-dependence of n 0,o . In the ε d ≤ −0.025 case, n 0,o has value almost 1.0 at ϕ ∼ 0. It initially decreases with increasing ϕ, and then increases gradually. In the ε d ≥ −0.020 case, n 0,o has very small value at ϕ ∼ 0. It increases with increasing ϕ. In contrast to the behavior of n 0,o , the occupation number n 0,e shows mild dependence on ε d and ϕ because the hybridization strength ∆ e is relatively large in 0 ≤ ϕ ≤ π region.InFig. 3, we present the numerical results of the conductance at zero temperature G F (ϕ) for various ε d . The AB oscillations vanish completely in the electron-hole symmetric case,25) The conductance at zero temperature, GF(ϕ), for various cases of ε d . The units of GF(ϕ) is 2e 2 /h. Inthe electron-hole symmetric case (ε d = −0.50U = −0.050), AB oscillations vanish completely. As ε d shifts from ε d = −0.050, the AB oscillations appear. The conductance GF(ϕ) near ϕ = 0 increases rapidly when ε d increases from ε d = −0.025 to −0.020. ε d = −0.050, because n 0,e = n 0,o = 1 as noted in Fig. 2. When ε d moves away from ε d = −0.050, the AB oscillations appear because the quantity n 0,e − n 0,o has non zero value and depends on the flux ϕ.We note that G F (ϕ) near ϕ ∼ 0 increases rapidly as ε d increases between ε d = −0.025 to −0.020.This phenomenon is reflecting the behavior of n 0,o . In the region of n 0,o ∼ 0, the Kondo resonance peak of the odd-channel moves away from the Fermi energy. Therefore, the contribution of the resonant tunneling via the Kondo resonance of the odd-channel becomes very small. From this we can see that the conductance G F (ϕ = 0) take value near 2e 2 /h when only the coherent resonant tunneling of the even-channel contributes.To conclude, the AB oscillations of the conductance at zero temperature show two types of ϕ dependence reflecting the change of the ground state properties of the system. In the ε d ≤ −0.025 case, G F (ϕ) is very small at ϕ ∼ 0. It decreases initially and soon increases with ϕ, and then it decreases near ϕ/π ∼ 1 after showing maximum. In other word, the AB oscillations have higher harmonics components. On the other hand in ε d ≥ −0.020 case, G F (ϕ) has large value at ϕ = 0, and it decreases with ϕ.
Fig. 4 .
4Examples of the spectrum, P ′′ (ω)/ω, calculated by using the NRG method. The parameters are ε d = −0.025 and T = T5(= 4.7 × 10 −5 ). The symbols give the NRG raw data points, and lines are given by spline interpolation.The edge of the lines give the lowest energy data points for the interpolation, |ωmin| = 4.0T5. The data points ω/T < α ∼ 1 should be discarded in the NRG calculation, see the text.
Fig. 5 .
5The conductance G(ϕ) at ε d = −0.025 for various temperatures. The regions given by the broken lines are the magnetic state and the regions given by the solid lines are the non-magnetic state. For the definition, see the text. The curves of the conductance given by the solid line region for T1 ≤ T ≤ T7 cases coincide to GF(ϕ) of ε d = −0.025 case inFig. 3. Tunneling processes are expected to be dominated by the coherent resonant tunneling process on solid lines. On broken lines, contributions from the process with the spin inelastic scattering will be not so small. The conductance rapidly changes at ϕ/π ∼ 0.10 in the temperature cases T1 ≤ T ≤ T7. For the origin of this phenomena, see also the text.
Fig. 6 .
6Excitation spectra π∆eρe(ω), π∆oρo(ω), and P ′′ (ω)/ω at T = T1 and ε d = −0.025. The Kondo resonance peaks on Fermi energy always exist in ρe(ω). On the other hand, There are no Kondo resonance peaks in ρo(ω) when ϕ/π < 0.1. The Kondo resonance peaks in ρo(ω) gradually grow up as flux increases from ϕ/π ∼ 0.1. The peaks appear completely in the ϕ/π > 0.15 region.
Fig. 7 .
7Excitation spectra π∆eρe(ω), π∆oρo(ω), and P ′′ (ω)/ω at T = T3 and ε d = −0.025. The Kondo resonance peaks in ρo(ω) appear in the ϕ/π > 0.25 region. The region of ϕ that the odd-channel being magnetic state becomes wider when the temperature rises as seen from comparison with Fig. 6.T K (ϕ) as an energy of the peak position of the magnetic excitation spectrum at zero temperature χ ′′ m,e (ω) given by eq.(2.19). The result is shown inFig. 8. In the figure we also show the spectrum χ ′′ m,e (ω) for several values of ϕ. The energy of the peak position is expected to reflect the characteristic energy of the spin fluctuation on the odd-channel. For ϕ/π = 0.125, we have peak at ω/D = 5.2 × 10 −6 and the energy of the peak position increases rapidly as ϕ increases. For the cases ϕ/π < 0.125, the peak position shift to very low energy, and thus we can not show it in the frame of the figure.
Fig. 8 .
8The Kondo temperature TK(ϕ) as a function of the magnetic flux ϕ at ε d = −0.025. Inset data are the magnetic excitation spectra at zero temperature. The energies ω of the peak positions of the spectra are used for the definition of TK(ϕ). TK(ϕ ∼ 0) is extremely low because the hybridization intensity of the odd-channel, ∆o, is
Fig. 9 .
9The excitation spectra P ′′ (ω)/ω at the temperature T = T1 = 5.8 × 10 −7 (left figure) and T = 2.1 × 10 −8 (right figure) at ϕ/π = 0.117, ε d = −0.025. The spectrum P ′′ (ω)/ω at T = 2.1 × 10 −8 gives consistent result with GF(ϕ/π = 0.117) = 2.4 × 10 −3 at ε d = −0.025.
n 0,o ∼ 1.0, the crossover of the spin state in the odd-channel occurred at a critical value of ϕ at low temperature. This is because the hybridization strength of odd-channel increases from the small value near ϕ ∼ 0 to large value with increasing ϕ. AB oscillations had strong higher harmonics components reflecting the crossover of the spin state of the system. This result was contrasted to the single cos ϕ dependence derived by Akera. 13) When the temperature rose up very high, the amplitude of the AB oscillations became smaller as expected. When the dots system were in the mixed valence regime near ϕ ∼ 0, i.e., n 0,o ∼ 0.0, the drastic change of the conductance as a function of ϕ did not occur.At this place, we remark limitation of the assumptions used to derive the simplified expressions, eqs. (2.5), (2.6) and (2.7). The even and odd channels were separated out, and the hybridization
Fig. 11 .
11The Kondo temperature TK(ϕ) as a function of the magnetic flux ϕ at ε d = 0.000 case. Inset data are the magnetic excitation spectra at zero temperature. The energies ω of the peak positions of the spectra are used for the definition of TK(ϕ). The variation of TK(ϕ) for the change of ϕ is smaller when compared with result for ε d = −0.025 case. of odd-channel is proportional to sin(ϕ/4). This led the drastic change of the electronic state of the odd-channel near ϕ ∼ 0. However, if the dot 1 and dot 2 are asymmetric under the interchange of them, electrons in both channels mix with each other. Therefore, the ϕ-dependence near ϕ ∼ 0 will be blurred in the asymmetric case. The experimental work of Yacoby et al. has been done
added to H which is given by eq. (2.1). The expectation values, Ṅ A and Ṅ B are given within the linear response theory,22)
σ α a,kσ+ sin
ϕ
4
d †
o,σ α s,kσ − h.c. .
(A . 18)
AcknowledgmentsThe authors would like to thank H. Akera for valuable discussions and information, and R.Takayama for helpful advice in numerical computation. The numerical computation was performed at the Computer Center of Institute for Molecular Science, the Computer Center of Tohoku University and the Supercomputer Center of Institute for Solid State.
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This result is critically dependent on the assumption that the hybridization of each channel has electron-hole symmetry. In general, there is no reason to be n0,e = n0,o = 1. In realistic cases, AB oscillations with small amplitude are expectedIn general, there is no reason to be n0,e = n0,o = 1. This result is critically dependent on the assumption that the hybridization of each channel has electron-hole symmetry. In realistic cases, AB oscillations with small amplitude are expected.
For the temperatures higher than TK, and in the parameters region ε d < εF < ε d + U. one can find that the conductance is proportional to (K 2 +4J 2 S1· S2 ) cos ϕ+C, where K = (1/ε d +1/(ε d +U )), J = (1/ε d −1/(ε d +UFor the temperatures higher than TK, and in the parameters region ε d < εF < ε d + U , one can find that the conductance is proportional to (K 2 +4J 2 S1· S2 ) cos ϕ+C, where K = (1/ε d +1/(ε d +U )), J = (1/ε d −1/(ε d +U ))
This expression is obtained following Akera's consideration. 13) ) Usually we can expect 4J 2 ≫ K 2 and S1 · S2 < 0, so the coefficient of cos ϕ could have the negative sign. The numerical results in T ≥ T11 might be shown in such a case. and C is constant in ϕ.and C is constant in ϕ. (This expression is obtained following Akera's consideration. 13) ) Usually we can expect 4J 2 ≫ K 2 and S1 · S2 < 0, so the coefficient of cos ϕ could have the negative sign. The numerical results in T ≥ T11 might be shown in such a case.
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"A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification",
"A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification"
]
| [
"Anastasios N Angelopoulos ",
"Stephen Bates "
]
| []
| []
| Black-box machine learning models are now routinely used in high-risk settings, like medical diagnostics, which demand uncertainty quantification to avoid consequential model failures. Conformal prediction (a.k.a. conformal inference) is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. Critically, the sets are valid in a distribution-free sense: they possess explicit, non-asymptotic guarantees even without distributional assumptions or model assumptions. One can use conformal prediction with any pre-trained model, such as a neural network, to produce sets that are guaranteed to contain the ground truth with a user-specified probability, such as 90%. It is easy-to-understand, easy-to-use, and general, applying naturally to problems arising in the fields of computer vision, natural language processing, deep reinforcement learning, and so on.This hands-on introduction is aimed to provide the reader a working understanding of conformal prediction and related distribution-free uncertainty quantification techniques with one self-contained document. We lead the reader through practical theory for and examples of conformal prediction and describe its extensions to complex machine learning tasks involving structured outputs, distribution shift, time-series, outliers, models that abstain, and more. Throughout, there are many explanatory illustrations, examples, and code samples in Python. With each code sample comes a Jupyter notebook implementing the method on a real-data example; the notebooks can be accessed and easily run by clicking on the following icons:. | null | [
"https://export.arxiv.org/pdf/2107.07511v6.pdf"
]
| 235,899,036 | 2107.07511 | c3ea8eb80bc8ca0b21efa273b9e4a9fd059c65be |
A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification
December 8, 2022
Anastasios N Angelopoulos
Stephen Bates
A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification
December 8, 2022
Black-box machine learning models are now routinely used in high-risk settings, like medical diagnostics, which demand uncertainty quantification to avoid consequential model failures. Conformal prediction (a.k.a. conformal inference) is a user-friendly paradigm for creating statistically rigorous uncertainty sets/intervals for the predictions of such models. Critically, the sets are valid in a distribution-free sense: they possess explicit, non-asymptotic guarantees even without distributional assumptions or model assumptions. One can use conformal prediction with any pre-trained model, such as a neural network, to produce sets that are guaranteed to contain the ground truth with a user-specified probability, such as 90%. It is easy-to-understand, easy-to-use, and general, applying naturally to problems arising in the fields of computer vision, natural language processing, deep reinforcement learning, and so on.This hands-on introduction is aimed to provide the reader a working understanding of conformal prediction and related distribution-free uncertainty quantification techniques with one self-contained document. We lead the reader through practical theory for and examples of conformal prediction and describe its extensions to complex machine learning tasks involving structured outputs, distribution shift, time-series, outliers, models that abstain, and more. Throughout, there are many explanatory illustrations, examples, and code samples in Python. With each code sample comes a Jupyter notebook implementing the method on a real-data example; the notebooks can be accessed and easily run by clicking on the following icons:.
A. 1
Conformal Prediction
Conformal prediction [1][2][3] (a.k.a. conformal inference) is a straightforward way to generate prediction sets for any model. We will introduce it with a short, pragmatic image classification example, and follow up in later paragraphs with a general explanation. The high-level outline of conformal prediction is as follows. First, we begin with a fitted predicted model (such as a neural network classifier) which we will callf . Then, we will create prediction sets (a set of possible labels) for this classifier using a small amount of additional calibration data-we will sometimes call this the calibration step.
Formally, suppose we have images as input and they each contain one of K classes. We begin with a classifier that outputs estimated probabilities (softmax scores) for each class:f (x) ∈ [0, 1] K . Then, we reserve a moderate number (e.g., 500) of fresh i.i.d. pairs of images and classes unseen during training, (X 1 , Y 1 ), . . . , (X n , Y n ), for use as calibration data. Usingf and the calibration data, we seek to construct a prediction set of possible labels C(X test ) ⊂ {1, . . . , K} that is valid in the following sense:
1 − α ≤ P(Y test ∈ C(X test )) ≤ 1 − α + 1 n + 1 ,(1)
where (X test , Y test ) is a fresh test point from the same distribution, and α ∈ [0, 1] is a user-chosen error rate. In words, the probability that the prediction set contains the correct label is almost exactly 1 − α; we call this property marginal coverage, since the probability is marginal (averaged) over the randomness in the calibration and test points. See Figure 1 for examples of prediction sets on the Imagenet dataset.
To construct C fromf and the calibration data, we will perform a simple calibration step that requires only a few lines of code; see the right panel of Figure 2. We now describe the calibration step in more detail, introducing some terms that will be helpful later on. First, we set the conformal score s i = 1 −f (X i ) Yi to be one minus the softmax output of the true class. The score is high when the softmax output of the true class is low, i.e., when the model is badly wrong. Next comes the critical step: defineq to be the (n+1)(1−α) /n empirical quantile of s 1 , ..., s n , where · is the ceiling function (q is essentially the 1 − α quantile, but with a small correction). Finally, for a new test data point (where X test is known but Y test is not), create a prediction set C(X test ) = {y :f (X test ) y ≥ 1 −q} that includes all classes with a high enough softmax output (see Figure 2). Remarkably, this algorithm gives prediction sets that are guaranteed to satisfy (1), no matter what (possibly incorrect) model is used or what the (unknown) distribution of the data is.
Remarks
Let us think about the interpretation of C. The function C is set-valued -it takes in an image, and it outputs a set of classes as in Figure 1. The model's softmax outputs help to generate the set. This method constructs a different output set adaptively to each particular input. The sets become larger when the model is uncertain or the image is intrinsically hard. This is a property we want, because the size of the set gives you an indicator of the model's certainty. Furthermore, C(X test ) can be interpreted as a set of plausible classes that the image X test could be assigned to. Finally, C is valid, meaning it satisfies (1). 1 These properties of C translate naturally to other machine learning problems, like regression, as we will see.
With an eye towards generalization, let us review in detail what happened in our classification problem. To begin, we were handed a model that had an inbuilt, but heuristic, notion of uncertainty: softmax outputs. The softmax outputs attempted to measure the conditional probability of each class; in other words, the jth entry of the softmax vector estimated P(Y = j | X = x), the probability of class j conditionally on an input image x. However, we had no guarantee that the softmax outputs were any good; they may have been arbitrarily overfit or otherwise untrustworthy. Therefore, instead of taking the softmax outputs at face value, we used the holdout set to adjust for their deficiencies.
The holdout set contained n ≈ 500 fresh data points that the model never saw during training, which allowed us to get an honest appraisal of its performance. The adjustment involved computing conformal scores, which grow when the model is uncertain, but are not valid prediction intervals on their own. In our case, the conformal score was one minus the softmax output of the true class, but in general, the score can be any function of x and y. We then tookq to be roughly the 1 − α quantile of the scores. In this case, the quantile had a simple interpretation-when setting α = 0.1, at least 90% of ground truth softmax outputs are guaranteed to be above the level 1 −q (we prove this rigorously in Appendix D). Taking advantage of this fact, at test-time, we got the softmax outputs of a new image X test and collected all classes with outputs above 1 −q into a prediction set C(X test ). Since the softmax output of the new true class Y test is guaranteed to be above 1 −q with probability at least 90%, we finally got the guarantee in Eq. (1).
Instructions for Conformal Prediction
As we said during the summary, conformal prediction is not specific to softmax outputs or classification problems. In fact, conformal prediction can be seen as a method for taking any heuristic notion of uncertainty from any model and converting it to a rigorous one (see the diagram below). Conformal prediction does not care if the underlying prediction problem is discrete/continuous or classification/regression. We next outline conformal prediction for a general input x and output y (not necessarily discrete).
1. Identify a heuristic notion of uncertainty using the pre-trained model. 2. Define the score function s(x, y) ∈ R. (Larger scores encode worse agreement between x and y.)
3. Computeq as the (n+1)(1−α) n quantile of the calibration scores s 1 = s(X 1 , Y 1 ), ..., s n = s(X n , Y n ).
Use this quantile to form the prediction sets for new examples:
C(X test ) = {y : s(X test , y) ≤q} .
(
As before, these sets satisfy the validity property in (1), for any (possibly uninformative) score function and (possibly unknown) distribution of the data. We formally state the coverage guarantee next.
Theorem 1 (Conformal coverage guarantee; Vovk, Gammerman, and Saunders [5]). Suppose (X i , Y i ) i=1,...,n and (X test , Y test ) are i.i.d. and defineq as in step 3 above and C(X test ) as in step 4 above. Then the following holds:
P Y test ∈ C(X test ) ≥ 1 − α.
See Appendix D for a proof and a statement that includes the upper bound in (1). We note that the above is only a special case of conformal prediction, called split conformal prediction. This is the most widely-used version of conformal prediction, and it will be our primary focus. To complete the picture, we describe conformal prediction in full generality later in Section 6 and give an overview of the literature in Section 7.
Choice of score function
Upon first glance, this seems too good to be true, and a skeptical reader might ask the following question:
How is it possible to construct a statistically valid prediction set even if the heuristic notion of uncertainty of the underlying model is arbitrarily bad?
Let's give some intuition to supplement the mathematical understanding from the proof in Appendix D. Roughly, if the scores s i correctly rank the inputs from lowest to highest magnitude of model error, then the resulting sets will be smaller for easy inputs and bigger for hard ones. If the scores are bad, in the sense that they do not approximate this ranking, then the sets will be useless. For example, if the scores are random noise, then the sets will contain a random sample of the label space, where that random sample is large enough to provide valid marginal coverage. This illustrates an important underlying fact about conformal prediction: although the guarantee always holds, the usefulness of the prediction sets is primarily determined by the score function. This should be no surprise-the score function incorporates almost all the information we know about our problem and data, including the underlying model itself. For example, the main difference between applying conformal prediction on classification problems versus regression problems is the choice of score. There are also many possible score functions for a single underlying model, which have different properties. Therefore, constructing the right score function is an important engineering choice. We will next show a few examples of good score functions.
Examples of Conformal Procedures
In this section we give examples of conformal prediction applied in many settings, with the goal of providing the reader a bank of techniques to practically deploy. Note that we will focus only on one-dimensional Y in this section, and smaller conformal scores will correspond to more model confidence (such scores are called nonconformity scores). Richer settings, such as high-dimensional Y , complicated (or multiple) notions of error, or where different mistakes cost different amounts, often require the language of risk control, outlined in Section A.
Classification with Adaptive Prediction Sets
Let's begin our sequence of examples with an improvement to the classification example in Section 1. The previous method produces prediction sets with the smallest average size [6], but it tends to undercover hard subgroups and overcover easy ones. Here we develop a different method called adaptive prediction sets (APS) that avoids this problem. We will follow [7] and [4].
As motivation for this new procedure, note that if the softmax outputsf (X test ) were a perfect model of Y test |X test , we would greedily include the top-scoring classes until their total mass just exceeded 1 − α. Formally, we can describe this oracle algorithm as
{π 1 (x), ..., π k (x)} , where k = sup k : k j=1f (X test ) πj (x) < 1 − α + 1,
and π(x) is the permutation of {1, ..., K} that sortsf (X test ) from most likely to least likely. In practice, however, this procedure fails to provide coverage, sincef (X test ) is not perfect; it only provides us a heuristic notion of uncertainty. Therefore, we will use conformal prediction to turn this into a rigorous notion of uncertainty.
To proceed, we define a score function inspired by the oracle algorithm:
s(x, y) = k j=1f (x) πj (x) , where y = π k (x).
In other words, we greedily include classes in our set until we reach the true label, then we stop. Unlike the score from Section 1, this one utilizes the softmax outputs of all classes, not just the true class.
The next step, as in all conformal procedures, is to setq = Quantile(s 1 , ..., s n ; (n+1)(1−α) n ). Having done so, we will form the prediction set {y : s(x, y) ≤q}, modified slightly to avoid zero-size sets:
C(x) = {π 1 (x), ..., π k (x)} , where k = sup k : k j=1f (x) πj (x) <q + 1.
(3) Figure 3 shows Python code to implement this method. As usual, these uncertainty sets (with tie-breaking) satisfy (1). See [4] for details and significant practical improvements, which we implemented here: . # Get scores cal_scores = np.maximum(cal_labels-model_upper(cal_X), model_lower(cal_X)-cal_labels) # Get the score quantile qhat = np.quantile(cal_scores, np.ceil((n+1)*(1-alpha))/n, interpolation='higher') # Deploy (output=lower and upper adjusted quantiles) prediction_sets = [val_lower -qhat, val_upper + qhat]
Conformalized Quantile Regression
We will next show how to incorporate uncertainty into regression problems with a continuous output, following the algorithm in [8]. We use quantile regression [9] as our base model. As a reminder, the quantile regression algorithm attempts to learn the γ quantile of Y test |X test = x for each possible value of x. We will call the true quantile t γ (x) and the fitted modelt γ (x). Since by definition Y test |X test = x lands below t 0.05 (x) with 5% probability and above t 0.95 (x) with 5% probability, we would expect the interval t 0.05 (x),t 0.95 (x) to have approximately 90% coverage. However, because the fitted quantiles may be inaccurate, we will conformalize them. Python pseudocode for conformalized quantile regression is in Figure 5.
After training an algorithm to output two such quantiles (this can be done with a standard loss function, see below), t α/2 and t 1−α/2 , we can define the score to be the difference between y and its nearest quantile,
s(x, y) = max t α/2 (x) − y, y −t 1−α/2 (x) .
After computing the scores on our calibration set and settingq = Quantile(s 1 , ..., s n ; (n+1)(1−α) n ), we can form valid prediction intervals by taking
C(x) = t α/2 (x) −q,t 1−α/2 (x) +q .(4)
Intuitively, the set C(x) just grows or shrinks the distance between the quantiles byq to achieve coverage.
q u a n t il e re gre s s io n p rediction s e t Figure 6: A visualization of the conformalized quantile regrssion algorithm in Eq. (4). We adjust the quantiles by the constantq, picked during the calibration step.
As before, C satisfies the coverage property in Eq. (1). However, unlike our previous example in Section 1, C is no longer a set of classes, but instead a continuous interval in R. Quantile regression is not the only way to get such continuous-valued intervals. However, it is often the best way, especially if α is known in advance. The reason is that the intervals generated via quantile regression even without conformal prediction, i.e. [t α/2 (x),t 1−α/2 (x)], have good coverage to begin with. Furthermore, they have asymptotically valid conditional coverage (a concept we will explain in Section 3). These properties propagate through the conformal procedure and lead to prediction sets with good performance.
One attractive feature of quantile regression is that it can easily be added on top of any base model simply by changing the loss function to a quantile loss (informally referred to as a pinball loss),
L γ (t γ , y) = (y −t γ )γ1 y >t γ + (t γ − y)(1 − γ)1 y ≤t γ .
The reader can think of quantile regression as a generalization of L1-norm regression: when γ = 0.5, the loss function reduces to L 0.5 = |t γ (x) − y|/2, which encouragest 0.5 (x) to converge to the conditional median. Changing γ just modifies the L1 norm as in the illustration above to target other quantiles. In practice, one can just use a quantile loss instead of MSE at the end of any algorithm, like a neural network, in order to regress to a quantile.
Conformalizing Scalar Uncertainty Estimates
The Estimated Standard Deviation
As an alternative to quantile regression, our next example is a different way of constructing prediction sets for continuous y with a less rich but more common notion of heuristic uncertainty: an estimate of the standard deviationσ(x). For example, one can produce uncertainty scalars by assuming Y test | X test = x follows some parametric distribution-like a Gaussian distribution-and training a model to output the mean and variance of that distribution. To be precise, in this setting we choose to model Y test | X test = x ∼ N (µ(x), σ(x)), and we have modelsf (x) andσ(x) trained to maximize the likelihood of the data with respect to E [Y test | X test=x ] and Var [Y test | X test = x] respectively. Then,f (x) gets used as the point prediction andσ(x) gets used as the uncertainty. This strategy is so common that it is commoditized: there are inbuilt PyTorch losses, such as GaussianNLLLoss, that enable training a neural network this way. However, we usually know Y test | X test isn't Gaussian, so even if we had infinite data,σ(x) would not necessarily be reliable. We can use conformal prediction to turn this heuristic uncertainty notion into rigorous prediction intervals of the formf (x)±qσ(x).
Other 1-D Uncertainty Estimates
More generally, we assume there is a function u(x) such that larger values encode more uncertainty. This single number can have many interpretations beyond the standard deviation. For example, one instance of an uncertainty scalar simply involves the user creating a model for the magnitude of the residual. In that setting, the user would first fit a modelf that predicts y from x. Then, they would fit a second model r (possibly the same neural network), that predicts y −f (x) . Ifr were perfect, we would expect the set f (x) −r(x),f (x) +r(x) to have perfect coverage. However, our learned model of the errorr is often poor in practice.
There are many more such uncertainty scalars than we can discuss in this document in detail, including 1. measuring the variance off (x) across an ensemble of models, 2. measuring the variance off (x) when randomly dropping out a fraction of nodes in a neural net, 3. measuring the variance off (x) to small, random input perturbations, 4. measuring the variance off (x) over different noise samples input to a generative model, 5. measuring the magnitude of change inf (x) when applying an adversarial perturbation, etc.
These cases will all be treated the same way. There will be some point predictionf (x), and some uncertainty scalar u(x) that is large when the model is uncertain and small otherwise (in the residual setting, u(x) :=r(x), and in the Gaussian setting, u(x) :=σ(x)). We will proceed with this notation for the sake of generality, but the reader should understand that u can be replaced with any function. Now that we have our heuristic notion of uncertainty in hand, we can define a score function,
s(x, y) = y −f (x) u(x)
. This score function has a natural interpretation: it is a multiplicative correction factor of the uncertainty scalar (i.e., s(x, y)u(x) = y −f (x) ). As before, takingq to be the (1−α)(n+1) n quantile of the calibration scores guarantees us that for a new example,
P [s(X test , Y test ) ≤q] ≥ 1 − α =⇒ P Y test −f (X test ) ≤ u(X test )q ≥ 1 − α.
Naturally, we can then form prediction sets using the rule
C(x) = f (x) − u(x)q,f (x) + u(x)q .(5)
pre dic t i o n s e t Let's reflect a bit on the nature of these prediction sets. The prediction sets are valid, as we desired. Due to our construction, they are also symmetric about the prediction,f (x), although symmetry could be relaxed with minor modifications. However, uncertainty scalars do not necessarily scale properly with α. In other words, there is no reason to believe that a quantity likeσ would be directly related to quantiles of the label distribution. We tend to prefer quantile regression when possible, since it directly estimates this quantity and thus should be a better heuristic (and in practice it usually is; see [10] for some evaluations). Nonetheless, uncertainty scalars remain in use because they are easy to deploy and have been commoditized in popular machine learning libraries. See Figure 7 for a Python implementation of this method.
Conformalizing Bayes
Our final example of conformal prediction will use a Bayesian model. Bayesian predictors, like Bayesian neural networks, are commonly studied in the field of uncertainty quantification, but rely on many unverifiable and/or incorrect assumptions to provide coverage. Nonetheless, we should incorporate any prior information we have into our prediction sets. We will now show how to create valid prediction sets that are also Bayes optimal among all prediction sets that achieve 1 − α coverage. These prediction sets use the posterior predictive density as a conformal score. The Bayes optimality of this procedure was first proven in [11], and was previously studied in [12,13]. Because our algorithm reduces to picking the labels with high posterior predictive density, the Python code will look exactly the same as in Figure 2. The only difference is interpretation, since the softmax now represents an approximation of a continuous distribution rather than a categorical one.
Let us first describe what a Bayesian would do, given a Bayesian modelf (y | x), which estimates the value of the posterior distribution of Y test at label y with input X test = x. If one believed all the necessary assumptions-mainly, a correctly specified model and asymptotically large n-the following would be the optimal prediction set:
S(x) = y :f (y | x) > t , where t is chosen so y∈S(x)f (y | x)dy = 1 − α.
However, because we cannot make assumptions on the model and data, we can only considerf (y | x) to be a heuristic notion of uncertainty.
Following our now-familiar checklist, we can define a conformal score,
s(x, y) = −f (y | x),
which is high when the model is uncertain and otherwise low. After computingq over the calibration data, we can then construct prediction sets:
C(x) = y :f (y | x) > −q .(6)
prediction set This set is valid because we chose the thresholdq via conformal prediction. Furthermore, when certain technical assumptions are satisfied, it has the best Bayes risk among all prediction sets with 1 − α coverage.
To be more precise, under the assumptions in [11], C(X test ) has the smallest average size of any conformal procedure with 1 − α coverage, where the average is taken over the data and the parameters. This result should not be a surprise to those familiar with decision theory, as the argument we are making feels similar to that of the Neyman-Pearson lemma. This concludes the final example.
Discussion
As our examples have shown, conformal prediction is a simple and pragmatic technique with many use cases. It is also easy to implement and computationally trivial. Additionally, the above four examples serve as roadmaps to the user for designing score functions with various notions of optimality, including average size, adaptivity, and Bayes risk. Still more is yet to come-conformal prediction can be applied more broadly than it may first seem at this point. We will outline extensions of conformal prediction to other prediction tasks such as outlier detection, image segmentation, serial time-series prediction, and so on in Section 4. Before addressing these extensions, we will take a deep dive into diagnostics for conformal prediction in the standard setting, including the important topic of conditional coverage.
Evaluating Conformal Prediction
We have spent the last two sections learning how to form valid prediction sets satisfying rigorous statistical guarantees. Now we will discuss how to evaluate them. Our evaluations will fall into one of two categories.
1. Evaluating adaptivity. It is extremely important to keep in mind that the conformal prediction procedure with the smallest average set size is not necessarily the best. A good conformal prediction procedure will give small sets on easy inputs and large sets on hard inputs in a way that faithfully reflects the model's uncertainty. This adaptivity is not implied by conformal prediction's coverage guarantee, but it is non-negotiable in practical deployments of conformal prediction. We will formalize adaptivity, explore its consequences, and suggest practical algorithms for evaluating it.
2. Correctness checks. Correctness checks help you test whether you've implemented conformal prediction correctly. We will empirically check that the coverage satisfies Theorem 1. Rigorously evaluating whether this property holds requires a careful accounting of the finite-sample variability present with real datasets. We develop explicit formulae for the size of the benign fluctuations-if one observes deviations from 1 − α in coverage that are larger than these formulae dictate, then there is a problem with the implementation.
Many of the evaluations we suggest are computationally intensive, and require running the entire conformal procedure on different splits of data at least 100 times. Naïve implementations of these evaluations can be slow when the score takes a long time to compute. With some simple computational tricks and strategic caching, we can speed this process up by orders of magnitude. Therefore to aid the reader, we intersperse the mathematical descriptions with code to efficiently implement these computations.
Evaluating Adaptivity
Although any conformal procedure yields prediction intervals that satisfy (1), there are many such procedures, and they differ in other important ways. In particular, a key design consideration for conformal prediction is adaptivity: we want the procedure to return larger sets for harder inputs and smaller sets for easier inputs. While most reasonable conformal procedures will satisfy this to some extent, we now discuss precise metrics for adaptivity that allow the user to check a conformal procedure and to compare multiple alternative conformal procedures.
Set size. The first step is to plot histograms of set sizes. This histogram helps us in two ways. Firstly, a large average set size indicates the conformal procedure is not very precise, indicating a possible problem with the score or underlying model. Secondly, the spread of the set sizes shows whether the prediction sets properly adapt to the difficulty of examples. A wider spread is generally desirable, since it means that the procedure is effectively distinguishing between easy and hard inputs. set size # set size # It can be tempting to stop evaluations after plotting the coverage and set size, but certain important questions remain unanswered. A good spread of set sizes is generally better, but it does not necessarily indicate that the sets adapt properly to the difficulty of X. Above seeing that the set sizes have dynamic range, we will need to verify that large sets occur for hard examples. We next formalize this notion and give metrics for evaluating it.
Conditional coverage. Adaptivity is typically formalized by asking for the conditional coverage [14] property:
P [Y test ∈ C(X test ) | X test ] ≥ 1 − α.(7)
That is, for every value of the input X test , we seek to return prediction sets with 1 − α coverage. This is a stronger property than the marginal coverage property in (1) that conformal prediction is guaranteed to achieve-indeed, in the most general case, conditional coverage is impossible to achieve [14]. In other words, conformal procedures are not guaranteed to satisfy (7), so we must check how close our procedure comes to approximating it.
The difference between marginal and conditional coverage is subtle but of great practical importance, so we will spend some time think about the differences here. Imagine there are two groups of people, group A and group B, with frequencies 90% and 10%. The prediction sets always cover Y among people in group A and never cover Y when the person comes from group B. Then the prediction sets have 90% coverage, but not conditional coverage. Conditional coverage would imply that the prediction sets cover Y at least 90% of the time in both groups. This is necessary, but not sufficient; conditional coverage is a very strong property that states the probability of the prediction set needs to be ≥ 90% for a particular person. In other words, for any subset of the population, the coverage should be ≥ 90%. See Figure 10 for a visualization of the difference between conditional and marginal coverage. Feature-stratified coverage metric. As a first metric for conditional coverage, we will formalize the example we gave earlier, where coverage is unequal over some groups. The reader can think of these groups as discrete categories, like race, or as a discretization of continuous features, like age ranges. Formally, suppose we have features X
(val) i,1
that take values in {1, . . . , G} for some G. (Here, i = 1, . . . , n val indexes the example in the validation set, and the first coordinate of each feature is the group.) Let I g ⊂ {1, . . . , n val } be the set of observations such that X (val) i,1 = g for g = 1, . . . , G. Since conditional coverage implies that the procedure has the same coverage for all values of X test , we use the following measure:
FSC metric : min g∈{1,...,G} 1 |I g | i∈Ig 1 Y (val) i ∈ C X (val) i
In words, this is the observed coverage among all instances where the discrete feature takes the value g. If conditional coverage were achieved, this would be 1 − α, and values farther below 1 − α indicate a greater violation of conditional coverage. Note that this metric can also be used with a continuous feature by binning the features into a finite number of categories.
Size-stratified coverage metric. We next consider a more general-purpose metric for how close a conformal procedure comes to satisfying (7), introduced in [4]. First, we discretize the possible cardinalities of C(x), into G bins, B 1 , . . . , B G . For example, in classification we might divide the observations into three groups, depending on whether C(x) has one element, two elements, or more than two elements. Let I g ⊂ {1, . . . , n val } be the set of observations falling in bin g for g = 1, . . . , G. Then we consider the following
SSC metric : min g∈{1,...,G} 1 |I g | i∈Ig 1 Y (val) i ∈ C X (val) i
In words, this is the observed coverage for all units for which the set size |C(x)| falls into bin g. As before, if conditional coverage were achieved, this would be 1 − α, and values farther below 1 − α indicate a greater violation of conditional coverage. Note that this is the same expression as for the FSC metric, except that the definition of I g has changed. Unlike the FSC metric, the user does not have to define an important set of discrete features a-priori-it is a general metric that can apply to any example.
See [15] and [16] for additional metrics of conditional coverage.
The Effect of the Size of the Calibration Set
We first pause to discuss how the size of the calibration set affects conformal prediction. We consider this question for two reasons. First, the user must choose this for a practical deployment. Roughly speaking, our conclusion will that be choosing a calibration set of size n = 1000 is sufficient for most purposes. Second, the size of the calibration set is one source of finite-sample variability that we will need to analyze to correctly check the coverage. We will build on the results here in the next section, where we give a complete description of how to check coverage in practice.
How does the size of the calibration set, n, affect conformal prediction? The coverage guarantee in (1) holds for any n, so we can see that our prediction sets have coverage at least 1 − α even with a very small calibration set. Intuitively, however, it may seem that larger n is better, and leads to more stable procedures. This intuition is correct, and it explains why using a larger calibration set is beneficial in practice. The details are subtle, so we carefully work through them here.
The key idea is that the coverage of conformal prediction conditionally on the calibration set is a random quantity. That is, if we run the conformal prediction algorithm twice, each time sampling a new calibration dataset, then check the coverage on an infinite number of validation points, those two numbers will not be equal. The coverage property in (1) says that coverage will be at least 1 − α on average over the randomness in the calibration set, but with any one fixed calibration set, the coverage on an infinite validation set will be some number that is not exactly 1 − α. Nonetheless, we can choose n large enough to control these fluctuations in coverage by analyzing its distribution.
In particular, the distribution of coverage has an analytic form, first introduced by Vladimir Vovk in [14], namely,
P Y test ∈ C (X test ) {(X i , Y i )} n i=1 ∼ Beta (n + 1 − l, l) , where l = (n + 1)α .
Notice that the conditional expectation above is the coverage with an infinite validation data set, holding the calibration data fixed. A simple proof of this fact is available in [14]. We plot the distribution of coverage for several values of n in Figure 11.
Inspecting Figure 11, we see that choosing n = 1000 calibration points leads to coverage that is typically between .88 and .92, hence our rough guideline of choosing about 1000 calibration points. More formally, we can compute exactly the number of calibration points n needed to achieve a coverage of 1 − α ± with probability 1 − δ. Again, the average coverage is always at least 1 − α; the parameter δ controls the tail probabilities of the coverage conditionally on the calibration data. For any δ, the required calibration set size n can be explicitly computed from a simple expression, and we report on several values in Table 1 for the reader's reference. Code allowing the user to produce results for any choice of n and α accompanies the table. Table 1: Calibration set size n( ) required for coverage slack with δ = 0.1 and α = 0.1.
Checking for Correct Coverage
As an obvious diagnostic, the user will want to assess whether the conformal procedure has the correct coverage. This can be accomplished by running the procedure over R trials with new calibration and validation sets, and then calculating the empirical coverage for each,
C j = 1 n val n val i=1 1 Y (val) i,j ∈ C j X (val) i,j , for j = 1, ..., R,
where n val is the size of the validation set, (X
(val) i,j , Y (val)
i,j ) is the ith validation example in trial j, and C j is calibrated using the calibration data from the jth trial. A histogram of the C j should be centered at roughly 1 − α, as in Figure 11. Likewise, the mean value,
C = 1 R R j=1 C j ,
should be approximately 1 − α.
With real datasets, we only have n + n val data points total to evaluate our conformal algorithm and therefore cannot draw new data for each of the R rounds. So, we compute the coverage values by randomly splitting the n + n val data points R times into calibration and validation datasets, then running conformal. Notice that rather than splitting the data points themselves many times, we can instead first cache all conformal scores and then compute the coverage values over many random splits, as in the code sample in Figure 12.
If properly implemented, conformal prediction is guaranteed to satisfy the inequality in (1). However, if the reader sees minor fluctuations in the observed coverage, they may not need to worry: the finiteness of n, n val , and R can lead to benign fluctuations in coverage which add some width to the Beta distribution in Figure 11. Appendix C gives exact theory for analyzing the mean and standard deviation of C. From this, we will be able to tell if any deviation from 1 − α indicates a problem with the implementation, or try: # try loading the scores first scores = np.load('scores.npy') except:
# X and Y have n + n_val rows each scores = get_scores(X,Y) np.save(scores, 'scores.npy') # calculate the coverage R times and store in list coverages = np.zeros((R,)) for r in range(R):
np.random.shuffle(scores) # shuffle calib_scores, val_scores = (scores[:n],scores[n:]) # split qhat = np.quantile(calib_scores, np.ceil((n+1)*(1-alpha)/n), method='higher') # calibrate coverages[r] = (val_scores <= qhat).astype(float).mean() # see caption average_coverage = coverages.mean() # should be close to 1-alpha plt.hist(coverages) # should be roughly centered at 1-alpha Figure 12: Python code for computing coverage with efficient score caching. Notice that from the expression for conformal sets in (2), a validation point is covered if and only if s(X, Y ) ≤q, which is how the third to last line is succinctly computing the coverage.
if it is benign. Code for checking the coverage at all different values of n, n val , and R is available in the accompanying Jupyter notebook of Figure 12.
Extensions of Conformal Prediction
At this point, we have seen the core of the matter: how to construct prediction sets with coverage in any standard supervised prediction problem. We now broaden our horizons towards prediction tasks with different structure, such as side information, covariate shift, and so on. These more exotic problems arise quite frequently in the real world, so we present practical conformal algorithms to address them.
Group-Balanced Conformal Prediction
In certain settings, we might want prediction intervals that have equal error rates across certain subsets of the data. For example, we may require our medical classifier to have coverage that is correct for all racial and ethnic groups. To formalize this, we suppose that the first feature of our inputs, X i,1 , i = 1, ..., n takes values in some discrete set {1, ..., G} corresponding to categorical groups. We then ask for group-balanced coverage:
P (Y test ∈ C(X test ) | X test,1 = g) ≥ 1 − α,(8)
for all groups g ∈ {1, . . . , G}. In words, this means we have a 1 − α coverage rate for all groups. Notice that the group output could be a post-processing of the original features in the data. For example, we might bin the values of X test into a discrete set.
Recall that a standard application of conformal prediction will not necessarily yield coverage within each group simultaneously-that is, (8) may not be satisfied. We saw an example in Figure 10; the marginal guarantee from normal conformal prediction can still be satisfied even if all errors happen in one group.
In order to achieve group-balanced coverage, we will simply run conformal prediction seperately for each group, as visualized below.
Making this formal, given a conformal score function s, we stratify the scores on the calibration set by group, s
(g) i = s(X j , Y j ), where X j,1 is the ith occurrence of group g.
Then, within each group, we calculate the conformal quantilê
q (g) = Quantile s 1 , ..., s n (g) ; (n (g) + 1)(1 − α) n (g) , where n (g) is the number of examples of group g.
Finally, we form prediction sets by first picking the relevant quantile,
C(x) = y : s(x, y) ≤q (x1) .
That is, for a point x that we see falls in group x 1 , we use the thresholdq (x1) to form the prediction set, and so on. This choice of C satisfies (8), as was first documented by Vovk in [14].
Proposition 1 (Error control guarantee for group-balanced conformal prediction).
Suppose (X 1 , Y 1 ), . . . , (X n , Y n ), (X test , Y test ) are an i.i.d.
sample from some distribution. Then the set C defined above satisfies the error control property in (8).
Class-Conditional Conformal Prediction
In classification problems, we might similarly ask for coverage on every ground truth class. For example, if we had a medical classifier assigning inputs to class normal or class cancer, we might ask that the prediction sets are 95% accurate both when the ground truth is class cancer and also when the ground truth is class normal. Formally, we return to the classification setting, where Y = {1, ..., K}. We seek to achieve class-balanced coverage,
P (Y test ∈ C(X test ) | Y test = y) ≥ 1 − α,(9)
for all classes y ∈ {1, . . . , K}.
To achieve class-balanced coverage, we will calibrate within each class separately. The algorithm will be similar to the group-balanced coverage of Section 4.1, but we must modify it because we do not know the correct class at test time. (In contrast, in Section 4.1, we observed the group information X test,1 as an input feature.) See the visualization below.
Turning to the algorithm, given a conformal score function s, stratify the scores on the calibration set by class, s
(k) i = s(X j , Y j ), where Y j is the ith occurrence of class k.
Then, within each class, we calculate the conformal quantile,
q (k) = Quantile s 1 , ..., s n (k) ; (n (k) + 1)(1 − α) n (k) , where n (k) is the number of examples of class k.
Finally, we iterate through our classes and include them in the prediction set based on their quantiles:
C(x) = y : s(x, y) ≤q (y) .
Notice that in the preceding display, we take a provisional value of the response, y, and then use the conformal thresholdq (y) to determine if it is included in the prediction set. This choice of C satisfies (9), as proven by Vovk in [14]; another version can be found in [6].
Proposition 2 (Error control guarantee for class-balanced conformal prediction).
Suppose (X 1 , Y 1 ), . . . , (X n , Y n ), (X test , Y test ) are an i.i.d.
sample from some distribution. Then the set C defined above satisfies the error control property in (9).
Conformal Risk Control
So far, we have used conformal prediction to construct prediction sets that bound the miscoverage,
P Y test / ∈ C(X test ) ≤ α.(10)
However, for many machine learning problems, the natural notion of error is not miscoverage. Here we show that conformal prediction can also provide guarantees of the form
E C(X test ), Y test ≤ α,(11)
for any bounded loss function that shrinks as C grows. This is called a conformal risk control guarantee. Note that (11) recovers (10) when using the miscoverage loss,
C(X test ), Y test = 1 {Y test / ∈ C(X test )}.
However, this algorithm also extends conformal prediction to situations where other loss functions, such as the false negative rate (FNR), are more appropriate.
As an example, consider multilabel classification. Here, the response Y i ⊆ {1, ..., K} a subset of K classes. Given a trained model f : X → [0, 1] K , we wish to output sets that include a large fraction of the true classes in Y i . To that end, we post-process the model's raw outputs into the set of classes with sufficiently high scores,
C λ (x) = {k : f (X) k ≥ 1 − λ}.
Note that as the threshold λ grows, we include more classes in C λ (x)-it becomes more conservative in that we are less likely to omit true classes. Conformal risk control can be used to find a threshold valueλ that controls the fraction of missed classes. That is,λ can be chosen so that the expected value of Cλ(X test ), Y test = 1 − |Y test ∩ C λ (X test )|/|Y test | is guaranteed to fall below a user-specified error rate α. For example, setting α = 0.1 ensures that Cλ(X test ) contains 90% of the true classes in Y test on average. We will work through a multilabel classification example in detail in Section 5.1.
Formally, we will consider post-processing the predictions of the model f to create a prediction set C λ (·). The prediction set has a parameter λ that encodes its level of conservativeness: larger λ values yield more conservative outputs (e.g., larger prediction sets). To measure the quality of the output of C λ , we consider a loss function (C λ (x), y) ∈ (−∞, B] for some B < ∞. We require the loss function to be non-increasing as a function of λ. The following algorithm picksλ so that risk control as in (11) holds:
λ = inf λ : R(λ) ≤ α − B − α n ,(12)
where R(λ) = C λ (X 1 ), Y 1 + . . . + C λ (X n ), Y n /n is the empirical risk on the calibration data. Note that this algorithm simply corresponds to tuning based on the empirical risk at a slightly more conservative level than α. For example, if B = 1, α = 0.1, and we have n = 1000 calibration points, then we selectλ to be the value where empirical risk hits levelλ = 0.0991 instead of 0.1.
Then the prediction set Cλ(X test ) satisfies (11).
Theorem 2 (Conformal Risk Control [17]). Suppose (X 1 , Y 1 ), . . . , (X n , Y n ), (X test , Y test ) are an i.i.d. sample from some distribution. Further, suppose is a monotone function of λ, i.e., one satisfying C λ1 (x), y ≥ C λ2 (x), y(13)
for all (x, y) and
λ 1 ≤ λ 2 . Then E Cλ(X test ), Y test ≤ α,
whereλ is picked as in (12).
Theory and worked examples of conformal risk control are presented in [17]. In Sections 5.1 and 5.2 we show a worked example of conformal risk control applied to tumor segmentation. Furthermore, Appendix A describes a more powerful technique called Learn then Test [18] capable of controlling general risks that do not satisfy (13).
Outlier Detection
Conformal prediction can also be adapted to handle unsupervised outlier detection. Here, we have access to a clean dataset X 1 , . . . , X n and wish to detect when test points do not come from the same distribution. As before, we begin with a heuristic model that tries to identify outliers; a larger score means that the model judges the point more likely to be an outlier. We will then use a variant of conformal prediction to calibrate it to have statistical guarantees. In particular, we will guarantee that it does not return too many false positives.
Formally, we will construct a function that labels test points as outliers or inliers, C : X → {outlier, inlier}, such that
P (C(X test ) = outlier) ≤ α,(14)
where the probability is over X test , a fresh sample from the clean-data distribution. The algorithm for achieving (14) is similar to the usual conformal algorithm. We start with a conformal score s : X → R (note that since we are in the unsupervised setting, the score only depends on the features). Next, we compute the conformal score on the clean data: s i = s(X i ) for i = 1, . . . , n. Then, we compute the conformal threshold in the usual way:q = quantile s 1 , . . . , s n ; (n + 1)(1 − α) n .
Lastly, when we encounter a test point, we declare it to be an outlier if the score exceedsq:
C(x) = inlier if s(x) ≤q outlier if s(x) >q .
This construction guarantees error control, as we record next.
Proposition 3 (Error control guarantee for outlier detection). Suppose X 1 , . . . , X n , X test are an i.i.d. sample from some distribution. Then the set C defined above satisfies the error control property in (14).
As with standard conformal prediction, the score function is very important for the method to perform well-that is, to be effective at flagging outliers. Here, we wish to choose the score function to effectively distinguish the type of outliers that we expect to see in the test data from the clean data. The general problem of training models to distinguish outliers is sometimes called anomaly detection, novelty detection, or one-class classification, and there are good out-of-the box methods for doing this; see [19] for an overview of outlier detection. Conformal outlier detection can also be seen as a hypothesis testing problem; points that are rejected as outliers have a p-value less than alpha for the null hypothesis of exchangeability with the calibration data. This interpretation is closely related to the classical permutation test [20,21]. See [22][23][24] for more on this interpretation and other statistical properties of conformal outlier detection.
Conformal Prediction Under Covariate Shift
All previous conformal methods rely on Theorem 1, which assumes that the incoming test points come from the same distribution as the calibration points. However, past data is not necessarily representative of future data in practice.
One type of distribution shift that conformal prediction can handle is covariate shift. Covariate shift refers to the situation where the distribution of X test changes from P to P test , but the relationship between X test and Y test , i.e. the distribution of Y test |X test , stays fixed. Imagine our calibration features {X i } n i=1 are drawn independently from P but our test feature X test is drawn from P test . Then, there has been a covariate shift, and the data are no longer i.i.d. This problem is common in the real world. For example,
• You are trying to predict diseases from MRI scans. You conformalized on a balanced dataset of 50% infants and 50% adults, but in reality, the frequency is 5% infants and 95% adults. Deploying the model in the real world would invalidate coverage; the infants are over-represented in our sample, so diseases present during infancy will be over-predicted. This was a covariate shift in age.
• You are trying to do instance segmentation, i.e., to segment each object in an image from the background. You collected your calibration images in the morning but seek to deploy your system in the afternoon. The amount of sunlight has changed, and more people are eating lunch. This was a covariate shift in the time of day.
To address the covariate shift from P to P test , one can form valid prediction sets with weighted conformal prediction, first developed in [25].
In weighted conformal prediction, we account for covariate shift by upweighting conformal scores from calibration points that would be more likely under the new distribution. We will be using the likelihood ratio
w(x) = dP test (x) dP(x) ;
usually this is just the ratio of the new PDF to the old PDF at the point x. Now we define our weights,
p w i (x) = w(X i ) n j=1 w(X j ) + w(x) and p w test (x) = w(x) n j=1 w(X j ) + w(x)
.
Intuitively, the weight p w i (x) is large when X i is likely under the new distribution, and p w test (x) is large when the input x is likely under the new distribution. We can then express our conformal quantile as the 1 − α quantile of a reweighted distribution,
q(x) = inf s j : j i=1 p w i (x)1 {s i ≤ s j } ≥ 1 − α ,
where above for notational convenience we assume that the scores are ordered from smallest to largest a-priori. The choice of quantile is the key step in this algorithm, so we pause to parse it. First of all, notice that the quantile is now a function of an input x, although the dependence is only minor. Choosing
p w i (x) = p w test (x) = 1 n+1
gives the familiar case of conformal prediction-all points are equally weighted, so we end up choosing the (n + 1)(1 − α) th-smallest score as our quantile. When there is covariate shift, we instead re-weight the calibration points with non-equal weights to match the test distribution. If the covariate shift makes easier values of x more likely, it makes our quantile smaller. This happens because the covariate shift puts more weight on small scores-see the diagram below. Of course, the opposite holds the covariate shift upweights difficult values of x: so the covariate-shift-adjusted quantile grows. By accounting for the covariate shift in our choice ofq, we were able to make our calibration data look exchangeable with the test point, achieving the following guarantee.
Theorem 3 (Conformal prediction under covariate shift [25]).
Suppose (X 1 , Y 1 ), ..., (X n , Y n ) are drawn i.i.d. from P × P Y |X and that (X test , Y test ) is drawn independently from P test × P Y |X . Then the choice of C above satisfies P (Y test ∈ C(X test )) ≥ 1 − α.
Conformal prediction under various distribution shifts is an active and important area of research with many open challenges. This algorithm addresses a somewhat restricted case-that of a known covariate shift-but is nonetheless quite practical.
Conformal Prediction Under Distribution Drift
Another common form of distribution shift is distribution drift: slowly varying changes in the data distribution. For example, when collecting time-series data, the data distribution may change-furthermore, it may change in a way that is unknown or difficult to estimate. Here, one can imagine using weights that give more weight to recent conformal scores. The following theory provides some justification for such weighted conformal procedures; in particular, they always satisfy marginal coverage, and are exact when the magnitude of the distribution shift is known.
More formally, suppose the calibration data {(X i , Y i )} n i=1 are drawn independently from different distributions {P i } n i=1 and the test point (X test , Y test ) is drawn from P test . Given some weight schedule w 1 , ..., w n , w i ∈ [0, 1], we will consider the calculation of weighted quantiles using the calibration data:
q = inf q : n i=1w i 1 {s i ≤ q} ≥ 1 − α ,
where thew i are normalized weights,w
i = w i w 1 + . . . + w n + 1 .
Then we can construct prediction sets in the usual way,
C(x) = {y : s(x, y) ≤q} .
We now state a theorem showing that when the distribution is shifting, it is a good idea to apply a discount factor to old samples. In particular, let i = d TV (X i , Y i ), (X test , Y test ) be the TV distance between the ith data point and the test data point. The TV distance is a measure of how much the distribution has shifted-a large i (close to 1) means the ith data point is not representative of the new test point. The result states that if w discounts those points with large shifts, the coverage remains close to 1 − α.
Theorem 4 (Conformal prediction under distribution drift [26]).
Suppose i = d TV (X i , Y i ), (X test , Y test ) . Then the choice of C above satisfies P (Y test ∈ C(X test )) ≥ 1 − α − 2 n i=1w i i .
When either factor in the productw i i is small, that means that the ith data point doesn't result in loss of coverage. In other words, if there isn't much distribution shift, we can place a high weight on that data point without much penalty, and vice versa. Setting i = 0 above, we can also see that when there is no distribution shift, there is no loss in coverage regardless of what choice of weights is used-this fact had been observed previously in [25,27].
The i are never known exactly in advance-we only have some heuristic sense of their size. In practice, for time-series problems, it often suffices to pick either a rolling window of size K or a smooth decay using some domain knowledge about the speed of the drift:
w fixed i = 1 {i ≥ n − K} or w decay i = 0.99 n−i+1 .
We give a worked example of this procedure for a distribution shifting over time in Section 5.3.
As a final point on this algorithm, we note that there is some cost to using this or any other weighted conformal procedure. In particular, the weights determine the effective sample size of the distribution:
n eff (w 1 , . . . , w n ) = w 1 + . . . + w n w 2 1 + . . . + w 2 n .
This is quite important in practice, since the variance of the weighted conformal procedure can explode when n eff is small; as in Section 3, the variance of coverage scales as 1/ √ n eff , which can be large if too many of the w i are small. To see more of the theory of weighted conformal prediction under distribution drift, see [26].
Worked Examples
We now show several worked examples of the techniques described in Section 4. For each example, we provide Jupyter notebooks that allow the results to be conveniently replicated and extended. In the multilabel classification setting, we receive an image and predict which of K objects are in an image. We have a pretrained modelf that outputs estimated probabilities for each of the K classes. We wish to report on the possible classes contained in the image, returning most of the true labels. To this end, we will threshold the model's outputs to get the subset of K classes that the model thinks is most likely, C λ (x) = {y :f (x) ≥ λ}, which we call the prediction. We will use conformal risk control (Section 4.3) to pick the threshold value λ certifying a low false negative rate (FNR), i.e., to guarantee the average fraction of ground truth classes that the model missed is less than α.
Multilabel Classification
More formally, our calibration set {(X i , Y i )} n i=1 contains exchangeable images X i and sets of classes Y i ⊆ {1, ..., K}. With the notation of Section 4.3, we set our loss function to be FNR (C λ (x), y) = 1−|C λ (x)∩y|/|y|. Then, pickingλ as in 12 yields a bound on the false negative rate, In the tumor segmentation setting, we receive an M × N × 3 image of a tumor and predict an M × N binary mask, where '1' indicates a tumor pixel. We start with a pretrained segmentation modelf that outputs an M × N grid of the estimated probabilities that each pixel is a tumor pixel. We will threshold the model's outputs to get our predicted binary mask, C λ (x) = {(i, j) :f (x) (i,j) ≥ λ}, which we call the prediction. We will use conformal risk control (Section 4.3) to pick the threshold value λ certifying a low FNR, i.e., guaranteeing the average fraction of tumor pixels missed is less than α.
E FNR Cλ(X test ), Y test ≤ α.
Tumor Segmentation
More formally, our calibration set {(X i , Y i )} n i=1 contains exchangeable images X i and sets of tumor pixels Y i ⊆ {1, . . . , M } × {1, . . . , N }. As in the previous example, we let the loss be the false negative proportion, FNR . Then, pickingλ as in 12 yields the bound on the FNR in 5.1. Figure 14 gives results and code on a dataset of gut polyps. On the left is a plot of coverage over time; 'weighted' denotes the procedure in Section 5.3 while 'unweighted' denotes the procedure that simply computes the conformal quantile on all conformal scores seen so far. Note that we compute coverage using a sliding window of 500 points, which explains some of the variability in the coverage. Running the notebook with a trailing average of 5000 points reveals that the unweighted version systematically undercovers before the change-point as well. On the right is a plot showing the intervals resulting from the weighted procedure.
Weather Prediction with Time-Series Distribution Shift
In this example we seek to predict the temperature of different locations on Earth given covariates such as the latitude, longitude, altitude, atmospheric pressure, and so on. We will make these predictions serially in time. Dependencies between adjacent data points induced by local and global weather changes violate the standard exchangeability assumption, so we will need to apply the method from Section 4.6.
In this setting, we have a time series (X t , Y t ) T t=1 , where the X t are tabular covariates and the Y t ∈ R are temperatures in degrees Celsius. Note that these data points are not exchangeable or i.i.d.; adjacent data points will be correlated. We start with a pretrained modelf taking features and predicting temperature and an uncertainty modelû takes features and outputs a scalar notion of uncertainty. Following Section 2.3, we compute the conformal scores
s t = Y t −f (X t ) û(X t ) .
Since we observe the data points sequentially, we also observe the scores sequentially, and we will need to pick a different conformal quantile for each incoming data point. More formally, consider the task of predicting the temperature at time t ≤ T . We use the weighted conformal technique in Section 5.3 with the fixed K-sized window w t = 1 {t ≥ t − K} for all t < t. This yields the quantileŝ
q t = inf q : 1 min(K, t − 1) + 1 t−1 t =1 s t 1 {t ≥ t − K} ≥ 1 − α .
With these adjusted quantiles in hand, we form prediction sets at each time step in the usual way,
C(X t ) = f (X t ) −q tû (X t ) ,f (X t ) +q tû (X t ) .
We run this procedure on the Yandex Weather Prediction dataset. This dataset is part of the Shifts Project [29], which also provides an ensemble of 10 pretrained CatBoost [30] models for making the temperature predictions. We take the average prediction of these models as our base modelf . Each of the models has its own internal variance; we take the average of these variances as our uncertainty scalarû. The dataset includes an in-distribution split of fresh data from the same time frame that the base model was trained and an out-of-distribution split consisting of time windows the model has never seen. We concatenate these datasets in time, leading to a large change point in the score distribution. Results in Figure 15 show that the weighted method works better than a naive unweighted conformal baseline, achieving the desired coverage in steady-state and recovering quickly from the change point. There is no hope of measuring the TV distance between adjacent data points in order to apply Theorem 4, so we cannot get a formal coverage bound. Nonetheless, the procedure is useful with this simple fixed window of weights, which we chose with only a heuristic understanding of the distribution drift speed. It is worth noting that conformal prediction for time-series applications is a particularly active area of research currently, and the method we have presented is not clearly the best. See [31][32][33] and [34] for two differing perspectives. We provide a type-1 error guarantee on a model that flags toxic online comments, such as threats, obscenity, insults, and identity-based hate. Suppose we are given n non-toxic text samples X 1 , ..., X n and asked whether a new text sample X test is toxic. We also have a pre-trained toxicity prediction model f (x) ∈ [0, 1], where values closer to 1 indicate a higher level of toxicity. The goal is to flag as many toxic comments as possible while not flagging more than α proportion of non-toxic comments.
Toxic Online Comment Identification via Outlier Detection
The outlier detection procedure in Section 4.4 applies immediately. First, we run the model on each calibration point, yielding conformal scores s i =f (X i ). Taking the toxicity thresholdq to be the (n + 1)(1 − α) -smallest of the s i , we construct the function
C(x) = inlierf (x) ≤q outlierf (x) >q.
This gives the guarantee in Proposition 3-no more than α fraction of future nontoxic text will be classified as toxic. Figure 16 shows results of this procedure using the Unitary Detoxify BERT-based model [35,36] on the Jigsaw Multilingual Toxic Comment Classification dataset from the WILDS benchmark [37]. It is composed of comments from the talk channels of Wikipedia pages. With a type-1 error of α = 10%, the system correctly flags 70% of all toxic comments.
Selective Classification
In many situations, we only want to show a model's predictions when it is confident. For example, we may only want to make medical diagnoses when the model will be 95% accurate, and otherwise to say "I don't know." We next demonstrate a system that strategically abstains in order to achieve a higher accuracy than the base model in the problem of image classification.
More formally, given image-class pairs
{(X i , Y i )} n i=1
and an image classifierf , we seek to ensure
P Y test = Y (X test ) P (X test ) ≥λ ≥ 1 − α,(15)
where Y (x) = arg max yf (x) y , P (X test ) = max yf (x) y , andλ is a threshold chosen using the calibration data. This is called a selective accuracy guarantee, because the accuracy is only computed over a subset of high-confidence predictions. This quantity cannot be controlled with techniques we've seen so far, since we are not guaranteed that model accuracy is monotone in the cutoff λ. Nonetheless, it can be handled with Learn then Test-a framework for controlling arbitrary risks (see Appendix A). We show only the special case of controlling selective classification accuracy here.
We pick the threshold using based on the empirical estimate of selective accuracy on the calibration set,
R(λ) = 1 n(λ) n i=1 1 Y i = Y (X i ) and P (X i ) ≥ λ , where n(λ) = n i=1 1 P (X i ) ≥ λ .
Since this function is not monotone in λ, we will chooseλ differently than in Section 4.3. In particular, we will scan across values of λ looking at a conservative upper bound for the true risk (i.e., the top end of a confidence interval for the selective misclassification rate). Realizing that R(λ) is a Binomial random variable with n(λ) trials, we upper-bound the misclassification error as R + (λ) = sup r : BinomCDF( R(λ); n(λ), r) ≥ δ for some user-specified failure rate δ ∈ [0, 1]. Then, scan the upper bound until the last time the bound exceeds α,λ = inf λ : R + (λ ) ≤ α for all λ ≥ λ .
Deploying the thresholdλ will satisfy (15) with high probability.
Proposition 4. Assume the {(X i , Y i )} n i=1 and (X test , Y test ) are i.i.d.
andλ is chosen as above. Then (15) is satisfied with probability 1 − δ. See results on Imagenet at level α = 0.1 in Figure 17. For a deeper dive into this procedure and techniques for controlling other non-monotone risks, see Appendix A.
Full conformal prediction
Up to this point, we have only considered split conformal prediction, otherwise known as inductive conformal prediction. This version of conformal prediction is computationally attractive, since it only requires fitting the model one time, but it sacrifices statistical efficiency because it requires splitting the data into training and calibration datasets. Next, we consider full conformal prediction, or transductive conformal prediction, which avoids data splitting at the cost of many more model fits. Historically, full conformal prediction was developed first, and then split conformal prediction was later recognized as an important special case. Next, we describe full conformal prediction. This discussion is motivated from three points of view. First, full conformal prediction is an elegant, historically important idea in our field. Second, the exposition will reveal a complimentary interpretation of conformal prediction as a hypothesis test. Lastly, full conformal prediction is a useful algorithm when statistical efficiency is of paramount importance.
Full Conformal Prediction
This topic requires expanded notation. Let (X 1 , Y 1 ), . . . , (X n+1 , Y n+1 ) be n + 1 exchangeable data points. As before, the user sees (X 1 , Y 1 ), . . . , (X n , Y n ) and X n+1 , and wishes to make a prediction set that contains Y n+1 . But unlike split conformal prediction, we allow the model to train on all the data points, so there is no separate calibration dataset.
The core idea of full conformal prediction is as follows. We know that the true label, Y n+1 , lives somewhere in Y -so if we loop over all possible y ∈ Y, then we will eventually hit the data point (X n+1 , Y n+1 ), which is exchangeable with the first n data points. Full conformal prediction is so-named because it directly computes this loop. For each y ∈ Y, we fit a new modelf y to the augmented dataset (X 1 , Y 1 ), . . . , (X n+1 , y). Importantly, the model fitting forf must be invariant to permutations of the data. Then, we compute a score function s y i = s(X i , Y i ,f y ) for i = 1,. . . ,n and s y n+1 = s(X n+1 , y,f y ). This score function is exactly the same as those from Section 2, except that the modelf y is now given as an argument because it is no longer fixed. Then, we calculate the conformal quantile, q y = Quantile s y 1 , . . . , s y n ;
(n + 1)(1 − α) n .
Then, we collect all values of y that are sufficiently consistent with the previous data (X 1 , Y 1 ), . . . , (X n , Y n ) are collected into a confidence set for the unknown value of Y n+1 :
C(X test ) = {y : s y n+1 ≤q y }.(16)
This prediction set has the same validity guarantee as before:
Theorem 5 (Full conformal coverage guarantee [1]). Suppose (X 1 , Y 1 ), ..., (X n+1 , Y n+1 ) are drawn i.i.d.
from P, and thatf is a symmetric algorithm. Then the choice of C above satisfies
P (Y n+1 ∈ C(X n+1 )) ≥ 1 − α.
More generally, the above holds for exchangeable random variables (X 1 , Y 1 ), ..., (X n+1 , Y n+1 ); the proof of Theorem 5 critically relies on the fact that the score s . We defer the proof to [1], and note that upper bound in (1) also holds when the score function is continuous.
What about computation? In principle, to compute (16), we must iterate over all y ∈ Y, which leads to a substantial computational burden. (When Y is continuous, we would typically first discretize the space and then check each element in a finite set.) For example, if |Y | = K, then computing (16) requires (n + 1) · K model fits. For some specific score functions, the set in (16) can actually be computed exactly even for continuous Y , and we refer the reader to [1] and [38] for a summary of such cases and [39,40] for recent developments. Still, full conformal prediction is generally computationally costly.
Lastly, we give a statistical interpretation for the prediction set in (16). The condition s y n+1 ≤q y is equivalent to the acceptance condition of a certain permutation test. To see this, consider a level α permutation test for the exchangeability of s y 1 , . . . , s y n and the test score s y n+1 , rejecting when the score function is large. The values of y such that the test does not reject are exactly those in (16). In words, the confidence set is all values of y such that the hypothetical data point is consistent with the other data, as judged by this permutation test. We again refer the reader to [1] for more on this viewpoint on conformal prediction.
Cross-Conformal Prediction, CV+, and Jackknife+
Split conformal prediction requires only one model fitting step, but sacrifices statistical efficiency. On the other hand, full conformal prediction requires a very large number of model fitting steps, but has high statistical efficiency. These are not the only two achievable points on the spectrum-there are techniques that fall in between, trading off statistical efficiency and computational efficiency differently. In particular, cross-conformal prediction [41] and CV+/Jackknife+ [42] both use a small number of model fits, but still use all data for both model fitting and calibration. We refer the reader to those works for a precise description of the algorithms and corresponding statistical guarantees.
Historical Notes on Conformal Prediction
We hope the reader has enjoyed reading the technical content in our gentle introduction. As a dénouement, we now pay homage to the history of conformal prediction. Specifically, we will trace the history of techniques related to conformal prediction that are distribution-free, i.e., (1) agnostic to the model, (2) agnostic to the data distribution, and (3) valid in finite samples. There are other lines of work in statistics with equal claim to the term "distribution-free" especially when it is interpreted asymptotically, such as permutation tests [43], quantile regression [9], rank tests [44][45][46], and even the bootstrap [47,48]-the following is not a history of those topics. Rather, we focus on the progenitors and progeny of conformal prediction.
Origins
The story of conformal prediction begins sixty-three kilometers north of the seventh-largest city in Ukraine, in the mining town of Chervonohrad in the Oblast of Lviv, where Vladimir Vovk spent his childhood. Vladimir's parents were both medical professionals, of Ukrainian descent, although the Lviv region changed hands many times over the years. During his early education, Vovk recalls having very few exams, with grades mostly based on oral answers. He did well in school and eventually took first place in the Mathematics Olympiad in Ukraine; he also got a Gold Medal, meaning he was one of the top graduating secondary school students. Perhaps because he was precocious, his math teacher would occupy him in class by giving him copies of a magazine formerly edited by Isaak Kikoin and Andrey Kolmogorov, Kvant, where he learned about physics, mathematics, and engineering-see Figure 18. Vladimir originally attended the Moscow Second Medical Institute (now called the Russian National Research Medical University) studying Biological Cybernetics, but eventually became disillusioned with the program, which had too much of a medical emphasis and imposed requirements to take classes like anatomy and physiology (there were "too many bones with strange Latin names"). Therefore, he sat the entrance exams a second time and restarted school at the Mekh-Mat (faculty of mechanics and mathematics) in Moscow State University. In his third year there, he became the student of Andrey Kolmogorov. This was when the seeds of conformal prediction were first laid. Today, Vladimir Vovk is widely recognized for being the co-inventor of conformal prediction, along with collaborators Alexander Gammerman, Vladimir Vapnik, and others, whose contributions we will soon discuss. First, we will relay some of the historical roots of conformal prediction, along with some oral history related by Vovk that may be forgotten if never written. Kolmogorov and Vovk met approximately once a week during his three remaining years as an undergraduate at MSU. At that time, Kolmogorov took an interest in Vovk, and encouraged him to work on difficult mathematical problems. Ultimately, Vovk settled on studying a topic of interest to Kolmogorov: algorithmically random sequences, then known as collectives, and which were modified into Bernoulli sequences by Kolmogorov.
Vladimir Vovk
Work on collectives began at the turn of the 20th century, with Gustav Fechner's Kollectivmasslehre [49], and was developed significantly by von Mises [50], Abraham Wald [51], Alonzo Church [52], and so on. A long debate ensued among these statisticians as to whether von Mises' axioms formed a valid foundation for probability, with Jean Ville being a notable opponent [53]. Although the theory of von Mises' collectives is somewhat defunct, the mathematical ideas generated during this time continue to have a broad impact on statistics, as we will see. More careful historical reviews of the original debate on collectives exist elsewhere [52,[54][55][56]. We focus on its connection to the development of conformal prediction.
Kolmogorov's interest in Bernoulli sequences continued into the 1970s and 1980s, when Vovk was his student. Vovk recalls that, on the way to the train station, Kolmogorov told him (not in these exact words), "Look around you; you do not only see infinite sequences. There are finite sequences."
Feeling that the finite case was practically important, Kolmogorov extended the idea of collectives via Bernoulli sequences.
Definition 1 (Bernoulli sequence, informal). A deterministic binary sequence of length n with k 1s is Bernoulli if it is a "random" element of the set of all n k sequences of the same length and with the same number of 1s. "Random" is defined as having a Kolmogorov complexity close to the maximum, log n k .
As is typical in the study of random sequences, the underlying object itself is not a sequence of random variables. Rather, Kolmogorov quantified the "typicality" of a sequence via Kolmogorov complexity: he asked how long a program we would need to write in order to distinguish it from other sequences in the same space [57][58][59]. Vovk's first work on random sequences modified Kolmogorov's [60] definition to better reflect the randomness in an event like a coin toss. Vovk discusses the history of Bernoulli sequences, including the important work done by Martin-Löf and Levin, in the Appendix of [61]. Learning the theory of Bernoulli sequences brought Vovk closer to understanding finite-sample exchangeability and its role in prediction problems.
We will make a last note about the contributions of the early probabilists before moving to the modern day. The concept of a nonconformity score came from the idea of (local) randomness deficiency. Consider the sequence 00000000000000000000000000000000000000000000000000000000000000000001.
With a computer, we could write a very short program to identify the '1' in the sequence, since it is atypical -it has a large randomness deficiency. But to identify any particular '0' in the sequence, we must specify its location, because it is so typical -it has a small randomness deficiency. A heuristic understanding suffices here, and we defer the formal definition of randomness deficiency to [62], avoiding the notation of Turing machines and Kolmogorov complexity. When randomness deficiency is large, a point is atypical, just like the scores we discussed in Section 2. These ideas, along with the existing statistical literature on tolerance intervals [63][64][65][66] and works related to de Finetti's theorems on exchangeability [67][68][69][70][71][72] formed the seedcorn for conformal prediction: the rough notion of collectives eventually became exchangeability, and the idea of randomness deficiency eventually became nonconformity. Furthermore, the early literature on tolerance intervals was quite close mathematically to conformal prediction-indeed, the fact that order statistics of a uniform distribution are Beta distributed was known at the time, and this was used to form prediction regions in high probability, much like [14]; more on this connection is available in Edgar Dobriban's lecture notes [73].
Enter Conformal Prediction
The framework we now call conformal prediction was hatched by Vladimir Vovk, Alexander Gammerman, Craig Saunders, and Vladimir Vapnik in the years 1996-1999, first using e-values [74] and then with pvalues [5,75]. For decades, Vovk and collaborators developed the theory and applications of conformal prediction. Key moments include:
• the 2002 proof that in online conformal prediction, the probability of error is independent across time-steps [76];
• the 2002 development, along with Harris Papadopoulos and Kostas Proedrou, of split-conformal predictors [2];
• Glenn Shafer coins the term "conformal predictor" on December 1, 2003 while writing Algorithmic Learning in a Random World with Vovk [1].
• the 2003 development of Venn Predictors [77] (Vovk says this idea came to him on a bus in Germany during the Dagstuhl seminar "Kolmogorov Complexity & Applications");
• the 2012 founding of the Symposium on Conformal and Probabilistic Prediction and its Applications (COPA), hosted in Greece by Harris Papadopoulos and colleagues;
• the 2012 creation of cross-conformal predictors [41] and Venn-Abers predictors [78];
• The 2017 invention of conformal predictive distributions [79].
Algorithmic Learning in a Random World [1], by Vovk, Gammerman, and Glenn Shafer, contains further perspective on the history described above in the bibliography of Chapter 2 and the main text of Chapter 10. Also, the book's website links to several dozen technical reports on conformal prediction and related topics. We now help the reader understand some of these key developments.
Conformal prediction was recently popularized in the United States by the pioneering work of Jing Lei, Larry Wasserman, and colleagues [3,[80][81][82][83]. Vovk himself remembers Wasserman's involvement as a landmark moment in the history of the field. In particular, their general framework for distribution-free predictive inference in regression [83] has been a seminal work. They have also, in the special cases of kernel density estimation and kernel regression, created efficient approximations to full conformal prediction [3,84]. Jing Lei also created a fast and exact conformalization of the Lasso and elastic net procedures [85]. Another equally important contribution of theirs was to introduce conformal prediction to thousands of researchers, including the authors of this paper, and also Rina Barber, Emmanuel Candès, Aaditya Ramdas, Ryan Tibshirani who themselves have made recent fundamental contributions. Some of these we have already touched upon in Section 2, such as adaptive prediction sets, conformalized quantile regression, covariate-shift conformal, and the idea of conformal prediction as indexing nested sets [86].
This group also did fundamental work circumscribing the conditions under which distribution-free conditional guarantees can exist [87], building on previous works by Vovk, Lei, and Wasserman that showed for an arbitrary continuous distribution, conditional coverage is impossible [3,14,83]. More fine-grained analysis of this fact has also recently been done in [88], showing that vanishing-width intervals are achievable if and only if the effective support size of the distribution of X test is smaller than the square of the sample size.
Current Trends
We now discuss recent work in conformal prediction and distribution-free uncertainty quantification more generally, providing pointers to topics we did not discuss in earlier sections. Many of the papers we cite here would be great starting points for novel research on distribution-free methods.
Many recent papers have focused on designing conformal procedures to have good practical performance according to specific desiderata like small set sizes [6], coverage that is approximately balanced across regions of feature space [4,7,15,27,87,89], and errors balanced across classes [6,23,90,91]. This usually involves adjusting the conformal score; we gave many examples of such adjustments in Section 2. Good conformal scores can also be trained with data to optimize more complicated desiderata [92].
Many statistical extensions to conformal prediction have also emerged. Such extensions include the ideas of risk control [4,18] and covariate shift [25] that we previously discussed. One important and continual area of work is distribution shift, where our test point has a different distribution from our calibration data. For example, [93] builds a conformal procedure robust to shifts of known f -divergence in the score function, and adaptive conformal prediction [31] forms prediction sets in a data stream where the distribution varies over time in an unknown fashion by constantly re-estimating the conformal quantile. A weighted version of conformal prediction pioneered by [26] provides tools for addressing non-exchangeable data, most notably slowly changing time-series. This same work develops techniques for applying full conformal prediction to asymmetric algorithms. Beyond distribution shift, recent statistical extensions also address topics such as creating reliable conformal prediction intervals for counterfactuals and individual treatment effects [94][95][96], covariate-dependent lower bounds on survival times [97], prediction sets that preserve the privacy of the calibration data [98], handling dependent data [99][100][101], and achieving 'multivalid' coverage that is conditionally valid with respect to several possibly overlapping groups [102,103].
Furthermore, prediction sets are not the only important form of distribution-free uncertainty quantification. One alternative form is a conformal predictive distribution, which outputs a probability distribution over the response space Y in a regression problem [79]. Recent work also addresses the issue of calibrating a scalar notion of uncertainty to have probabilistic meaning via histogram binning [104,105]-this is like a rigorous version of Platt scaling or isotonic regression. The tools from conformal prediction can also be used to identify times when the distribution of data has changed by examining the score function's behavior on new data points. For example, [24] performs outlier detection using conformal prediction, [61,106] detect change points in time-series data, [107] tests for covariate shift between two datasets, and [108] tracks the risk of a predictor on a data-stream to identify when harmful changes in its distribution (one that increases the risk) occur.
Developing better estimators of uncertainty improves the practical effectiveness of conformal prediction. The literature on this topic is too wide to even begin discussing; instead, we point to quantile regression as an example of a fruitful line of work that mingled especially nicely with conformal prediction in Section 2.2. Quantile regression was first proposed in [9] and extended to the locally polynomial case in [109]. Under sufficient regularity, quantile regression converges uniformly to the true quantile function [109][110][111][112][113]. Practical and accessible references for quantile regression have been written by Koenker and collaborators [114,115]. Active work continues today to analyze the statistical properties of quantile regression and its variants under different conditions, for example in additive models [116] or to improve conditional coverage when the size of the intervals may correlate with miscoverage events [16]. The Handbook of Quantile Regression [115] includes more detail on such topics, and a memoir of quantile regression for the interested reader. Since quantile regression provides intervals with near-conditional coverage asymptotically, the conformalized version inherits this good behavior as well.
Along with such statistical advances has come a recent wave of practical applications of conformal prediction. Conformal prediction in large-scale deep learning was studied in [4], focusing on image classification.
One compelling use-case of conformal prediction is speeding up and decreasing the computational cost of the test-time evaluation of complex models [117,118]. The same researchers pooled information across multiple tasks in a meta-learning setup to form tight prediction sets for few-shot prediction [119]. There is also an earlier line of work, appearing slightly after that of Lei and Wasserman, applying conformal prediction to decision trees [120][121][122]. Closer to end-users, we are aware of several real applications of conformal prediction. The Washington Post estimated the number of outstanding Democratic and Republican votes in the 2020 United States presidential election using conformal prediction [123]. Early clinical experiments in hospitals underscore the utility of conformal prediction in that setting as well, although real deployments are still to come [124,125]. Fairness and reliability of algorithmic risk forecasts in the criminal justice system improves (on controlled datasets) when applying conformal prediction [125][126][127]. Conformal prediction was recently applied to create safe robotic planning algorithms that avoid bumping into objects [128,129]. Recently a scikit-learn compatible open-source library, MAPIE, has been developed for constructing conformal prediction intervals. There remains a mountain of future work in these applications of conformal prediction and many others.
Today, the field of distribution-free uncertainty quantification remains small, but grows rapidly year-onyear. The promulgation of machine learning deployments has caused a reckoning that point predictions are not enough and shown that we still need rigorous statistical inference for reliable decision-making. Many researchers around the world have keyed into this fact and have created new algorithms and software using distribution-free ideas like conformal prediction. These developments are numerous and high-quality, so most reviews are out-of-date. To keep track of what gets released, the reader may want to see the Awesome Conformal Prediction repository [130], which provides a frequently-updated list of resources in this area. Figure 19: Object detection with simultaneous distribution-free guarantees on the expected intersection-over-union, recall, and coverage rate.
A Distribution-Free Control of General Risks
For many prediction tasks, the relevant notion of reliability is not coverage. Indeed, many applications have problem-specific performance metrics-from false-discovery rate to fairness-that directly encode the soundness of a prediction. In Section 4.3, we saw how to control the expectation of monotone loss functions using conformal risk control. Here, we generalize further to control any risk and multiple risks in a distribution-free way without retraining the model. As an example, in instance segmentation, we are given an image and asked to identify all objects in the image, segment them, and classify them. All three of these sub-tasks have their own risks: recall, intersection-over-union (IOU), and coverage respectively. These risks can be automatically controlled using distribution-free statistics, as we preview in Figure 19.
We first re-introduce the theory of risk control below, then give a list of illustrative examples. As in conformal risk control, we start with a pretrained modelf . The model also has a parameter λ, which we are free to choose. We usef (x) and λ to form our prediction, T λ (x), which may be a set or some other object. For example, when performing regression, λ could threshold the estimated probability density, as below.
We then define a notion of risk R(λ). The risk function measures the quality of T λ according to the user. The goal of risk control is to use our calibration set to pick a parameterλ so that the risk is small with high probability. In formal terms, for a user-defined risk tolerance α and error rate δ, we seek to ensure
P R λ < α ≥ 1 − δ,(17)
where the probability is taken over the calibration data used to pickλ. Note that this guarantee is highprobability, unlike that in Section 4.3, which is in expectation. We will soon introduce a distribution-free technique called Learn then Test (LTT) for findingλ that satisfy (17). Below we include two example applications of risk control which would be impossible with conformal prediction and conformal risk control.
• Multi-label Classification with FDR Control: In this setting, X test is an image and Y test is a subset of K classes contained in the image. Our modelf gives us the probability each of the K classes is contained in the image. We will include a class in our estimate of y iff k > λ -i.e., the parameter λ thresholds the estimated probabilities. We seek to find theλs that guarantees our predicted set of labels is sufficiently reliable as measured by the false-discovery rate (FDR) risk R(λ).
• Simultaneous Guarantees on OOD Detection and Coverage: For each input X test with true class Y test , we want to decide if it is out-of-distribution. If so, we will flag it as such. Otherwise, we want to output a prediction set that contains the true class with 90% probability. In this case, we have two models: OOD(x), which tells us how OOD the input is, andf (x), which gives the estimated probability that the input comes from each of K classes. In this case, λ has two coordinates, and we also have two risks. The first coordinate λ 1 tells us where to threshold OOD(x) such that the fraction of false alarms R 1 is controlled. The second coordinate λ 2 tells us how many classes to include in the prediction set to control the miscoverage R 2 among points identified as in-distribution. We will findλs that control both R 1 (λ) and R 2 (λ) jointly.
We will describe each of these examples in detail in Section B. Many more worked examples, including the object detection example in Figure 19, are available in the cited literature on risk control [18,131]. First, however, we will introduce the general method of risk control via Learn then Test.
A.1 Instructions for Learn then Test
First, we will describe the formal setting of risk control. We introduce notation and the risk-control property in Definition 2. Then, we describe the calibration algorithm.
Formal notation for error control
Let (X i , Y i ) i=1,.
..,n be an independent and identically distributed (i.i.d.) set of variables, where the features X i take values in X and the responses Y i take values in Y. The researcher starts with a pre-trained predictive modelf . We show how to subsequently create predictors fromf that control a risk, regardless of the quality of the initial model fit or the distribution of the data.
Next, let T λ : X → Y be a function with parameter λ that maps a feature to a prediction (Y can be any space, including the space of responses Y or prediction sets 2 Y ). This function T λ would typically be constructed from the predictive model,f , as in our earlier regression example. We further assume λ takes values in a (possibly multidimensional) discrete set Λ. If Λ is not naturally discrete, we usually discretize it finely. For example, Λ could be the set {0, 0.001, 0.002, ..., 0.999, 1}.
We then allow the user to choose a risk for the predictor T λ . This risk can be any function of T λ , but often we take the risk function to be the expected value of a loss function,
R(T λ ) = E L T λ (X test ), Y test Loss function .(18)
The loss function is a deterministic function that is high when T λ (X test ) does badly at predicting Y test . The risk then averages this loss over the distribution of (X test , Y test ). For example, taking
R miscoverage T λ ) = E 1 {Y test / ∈ T λ (X test )} = P (Y test / ∈ T λ (X test ))
gives us the familiar case of controlling miscoverage.
To aid the reader, we point out some facts about (18) that may not be obvious. The input T λ to the risk is a function; this makes the risk a functional (a function of a function). When we plug T λ into the risk, we take an expectation of the loss over the randomness in a single test point. At the end of the process, for a deterministic λ, we get a deterministic scalar R(T λ ). Henceforth, for ease of notation, we abbreviate this number as R(λ) := R(T λ ).
Our goal is control the risk in the following sense: Definition 2 (Risk control). Letλ be a random variable taking values in Λ (i.e., the output of an algorithm run on the calibration data). We say that Tλ is a (α, δ)-risk-controlling prediction (RCP) if, with probability at least 1 − δ, we have R λ ≤ α.
In Definition 2, we plug in a random parameterλ which is chosen based on our calibration data; therefore, R(λ) is random even though the risk is a deterministic function. The high-probability portion of Definition 2 therefore says thatλ can only violate risk control if we choose a bad calibration set; this happens with probability at most δ. The distribution of the risk over many resamplings of the calibration data should therefore look as below.
α risk # δ The Learn then Test procedure Recalling Definition 2, our goal is to find a set function whose risk is less than some user-specified threshold α. To do this, we search across the collection of functions {T λ } λ∈Λ and estimate their risk on the calibration data (X 1 , Y 1 ), . . . , (X n , Y n ). The output of the procedure will be a set of λ values which are all guaranteed to control the risk, Λ ⊆ Λ. The Learn then Test procedure is outlined below.
1. For each λ ∈ Λ, associate the null hypothesis H λ : R(λ) > α. Notice that rejecting the H λ means you selected λ as a point where the risk is controlled. Here we denote each null with a blue dot; the yellow dot is highlighted, so we can keep track of it as we explain the procedure. For example, the Bonferroni correction yields Λ = λ : p λ < δ |Λ| . We define the FWER and preview ways to design good FWER-controlling procedures in Section A.1.2. The nulls with red crosses through them below have been rejected by the procedure; i.e., they all control the risk with high probability. 0.0 0.5 1.0 By following the above procedure, we get the statistical guarantee in Theorem A.1.
Theorem A.1. The Λ returned by the Learn then Test procedure satisfies
P sup λ∈ Λ {R(λ)} ≤ α ≥ 1 − δ.
Thus, selecting anyλ ∈ Λ, Tλ is an (α, δ)-RCP.
The LTT procedure decomposes risk control into two subproblems: computing p-values and combining them with multiple testing. We will now take a closer look at each of these subproblems.
A.1.1 Crash Course on Generating p-values
What is a p-value, and why is it related to risk control? In Step 1 of the LTT procedure, we associated a null hypothesis H λ to every λ ∈ Λ. When the null hypothesis at λ holds, the risk is not controlled for that value of the parameter. In this reframing, our task is to automatically identify points λ where the null hypothesis does not hold-i.e., to reject the null hypotheses for some subset of λ such that R(λ) ≤ α. The process of accepting or rejecting a null hypothesis is called hypothesis testing.
Rejecting the null hypothesis H λ → the risk is controlled at λ.
Accepting the null hypothesis H λ → the risk is not controlled at λ.
In order to reject a null hypothesis, we need to have empirical evidence that at λ, the risk is controlled. We use our calibration data to summarize this information in the form of a p-value p λ . A p-value must satisfy the following condition, which we sometimes refer to as validity or super-uniformity,
∀t ∈ [0, 1], P H λ (p λ ≤ t) ≤ t,
where P H λ refers to the probability under the null hypothesis. Parsing the super-uniformity condition carefully tells us that when p λ is low, there is evidence against the null hypothesis H λ . In other words, for a particular λ, we can reject H λ if p λ < 5% and expect to be wrong no more than 5% of the time. This process is called testing the hypothesis at level δ, where in the previous sentence, δ = 5%. More powerful p-values based on tighter concentration bounds are included in [18]. In particular, many of the practical examples in that reference use a stronger p-value called the Hoeffding-Bentkus (HB) p-value, p HB λ = min exp{−nh 1 ( R(λ) ∧ α, α)}, eP Bin(n, α) ≤ n R(λ) ,
where h 1 (a, b) = a log a b + (1 − a) log 1 − a 1 − b .
Note that any valid p-value will work-it is fine for the reader to keep p Hoeffding λ in mind for the rest of this manuscript, with the understanding that more powerful choices are available.
A.1.2 Crash Course on Familywise-Error Rate Algorithms
If we only had one hypothesis H λ , we could simply test it at level δ. However, we have one hypothesis for each λ ∈ Λ, where |Λ| is often very large (in the millions or more). This causes a problem: the more hypotheses we test, the higher chance we incorrectly reject at least one hypothesis. We can formally reason about this with the familywise-error rate (FWER).
Definition 3 (familywise-error rate). The familywise-error rate of a procedure returningΛ is the probability of making at least one false rejection, i.e., FWER Λ = P ∃λ ∈ Λ : R(λ) > α .
As a simple example to show how naively thresholding the p-values at level δ fails to control FWER, consider the case where all the hypotheses are null, and we have uniform p-values independently tested at level δ. The FWER then approaches 1; see below.
If we take Λ = {λ : p λ < δ}, then FWER( Λ) = 1 − (1 − δ) |Λ| .
This simple toy analysis exposes a deeper problem: without an intelligent strategy for combining the information from many p-values together, we can end up making false rejections with high probability. Our challenge is to intelligently combine the p-values to avoid this issue of multiplicity (without assuming the p-values are independent). This fundamental statistical challenge has led to a decades-long and continually rich area of research called multiple hypothesis testing. In particular, a genre of algorithms called FWER-controlling algorithms seek to select the largest set of Λ that guarantees FWER( Λ) ≤ δ. The simplest FWER-controlling algorithm is the Bonferroni correction,
Λ Bonferroni = λ ∈ Λ : p λ ≤ δ |Λ| .
Under the hood, the Bonferroni correction simply tests each hypothesis at level δ/|Λ|, so the probability there exists a failed test is no more than δ by a union bound. It should not be surprising that there exist improvements on Bonferroni correction.
First, we will discuss one important improvement in the case of a monotone loss function: fixed-sequence testing. As the name suggests, in fixed-sequence testing, we construct a sequence of hypotheses {H λj } N j=1 where N = |Λ|, before looking at our calibration data. Usually, we just sort our hypotheses from most-to least-promising based on information we knew a-priori. For example, if large values of λ are more likely to control the risk, {λ j } N j=1 just sorts Λ from greatest to least. Then, we test the hypotheses sequentially in some fixed order at level δ, including them in Λ as we go, and stopping when we make our first acceptance: person backpack handbag suitcase chair dining table potted plant dining table potted plant handbag person backpack suitcase chair zebra zebra car parking meter car parking meter bicycle car bench dog dining table dining table bicycle car bench dog person bicycle car skateboard bench bench bicycle person car skateboard cup potted plant dining table vase cup potted plant dining table vase train clock train clock person horse carrot car car person horse carrot giraffe giraffe potted plant vase potted plant vase This sequential procedure, despite testing all hypotheses it encounters at level δ, still controls the FWER. For monotone and near-monotone risks, such as the false-discovery rate, it works quite well.
Λ FST = {λ j , j ≤ T }, where T = max t ∈ {1, ..., N } : p λ t ≤ δ, for all t ≤ t .
It is also possible to extend the basic idea of fixed-sequence testing to non-monotone functions, creating powerful and flexible FWER-controlling procedures using an idea called sequential graphical testing [132]. Good graphical FWER-controlling procedures can be designed to have high power for particular problems, or alternatively, automatically discovered using data. This topic is given a detailed treatment in [18], and we omit it here for simplicity.
We have described a general-purpose pipeline for distribution-free risk control. It is described in PyTorch code in Figure 20. Once the user sets up the problem (i.e., picks Λ, T λ , and R), the LTT pipeline we described above automatically produces Λ. We now go through three worked examples which teach the reader how to choose Λ, T and R in practical circumstances.
B Examples of Distribution-Free Risk Control
In this section, we will walk through several examples of distribution-free risk control applied to practical machine learning problems. The goal is again to arm the reader with an arsenal of pragmatic prototypes of distribution-free risk control that work on real problems.
B.1 Multi-label Classification with FDR Control
We begin our sequence of examples with a familiar and fundamental setup: multi-label classification. Here, the features X test can be anything (e.g. an image), and the label Y test ⊆ {1, ..., K} must be a set of classes (e.g. those contained in the image X test ). We have a pre-trained machine learning modelf (x), which gives us an estimated probabilityf (x) k that class k is in the corresponding set-valued label. We will use these probabilities to include the estimated most likely classes in our prediction set, 1} (a discretization of [0, 1]). However, one question remains: how do we choose λ?
T λ (x) = k :f (x) k > λ , λ ∈ Λ where Λ = {0, 0.001, ...,
LTT will allow us to identify values of λ that satisfy a precise probabilistic guarantee-in this case, a bound on the false-discovery rate (FDR),
R FDR (λ) = E 1 − |Y test ∩ T λ (X test )| |T λ (X test )| LFDP(T λ (Xtest),Ytest) .
As annotated in the underbrace, the FDR is the expectation of a loss function, the false-discovery proportion (FDP). The FDP is low when our prediction set T λ (X test ) contains mostly elements from Y test . In this sense, the FDR measures the quality of our prediction set: if we have a low FDR, it means most of the elements in our prediction set are good. By setting α = 0.1 and δ = 0.1, we desire that
P R FDR (λ) > 0.1 < 0.1,
where the probability is over the randomness in the calibration set used to pickλ. Now that we have set up our problem, we can just run the LTT procedure via the code in Figure 23. We use fixed-sequence testing because the FDR is a nearly monotone risk. In practice, we also wish to use the HB p-value, which is stronger than the simple Hoeffding p-value in Figure 23. The result of this procedure on the MS-COCO image dataset is in Figure 22.
B.2 Simultaneous Guarantees on OOD Detection and Coverage
In our next example, we perform classification with two goals: high probability.
Part of the purpose of this example is to teach the reader how to deal with multiple risk functions (one of which is a conditional risk) and a multi-dimensional parameter λ.
Our setup requires two different models. The first, OOD(x), outputs a scalar that should be larger when the input is OOD. The second,f (x) y , estimates the probability that input x is of class y; for example, f (x) could represent the softmax outputs of a neural net. Similarly, the construction of T λ (x) has two substeps, each of which uses a different model. In our first substep, when OOD(x) becomes sufficiently large, exceeding λ 1 , we flag the example as OOD by outputting ∅. Otherwise, we essentially use the APS method from Section 2.1 to form prediction sets. We precisely describe this procedure below:
T λ (x) = ∅ OOD(x) > λ 1 {π 1 (x), ..., π K (x)} else,
where K = inf{k : k j=1f (x) πj (x) > λ 2 } and π(x) sortsf (x) from greatest to least. We usually take Λ = {0, 1/N, 2/N, ..., 1} 2 , i.e., we discretize the box [0, 1] × [0, 1] into N 2 smaller boxes, with N ≈ 1000. The intuition of T λ (x) is very simple. If the example is sufficiently atypical, we give up. Otherwise, we create a prediction set using a procedure similar to (but not identical to) conformal prediction.
OOD?
{ squirrel, chipmunk { yes no O "I've never seen anything like this before!" "I'm 90% certain this is a squirrel or a chipmunk."
Along the same lines, we control two risk functions simultaneously, R 1 (λ) = P (T λ (X test ) = ∅) and R 2 (λ) = P Y test / ∈ T λ (X test ) T λ (X test ) = ∅ .
The first risk function R 1 is the probability of a false flag, and the second risk function R 2 is the coverage conditionally on being deemed in-distribution. The user must define risk-tolerances for each, so α is a twovector, where α 1 determines the desired fraction of false flags and α 2 determines the desired miscoverage rate. Setting α = (0.05, 0.1) will guarantee that we falsely throw out no more than 5% of in-distribution data points, and also that among the data points we claim are in-distribution, we will output a prediction set containing the correct class with 90% probability. In order to control both risks, we now need to associate a composite null hypothesis to each λ ∈ Λ. Namely, we choose
λ : R 1 (λ) > α 1 and H (2) λ : R 2 (λ) > α 2 .
We summarize our setup in the below table.
# ood is an OOD detector, model is classifier with softmax output lambda1s = torch.linspace(0,1,N) # Usually N~= 1000 lambda2s = torch.linspace(0,1,N) losses = torch.zeros((2,n,N,N)) # 2 losses, n data points, N x N lambdas # The following loop can be massively parallelized (and GPU accelerated) for (i,j,k) in [(i,j,k) for i in range(n) for j in range(N) for k in range(N)]: softmaxes = model(X[i].unsqueeze(0)).softmax (1)
Goal
Null hypothesis Parameter Do not incorrectly label too many images as OOD.
H (1) λ : R 1 (λ) > α 1 λ 1 Return a set of labels guaranteed to contain the true one. H (2) λ : R 2 (λ) > α 2 λ 2
Having completed our setup, we can now apply LTT. The presence of multiple risks creates some wrinkles, which we will now iron out with the reader. The null hypothesis H λ has a different structure than the ones we saw before, but we can use the same tools to test it. To start, we produce p-values for the intermediate nulls,
1 {Y i / ∈ T λ (X i ), T λ (X i ) = ∅} − α 2 1 {T λ (X i ) = ∅} . 2
Since the maximum of two p-values is also a p-value (you can check this manually by verifying its superuniformity), we can form the p-value for our union null as
p λ = max p (1) λ , p (2) λ .
In practice, as before, we use the p-values from the HB inequality as opposed to those from Hoeffding. Then, instead of Bonferroni correction, we combine them with a less conservative form of sequential graphical testing; see [18] for these more mathematical details. For the purposes of this development, it suffices to return the Bonferroni region,
Λ = λ : p λ ≤ δ |Λ| .
Then, every element of Λ controls both risks simultaneously. See Figure 24 for a PyTorch implementation of this procedure.
C Concentration Properties of the Empirical Coverage
We adopt the same notation as Section 3.
The variation in C has three components. First, n is finite. We analyzed how this leads to fluctuations in the coverage in Section 3.2. The second source of fluctuations is the finiteness of n val , the size of the validation set. A small number of validation points can result in a high-variance estimate of the coverage. This makes the histogram of the C j wider than the beta distribution above. However, as we will now show, C j has an analytical distribution that allows us to exactly understand the histogram's expected properties.
We now examine the distribution of C j . Because C j is an average of indicator functions, it looks like it is a binomially distributed random variable. This is true conditionally on the calibration data, but not marginally. This is because the mean of the binomial is beta distributed; as we showed in the above analysis, E C j {(X i,j , Y i,j )} n i=1 ∼ Beta(n + 1 − l, l), where (X i,j , Y i,j ) is the ith calibration point in the jth trial. Conveniently, binomial random variables with beta-distributed mean, C j ∼ 1 n val Binom(n val , µ) where µ ∼ Beta(n + 1 − l, l), are called beta-binomial random variables. We refer to this distribution as BetaBinom(n val , n + 1 − l, l); its properties, such as moments and probability mass function, can be found in standard references.
Knowing the analytic form of the C j allows us to directly plot its distribution. After a sufficient number of trials R, the histogram of C j should converge almost exactly to its analytical PMF (which is only a function of α, n, and n val ). The plot in Figure 25 shows how the histograms should look with different values of n val and large R. Code for producing these plots is also available in the aforementioned Jupyter notebook. Distribution of coverage with n val validation points (n = 1000) n val =100 n val =1000 n val =10000 n val =100000 1 Figure 25: The distribution of empirical coverage converges to the Beta distribution in Figure 11 as n val grows. However, for small values of n val , the histogram can have an inflated variance.
The final source of fluctuations is due to the finite number of experiments, R. We have now shown that the C j are independent beta-binomial random variables. Unfortunately, the distribution of C-the mean of R independent beta-binomial random variables-does not have a closed form. However, we can simulate the distribution easily, and we visualize it for several realistic choices of R, n val , and n in Figure 26. Furthermore, we can analytically reason about the tail properties of C. Since C is the average of R i.i.d. beta-binomial random variables, its mean and standard deviation are E C = 1 − l n + 1 and Var C = l(n + 1 − l)(n + n val + 1) n val R(n + 1) 2 (n + 2) = O 1 R min(n, n val ) .
The best way for a practitioner to carefully debug their procedure is to compute C empirically, and then cross-reference with Figure 26. We give code to simulate histograms with any n, R, and n val in the linked notebook of Figure 26. If the simulated average empirical coverage does not align well with the coverage observed on the real data, there is likely a problem in the conformal implementation.
D Theorem and Proof: Coverage Property of Conformal Prediction
This is a standard proof of validity for split-conformal prediction first appearing in [2], but we reproduce it here for completeness. Let us begin with the lower bound.
Theorem D.1 (Conformal calibration coverage guarantee). Suppose (X i , Y i ) i=1,...,n and (X test , Y test ) are i.i.d. Then defineq asq = inf q : |{i : s(X i , Y i ) ≤ q}| n ≥ (n + 1)(1 − α) n .
and the resulting prediction sets as C(X) = {y : s(X, y) ≤q} .
Then, P Y test ∈ C(X test ) ≥ 1 − α.
This is the same coverage property as (1) in the introduction, but written more formally. As a technical remark, the theorem also holds if the observations to satisfy the weaker condition of exchangeability; see [1]. Below, we prove the lower bound.
Proof of Theorem 1. Let s i = s(X i , Y i ) for i = 1, . . . , n and s test = s(X test , Y test ). To avoid handling ties, we consider the case where the s i are distinct with probability 1. See [25] for a proof in the general case.
Without loss of generality we assume the calibration scores are sorted so that s 1 < · · · < s n . In this case, we have thatq = s (n+1)(1−α) when α ≥ 1 n+1 andq = ∞ otherwise. Note that in the caseq = ∞, C(X test ) = Y, so the coverage property is trivially satisfied; thus, we only have to handle the case when α ≥ 1 n+1 . We proceed by noticing the equality of the two events
{Y test ∈ C(X test )} = {s test ≤q}.
Combining this with the definition ofq yields
{Y test ∈ C(X test )} = {s test ≤ s (n+1)(1−α) }.
Now comes the crucial insight. By exchangeability of the variables (X 1 , Y 1 ), . . . , (X test , Y test ), we have P (s test ≤ s k ) = k n + 1 for any integer k. In words, s test is equally likely to fall in anywhere between the calibration points s 1 , . . . , s n . Note that above, the randomness is over all variables s 1 , . . . , s n , s test From here, we conclude P (s test ≤ s (n+1)(1−α)) ) = (n + 1)(1 − α) (n + 1)
≥ 1 − α,
which implies the desired result. Now we will discuss the upper bound. Technically, the upper bound only holds when the distribution of the conformal score is continuous, avoiding ties. In practice, however, this condition is not important, because the user can always add a vanishing amount of random noise to the score. We will state the theorem now, and defer its proof.
Theorem D.2 (Conformal calibration upper bound). Additionally, if the scores s 1 , ..., s n have a continuous joint distribution, then P Y test ∈ C(X test , U test ,q) ≤ 1 − α + 1 n + 1 .
Proof. See Theorem 2.2 of [83].
Figure 1 :
1Prediction set examples on Imagenet. We show three progressively more difficult examples of the class fox squirrel and the prediction sets (i.e., C(X test )) generated by conformal prediction.
Figure 2 :
2Illustration of conformal prediction with matching Python code.
Figure 3 :
3Python code for adaptive prediction sets.
Figure 4 :
4A visualization of the adaptive prediction sets algorithm in Eq. (3). Classes are included from most to least likely until their cumulative softmax output exceeds the quantile.
Figure 5 :
5Python code for conformalized quantile regression.
Figure 7 :
7Python code for conformalized uncertainty scalars.
Figure 8 :
8A visualization of the uncertainty scalars algorithm in Eq. (5). We produce the set by adding and subtractingqu(x). The constantq is picked during the calibration step.
Figure 9 :
9A visualization of the conformalized Bayes algorithm in Eq.(6). The prediction set is a superlevel set of the posterior predictive density.
Figure 10 :
10Prediction sets with various notions of coverage: no coverage, marginal coverage, or conditional coverage (at a level of 90%). In the marginal case, all the errors happen in the same groups and regions in X-space. Conditional coverage disallows this behavior, and errors are evenly distributed.
Figure 11 :
11The distribution of coverage with an infinite validation set is plotted for different values of n with α = 0.1. The distribution converges to 1 − α with rate O n
this quantile function in hand, we form our prediction set in the standard way, C(x) = {y : s(x, y) ≤q(x)} .
Figure 13 :
13Examples of false negative rate control in multilabel classification on the MS COCO dataset with α = 0.1. False negatives are red, false positives are blue, and true positives are black.
Figure 13
13gives results and code for FNR control on the Microsoft Common Objects in Context dataset[28].
Figure 14 :
14Examples of false negative rate control in tumor segmentation with α = 0.1. False negatives are red, false positives are blue, and true positives are black.
Figure 15 :
15Conformal prediction for time-series temperature estimation with α = 0.1.
Figure 16 :
16УЗНАТЬ ,КАК В ЭТОМ ГОДУ БУДУ ТПРОИЗВОДИТЬ СПУСК ВОДЫ И БУДУТ ЛИ ЗАТОПЛЕНИЯ I would like to know how water will be drained this year and whether there will be flooding.True Positive: CALLATE BOT DE M****A SIN VIDA SOCIAL SHUT UP, BOT OF S**T WITHOUT A SOCIAL LIFE False Positive: Bah, sono completamente d accordo con te, e questa dovrebbe essere un enciclopedia libera???? veramente ridicolo! Bah, I completely agree with you, and this is supposed to be a free encyclopedia ???? False Negative: votre petit discours provocateur à souhait n a pas plus d effet sur moi qu un pet de lapin sur une toile cirée your provocative little speech has no more effect on me than a rabbit fart on an oilcloth Examples of toxic online comment identification with type-1 error control at level α = 0.1 on the Jigsaw Multilingual Toxic Comment Classification dataset.
Figure 17 :
17Results using selective classification on Imagenet with α = 0.1.
Figure 18 :
18Pages from the 1976 edition of Kvant magazine.
: R(0.38) > 2. For each null hypothesis, compute a p-value using a concentration inequality. For example, Hoeffding'sinequality yields p λ = e −2n(α− R(λ)) 2 + , where R(λ) T λ (X i ), Y i ).We remind the reader what a p-value is, why it is relevant to risk control, and point to references with stronger p-values in A.Λ = A {p λ } λ∈Λ , where A is an algorithm that controls the familywise-error rate (FWER).
#
Implementation of LTT. Assume access to X, Y where n=X.shape[0]=Y.shape[0] lambdas = torch.linspace(0,1,N) # Commonly choose N=1000 losses = torch.zeros((n,N)) # Compute the loss function next for (i,j) in [(i,j) for i in range(n) for j in range(N)]:prediction_set = T(X[i],lambdas[j]) # T ( ) is problem depemdent losses[i,j] = get_loss(prediction_set,Y[i])# Loss is problem dependent risk = losses.mean(dim=0) pvals = torch.exp(-2*n*(torch.relu(alpha-risk)**2)) # Or any p-value lambda_hat = lambdas[pvals<delta/lambdas.shape[0]] # Or any FWER-controlling algorithm
Figure 20 :
20PyTorch code for running Learn then Test.
the key ingredients in Learn then Test is a p-value with distribution-free validity: it is valid under without assumptions on the data distribution. For example, when working with risk functions that take values in [0, 1]-like coverage, IOU, FDR, and so on-the easiest choice of p-value is based on Hoeffding's inequality:
Figure 22 :
22Examples of multi-label classification with FDR control on the MS-COCO dataset. Black classes are true positives, blue classes are spurious, and red classes are missed. The FDR is controlled at level α = 0.1, δ = 0.1.
λ 1 λ 2 λ 3 λ 4 λ 5 λ 6 λ 7 λ 8 λ 9 Figure 21 :
921An example of fixed-sequence testing with δ = 0.05. Each blue circle represents a null, and each row a step of the procedure. The nulls with a red cross have been rejected at that step.
#
model is a multi-class neural network, X.shape[0]=Y.shape[0]=n lambdas = torch.linspace(0,1,N) # N can be taken to infinity without penalty losses = torch.zeros((n,N)) # loss for example i with parameter lambdas[j] for i in range(n): # In reality we parallelize these loops massively sigmoids = model(X[i].unsqueeze(0)).sigmoid().squeeze() # Care with dims for j in range(N): T = sigmoids > lambdas[j] # This is the prediction set set_size = T.float().sum() if set_size != 0: losses[i,j] = 1 -(T[Y] == True).float().sum()/set_size risk = losses.mean(dim=0) pvals = torch.exp(-2*n*(torch.relu(alpha-risk)**2)) # Or the HB p-value # Fixed-sequence test starting at lambdas[-1] and ending at lambdas[0] below_delta = (pvals <= delta).float() valid = torch.tensor([(below_delta[j:].mean() == 1) for j in range(N)]) lambda_hat = lambdas[valid]
Figure 23 :
23PyTorch code for performing FDR control with LTT.
where H λ is the union of two intermediate null hypotheses, H
Figure 24 :
24PyTorch code for simultaneously controlling the type-1 error of OOD detection and prediction set coverage.
1
{T λ (X i ) = ∅} and R 2
Figure 26 :
26The distribution of average empirical coverage over R trials with n calibration points and n val validation points.
.1 Crash Course on Generating p-values . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.1.2 Crash Course on Familywise-Error Rate Algorithms . . . . . . . . . . . . . . . . . . . 44 B Examples of Distribution-Free Risk Control 45 B.1 Multi-label Classification with FDR Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 B.2 Simultaneous Guarantees on OOD Detection and Coverage . . . . . . . . . . . . . . . . . . . 46C Concentration Properties of the Empirical Coverage
49
D Theorem and Proof: Coverage Property of Conformal Prediction
50
# Get scores. calib_X.shape[0] == calib_Y.shape[0] == n cal_pi = cal_smx.argsort(1)[:,::-1]; cal_srt = np.take_along_axis(cal_smx,cal_pi,axis=1).cumsum(axis=1) cal_scores = np.take_along_axis(cal_srt,cal_pi.argsort(axis=1),axis=1)[range(n),cal_labels] # Get the score quantile qhat = np.quantile(cal_scores, np.ceil((n+1)*(1-alpha))/n, interpolation='higher') # Deploy (output=list of length n, each element is tensor of classes) val_pi = val_smx.argsort(1)[:,::-1]; val_srt = np.take_along_axis(val_smx,val_pi,axis=1).cumsum(axis=1) prediction_sets = np.take_along_axis(val_srt <= qhat,val_pi.argsort(axis=1),axis=1)
.squeeze() # Care with dims cumsum = softmaxes.sort(descending=True)[0].cumsum(0)[Y[i]] if odd(X) > lambda1s[j]: losses[0,i,j,k] = 1 continue losses[1,i,j,k] = int(cumsum > lambda2s[k]) risks = losses.mean(dim=1) # 2 x N x N risks[1] = risks[1] -alpha2*risks[0] pval1s = torch.exp(-2*n*(torch.relu(alpha1-risks[0])**2)) # Or HB p-value pval2s = torch.exp(-2*n*(torch.relu(alpha2-risks[1])**2)) # Ditto pvals = torch.maximum(pval1s,pval2s) # Bonferroni can be replaced by sequential graphical test as in LTT paper valid = torch.where(pvals <= delta/(N*N)) lambda_hat = [lambda1s[valid[0]], lambda2s[valid[1]]]
Due to the discreteness of Y , a small modification involving tie-breaking is needed to additionally satisfy the upper bound (see[4] for details; this randomization is usually ignored in practice). We will henceforth ignore such tie-breaking.
The second empirical risk, R 2 , looks different from a standard empirical risk because of the conditioning. In other words, not all of our calibration data points have nonempty prediction sets; see Section 4 of[18] to learn more about this point.
We will end our Gentle Introduction with a personal note to the reader-you can be part of this story too. The infant field of distribution-free uncertainty quantification has ample room for significant technical contributions. Furthermore, the concepts are practical and approachable; they can easily be understood and implemented in code. Thus, we encourage the reader to try their hand at distribution-free uncertainty quantification; there is a lot more to be done!
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Adaptive conformal prediction for motion planning among dynamic agents. A Dixit, L Lindemann, M Cleaveland, S Wei, G J Pappas, J W Burdick, arXiv:2212.00278arXiv preprintA. Dixit, L. Lindemann, M. Cleaveland, S. Wei, G. J. Pappas, and J. W. Burdick, "Adaptive conformal prediction for motion planning among dynamic agents," arXiv preprint arXiv:2212.00278, 2022.
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Flag out-of-distribution (OOD) inputs without too many false flags. Flag out-of-distribution (OOD) inputs without too many false flags.
If an input is deemed in-distribution. output a prediction set that contains the true class withIf an input is deemed in-distribution (In-D), output a prediction set that contains the true class with
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| We present a theoretical study of the processes d(d, p) 3 H and d(d, n) 3 He at energies of interest for energy production and for big-bang nucleosynthesis. We accurately solve the four body scattering problem using the ab-initio hyperspherical harmonic method, starting from nuclear Hamiltonians which include modern two-and three-nucleon interactions, derived in chiral effective field theory. We report results for the astrophysical factor, the quintet suppression factor, and various single and double polarized observables. An estimate of the "theoretical uncertainty" for all these quantities is provided by varying the cutoff parameter used to regularize the chiral interactions at high momentum. | 10.1103/physrevlett.130.122501 | [
"https://export.arxiv.org/pdf/2207.01433v1.pdf"
]
| 250,265,016 | 2207.01433 | de9421b747f4bf298a3813d76703a725a4ee569f |
M Viviani
Istituto Nazionale di Fisica Nucleare
Sezione di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
L Girlanda
Department of Mathematics and Physics
University of Salento
Via ArnesanoI-73100LecceItaly
INFN-Lecce
Via ArnesanoI-73100LecceItaly
A Kievsky
Istituto Nazionale di Fisica Nucleare
Sezione di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
D Logoteta
Department of Physics "E. Fermi"
University of Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
L E Marcucci
Istituto Nazionale di Fisica Nucleare
Sezione di Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
Department of Physics "E. Fermi"
University of Pisa
Largo B. Pontecorvo 3I-56127PisaItaly
arXiv:2207.01433v1 [nucl-th] 4 Jul 2022Theoretical study of the d(d, p) 3 H and d(d, n) 3 He processes at low energies
We present a theoretical study of the processes d(d, p) 3 H and d(d, n) 3 He at energies of interest for energy production and for big-bang nucleosynthesis. We accurately solve the four body scattering problem using the ab-initio hyperspherical harmonic method, starting from nuclear Hamiltonians which include modern two-and three-nucleon interactions, derived in chiral effective field theory. We report results for the astrophysical factor, the quintet suppression factor, and various single and double polarized observables. An estimate of the "theoretical uncertainty" for all these quantities is provided by varying the cutoff parameter used to regularize the chiral interactions at high momentum.
I. INTRODUCTION
The fusion reactions d(d, p) 3 H and d(d, n) 3 He are critical processes for our understanding of Big-Bang nucleosynthesis (BBN) and for new designs of fusion reactors. In fact, the uncertainties in the prediction of the deuteron abundance [D/H] in BBN models is currently dominated by the lack of precise knowledge of the astrophysical Sfactor S(E) of these processes [1,2]. Therefore, accurate calculations of S(E) could be very helpful in reducing the uncertainty of the [D/H] estimate.
Moreover, it has been speculated that the rate of d(d, p) 3 H and d(d, n) 3 He would be reduced preparing the initial deuterons with parallel spins (i.e. being in the "quintet" spin state) [3,4]. This suppression is referred as the quintet suppression. The interest on this suppression is related to the construction of "neutron lean reactors"with a d + 3 He plasma, which would produce energy via the reaction d+ 3 He → p+ 4 He. However, the neutrons from the process d+d → n+ 3 He would be always present. Hence, the interest in the use of polarized fuel [5] and in the quintet suppression. Naively, the suppression of the d( d, n) 3 He (and of d( d, p) 3 H) rate is expected when one assumes the capture to take place in S-wave. Then, the process would require a spin-flip to produce either a 3 H or 3 He nucleus, a process generally suppressed. However, this argument does not take into account the presence of the deuteron D-state or the possible capture in P-and D-waves, whose importance has been already established also at low energy [4]. The suppression factor of the reaction rate when the two deuterons are in the total spin S = 2 quintet state with respect to the unpolarized case is referred as the quintet suppression factor (QSF). No experimental study of the QSF has been reported so far. From the theoretical point of view, different predictions for the QSF have been reported, all at variance between each other [6]. The most accurate calculations predict a mild rate reduction using a polarized beam of laboratory energy above 50 keV, and even a rate increase at lower energy (i.e. QSF > 1) [7]. Clearly, further studies are necessary to better clarify this issue.
Another advantage advocated for the use of polarized fuels in reactors, is related to the possibility of handling the emission directions of reaction products, in particular the neutrons [5]. This could have an important impact on cost and safety of future fusion reactors, having the possibility to design fusion chambers where less parts of the walls are bombarded by neutrons [4]. The PolFusion experiment is currently being designed to study these processes using polarized deuterons for beam and target [8,9].
The d(d, n) 3 He reaction is also used as a source of neutrons, subsequently employed to produce innovative medical radioisotopes. For example, the SORGENTINA-RF project [10] has been designed to use these neutrons to produce 99 Mo from the stable isotope 100 Mo, via the reaction 100 Mo(n, 2n) 99 Mo. From 99 Mo is then possible to produce 99m Tc, a radio-tracer used in single photon emission computed tomography. Again, it is important to know accurately the corresponding d(d, n) 3 He cross section in the energy range more relevant for this application.
The general spin formalism for the scattering of two (identical) spin-one particles can be found in Ref. [4]. There are one unpolarized cross section, one vector analyzing power, three tensor analyzing powers and 19 correlation coefficients. For future reference, we consider the case of a deuteron beam of energy T d (in the lab. system), impinging on a deuteron target at rest. The energy of interest for energy production is in the range T d = 10 − 50 keV, while for BBN T d = 100 ÷ 400 keV. For the production of 99 Mo, a beam energy in the range T d = 200 ÷ 300 keV is considered optimal.
The total cross section (or equivalently, the astrophysical S-factor) has been studied with great detail, in view of its importance for BBN and energy production. The most recent measurements are reported in Refs. [11][12][13][14][15][16][17][18][19]. However, as discussed earlier, the different sets of data show a fairly large scatter [2]. The d(d, n) 3 He astrophysical S-factor has been experimentally investigated also using laser induced fusion in plasmas [20]. The unpolarized differential cross section measurements reported in the literature are somewhat older (and with a gap around T d ∼ 200 keV) [13,[21][22][23][24][25]. Noticeably, there exist a few accurate measurements of vector and tensor analyzing observables below T d < 100 keV. In particular, very precise data for the tensor analyzing powers A zz,0 and A xx,0 − A yy,0 for both reactions d(d, p) 3 H and d(d, n) 3 He have been reported [26]. Moreover, precise measurements of the d(d, p) 3 H iT 11 , T 20 , T 21 , and T 22 observables have been performed at the Tandem Accelerator Center at Tsukuba [27]. In all these cases, only the deuterons in the beam were polarized. The already cited PolFusion experiment is planned to measure double-polarized observables, in particular A z,z and A zz,zz [9].
The study of these processes demands accurate solution of the four nucleon scattering problem, as S-, P-, and D-waves have been found to give important contributions, at low energy as well [4]. The importance of Pand D-waves may be understood by taking into account the large extension of the deuteron wave functions (still sizable at interparticle distances of 6 fm). Therefore, the two entrance particles will interact also at a relatively large impact parameter.
From the theoretical side, there are a few accurate calculations reported in literature, such as those obtained from the solution of the Faddeev-Yakubovsky (FY) equations [7] and using the Correlated Gaussian method [28]. Other calculations can be found in Refs. [29][30][31].
In the present paper, we study these processes using the hyperspherical harmonics (HH) expansion method [32,33]. The potentials considered in this study are the chiral nucleon-nucleon (NN) interactions derived at next-to-next-to-next-to-leading order (N3LO) by Entem and Machleidt [34,35], with cutoff Λ = 500 and 600 MeV. We include in the Hamiltonian also a chiral three-nucleon (3N) interaction, derived at next-to-nextto leading order (N2LO) in Refs. [36,37]. The two free parameters in this N2LO 3N potential, denoted usually as c D and c E , have been fixed in order to reproduce the experimental values of the A = 3 binding energies and the Gamow-Teller matrix element (GTME) of the tritium β decay [38][39][40][41]. Such interactions will be labeled as N3LO500/N2LO500 and N3LO600/N2LO600.
We report here the results obtained for a selected set of observables and compare them with the available experimental data and other theoretical calculations. We also provide a preliminary estimate of the associated "theoretical uncertainty", calculated from the difference of the results obtained with the two values of cutoff Λ. In future, we plan to perform a better estimate of this uncertainty following the procedure of Ref. [42]. However, we are confident that the reported theoretical uncertainty be of the correct order of magnitude. This uncertainty takes into account our incomplete knowledge of the nuclear dynamics.
The paper is organized as follows. In Section II a brief description of the method is given, while in Section III the results of the calculations are reported and compared with a selected set of available experimental data. The conclusions and the perspectives of this approach will be given in Section IV.
II. THEORETICAL ANALYSIS
In the following, we will denote with the index γ a particular clusterization A + B of the four-nucleon system in the asymptotic region. More specifically, γ = 1, 2, 3 will correspond to the p + 3 H, n + 3 He, and d + d clusterization, respectively. Please note that at the energies considered here, all these three asymptotic channels are open, while breakup channels are closed. Let us consider a scattering state with total angular momentum quantum number JJ z , and parity π. The wave function Ψ γLS,JJz describing a state with incoming clusters γ in a relative orbital angular momentum L and channel spin S [note that π ≡ (−) L ] can be written as
Ψ γLS,JJz = Ψ C γLS,JJz + Ψ A γLS,JJz ,(2.1)
where the core part Ψ C γLS,JJz vanishes in the limit of large inter-cluster separations, and hence describes the system where the particles are close to each other and their mutual interactions are strong. We compute Ψ C γLS,JJz by expanding it over the HH basis [32,33]. On the other hand, Ψ A γLS,JJz describes the wave function in the asymptotic regions, where the mutual interaction between the clusters is negligible (except for the long-range Coulomb interaction). In the asymptotic region therefore the wave functions Ψ γLS,JJz reduces to Ψ A γLS,JJz , which must be the appropriate asymptotic solution of the Schrödinger equation. The functions Ψ A γLS,JJz depend on the Tmatrix elements (TMEs) J T γ,γ ′ LS,L ′ S ′ , which are the amplitudes for the transition between the initial state γ, L, S to the final state γ ′ , L ′ , S ′ for the wave with the specified value of J. Clearly, we are interested in the terms J T γ=3,γ ′ =1 LS,L ′ S ′ and J T γ=3,γ ′ =2 LS,L ′ S ′ . Full detail of the procedure adopted to determine Ψ C γLS,JJz and the TMEs is reported in Refs. [32,33].
III. RESULTS
First of all, let us consider the unpolarized total cross section, which is simply given by with S = 0, 1 (the TME 2 T (3,γ ′ ) 02, 21 gives the largest contribution). However, there is also a sizable contribution from the TME 1 T (3,γ ′ ) 11,11 , which, as the energy increases (T d > 100 keV) becomes dominant. Other L = 1 TMEs contribute only marginally, while the L ≥ 2 TMEs are much smaller and become sizable only at T d ≥ 1 MeV. From the total cross section, we have calculated the astrophysical S-factor, defined as S (γ) (E cm ) = E cm σ (γ) e 2πη , where E cm = T d /2 ≡ q 2 /2m, m being the nucleon mass and η = me 2 /q the Sommerfeld parameter. The calculated S-factors S (γ) (E) for γ = 1, 2 are reported in Fig. 1, where they are compared with recent experimental data [15,17,20]. The calculations have been performed using the N3LO500/N2LO500 and N3LO600/N2LO600 interactions and the results are shown as bands, their width reflecting the spread of theoretical results using Λ = 500 or 600 MeV cutoff values. As it can be seen from the figure, the calculations correctly reproduce the energy dependence of the data. The astrophysical S-factor for d(d, n) 3 He results to be larger than that of d(d, p) 3 H for E cm > 0.1 MeV. The calculations are well in agreement with the data of Ref. [17], while, the data of Ref. [15] are slightly underpredicted, especially at low energy.
σ (γ ′ ) = 1 6 4π q 2 3 J,LS,L ′ S ′ (2J + 1)| J T (3,γ ′ ) LS,L ′ S ′ | 2 ,(3.
Next we consider the QSF. We compute σ (γ) 11 as the total cross section for both deuterons polarized along the beam direction. Then, QSF=σ Fig. 1. We report also the results obtained with other theoretical approaches: T-matrix [43]; R-matrix [26]; RRGM [29,30] estimates obtained using various methods [7,15,[29][30][31]43]. As it can be seen, our calculations agree fairly well with the results of the FY calculation of Ref. [7] and with those obtained from the R-matrix analysis reported in Ref. [15]. Therefore, the trend with energy appears to be well consolidated: the QSF is close to unity at small energies and then slowly decreases. At T d = 1 MeV (not shown in the figure), it reaches a sort of plateau. These findings are at variance, however, with what found by other analyses [29][30][31]43].
The calculated unpolarized differential cross sections up to T d < 1 MeV, are generally in good agreement with the experimental data [13,[21][22][23]. More interesting is the comparison with the measured polarization observables below T d < 100 KeV. For example, we report in Fig. 3, the comparison between our theoretical results and the observables measured at T d = 21 keV in Ref. [26]. The results of our calculations are again shown as bands and they turn out to be in good agreement with these experimental data.
We have performed other comparisons with the available experimental data in this range of energies and a good agreement between theory and measurements has always been found. We are therefore confident of the accuracy of the calculations and we can make (sound) predictions for other observables. For example, in Fig. 4, we show the prediction for the observables A z,z and A zz,zz , which will be studied in the near future by the experiment PolFusion [9]. The error estimated from the variation of the cutoff in these cases is of the order of 5%.
IV. CONCLUSIONS
In this work, we have studied the d(d, p) 3 H and d(d, n) 3 He processes at energies of interest for BBN and for energy production in fusion reactors. The results of the calculations have been presented as bands, being their width a preliminary estimate of the theoretical uncertainty related to our incomplete knowledge of the nuclear dynamics. In practice, the width of the bands reflects the difference between the theoretical results obtained with the two values Λ = 500 and 600 MeV of the cutoff parameter in the nuclear interaction. By taking into account the width of the bands, we can conclude that the theoretical results and the data well agree. We have also presented predictions for the QSF and for some double-polarized observables, which will be the object of a future campaign of measurements by the PolFusion experiment. The d(d, p) 3 H [d(d, n) 3 He] astrophysical Sfactor at zero energy is estimated to be S(0) = 50.8 ± 1.9 keV b (51.0 ± 1.4 keV b). The analysis of the consequences of these values for the cosmological models is currently underway.
In future, we plan to perform a better estimate of the theoretical uncertainties, in particular, using the new χEFT interactions derived up to next-to-next-tonext-to-next-to-leading order [44] and the procedure of Ref. [42]. We plan also to study the changes in the fusion rates induced by the presence of strong high-frequency electromagnetic fields, as there are suggestions that the Coulomb barrier penetrability could increase significantly in certain configurations [45][46][47].
Acknowledgments The Authors thank H. Karwoski and K. Fletcher for providing them with their data. The calculations were made possible by grants of computing time from the Italian National Supercomputing Center CINECA and from the National Energy Research Supercomputer Center (NERSC). We also gratefully acknowledge the support of the INFN-Pisa computing center. D.L. acknowledges the support of ACTA Srl, and in particular of its CEO Dr. Eng. Davide Mazzini. This ar-ticle is supported by the Ministry of University and Research (MUR) as part of the PON 2014-2020 "Research and Innovation" resources -Green Action -DM MUR 1062/2021 of title "Study of nuclear reactions of interest for the "green" energy production from nuclear fusion".
FIG. 1 .
13 is the relative momentum between the two deuterons and γ ′ = 1 (2) for the d(d, p) 3 H [d(d, n) 3 He] reaction. We have calculated it including all waves up to L = 4. At T d < 100 keV, the dominant contributions comes from the L = 0 TMEs, 0 T (3,γ ′ ) 00,00 and 2 T (color online) The astrophysical S-factor for the processes d(d, n) 3 He (left panel) and d(d, p) 3 H (right panel) calculated with the N3LO500/N2LO500 and N3LO600/N2LO600 interactions. The width of the bands reflects the spread of theoretical results using Λ = 500 or 600 MeV cutoff values. See the main text for more details. The experimental values are from Refs.[14,15,[17][18][19][20].
σ (γ) . We report the calculated QSF inFig. 2, together with other theoretical . 2. (color online) The QSF for the processes d(d, n) 3 He and d(d, p) 3 H shown as bands, in analogy of
FIG. 3 .
3(color online) The observables Azz,0 and Axx,0 −Ayy,0 for the d(d, p) 3 H and d(d, n) 3 He processes at T d = 21 keV. The (cyan) bands show the results of the present calculations. The experimental values are taken from Ref. [26]. . 4. (color online) The polarization observables Az,z and Azz,zz calculated for the d( d, p) 3 H and d( d, n) 3 He processes at various laboratory energies. The calculations have been performed for the N3LO500/N2LO500 interaction. The associated theoretical error is of the order of 5%
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| []
|
[
"Auto-STGCN: Autonomous Spatial-Temporal Graph Convolutional Network Search Based on Reinforcement Learning and Existing Research Results",
"Auto-STGCN: Autonomous Spatial-Temporal Graph Convolutional Network Search Based on Reinforcement Learning and Existing Research Results"
]
| [
"Chunnan Wang [email protected] \nHarbin Institute of Technology\n\n",
"Kaixin Zhang \nHarbin Institute of Technology\n\n",
"Hongzhi Wang [email protected] \nHarbin Institute of Technology\n\n\nPeng Cheng Laboratory\n\n",
"Bozhou Chen [email protected] \nHarbin Institute of Technology\n\n"
]
| [
"Harbin Institute of Technology\n",
"Harbin Institute of Technology\n",
"Harbin Institute of Technology\n",
"Peng Cheng Laboratory\n",
"Harbin Institute of Technology\n"
]
| []
| In recent years, many spatial-temporal graph convolutional network (STGCN) models are proposed to deal with the spatial-temporal network data forecasting problem. These STGCN models have their own advantages, i.e., each of them puts forward many effective operations and achieves good prediction results in the real applications. If users can effectively utilize and combine these excellent operations integrating the advantages of existing models, then they may obtain more effective STGCN models thus create greater value using existing work. However, they fail to do so due to the lack of domain knowledge, and there is lack of automated system to help users to achieve this goal. In this paper, we fill this gap and propose Auto-STGCN algorithm, which makes use of existing models to automatically explore highperformance STGCN model for specific scenarios. Specifically, we design Unified-STGCN framework, which summarizes the operations of existing architectures, and use parameters to control the usage and characteristic attributes of each operation, so as to realize the parameterized representation of the STGCN architecture and the reorganization and fusion of advantages. Then, we present Auto-STGCN, an optimization method based on reinforcement learning, to quickly search the parameter search space provided by Unified-STGCN, and generate optimal STGCN models automatically. Extensive experiments on real-world benchmark datasets show that our Auto-STGCN can find STGCN models superior to existing STGCN models with heuristic parameters, which demonstrates the effectiveness of our proposed method. | 10.1145/3571285 | [
"https://arxiv.org/pdf/2010.07474v1.pdf"
]
| 222,379,299 | 2010.07474 | 1e1f2753a5d8965d7bdc528a322241d042b03994 |
Auto-STGCN: Autonomous Spatial-Temporal Graph Convolutional Network Search Based on Reinforcement Learning and Existing Research Results
Chunnan Wang [email protected]
Harbin Institute of Technology
Kaixin Zhang
Harbin Institute of Technology
Hongzhi Wang [email protected]
Harbin Institute of Technology
Peng Cheng Laboratory
Bozhou Chen [email protected]
Harbin Institute of Technology
Auto-STGCN: Autonomous Spatial-Temporal Graph Convolutional Network Search Based on Reinforcement Learning and Existing Research Results
In recent years, many spatial-temporal graph convolutional network (STGCN) models are proposed to deal with the spatial-temporal network data forecasting problem. These STGCN models have their own advantages, i.e., each of them puts forward many effective operations and achieves good prediction results in the real applications. If users can effectively utilize and combine these excellent operations integrating the advantages of existing models, then they may obtain more effective STGCN models thus create greater value using existing work. However, they fail to do so due to the lack of domain knowledge, and there is lack of automated system to help users to achieve this goal. In this paper, we fill this gap and propose Auto-STGCN algorithm, which makes use of existing models to automatically explore highperformance STGCN model for specific scenarios. Specifically, we design Unified-STGCN framework, which summarizes the operations of existing architectures, and use parameters to control the usage and characteristic attributes of each operation, so as to realize the parameterized representation of the STGCN architecture and the reorganization and fusion of advantages. Then, we present Auto-STGCN, an optimization method based on reinforcement learning, to quickly search the parameter search space provided by Unified-STGCN, and generate optimal STGCN models automatically. Extensive experiments on real-world benchmark datasets show that our Auto-STGCN can find STGCN models superior to existing STGCN models with heuristic parameters, which demonstrates the effectiveness of our proposed method.
Introduction
Spatial-temporal network data forecasting (Spatial-Temporal NDF), which aims at predicting the future observations of a spatial-temporal network according to its historical series, is a fundamental research problem in spatial-temporal data mining. This problem has numerous real applications such as traffic speed forecasting , driver maneuver anticipation (Jain et al. 2016) and human action recognition (Yan, Xiong, and Lin 2018), and has attracted considerable research interest due to its importance. Researchers have proposed many methods to deal with it, and among them spatial-temporal graph convolutional network (STGCN) models are the most popular and effective solutions. STGCN models introduce graph convolutional network (GCN) (Bacciu, Errica, and Micheli 2018;Liu et al. 2019;Xu et al. 2019;Chiang et al. 2019), a powerful deep learning approach for graph-structured data, to learn high-level node representations (Velickovic et al. 2018;Hamilton, Ying, and Leskovec 2017;Zhang et al. 2020), and combine GCN with other models or methods which are capable of modeling the temporal dependency, to extract high-quality spatial-temporal features directly from spatial-temporal network data. Compared with other solutions to the Spatial-Temporal NDF problem, which only take temporal information into account (Williams and Hoel 2003;Drucker et al. 1996) or can only process standard grid structures rather than general domains (Guo et al. 2019b;Shi et al. 2015;Yao et al. 2018), STGCN models can analyze the graph-structured time series more effectively, and thus make more accurate predictions .
Recently, many STGCN models (Yu, Yin, and Zhu 2018;Song et al. 2020;Guo et al. 2019a;Bai et al. 2019a) are proposed to deal with Spatial-Temporal NDF problems. We notice that these STGCN models have their own advantages, i.e., each of them puts forward many effective operations and achieves good prediction results in the real applications. If we can break the original combinations making excellent operations of different models capable of being combined together, then we can gain the following two benefits. (1) Obtaining novel and more powerful STGCN models. We may create more effective STGCN models by integrating the advantages of different models. (2) Promoting to realize the autonomous STGCN search. The effective operations provided by existing STGCN models constitute the search space of STGCN model, a key factor in achieving autonomous search of the STGCN model. We can utilize the effective optimization approach to explore this search space, and thus automatically design powerful STGCN models for non-experts according to their specific scenarios.
However, achieving these benefits also brings two challenges. On the one hand, there is no unified framework to describe the design flow of STGCN models and lack of effective ways of representing various STGCN models. We need to fill this gap by figuring out the overall operation process of STGCN models, and thus provide guidance on how to collect operations from existing STGCN models and how to combine these operations. With the complete operation process and operation options, then we can find effective method of representing various STGCN models, thus realize automated STGCN search. On the other hand, there is lack of autonomous search methods designed for the STGCN model. We need design effective search method according to the characteristics of STGCN models, and thus quickly explore the huge search space of the STGCN model and discover powerful STGCN models.
In this paper, we overcome these challenges, and propose Auto-STGCN algorithm to make use of existing excellent models to automatically explore high-performance STGCN models. Specifically, we present Unified-STGCN framework, which reveals the overall operation process of STGCN models and summarizes operations of existing architectures. We use parameters to control the usage and characteristic attributes of each operation in Unified-STGCN, so as to realize the parameterized representation of the STGCN model and the reorganization and fusion of advantages. Then, we propose Auto-STGCN, an effective optimization method based on reinforcement learning, to quickly search the parameter search space provided by Unified-STGCN, and generate the optimal STGCN models automatically. Auto-STGCN considers both architecture-related parameters and trainingrelated parameters during the optimization phase, and therefore can provide a complete solution, i.e., optimal STGCN structure combined with its optimal training setting, for the given Spatial-Temporal NDF problem.
Main contributions of our paper are concluded as follows:
• Unification: We unify various STGCN models under our Unified-STGCN framework, achieving the parameterized representation of the STGCN model. Unified-STGCN provides the parameter search space necessary for optimization method and deepens our understanding of popular STGCN models.
• Automation: Our Auto-STGCN is an automated system for STGCN model development. It empowers non-experts to deploy STGCN models optimized for their specific scenarios. To the best of our knowledge, this is the first automated system in the field of Spatial-Temporal NDF.
• Effectiveness: Extensive experiments on real-world benchmark datasets show that our Auto-STGCN can find STGCN models superior to existing STGCN models with heuristic parameters, which demonstrates the effectiveness of our proposed method.
Prerequisite
In this section, we give the related concepts of Spatial-Temporal Network Data Forecasting (Section 2.1), and introduce the state-of-the-art STGCN models (Section 2.2).
Spatial-Temporal Network Data Forecasting
We firstly define spatial network and graph signal matrix, then describe Spatial-Temporal NDF problem using them. Definition 1: Spatial network G. We use G = (V, E, A) to denote the spatial information of a network, where V is the set of vertices, |V | = N denotes the number of vertices, E denotes the set of edges, and A ∈ R N ×N is the adjacency matrix of G. A spatial network G can be either directed or undirected and its structure does not change with time.
Definition 2: Graph signal matrix χ t G . We use χ t G = (χ t G,v1 , . . . , χ t G,vn ) T ∈ R N ×C to denote the observations of the spatial network G = (V, E, A) at the time step t, where C is the number of attribute features and χ t G,vi denotes the values of all the features of node v i ∈ V at time step t.
Definition 3: Spatial-Temporal NDF Problem. Given a spatial network G and its historical graph signal matrices X = (χ t−T +1 G , χ t−T +2 G , . . . , χ t G ) ∈ R T ×N ×C , the Spatial-Temporal Network Data Forecasting (Spatial-Temporal NDF) problem aims at predicting the future observations of G: Y = (χ t+1 G , χ t+2 G , . . . , χ t+T G ) ∈ R T ×N ×C , where T and T denote the length of the historical sequences and the target sequences to forecast respectively.
Spatial-Temporal Graph Convolution Network Models
The key to solve Spatial-Temporal NDF problems is to capture spatial dependencies and temporal dependencies from spatial-temporal network data, and utilize these spatialtemporal features to make prediction.
Recently, many Spatial-temporal graph convolutional network (STGCN) models are proposed to effectively deal with Spatial-Temporal NDF problems. They present various methods to capture dynamic spatial-temporal features of graph-structured time series. For example, (Yu, Yin, and Zhu 2018) uses a GCN in spatial dimension to capture spatial dependencies from neighborhood and a gated CNN along temporal dimension to exploit temporal dependencies from nearby times. ) constructs localized spatialtemporal graphs which connect individual spatial graphs of adjacent time steps into one graph, then synchronously captures localized spatial-temporal correlations in these localized spatial-temporal graphs using GCNs. (Bai et al. 2019a) assumes that spatial correlations only depend on nodes with similar patterns and redefines connectivity of the graph according to the similarity among nodes. It utilizes GCN and the newly defined adjacency matrix to capture spatial correlations from most related regions, then uses multi-layer LSTM network to capture temporal relationships.
Existing STGCN models have their own advantages, and have achieved extraordinary performance on many real applications . In this paper, we try to design more effective STGCN models by making use of valuable methods provided by them, achieving the fusion of advantages of existing STGCN models.
Unified-STGCN: Unified STGCN Framework
We first propose our unified framework Unified-STGCN, and further explain how existing STGCN models fit in the framework (Section 3.1). Based on these example models, we outline the shortcomings of existing STGCN models and give the parameterized representation of STGCN models on the basis of Unified-STGCN (Section 3.2). Existing work generally uses several consecutive ST-blocks with same structure to achieve this goal. The right column shows the 3 main parts that decide the structure of a ST-block.
Temporal Information Processing
Method (TIPM)
• T IP M1: Temporal attention (Model2) • T IP M2: Add learnable spatial and temporal embedding to the input series (Model4)
• T IP M3: None (Model1, Model3) GCN-based Feature Embedding Structure (FES) • F ES1: TST-Sandwich Structure (Model1) • F ES2: GCN Layer (Model3) • F ES3: ST-Linear Structure (Model2) • F ES4: TS-Sliding Window Structure (Model4) Stage3: Output Trans- form Stage
Transform the output of the Stage2 (S2) into the expected prediction.
Output Structure (OS) Batch Size (BS)
• BS1: 32 (Model3, Model4) • BS2: 50 (Model1) • BS3: 64 (Model2) Initial Learning Rate (ILR) • ILR1: 1e-3 (Model1, Model3, Model4) • ILR2: 7e-4 (Model3) • ILR3: 1e-4 (Model3, Model2)
Optimization Function (OF)
• OF1: RMSprop + StepDecay (decay rate of 0.7 every 10 epochs) (Model1)
• OF2: Adam (Model2, Model3) • OF3: Adam + PolyScheduler (Model4)
Unified-STGCN
X = (χ t−T +1 G , χ t−T +2 G , . . . , χ t G ) ∈ R T ×N ×C of a Spatial-Temporal NDF problem.
Model4 presents to use a fully connected layer at the top of the network to transform X into a high-dimension space so as to achieve this goal, whereas the other 3 models do not include this stage. In Unified-STGCN, we utilize parameter: Input Structure (IS) to describe the detailed operation of this stage, and IS 1 and IS 2 denote the solution provided by Model4 and the other 3 models respectively.
IS 1 : X = F ullyConnected(X) ∈ R T ×N ×C IS 2 : X = X ∈ R T ×N ×C (1)
Stage2: Spatial-Temporal Embedding Stage. This is the most important stage in STGCN model design. It aggregates high-level spatial-temporal correlations of the entire network series for final prediction. It takes the X provided by Stage1 and the input adjacency matrix A of the Spatial-Temporal NDF problem as inputs, and outputs high-level representations of nodes in the network by encoding local graph structures and node attributes at different time steps.
Existing work generally uses several consecutive Spatial-Temporal blocks (ST-blocks) with same structure to achieve this goal. The structure of a ST-block is decided by 3 parameters: Spatial Information Processing Method (SIPM), Temporal Information Processing Method (TIPM) and GCNbased Feature Embedding Structure (FES). A ST-block firstly adjusts input network series and adjacency matrix according to SIPM and TIPM, then utilizes FES to process them and thus gets high-quality spatial-temporal features. Equation (2) describes its workflow. Since one ST-block can only capture low-level spatial-temporal features, existing models generally stack multiple ST-blocks to form deep models for more complicated features .
A = SIP M (X , A), X = T IP M (X ) X = F ES(X , A , A) ∈ R T ×N ×F I (T ≤ T )(2)
We summarize the ST-blocks used in existing STGCN models, and find that their structures are quite different. Model1 skips SIPM and TIPM steps, and directly uses a TST-Sandwich Structure (as is shown in Figure 1 (a)), where GCN is used in the middle for extracting spatial features and two gated CNNs are applied for extracting temporal features, to jointly process graph-structured time series. Model2 introduces attention mechanism to adaptively capture dynamic correlations on the given network. It uses spatial attention to dynamically adjust impacting weights between nodes and uses temporal attention to dynamically adjust the input by merging relevant information, then feeds the adjusted input and matrix into convolution layers for getting high-quality features. As for Model3, it assumes that spatial correlations only depend on the nodes with similar patterns (Bai et al. 2019b). Its ST-block constructs a new adjacent matrix according to node similarity, then apply new matrix to GCN layers to extract high-level features. Figure 1 (c) gives the structure of this ST-block. The ST-block of Model4 equips position embedding to the input spatial-temporal network series so that each node contains time attributes, and adds a learnable mask matrix to the original adjacency matrix to adjust the aggregation weights so that the aggregation becomes more reasonable. Then, it utilizes a TS-Sliding Window Structure, which deploys multiple individual STS-GCMs 1 on different time periods (as is shown in the top of Figure 1 (d)), to extract long-range spatial-temporal features.
Existing ST-blocks have provided us with many effective methods for designing ST-blocks. We collect the options for SIPM, TIPM and FES parameters from state-of-the-art STblocks (as is shown in Table 1), and thus construct a powerful search space of the ST-block structure. Later, we will utilize this search space to realize the automatic design of ST-blocks, constructing more powerful STGCN models.
Stage3: Output Transform Stage. Stage3 is the last step of STGCN structure design. This stage aims at transforming the output of Stage2 into the expected prediction. Existing STGCN models propose many effective solutions. For example, Model1 and Model2 directly apply a fully connected layer to achieve the final transformation. Model3 utilizes a LSTM-based encoder-decoder method to generate the multistep prediction. Model4 notices that there is heterogeneity in spatial-temporal data, i.e., each node may exhibit different properties at different time steps, and deploys multiple twofully-connected layers to generate predictions of each time step to further improve the prediction performance.
In Unified-STGCN, we utilize parameter: Output Structure (OS) to describe the detailed operation of this stage, and OS 1 , OS 2 and OS 3 denote the solution provided by Model3, Model1 and Model2, Model4 respectively.
Y = OS(X ) ∈ R T ×N ×C(3)
Stage4: Training Stage. The performance of a STGCN model depends not only on the network structure but also on the training setting. Therefore, finding a suitable training method for the STGCN model is also an important task, and we consider the model training as the fourth stage of STGCM model design. In Unified-STGCN, we summarize four parameters to describe the training setting of a STGCN model: Loss Function (LF), Batch Size (BS), Initial Learning Rate (ILR) and Optimization Function (OF). Existing works find out some training settings which are suitable for training STGCN models through repeated attempts. We collect the options for LF, BS, ILR and OF parameters from existing work (as is shown in Table 1), and thus construct a powerful search space of the STGCN model training. Later, we will use this search space to automatically design optimal training setting for our newly designed STGCN models.
Parameterized Representation of STGCN
Nine parameters defined in Unified-STGCN clearly describe the design process of existing STGCN models, and enable us to realize the parameterized representation of STGCN models. However, if we only consider these parameters in the STGCN search space, then we may miss many powerful models. Specifically, we observe that structures of existing STGCN models, where ST-blocks are connected linearly and share the same structure (as is shown in Figure 2 This rule may also apply to our STGCN study, which focuses on similar neural networks as NAS. We may discover more effective STGCN models if we break the existing structural mode. Motivated by this, we introduce more parameters to consider more flexible and diverse STGCN structures. Suppose there are N ST-block in a STGCN model, we allow each ST-block have different structural settings, so as to increase the structural diversity. That is to say we use totally N groups of parameters (SIP M bi , T IP M bi , F ES bi ) (i = 1, . . . , N ) 2 to determine the structure of N ST-blocks respectively. Besides, we design the following parameters to describe the flexible connection method among ST-blocks and the way of generating the final output of Stage2 in our Unified-STGCN utilizing these ST-blocks.
(1) Pre Block Index (PBIndex): Different from previous STGCN models, we do not restrict to use the sequential connection method, but allow each ST-block connect with any one of its previous ST-blocks instead, and thus construct more flexible STGCN structures. We introduce parameters P LIndex bi (i = 1, . . . , N ) to clarify the inputs of ST- P LIndex bi ∈ {b 1 , . . . , b i−1 } (4) Setting P LIndex bi to b j means that taking the output of j th ST-block as the input of i th ST-block.
(2) Multiple ST-blocks Output Fusion method (MBOF): The flexible connection method in our STGCN models may result in multiple output of Stage2, i.e., there may have multiple ST-blocks which are not considered as the preceding ST-block of any other ST-blocks. Under such circumstance, we aggregate multiple outputs to one using add or concentration approach, and we use MBOF to describe this operation.
• Add Aggregation Approach (denoted by M BOF 1 ): Adjust the shape of multiple outputs to be the same by using fully connected operation, then add them together.
• Concentration Aggregation Approach (denoted by M BOF 2 ): Adjust the feature dimension of multiple outputs to be the same by using fully connected operation, then concentrate them along the temporal dimension.
With the usage of these new parameters, we can obtain more flexible and diverse STGCN models (Figure 2 (b) is an example). Note that, filter size of convolution operations in ST-blocks is a hyperparameter of the STGCN model, which has great importance on model size. In this paper, we use parameter: Filter Size of Convolution (FSC)∈ {16, 32, 64} to control this hyperparameter, so as to obtain STGCN models with different sizes. Overall, we utilize 8+4×N parameters to describe the design process of a STGCN model with N ST-blocks. We then denote the configuration space of these parameters as S ST GCN , where each configuration scheme m ∈ S ST GCN corresponds to a STGCN model.
Constraint-Aware Objective Function
Given a STGCN model m ∈ S ST GCN , let M AE(m) denote its Mean Absolute Errors score on the target Spatial-Temporal NDF task, T (m) denote the inference time, and T max is the maximum time constraint. Formally, our research target is defined as follows. min (5) We hope to discover effective STGCN models whose prediction speed is not too slow. Therefore, we set T max to twice the inference time of ), a state-of-theart STGCN model, on the target task. Note that, reinforcement learning fails to deal with such constraint. To guide reinforcement learning approach to discover STGCN models that satisfy the requirements, we construct a log barrier function to quantify the time constraint in Equation (5), and define a new constraint-aware objective function as follows.
m∈S ST GCN M AE(m) subject to T (m) ≤ T maxmin m∈S ST GCN M AE(m) − λlog( T max T (m) )(6)
where λ is set to e −19 , a very small value to make the constraint tight. As we can see, if T (m) ≤ T max , barrier function is close to being violated, whereas, the value of log barrier function approaches infinity when the constraint is violated. Equation (6) is a smooth approximation of the Equation (5), and reasonably describes our optimization target by using only one function. Later, we will apply this constraintaware objective function to our Auto-STGCN, thus search powerful STGCN models which satisfy the time constraint.
Auto-STGCN Algorithm
In Autonomous STGCN Search (Auto-STGCN) algorithm, we employ the well-known Q-learning (Kröse 1995), a value-based reinforcement learning algorithm which uses Q function to find the optimal action-selection policy, with epsilon-greedy strategy (Mnih et al. 2015) to effectively and automatically search the optimal STGCN model. State and actions in Auto-STGCN are defined as follows. We utilize N + 3 states (s −2 , s −1 , . . . , s N ) s i ∈ State i (as is shown in Table 2) to describe the setting of 8 + 4 × N parameters designed in Section 3.2, thus completely and clearly describe a STGCN model. Specifically, s −2 =[-2,-1,-1,-1,-1] is an initial state; s −1 ∈ State −1 ={-1, LF, BS, ILR, OF} shows the training details of the STGCN model; s 0 ∈ State 0 ={0, IS, OS, FSC, MBOF} determines the input structure and output structure applied in STGCN, and sets the filter size and the method of dealing with multiple outputs of ST-blocks in STGCN model; s i ∈ State i ={i, SIPM, TIPM, FES, PBIndex} ∪ {[i,-1,-1,-1,-1]} (i = 1, . . . , N ) 3 elaborates on the connection details and structure details of ST-blocks in STGCN. For each state s i ∈ State i (i = −2,. . . ,N − 1), we define its action space as Action i = State i+1 , and use A(s i )∈Action i to decide for its next successive state. Accordingly, the design process of a STGCN model can be considered as an action selection trajectory, and our state transition process is described as follows.
s −2 = [ − 2, −1, −1, −1, −1] (s i , A(s i )) → s i+1 A(s i ) ∈ Action i , s i ∈ State i , i = −2, . . . N − 1(7)
We model the action selection process as a Markov Decision Process (Puterman 1994). In order to find the optimal STGCN model, we ask the agent to maximize its expected reward over all possible trajectories, and utilize recursive Bellman Equation (Bellman and Kalaba 1957) to deal with this maximization problem. Given a state s i ∈ State i , and subsequent action A(s i ) ∈ Action i , we denote the maximum expected accumulative reward that the agent would receive as Q * (s i , A(s i )), and the recursive Bellman Equation can be written as follows.
Q * (si, A(si)) =E s i+1 |s i ,A(s i ) [E r|s i ,A(s i ),s i+1 [r|si, A(si), si+1] + γ max a∈Action i+1 Q * (si+1, a)](8)
where γ is the discount factor which measures the importance of the future rewards. Formulating the above equation as an iterative update, then we get the following equations:
Q(s T , N one) = 0 Q(s T −1 , A(s T −1 )) = (1 − α)Q(s T −1 , A(s T −1 )) + αr T −1 Q(si, A(si)) = (1 − α)Q(si, A(si)) + α[ri +γ max a∈Action i+1 Q(si+1, a)], i ∈ {−2, . . . , T − 2}(9)
where α is the learning rate which determines how the newly acquired information overrides the old Q-value, r i . . ,N −1}∪ State N refers to a terminal state. Note that rewards: r −2 , . . . , r T −2 cannot be explicitly measured in our task. Ignoring them in the iterative process by setting them to 0, may causes a slow convergence in the beginning (Zhong et al. 2018) and thus makes Auto-STGCN time consuming. To speed up the learning process of agent, we introduce reward shaping (Ng, Harada, and Russell 1999) method and apply the following shaped intermediate reward instead in our Auto-STGCN algorithm 4 .
r i = R s−2∼s T T + 1 = −M AE(m) + λlog( Tmax T (m) ) T − 1(10)
where R s1∼s T is the validation performance of corresponding STGCN model trained convergence on training set for the trajectory (s 1 , . . . , s T ). In Auto-STGCN, we utilize Qvalue combined with epsilon-greedy strategy to select the STGCN model to be evaluated for each iteration, and update Q-value according to the evaluation information and Equation (9). We provide the pseudo-code and convergence proof of Auto-STGCN in the supplementary material.
Experiments
In this section, we evaluate the Auto-STGCN algorithm. We implement all experiments using MXNet .
Experimental Setting
Datasets. In the experiment, we use four high-way traffic datasets: PEMS03, PEMS04, PEMS07 and PEMS08, which are collected by the Caltrans Performance Measurement System (PeMS) (Chen et al. 2001). For each dataset, we split all datasets with ratio 6:2:2 into training sets, validation sets and test sets. We use the past 12 continuous time steps to predict the future 12 continuous time steps.
Implementation details of Auto-STGCN. The maximum number of ST-blocks N is set to 4. In the Q-value update process, learning rate α is set to 0.001, and discount factor γ is set to 0.9. During the searching phase, we train the agent with 2000 episodes, i.e., sampling 2000 STGCN models in total. For each generated STGCN model, we train it for a fixed 5 epochs on PEMS03 dataset, and measure its performance according to its MAE score and inference latency on validation sets. As for the epsilon-greedy strategy applied in Auto-STGCN, we decrease ε from 0.9 to 0.0 following the epsilon schedule as shown in Figure 3. Our Auto-STGCN takes about 4.75 GPU days to accomplish the search phase on a single NVIDIA Tesla V100 GPU. After obtaining the best auto-generated STGCN model searched on PEMS03, which is denoted by AutoSTGCNM, we train it for 50 epochs on PEMS03 dataset, and report its performance scores on the test set. We also evaluate the transfer ability of AutoSTGCNM to the other 3 datasets, i.e., PEMS04, PEMS07 and PEMS08. Figure 4 shows the details of AutoSTGCNM.
Effectiveness of Auto-STGCN
In this part, we examine the effectiveness of Auto-STGCN. We compare AutoSTGCNM with 4 state-of-the-art STGCN models discussed in Section 3.1: STSGCN , ASTGCN (Guo et al. 2019a), STGCN(2018) (Yu, Yin, and Zhu 2018), STGCRN (Bai et al. 2019a), using four Spatial-Temporal NDF tasks. Results are shown in Table 3. Our AutoSTGCNM consistently outperforms existing STGCN methods on three datasets except for PEMS08. In PEMS08, AutoSTGCNM has the best MAPE and RMSE, except for MAE which is slightly larger than that of STS-GCN. Taking operations of existing STGCN models as components, Auto-STGCN designs a more powerful STGCN model by integrating advantages of different models, which demonstrates the effectiveness of our approach.
Importance of Diversity and Flexibility
We further investigate the effect of diverse ST-block structures and flexible connection method on the performance of STGCN models using the following three variants of Au-toSTGCNM, thus examine the reasonability of search space designed in our Auto-STGCN. As Figure 5 illustrates, AutoSTGCNM has much better performance than -Diversity and -Connection Flexibility. This result shows us that applying diverse ST-block structures and the flexible connection method among ST-blocks can effectively improve the performance of the STGCN model, which coincides with our discussion in Section 3.2 and demonstrates the reasonability of search space designed in our Auto-STGCN. Besides, we observe that -Multiple Source performs the worst among them. This result shows us that the significance and necessity of breaking the original combinations of existing STGCN models. More powerful STGCN models can be found by combining excellent operations of different STGCN models, which demonstrates the reasonability of our approach.
Conclusion and Future Works
In this paper, we propose Auto-STGCN to help users automatically design high-performance STGCN models using existing work, and thus effectively deal with practical Spatial-Temporal NDF problems. Our approach breaks the original combinations making excellent operations of different STGCN models capable of being combined together, and discovers more powerful STGCN models by integrating advantages of multiple models and applying effective STGCN optimization method. Extensive experiments on real-world benchmark datasets show that our Auto-STGCN can find STGCN models superior to existing STGCN models with heuristic parameters, which demonstrates the effectiveness of our proposed method. In the future work, we will try to propose effective methods to quickly evaluate candidates STGCN models analyzed in the optimization process to further accelerate the optimization speed.
Figure 1 :
1The structures of ST-blocks of existing STGCN models.
(a)), are not flexible enough. Previous works on Neural Architecture Search (NAS) (Tan et al. 2019; Zoph et al. 2018; Yang et al. 2020) point out that block diversity and flexible connection method have great importance on model performance.
Figure 2 :
2The structures of existing STGCN models and our STGCN models. The structures of existing STGCN models and our STGCN models. ST-blocks with different texture patterns have different structural settings. blocks in STGCN, and their options are as follows.
3
The [i,-1,-1,-1,-1] (i = 1,. . ., N ) are terminal states. The si+1,. . .,sN becomes invalid when si is set to [i,-1,-1,-1,-1]. (i = −2, . . . , T − 1) denotes the intermediate reward observed for the current state s i after taking action A(s i ), and s T ∈ {[i,-1,-1,-1,-1]|i=1,.
Figure 3 :
31. -Diversity: This model changes the structures of all ST-blocks in AutoSTGCNM model to be the same as that of the third ST-block in AutoSTGCNM, i.e., (SIP M 2 ,T IP M 2 ,F ES 3 ). 2. -Connection Flexibility: This model changes the connection method of ST-blocks in AutoSTGCNM model to ST-block1→ST-block2→ST-block3. 3. -Multiple Source: This model changes the structures of all ST-blocks in -Connection Flexibility model to Epsilon values applied in Auto-STGCN.
Figure 4 :
4Details of the AutoST-GCNM searched by Auto-STGCN.
Figure 5 :
5Performance of three variants of AutoSTGCNM on PEMS03 dataset. (SIP M 3 ,T IP M 2 ,F ES 4 ), where all operations related to ST-blocks come from the same paper, i.e., STS-GCN (Song et al. 2020).
Table 1 :
1Four stages that need to be gown through to get a STGCN model and its performance. Each stage needs to be welldesigned so as to get a high-performance STGCN model.• SIP M1: Get new adjacent matrix using Pearson Coefficient (Model3) • SIP M2: Spatial attention (Model2) • SIP M3: Add mask weight to original adjacency matrix (Model4) • SIP M4: None (Model1)Stage
Stage Function Description
Detail Contents
Solutions or Settings used in Existing Work
Stage1: Input Trans-
form Stage
Enhance representation power of
the input historical sequence.
Input Structure (IS)
• IS1: Add dimension using fully-connected layer (Model4)
• IS2: None (Model1, Model3, Model2)
Stage2: Spatial-
Temporal Embedding
Stage
Capture spatial-temporal dependen-
cies from neighborhood or nearby
times.
Spatial
Information
Processing
Method
(SIPM)
Spatial-Temporal Embedding Stage, Output Transform Stage and Training Stage. We define 9 parameters to describe the main content of these stages and thus realize the parameterized representation of the STGCN model. In addition, taking 4 state-of-the-art STGCN models, which are denoted by Model1(Yu, Yin, and Zhu 2018), Model2 (Guo et al. 2019a), Model3 (Bai et al. 2019a) and Model4 (Song et al. 2020) respectively, as example, we provide the options for each parameter.Table 1summarizes the main contents of this part, and detailed introductions are as follows.Stage1: Input Transform Stage. The target of this stage is to improve the representation power of the input historical sequenceWe analyze the workflow of existing STGCN models and
sum up 4 stages of STGCN model design: Input Transform
Stage,
Table 2 :
2Four types of states that are used to describe the coding of a STGCN model. Considering a STGCN model which consists of n ST-blocks, (N+3) 5-D-vectors are used to describe its details, including its structure and its training setting.State Space
Symbol
State Space Meaning
State Space Description (5-D-vector): Contents and Their Settings
Action Space
State−2
The start state of a Spatial-
Temporal GCN model
[−2, −1, −1, −1, −1]
Actioni = Statei+1
i = {−2, . . . , N − 1}
State−1
Set the details of the
• Stage4: Training Stage
State
Index
LF
BS
ILR
OF
-1
LF1, LF2
BS1,...,BS3
ILR1,...,ILR3
OF1,...,OF3
State0
Set the details of the
• Stage1: Input Transform Stage
State
Index
IS
OS
FI
MBOF
• Stage3: Output Transform Stage
Set FI and MBOF of the
• Stage2: Spatial-Temporal Embe-
dding Stage
0
IS1, IS2
OS1,...,OS3
16, 32, 64
M BOF1, M BOF2
Statei
i = {1, . . . , N }
Set details of multiple ST-blocks
and their internal connections in
• Stage2: Spatial-Temporal Embe-
dding Stage
State
Index
SIPM
TIPM
FES
PBIndex
i
SIP M1,...,SIP M4 T IP M1,...,T IP M3 F ES1,...,F ES4
{1,2,...,i − 1} ∪ {0}
Layer index 0
represents the
output of Stage2
Terminal State: [i, −1, −1, −1, −1]
Table 3 :
3Performance comparison of different STGCN models. Time refers to the Inference time on test sets. 23±1.02 17.69±1.43 17.49±0.46 17.48±0.15 16.43±0.11 MAPE 21.44±0.62 19.40±2.24 17.15±0.46 16.78±0.20 15.33±0.41 RMSE 47.71±3.11 29.66±1.68 30.12±0.70 29.21±0.56 25.65±0.17 PEMS04 MAE 27.27±2.15 22.93±1.29 22.70±0.64 21.19±0.10 20.63±0.22 MAPE 17.31±1.12 16.56±1.36 14.50±0.21 13.90±0.05 13.69±0.30 RMSE 50.88±7.29 35.22±1.90 35.55±0.75 33.65±0.20 31.30±0.32 26±0.14 23.47±0.12 MAPE 22.61±2.65 13.92±1.65 11.08±0.18 10.21±0.05 10.09±0.15 RMSE 79.34±4.59 42.57±3.31 38.78±0.58 39.03±0.27 35.97±0.17 PEMS08 MAE 23.02±0.77 18.61±0.40 18.02±0.14 17.13±0.09 17.16±0.31 MAPE 13.83±0.36 13.08±1.00 11.40±0.10 10.96±0.07 10.60±0.19 RMSE 37.66±3.39 28.16±0.48 27.83±0.20 26.80±0.18 25.84±0.44Datasets Metrics
Algorithm
STGCRN
ASTGCN STGCN(2018) STSGCN
AutoSTGCNM
PEMS03
MAE
24.PEMS07
MAE
52.01±5.66 28.05±2.34 25.38±0.49
24.
STSGCM used in Model4 constructs localized spatialtemporal graphs which connect individual spatial graphs of adjacent time steps into one graph, then uses GCN operations to simultaneously capture localized spatial-temporal correlations.
SIP M b i , T IP M b i , F ES b i represent the SIPM, TIPM and FES of i th (i = 1, . . . , N ) ST-block in the STGCM model, and have the same value spaces as SIP M , T IP M , F ES respectively.
Auto-STGCN: Autonomous STGCN SearchIn this section, we propose Auto-STGCN to optimize parameters provided by Section 3, and design the optimal STGCN model automatically. Section 4.1 gives our optimization target and Section 4.2 introduces Auto-STGCN in detail.
This method has been proved to be able to accelerate the learning process of agent in(Zhong et al. 2018)
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| []
|
[
"Neural Forecasting at Scale",
"Neural Forecasting at Scale"
]
| [
"Philippe Chatigny \nUniversity of Sherbrooke\nSherbrookeQCCanada\n",
"Shengrui Wang \nUniversity of Sherbrooke\nSherbrookeQCCanada\n",
"Jean-Marc Patenaude \nLaplace Insights\nSherbrookeQCCanada\n",
"Boris N Oreshkin \nUnity Technologies\nLabs, MontrealQCCanada\n"
]
| [
"University of Sherbrooke\nSherbrookeQCCanada",
"University of Sherbrooke\nSherbrookeQCCanada",
"Laplace Insights\nSherbrookeQCCanada",
"Unity Technologies\nLabs, MontrealQCCanada"
]
| []
| We study the problem of efficiently scaling ensemble-based deep neural networks for multi-step time series (TS) forecasting on a large set of time series. Current state-of-the-art deep ensemble models have high memory and computational requirements, hampering their use to forecast millions of TS in practical scenarios. We propose N-BEATS(P), a global parallel variant of the N-BEATS model designed to allow simultaneous training of multiple univariate TS forecasting models. Our model addresses the practical limitations of related models, reducing the training time by half and memory requirement by a factor of 5, while keeping the same level of accuracy in all TS forecasting settings. We have performed multiple experiments detailing the various ways to train our model and have obtained results that demonstrate its capacity to generalize in various forecasting conditions and setups. | null | [
"https://arxiv.org/pdf/2109.09705v4.pdf"
]
| 237,571,900 | 2109.09705 | c5b781f79b380107c466e373df242a181318bc2f |
Neural Forecasting at Scale
Philippe Chatigny
University of Sherbrooke
SherbrookeQCCanada
Shengrui Wang
University of Sherbrooke
SherbrookeQCCanada
Jean-Marc Patenaude
Laplace Insights
SherbrookeQCCanada
Boris N Oreshkin
Unity Technologies
Labs, MontrealQCCanada
Neural Forecasting at Scale
Univariate time series forecastingdeep neural networksN-BEATSensemble models
We study the problem of efficiently scaling ensemble-based deep neural networks for multi-step time series (TS) forecasting on a large set of time series. Current state-of-the-art deep ensemble models have high memory and computational requirements, hampering their use to forecast millions of TS in practical scenarios. We propose N-BEATS(P), a global parallel variant of the N-BEATS model designed to allow simultaneous training of multiple univariate TS forecasting models. Our model addresses the practical limitations of related models, reducing the training time by half and memory requirement by a factor of 5, while keeping the same level of accuracy in all TS forecasting settings. We have performed multiple experiments detailing the various ways to train our model and have obtained results that demonstrate its capacity to generalize in various forecasting conditions and setups.
Introduction
In the past few years, abundant evidence has emerged suggesting that deep neural networks (DNN) constitute an effective modeling framework for solving time series (TS) forecasting problems. DNN models have been shown to produce state-of-the-art forecasts when large homogeneous datasets with multiple observations are available [1]. The success of DNN is largely accounted for by two factors: (i) the cross-learning on multiple time-series 1 [3,4,5] and (ii) the use of over-specified large capacity ensemble models 2 . However, the high computational requirements of such models in comparison to statistical models have raised concerns regarding their applicability in practical scenarios [3]. Indeed, the deployment of a reliable DNN with an automatic training procedure is far more challenging because of this cost and other factors such as optimal architecture and hyperparameter tuning which various authors discused in previous studies [10,11]. These factors can be summarized by the following.
The need to render these methods more efficient has been pointed out multiple times [3,12] and is one of the core challenges that must be solved to democratize their use. Currently, they require much time, specialized hardware and energy to train and deploy. Besides their model size, which can render their use cumbersome, re-training these models at every forecast for different TS is not viable for most organizations given their training time. Reducing memory requirements and computational cost, as well as offering model that are "ready-to-use" once trained are key aspects to improve upon to make these model more accessible to smaller organizations who neither have the money nor the data to support frequent retraining.
Only recently has some work been done to evaluate how to generalize these models to multiple types of TS when trained on public datasets that cover various TS settings while maintaining an acceptable level of accuracy within a zero-shot regime [13], i.e. to train a neural network on a source TS dataset and deploy it on a different target TS dataset without retraining, which provides a more efficient and reliable solution to forecast at scale than its predecessor even in difficult forecasting conditions, or in a few-shot learning regime, i.e., by fine-tuning the model to the target dataset of interest [14,15]. In the ensemble case, the cost of runing these models is amplified since most of the current top-performing models rely on independent training of ensemble 1 Cross-learning is the approach were a single model is trained on multiple TS. The model assumes that all TS follow the same process and that each TS are independent samples of this process. A notable instance of such model is the FFORMA model [2]. 2 Over-specified large capacity ensembles refers to ensemble of models (DNN here) where each model have high number of parameters, perhaps larger than what would be needed to get a good training error. We refer the reader to recent empirical [6,7] and theoretical [8,9] evidences that indicates that larger networks may indeed be easier to train to achieve better results. members. Producing forecasts with a small ensemble size without affecting accuracy is of great interest for smaller organization.
On the other hand, there are plentiful examples of successful deployment of neural networks in large-scale TS forecasting. It appears that the benefits of using such models in practice definitely outweigh the associated costs and difficulties [16]. First, such models are scalable: a neural TS forecasting model usually performs better as the scale of the data used to train it increases. This has been observed in various TS competitions [3,17,18]. Second, they can be reusable: we can reuse a model to produce forecasts over multiple TS [13] not observed during training faster and produce forecast in real-time. They also offer flexibility: It is typicaly easier to adjust a DNN-based model for handling missing values [19], adjusting its parameters based on custom business/scientific objectives [20] or considering multi-modal (various source and representation of data such as text, video, etc.) within a single end-to-end model [21].
These benefits made neural TS forecasting models popular and even mainstream in various settings. In fact, the prevalent use of neural networks manifests a paradigm shift in data-driven forecasting techniques, with fullyautomated models being the de-facto standard in organizations that that can afford DNN based forecasting workflows. Many examples of DNN for TS exist. Some of the largest online retail platforms are using neural networks to forecast product demand for millions of retail items [22,23]. AutoML solutions with heavy use of DNNs like [24] are being used in various settings and have been demonstrated to be very competitive [3] with almost no human involvement. Some companies that need to allocate a large pool of resources in different environments are using neural networks to anticipate required resources for different periods of the day [25,26]. Large capital markets companies are using neural networks to predict the future movement of assets [27] via a process that links the trade-generating strategies with notifications and trade automation from these forecasts. 3 However, these approaches are often not accessible to smaller organization because of their cost to opperate.
We propose to tackle the problems within a single approach to facilitate the use of DNN for TS forecasting at scale.
Related Work
TS forecasting models: Traditional local, univariate models for TS forecasting include the autoregressive integrated moving average (ARIMA) model [30], exponential smoothing methods (HOLT, ETS, DAMPED, SES) [31,4] decomposition-based approaches, including the THETA model [32], and autoregressive (AR) models with time-varying coefficients as in [33,34]. Global univariate TS models that rely on deep neural networks (DNNs) have been proposed recently as alternatives to these models such as DEEP-STATE [35], DEEP-AR [22] and more recently Transformer -based models [36,37]. In contrast to the traditional approaches, they can be trained with multiple independent TS simultaneously and handle non-stationary TS without preprocessing steps. One of the key difference between these two class of model is are how they approach the forecasting problem. Traditonal models typically learn from TS locally, by considering each TS as a separate regression task and fitting a function to each (local model) whereas DNNs do so by fitting a single function to multiple TS (global model) [16].
Some concerns have been raised regarding machine learning (ML) publications claiming satisfactory accuracy without adequate comaprison with the well-established statistical baselines and using inappropriate criteria often leading to misleading results [38]. It is inspiring to see that recent ML publications such as [35,22,39] have largely solved these problems by following more rigorous evaluation protocols and baseline comparisons.
Ensemble methods: Combining multiple models is often a more straightforward strategy to produce accurate forecasts than finding the best parameterization for one particular model [40,41]. Recently, both M4 and M5 forecasting competitions have empirically confirmed the accuracy of ensemble methods [17,3]. Notable instances of model for univariate TS forecasting include that use this method FFORMA [2] (second entry in M4), ES-RNN [42] (first entry in M4) and subsequently N-BEATS 4 [39]. Because they use en-sembling, these models, especially N-BEATS, have high computational and memory complexities, which require specialized infrastructure to accelerate their training and store the trained models [4]. For example, the full N-BEATS models consists of 180 individual models. It takes around 11'755 hours to train on the full M4 dataset using 1 NVIDIA GTX 2080Ti GPU. Furthermore, the total size of the models in ensemble is 160 GB, which, depending on the number of training logs and saved snapshots of the model, can increase to over 450 GB. In comparison, the Theta method takes around 7 min to do the same.
N-BEATS:
The overall N-BEAT model [39] is designed to apply signal decomposition of the original TS similar to the "seasonality-trend-level" approach of [43] but using a fully connected neural networks organized into a set of blocks. Each blocks applies a decomposition of the signal it is given as input and make a forecast from this signal and pass the reminder of the signal to the other block. Beside the parmaterization of each block, one as to specify the number of past observations all blocks must consider which we refer as the lookback windows. When trained on large datasets, N-BEATS is trained with a bagging proeceduce [44] on various lookback windows, losses, and subpopulations to produce an ensemble of models. All of these models are independently trained and inflate the parameter size of the ensemble and thus the time to train the complete model.
Hence, the major issues of using these model at scale come down to parameter size of the model, time to train the model and whether or not we can offset the operating costs of DNN based forecasting workflows for ensemble models. This paper seek to reduce the computational complexity gap between classical and neural TS models by proposing a more memory-and computation -efficient version of the N-BEATS model [39]. Our approach achieves this by re-formulating the original fully-connected N-BEATS architecture as a single kernel convolution, which allows for training multiple models, each with different lookback windows, in parallel on the same GPU while sharing most of the parameters in the network. This leads to reduced ensemble training time and memory footprint as well as reduced ensemble model size, which positively affects the costs of training, querying and storing the resulting ensemble without compromising its accuracy.
Our contributions can be summarized as follows: [1] We introduce N-BEATS(P), a multi-head parallelizable N-BEATS architecture that permits the simultaneous training of multiple global TS models. Our model is twice as fast as N-BEATS, has 5 times fewer parameters than its predecessor, and performs at the same level of accuracy on M4 dataset than N-BEATS and generalize well in other TS forecast condition.
[2] Our model is faster to train and more accurate than the top-scoring models of the M4 competition (ES-RNN [42] FFORMA [2]).
The remainder of this paper is organized as follows. Section 3 describes the univariate TS forecasting problem. Section 4 presents our modeling approach. Section 5 outlines empirical evaluation setup and our results. Finally, Section 6 presents our conclusions.
Problem Statement
We consider the univariate point forecasting problem in discrete time where we have a training dataset of N TS, D train = {X
T i +1:T i +H } N i=1 .
The task is to forecast future values of the series, Y (i) T i +1:H ∈ R H , given a regularly-sampled sequence of past observations, X (i) 1:T i ∈ R T (i) . We use the bold notation to define vectors, matrix and tensor. To solve the task, we define a forecasting function f θ : R l → R H , parameterized with a set of learnable parameters θ ∈ Θ ⊂ R M where l ≤ T i The parameters of the forecasting function can be learned using an empirical risk minimization framework based on the appropriate samples of forecasting function inputs, Z in ∈ R l , and outputs, Z out ∈ R H , taken from the training set:
θ = arg min θ∈Θ Z in ,Zout∈D train L(Z out , f θ (Z in ))(1)
A few remarks are in order regarding the selection of the model input window size l. The optimal choice of l is highly data-dependent. In terms of general guidelines, TS with a swiftly changing generating process [45] will favor small values of l, as historical information quickly becomes outdated. TS with long seasonality periods will favor larger l, as observing at least one and maybe a few seasonality periods may be beneficial for making a more informed forecast. Obviously, several conflicting factors can be at play here and finding a universally optimal solution for all TS does not seem viable. Therefore, l can be treated as a hyperparameter selected on a TS-specific validation set. A more productive and accurate solution would entail using an ensemble of several models, each trained with its own l, as in [39]. However, this solution tends to inflate the ensemble size, and that is the problem we aim to address in this paper. In general, increasing the diversity of an ensemble [46] with different forecasting models usually results in the inflation of the ensemble size and computational costs. Therefore, we focus on providing a solution to more effectively parallelize training of the N-BEATS ensemble, which is obviously applicable to situations other than using multi-l ensembles.
Model
Model architecture
The basic building block of the proposed model has a multi-head architecture and is depicted in Fig. 1 (left). Each -th block can take as input up to W input signals x ( ) lw ; w ∈ {1, . . . , W } of the same TS with different lookback windows l = [l 1 , · · · l W ], and generates two output vectors for each of the input signals provided: the backcast signalx ( ) lw of length l w and the forecast signalỹ ( ) lw of length equals to the forecast horizon H. We set each l w to a multiple of H ranging from 2H to 7H.x ( ) lw is fed to the next block for its input andỹ ( ) lw is added to the previous forecast from the previous block. Internally, the basic building block is divided into four parts.
The first part consists of W independent FC input layers that project the signal into a fixed higher-dimensional representation z ( ) lw ∈ R + . This is done by mapping the w-th model with φ w : R lw → R + such as z ( )
lw = FC lw (x ( ) lw ).
To achieve parallelization in practice, we do this mapping with φ w : R L → R + where L = max(l) instead and pad missing values of the W − 1 signals with 0 that have smaller lookback windows. We set the padding to 0 for the missing values of the lookback and and make sure the FC doesn't have bias to guarantee obtaining the same result as mapping W times the input of each models sequentialy with their respective φ w . Figure 1: Illustration of the proposed model. The basic block consists of multi-head and multi-output fully connected (FC) layers with ReLU non-linear activations, where some layers are shared between the W models. Each block input X l ∈ R N ×L×W contains the same input signal at different lookback windows l 1 · · · l W , where for each of the W representations of the signal, missing values are padded with 0. The multi-output part of the block consists of W independent layers (represented by the blue cube in the figure) that predict basis expansion coefficients both forward θ f lw (Forecast) and backward θ b lw (Backcast) for each of the W models. A stack can have layers with shared g b lw and g f lw . Forecasts are aggregated by summing over all partial forecasts of each block, enabling us to retrieve which block had the most impact in making the forecast. Parallelization is achieved by forcing head layers of each block to have the same input size, by using mask layers in the input layer to consider only the T w first observations of input signals and reshaping the tensor to force computation in parallel instead of sequentially applying computation in a loop for each of the W models.
The second part consists of a shared FC stack that takes as input the TS representation produced in the first part and outputs forward θ This approach allows us to parallelize the computation of the forecast by considering an input X ∈ R N ×L×W and producing outputỸ ∈ R N ×H×W and obtain W forecasts for N TS simulatenously for each of the W lookback windows.
These opperations are repeated iteratively over all blocks across all stacks of the model. Thus, the computation of the forecast and backcast for the -th block given the w-th signal, is described by the following equations:
z ( ) lw = FC(FC(· · · (FC ,lw (x ( ) lw ) (2) θ f,( ) lw = FC f lw (z ( ) lw ) (3) θ b,( ) Tw = FC b lw (z ( ) lw ) (4) y ( ) lw = g b,( ) lw (θ f,( ) lw ) = dim θ f,( ) lw i=1 θ f,( ) i,lw v f,( ) i,lw(5)
x ( )
lw = g f,( ) lw (θ b,( ) Tw ) = dim θ b,( ) lw i=1 θ b,( ) i,Tw v b,( ) i,Tw(6)
Here, FC corresponds to a fully connected layer with ReLU non-linearity activation [47], and v f,( ) i,lw and v b,( ) i,lw are forecast and backcast basis vectors for the -th block. These vectors can either be chosen to be learnable parameters or can be set to specific functional forms that are fixed prior to training the model. In Eq. 6, the number of time FC is applied is based on the number of layer and is part of the specification of the model.
Eqs.2-6 are then repeated iteratively for all blocks, following the same architecture topology as N-BEATS [39]. The individual blocks are stacked using two residual branches. The first branch, illustrated in Fig. 1 (middle), runs over the backcast signal produced by each block and iteratively decomposes the initial TS signal such that the subsequent block consider the residual of its preceding block. The second branch, illustrated in Fig. 1 (right), aggregates the partial forecast of each block. These operations are described by the following equations:
x ( +1) lw = x ( ) lw −x ( ) lw (7) y lw = ỹ ( ) lw(8)
Generic and Interpretable Model Version
Multiple versions of this approach can be provided to parameterize each of the W models. For instance, both the generic and interpretable versions of N-BEATS proposed in [39] are compatible with our model. We will briefly describe these two extensions; we refer the reader to the original paper for more details [39].
The generic architecture: in this version, g b lw and g f lw are specified as a linear projection of the previous layer output such that the outputs of the -th block are described as follows: The interpretable architecture: Similar to the traditional TS decomposition into trend and seasonality found in [43,48], trend and seasonality decomposition can be enforced in V f,( ) lw and V b,( ) lw . [39] proposed to do this by conceptually separating the set blocks into two stacks such that one stack of blocks is parameterized with a trend model (T ) and the other with a seasonal model (S). All block in a stack shared the same parameters. The trend model consists of constraining the basis function to modelize a trend signal, i.e., using a function polynomial of small degree p as follows:
y ( ) lw = V f,( ) lw θ f,( ) lw + B f,( ) lwx ( ) lw = V b,( ) lw θ b,( ) lw + B b,( ) lw (9) where V f,( ) lw ∈ R H×dim θ f,( ) lw , B b,( ) lw ∈ R H and V b,( ) lw ∈ R L×dim θ b,( ) lw , B b,( ) lwy ( ) lw = g f,( ) lw,trend (θ f,( ) lw ) = Tθ f,( ) lw ; T = [1, t, · · · t p ](10)
where T is a matrix of powers of p. Thus the waveform extracted will follow a monotonic or a slowly varying function. The seasonal model constrains the basis function to modelize periodic functions, i.e, g f,( )
Tw (θ f,( ) lw ; V f,( )
t,lw ), using Fourier series as follows:
y ( ) lw = g f,( ) lw,seas. (θ f,( ) lw ) = Sθ f,( ) lw ;(11)S = [1, cos(2πt, · · · , cos(2π H/2 − 1 t), sin(2π H/2 − 1 t)]
Thus, by first (1) applying the trend model and then (2) lw . In any configuration of the model, estimating the parameters of the model problem is done by maximum likelihood estimation (MLE). To simplify the notation, we consider eq. 12 as the function that establishes the forecast, where θ NBEATS is the set of all parameters of each block and x i lw is the i-th TS considered with input size of length l w .
y lw Y i = NBEATS(x i lw ; θ NBEATS )(12)
Thus, optimizing the model consists of optimizing eq. 13. We use a stochastic gradient descent optimization with Adam [49] over a fixed set of itterations and a three-steps learning rate schedule. Here L(N BEAT S(x n Tw ; θ NBEATS ), y (n) ) corresponds to some metric function that measures the quality of the forecast to the ground truth Y. Note that we combine the losses of the forecasts of all models, using the mean values to promote cooperation between the different models. Following the same training framework as [39], we used the MAPE, MASE and SMAPE losses to build the ensemble, all of which are detailed in the following section. We refer the reader to [39] for design choice of the model and a exhaustive discussion on the parameter choice of this model. In our work we reuse the same set of parameters and do not apply hyper-parameters search at the exception of the yearly TS where our model converge to a stable results earlier (10k itterations instead of 15k).
θ * NBEATS = argmin θ * NBEATS 1 N N i=0 1 W W w=1 L(NBEATS(x i lw ; θ NBEATS ), y i )(13)
Experimental setup
We conducted the experimental evaluation of the forecasting methods on 6 datasets which include a total of 105'968 unique TS when combined and over 2.5 million forecasts to produce on these TS. We report the accuracy of our model on the first and dataset and consider the rest to assess the model ability to generalize in other settings. We details all datasets here and report our generalization results on zero-shot forecasting in Appendix C. The datasets are the following:
(1) (public) M4: 100'000 heterogeneous TS from multiple sectors that include economic, finance, demographics and other industry used in the M4 TS competition [17,4].
(2) (public) M3: 3003 heterogeneous TS from derived from mostly from financial and economic domains [50].
(3) (public) Tourism: 1311 TS of indicators related to tourism activities sampled monthly, quarterly and yearly [51,52].
(4) (public) Electricity: 370 TS of the hourly electricity usage of 370 customers over three years [53,54]. S. financial markets, each covering different types of asset classes including stocks, bonds, commodities, currencies and market indexes, or a proxy for a market index covering a larger set of financial asset than the dataset used in [55].
For the M4, M3 and Tourism datasets, target TS trajectories were specified by the competition's organizers with each subpopulation of TS with the same frequency (Hourly, Quarterly, etc..) having its own horizon ( We compared the forecast accuracy of our approaches with the reported accuracy of other TS models in the M4 TS competition including FFORMA [2] and ES-RNN [42]. In reporting the accuracy of these models, we relied upon the accuracy and the pre-computed forecasts reported in their respective original paper. The statistical models were produced on R using the forecast package [56] and we measured the training time to train and produce each forecast of our model as well as the Theta method. We also relied upon the reported running time of the implementation provided in [4]. Finally, all models were compared on a naive forecast, i.e., a random walk model or a seasonally adjusted random walk, that assumes all future values will be the same as the last known one(s). This was done to assess whether the forecasts of these models are accurate in the first place.
M AP E(x, x) = 100 H H i=1 |x T +i − x T +i | x T +i (14) M ASE(x, x) = 1 H H i=1 |x T +i −x T +i | 1 T −m T t=m+1 |x t − x t−m | (15) SM AP E(x, x) = 200 H H i=1 |x T +i −x T +i | |x j |+|x T +i | (16) OW A(x, x) = 1 2 SM AP E SM AP E N AIV E2 + M ASE M ASE N AIV E2 (17) N D(x, x) = H i=1 |x T +i − x T +i | H i=1 |x T +i | (18) M DA(x, x) = 1 H H i=1 sign(x T +i − x T ) = sign(x T +i − x T )(19)
We evaluated the forecast accuracy using 8 standard TS metrics: the mean absolute percentage error (MAPE) used in the Tourism compeition [51], the mean absolute scaled error (MASE) [57], the scaled mean absolute percentage error (SMAPE) used in the M3 competition [50], the normalized deviation (ND) used in [22] and the mean directional accuracy (MDA). Additionally for the M4 competition, we evaluated the model on the overall weighted average (OWA) between the SMAPE and the MASE such that a seasonally-adjusted naive (NAIVE2) forecasting model obtains a score of 1.0 [4]. For instance, an OWA of 0.90 means that the forecast is on average 10% better than a NAIVE2 model with respect to both the SMAPE and MASE metrics. The MDA measures the model's ability to produce forecasts where the trajectory follows the actual change of the TS relative to the last known value: the higher the MDA is, the better a model predicts the trend of a TS at any given time. For all other metrics, the lower the value, the better a model predicts the TS. For the M4 dataset, we only consider the OWA, the MASE and the MDA. The other metrics were used in order to compare ourselve with other methods and other datasets as detailed in Appendix C.
Eq. 14-19 describes how these metrics are computed.x is the forecast, x is the ground truth and m is the time interval between successive observations considered by the organizers for each data frequency, i.e., 12 for monthly, four for quarterly, 24 for hourly and one for yearly, weekly and daily data. Without loss of generality to previous equatios T is the number of point in-sample observed to make the forecastushe and H is the forecast horizon. We present the results of the baseline and benchmark accuracies for the M4 dataset in Table 1. The table gives the reported accuracy of N-BEATS reported in the original papers [39], the replicated results using the publicly accessible implementation provided by the original authors along with their scaled versions [13] based upon their implementation and our model NBEATS(P). Three main conclusions can be drawn:
Baseline and Benchmark
(1) Scaling TS to allow generalization on other datasets for the N-BEATS model, as presented in [13], adds a penalty on the OWA metrics for the M4 dataset, which suggests that there is a trade-off between accuracy and generalization on other datasets for DNN-based models.
(2) Figure 2 details how sensemble size has an impact on computational time to train. It can be seen that applying a bagging procedure [44] 3 to 4 times is sufficient to get an accurate ensemble for both the NBEATS and NBEATS(P) model but NBEATS(P) is more efficient the larger the ensemble size is.
(3) The top-performing models do not differ significantly with respect to the coverage of the TS forecasted and the mean directional accuracy (MDA). This provides an argument that if one is mainly interested in predicting the TS variation from the forecast origin, relying on the fastest implementation of the top-performing models for a first initial prediction is a cost-effective solution.
. Regarding (2), we illustrate this phenomenon in fig. 3 by plotting the TSNE embedding of each series of M4 computed from the same set of features used in the FFORMA [2] by comparing the top performing model with the N-BEATS(i) model by coloring each TS with its individual MASE accuracy. We refer the reader to table.A.5 and [2,59] for a detailed overview of the 42 features used and their interpretation. Appendix A details the distribution of these features over all datasets we considered. Note that there are no substantial differences between the approaches, despite some subtle regions of the graph where we can observe N-BEATS(i) performing better overall than ES-RNN. Table 2 presents the time to train each model accuracy as well as the average pairwise absolute percentage error correlation of the forecast residuals between ensemble members, in a way to similar to the experimental evaluation of M4 submission performed in [64]. At first glance, we see that the training time of these models be as long as multiple weeks. N-BEATS models timing are reported with a single GPU. In comparison to N-BEATS, N-BEATS(P) takes less time regardless of the variant used but the gain is observed especially Table 1: Averaged forecasting results of the M4 competition for the evaluated models. The OWA metric is presented for each seasonal pattern observed. Forecasts from models in italics were pre-computed except for the N-BEATS models. We replicate the results with the implementation provided by the authors, e.g: N-BEATS (I) (original) vs N-BEATS (I) (our). MLP ' and RNN ' models are appended with "'" to signify that these model were trained per TS using a seasonal and trend decomposition with manual pre-and post processing steps [4]. We also considered a coverage indicator which measures the number of series that a model forecasts better than an arbitrary MASE accuracy threshold of τ = 1.0. We also added the MDA of the forecast.
Training Time and Number of Parameters
for the generic architecture. Both the original approach and ours are at the same level of correlation and, while ours is slightly less diverse, it is roughly twice as efficient and achieves the same level of accuracy. We can observe that using the scaled version has little impact in terms of diversity. In our preliminary result, we also observed that there was no significant difference in terms of the TS samplers used to train N-BEATS(P) where for instance, different TS were sampled for the W model.
In practice, computational time remains more than significant for training these models on a single GPU. However, we can speed up the training by training model simulatenously. If we consider N-BEATS(G) vs N-BEATS(P, G), each ensemble member trained on a single TS frequency takes on average 12 & 27 minutes, and at worst, 19 & 57 minutes respectively. N-BEATS(G)'s time is for a single lookback and thus requires 1080 (6 lookback × 3 losses × 6 frequencies × 10 repeats) independent models to be trained whereas N-BEATS(P, G) requires only (3 losses×6 frequencies×10 repeats) independent models. Thus, if one would have access to 1080 GPUS, the total training time of N-BEATS(G) could be done in 20 min, but this is an unrealistic amount of ressources for most organization. Our approach cuts down that cost: given 10 GPUs, the N-BEATS(G) and a greedy schedudling of model training, it would take roughly 35h to train whereas our would only take 16h with all 10 repeats. Using 20 gpus, our model would achieve it in 8 hours whereas the previous model would take 18. When forecasting larger amount of TS, say 1 billions monthly distinct TS, estimating the cost can be difficult. We can make a reasonable assumption that the computational time required to train a model scales linearly with the number of time series to forecast altought it takes roughly the same to train on different subpopulations or the other based on the number of itterations. Hence assuming we are using the same 3 losses and bagging procedure and it takes a single model to train the monthly TS where the number of itterations required linearly scaled from the one used in M4 (see B.9) and require 75k itterations instead, it would take 1878h and 1551h for the generic and interetable version of our model on 30 GPUS. Altough a larger dataset may benefit from a deeper & wider model further inflating the cost, the number of itterations might not need to be this high. However, our approach will result in similar gain in the scenario of deeper & wider model. Regardless, training these model for everyday usage requires a lot of computational ressouce. In order for the cost to be kept low at this scale, training would need to be done less frequently and models would have to remain outdated to some extent as recent trends and structural changes in the data wouldn't be used to update the model parameters.
To further show the performance of our model, we show in Table. 3 that one could have achieved the same average accuracy as the top M4 competitions entries by training our for approximatively 2h on a single GPU without bagging. Thus, even with minimal amount of ressources, smaller organizations can train 54 DNN-based models, each on TS of different freqencies, losses and lookback windows very fast making our model far more accessible to small organization who doesn't have dozens of gpus available.
Since the training procedure of our approach takes roughly a fixed amount of time to train regardless of the number of TS to forecast 5 , forecasting more TS might requires more itterations and/or more parameters for the model to capture the dynamics of these additonal TS. Therefore the training time is expected to increase the more TS we want to forecast in the training regime. However, the overall training time doesn't increase linearly with the number of TS to forecast as increasing the number of itteration is a fixed cost and the number of itterations to train Quarterly (24K TS) or Monthly (48K TS) was the same in our setting. This leaves, the total number of itterations to train all models, the forecast horizons, the number of parameters and the number of models trained simultanously to have effect on training time.
The difference in improvement factor between parallelized generic and interpretable versions of N-BEATS(P) is due to the hidden layer sizes between the two versions. Having a higher number of hidden neurons reduce the computational gain of training multiple models conjointly as it saturate GPU usage. If we have a sufficiently expressive model without requiring too many hidden neurons, N-BEATS(P) is expected to produce accurate forecasts at a fraction of the cost. Otherwise the gain will be diminished. Regardless, these results show that ensemble diversity and accurate forecasts could have been achieved with reduction in resources and computation time.
Given the increasing trend of top-performing models requiring ever more
Conclusion
We proposed an efficient novel architecture for training multiple TS models conjointly for univariate TS forecasting. We empirically validated the flexibility of our approach on the M4 TS datasets as well as assessing its generalizability to other domains of application, using 5 other datasets which, combined, cover over 2.5 million forecasts. We provided forecasts in various TS settings at the same level of accuracy as current state-of-the-art models with a model that is twice as fast while requiring 5 times fewer parameters than the top performing model. We highlighted both stylized facts and limitations of the performance of the model studied, in an effort to provide insights to TS practitioners for operating DNN-based models at scale. Our results suggest that training global univariate models conjointly by sharing parts of their parameterizations yield competitive forecasts in a fraction of the time and does not significantly impair either forecast accuracy or ensemble diversity.
and Traffic datasets exhibit multiple seasonal patterns, whereas datasets like Finance exhibit large difference from other datasets in terms of high order autocorrelation (x_acf10), autoregressive conditional heteroscedasticity (archlm, garch_r2), strength of trend (trend) and high variance of the mean of observation from non-overlapping windows (stability). All sampled TS from all datasets are summarized in Fig. A.5 using the T-SNE algorithm [69]. Each point of this graph correspond to the 2-dimensional embedded space of a single TS computed from the same set of endogenous statistical features [2] . One can observe the heterogeneity of these datasets and note that the subpopulations of TS within a dataset can have high variance in their statistical properties while being similar to other subpopulations of other datasets. When considering the Finance dataset, we can see how the [59]. Default values when test failed was 0. The M4 7 dataset is a publicly accessible dataset that contains a large set of 100'000 heterogenous TS sampled from the ForeDeCk database for the M4 competition [17]. The database is compiled at the National Technical University of Athens and is built from multiple diverse and publicly accessible sources. It includes TS frequently encountered in business domains such as industries, services, tourism, imports/exports, demographics, education, labor & wage, government, households, bonds, stocks, insurances, loans, real estate, transportation, and natural resources & environment. TS were sampled at different frequencies [Yearly, Quarterly, Monthly, Weekly, Daily and Hourly] each with different forecast horizons, i,e, [6,8,18,13,14,48] according to the competition organizer. Table A.6 outlines the composition of the M4 dataset across domains and forecast horizons.
All TS were provided with a prepossessing scaling procedure to ensure positive observed values at all time-steps with minimum observed values greater than or equal to 10. The scaling was applied only to sampled TS whose minimum oberved value was smaller than 10 by adding a per-TS constant to all TS to ensure that the minimal values was positive. All other TS were unaltered by any preprocessing step. The dataset was subdivized into a training and a test dataset by the M4 TS competition organizers. For further details on this dataset, we refer the reader to the following: [4,17]. We relied on the pre-computed forecasts and PI available at https: The Tourism 9 dataset is a publicly accessible dataset that contains TS collected by [51] from tourism government agencies and academics who had used them in previous tourism forecasting studies. The TS of this dataset are highly variable in length. It includes yearly, quarterly and monthly TS. Table. A.8 details the proportion of TS from each frequency. For further detail on this dataset, we refer the reader to [51]. This dataset was considered for zero-shot forecasting, to examine a case where the target dataset comes from domains that are not present in the M4 dataset.
/
Appendix A.4. Electricity and Traffic Datasets Details
Electricity 10 and Traffic [70] are two publicly available datasets from the Univeristy of California Irvine Machine Learning repository. The Electricity dataset contains the hourly electricity usage monitoring of 370 customers over three years, with some clients being added during the the observation periods creating cold-start conditions for producing some forecasts. The Traffic dataset contains TS of the hourly occupancy rates, scaled in the (0,1) range for 963 lanes of freeways in the San Francisco Bay area over a period of slightly more than a year. Both of these dataset exhibit strong seasonal patterns due to their nature and are mostly homogeneous. These two TS datasets are used de facto to evaluate the quality of DNN-based TS models as in [22,35,71]. We included these two datasets as a sanity check for zero-shot forecasting, to ensure that zero-shot forecasts were accurate in a setting where it is relatively easy to produce accurate forecasts.
Appendix A.5. Finance dataset
The Finance dataset contains daily closing prices of U.S. MFs and ETFs observed between 2005-07-01 and 2020-10-16 and traded on U.S. financial markets, each covering different types of asset classes including stocks, bonds, commodities, currencies and market indexes, or a proxy for a market index. The dataset was obtained through three data providers: (1) Fasttrack 11 , a professional-grade data provider for financial TS, (2) Yahoo Finance API and (3) the Federal Reserve of Saint-Louis (FRED) database. Part of this dataset is proprietary, so we do not have permission to share that part publicly. However, the list of securities is given in Table. D.13 to help interested readers reconstruct the dataset from public data sources.
We considered this dataset in our zero-shot experiments by sampling the TS at three different frequencies [dDaily, weekly and monthly] and specifying the same forecast horizon as that of the M4 dataset. We used this dataset to present a worst-case scenario for zero-shot. First this is a case where the forecasting application is notorious for its forecasting difficulty. Moreover, the source dataset on which we train our model has at most 10K TS to train from and at worst 164 TS, which force zero-shot generalization with very few training data. Also, by sampling the TS at large scale, we emulated how zero-shot could be applied on the whole history of the TS, similar to the procedure carried out by portfolio managers and quantitative analyst to backtest the validity of their investment strategies. The TS were split into We used the same overall training framework as [39] including the stratified uniform sampling of TS in the source dataset to train the model. Training N-BEATS and N-BEATS(P) was done by first segmenting the training dataset into non-overlapping subsets based on the TS frequency they were observed in. Then, independent training instances were trained, one each group by specifying the forecast horizon of each instance based on the common forecast horizon of the subset. Table B.9 presents the HP settings used to train all N-BEATS and N-BEATS(P) models on the different subsets of M4. Except for the number of iterations on the yearly TS, all other HPs are the same. We did not proceed with an exhaustive parameter search since this was not the focus of our work. We were interested in whether or not we could make the N-BEATS model model faster and more usable in practical scenarios.
For zero-shot application, we relied on the scaled version of each model, i.e. where the TS is scaled based on its maximum observed value over its lookback periods. With one exception, the model trained on a given frequency split of a source dataset is used to forecast the same frequency split on the target dataset. The only exception is follows: when transferring from M4 to M3, the Other subpopulation of M3 is forecast with the model trained on the Quarterly subpopulation of M4. Table B.10 describe the different zero-shot training regimes on which the model was trained on the source dataset.
The source code to replicate the experiments for both traditional forecasting regime and zero-shot forecasting of Appendix C is available at: https://anonymous.4open.science/r/actm-7F90.
Appendix C. Zero-shot forecasting
To test whether our model can generalize to other datasets, we evalate its capacity to support zero-shot TS forecasting, i.e., to train a neural network on a source TS dataset and deploy it on a different target TS dataset without retraining, which provides a more efficient and reliable solution to forecast at scale than its predecessor. We present a flowchart of the zero-shot forecasting regime in fig. C.6. In this setting a single model is trained once on a source datasets and can be used to forecast multiple target datasets without retraining as in [13]. We demonstrate that N-BEATS(P) has comparable level of accuracy than N-BEATS for zero-shot generalization ability in various settings. It can operate on various domains of applications and on target datasets that are out-of-distribution of the source dataset it was trained on, i.e. on dataset from other dommains, settings and/or that have different statistical properties than the dataset it was trained on. We evaluated their performance in the zero-shot regime on all other datasets (M3, Tourism, Electricty, Traffic, Finance) by training models on the M4 dataset only using scaled TS as in [13]. The reason for this preprocessing step was to prevent catastrophic failure when the target dataset scale is different from that of the source dataset.
We tested 3 different setups for zero-shot forecasting, which we denote by R O , R SH,LT and R SH . R O is a setup where we use the same model to produce results on M4 (Table 1) and apply it on the target dataset. This required us to truncate the forecast or apply the model iteratively on the basis of previous forecasts to ensure the forecast size is the same as the target dataset. The model was not trained to operate when this condition occurs. R SH is a setup where the model is trained with the same number of iterations as R O but we specified the model's forecast horizon to be the same that of the target datasets. R SH,LT is the same training regime as R SH , but we allowed the model to consider TS samples from further in the past during training and trained the model with more iterations. The rationale of testing these training setup is to evaluate the impact of training the model longer for generalization and to to test the model in forecast condition it wasn't trained to do (e.g. in R O when the forecast horizon of the target dataset exceed or is inferior to the forecast horizon of the target dataset). When training the model, we consider an hyper-parameter L H which is a coefficient defining the length of training history immediately preceding the last point in the train part of the TS that is used to generate training sample. This coefficient multiplied by the forecast horizon determined the maximum number of most recent points in the train dataset for each TS to generate training sample. The higher that coeffcient, the further in the past we consider TS observation to train this model. To produce forecasts, we used the subset of the ensemble models trained on the same TS frequency to produce the multiple forecasts and combined them by median aggregation. Detailed explanations of this aggregation, selection of L H and the ensemble parameters used are given in Appendix B.
We compared against statistical baselines and other ML models such as DEEP-STATE [35], N-BEATS [39,13], DEEP-AR [22], FFORMA [2], ES-RNN [42], Deep Factors [72] and many others including statistical baselines already evaluated on theses datasets. In reporting the accuracy of these models, we relied upon the accuracy and the pre-computed forecasts reported in their respective original paper. (Table. 1), which required to truncation of the forecast or applying the model iteratively at the basis of previous forecasts to ensure the forecast size was the same that of the target dataset. R SH is trained in the same fashion as R O but we specified the model's forecast horizon to be the same as that of the target datasets. R SH,LT is the same training regime as R SH except that the model is allowed to consider TS samples from further in the past while training: See Table. B.10 for more detail. Results for models with * appended to their names are replicated from the original papers and signifies an anonymous submission for which we do not know the methodology. (2) Comparing with [13], where a different training regime was used, the difference between their results and ours highlights the importance of the optimization procedure to facilitate transfer to another dataset. In certain cases, some datasets (e.g., Tourism, will benefit from a longer training, to the detriment of the forecast accuracy on the source dataset. In other cases, like the Electricity dataset, no adjustments are required between the source and the target dataset.
(3) The case of the Tourism dataset highlights the importance of ensuring that the forecast horizon of the source dataset used to train the model is longer than or equal of the target dataset; this is a key factor in producing reliable zero-shot forecasts.
Considering that the M4 dataset includes a large number of heterogenous TS that contain at least some TS with similar statistical properties to those present in the target dataset, zero-shot forecasting can be easily deployed with a pre-trained DNN model and can produce initial forecasts that are on par with the level of accuracy of multiple baselines, and sometimes benchmarks, very quickly at a minimal cost.
However, not all settings will benefit equally from this approach. The Finance dataset is a prime example of a setting where zero-shot forecasting produce mixed results. In this setting, the source TS dataset has very few TS to train from in comparison to the test sets. Also, these TS are very difficult to forecast in a univariate setting since they are almost all non-ergodic, heteroscedastic, and have high noise-to-signal ratio. Despite these added difficulties, both N-BEATS and N-BEATS(P) can produce forecasts at a comparable level to a single statistical model in term of MDA when using the R O training setup. However, the zero-shot regime achieved forecasts better than a naive one only by sampling these TS at a monthly frequency, which coincidentally is the largest pool of TS in M4 (48'000.) In comparison, the daily (3594) and weekly (227) subsets contain fewer TS. Hence, even under poor conditions of application for zero-shot N-BEATS(P), we can still produce preliminary forecasts quickly. These results highlight the importance of selecting a good source dataset but even in subpar conditions, our approach can still generalize well with respect to the MDA metric.
T i } N i=1and a test dataset of future values of these TS D eval = {Y (i)
predictors of expansion coefficients for each of the W lookback periods.The third part consists of the W independent backward g
∈
R L are basis vectors learned by the model, which can be taught as waveforms. Because no additional constraints are imposed on the form of V f,( ) lw the waveforms learned do not have inherent structure on how they should look.
applying the seasonal model within the doubly residual stacking topology of the model, we obtain a model that applies TS component decomposition in a similar way to than traditional decomposition approaches. Basis functions are a generalization of linear regression where we replace each input with a function of the input. Here, the polynomial and the Fourier series are functions that model uses the trend and seasonality and take as input the embedding computed from the TS at each block ant not the raw TS values. In the case of the generic version, the basis functions for the forecasts and the backast are respectively the vectors V f,( ) lw and V b,( )
( 5 )
5(public) Traffic: 963 TS of the hourly occupancy rates on the San Francisco Bay Area freeways scaled between 0 and 1[53,54].(6) (proprietary) Finance: 321 TS observed between 2005-07-01 and 2020-10-16 of the adjusted daily closing price of various U.S. mutual funds and exchange traded funds traded on U.
Figure 2 :
2OWA metric (left) and Time (right) in minutes for the different N-BEATS models configurations as a function of ensemble size.
Figure 3 :
3MASE coverage for the ES-RNN (left) and N-BEATS(i) (right) over the M4 dataset. Each point on the graphs corresponds to a single TS and the darker its color, the better the given model according to the MASE. Horizontal and vertical coordinates represent the value of the two-dimensional embedding computed with TSNE[58] from the statistical features of the series which we detail in Appendix A. All TS with MASE values over 3 where assigned the same color to facilitate visualization.
85 ± 0.02 N-BEATS(G) scaled 11755 0.84 ± 0.04 N-BEATS(I) [39] 7437 0.89 ± 0.02 N-BEATS(I) scaled 6607 0.85 ± 0.03 N-BEASTS(I+G) [39] 19211 0.84 ± 0.03 N-BEATS(I+G) scaled 19170 0.84 ± 0.03 N-BEATS(P, G) (our) 5301 0.87 ± 0.02 N-BEATS(P,G) scaled (our) 6157 0.90 ± 0.02 N-BEATS(P, I) (our) 6990 0.89 ± 0.03 N-BEATS(P,I) scaled (our) 4785 0.89 ± 0.08 N-BEATS(P, I+G) (our) 11840 0.88 ± 0.02 N-BEATS(P,I+G) scaled (our) 10943 0.89 ± 0.06
Figure A. 4 :
4Cumulative distribution function plot for TS datasets over 42 statistical TS features and TS count by dataset and frequency.
Figure A. 5 :
5TSNE embedding of all TS forecasted with their different subpopulations. behavior of the 321 TS changes is heterogenou over time in comparison to the other dataset. The Electricity & Traffic TS share almost the same statistical properties as both populations of TS are concentrated in the same region of the graph.
)
11 https://investorsFasttrack.com chunks of the maximum lookback period of the N-BEATS model as training sample, and H steps-ahead as testing sample. Quarterly (h = 8) Monthly (h = 18) Weekly (h = 13) Daily (h = 14) Hourly (
Figure C. 6 :
6Flowchart of the zeros-shot forecasting regime on D target datasets from one source datasets (D source ). The blue square (left) represent the traditional setup of training a model and forecasting on such dataset. The red squares (red) represent the process of loading a pre-trained models and forecasting TS from a different TS dataset than the one used for training. In term of time, the time to train a model is almost always greater than the one from infering a target datasets.
We trained our model on the M4 dataset on the TS each subpopulations, i.e. [Yearly, Quarterly, Monthly, Weekly, Daily, Hourly]. We replicated the results from [39] by training the two N-BEAT model variants discussed in Sec. 4.2 using the implementation provided by the original authors and with scaled TS where we divided all TS observations by the maximum values observed. This scaling was done per TS with respect to the lookback window. For our model, the scaling was done on by dividing all lookback windows by the maximum value observed over all lookback windows.6, 8, etc...). For
the Electricity and Traffic datasets, the test was set using rolling window
operation as described in Appendix A.4. For the Finance dataset, the
forecast was evaluated on three rolling forecast setups by sampling the TS
on different frequencies, i.e.: daily, weekly and monthly. In total there are
2'602'878 individual TS that were sampled from the 321 original ones across
3 forecast horizons. Despite the dataset being collected from proprietary data
sources which we cannot redistribute, we provide the necessary details to help
interested readers reconstruct the datasets in Appendix A.5. A summary of
the statistical properties, forecast horizons and metadata of these dataset are
presented in Appendix A.
Yearly Quarterly Monthly Others AverageCoverage
OWA
MASE % (< T = 1.0) MDA(%)
NAIVE
1.000
1.066
1.095
1.335
1.058
40.299
3.2
NAIVE2
1.000
1.000
1.000
1.000
1.000
43.288
33.1
SNAIVE
1.000
1.153
1.146
0.945
1.078
36.095
42.7
ARIMA [30]
0.892
0.898
0.903
0.967
0.903
51.145
53.8
HOLT [60]
0.947
0.932
0.988
1.180
0.971
48.659
61.7
ETS [61]
0.903
0.890
0.914
0.974
0.908
50.987
48.6
THETA [32]
0.872
0.917
0.907
0.995
0.897
48.686
61.7
SES [31]
1.002
0.970
0.951
0.995
0.975
44.719
35.3
DAMPED [4]
0.890
0.893
0.924
1.005
0.907
49.838
61.1
COMB [4]
0.867
0.890
0.920
1.039
0.898
49.784
61.3
MLP ' [4, 62]
1.288
1.684
1.749
3.028
1.642
26.603
60.6
RNN ' [4, 62]
1.308
1.508
1.587
1.702
1.482
28.437
59.8
ProLogistica [63]
0.820
0.855
0.867
0.742
0.841
53.620
62.6
FFORMA [2]
0.799
0.847
0.858
0.914
0.838
53.418
63.7
ES-RNN [42]
0.778
0.847
0.836
0.920
0.821
53.271
63.2
N-BEATS (I) [39]
0.765
0.800
0.820
0.822
0.797
-
-
N-BEATS (G) [39]
0.758
0.807
0.824
0.849
0.798
-
-
N-BEATS (I+G) [39]
0.758
0.800
0.819
0.840
0.795
-
-
Ours:
N-BEATS (G) [39]
0.770
0.793
0.818
0.832
0.798
55.576
64.6
N-BEATS (I) [39]
0.763
0.797
0.817
0.838
0.795
55.600
63.7
N-BEATS (I+G) [39]
0.761
0.792
0.814
0.834
0.793
55.868
64.6
N-BEATS (G) scaled [13]
0.784
0.810
0.827
0.836
0.809
54.960
64.5
N-BEATS (I) scaled [13]
0.773
0.817
0.826
0.843
0.806
54.919
63.7
N-BEATS (I+G) scaled [13]
0.778
0.814
0.824
0.836
0.806
55.109
64.4
N-BEATS parallel (G)
0.764
0.804
0.820
0.855
0.799
55.332
64.4
N-BEATS parallel (I)
0.759
0.817
0.824
0.850
0.801
54.966
63.7
N-BEATS parallel (I+G)
0.757
0.806
0.820
0.851
0.796
55.375
64.5
N-BEATS parallel (G) scaled
0.775
0.829
0.833
0.851
0.812
54.506
63.8
N-BEATS parallel (I) scaled
0.772
0.845
0.844
0.867
0.819
53.772
63.6
N-BEATS parallel (I+G) scaled
0.771
0.834
0.835
0.854
0.813
54.344
63.9
Table 2 :
2Time required to train to train all members of the ensemble of our models vs other and average & standard deviation of the absolute percentage correlation between ensemble members on the test sets. We include the total time to produce a forecast for the theta method for comparison. Except for Prologistica, FFORMA and ES-RNN whose training time was replicated in[4], the total time presented is with all model are for single instance and do not consider the speedup that can be achieved based when training the whole ensemble on multiple GPUs.
Table 3 :
3Performance of a small ensemble only trained on the MAPE loss for all lookback without bagging and time to train on a single GPU.training time [4], training and deploying state-of-the-art models in real-case
scenarios can entail high costs for organizations -costs that are avoidable.
For instance, on Google's cloud platform, the estimated cost of training
N-BEATS(P) would drop to 530.11 USD$ instead of the 860.13 USD$ their
price simulator gives for N-BEATS 6 . Thus, in terms of both cost and time
saved, our work provides encouraging results that suggest how multiple TS
ensemble models can be accelerated without any great drawback by sharing
a subset of their parameterization.
Model name # of parameters
Model name
# of parameters
N-BEATS(G)
28'508'265'900 N-BEATS(P, G)
5'972'957'400
N-BEATS(I)
42'288'737'310 N-BEATS(P, I)
8'102'076'930
N-BEATS(I+G)
70'797'003'210 N-BEATS(P, I+G)
14'075'034'330
Table 4 :
4Number of parameters for the whole ensemble for N-BEATS and N-BEATS (parallel) trained on the M4 dataset with 6 lookback windows.
Table A.5: List of features used to compare datasets. The functions for calculating these features are implemented in the tsfeatures R package byFeatures
Description
Seasonal Non-Seasonsal
1 T
length of time series
2 trend
strength of trend
3 seasonality
strength of seasonality
-
4 linearity
Linearity
5 curvature
Curvature
6 spikiness
Variance of the leave-one-out variances of the remainder component
in STL decomposition
7 e_acf1
first autocorrelation function (ACF) value of remainder series
8 e_acf10
sum of squares of first 10 ACF values of remainder series
9 stability
sum of squares of first 10 ACF values of remainder series
10 lumpiness
Variance of the means produced for tiled (non-overlapping) windows
11 entropy
Spectral entropy (Shannon entropy) of the TS
12 hurst
Hurst exponent from [65]
13 nonlinearity
Teraesvirta modified test [66]
13 alpha
ETS(A,A,N)α
14 beta
ETS(A,A,N)β
15 hwalpha
ETS(A,A,A)α
16 hwbeta
ETS(A,A,A)β
-
17 hwgamma
ETS(A,A,A)γ
-
18 ur_pp
Test statistic based on Phillips-Perron test [67]
19 ur_kpss
test statistic based on KPSS test [68]
20 y_acf1
first ACF value of the original series
21 diff1y_acf1
First ACF value of the differenced series
22 diff2y_acf1
First ACF value of the twice-differenced series
23 y_acf10
Sum of squares of first 10 ACF values of original series
24 diff1y_acf10
Sum of squares of first 10 ACF value of the differenced series
25 diff2y_acf10
Sum of squares of first 10 ACF value of the twice-differenced series
26 seas_acf1
autocorrelation coefficient at first seasonal lag
-
27 sediff_acf1
first ACF value of seasonally differenced series
-
28 y_pacf5
sum of squares of first 5 PACF values of original series
29 diff1y_pacf5
sum of squares of first 5 PACF values of original series
30 diff2y_pacf5
sum of squares of first 5 PACF values of twice-differenced series
31 seas_pacf
partial autocorrelation coefficient at first seasonal lag
32 crossing_points number of times the time series crosses the median
33 flat_spots
number of flat spots, calculated by discretizing the series into 10
equal-sized intervals and counting the maximum run length within
any single interval
34 nperiods
number of seasonal periods in the series
-
35 seasonal_period length of seasonal period
-
36 peak
strength of peak
37 trough
strength of trough
38 ARCH.LM
ARCH.LM statistic
39 arch_acf
sum of squares of the first 12 autocorrelations of z 2
40 garch_acf
sum of squares of the first 12 autocorrelations of r 2
41 arch_r2
R 2 value of an AR model applied to z 2
42 garch_r2
R 2 value of an AR model applied to r 2
Table A . 6 :
A6Composition of the M4 TS dataset: number of time series based on their sampling frequency and type.
/github.com/Mcompetitions/M4-methods.Table A.7: Composition of the M3 TS dataset: the number of TS based on sampling frequency and type.Appendix A.2. M3 Dataset Details
Frequency/Horizon
Type
Yearly (h = 6) Quarterly (h = 8) Monthly (h = 18) Other (h = 8) Total
Demographic
245
57
111
0
413
Finance
58
76
145
29
308
Industry
102
83
334
0
519
Macro
83
336
312
0
731
Micro
146
204
474
4
828
Other
11
0
52
141
204
Total
645
756
1428
174
3'003
The M3 8 dataset is a publicly accessible dataset that is smaller than
8 https://forecasters.org/resources/time-series-data/m3-competition/
the M4 dataset but remains relatively large and diverse. Similarly to the
M4 dataset, it contains TS frequently encountered in business, financial and
economic forecasting. It include yearly, quarterly, monthly, weekly, daily
and hourly time series, each with different forecast horizons, i,e, [6, 8, 18,
13, 14, 48]. All series have positive observed values at all time-steps. The
dataset was subdivided into a training and a test dataset by the M3 TS
competition organizers. Table A.7 outlines the composition of the M3 dataset
across domains and forecast horizons. For further details on this dataset, we
refer the reader to [50]. This dataset was considered for zero-shot forecasting,
to examine a case where the target dataset is from the same domains of
application but with other TS.
Appendix A.3. Tourism Dataset Details
Frequency/Horizon
Type
Yearly (h = 4) Quarterly (h = 8) Monthly (h = 24) Total
Tourism
518
427
366
1311
Table A.8: Composition of the Tourism TS dataset: number of time series based on
sampling frequency and type.
Table B . 9 :
B9Hyper parameters used to produce results on the M4 TS dataset
Native zero-shot (R O )Frequency
Yearly
Quarterly
Monthly
Weekly
Daily
Hourly
horizon
(h = 6)
(h = 8)
(h = 18)
(h = 13)
(h = 14)
(h = 48)
L H
1.5
1.5
1.5
10
10
10
Iterations N-BEATS
15k
15k
15k
5k
5k
5k
Iterations N-BEATS (P)
10k
15k
15k
5k
5k
5k
Native Zero-Shot with equal forecast horizon (R SH )
horizon
(h = h
(Dtgrt.)
Yearly ) (h = h
(Dtgrt.)
Quarterly ) (h = h
(Dtgrt.)
Monthly ) (h = h
(Dtgrt.)
Weekly ) (h = h
(Dtgrt.)
Daily ) (h = h
(Dtgrt.)
Hourly )
L H
1.5
1.5
1.5
10
10
10
Iterations
15k
15k
15k
5k
5k
5k
Native Zero-Shot with equal forecast horizon(R SH,LT )
horizon
(h = h
(Dtgrt.)
Yearly ) (h = h
(Dtgrt.)
Quarterly ) (h = h
(Dtgrt.)
Monthly ) (h = h
(Dtgrt.)
Weekly ) (h = h
(Dtgrt.)
Daily ) (h = h
(Dtgrt.)
Hourly )
L H
10
10
10
10
10
10
Iterations
15k
15k
15k
15k
15k
15k
Table B .
B10: HP differences between the different zero-shot strategies. All models were
trained on the M4 TS dataset
Table C .
C11 describes the zero-shot performance of N-BEATS and N-BEATS(P). Several observations can be made: we show the metrics for three training regimes: R SH,LT /R SH /R O . R O is the same model used to produce the results on M4(1) N-BEATS(P) produces comparable zero-shot results to previous state-of-
the-art models for all datasets. In other training regimes, where models
trained with the same forecast horizon or longer ones, comparable levels
of accuracy were observed.
M3, SMAPE
Tourism, MAPE
Electricity, ND
Traffic, ND
N-SHOT:
Naive
16.59
SNaive
24.80
Naive
0.37
0.57
Comb [4]
13.52
ETS [61]
20.88
MatFact [54]
0.16
0.17
ForePro [52]
13.19
Theta [32]
20.88
DeepAR [22]
0.07
0.17
Theta [32]
13.01
ForePro [52] 19.84
DeepState [35] 0.08
0.17
DOTM [73]
12.90
Strato
19.52
Theta [32]
0.08
0.18
EXP [74]
12.71
LCBaker [75] 19.35
ARIMA [30]
0.07
0.15
N-BEATS [39]
12.37
18.52
0.07
0.11
DEEP-AR* [22, 13]
12.67
19.27
0.09
0.19
ZERO-SHOT: (RSH,LT /RSH/RO)
M4 N-BEATS (G) scaled * [39]
12.36/12.67/12.72
18.90/20.16/24.14
0.09/0.09/0.08 0.16/0.16/0.14
M4 N-BEATS (I) scaled * [39]
12.43/12.63/12.66
19.43/20.58/23.26
0.10/0.09/0.08 0.16/0.16/0.14
M4 N-BEATS (g+i) scaled * [39] 12.38/12.61/12.64
19.04/20.22/23.43
0.10/0.09/0.08 0.16/0.16/0.14
M4 N-BEATS (P+G) scaled
12.48/12.65/12.65
18.99/19.98/22.85
0.09/0.09/0.08 0.16/0.18/0.14
M4 N-BEATS (P+I) scaled
12.69/12.76/12.72
20.54/20.97/23.18
0.09/0.10/0.09 0.17/0.17/0.16
M4 N-BEATS (P+G&I) scaled
12.56/12.67/12.64
19.50/20.24/22.79
0.09/0.09/0.08 0.16/0.16/0.14
Table C.11: Averaged forecasting for the zero-shot regime for each dataset; lower values are
better. Zero-shot forecasts are compared for N-BEATS and our approach. For the models
in italic using the following references, we relied on their reported accuracy. For zero-shot
results,
Daily (H = 14)N = (1 222 866) Weekly (H = 13, N = 1 091 898) Monthly (H = 18, N = 288 114) Table C.12: Comparison between statistical baselines and zero-shot application of the N-BEATS model in terms of OWA, MDA and time to produce forecast. Forecasts were made with the native zero-shot approach (R O ).Models
OWA MDA
Time (min.)
OWA MDA
Time (min.)
OWA MDA
Time (min.)
N-SHOT:
Naive
1.000 07.1
-
1.00
02.0
-
1.000 00.5
-
ARIMA [30]
1.041 27.1
4685
1.059 28.9
3597
0.891 40.3
816
THETA [32]
0.995 49.4
241
0.993 53.5
262
0.913 61.4
49
SES [31]
1.000 09.3
174
1.001 05.2
160
1.000 02.9
25
HOLT [60]
1.081 49.6
167
1.116 53.9
160
0.931 60.3
42
ETS [61]
1.019 20.4
969
1.044 18.8
512
0.940 29.2
181
ZERO-SHOT:
M4 N-BEATS (G) [39]
1.165 50.3
24
1.078 50.9
21
0.963 55.1
6
M4 N-BEATS (I) [39]
1.222 50.5
26
1.045 51.3
23
0.962 54.9
6
M4 N-BEATS (I+G) [39] 1.191 50.7
50
1.056 51.1
44
0.961 55.4
12
M4 N-BEATS (P+G)
1.210 48.5
25
1.098 51.6
21
0.973 54.4
6
M4 N-BEATS (P+I)
1.135 48.5
26
1.055 52.1
24
0.975 54.1
6
M4 N-BEATS (P+I&G)
1.139 49.0
51
1.044 52.0
45
0.973 54.4
12
Appendix D. Finance dataset: List of Securities Considered Data Source: Yahoo US Sector Stock Index XLE StateSt ETF Energy Select Sector SPDR Fd US Sector Stock Index XLF StateSt ETF Financial Select Sector SPDR US Sector Stock Index XLI StateSt ETF Industrial Sel Sector SPDR US Sector Stock Index XLK StateSt ETF Tech Select Sector SPDR US Sector Stock Index US Sector Stock Index IYR iShares U.S. Real Estate ETF US Sector Stock Index XOI-I AMEX Oil Ix Commodities US Bonds -Corp Invst MBOA-BofAML US Corporate 15 Year Semi-Annual US Bonds -Corp Invst QQQ Nasdaq 100 ETF Stock Index (US) SP-GB S&P Global BMI Idx DivAdj Global Stock Index SP-GL S&P Global 1200 Idx DivAdj Global Stock Index SP-HB S&P 500 High Beta Idx DivAdj Global Stock Index SP-IO S&P Global 100 Idx DivAdj Global Stock Index SP-L4 S&P Latin America 40 Idx Di-vAdj Table D.13: List of US traded funds used to create the finance dataset. The class columns correspond to the type of securities and the source columns specify where the TS was collected. See Tab. D.14 for a brief description of the asset classes TS type Description US Stock Index Index of US stocks, such as the S&P500 US Stock A fund (ETF or mutual funds) made up primarily of US stocks US Sector Stock Index US stock industry sector index Regional Stock Index Global region stock index, such as Europe National Stock Index Country stock index Global Stock Index Global / world stock index US Bonds -Gvmnt US treasury funds US Bonds -Corp Invst US corporate bond funds, investment grade US Bonds -Corp HY US bond funds, high yield Table D.14: Brief description of the typea of TS used in the Finance dataset.Ticker
Description
Class
DJAT
Dow Jones Asian Titan 50 In-
dex
Regional Stock Index
DJI
Dow Jones Industrial Average Stock Index (US)
DJT
Dow Jones Transportation Av-
erage
Stock Index (US)
DJU
Dow Jones Utility Average
Stock Index (US)
GSPC
S&P 500
Stock Index (US)
IXIC
NASDAQ Composite
Stock Index (US)
NDX
NASDAQ-100
Stock Index (US)
OEX
S&P 100
Stock Index (US)
XMI
NYSE Arca Major Market In-
dex
Stock Index (US)
DX-Y.NYB
US Dollar/USDX -Index -
Cash
Forex
FDCPX
Fidelity Select Computers
US Sector Stock Index
HSI
HANG SENG INDEX (Cur-
rency in HKD)
National Stock Index
Data Source: Fred
GOLDPMGBD228NLBMGold Fixing Price 3:00 P.M.
(London time) in London Bul-
lion Market & based in U.S.
Dollars
Others
WILL4500IND
Wilshire 4500 Total Market In-
dex
Stock Index (US)
WILL4500PR
Wilshire 4500 Price Index
Stock Index (US)
WILL5000IND
Wilshire 5000 Total Market In-
dex
Stock Index (US)
WILL5000INDFC
Wilshire 5000 Total Market
Full Cap Index
Stock Index (US)
WILL5000PR
Wilshire 5000 Price Index
Stock Index (US)
Data Source: FastTrack
FPX1
CAC 40 Ix
National Stock Index
SHCP
Shanghai Composite Ix
National Stock Index
SPXX
STOXX Europe 600 Ix
Regional Stock Index
SX5P
STOXX Europe 50 Ix
Regional Stock Index
A-CWI
MSCI ACWI DivAdj Idx
Global Stock Index
A-XUS
MSCI ACWI xUS DivAdj Idx Global Stock Index
AUD-
US / Australia Foreign Ex-
change Rate
Forex
BBG-
CBOE US T-Bill 13-Week Yld
Bd Ix
US Bonds -Gvmnt
BBG-9
BBG Barclay Agg Bond-US
Universal TR Ix
US Bonds -Gvmnt
BBG-G
BBG Barclay Agg Bond-US
Corp IG TR Ix
US Bonds -Gvmnt
BBG-H
ML US HY Bb-B Ix
US Bonds -Corp HY
BBG-I
BBG Barclay Agg Bond-US
Agency Long Ix
US Bonds -Gvmnt
BBG-O
BBG Barclay Agg Bond-Yan-
kee Ix
US Bonds -Gvmnt
BBG-S
BBG Barclay Agg Bond-US
MBS Agncy TR Ix
US Bonds -Gvmnt
BBG-T
BBG Barclay Agg Bond-US
MBS Agncy TR Ix
US Bonds -Gvmnt
BBG-U
BBG Muni Bond 3yr Idx
US Bonds -Gvmnt
BBG-Y
BBG Muni Bond 20yr Idx
US Bonds -Gvmnt
BBM-2
BBG Muni Bond 5yr Idx
US Bonds -Gvmnt
BBM-3
BofAML US Corp 5-7yr Total
Return Ix
US Bonds -Corp Invst
BBM-5
BBG Muni Bond Composite
Idx
US Bonds -Gvmnt
BBM-I
BBG Muni Bond Long Term
Idx
US Bonds -Gvmnt
BBM-L
BBG Muni Bond 10yr Idx
US Bonds -Gvmnt
BBM-T
BBG Barclay Agg Bond-US
Composite TR Ix
US Bonds -Gvmnt
CAD-
Canada / US Foreign Exchange
Rate Ix
Forex
CDN-X
Canadian Dollar For 100 CDN
Ix
Forex
CHF-
Switzerland/ US Foreign Ex-
change Rate Ix
Forex
CNY-
China / US Foreign Exchange
Rate Ix
Forex
CR-TR
CRB Total Return Ix
Commodities
DBC
Invesco DB Commodity Index
Tracking Fund
Commodities
DKK-
Denmark / US Foreign Ex-
change Rate Ix
Forex
DXY-Z
US Dollar Ix
Forex
EFA
iShares MSCI EAFE ETF
Regional Stock Index
EURO-
US/Euro Foreign Exchange
Rate Ix
FBMPX
Fidelity Select Communication
Services Portfolio
US Sector Stock Index
FCYIX
Fidelity Select Industrials
US Sector Stock Index
FDAC-
Frankfurt Dax Ix
National Stock Index
FDFAX
Fidelity Select Consumer Sta-
ples
US Sector Stock Index
FDLSX
Fidelity Select Leisure
US Sector Stock Index
FEZ-X
Europe 50 STOXX stTr Ix
Regional Stock Index
FIDSX
Fidelity Select Financial Ser-
vices
US Sector Stock Index
FNARX
Fidelity Select Natural Re-
sources
US Sector Stock Index
FNMIX
Fidelity New Markets Income Regional Stock Index
FRESX
Fidelity Fidelity Real Estate In-
vestment Portfolio
Others
FSAGX
Fidelity Select Gold
US Sector Stock Index
FSAIX
Fidelity Select Air Transporta-
tion
US Sector Stock Index
FSAVX
Fidelity Select Automotive
US Sector Stock Index
FSCHX
Fidelity Select Chemicals
US Sector Stock Index
FSCPX
Fidelity Select Consumer Dis-
cretion
US Sector Stock Index
FSCSX
Fidelity Select software &
Comp Service
US Sector Stock Index
FSDAX
Fidelity Select Defense &
Aerospace
US Sector Stock Index
FSDCX
Fidelity Select Commun Equip-
ment
US Sector Stock Index
FSDPX
Fidelity Select Materials
US Sector Stock Index
FSELX
Fidelity Select Semiconductors US Sector Stock Index
FSENX
Fidelity Select Energy
US Sector Stock Index
FSESX
Fidelity Select Energy Service US Sector Stock Index
FSHCX
Fidelity Select Health Care Ser-
vice
US Sector Stock Index
FSHOX
Fidelity Select Const & Hous-
ing
US Sector Stock Index
FSLBX
Fidelity Select Brokrg & INV
Mgt
US Sector Stock Index
FSLEX
Fidelity Select Environmental
& Alt
US Sector Stock Index
FSNGX
Fidelity Select Natural Gas
US Sector Stock Index
FSPCX
Fidelity Select Insurance
US Sector Stock Index
FSPHX
Fidelity Select Health Care
US Sector Stock Index
FSPTX
Fidelity Select Technology
US Sector Stock Index
FSRBX
Fidelity Select Banking
US Sector Stock Index
FSRFX
Fidelity Select Transportation US Sector Stock Index
FSRPX
Fidelity Select Retailing
US Sector Stock Index
FSTCX
Fidelity Select Telecommunica-
tions
US Sector Stock Index
FSUTX
Fidelity Select Utilities
US Sector Stock Index
FSVLX
Fidelity Select Consumer Fi-
nance
US Sector Stock Index
FTSE-
London FT-SE 100 Ix
National Stock Index
GBP-
US / UK Foreign Exchange
Rate Ix
UUP
Invesco DB US Dollar Index
Bullish Fund
Forex
VASVX
Vanguard Selected Value Fund Stock Index (US)
VBISX
Vanguard Short Term Bond In-
dex
US Bonds -Gvmnt
VEIEX
Vanguard Emerging Market
Stock Index INV
Regional Stock Index
VEXMX
Vanguard Extended Market In-
dex Fund
Global Stock Index
VEXPX
Vanguard Explorer Fund INV Stock Index (US)
VFICX
Vanguard Int. Term Invest-
ment Grade Bond Fund
US Bonds -Corp Invst
VFIIX
Vanguard GNMA INV
US Bonds -Gvmnt
VFISX
Vanguard Short-Term Treasury
INV
US Bonds -Gvmnt
VFITX
Vanguard Intermediate Term
Treasury Fund
US Bonds -Gvmnt
VFSTX
Vanguard Short-Term INV
Growth Incm INV
US Bonds -Corp Invst
VGENX
Vanguard Energy INV
National Stock Index
VGHCX
Vanguard Health Care INV
National Stock Index
VGPMX
Vanguard Global Capital Cy-
cles Fund
Stock Index (US)
VGSIX
Vanguard REIT Index INV
Others
VINEX
Vanguard International Ex-
plorer Fund
Global Stock Index
VNQ
Vanguard Real Estate Index
Fund ETF Shares
Others
VTRIX
Vanguard International Value
Fund
Global Stock Index
VTSMX
Vanguard Total Stock Markets
Index INV
Global Stock Index
VUSTX
Vanguard Long-Term Treasury
INV
US Bonds -Gvmnt
VWEHX
Vanguard Hi-Yield Corporate
INV
US Bonds -Corp HY
VWESX
Vanguard Long-Term INV
Growth Income INV
US Bonds -Corp Invst
VWIGX
Vanguard
International
Growth INV
Others
VWINX
Vanguard Wellesley Income
INV
US Bonds -Gvmnt
VWO
Vanguard FTSE Emerging
Markets Index Fund ETF
Shares
Global Stock Index
VWUSX
Vanguard US Growth INV
Stock Index (US)
VXF
Vanguard Extended Market In-
dex Fund ETF Shares
Global Stock Index
WDG-X
MSCI Germany iShr Ix
National Stock Index
WPB-X
MSCI Canada iShr Ix
National Stock Index
XLB
StateSt ETF Materials Select
Sector SPDR
XLP
StateSt ETF Consumer Staples
SelSctrSPDR
US Sector Stock Index
XLU
StateSt ETF Utilities Select
Sector SPDR
US Sector Stock Index
XLV
StateSt ETF Health Care Sel
Sector SPDR
US Sector Stock Index
XLY
StateSt ETF Consumer Dis-
cretnrySlSctSPDR
US Sector Stock Index
XLC
StateSt ETF Communication
Service SlSctSPDR
US Sector Stock Index
XLRE
StateSt ETF Real Estate SlSct-
SPDR
US Sector Stock Index
VOX
Vanguard Communication Ser-
vices Index Fund ETF Shares
Regional Stock Index
SPY
StateSt ETF SPDR S&P 500 Stock Index (US)
ST-AG
Silver Spot
Commodities
ST-AU
Gold Spot
Commodities
ST-BC
Brent Crude Spot
Commodities
ST-CA
Cocoa Spot
Commodities
ST-CF
Coffee Bushel Spot
Commodities
ST-CO
Crude Oil Spot
Commodities
ST-CT
Cotton Bushel Spot
Commodities
ST-CU
Copper Spot
Commodities
ST-HO
Heating Oil Spot
Commodities
ST-NG
Natural Gas Spot
Commodities
ST-PD
Palladium Spot
Commodities
ST-PL
Platinum Spot
Commodities
WTI-B
Blmbrg WTI Crude Oil Sub Ix
Total Return
Commodities
VIPSX
Vanguard Inflation-Protected
Securities Fund Investor Shares
US Bonds -Gvmnt
Country Funds
Country index fund
Forex
Foreign Exchange
Commodities
Commodities tracking fund
Real Estate
Real estate fund
Other
Other fund or index
For interested reader, this special kind of forecasting is known as asset pricing[28,29]. In this setup we are interested in modelizing the relationship between systematic risk factor and expected excess return of assets over the market and ultimately build an optimal portfolio. This goes out of the scope of this paper. We focus on forecasting any TS, regardless if is an asset, solely based on its historical values.
N-BEATS was not part of the M4 competition, and attained state-of-the-art results on M4 benchmark ex post facto. N-BEATS was the core part of the second-entry solution in
M5 competition[3].
This is because the procedure to train our model is itteration-based and not epochbased. The term "epochs" refers to the number of passes of the entire training dataset our models has seen. Our approach differs in that we itterate on batches of TS sampled and sliced randomly at different cut-off points.
Prices are at the rate calculated using their cost estimator on 04-08-2021, employing their "AI Platform" configuration with a single NVIDA P100 GPU
https://github.com/Mcompetitions/M4-methods
https://robjhyndman.com/data/27-3-Athanasopoulos1.zip 10 https://archive.ics.uci.edu/ml/datasets/PEMS-SF
Appendix B.1. Forecasting CombinationForecast combination with N-BEATS(P) and N-BEATS was done as follows: to produce a forecast from the ensemble, all forecasts of ensemble members were considered and the median was computed for every forecast for all time t per TS forecast. When the forecast horizon of the model was shorter than the forecast horizon of the target dataset, we iteratively appended the forecast to the original TS signal and based our forecasts upon the transformed signal until the total forecast was longer than or equal to the forecast horizon of the target dataset. In cases where the forecast produced was longer than the forecast horizon, we truncated the forecast to keep only the H first observations.
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Dataset Fig. A.4 illustrates the difference between the statistical properties of all 6 datasets, employing the same set of TS features used in the FFORMA model. A Appendix, Appendix A. Dataset Fig. A.4 illustrates the difference between the statistical properties of all 6 datasets, employing the same set of TS features used in the FFORMA model
As an example of the observations that can be drawn from this figure: it can bee seen that both the Electricy ML-10 M2EY-BofAML US Corporate AAA Semi-Annual Yiel US Bonds. Corp Invst M3OA-BofAML US Corporate BBB Semi-Annual Yiel US Bonds -Corp Invst M4EY-BofAML US Corporate. 2] for a detailed overview of the 42 features used and their interpretationWe refer the reader to table.A.5 and [2] for a detailed overview of the 42 features used and their interpretation. As an example of the observations that can be drawn from this figure: it can bee seen that both the Electricy ML-10 M2EY- BofAML US Corporate AAA Semi-Annual Yiel US Bonds -Corp Invst M3OA- BofAML US Corporate BBB Semi-Annual Yiel US Bonds -Corp Invst M4EY- BofAML US Corporate 1-3
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| Knowledge Graph Question Answering (KGQA) has become a prominent area in natural language processing due to the emergence of large-scale Knowledge Graphs (KGs). Recently Neural Machine Translation based approaches are gaining momentum that translates natural language queries to structured query languages thereby solving the KGQA task. However, most of these methods struggle with out-of-vocabulary words where test entities and relations are not seen during training time. In this work, we propose a modular two-stage neural architecture to solve the KGQA task. The first stage generates a sketch of the target SPARQL called SPARQL silhouette for the input question. This comprises of (1) Noise simulator to facilitate out-of-vocabulary words and to reduce vocabulary size (2) seq2seq model for text to SPARQL silhouette generation. The second stage is a Neural Graph Search Module. SPARQL silhouette generated in the first stage is distilled in the second stage by substituting precise relation in the predicted structure. We simulate ideal and realistic scenarios by designing a noise simulator. Experimental results show that the quality of generated SPARQL silhouette in the first stage is outstanding for the ideal scenarios but for realistic scenarios (i.e. noisy linker), the quality of the resulting SPARQL silhouette drops drastically. However, our neural graph search module recovers it considerably. We show that our method can achieve reasonable performance improving the state-of-art by a margin of 3.72% F1 for the LC-QuAD-1 dataset. We believe, our proposed approach is novel and will lead to dynamic KGQA solutions that are suited for practical applications. | null | [
"https://arxiv.org/pdf/2109.09475v1.pdf"
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| 237,572,091 | 2109.09475 | 294311422b802a361c125fbb73f434e274b0cc9e |
Knowledge Graph Question Answering via SPARQL Silhouette Generation
Sukannya Purkayastha [email protected]
TCS Research
Saswati Dana
IBM Research
Dinesh Garg [email protected]
IBM Research
Dinesh Khandelwal
IBM Research
G P Shrivatsa Bhargav
IBM Research
Knowledge Graph Question Answering via SPARQL Silhouette Generation
Knowledge Graph Question Answering (KGQA) has become a prominent area in natural language processing due to the emergence of large-scale Knowledge Graphs (KGs). Recently Neural Machine Translation based approaches are gaining momentum that translates natural language queries to structured query languages thereby solving the KGQA task. However, most of these methods struggle with out-of-vocabulary words where test entities and relations are not seen during training time. In this work, we propose a modular two-stage neural architecture to solve the KGQA task. The first stage generates a sketch of the target SPARQL called SPARQL silhouette for the input question. This comprises of (1) Noise simulator to facilitate out-of-vocabulary words and to reduce vocabulary size (2) seq2seq model for text to SPARQL silhouette generation. The second stage is a Neural Graph Search Module. SPARQL silhouette generated in the first stage is distilled in the second stage by substituting precise relation in the predicted structure. We simulate ideal and realistic scenarios by designing a noise simulator. Experimental results show that the quality of generated SPARQL silhouette in the first stage is outstanding for the ideal scenarios but for realistic scenarios (i.e. noisy linker), the quality of the resulting SPARQL silhouette drops drastically. However, our neural graph search module recovers it considerably. We show that our method can achieve reasonable performance improving the state-of-art by a margin of 3.72% F1 for the LC-QuAD-1 dataset. We believe, our proposed approach is novel and will lead to dynamic KGQA solutions that are suited for practical applications.
Introduction
In recent years, there is an increasing interest in the Knowledge Graph Question Answering (KGQA) (Diefenbach et al. 2018) task in Natural Language Processing community due to its applicability in various real life and practical business applications. Availability of large-scale knowledge graphs, such as Freebase (Bollacker et al. 2008), DBpedia (Lehmann et al. 2015), YAGO (Pellissier Tanon, Weikum, and Suchanek 2020), NELL (Mitchell et al. 2015), and Google's Knowledge Graph (Steiner et al. 2012) made this possible. The KGQA task requires a system to answer a natural language question leveraging facts present in a given KB. Mainly two streams of approaches are followed by * Work done while the author was an intern at IBM Research KGQA community (1) semantic parse based (2) information extraction based. In Semantic parsed based approach, the task can be accomplished by translating the natural language question into a structured query languages or logic form such as SPARQL, SQL, λ-DCS (Liang 2013), CCG (Zettlemoyer and Collins 2005), etc. Generated query is then executed over the given KG to finally arrive to the answer. Information extraction based approaches are primarily concerned with final answer but not intermediate logic form. In semantic parsed based approaches, main challenges in obtaining correct form of logic/SPARQL is getting the right structure along with specific entities and relations in the knowledge graph. Performance of existing off-the-shelf entity-relation linkers is not encouraging enough in KGQA dataset to adapt them in this task. Therefore, most of the state-of-art systems follow Pipeline-based approaches with inbuilt entityrelation linker. These pipeline based approaches Kapanipathi et al. 2020;Liang et al. 2021) are a popular way to handle questions that requires multiple entities and relations to answer a given question Usbeck et al. 2017;Trivedi et al. 2017). The error introduced by inbuilt linkers is a major bottleneck and reduces the overall pipeline performance.
With progress of neural network models, KGQA community aspires to perform the task by leveraging neural network. However to do so, we need large-scale training data which is a challenge. These challenges limit the applicability of Deep Neural Network (DNN) based approaches on KGQA task. Existing neural approaches however, are currently limited to answering questions that require single relation from KG (He and Golub 2016;Dai, Li, and Xu 2016;Hao et al. 2017;Lukovnikov, Fischer, and Lehmann 2019;Lukovnikov et al. 2017). Some neural approaches (Maheshwari et al. 2019) assume a noise-free entity linker or they mainly focus on relation linking sub-task (Yu et al. 2017). Hao et al. (2017) follows information extraction based approaches and leverages universal KG information to arrive at the final answer more accurately. Cheng and Lapata (2018) develops a system based on sequence-to-tree model where logic is in the latent form and supervision is in the form of final answer entity. Advances of translating natural language query to structured languages using NMT models (Yin, Gromann, and Rudolph 2021;Cai et al. 2017) is emerging in recent years. In case of KGQA task, these NMT based Figure 1: A high level view of our proposed two stage neural architecture for KGQA. methods suffer from out-of-vocabulary words and there is no explicit provision to handle unseen entities/relations at testtime. However, we believe that the core challenges involved in performing KGQA task via NMT methods is not explored fully and there is a significant scope for further investigation. Motivated by these observations, in this paper, we propose a novel two-stage neural architecture (see Figure 1) to answer KG based questions that need multiple entities and relations to answer them.
The main contributions of this work are as follows:
1. In Stage-I, a sketch of SPARQL called SPARQL silhouette is generated for input question. A noise simulator is designed in this module to devise three different kinds of masking strategies to simulate varying levels of noise introduced in entity/relation linking process. 2. In Stage-II, a simple and novel BERT based neural graph search module (NGS) is proposed which corrects predicted relations in the SPARQL silhouette. Purpose of having this module is to overcome performance limitation that arises due to the weaknesses of entity/relation linker present in the first stage. 3. An ideal entity/relation linker having 100% F 1 score is simulated and shown that the quality of generated SPARQL silhouette is high -83.08% F 1 for LC-QuAD-1 and 55.3% Macro F 1 QALD for QALD-9. 4. Realistic scenario is simulated and shown that as F 1 of the linker goes down, quality of the resulting SPARQL silhouette drops drastically. Finally, integrating Stage-II module with Stage-I boosts the performance significantly and improves the SOTA by a margin of 3.72% F 1 for LC-QuAD-1 dataset.
Related Work
In the beginning KGQA task was centralized in two directions either semantic parsed based (Unger et al. 2012;Berant et al. 2013;Reddy, Lapata, and Steedman 2014;Bast and Haussmann 2015;Abujabal et al. 2017) approaches or information retrieval based (Bast and Haussmann 2015;Yao and Van Durme 2014;Dong et al. 2015). Most of the earlier semantic parsed based approaches used handcrafted rules. We limit our discussion only to end-to-end neural approaches.
Deep Neural Network Based Approaches
With availability of large-scale datasets, DNN based techniques have made huge improvements in machine reading comprehension tasks (Nguyen et al. 2016;Rajpurkar et al. 2016;Joshi et al. 2017). This motivated NLP researchers to apply DNN technique to translate natural language question to structure database query languages (Yu et al. 2018;Wang et al. 2020;Hosu et al. 2018;Choi et al. 2021). For KGQA, datasets like SimpleQuestions (Bordes et al. 2015;He and Golub 2016), where only one entity and one relation are required to answer a question, performance of DNN models is already approaching the upper bound (Petrochuk and Zettlemoyer 2018 comparison of our approach with the neural network based previous approaches. To the best of our knowledge, our work is the first of its kind of solving KGQA task which considers multiple relations and used NMT method that handle outof-vocabulary situation by designing noise simulator with masking strategy.
The KGQA Task
In KGQA, we are given a Knowledge Graph G comprising of an entity set E, a relation set R, and a set of knowledge facts F. The knowledge facts are expressed in the form of triples; F = { e s , r, e o } ⊆ E × R × E, where e s ∈ E is known as subject or head entity, e o ∈ E is known as object or tail entity, and r is a relation which connects these two entities. These entities (relations) form the nodes (edges) of the KG. The task now is to identify the subset of entities from E that constitute the answer of a given question Q in the natural language form. The most common family of approaches for the KGQA task is semantic parsing where, the given question Q is first translated into an SPARQL query S which is then executed over the KG so as to get the answer set. For developing a system to convert a question into the corresponding SPARQL query, we are given a set of train-
ing data {Q i , S i , A i } n i=1 ,
where Q i is a question (in natural language text), S i is the SPARQL query, and A i is the answer set obtained by executing S i on G. The proposed system consists of two stage neural modules. In the Stage-I, seq2seq module generates a SPARQL silhoutte with specific entities. Relations predicted in this module are corrected by the Stage-II, neural graph search module.
Stage-I: Seq2Seq Model
Sequence-to-sequence model have achieved state-of-the-art performance in machine translation (Yin and Neubig 2017) task. Encoder-decoder architecture of seq2seq models can vary from RNN, CNN based to transformer models. Prior research shows (Yin, Gromann, and Rudolph 2021) that CNN based seq2seq model performs best among these for translating natural language to SPARQL query. Our preliminary experimental results were consistent with this fact since the CNN based model performed the best. Hence, we moved ahead with the CNN based seq2seq model as our base model for Stage I. Figure 2 shows the architecture of Stage-I. An external entity/relation linker is used to detect surface form mentions of the entities/relations in the question text and linking the same to the underlying KG (DBpedia here). We designed a noise simulator for adapting the data to be in necessary format for seq2seq model.
Noise Simulator
Purpose of designing noise simulator is twofold: (i) To simulate varying levels of noise in the entity/relation linking process (ii) To mask mentions and entities/relations in the question text and SPARQL.
[Need for Masking] Masking helps in two ways: (1) handling test entities/relations that are unseen during training (2) reducing vocabulary size as KGs contain a large number of entities and relations. A simple neural seq2seq model which translates natural language question into a SPARQL query will struggle to output some of the entities/relations during test time that are unseen during training time and hence will not be available in the output vocabulary. In the absence of linking and masking, our elementary experiments shows the performance of seq2seq model bo be very low with F1 score 16% which was expected. This outcome is in DBpedia) in validation and test sets that are available in the training set. This suggest that entity and relation linker is must for any neural model. Even if we use only neural models with perfect linkers, our SPARQL vocabulary dictionary will be over growing which becomes difficult to manage. To handle the situation of increasing SPARQL vocabulary dictionary, we need masking/tagging techniques to mask entities and relations.
[Scenario 'A': Noise-Free Linking] In this scenario, we simulate an entity and relation linker that has 100% F 1 . For this, we pick all entities/relations from the gold SPARQL and pretend as if they were the output of the linker (see Figure 7 in appendix). We begin with extracting all the entities and relations from the gold SPARQL using their prefixes (dbr for entities and dbp or dbo for relations). Next, we pick these entities and relations, and align the same with surface-form mention text in the given question. We observe that entities match exactly with substrings in the questions most of the time (e.g. Austin College in Figure 7 of the appendix). For relations, an exact match is not always possible, e.g., a given relation dbo:film is semantically best aligned to word movies in the question. We use pre-trained fastext embeddings (Bojanowski et al. 2017) to represent words and relation and compute cosine similarity between each word in the question and the given relation. The highest-scoring word is considered as the aligned word. After identifying mentions of entities/relations, we mask them in question text and the corresponding gold SPARQL query. This masked pair is subsequently supplied to the seq2seq module as a training example.
[Scenario 'B': Partly Noisy Linking] Purpose of this scenario is to allow partial noise in the entity/relation linking process. For this, we first feed the natural language question into an external entity/relation linker. The linker returns two things: (i) A set of surface form mentions for entities/relations in the question text, and (ii) Linked entities/relations for these mentions. We take linker's output and find intersection of these entities/relations with the entities/relations present in the gold SPARQL. These common entities/relations are masked in the SPARQL query. Also, their corresponding surface forms are masked in the question text. In order to mask the surface forms in the question, we use exact match and string overlap based Jaccard similarity. Figure 8 in appendix illustrates this scenario.
[Scenario 'C': Fully Noisy Linking] Goal here is to simulate a completely realistic scenario where we rely entirely on an external entity/relation linker. For this, we feed input question to the entity/relation linker and get the suggested surface form mentions and linked entities/relations. We mask each of these suggested mentions using exact match and partial match. Corresponding SPARQL query's entities and relations are also masked based on the suggestions. This scenario is depicted in Figure 3.
Convolutional Seq2Seq Model
The pair of masked question and SPARQL query obtained from the noise simulator, under any noise scenario, is fed to a Convolutional Neural Network (CNN) based seq2seq model (Gehring et al. 2017). As shown in Figure 4, this model reads the entire masked question and then predicts the corresponding masked SPARQL query token-by-token in a left-to-right manner. This seq2seq model consists of the following key components.
[Input Embedding Layer] Both encoder and decoder consist of an embedding layer that maps each input token to a point-wise summation of its word embedding and positional embedding. The embedding of each word is initialized randomly. In order to capture the sense of order, the model is provisioned with the positional embedding.
[Convolution + Pooling Layers] The token embeddings obtained from the previous layer are fed to the multiple convolution and pooling layers. Each convolution layer consists of a 1-dimensional convolution followed by Gated Linear Units (GLU) . Residual connections ) are added from input to the output of each convolution layer.
[Multi-
Step Attention] Each decoder layer comprises a convolution layer followed by a multi-step attention layer. This multi-step attention is used to find the attention scores from a particular decoder state to the source tokens. Attention between decoder state d i (after i th layer) of the last token in generated sequence so far and state z j of the j th source element (after last encoder layer) is computed as:
a i j = exp(d i · z j )/ m t=1 exp(d i · z t )
where, m is the number of source elements. The context vector, c i , is now computed as, c i = [ m j=1 a i j (z j +e j )]+d i where, e j is the input embedding for the source element j.
[Output Layer] Finally, output at a particular time step is calculated over all the Z possible tokens, P (z t+1 |z 1 , . . . , z t , X) = sof tmax(W d L + b) where P (z t+1 |·) ∈ R Z , and W , b are trainable parameters. d L is the decoder state of last target element at the last layer L. X is the input sequence.
[Training Loss:] The model is trained using label smoothed cross-entropy loss given by following expression (for single training example) L(θ) = −(1/N ) · N n=1 Z z=1 q(y n = z|y n−1 ) · log P θ (y n = z|y n−1 ) where, N is the number of words in output sequence and y n is the first n tokens of output sequence. P θ (y n = z|y n−1 ) is model's probability to output token z given y n−1 sequence generated so far. The quantity q(y n = z|y n−1 ) is equal to γ if f (y n ) = z and (1 − γ)/(Z − 1) o/w, where γ ∈ [0, 1], γ > 1/Z.
Stage-II: Neural Graph Search Module
While working with LC-QuAD-1 and QALD-9 datasets, our error analysis on output of Stage-I revealed that entity linking performance is reasonably good but the same is not true for relation linking. Existing literature (Wu et al. 2020;Li et al. 2020) also show enough evidences of achieving high performance on the entity linking task, whereas relation linking turns out to be harder due to complexity of natural language. Because of this, we have most of the entities within a SPARQL silhouette generated by Stage-I as correct but the relations are incorrect. Graph search module in Stage-II takes a SPARQL silhouette as input and produces an improved version of the same by replacing incorrect relations(See Figure 10 in appendix for an example). 1 This is a BERT-based module and its architecture is shown in Figure 5. This module works as follows.
1. One-by-one, we consider each triple e s , r, e o in the SPARQL silhouette and try correcting its relation r through this module. Note, in triple e s , r, e o , at least one of the entity must be an existential variable unless it is an rdf:type relation, which we handle separately. We consider this triple for the correction only if the other entity is grounded to some KB entity and that grounded entity could be in subject (or object) position. 2. For each such triple identified in the previous step, we prepare input in the following format: [CLS] Q [SEP] [SUB (or OBJ)] [SEP] e s (or e o ). Here, Q is token sequence of input question text and [SUB (or OBJ)] is special token depending on whether the grounded entity is in subject (or object) position (refer Figure 5a). We also pass grounded entity (e s or e o ) as the last element of this input.
[CLS] and [SEP] are special tokens from BERT vocabulary. 3. We feed above input sequence of tokens into the BERT layer of graph search module. The output is passed through a linear layer followed by a softmax layer. This softmax layer induces a probability score p r for each relation r ∈ R in the given KG. While training, we use the following loss function (given for single example):
= (1 − α) * ( c ) + (α) * ( gs ).
Here, c denotes standard cross entropy loss between predicted probabilities {p r } r∈R and the gold relation. The graph search loss term gs forces the predicted probabilities to be low for all those relations which are invalid relations (in the given KG) for corresponding input entity e s (or e o ) in the input position (subject or object). For this, we assume a uniform probability distribution over all such valid relations and compute its cross entropy loss with {p r } r∈R . α is a hyperparameter. 4. During inference, at softmax layer, we restrict the outputs only to those relations r ∈ R which are valid relation for the input entity as being subject or object. For example, if input grounded entity is e s then we restrict prediction to only those relations r for which e s , r, ?x is a valid triple for some grounding of ?x. In DBpedia same relation can exist in the form of 'dbo' and 'dbp' for a specific entity. In such cases, we always pick the 'dbo' version.
Prediction is made based out of 61623 relations available in DBpedia. 5. If a relation r in a given triple is rdf:type then we handle them little differently. Note, in DBpedia, a triple containing rdf:type relation looks like this ?x, rdf:type, dbo:type where, ?x is a variable and dbo:type is the DBpedia ontology class of the entity ?x. For such triples, we maintain a separate version of the neural graph module (refer Figure 5b). Input to this module is [CLS] Q. We need to predict the corresponding ontology class dbo:type. DBpedia ontology contains 761 classes and hence, in this model, prediction is one of these 761 classes. This module is trained with standard cross-entropy loss. An example of the rdf:type classification would be to predict dbo:Country for the question 'Name the country with currency as Aureus?'.
Experiments
Datasets: We work with two different KGQA datasets based on DBpedia: LC-QuAD-1 (Trivedi et al. 2017) and QALD-9 (Ngomo 2018). LC-QuAD-1 contains 5000 examples and is based on the 04-2016 version of the DBpedia. We split this dataset into 70% training, 10% validation, and 20% test sets (same as the leaderboard). QALD-9 is a multilingual dataset and is based on the 10-2016 version of the DBpedia. Questions in this dataset vary in terms of reason-ing nature (e.g. counting, temporal, superlative, comparative, etc.) and therefore, in terms of the SPARQL aggregation functions as well. This dataset contains 408 training and 150 test examples. We split the training set into 90% training and 10% validation sets.
Evaluation Metric: Performance is evaluated based on the standard precision, recall, F 1 score for KGQA systems. For more detail please refer to . Baseline: We compare our approach with three baselines: WDAqua (Diefenbach et al. 2020), QAmp (Vakulenko et al. 2019) and gAnswer (Zou et al. 2014). WDAqua is a graph based approach where authors first develop SPARQL query based on four predefined patterns. In the second step they rerank generated candidates. QAmp used text similarity and graph structure based on an unsupervised message-passing algorithm. gAnswer is graph data driven approach and generate query graph to represent user intention.
Experimental Setup: 1) Stage-I: We use Falcon (Sakor, Singh, and Vidal 2019) for entity/relation linking and experiment with all 3 noise scenarios. We use fairseq 2 library for implementation of CNN based seq2seq model (Gehring et al. 2017) comprising of 15 layers 3 . and used Nesterov Accelerated Gradient (NAG) optimizer. We experimented with different values of hyperparameters and report results for the values yielding the best performance on the validation set. Details about tuning ranges and optimal values of all these hyperparameters are given in Table 6 and Figure 9 of appendix. We used 2 Tesla v100 GPUs for training seq2seq model.
2) Stage-II: For neural graph search module, we work with a pre-trained BERT-base uncased model. It consists of 12 transformer layers, 12 self-attention heads, and 768 hidden dimension. We used 1 Tesla v100 GPU for training. Table 3 compares the performance of our model with baseline models for the LC-QuAD-1 dataset. The first two rows are top entries in the LC-QuAD-1 leaderboard 4 . The next set of rows show result of our approach. Our results of stage-II are under realistic scenario or full noise setting for entity/relation linking. Table 4 captures the performance of our approach on QALD-9 dataset. The first two rows in Table 4 correspond to a baseline model and a top entry in the QALD-9 challenge (Ngomo 2018 of entity/relation linker impacts the overall performance of KGQA. Further, the performance of Stage-II demonstrates how one can improve the lower bound numbers by adopting our proposed graph search module. For LC-QuAD-1, we gain 11.46% in F 1 in Stage-II whereas, for QALD-9 this gain is only 0.2% in Macro F 1 QALD. For QALD-9 dataset, the numbers in last two rows are same because we have only two questions with rdf:type and their classes belong to YAGO ontology so our model does not predict them. One may also observe that the overall performance after Stage-II improves the respective baseline in case of LC-QuAD-1 dataset but QALD-9 dataset it struggles. Error Analysis: Reason for QALD-9 having low upper bound is its training set size being too small (367). Further analysis reveals that QALD-9 dataset has large variety of SPARQL keywords from a small train set. Figure 6 captures the distribution of SPARQL keywords in QALD-9 dataset (excluding SELECT and DISTINCT keywords as they appear in almost all the questions). From this figure, its clear that number of questions varies from 3 to 37 for each category of SPARQL keywords which is too less for any neural model to learn from. We also trained our model with combining LC-QuAD-1 and QALD-9 dataset in both the stages. But it did not improve the performance of QALD-9 dataset because the nature of the SPARQL is very different in both the datasets. Because of these reasons, unlike LC-QuAD-1, the generated SPARQL silhouette in QALD-9 dataset has errors other than incorrect entity/relation. Therefore, Stage-II offers much smaller gain for QALD-9 relative to LC-QuAD-1. Lastly, Table 5 which demonstrates the performance of Falcon linker on test set for both the datasets rules out the possibility of systematic data bias in terms of entity/relation linking. Though entity linking performance is reasonable for both the datasets, relation linking is consistently substandard for both the datasets. The poor performance of Falcon on relation linking also justifies a substantial gap between upper and lower bounds for both datasets. Anecdotal Examples: Table 7 of appendix shows examples from LC-QuAD-1 test set where our neural graph search module struggles to disambiguate between two very similar looking relations that exist in DBpedia for an entity. Table 8 captures examples from QALD-9 test set where gold SPARQL have an intrinsic structure because the way in which corresponding facts are being captured within DBpedia. This makes it difficult for any KB agnostic techniques (such as seq2seq) to output such structures. Finally, Table 9 shows examples from QALD-9 test set where gold SPARQL comprises infrequent SPARQL keywords making it hard for seq2seq model to learn about them.
Results
Conclusions
We propose a simple sequential two-stage purely neural approach to solve the KGQA task. We demonstrate that, if entity/relation linking tasks are done perfectly, then Stage-I, vanilla seq2seq neural module can produce impressive performance on KGQA task. However, in noisy realistic scenarios, it performs differently. We have proposed a novel Stage-II, a neural graph search module to overcome noise introduced by entity/relation modules. Our approach improves state-of-art performance for LC-QuAD-1 dataset. Though, for QALD-9 dataset due to the small training size and in-trinsic nature of facts in DBpedia, our model struggles to improve state-of-art, we believe, this research demonstrates great potential of pure neural approaches to solve the KGQA task and opens up a new research direction. Figure 7: An illustrative example for Scenario 'A': Noise-Free Linking. To align the surface forms of the entities/relations mentions in the given question text, we used word embedding as it offers higher alignment F 1 . We used Falcon as a linker. To align the surface forms of the entities/relations mentions in the given question text, we used exact match as well as string overlap based Jaccard similarity with a threshold of 0.7. We used Falcon as a linker.
Evaluation Metric
1) Precision, Recall, and F 1 for Single Question: For single question Q, we compute precision P , recall R, and F 1 using the set of gold answer entities S g and predicted answer entities S p . While computing these metrics, we handle boundary cases as follows. If S g = S p = ∅ then we take P = R = F 1 = 1. If only S g = ∅ then we take R = F 1 = 0.
2) Macro Precision, Macro Recall, Macro F 1 , and Macro F 1 QALD: These metrics are defined for the whole dataset. For this, we first compute P , R, and F 1 at individual question level and average of these numbers across entire dataset gives us the macro version of these metrics. For F 1 , if use the boundary condition of having P = 1 when S p = ∅, S g = ∅ then such a Macro F 1 is called as Macro F 1 QALD as per Ngomo (2018). But if we instead use P = 0 then it is called Macro F 1 .
3) Precision, Recall, and F 1 for the whole set: For whole set, P and R are same as macro version of these metrics. F 1 , however, is computed by taking Harmonic mean of these P and R. The reported metrics for the LC-QuAD-1 dataset were computed in this manner. 4) Answer Match (AM): For a question Q, when executing the predicted SPARQL, if we have S p = S g then we say AM=1 otherwise AM=0. Table 7 shows examples from LC-QuAD-1 test set where our neural graph search module is unable to disambiguate between two very similar looking relations that exist in DBpedia for an entity. Table 8 captures examples from QALD-9 test set where gold SPARQL have a peculiar structure just because the way in which corresponding facts are being captured within in the DBpedia and that makes it almost impossible for any KB agnostic techniques (such as seq2seq) to output such structures. The first two rows of Table 7 shows examples where gold SPARQL queries of two very similar questions is quite different. Even though Falcon picks correct entities, our SPARQL silhouette struggle to yield two differently structured SPARQL queries for two very similar looking natural language questions. Third row of the table contains some entities/relations containing dct, dbc, etc. Falcon linker does not tag these kinds of entity/relation, so we miss out correctly predicting the sketch in Stage-I and so in Stage-II as well. Table 9 shows various examples from QALD-9 test set where we miss predicting the correct sketch of SPARQL be- α for LC-QuAD-1 [1 × 10 −1 , 4 × 10 −1 , 6 × 10 −1 , 7 × 10 −1 ] 4 × 10 −1 α for QALD-9 [1 × 10 −1 , 4 × 10 −1 , 6 × 10 −1 , 7 × 10 −1 ] 6 × 10 −1 placeOfDeath deathPlace Name the rivers who originate from Essex? mouthPlace sourceRegion Name the artist who made Dream Dancing and is often associated with Joe Pass. associatedBand associatedMusicalArtist
Anecdotal Examples
What is used as money for French Southern and Antarctic Lands is also the product of the Karafarin Bank ?
product products Table 8: Anecdotal examples from QALD-9 test set where gold SPARQL have a peculiar structure just because the specific way in which the corresponding facts are present in the DBpedia.
Figure 2 :
2A detailed architecture of Stage-I.
Figure 3 :
3An illustrative example for Scenario 'C'.
Figure 5 :
5Architecture of neural graph search module. (a) Relation Classifier. This module predicts relation for a given entity (b) Ontology Type Classifier. This module predicts rdf:type ontology class.
Figure 8 :
8An illustrative example for Scenario 'B': Partly Noisy Linking.
cause of very few number of such examples present in the training set. These are examples where SPARQL contains infrequent keywords such as GROUP BY, UNION,
Figure 10 :
10An example of input to the neural graph search module. the person born who died in Bryn Mawr Hospital?
Table 2 :
2% of the entities and relations in val and test sets that are available within train set's gold SPARQLs.obvious given the statistics inTable 2which captures percentage of entities and relations (i.e. properties and ontology Who all are known to play the Gibson Guitar Corporation?CNN-based
Seq2Seq Model
Figure 4: A CNN-based Seq2Seq model for KGQA. We have assumed noise-free linking scenario here.Attention
Scores
context vector
Linear Layer
encoder states
decoder states
0.1
0.01
0.5
0.1
0.1
SELECT
DISTINCT
var_uri
0
0
Natural Language
Question
Input token seq
Output sequence
generated so far
Next
predicted
token
Initial (token + position)
embeddings
(Convolution + pooling)
layer #1
(Convolution + pooling)
layer #15
Output vocabulary
BERT
CLS Question SEP
SUB
SEP
Stanley_
Kubrick
Linear Layer
Softmax Layer
, ∀ ∈ ℛ(
,
/ )
BERT
Question
Linear Layer
Softmax Layer
, ∀ ∈
(a)
(b)
director
How many movies did
Stanley Kubrick direct ?
Name the country
with currency as
Aureus.
dbo:Country
Stage-I under No Noise linking becomes an upper bound on the performance of seq2seq model. This means seq2seq model can achieve upto 83.08% F 1 for LC-QuAD-1 and 55.3% Macro F 1 QALD for QALD-9 dataset if the entity/relation linker were to be 100% correct. The gap between the performance of No Noise linking (upper bound) and Full Noise linking (lower bound) illustrates how the performance). The next set of rows show performance of
our model.
Insights: From Tables 3 and 4, one can observe that per-
formance of Model
Type
Model
Name
AM Prec. Recall
F 1
Baseline
WDAqua
-22.00 38.00 28.00
QAmp
-25.00 50.00 33.33
Stage-I
(Ours)
No Noise 82.88 83.11 83.04 83.08
Part Noise 41.34 42.40 42.26 42.33
Full Noise 24.92 25.54 25.64 25.59
Stage-II
(Ours)
w/o type 30.63 32.17 32.20 32.18
w/ type
34.83 37.03 37.06 37.05
Table 3 :
3Test set performance on LC-QuAD-1 dataset.Model
Type
Model
Name
AM
Mac.
Prec.
Mac.
Rec.
Mac.
F 1
Mac.
F 1
QALD
Baseline
WDAqua -
26.1 26.7 25.0 28.9
gAnswer -
29.3 32.7 29.8 43.0
Stage-I
(Ours)
No Noise 29.9 80.4 42.1 40.9 55.3
Part Noise 13.1 63.9 28.7 22.4 39.6
Full Noise 11.1 82.6 23.0 20.6 36.0
Stage-II
(Ours)
w/o type 15.3 59.4 26.1 23.3 36.2
w/ type
15.3 59.4 26.1 23.3 36.2
Table 4 :
4Test set performance on QALD-9 dataset. Here Mac. means Macro and Rec. means Recall.
Table 5 :
5Falconperformance on entity (E) and relation (R)
linking on test sets.
[ Q ]
QName the mascot of Austin College? Entity and Relation Masking [Q m ] name the mascot of <e0> ? [S m ] SELECT var_uri WHERE brack_open <e0> <dbp_mascot> var_uri brack_close[S] SELECT ?uri WHERE
{ dbr:Austin_College dbp:mascot ?uri }
Entity and Relation Linker
Entity and Relation Extraction
dbr:Austin_College
dbp: name
dbo:mascot
dbr:Austin_College
dbp:mascot
Intersection
dbr:Austin_College
Table 6 :
6Tuning range and the final chosen best values of various hyperparameters. η means learning rate and b means batch size.Figure 9: Change in validation set accuracy with hyperparameter α Which architect of Marine Corps Air Station Kaneohe Bay was also tenant of New Sanno hotel [SEP] SUB [SEP] Marine_Corps_Air_Station_Kaneohe_Bay Input for Stage-II for the first triple in SPARQL Silhouette dbo: architect0.2
0.4
0.6
78
80
82
84
86
88
90
Validation Accuracy
LC-QuAD-1
0.2
0.4
0.6
75.4
75.6
75.8
76.0
76.2
76.4
QALD-9
SELECT DISTINCT ?uri WHERE
{
dbr:Marine_Corps_Air_Station_Kaneohe_Bay
dbo:tenant ?uri .
dbr:New_Sanno_Hotel dbo:architect ?uri
}
Which architect of
Marine Corps Air
Station Kaneohe Bay
was also tenant of
New Sanno hotel?
Input Question
SPARQL Silhouette
Stage-I
Seq2Seq Model
Stage-II
Neural Graph
Search Module
Corrected Relation
Table 7 :
7Anecdotal examples from LC-QuAD-1 test set where graph search module is unable to disambiguate between two closely related relations (gold and predicted) that are available for the highlighted entities in DBpedia. SELECT DISTINCT ?uri WHERE { ?uri <http://xmlns.com/foaf/0.1/nick> "Rodzilla"@en } SELECT DISTINCT ?uri WHERE { dbr:Rodzilla dbo:alias ?uri } Give me all gangsters from the prohibition era. SELECT DISTINCT ?uri WHERE { ?uri dbo:occupation dbr:Gangster ; dct:subject dbc:Prohibition-eragangsters } SELECT DISTINCT ?uri WHERE { ?uri a dbo:Film ; dbo:time dbr:Gangsters of the Frontier }Question
Gold SPARQL
SPARQL silhouette
Who was called Scar-
face?
SELECT ?uri WHERE
{
?uri dbo:alias ?alias
FILTER contains(lcase(?alias),
"scarface")
}
SELECT DISTINCT ?uri WHERE
{
dbr:Scarface dbo:alias ?uri
}
Who
was
called
Rodzilla?
It is easy to extend this idea and perform an iterative graph searching when entity linker performance is also low.
https://github.com/pytorch/fairseq 3 We will release our code after the review period. 4 http://lc-quad.sda.tech/lcquad1.0.html
Table 9: Anecdotal examples from QALD-9 test set where gold SPARQL comprises infrequent SPARQL keywords. The corresponding SPARQL Silhouette predicted by our Stage-I is also shown for these examples.
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"Matryoshka Policy Gradient for Entropy-Regularized RL: Convergence and Global Optimality",
"Matryoshka Policy Gradient for Entropy-Regularized RL: Convergence and Global Optimality"
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"François G Ged \nChair of Statistical Field Theory\nÉcole Polytechnique Fédérale de Lausanne\nLausanneSwitzerland\n",
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| A novel Policy Gradient (PG) algorithm, called Matryoshka Policy Gradient (MPG), is introduced and studied, in the context of max-entropy reinforcement learning, where an agent aims at maximizing entropy bonuses additional to its cumulative rewards. MPG differs from standard PG in that it trains a sequence of policies to learn finite horizon tasks simultaneously, instead of a single policy for the single standard objective. For softmax policies, we prove convergence of MPG and global optimality of the limit by showing that the only critical point of the MPG objective is the optimal policy; these results hold true even in the case of continuous compact state space. MPG is intuitive, theoretically sound and we furthermore show that the optimal policy of the standard maxentropy objective can be approximated arbitrarily well by the optimal policy of the MPG framework. Finally, we justify that MPG is well suited when the policies are parametrized with neural networks and we provide an simple criterion to verify the global optimality of the policy at convergence. As a proof of concept, we evaluate numerically MPG on standard test benchmarks. | 10.48550/arxiv.2303.12785 | [
"https://export.arxiv.org/pdf/2303.12785v1.pdf"
]
| 257,663,551 | 2303.12785 | 299ea87435c8a9f0edede2c35a71b19aaef9fa99 |
Matryoshka Policy Gradient for Entropy-Regularized RL: Convergence and Global Optimality
François G Ged
Chair of Statistical Field Theory
École Polytechnique Fédérale de Lausanne
LausanneSwitzerland
Maria Han Veiga
Department of Mathematics
University of Michigan
USA
Michigan Institute for Data Science
University of Michigan
USA
Matryoshka Policy Gradient for Entropy-Regularized RL: Convergence and Global Optimality
A novel Policy Gradient (PG) algorithm, called Matryoshka Policy Gradient (MPG), is introduced and studied, in the context of max-entropy reinforcement learning, where an agent aims at maximizing entropy bonuses additional to its cumulative rewards. MPG differs from standard PG in that it trains a sequence of policies to learn finite horizon tasks simultaneously, instead of a single policy for the single standard objective. For softmax policies, we prove convergence of MPG and global optimality of the limit by showing that the only critical point of the MPG objective is the optimal policy; these results hold true even in the case of continuous compact state space. MPG is intuitive, theoretically sound and we furthermore show that the optimal policy of the standard maxentropy objective can be approximated arbitrarily well by the optimal policy of the MPG framework. Finally, we justify that MPG is well suited when the policies are parametrized with neural networks and we provide an simple criterion to verify the global optimality of the policy at convergence. As a proof of concept, we evaluate numerically MPG on standard test benchmarks.
Introduction
Policy gradient and max-entropy reinforcement learning
Reinforcement Learning (RL) tasks can be informally summarized as follows: sequentially, an agent is located at a given state s, takes an action a, receives a reward r(a, s) and moves to a next state s ∼ p(s, a, ·), where p is a transition probability kernel. The agent thus seeks to maximize the cumulative rewards from its interactions with the environment, that is, it optimizes its policy π by reinforcing decisions that led to high rewards, where π(a|s) is the probability for the agent to take action a while at state s.
Policy Gradient (PG) methods are model-free algorithms that aim at solving such RL tasks; model-free refers to the fact that the agent tries to learn (i.e. improve its policy's performance) without learning the dynamics of the environment governed by p, nor the reward function r. Though their origins in RL can be dated from several decades ago with the algorithm REINFORCE [37], the name Policy Gradient appearing only in 2000 in [34], they recently regained interest thanks to many remarkable achievements, to name a few: in continuous control [26,31,32] and natural language processing with GPT-3 [8] 1 . See the blog post [36] that lists important PG methods and provides a concise introduction to each of them.
Max-entropy RL. More generally, PG methods are considered more suitable for large (possibly continuous) state and action spaces than other nonetheless important methods such as Q-learning and its variations. However, for large spaces, the exploitation-exploration dilemma becomes more challenging. In order to enhance exploration, it has become standard to use a regularization to the objective, as in max-entropy RL [28,29,30], where the agent maximizes the sum of its rewards plus a bonus for the entropy of its policy 2 . Not only does max-entropy RL boost exploration, it also yields an optimal policy that is stochastic, in the form of a Boltzmann measure, such that the agent keeps taking actions at random while maximizing the regularized objective. This is sometimes preferable than deterministic policies. In particular, [16] shows that the max-entropy RL optimal policy is robust to adversarial change of the reward function (their Theorem 4.1) and transition probabilities (their Theorem 4.2); see also references therein for more details on that topic. Finally, max-entropy RL is appealing from a theoretical perspective. For example, soft Q-learning, introduced in [19] (see also [21,20] for implementations of soft Q-learning with an actor-critic scheme), strongly resembles PG in max-entropy RL [30]; max-entropy RL has also been linked to variational inference in [24].
Other appealing features of max-entropy RL are discussed in [15] and references therein.
Convergence guarantees of PG. The empirical successes motivated the RL community to attempt building the solid theory for PG methods that has been lacking. Indeed, besides the well-known Policy Gradient Theorem (see Chapter 13 in [33]) that can imply convergence of PG (provided good learning rate and other assumptions), for many years, not much more was known about the global convergence of PG (i.e. convergence to an optimal policy) until recently. Despite the numerous gaps that remain, some important progress have already been made. In particular, the global convergence of PG methods has been studied and proved in specific settings, see for instance [17,1,5,27,38,39,9,13,35,2,6]. Convergence guarantees often come with convergence rates (with or without perfect gradient estimates). Though strengthening the trust in PG methods for practical tasks, most of the theoretical guarantees that have been obtained in the literature so far require rather restrictive assumptions, and often assume that the action-state space of the MDP is finite (but not always, e.g. [2] addresses continuous action-state space for neural policies in the mean-field regime). In particular, [25] shows that many convergence rates that have been obtained in the literature ignore some parameters such as the size of the state space. After making explicit the dependency of the bounds on these parameters, they manage to construct environments where the rates blow up and convergence takes super-exponential time.
Contributions
We consider parametric policies constructed as the softmax of linear models. The main contributions of this work are:
(i) We reshape the max-RL objective function and define a new algorithm (Equation (10)), named Matryoshka Policy Gradient (MPG).
(ii) After showing a policy gradient theorem (Theorem 2) that ensures convergence of MPG, we establish global convergence under few mild assumptions (Theorem 3); in particular, we obtain global convergence for continuous compact state space.
(iii) The optimal policy for the standard objective can be approximated arbitrarily well by the optimal policy of the MPG objective (Proposition 2).
(iv) In the case where the policy is parametrized as the softmax of a (deep) neural network's output, we provide a simple criterion on the so-called Neural Tangent Kernel of the neural network at the end of training, that is sufficient to guarantee global optimality of the limit (Corollary 1).
(v) Numerically, we successfully train agents on standard simple tasks without relying on RL tricks (see Section 4).
The techniques employed in the above cited papers to study global convergence often come from non-convex optimization; for example, if one shows that the so-called gradient domination property is satisfied, one can then show global convergence and deduce the rate of convergence, see e.g. Section 4 in [1]. Though MPG is very intuitive, the MPG objective is still non-convex. Nevertheless, we show global convergence of MPG without relying on standard techniques: for softmax parametrizations, the only critical point of the MPG objective is the optimal policy. This makes the proof very direct and adapted to continuous compact state space, contributing to the theoretical attractiveness of MPG. On a more practical point of view, besides contribution (iv), it turns out that MPG is well suited for policies parametrized by a neural network. Indeed, MPG relies on the fact that the agent's policy learns to optimize the cumulative rewards for several horizons. One assumption to ensure global convergence of MPG is that the policies for different horizons do not share any parameters. However, it is intuitive that optimizing for the next 50 rewards or optimizing for the next 51 rewards should yield similar policies (for smooth enough, relevant environments). Hence, our heuristic is that plugging the horizon as an input in a single neural network encoding all different horizons' policies should enhance rather than harm training. More details on neural policies in Appendix A. 2 In our numerical experiments described in Section 4, we consider standard benchmarks from OpenAI. Namely, the Frozen-Lake game and Pendulum. We obtain successful policies for both benchmarks with a very simple implementation of the MPG algorithm. Rather than competing with the state-of-the-art algorithms, our aim is to provide a proof of concept by showing that successful training can be obtained without using standard RL tricks that are known to improve training performance. Standard tricks are nevertheless very straightforward to apply to MPG and we hope that more general and bigger scale experiments implementing variations of MPG will follow the present work.
Works related to nested policies. The main idea of MPG is to use a sequence of policies (that we call extended policy) π (1) , . . . , π (n) to choose a sequence of actions. Policy π (i) is optimized to maximize the next i regularized rewards and is thus called the i-step policy. This makes the analysis of MPG easier, allowing us to establish global convergence in a "cascade" manner, from small to large horizons. Note that having different policies for different step is standard in dynamic programming [4], where they are called non-stationary policies.
In the standard finite horizon RL objective -also called the episodic case -the length of an episode is a finite random variable. What distinguishes the MPG framework (for which an episode's length is fixed a priori) with the episodic case is that with MPG, we train an extended policy to learn all finite horizon objectives up to a maximal horizon. The idea to look at such finite horizons and let the maximal horizon grow to infinity was already used two decades ago in [14]. The focus of that paper is to exploit that the optimal policies of the finite horizon objectives converge, as the maximal horizon tend to infinity, to the infinite horizon objective; they do not propose (and therefore study) an algorithm that trains an extended policy.
Closer to the idea behind MPG, the construction of the value function in [3] resembles that of the nested structure of extended policies, where a sequence of value and Q-functions are trained to solve finite horizon tasks. Training is done with a Temporal Difference (TD) algorithm and thus differs from MPG, which is a PG method. TD involves bootstrapping, and when it is used offline (off-policy) together with function approximation, it encounters the well-known stability issue called the deadly triad, see [33] Section 11.3. By using different value functions for different horizons, they do not rely on bootstrapping, getting rid of one element of the triad, thus ensuring more stability.
Even more recently, the authors of the preprint [18] exploited a similar idea with an actor-critic method for constrained RL, where the agent aims at maximizing the cumulative rewards while satisfying some given constraints. They investigate convergence rates in both constrained and unconstrained settings. To guarantee convergence, they assume a condition given by their Equation (20). Roughly speaking, this condition is that smaller horizon policies are closer to convergence than larger horizon policies. Therefore, they prove convergence of the training algorithm through a cascade of convergence. The MPG setting that we introduce differs from their work in that the MPG objective is not constrained and includes an entropy regularization. In particular, specific properties of entropy regularization yields that we do not need to assume that smaller horizon policies are closer to convergence than larger horizon policies during training. Not only is convergence guaranteed for MPG, but the global optimality is also established under appropriate assumptions; it is interesting to note that we also make use of a "cascading" argument to ensure global optimality. Note however that, unlike [18], we do not provide convergence rate (though standard techniques should apply and is left for future work).
Single state case
In the single state case (also known as the bandit problem), MPG is identical to the usual PG for the max-entropy objective. Nevertheless, we study this case first as it helps understanding the general state space case.
Definitions
Consider the the case when the state space is restricted to a single state S = {s 0 }. We suppose moreover that the action space A is finite -the argument for A a continuous compact real subset requires some technical adaptation that we believe are unnecessary for our presentation. In the bandit problem, at each discrete time step, an agent takes an action a ∈ A and receives a random reward R(a) with law p rew (·|a) on R. We assume that rewards are bounded, and denote the expected reward map r(a) := E[R(a)]. The agent chooses actions according to a policy, denoted by π, which is a probability distribution on A. If π = δ a for all a ∈ A, we say that π is a stochastic policy and at a given time, we denote by A the random action the agent selects. The objective function J to maximize maps policies π to
J(π) := E π [r(A)] − τ D KL (π||π),
where D KL (p||q) := a∈A p(a) log p(a) q(a) is the Kullback-Leibler divergence, τ > 0 is the temperature parameter governing the strength of the entropy regularization and π is a baseline policy with full support; for example the uniform measure on A gives entropy bonuses shifted by − log |A|. Regularizing with the Kullback-Leibler divergence is thus more general than with entropy bonuses and this is the regularization that we consider in this paper, akin [30]. The optimal policy π * = argmax π J(π) is given by
π * (a) = π(a) exp(r(a)/τ ) E π [exp(r(A)/τ )] ,(1)
see e.g. Equation (2) in [30].
Parametrization of the agent
Applying the softmax function (with baseline measure π and temperature parameter τ ) to a map h : A → R yields the Boltzmann measure a → π(a) exp(h(a)/τ ) a ∈A π(a ) exp(h(a )/τ )
.
In this context, h(a) is called the preference of a of the agent. We choose a linear model for the preferences: let Θ : A×A → R be a symmetric positive semi-definite kernel and denote by H the induced reproducible kernel Hilbert space (RKHS). For all parameters θ ∈ R P , where P denotes the number of parameters of the model, we define
h θ (a) := θ · ψ(a),
where ψ(a) is a feature map associated with the kernel Θ. Then, we define the agent's policy by π θ : a → π(a) exp(h θ (a)/τ ) a ∈A π(a ) exp(h θ (a )/τ ) .
Policy Gradient
Let t ∈ N be the time of training, starting from t = 0 at initialisation. At step t, we denote the parameters by θ t , and we write π t , h t respectively for π θt , h θt . Without loss of generality, we assume that the agent's policy is updated at every step; we thus write A t for the (random) action taken at step t under π t and R t the corresponding reward. Let (F t ) t≥0 be the natural filtration of the agent, that is generated by (π t , A t , R t ) t≥0 . We denote by E πt the conditional expectation given F t under π t . Let η be the learning rate. The ideal PG updates are as follows:
θ t+1 = θ t + ηE πt r(A t ) − D KL (π t ||π)τ ∇ θ π t (A t ) π t (A t ) .(2)
In practice, one can use the estimate
θ t+1 = θ t + η R t − τ log π t (A t ) π(A t ) − v t ∇ θ π t (A t ) π t (A t ) ,
where v t is a baseline, chosen by the user, at time t that does not depend on the action, whose role is to reduce the variance, without changing the expectation of the update. In particular, this estimate is unbiased. Let us consider the ideal PG update (2). Note that θ t+1 − θ t = η∇ θ J(π t ) since we have
∇ θ D KL (π t ||π) = ∇ θ |A| =1 π t (a ) log π t (a ) π(a ) = |A| =1 log π t (a ) π(a ) + 1 ∇ θ π t (a ) = |A| =1 log π t (a ) π(a ) ∇ θ π t (a ),(3)
by a basic property of softmax policies, see (11) in the Appendix.
Convergence
The action space being finite, the uniform convergence of a probability measure on A corresponds to the pointwise convergence. This is the notion of convergence that we consider in the next statement, that is, we say that π t → π ∞ if and only if sup
a∈A |π t (a) − π ∞ (a)| → 0 as t → ∞ Recall that θ t+1 − θ t = η∇ θ J(π t )
; the next result is straightforward.
Lemma 1. There exists η 0 > 0 such that if η < η 0 , then π t converges to some policy π ∞ , as t → ∞.
It is not clear a priori that π ∞ is close to π * . We address this question in the next section.
Global optimality
In order to guarantee global optimality, we will assume (explicitly recalled when needed) that A1. Θ is strictly positive definite, i.e. (Θ(a, a )) a,a ∈A is invertible.
In general, when the kernel Θ is positive-semidefinite, the so-called Mercer's Theorem entails that for all a, a ∈ A,
Θ(a, a ) = |A| i=1 λ i e i (a)e i (a ),
where (e i (a), λ i ) i=1,...,|A| are eigenvector/eigenvalue pairs of (Θ(a, a )) a,a ∈A , {(e i (a)) a∈|A| ; i = 1, . . . , |A|} being an orthonormal basis of R |A| . The eigenvalues are all non-negative and assuming moreover A1., each is also non-zero. We write u ⊥ v to denote that two vectors u and v are orthogonal.
Theorem 1 (Global optimality). Let π ∞ be the policy from Lemma 1 such that π t → π ∞ as t → ∞. Define the vector d ∈ R |A| by
d a := π ∞ (a) log π ∞ π * (a) − D KL (π ∞ ||π * ) , a ∈ A. Then for all i ∈ {1, . . . , |A|}, we have λ i > 0 ⇒ d ⊥ e i .
In particular, if A1. holds, then d = 0, and π ∞ ≡ π * .
As we explained in Introduction, global optimality of PG for finite state space (here |S| = 1) however we will extend the ideas in the proof to the more complex and interesting case of a continuous compact state space, where the environment does not only generate a reward from an action, but also sends the agent to a new state, influencing its future rewards as well. We also note that the theorem extends to the case of A continuous and compact, for which Mercer's theorem applies similarly.
General state space
In this section, we introduce Matryoshka Policy Gradient (MPG) for max-entropy RL, and establish its convergence guarantees.
Definitions
Besides the reward, the environment now includes a state space S such that when the agent chooses an action, the induced reward depends also on the current state of the agent, which then transitions to a next state that depends on the state and action it just left and took.
More formally, the agent evolves according to a Markov Decision Process (MDP) characterized by the tuple (S, A, p, p rew ) 3 . We assume that the action space A is finite and that the state space S ⊂ R d is a compact set. Let s → p(s, a, s ) be the probability (or the density if S is continuous) that the agent moves from s ∈ S to s ∈ S after taking action a ∈ A. When p(s, a, s ) = δ s ,f (s,a) for some f : S × A → S, then we say that the transitions are deterministic. As opposed to the bandit setting, the reward does not solely depend on the action, but also on the current state, and its law is p rew (·|s, a). To ease the presentation, we assume that the rewards are uniformly bounded and for all (s, a) ∈ S × A, we denote by r(a, s) the mean reward after taking action a at state s. All random variables are such that the process is Markovian.
A policy π : A × S → [0 , 1] is a map such that for all s ∈ S, π(·|s) is a probability distribution on A that describes the law of the action taken by the agent at state s. We denote by P the set of policies. For every s ∈ S and π, π ∈ P, we denote by D KL (π||π )(s) = D KL (π(·|s)||π (·|s)).
Henceforth, we assume that S is continuous, the results identically holding true when S is finite. The case where S is infinite countable requires more technical adjustments that we believe are possible but the presentation would not benefit from it.
Henceforth, we assume that
• the MDP is irreducible, in the sense given in (18) in Appendix;
• For any π ∈ P n , the law of S i is absolutely continuous w.r.t. the Lebesgue measure, for all i = 0, · · · , n, with S 0 ∼ Unif(S).
• for all a ∈ A, the reward function s → r(a, s) and the transitions (s, s ) → p(s, a, s ) are continuous.
The second and third items are to avoid unnecessary technicalities.
Extended policies. As opposed to standard policies as studied and used in RL, we construct a chain of policies to define an extended class of policies as follows. Let n ∈ N. Let π = (π (1) , π (2) , . . . , π (n) ), where each π (i) ∈ P and let P n be the set of such policies. We say that the agent follows the (extended) policy π if it chooses its first action according to π (n) , then its second according to π (n−1) , and so on and so forth until it has chosen n actions, the last one according to π (1) . Note that usual policies correspond to {(π, π, π, · · · ); π ∈ P} ⊂ P ∞ , and we write P for the set of policies in the usual sense. As before, we denote by π ∈ P the arbitrary baseline policy (our study would not change if it were an extended policy). We define the n-step value function V (n) π : S → R induced by a policy π ∈ P n as
V (n) π : s → E π n k=0 (R k − τ D KL (π (n−k) ||π)(S k )) S 0 = s ,
where the expectation is along the trajectory of length n sampled under policy π. Note that we have
V (n) π (s) = E π (n) [R 0 ]−τ D KL (π (n) ||π)(s) + E π (n) [V (n−1) π (S 1 )],(4)
where π = (π (1) , . . . , π (n−1) ) ∈ P n−1 , and S 1 ∼ a∈A π (n) (a)p(s, a, ·). It is common to add a discount factor γ ∈ (0 , 1] to the rewards such that the last term of the equation above is multiplied by γ (often chosen close but not equal to 1). As a consequence, the agent favors more the immediate (or quickly obtainable) rewards, since those obtained after 100 steps are then multiplied by γ 100 and so on. In the infinite horizon case (n = ∞), this ensures that the cumulative reward is finite a.s. (provided finite first moment). Our study trivially applies to the case where the rewards are discounted, although we make the choice to continue without including it in the model, in order to ease the expressions. The n-step entropy regularized Q-function induced by π is defined as
Q (n) π : (a, s) → r(a, s) + s ∈S p(s, a, s )V (n−1) π (s ).(5)
Notation: Henceforth, for a policy π ∈ P n , we use the abuse of notation V (i) π for i < n for the i-step value function associated with (π (1) , . . . , π (i) ), and similarly for the Q functions and other quantities of interest, when the context makes it clear which policy is used.
Objective and optimal policy
The standard discounted max-entropy RL objective is defined for π ∈ P by
J(π) := S E πt T k=0 γ k (R k − D KL (π t ||π)(S k )) S 0 = s ν 0 (ds),
where the horizon T ∈ N ∪ {∞} and ν 0 is the initial state distribution.
Before we redefine the objective function for extended policies, let us discuss the initial state distribution. Suppose that the initial state is deterministic. In our case, this would have the effect to optimize our n-step policy only on that state, and then the (n − 1)-step policy only on the states reachable in one step from that state, and so on and so forth. Therefore, to avoid technicalities (such as non-uniqueness of the optimal policy on unreachable states), we assume that the initial state policy is the uniform measure on S (well defined by compactness). We thus define the objective function as follows:
J n (π) := S V (n) π (s)ds.(6)
Since we assume that the rewards are bounded and by compactness of S, the objective function above is itself bounded.
Let L 2 be the space of square-integrable real maps f defined on A × S, that is, such that S a∈A f (a, s) 2 ds.
With a slight abuse of notation, we write π = π in L 2 to say that π (i) = π (i) in L 2 , for all i = 1, . . . , n.
We say that a policy π ∈ P n is optimal if and only if J n (π) ≥ J n (π ) for all π ∈ P n . The existence and unicity of the optimal policy is established by the next proposition, providing in passing its explicit expression. Proposition 1. There exists a unique optimal policy in the L 2 sense, denoted by π * = (π (1) * , . . . , π (n) * ) ∈ P n . The i-step optimal policies, i = 1, . . . , n, can be obtained as follows: for all a ∈ A, s ∈ S,
π (1) * (a|s) = π(a|s) exp(r(a, s)/τ ) E π [exp(r(A, s)/τ )] , π (i+1) * (a|s) = π(a|s) exp Q (i+1) * (a, s)/τ E π exp Q (i+1) * (A, s)/τ , where Q (i+1) *
is a short-hand notation for Q (i+1) π * recursively defined in (5).
Lemma 2. For all s ∈ S and n ≥ 1, it holds that
V (n) * (s) = τ log E π exp Q (n) * (A, s)/τ , where V (0) * (s ) = 0.
Thanks to Lemma 2, we can write more concisely
π (i) * (a|s) = π(a|s) exp Q (i) * (a, s) − V (i) * (s) /τ .(7)
For all n, m ∈ N such that n > m, we define the operator T n,m : (π 1 , . . . , π n ) → (π 1 , . . . , π m ) (8)
P n → P m .
In Proposition 2 below, for all n ∈ N, we denote by π * ,n ∈ P n the optimal policy for J n . We write the standard discounted, infinite horizon entropy-regularized RL objective J ∞ , with discounted factor γ ∈ (0 , 1) (i.e. V π (s) = E[R 0 − D KL (π||π) + γV π (S 1 )] with π ∈ P 1 ). We denote by π * ,∞ the optimal policy of J ∞ .
Proposition 2. We have:
(i) As n → ∞, the policy π (n) * ,n converges to π * ,∞ , in the sense that lim n→∞ S a∈A π (n) * ,n (a|s) − π * ,∞ (a, s) ds = 0.
(ii) for all n, m ∈ N such that n > m, it holds that T n,m (π * ,n ) = π * ,m .
The above Proposition 2 is intuitive: item (i) shows that one can learn the standard discounted entropy-regularized RL objective by incrementally extending the agent's horizon; item (ii) goes the other way and shows that the optimal policy for large horizon is built of smaller horizons policies in a consistent manner.
Policy parametrization
Let Θ (i) : (A × S) 2 → R be a positive-semidefinite kernel. For i ∈ {1, . . . , n}, let θ (i) ∈ R P be the parameters of a linear model h a, s) for action a at state s, that is,
(i) θ (i) : A × S → R, that outputs for all (a, s) ∈ A × S the i-step preference h (i) θ (i) (h (i) θ (i) (a, s) := θ (i) · ψ (i) (a, s),
where ψ (i) : A × S → R P is a feature map associated with the kernel Θ (i) . The i-step policy π (i) θ (i) is defined as the Boltzmann policy according to h (i) , that is, for all (a, s) ∈ A × S,
π (i) θ (i) (a|s) := π(a|s) exp(h (i) θ (i) (a, s)/τ ) a ∈A π(a |s) exp(h (i) θ (i) (a , s)/τ ) .
The gradient of the policy thus reads as
∇π (i) θ (i) (a|s) = π (i) θ (i) (a|s) a ∈A δ a,a − π (i) θ (i) (a |s) ∇h (i) θ (i) (a , s)/τ.(9)
Note that when S is finite with Kronecker delta kernels Θ (i) ((a, s), (a , s )) = δ a,a δ s,s , we retrieve the so-called tabular case. On the other hand, in full generality for S continuous, we could have P = ∞, in particular when Θ (i) is strictly positive-definite. See Appendix A.3 to approximate Θ (i) with finitely many parameters.
Matryoshka Policy Gradient
In our setting, the ideal PG update would be such that θ t+1 − θ t = η∇ θ J n (π t ). We introduce Matryoshka Policy Gradient (MPG), as a practical algorithm that produces unbiased estimates of the gradient (see Theorem 2 below). Suppose that at time t ∈ N of training, the agent starts at a uniformly sampled state S 0 ∈ S. To lighten the notation, we write π
(i) t (·|S 0 ) := π (i) θ (i) t
(·|S 0 ). We assume that the agent samples a trajectory according to the policy π t , defined as follows:
• sample action A 0 according to π (n) t (·|S 0 ), • collect reward R 0 ∼ p rew (·|S 0 , A 0 ) and move to next state S 1 ∼ p(S 0 , A 0 , ·),
• sample action A 1 according to π
(n−1) t (·|S 0 ), • · · · • stop at state S n .
The MPG update is as follows: for i = 1, . . . , n,
θ (i) t+1 = θ (i) t + η n−1 =n−i R − τ log π (n− ) t π (A |S ) ∇ log π (i) t (A n−i |S n−i ) = θ (i) t + ηC i ∇ log π (i) t (A n−i |S n−i ),(10)
where we just introduced the shorthand notation C i . We see that the i-step policy π (i) is updated using the (i − )-step policies.
Matryoshka Policy Gradient Theorem
Recall that we suppose that A is finite and that S is compact. We say that a sequence of policies (π t ) t∈N ⊂ P n converges to π ∈ P n if and only if sup i∈{1,...,n} S sup a∈A π (i) t (a|s) − π (i) (a|s) ds → 0, as n → ∞, where the suprema can be replaced by maxima by compacity.
With PG, the so-called Policy Gradient Theorem (see Section 13.2 in [33]) provides a direct way to guarantee convergence of the algorithm. Our next theorem shows that MPG also satisfies a Policy Gradient Theorem for extended policies.
Theorem 2. With MPG as defined in (10), it holds that E[θ t+1 − θ t ] = η∇ θ J n (π t ). In particular, assuming ideal MPG udpate, that is, θ t+1 − θ t = η∇ θ J n (π t ), there exists η 0 > 0 such that if 0 < η < η 0 , then π t converges as t → ∞ to some π ∞ ∈ P n .
Global optimality
As in the bandit case, the positive-semidefiniteness of Θ (i) for all i = 1, . . . , n implies by Mercer's Theorem that
Θ (i) ((a, s), (a , s )) = j≥1 λ (i) j e (i) j (a, s)e (i) j (a , s ),
where (e Furthermore, {e (i) j ; j ≥ 1} is an orthonormal basis of L 2 . We will specify in our statements when we assume the following:
A2. For all i ≥ 1, Θ (i) is strictly positive definite, i.e. λ (i) j > 0, ∀j ≥ 1.
Henceforth, we write m πt at s. In the theorem below, we assume ideal MPG update, that is θ t+1 − θ t = η∇ θ J n (π t ). Then, for all j ≥ 1, we have λ
Neural MPG
Suppose that instead of a linear model, the policy's preferences h (i) θ , i = 1, . . . , n, are parametrized by deep neural networks. It is immediate from the proofs that the policy gradient theorem holds true, that is, θ t+1 − θ t = η∇ θ J n (π t ) for the ideal MPG update. Furthermore, we deduce a simple criteron that is sufficient for the policy at the end of training to be optimal. This criterion is expressed in terms of the Neural Tangent Kernels (NTKs) of the neural networks. The NTK of the i-step policy at time t of training is defined for all (a, s), (a , s ) ∈ A × S as , s), (a , s )
Θ (i) t ((a) := ∇ θ (i) h (i) t (a, s) · ∇ θ (i) h (i) t (a , s ).
Corollary 1. Let π t ∈ P n be parametrized by a neural network. Suppose that θ t+1 − θ t = η∇ θ J n (π t ) and that π ∞ = lim t→∞ π t . If Θ (i) ∞ is strictly positive definite for all i = 1, . . . , n, then π ∞ = π * in L 2 .
In the above corollary, assuming moreover that the neural networks are in the so-called NTK regime (a.k.a. lazy regime, kernel regime, ...), ideal MPG converges to the optimal policy. Outside of this regime, it suggests that training failure can be investigated through the lens of the NTK at the end of training.
Numerical experiments
This section summarizes the performance of the MPG framework using a deep neural network. Our current implementation of the MPG is a very simple one, without standard RL tricks, such as replay buffer, gradient clipping, etc. Details on the implementation and experimental setup, as well as additional results can be found in Appendix E.
First, we considered the FrozenLake benchmark [7], which is a maze-like k × k grid world composed of safe cells, holes and one treasure. The goal is for the agent to reach the treasure while avoiding holes. It features a finite environment and discrete action space. For k = 4, the MPG obtained policies were optimal or very close to optimal. For k = 8, we were able to obtain close-to-optimal policies in the sense that the treasure was consistently found after an average of 16.7 steps (the optimal deterministic path contains 14 steps). This near optimal aspect could be due to the stochasticity of the policy.
The second benchmark is the Cart Pole [7], which is a classical control problem. A pole is attached by an un-actuated joint to a cart, which moves along a frictionless track. The pole is placed upright on the cart, and the goal is to balance the pole by moving the cart to the left or right for some finite horizon time. It features a continuous environment and a discrete action space. Considering a horizon of n = 100, we obtained agents for which 77% of the games played (500 games) balanced the pole for the full horizon, with an average length for the game of 96 steps.
Conclusion
In this paper, we have introduced a new framework together with Matryoshka Policy Gradient, to solve a reshaped version of the max-entropy RL objective. We proved the global convergence of the ideal MPG update under mild assumptions and showed that the standard max-entropy objective optimal policy is retrieved by extending the maximal horizon of MPG. For neural policies, we found an easy-to-check criterion of the neural tangent kernel at convergence that guarantees the global optimality of the limit. The MPG framework is intuitive, theoretically sound and it is easy to implement without standard RL tricks, as we verified in numerical experiments.
The simplicity of the framework and guarantees we have derived leave open many perspectives:
• We have not studied the rate of convergence of MPG; it would be interesting to understand how the fixed maximal horizon n of MPG influences it, prescribing an optimal increasing schedule for n in terms of the environment's parameters.
• We guaranteed global convergence of the ideal MPG update; we did not dwell on the stability of training with the estimate MPG update (10).
• Additionally to MPG as defined in this paper, we expect to have nice theoretical properties of variations of MPG that are used for other PG algorithms. E.g. one can think of natural MPG, actor-critic MPG, path consistency MPG (see [28] for the definition of the path consistency learning algorithm).
• We motivated the use of MPG with neural softmax policies by some theoretical, practical, and heuristic arguments; we believe that more can be said on the use of neural policies with MPG.
• How does the MPG framework compares to the standard max-entropy RL framework in terms of exploration, adversarial robustness, ...?
• We limited ourselves to a simple, trickless implementation of MPG for our numerical experiments, as a proof of concept. We expect to see better performances of MPG using standard RL tricks such as replay buffer, gradient clipping, etc.
Appendix
The appendix is organized as follows:
• A: we recall basic properties of softmax policies, then discuss the potential benefits to using a single neural network for the preferences of all i-step policies. This section ends with an explanation on how to approximate a kernel with finitely many features.
• B: we prove Theorem 1 with techniques that will extend to the general state space case.
• C: we prove the Matryoshka Policy Gradient Theorem (Theorem 2), Proposition 2 that shows that the infinite horizon optimal policy can approximated arbitrarily well by finite horizon optimal policies, and Theorem 3 that shows global convergence of MPG.
• D: we list and discuss the main assumptions of the present work.
• E: we provide more detailed numerical experiments implementing MPG.
• F: we propose a variation of MPG that could potentially accelerate training at the cost of stability.
A More on the parametrization A.1 Softmax policy
Softmax policies enjoy the two following properties:
• For all s ∈ S, it holds that
E π θ [∇ θ log π θ (A|s)] = a∈A ∇ θ π θ (a|s) = 0.(11)
• As long as preferences are finite, it holds that π θ (a|s) > 0 for all (a, s) ∈ A × S.
In order to compute the learning rate's value below which training converge, we use the following:
Lemma 3.
For the softmax policy with linear preferences, it holds that θ → ∇ θ J n (π θ ) is L-Lipschitz for some L > 0.
We refer to [39] Lemma 3.2 and the discussion after Assumption 3.1 therein for the proof of this fact in the standard setting; the proof straightforwardly adapts to the extended policy setting.
A direct consequence of Lemma 3 is that gradient ascent on J n (π θ ) converges as soon as the learning rate is smaller than 1/L.
A.2 Neural networks
Neural Tangent Kernel. For neural policies, by Corollary 1, if the NTKs at convergence are positive definite, we get global optimality. For infinitely wide neural networks in the supervised learning setting, some global convergence guarantees have been obtained (e.g. NTK or lazy regime [22,10], mean-field regime [11]). In particular, in the NTK regime, the NTK is fixed throughout training, thus ensuring that it remains pd, as long as it was initialized pd. Note that Corollary 1 gives a necessary condition, but does not say that the limit is suboptimal when the NTK is only semi-pd.
Extended policy parametrized by a single neural network. One of the assumptions of MPG is that for any i = j, the policies π (i) θ (i) and π (j) θ (j) do not share parameters. Using one neural network per horizon becomes quickly costly as the maximal horizon increases. In order to avoid this issue, one can use a single neural network h θ to parametrize all i-step policies by using i as an input such that π (i) θ (a|s) ∝ π(a|s) exp(h θ (a, s, i)/τ ). By deviating from the theory, we nonetheless expect that the performance of the model is enhanced: intuitively, the i-step optimal policy should be close to the i + 1-step policy. It is even more true as i gets large, so one could also use 1 − 1 i as an input to the network (or any increasing map g : N → [0 , 1] such that i → g(i + 1) − g(i) is decreasing).
A.3 Kernel methods
Suppose that Θ is a strictly pd kernel with P positive eigenvalues. Recall the linear model a → h θ (a) = θ · ψ(a), with parameters θ ∈ R P , such that ψ is a feature map associated with Θ. Then if P = ∞, one can use random features, i.e. sample g 1 , . . . , g P i.i.d. Gaussian processes with covariance kernel Θ, then h θ := 1 √ P P i=1 θ i g i . One can thus approximate the true kernel predictor using a finite number of features, see [23].
Another way to approximate the kernel predictor with finitely many features is to use the spectral truncated kernel Θ of rank P ∈ N, by cutting off the smallest eigenvalues. That is, if (e i , λ i ) i≥1 are the eigenfunction/eigenvalue pairs of Θ ranked in the non-increasing order of λ i , one can use
Θ(x, x ) := P i=1 λ i e i (x)e i (x ).
B Bandit case
The following result on the dynamics of the policy during training is essential to proving Theorem 1:
Lemma 4. For all a ∈ A and all t ∈ N, it holds that
log π t+1 (a) − log π t (a) = −ητ a ∈A π t (a ) log π t π * (a ) − D KL (π t ||π * ) × Θ(a, a ) − E πt [Θ(A, a )] + o (ηC(θ t )) ,
where the constant C(θ t ) does not depend on the learning rate η.
Proof. We have,
∇ θ π t (a) = 1 τ π t (a) a ∈A (δ a, − π t (a ))∇ θ h t (a ).(12)
Let B t := R t − τ log πt(At) π(At) . By Equation (2), and Equation (12), using a first order Taylor approximation, we get for all a ∈ A that log π t+1 (a) − log π t (a) = (θ t+1 − θ t ) · ∇ θ π t (a) π t (a) + o (ηC(θ t )) .
The first order term of the right-hand side reads as
ηE πt B t ∇ θ π t (A t ) · ∇ θ π t (a) π t (A t )π t (a) = η τ 2 E πt B t a ,a ∈A (δ At,a − π t (a ))(δ a,a − π t (a ))∇ θ h t (a ) · ∇ θ h t (a ) = η τ 2 E πt B t
a ,a ∈A (δ At,a − π t (a ))(δ a,a − π t (a ))Θ(a , a )
= η τ 2 E πt B t Θ(A t , a) + E πt⊗πt [Θ(A, A )] − E πt [Θ(A, a)] − E πt [Θ(A t , A)] ,
where the inner expectations are for A and A , and the outer one is for A t . Using that
E[X(Y − E[Y ])] = E[(X − E[X]
)Y ] for any two integrable real random variables, we then get that
log π t+1 (a) − log π t (a) = η τ 2 E πt (B t − E πt [B t ]) (Θ(A t , a) − E πt [Θ(A, a)]) + o (ηC(θ t )) .
To conclude the proof, it suffices to note that by Equation (1),
B t − E πt [B t ] = R t − τ log π t π (A t ) − E πt [r(A t )] + D KL (π t ||π) = log π * π (A t ) − τ log π t π (A t ) − E πt τ log π * π (A) + D KL (π t ||π) = −τ log π t π * (A t ) + τ D KL (π t ||π * ),
which yields the claim.
Proof of Theorem 1. Let p be any probability measure on A. Suppose that θ t is a critical point of θ → J(π θ ). Since θ t+1 − θ t = η∇ θ J(θ t ) = 0, by Lemma 4, we have that
0 = a∈A p(a) (log π t+1 (a) − log π t (a)) = −ητ a ∈A π t (a ) log π t π * (a ) − D KL (π t ||π * ) E p [Θ(A, a )] − E πt [Θ(A, a )] + o (ηC(θ t )) .
In particular, since the above must hold for all η > 0, we deduce that a ∈A π t (a ) log π t π * (a ) − D KL (π t ||π * )
× E p [Θ(A, a )] − E πt [Θ(A, a )] = 0(13)
We now show that if d ⊥ e i , then one can choose p such that the above is non-zero. If the map
f : a → a ∈A π t (a ) log π t π * (a ) − D KL (π t ||π * ) Θ(a, a )
is not constant in a, then take p = δ a min with a min := argmin a∈A f (a). Clearly, the left-hand side of (13) is (strictly) positive, since π t (a) > 0 for all a ∈ A, which is a contradiction. Therefore, f must be constant. We write
f (a) = |A| i=1 λ i e i (a) a ∈A π t (a ) log π t π * (a ) − D KL (π t ||π * ) e i (a ) = |A| i=1 λ i d, e i e i (a).
On the other hand, since a∈A d a = 0, we have that
0 = a∈A f (a)d(a) = |A| i=1 λ i d, e i a∈A d(a)e i (a) = |A| i=1 λ i d, e i 2 ,
which yields the claim since λ i ≥ 0 for all i ∈ {1, . . . , |A|}.
C General state space
C.1 Matryoshka Policy Gradient Theorem
Proof of Theorem 2. Let m (i) π denote the law of S n−i , that is, the (n − i)-th visited state under π. The distribution of the sequence S 0 , A 0 , . . . , A n−i−1 , S n−i is not influenced by the parameters θ (i) , thus we can write
∇ θ (i) J n (π t ) = S ∇ θ (i) V (n) πt (s)ds = S ∇ θ (i) E πt n−i−1 =0 R − τ log π (n− ) t π (A |S ) + E S n−i ∼m (i) π t E πt n =n−i R − τ log π (n− ) t π (A |S ) S n−i ds = S 0 + E S n−i ∼m (i) π t ∇ θ (i) V (i) πt (S n−i ) ds,
where we have used the Markov property. We then have that
∇ θ (i) V (i) πt (S n−i ) = ∇ θ (i) E T n,i (πt) n =n−i R − τ log π (n− ) t π (A |S ) S n−i = E T n,i (πt) n =n−i R − τ log π (n− ) t π (A |S ) − τ ∇ log π (i) t (A n−i |S n−i ) S n−i = E T n,i (πt) n =n−i R − τ log π (n− ) t π (A |S ) ∇ log π (i) t (A n−i |S n−i ) S n−i ,
where we have used (11) to get rid of τ .
Recalling the MPG update (10), we thus have proved that E[θ t+1 − θ t ] = η∇ θ J n (π t ); the convergence follows from Lemma 3.
C.2 On the optimal policy
Proof of Lemma 2. By definition, we write
V (n) * (s) = τ a∈A π (n) * (a|s) Q (n) * (a, s) − τ log π (n) * π (a|s) = τ log E π exp(Q (n) * (A, s)/τ ) a∈A π(a|s) exp Q (n) * (a, s)/τ E π exp Q (n) * (A, s)/τ = τ log E π exp Q (n) * (A, s)/τ ,
as claimed, which concludes the proof.
Proof of Proposition 2. (i) Let π ∈ P be any standard policy, and let π n = (π, . . . , π) ∈ P n . By definition of the standard discounted objective J ∞ , using the dominated convergence theorem (rewards are bounded), we have that J n (π n ) → J ∞ (π). In particular, we get that π (n) * ,n achieves a performance arbitrarily close to that of π * ,∞ in the infinite horizon discounted setting, and since the optimal policy of J ∞ is unique (Lebesgue almost everywhere), we deduce that π (n) * ,n → π * ,∞ as n → ∞.
(ii) Suppose that J 1 (π * ,1 ) > J 1 (T n,1 (π * ,n )), that is
S V (1) π * ,1 (s)ds > S V (1) π * ,n (s)ds.
In particular, the set S ⊂ S such that s ∈ S if and only if V π * ,n (s). Let π * ,n ∈ P n be identical to π * ,n except for the 1-step policy where π (1) * ,n is replaced by π * ,1 . Then, the recursive structure of the value function (4) entails that J n ( π * ,n ) > J n (π * ,n ) (we implicitly use that the MDP preserves the absolute continuity of its state's law), which is a contradiction. Therefore, T n,1 (π * ,n ) = π * ,1 .
Then, by induction and using the recursive structure of the value function, the same argument shows that T n,m (π * ,n ) = π * ,m for all m = 2, . . . , n−1, which concludes the proof.
C.3 Global optimality of MPG
Lemma 5. For all n ≥ 1, all π ∈ P n and all s ∈ S, it holds that
V (n) π (s) − V (n) * (s) = −E π n−1 i=0 D KL (π (n−i) ||π (n−i) * )(S i ) S 0 = s .
Proof. Recall (4) and write V (n) π (s) = a∈A π (n) (a|s) r(a, s) − τ log π (n) π (a|s) + where we plugged in the expression of the optimal policy (7). We can rewrite the above as
V (n) π (s) − V (n) * (s) = −D KL (π (n) ||π (n) * ) + E V (n−1) π (S 1 ) − V (n−1) * (S 1 ) S 0 = s .
The claim follows by induction.
Proof of Proposition 1. The Kullback-Leibler divergence being non-negative, it is readily seen that for all s ∈ S, the maximal value of π → V (n) π (s) is obtained for π = π * . It is then immediate that π * is the unique optimal policy (outside a set of Lebesgue measure 0) for the objective J n given in (6). For all a ∈ A and s ∈ S, it holds that
log π (m) t+1 (a|s) − log π (m) t (a|s) = −ητ S m (m) πt (ds ) a ∈A π (m) t (a |s ) log π (m) t π (m) * (a |s ) − D KL (π (m) t ||π (m) * )(s ) × Θ (m) ((a, s), (a , s )) − E π (m) t [Θ (m) ((A, s), (a , s ))] + o (ηC(θ t )) , where m (m)
πt is the law of S n−m under policy π t and the constant C(θ t ) does not depend on η.
Proof. We follow the same steps as in the single state case treated in Lemma 4. The gradient of the policy reads as
∇ θ π (m) t (a|s) = 1 τ π (m) t (a|s) a ∈A δ a,a − π (m) t (a |s) ∇ θ h (m) t (a, s).(14)
Let (a, s) ∈ A × S. Using (10) and a first order Taylor approximation, we write log π
(m) t+1 (a|s) − log π (m) t (a|s) = (θ (m) t+1 − θ (m) t ) · ∇ θ π (m) t (a|s) π (m) t (a|s) + o (ηC(θ t )) = η τ 2 E πt C m a ,a ∈A δ a,a − π (m) t (a |s) × δ A n−m ,a − π (m) t (a |S n−m ) Θ (m) ((a , s), (a , S n−m )) + o (ηC(θ t )) .(15)
We focus on the expectation. It is equal to Therefore, whenever λ j > 0, we must have d ⊥ e (m) j in L 2 . This concludes the proof.
E πt C m Θ (m) ((a,
D Assumptions
We say that the MDP is irreducible if and only if ∀s, s ∈ S, ∀ > 0, ∃k ∈ N, ∃a 0 , . . . , a k−1 :
B(s, ) ds 0 S ds 1 · · · S ds k−1 B(s , ) ds k k−1 m=0 p(s m , a m , s m+1 ) > 0,(18)
where B(s, ) denotes the ball { s ∈ S : ||s − s|| < }. In words, this means that any neighborhood of any state is reachable from any neighborhood of any state. We now list the assumptions and briefly mention their roles in this work:
• The initial state distribution is uniform on S: it is not restrictive, as its role is to ensure that the optimal policies for all horizons visit (Lebesgue almost all) the whole state space, thus avoiding considerations about reachable states.
• Continuous compact S: this allows to safely apply Mercer's Theorem. Extensions to non-compact spaces should be possible by applying other versions of Mercer's Theorem, see e.g. the introduction of [12].
• Finite A: simplifies the presentation. Extensions to continuous action spaces should be possible similarly as for the previous item.
• Continuous reward and transition functions: imply measurability of the variables generated by the MDP and avoid pathological cases.
• The MDP is irreducible: avoid technicalities.
• The MDP preserves absolute continuity: avoid technicalities, some statements and proofs have to be adapted if that is not the case.
• Rewards are bounded: ensures that value functions are well defined.
• The temperature τ is the same for all steps: it is unnecessary, as one could regularize the objectives according to the number of remaining steps. That is, π (i) could be regularized with τ i for all i = 1, . . . , n with τ i = τ j if i = j, and the theory would still be valid.
E Numerical experiments
We apply MPG on a number of case studies. The MPG is implemented as in algorithm 1.
Algorithm 1: MPG implementation
for t = 1, ... , horizon do generate trajectory {(s i , s i+1 , a i , r i )} n−1 i=0 from π t for i = 1, · · · , n do π t+1 ← update policy as in (10) end decay τ, η end
We use exponential decays for temperature τ and learning rate η, with prescribed terminal temperature τ T and learning rate η T . For example, for τ the decay rate is computed as
d τ = τ T τ 0 1/ngames .
E.1 Frozen lake
The FrozenLake benchmark [7] is a k × k grid composed of cells, holes and one treasure. It features a discrete action space, namely, the agent can move in four directions (up, down, left, right). The episode terminates when the agent reaches the treasure. We consider a k = 4, 8 for the numerical experiments. It is well-known that reshaping the reward function can change the performance of the algorithm. The original reward function does not discriminate between losing the game (falling into a hole), not moving and moving, so we use a reshaped reward function: losing the game (−1), moving against a wall (−0.05), moving (+0.05) and reaching the treasure (+10.0).
For k = 4, the optimal number of steps is 6. We define a terminal τ T = 0.03 and terminal learning rate η T = 3 × 10 −6 , vary the initial learning rates η, temperatures τ and horizon to see the impact of these on the success of the agents. We train sets of 10 agents on 1000 episodes. Then, the trained agents play 100 games. A summary of the results for horizons n = 10, 15 is given in table 1, showing that the policies obtained are optimal or very close to optimal. The column Failed to train denotes the policies that failed to converge (out of 10).
Considering a larger map, for k = 8, the optimal path contains 14 steps. We vary the terminal temperature τ T , with the intuition that the stochasticity of the policy can deteriorate the performance of the agent over longer horizons. We also consider a longer horizon of h = 100. Furthermore, we further modify the reward function on the states: moving against a wall (−0.1) and moving (0.01). We train sets of 5 agents on 2000 episodes. A summary is given in table 2, showing that near optimal policies obtained are attainable with our algorithm. However, the convergence and training stability of our method depends heavily on training hyperparameters, such as the terminal temperature τ T , initial temperature τ 0 and learning rates η 0 , η T .
E.2 Cart Pole
The Cart Pole benchmark is a classical control problem. A pole is attached by an un-actuated joint to a cart, which moves along a frictionless track. The pole is placed upright on the cart, and the goal is to balance the pole by moving the cart to the left or right for some finite horizon time. It features a continuous environment and a discrete action space. The original reward function gives a +1 reward for each time that the pole stays upright, and the task finishes if the cart leaves the domain or if the pole is far enough from being upright. We reshape the reward function to give a penalty (−10) if the task is unsuccessful (i.e. the pole falls before reaching the target horizon).
We set the terminal τ = 0.01 and terminal learning rate η = 5 × 10 −8 . We train sets of 5 agents on 2000 episodes. Then, we play 100 games with the trained agents and record the performance of the policies, as shown on table 3. In practice, in order to achieve a more robust implementation, one could consider various stabilisation techniques such as gradient clipping to avoid updates which are too large or usage of off-policy updates, for example, through the use of a replay buffer. Other improvements to the training strategy can be adopted, such as: adaptive techniques to choose τ and η during training can be considered, beyond the chosen continuous decay, batched trajectory updates, etc.
F Multiple updates per path
In Theorem 3, global optimality of π (i) for all 1 ≤ i ≤ n is shown using an inductive argument on i. It is therefore desirable to train the 1-step policy faster than the 2-step policy, itself faster than the 3-step policy, and so on. The following observation allows one to maximize the use of one path for training: since a path for one MPG update is of length n, the 1-step policy can be trained n times, the 2-step policy n − 1 times, and so on until the n-step policy that can be updated once.
t+1 = θ (i) t + η n−i k=0 ρ i,k i−1 =0 R +k − τ log π (i− ) t π (A k+ |S k+ ) ∇ log π (i) t (A k |S k ) = θ (i) t + η n−i k=0 C k,i ∇ log π (i) t (A k |S k ),
The scaling factors ρ i,k are used so that in expectation, the update of π (i) t is done with actions sampled from itself. However, the denominator in ρ i,k can lead to training instability: we naively implemented this trick on the 4 × 4 maze of E and we observed that with the trick, training failed to find the optimal policy where it was found without it, using the same set of hyperparameters.
j≥1 are eigenfunction/eigenvalue pairs, ranked in the non-increasing order of the non-negative eigenvalues, of the integral operator I (i) πt : L 2 → L 2 defined by I (i) πt f : (a, s) → S ds a ∈A f (a , s )Θ (i) ((a, s), (a , s )).
Theorem 3 .
3Let m ∈ {1, . . . , n}. Suppose that for all i = 1, . . . , m − 1, the policy π
t → ∞. Suppose moreover that π (m) converges too, and let π(m) ∞ be its limit. Define the map d : A × S → R by d(a, s) := m (m) π∞ (s)π (m) ) − D KL (π (m) ∞ ||π (m) * )(s) .
> 0
0⇒ d ⊥ e (m) j in L 2 . In particular, if A2. holds, then π (m) ∞ = π (m) * in L 2 .
n (s) is non-empty and has a positive Lebesgue measure. Furthermore, by optimality, s ∈ S \ S if and only if V
π
(n) (a|s) V (n) * (s) − τ log π (n) π * (a|s) + s ∈S p(s, a, s )(V (n−1) π (s ) − V (n−1) * (s )) ,
Lemma 6 .
6Let t ∈ N and m ∈ {1, . . . , n}. Suppose that π ) * for all k = 1, . . . , m − 1.
0 j
0s), (A n−m , S n−m )) − E A Θ (m) ((A, s), (A n−m , S n−m )) − E A Θ (m) ((a, s), (A , S n−m )) + E A,A Θ (m) ((A, s), (A , S n−m )) ,where A, A have respective laws π ·|S n−m ) and are mutually independent of all other variables (conditionally given S n−m for A ). Using the trickE[X(Y − E[Y ])] = E [ (X − E[X])Y ], we obtainSince the above must be true for all η > d a ,s Θ (m)((a, s), (a , s )).Let s 0 ∈ S such that a → f (a, s 0 ) is not constant and let a min := argmin a∈A f (a, s 0 ). Then, because π (m) t is a stochastic policy, the left-hand side of (17) with (a, s) = (a min , s 0 ) must be strictly positive, which is a contradiction. Hence, for every fixed s ∈ S, the map a → f (a, s) is constant.We then use Mercer's Theorem to write f (a, (a , s ).On the other hand, a∈A d a,s = 0 for all s ∈ S,
:
Performance of trained agents on the Frozen Lake task on 8 × 8 grid, for horizon n = 100 and varying terminal τ T .
Table 3 :
3CartPole task performance after training, for horizon with length 100. and more generally for i = 2..n,θ
(i)
instructGPT and chatGPT are trained with Proximal Policy Optimization, see https://openai.com/ blog/chatgpt/.2 Other regularization techniques are used and studied in the literature, we focus on entropy regularized RL in this paper.
Implicitely assumed in the MDP definition is the fact that all variables such that actions, visited states and rewards are measurable, so that they are well-defined random variables.
t (A k |S k ),
.We writewhere we used Lemma 5 and the fact that π (i) t = π (i) * for all i = 1, . . . , m − 1. Similarly and using the expression (7) of the optimal policy, we haveHence, the expression in(16)becomeswhich corresponds to the first order term in right-hand side of the equation in the Lemma. Coming back to(15), this concludes the proof.is a critical point of (θ (1) , . . . , θ (m) ) → J m ( π θ ). By Lemma 6, we have that 0 = log π 100.00 6.00 1 τ 0 = 0.6, η = 0.0005 100.00 6.07 0Table 1: Performance of trained agents on the Frozen Lake task on 4 × 4 grid, for horizons n = 10, 15.Let ρ i,k := i−1 =0 π (i− ) t π (n−k− ) t (A k+ |S k+ ). We propose the following variation of MPG:
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"A system for exploring big data: an iterative k-means searchlight for outlier detection on open health data"
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"PhD, Fellow, IEEEA Ravishankar Rao ",
"Subrata Garai [email protected] ",
"M.SDaniel Clarke [email protected] ",
"PhD.Soumyabrata Dey [email protected] ",
"\nIT Software Engineer\nFairleigh Dickinson University\nNJUSA\n",
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"IT Software Engineer\nFairleigh Dickinson University\nNJUSA",
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| The interactive exploration of large and evolving datasets is challenging as relationships between underlying variables may not be fully understood. There may be hidden trends and patterns in the data that are worthy of further exploration and analysis. We present a system that methodically explores multiple combinations of variables using a searchlight technique and identifies outliers. An iterative k-means clustering algorithm is applied to features derived through a split-apply-combine paradigm used in the database literature.Outliers are identified as singleton or small clusters. This algorithm is swept across the dataset in a searchlight manner. The dimensions that contain outliers are combined in pairs with other dimensions using a susbset scan technique to gain further insight into the outliers. We illustrate this system by anaylzing open health care data released by New York State. We apply our iterative k-means searchlight followed by subset scanning. Several anomalous trends in the data are identified, including cost overruns at specific hospitals, and increases in diagnoses such as suicides. These constitute novel findings in the literature, and are of potential use to regulatory agencies, policy makers and concerned citizens. | 10.1109/ijcnn.2018.8489448 | [
"https://export.arxiv.org/pdf/2304.02189v1.pdf"
]
| 52,985,489 | 2304.02189 | 406007ea07807fb108234d0d072e08a7353a155c |
A system for exploring big data: an iterative k-means searchlight for outlier detection on open health data
PhD, Fellow, IEEEA Ravishankar Rao
Subrata Garai [email protected]
M.SDaniel Clarke [email protected]
PhD.Soumyabrata Dey [email protected]
IT Software Engineer
Fairleigh Dickinson University
NJUSA
Fairleigh Dickinson University
NJUSA
A system for exploring big data: an iterative k-means searchlight for outlier detection on open health data
When presented with a vast amount of data, important questions to consider concern the regularity of the data, and conversely any types of irregularities [10]. Both these
The interactive exploration of large and evolving datasets is challenging as relationships between underlying variables may not be fully understood. There may be hidden trends and patterns in the data that are worthy of further exploration and analysis. We present a system that methodically explores multiple combinations of variables using a searchlight technique and identifies outliers. An iterative k-means clustering algorithm is applied to features derived through a split-apply-combine paradigm used in the database literature.Outliers are identified as singleton or small clusters. This algorithm is swept across the dataset in a searchlight manner. The dimensions that contain outliers are combined in pairs with other dimensions using a susbset scan technique to gain further insight into the outliers. We illustrate this system by anaylzing open health care data released by New York State. We apply our iterative k-means searchlight followed by subset scanning. Several anomalous trends in the data are identified, including cost overruns at specific hospitals, and increases in diagnoses such as suicides. These constitute novel findings in the literature, and are of potential use to regulatory agencies, policy makers and concerned citizens.
I. INTRODUCTION AND MOTIVATION
The intersection of the areas of big data and health care provide many opportunities for innovation. One of the interesting directions concerns open health data initiatives, which have been underway at several national and regional agencies worldwide, including the NHS in the UK, the Center for Medicare and Medicaid Services (CMS) in the USA, and New York State Statewide Planning and Research Cooperative System (SPARCS) [1][2][3]. This effort is driven by transparency as citizens are demanding more accountability from their governments.
Rao et al pointed out many challenges that need to be overcome [4][5][6][7][8], including the cleaning of these datasets, the difficulty in combining information from multiple agencies and sources, and the lack of a single platform to perform end-to-end data analytics. Though open data initiatives are expected to help the citizenry, there is a large gap between the ability of citizens to use existing tools and the sophisticated types of analyses that need to be performed. For instance, without aggregating information from multiple years, it is not easy to determine cost trends for different procedures, or to be able to identify individual hospitals that may be good for treatment of a specific condition.
Given the proliferation of different news sources and articles for information [9] it is becoming increasingly difficult to verify the "ground truth" in several domains. For instance, in the area of health care, we may want to know the expected cost of a given procedure in a given geographic area, such as hip replacement in New York city. Such cost information is difficult to obtain directly from the providers in countries like the US. However, open health data provides a source of relatively unbiased information [1][2][3]. This enables researchers and citizens to probe the the health care system and gain an unbiased view of costs. questions lead to interesting insights, and are required for model building and machine learning [11].
In a companion paper submitted to IJCNN 2018 [12], we investigate a model building exercise for the prediction of costs based on open health data. The focus of the current paper is on irregularities or outliers. Both types of analyses are important to gain a complete perspective about the data. The detection of outliers is particularly challenging as there is no standardized definition of what an outlier is [13].
The building of models for health data is challenging as the distribution of many variables such as cost in health care is heavily tailed [14], and different diseases may be associated with their own unique distribution types, e.g. Weibull, Cox or Tobit. Consequently, some researchers have thresholded the cost data, and ignore costs above say $50,000 in order to fit an appropriate model such as linear regression to the data [15]. If we simply apply such a technique and threshold the data to produce outliers, we could generate far too many possibilities that can be screened carefully.
There are many reasons that make outlier detection an important problem. Firstly, there may be trending health conditions that are not receiving sufficient attention from health providers or parents or educators. An example of this is the high increase in mental health issues with the teenage population in New York state [7]. Secondly, there may be specific hospitals that are underperforming relative to their peers. Identification of anomalous cost patterns could alert regulatory authorities, and also potential patients who may be considering treatment at these hospitals. Thirdly, outliers may be an indication of outright fraud such as gross overcharging for certain medical procedures. Fraud identification is an important application area for outlier detection [16].
We caution that the detection of an outlier does not directly imply that it represent fraudulent activity. Rather, it is merely a candidate for further consideration. Leite [17] presented a visual analytics approach to identify outliers in the financial domain, where candidates are presented to a human to enable fine tuning and generation of automatic alarms. Our approach is similar, where we utilize an outlier detection algorithm embedded in a human-in-the-loop system for verification.
In our earlier work, [8] we combined techniques from database analysis and machine learning to derive an iterative k-means technique to identify outliers. We developed a proof of concept for the technique, where it was applied successfully to understand graduation trends in different healthcare specialties using data from CMS
[18] for 800,000 practitioners. The main contribution of the current paper is to enhance the iterative k-means algorithm by using searchlight and subset scanning techniques, and to apply the enhanced algorithm on much larger datasets, containing de-identified data from approximately 15 million patients a year over a 5 year period. This data is provided by the SPARCS program, and contains discharge information about disease diagnoses and costs [3]. Our technique is described in detail in Section IV.
II. BACKGROUND AND RELATED WORK Gregory et al [14] reviewed several models for healthcare costs, including Weibull, Cox, Aalen and Tobit. They determined that it is not possible for a single model to deal with the variety that is encountered in cost data, and consequently developed separate models for different disease conditions. Mihaylova [19] presents an overview of different statistical methods to analyze data in the area of healthcare, and reaches a similar conclusion to Gregory et al [14] in that different models should be applied in different scenarios. Hence there is a need to investigate an approach that does not assume a prior statistical distribution for the cost values. The approach we present in the current paper is model-free.
Chen et al. [20] advocate a visualization process to scan data and understand relationships between the existing variables. A purely visual exploration approach is limited, as many combinations of variables need to be explored manually. We facilitate exploration by applying a clustering algorithm to automatically identify outliers and highlight interesting relationships.
Goldstein and Uchida [21] review techniques for unsupervised anomaly detection. Gupta et al. [22] present an overview of outlier detection techniques. Hauskrecht [23] presents a system developed at the University of Pittsburgh for monitoring individual patient data and issuing alerts about unusual case decisions. For this, nerarly 4500 electronic health records (EHRs) of patients were utilized. In contrast, we use a significantly larger dataset covering several million patient records, but at a coarser scale than their EHRs. The research presented in the current paper shows that one is able to extract meaningful outlier information from large public healthcare datasets. Krumholz [24] has pointed out that analysis is the bottleneck in the learning process in health-care. Hence, there is a need for a technique to quickly detect meaningful trends and outliers, and to present them to decision makers for further action. Our current paper presents a technique to address this need.
III. DESIGN
In our previous work, we investigated the architecture of an open-source system for the analysis of open health data [5,6]. We used a Python-based solution consisting of the following components: Python Pandas, Scikit-Learn and Matplotlib [5]. The Scikit-Learn Python library [25] provides several machine-learning capabilities such as clustering, classification and prediction.
In previous work [8], we presented an iterative k-means algorithm for outlier detection. We extend this work with the following additional enhancements, as shown in Figure 1. Firstly, we apply the iterative k-means algorithm in a searchlight manner, where the outlier detector is swept across pairwise aggregations of variables. This is similar to the use of searchlights in the functional magnetic resonance imaging literature in order to identify interesting patterns of brain activity, as explored by Rao [26]. Secondly, we apply a subset scan technique to further elaborate any dimensions that contain outliers. This allows more detailed relationships between the variables to be teased out of the data. In this paper we apply this algorithm to the SPARCS dataset and examine the results generated by using different aggregation possibilities, denoted by "Aggregate 1", "Aggregate 2" and so on, shown in The availability of a fast and robust outlier detection technique like the iterative k-means algorithm enables several interesting use cases, two of which are shown in Figure 2. Regulators can use the outliers as hypotheses for further analysis that needs to be investigated. Policy makers can use spikes in occurrences of certain disease patterns to allocate resources to treat them. Individuals can use the information on outliers as cautionary flags to consider when considering their medical treatment options. They could use the identification of outliers as a basis to conduct further investigation, say by searching news sources for further information about the outliers. We use the SPARCS dataset, consisting of de-identified in-patient discharge information about disease diagnoses and costs [3]. The aggregate data [2] contains 15,213,123 rows of patient data from 254 hospitals and 58 counties. There are a total of 264 different diagnosis descriptions.
In order to perform the aggregation in Figure 1 we can use different features in the original dataset, such as hospital county, CCS Diagnosis Description, or Facility Name, which are illustrated in Figure 3. We review the iterative k-means algorithm from Rao and Clarke [8] as shown in Figure 4. The algorithm repeatedly runs the k-means clustering technique. During each run, smaller clusters of points are treated as outliers and removed. This process is repeated until no small clusters are present, or until a fixed number of iterations are executed. The resulting clusters capture the most relevant groups after outlier removal. Figure 4 illustrates this procedure graphically through hypothetical data and shows four iterations labeled Step (a) -(d). We first select the number of clusters, e.g. k=4 and proceed as below.
Hospital County
1. Apply the k-means clustering algorithm.
2. If single-element or substantially small clusters (e.g. size < 2 or 3) exist, treat these as outliers and remove them from the dataset. Continue the computation with the remaining data.
3. If no substantially small clusters remain, the iterative k-means algorithm is terminated.
This approach exploits the sensitivity of the k-means algorithm towards outliers in order to isolate them. Our approach requires the number of clusters to be specified, but there is no limit on the number of outliers that can exist at each iteration. Our algorithm is executed for multiple iterations until the user is satisfied with the results. Sample code is presented in Rao [8]. The processing pipeline in Figure 1 consists of the following steps:
1. Use the split-apply-combine paradigm [27]. We use the Pandas pivot table functionality to group the data by different feature columns in the database, as follows.
2. This produces grouped items' counts for each "CCS Diagnosis Description", which are then further binned over the field "Discharge Year". The result is a feature vector for each specialty which consists of the count of hospitalization incidents binned into 'Year of the incident'. Then this feature vector is scaled to find the percentage of change with respect to base year (2009) values. Below is the graphical representation of the matrix created.
3. The feature vectors for each specialty are then processed by the iterative k-means algorithm outlined in Figure 4.
4.
We use interactive visualization to present different graphs to the user, which permits an understanding of unusual trends in the data. This approach is illustrated in Figure 5.
Aggregation with respect to a single field can hide interesting data patterns which may be visible when analyzed with respect to the other fields. At the other extreme, aggregation with respect to all possible fields subdivides the data into the highest level of granularity, and significantly increases the search space. In search of a balance we propose a novel algorithm combining a subset scan [28] with the iterative k-means outlier algorithm. If an outlier is found for the aggregation over a given field, say N, then we examine subsets that include the field N. For instance, if we find an outlier in counts for Diagnoses codes for "Suicides", we then search for outliers in "Suicides" over "Age Groups", "Ethnicity", "County" and so on. For each of these subsets, the iterative K-means algorithm is used. In summary, we starting with the entire data set, and hierarchically focus our search on only the outlier points to investigate the actual subset of fields responsible for the unusual patterns.
The current paper presents significant improvements over [8] in the size of the dataset considered (more than 15 million entries), a larger number of descriptive features, different choices for aggregation of the data, and a subset scan procedure to investigate outliers in more detail.
V. RESULTS
We applied the iterative k-means algorithm described in Section IV with k=8 on the SPARCS dataset. We explored different aggregations including total costs by hospital, or by county. Similarly, we aggregated total costs according to the individual CCS Diagnosis Descriptions. Trends in cost increases can also be computed. The results are organized by the types of aggregation applied, a few of which are shown below.
A. Percentage increase in counts of incidences per year, aggregated over CCS Diagnosis Descriptions
We computed the percentage change in total counts of incidences for medical procedures reported to New York State from the years 2009-2014 by using the year 2009 as the baseline. The results are shown in the order of sequential outlier detection, followed by plots of the remaining clusters. The significance of these results is reviewed in the discussion section.
B. Subset Scan
Once interesting outliers are identified, we perform a subset scan where we further refine the dimensions over which aggregation is performed. We use the same 'iterative k-means' method to include other dimensions of the dataset, one at a time, along with the current feature space. The following 2D matrix is created, where the 'Age Group' is considered as the second dimension.
We note that if we had N distinct values of Diagnosis Codes and T number of years, the input used in the first experiment would contain a [N x T] matrix. Whereas using another dimension (e.g. -'Age Group', 'Race', 'Ethnicity' etc.) along with the feature space would produce a [N x T x S] matrix, considering the third dimension has S unique values. Hence, we created a list of the other dimensions which might contribute to the outliers already found and developed count matrices for each aggregation in the list. The results obtained are shown in Figure 8.
(a) Subsetting the aggregate counts by Age group.
(b) Subsetting the aggregate counts by Race.
(c) Subsetting the aggregate counts by Ethnicity. Figure 8: This illustrates the subset scan procedure where we explore additional dimensions related to existing outliers. Examples are shown for subsetting by Age group, Race and Ethnicity, which are the subsets that produce outliers. Figure 9 shows the results of aggregating costs by CCS Diagnosis Description over all the hospitals, followed by the iterative k-means algorithm. The results are shown in the order of sequential outlier detection, followed by plots of the remaining clusters. From Figure 9 we can see that outliers consist of combinations of spikes in various dimensions. Though we could identify spikes in one dimension at a time, the advantage of the iterative k-means clustering algorithm is that it looks for outliers in the pattern of the entire cost distribution across all CCS Diagnosis Descriptions. We used 6 iterations of the iterative k-means algorithm to identify these outliers.
C. Total costs aggregated by CCS Diagnosis Description over all the Hospitals
We have applied our algorithm to detect outliers in several additional combinations of variables, including costs across counties, which identified Westchester county as having the largest cost increase for mental health diseases [7]. These results cannot be shown due to space limitations.
VI. DISCUSSION
From Figure 5 we observe that the following Diagnosis descriptions are outliers: 'Administrative/social admission', 'Immunity disorders', 'Suicide and intentional self-inflicted injury' and 'Influenza'. Specifically, 'Suicide and intentional self-inflicted injury' has a sharp spike for year 2015 compared to other years. Figure 8 is produced by adding three new dimensions ('Age Group', 'Race', 'Ethnicity') to the outlier detection method. This provides a more nuanced analysis of the trends in the data. Specifically, we note that 'Suicide and intentional self-inflicted injury' was high in Age Group '50 to 69' in the year 2015. The analysis also shows that this same diagnosis is high in the year 2015 for 'Spanish/Hispanic' and 'Non Spanish/Hispanic' ethnicities. Furthermore, there is an anomaly for 'Administrative/social admission' when viewed for the 'white' race. Figure 9 identifies four hospitals where the average costs of different diagnoses are anomalous with respect to the rest of the hospitals. By zooming into the diagnoses descriptions, we can see that "North Central Bronx Hospital" has very high average cost for diagnosis 'Cancer of Bronchus; Lung'. Similarly, 'Summit Park Hospital' has very high average cost for 'personality disorder' treatment. Also, 'Bon Secours Community Hospital' has high treatment cost for a several diagnoses. This suggests that a targeted intervention to educate the public in the North Bronx area about lung cancer issues may be useful in preventing incidences of this disease.
It is interesting to observe that a Big Data approach with interactive visualization tools can provide researchers with a ready capacity to start with the raw data provided by the government agencies and quickly identify and explore interesting trends. We expect the widespread adoption of such tools to enable concerned citizens to draw their own conclusions from important national data sources without having to deal with potential biases in reporting.
The Summit Park Hospital identified earlier was closed in 2015 due to cost overruns [29]. A recent news article mentions a hospital consolidation between Westchester Medical Center and the Bon Secours Health System [30]. It is likely that the outlier cost distribution in Figure 9 was an indication of issues at the Bon Secours Health System, and this hospital became a candidate for merger talks. This is merely a hypothesis at this point, driven directly by the data. Further verification is required. Nevertheless, the data provides a good source of possible hypotheses about the performance of different hospitals, which can then be subject to further scrutiny. We are using these examples to illustrate the verification process outlined in [31]. This hospital was formed to provide long-term acute care for physically disabled and medically fragile individuals. Given the type of patients that this hospital is designed to serve, it appears reasonable that their cost structure will be different from other hospitals in New York state.
These results demonstrate that we can quickly determine interesting and relevant trends in large health-care related datasets. This capability could provide concerned citizens with an unbiased data-driven interpretation of breaking news events in their regions as well as nationally.
In order to facilitate widespread adoption of the techniques presented in this paper, we have made our framework and code available freely to the research community at github.com/fdudatamining/.
VII. CONCLUSION
We presented an open-source toolkit based on Python that can be utilized to analyze and interpret large datasets thereby driving insight. Many government agencies, such as Medicare in the US release detailed data about their inner workings, but the capabilities of tools to interpret this data has not kept pace. Since the relationships between variables in these large datasets are not fully known, users typically engage in visual exploration, which tends to be slow and manually intensive. We have developed a machine learning approach, called iterative k-means, where clusters and outliers are automatically identified and presented to the user. This facilitates rapid visual exploration of new datasets.
We applied our toolkit to analyze health care data released by New York State SPARCS. Our technique identified interesting and meaningful trends in counts and cost increases for different diagnoses such as lung cancer and suicide rates. This information can be utilized by policy makers for targeted interventions to improve public health in specific regions. Our approach should also prove valuable to other researchers, and concerned citizens who are interested in exploring open health data.
Figure 1 .
1
Figure 1 :
1Proposed pipeline for data analysis involving outlier detection using a searchlight, followed by a subset scan and post-processing.
Figure 2 :
2Use case scenarios enabled by the outlier detection algorithm.
Figure 3 :
3An example of the data fields in SPARCS, spanning multiple features. A few selected fields are shown, drawn from a total of 35 such fields. Sample Diagnosis Descriptions consist of strings such as ABDOMINAL HERNIA, ABDOMINAL PAIN, ACQUIRD FOOT DEFORMIT and ACUTE CVD. IV. METHODS
Figure 4 :
4The iterative k-means algorithm works sequentially on the original data, shown in steps (a)-(d). Individual data points are shown as blue dots. The red circles are outliers that are not considered for further analysis. The blue ellipses identifiy the remaining clusters that are then processed in the next iteration.
Figure 5 :
5Each curve in this figure represents the percentage change in count for treatment of a specific CCS Diagnosis Description relative to the year 2009. The outliers detected at each iteration are shown in blue.
Figure 6 :
6This figure shows the clusters in the count trends for different CCS Diagnosis Descriptions. The top 4 outliers are shown in dotted red lines, and the k-means algorithm is run on the remaining trend curves, producing the color-coded clusters.
Figure 7 :
7This figure shows the largest cluster, with 66 members, when running the k-means algorithm after the first five outliers are removed.
Figure 9 :
9The x-axis consists of the index number assigned to each CCS Diagnosis Description. Each continuous curve represents a single hospital. (a) shows outliers in the distribution of "Total Costs", which identifies hospitals such as the Bon Secours Community Hospital, North Central Bronx Hospital and Henry Carter Specialty Hospital. (b) shows the pattern of cost for an individual hospital, the North Central Bronx Hospital. (c ) shows a zoomed-in version of the costs for North Central Bronx Hospital, showing a spike for lung cancer.
Figure 2 .
2Figure 2.
Figure 9
9Figure 9 sheds light on the developments surrounding the creation of the new Henry J. Carter Specialty Hospital in 2011 [31]. This hospital was formed to provide long-term acute care for physically disabled and medically fragile individuals. Given the type of patients that this hospital is designed to serve, it appears reasonable that their cost structure will be different from other hospitals in New York state.
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Data, information, and knowledge in visualization. M Chen, D Ebert, H Hagen, R S Laramee, R Van Liere, K.-L Ma, IEEE Computer Graphics and Applications. 29M. Chen, D. Ebert, H. Hagen, R. S. Laramee, R. Van Liere, K.-L. Ma, et al., "Data, information, and knowledge in visualization," IEEE Computer Graphics and Applications, vol. 29, pp. 12-19, 2009.
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| []
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[
"Atlantic Ranges & Targets, Electro-Optical Tracking Systems, 23013 Cedar Point Road, Bldg. 2118",
"Atlantic Ranges & Targets, Electro-Optical Tracking Systems, 23013 Cedar Point Road, Bldg. 2118"
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| [
"Aaron Hendrickson ",
"David P Haefner "
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| []
| [
"Research & Technology Integration"
]
| A comparative study of methods to estimate conversion gain in sub-electron and multi-electron read noise regimes ABSTRACT Of all sensor performance parameters, the conversion gain is arguably the most fundamental as it describes the conversion of photoelectrons at the sensor input into digital numbers at the output. Due in part to the emergence of deep sub-electron read noise image sensors in recent years, the literature has seen a resurgence of papers detailing methods for estimating conversion gain in both the sub-electron and multi-electron read noise regimes. Each of the proposed methods work from identical noise models but nevertheless yield diverse procedures for estimating conversion gain. Here, an overview of the proposed methods is provided along with an investigation into their assumptions, uncertainty, and measurement requirements. A sensitivity analysis is conducted using synthetic data for a variety of different sensor configurations. Specifically, the dependence of the conversion gain estimate uncertainty on the magnitude of read noise and quanta exposure is explored. Guidance into the trade-offs between the different methods is provided so that experimenters understand which method is optimal for their application. In support of the reproducible research effort, the MATLAB functions associated with this work can be found on the Mathworks file exchange. | null | [
"https://export.arxiv.org/pdf/2304.14164v1.pdf"
]
| 258,352,343 | 2304.14164 | 9d3f5c6a2e0b83b00d15314c400e9a17d8167d73 |
Atlantic Ranges & Targets, Electro-Optical Tracking Systems, 23013 Cedar Point Road, Bldg. 2118
27 Apr 2023
Aaron Hendrickson
David P Haefner
Atlantic Ranges & Targets, Electro-Optical Tracking Systems, 23013 Cedar Point Road, Bldg. 2118
Research & Technology Integration
Patuxent River MD, 20670 b U.S27 Apr 2023conversion gainDSERNphoton counting distributionphoton transferQISread noisesensor characterizationsub-electron noise
A comparative study of methods to estimate conversion gain in sub-electron and multi-electron read noise regimes ABSTRACT Of all sensor performance parameters, the conversion gain is arguably the most fundamental as it describes the conversion of photoelectrons at the sensor input into digital numbers at the output. Due in part to the emergence of deep sub-electron read noise image sensors in recent years, the literature has seen a resurgence of papers detailing methods for estimating conversion gain in both the sub-electron and multi-electron read noise regimes. Each of the proposed methods work from identical noise models but nevertheless yield diverse procedures for estimating conversion gain. Here, an overview of the proposed methods is provided along with an investigation into their assumptions, uncertainty, and measurement requirements. A sensitivity analysis is conducted using synthetic data for a variety of different sensor configurations. Specifically, the dependence of the conversion gain estimate uncertainty on the magnitude of read noise and quanta exposure is explored. Guidance into the trade-offs between the different methods is provided so that experimenters understand which method is optimal for their application. In support of the reproducible research effort, the MATLAB functions associated with this work can be found on the Mathworks file exchange.
INTRODUCTION
Since the advent of the Charge-Coupled Device (CCD) in the early 1970s, methods for characterizing electrooptical image sensors have continued to adapt to emerging technologies. Of particular importance in image sensor characterization is the measurement of conversion gain, which describes an intrinsic conversion constant relating arbitrary units of Digital Numbers (DN) at the sensor output back to a physically meaningful quantity of electrons (e-) at the sensor input. Generally speaking, each pixel in an image sensor array will have a unique conversion gain and this gain nonuniformity corrupts the output imagery. For this reason, a precise estimate of each pixel's conversion gain is needed to correct the image degrading effects of gain nonuniformity and calibrate the sensor in terms of absolute units. For many decades, the Photon Transfer (PT) method has been the standard approach to conversion gain estimation. [1][2][3][4] Since the arrival of photons at a sensor is accurately modeled by the Poisson distribution, the moments of the sensor input are known allowing PT to treat the sensor as a black box, only observing the statistical moments of the output, to determine the conversion gain.
In 2015, the first Deep Sub-Electron Read Noise (DSERN) image sensor was reported in the literature, carrying with it promising applications in low-light imaging and quantum technologies. 5 As a result of subelectron read noise, DSERN devices could discern the number of electrons generated in each pixel leading to never before seen structure in the data produced by such devices. As an example, Figure 1 shows histograms produced by a traditional scientific grade CCD pixel (left) and DSERN CMOS pixel (right) exposed to constant illumination. While it is not possible to observe electron events in the CCD data (since the read noise σ R is too large), the DSERN produced histogram shows distinct peaks where zero, one, two, etc. free-electrons have been detected within the pixel.
The additional structure observed in DSERN sensor data has led to the development of several new methods of conversion gain estimation, which leverage the additional structure to produce lower uncertainty estimates in comparison to the traditional PT method. [6][7][8] What is not clear in this body of research is that all of the proposed methods are derived from the same statistical model, which is valid for sensors with sub-electron and multi-electron read noise. Furthermore, while these newly proposed methods were designed to leverage the additional structure in data produced by DSERN capable devices, some show promise in characterizing sensors outside the DSERN regime; thus, serving as a general estimation procedure to supersede the legacy PT method. In this work, a comprehensive overview of all currently available methods for conversion gain estimation will be discussed using a unified model and notation to facilitate comparison between each method. This will be accomplished by first describing the unifying model of sensor noise and then using the framework of this model to describe each method in detail. With a full description of each method at hand, Monte Carlo simulations will be carried out to determine which method is best under a variety of sensor parameters. The authors aim to implement each method as faithfully as possible, and to this end, all of the code, including the implementation of each method, is available on the Mathworks file exchange.
THE PHOTON COUNTING DISTRIBUTION MODEL
The Photon Counting Distribution (PCD) model represents a single observation from a pixel (a digital gray value) as the random variable X given by 8,9
X = ⌈(K + R)/g + µ⌋ K ∼ Poisson(H) R ∼ N (0, σ 2 R ).(1)
The variables used in this model are defined as follows: K represents the electron number, H represents the expected number of electrons generated (thermally or otherwise) per integration time and is expressed in units of (e-), σ R represents the input referred analog read noise in units of (e-), g represents the conversion gain in units of (e-/DN), µ represents the pixel bias or DC offset in units of (DN), and ⌈·⌋ denotes rounding to the nearest integer. In all, the random variable X captures the process of adding noise (R) to a number of electrons (K) followed by the application of gain, offset, and finally quantization.
The quantization (rounding) defining the PCD model in (1) adds significant complexity to the distribution of X. If, however, g ≪ σ R so that the quantization bins are sufficiently small, the quantization process can be modeled as an additive noise source so that a continuous distribution still provides an adequate model. Under this assumption the distribution of X is modeled by the PCD
f X (x|θ) = ∞ k=0 e −H H k k! φ(x; µ + k/g, σ 2 ),(2)
where θ = (H, g, a, b 2 ) denotes the PCD parameter vector, φ(x; α, β 2 ) is the Gaussian probability density with mean α and variance β 2 , and σ = (σ 2 R /g 2 + σ 2 Q ) 1/2 represents the combined read and quantization noise in units of (DN). For most applications the series representation (2) works best since only a few terms are needed to get a good approximation for f X ; however, through the use of characteristic functions an integral representation can also be derived in the form
f X (x|θ) = 1 π ∞ 0 exp(H(cos(t/g) − 1) − σ 2 t 2 /2) cos((µ − x)t + H sin(t/g)) dt.(3)
Furthermore, (2) suggests a Monte Carlo estimator of the form
f X (x|θ) = Eφ(x; µ + K/g, σ 2 ) ≈ 1 n n k=1 φ(x; µ + K k /g, σ 2 ),(4)
where {K 1 , . . . , K n } are i.i.d. Poisson(H) random variables. For notational purposes the shorthand X ∼ PCD(H, g, µ, σ 2 ) will be used to denote a random variable distributed according to the PCD. Two special cases of the PCD occur as H → 0 and σ → ∞ giving PCD(H, g, µ, σ 2 ) → N (µ, σ 2 ) and PCD(H, g, µ, σ 2 ) → N (µ + H/g, σ 2 + H/g 2 ), respectively.
As far as the author's know, the first known mention of the PCD (albeit not by this name), in the context of image sensors, can be found is James Janesick's Photon Transfer (pg. 26, Figure 3.7), which showed simulated data for the standardized version PCD(1, 1, 0, σ 2 ). 3 Later papers by Fossum, Starkey, and Ma 6, 10-13 wrote down a more complete mathematical description of PCD(H, 1, 0, σ 2 ), which included a parameter for the quanta exposure. Furthermore, Nakamoto and Hotaka 7 included a parameter for the gain in the form PCD(H, g, 0, σ 2 ) but never in the full form accounting for the offset as seen in (2). What makes (2) a complete description is the fact that it provides all the parameters necessary to fit the PCD to raw sensor data. Depending on the specified parameters, the shape of the PCD can vary from a simple Gaussian bell-curve to a more complicated form involving many local maxima (peaks). The parameter µ acts as a location parameter shifting the PCD on the x-axis, while g acts as a scaling factor that controls the distance between adjacent peaks. Changing either of these parameters does not drastically change the overall look of the PCD. On the other hand, the parameters H and σ 2 play a significant role in the shape of the PCD. Figure 2 plots the PCD for various H and σ 2 (fixing µ = 0 and g = 1) to show how these parameters change the appearance of the probability density. In particular, for small enough σ 2 , the PCD oscillates showing many local maxima. Sensors that exhibit clearly resolved peaks like this are said to belong to the DSERN (a.k.a. sub-electron noise) regime. Likewise, the parameter H changes the overall envelope of the PCD from a highly skewed form at small H to a more Gaussian profile at large H.
METHODS FOR ESTIMATING CONVERSION GAIN
In recent years several novel methods have emerged for estimating conversion gain in the sub-electron and multielectron noise regimes. What is not immediately clear in the literature is that all of these newly proposed methods, as well as the traditional PT method, can all be fully described in the context of the PCD model introduced in the previous section. As such, the goal of this section is to explain each method, using a unified notation, in the PCD framework. Supporting theory for each method will be presented followed by a discussion of the associated pros and cons.
Photon Transfer Method
Photon Transfer (PT) is a classic method for measuring conversion gain that has been around for several decades in various forms. 1,3,14 To derive the PT estimator of the conversion gain, first let X ∼ PCD(H, g, µ, σ 2 ) and define v(H) := VarX = σ 2 + H/g 2 to be the variance of X as a function of the quanta exposure H. Here, the symbol E is used to denote the expectation operator so that the variance is defined as VarX = EX 2 − (EX) 2 . It follows from the definition of the derivative that
1/g ∂ H v(H) = lim ∆H→0 ∆H/g v(H + ∆H) − v(H) = lim ∆H→0 g = g.(5)
Notice that the fraction defining the derivative in (5) is independent of ∆H; thus, the limit ∆H → 0 is not needed and nonzero values of ∆H may be used to compute g. With this information now suppose X ∼ PCD(H + ∆H, g, µ, σ 2 ) and Y ∼ PCD(H, g, µ, σ 2 ) are independent PCD random variables. Consequently, ∆H/g = EX − EY and v(H + ∆H) − v(H) = VarX − VarY leading to the classic PT relation
g = EX − EY VarX − VarY .(6)
The PT method for conversion gain estimation replaces the populations means and variances in (6) with their respective unbiased estimators to obtain an estimator for g. Specifically, let x = {x 1 , . . . , x n1 } with x k ∼ PCD(H + ∆H, g, µ, σ 2 ) denote a random sample of n 1 observations at a quanta exposure of H + ∆H and y = {y 1 , . . . , y n2 } with y k ∼ PCD(H, g, µ, σ 2 ) denote a second (independent) random sample of n 2 observations at a quanta exposure of H. Denotingx = 1 n1 n1 k=1 x k as the sample mean andx = 1 n1−1 n1 k=1 (x k −x) 2 as the sample variance of the x-sample (and likewise forȳ andŷ), the PT estimate for the conversion gain is given by 14
, 15g =x −ȳ x −ŷ .(7)
Since this is an estimator of two independent samples, Hendrickson et. al. (2022) derived approximate optimal sample size pairs (n opt 1 , n opt 2 ) of the form 15 n opt
1 ∼ 2(1 + ζ) acv 2 0 (1 − ζ) 2 + 5 n opt 2 ∼ 2ζ(1 + ζ) acv 2 0 (1 − ζ) 2 + 1,(8)
where
ζ = VarY VarX = σ 2 + H g 2 σ 2 + H g 2 + ∆H g 2(9)
and acv 0 denotes the desired relative uncertainty of the final estimate, e.g. acv 0 = 0.05 corresponds to 5% estimator uncertainty. 15 These approximate optimal sample size pairs allow an experimenter to achieve the desired estimate uncertainty (acv 0 ) with the fewest total number of samples possible. 15 In practice, most image sensors are not perfectly linear (g is dependent on H) so that a small ∆H (ζ ≈ 1) is needed to obtain a meaningful estimate of g at the chosen illumination level. This, however, can cause instability in the estimator due to the fact that statistical uncertainty in the quantityx −ŷ can lead to division by zero type errors * . In fact, for even moderately large n 1 and n 2 , the sampling distribution of (x−ŷ) −1 is accurately modeled by the inverse gamma-difference distribution, which is known to have undefined moments in a similar manner as the Cauchy distribution. 16 This lack of well-defined moments leads to the PT estimatorg having ill-behaved statistical properties, most of which can be mitigated by using very large sample sizes (notice that n opt i → ∞ and ∆H → 0). Additionally, a disadvantage of this estimator is that it utilizes only the first two moments of the PCD. Since the PCD is not fully described by these first two moments, the PT estimator does not fully utilize all the information about g contained in the sample leading to larger estimator uncertainty compared to other techniques. Despite these disadvantages, the PT estimator is still attractive as it provides useful estimates of g in both the sub-electron and multi-electron read noise regimes and is calculated from basic sample moments; rendering it the most computationally inexpensive estimator of g.
Photon Counting Histogram Method
In response to the emergence of DSERN capable image sensors, the Photon Counting Histogram (PCH) method, developed at Dartmouth University, was the first documented method to explicitly incorporate the PCD model into the estimation of the sensor performance parameters. 5, 6 PCH is primarily a method for estimating conversion gain and read noise by detecting the locations of local maxima and minima observed in an experimentally generated histogram. To perform PCH characterization, a sample x = {x 1 , . . . , x n1 } with x k ∼ PCD(H, g, µ, σ 2 ) is captured and binned as a histogram, which is the experimental PCH. Each bin count is divided by the sample size n 1 to normalize the histogram so that it represents an approximation of the pixel's PCD at the chosen value of H. Assuming the read noise is small enough, many peaks (local maxima) in the experimental PCH should be present and an algorithm for detecting these peaks is deployed. Figure 3 shows a simulated PCH with the locations of ten detected peaks.
To estimate the conversion gain, let {(p x1 , p y1 ), . . . , (p xm , p ym )} denote a sequence of m consecutive peak locations detected in the experimental PCH. According to the PCD model, the abscissas of the peaks locations {p xk }, in units of (DN), are approximately located at equally spaced intervals of the form p xk = µ + k/g with k ∈ N 0 . As such, fitting a line to the data {(1, p x1 ), . . . , (m, p xm )} and extracting the reciprocal slope of the fit yields an estimateg for the conversion gain. If the electron number associated with each peak location is also known, one may instead fit a line to the data {(k 1 , p x1 ), . . . , (k m , p xm )} with the reciprocal slope again yielding g and the y-intercept yielding an estimate for the biasμ.
Estimation of the quanta exposure requires the two most prominent peak locations and their electron numbers denoted by {(p xk * , p yk * ), (p x(k * +1) , p y(k * +1) )} and (k * , k * + 1), respectively (see Figure 4). For small read noise values the ordinates of these peaks are approximated by
p yk ∼ 1 √ 2πσ e −H H k k! .
(10) * The numerical instability of this estimator for g is similar to the numerical instability of numerical derivatives. Taking the ratio of the two most prominent peaks and solving for H subsequently gives the estimate (see Figure 4)H
= (k * + 1) p y(k * +1) p yk * .(11)
In a similar manner, the read noise is calculated by first locating the valley (local minima) between the two most prominent peaks, denoted (v x * , v y * ), and then computing the Valley Peak Modulation (VPM)
VPM = 1 − v y * 1 2 (p yk * + p y(k * +1) )
.
The VPM is independent of the parameters g and µ so that a lookup table can be generated containing the VPM for various values of σ R and H. Using the estimateH, one can then lookup the value of σ R corresponding to the estimated VPM to obtain the PCH estimate of read noise. Assuming one is able to obtain estimates for all four parameters (which requires knowing the electron numbers for the detected peaks), the PCH estimates can be refined by fitting the PCD to the experimental PCH using nonlinear least squares with the initial parameter estimates as starting points.
PCH provides an intuitive graphical approach to sensor characterization and only requires a single sample of data. Because PCH incorporates the full description of the PCD model into the estimation procedure, it leverages the structure of DSERN data resulting in estimates of conversion gain with less uncertainty compared to the PT method. Furthermore, a unique feature of PCH is that the initial estimate of g obtained from peak locations does not assume a Poissonian light source so that sources of an arbitrary probabilistic nature can in theory be used. 7 The biggest challenge with this method is the need to reliably detect peak and valley locations, which requires both large sample sizes and sufficiently small read noise so that the peaks can be observed. For this reason, the applicability of the PCH method is restricted to the DSERN regime.
Fourier Transform Method
Initially, the Fourier-based approach for sensor characterization was not developed as a self-contained method, but rather as a means of deriving starting points for the PCH-EM algorithm discussed in Section 3.6. Nevertheless, this approach can be regarded as a characterization method in its own right. 8 The idea behind this technique, similar to the PCH method, stemmed from the fact that the PCD exhibits periodic oscillations when the read noise is sufficiently small. This approach revolves around the analytical expression for the magnitude of the PCD Fourier transform given by
|f X (ω)| := |E exp(−2πiωX)| = exp(H(cos(2πω/g) − 1) − 2π 2 σ 2 ω 2 ).(13)
Recall that when the read noise is sufficiently small, the PCD has local maxima occurring with a period of approximately 1/g (frequency of g). This property means the magnitude function should exhibit local maxima at frequencies approximately located at integer multiple of g as is seen in Figure 5 (blue curve). Let ω * denote the frequency corresponding to the local maxima near g, which is the second most prominent peak of the magnitude function after the primary peak at ω = 0. We note that in order for this secondary peak to exist we must have σ 2 e-/H < | min x>0 sinc x| = 0.217 . . . , where σ e-= σ × g is the read plus quantization noise in units of electrons and sinc x = sin x/x. Using Lagrange inversion and defining z = −σ 2 e-/H we compute
ω * = g + ∞ n=1 lim ω→g ∂ n−1 ω ω − g sinc(2πω/g) n z n n! = g(1 + z + z 2 + O(z 3 )),(14)
which shows that for small |z|, ω * is very well approximated by g.
With this information we can approximate the magnitude function |f X (ω)| near the secondary peak by considering the following asymptotic approximation as ω → g:
|f X (ω)| ∼ a exp(−2π 2 v(ω − b) 2 ),(15)
where Figure 5 shows the exact magnitude function (blue) compared to the asymptotic approximation (purple) along with the location of the secondary peak (ω * , |f X (ω * )|). As can be observed, the peak of the asymptotic approximation, (b, a), provides an excellent approximation to the location of the exact peak. To understand why this is, notice that this asymptotic expression approximates ω * as which shows agreement with the first three terms of the exact expansion for ω * obtained in (14).
v = VarX = σ 2 + H/g 2 , a = exp −2π 2 H − (H/g) 2 v ,(16)and b = H/g v .(17)ω * ∼ b = g 1 + σ 2 e-/H = g 1 − z = g(1 + z + z 2 + O(z 3 )),(18)
The system of three equations given by v, a, and b can be inverted to give the following approximations for H, g, and σ 2 :
H(v, a, b) ∼ vb 2 − log a 2π 2 g(v, a, b) ∼ b − log a 2π 2 vb σ 2 (v, a, b) ∼ v − v − log a 2π 2 b 2 −1 .(19)
Equipped with these details, the Fourier based method of characterization is as follows. First, a sample x = {x 1 , . . . , x n1 } with x k ∼ PCD(H, g, µ, σ 2 ) is captured and the sample meanx = 1 n1 n1 k=1 x k and sample variancex = 1 n1−1 n1 k=1 (x k −x) 2 are computed. Since the sample x is integer-valued, it is binned in bins centered on the integers and the counts c k for each bin location b k are normalized via p k = c k /n 1 so that we obtain a density normalized experimental PCH. The Discrete Fourier Transform (DFT) of the density normalized experimental PCH is then calculated and the location of the secondary peak in the DFT is detected yielding an estimate (b,ã). Initial estimates for H, g and σ 2 are then found from (19) giving (H 0 , g 0 , σ 2 0 ) = (H(x,ã,b), g(x,ã,b), σ 2 (x,ã,b)). Final estimatesH,g,σ 2 are then found by fitting the magnitude function (13) to the experimental PCH DFT using nonlinear least squares and the starting points (H 0 , g 0 , σ 2 0 ). Using the fact that EX = µ + H/g we then obtain an estimate for µ in the formμ =x−H/g. Further improvements onμ are possible using autocorrelation. 8 The Fourier based method echos that of the PCH method in that it requires sufficiently small read noise to work. It also requires some implementation of peak detection and refines the initial estimates with nonlinear least squares. In fact, the PCH and Fourier methods are essentially the same method with PCH operating in the original data space and Fourier operating in the frequency space. What makes the Fourier method attractive as that it obtains estimates of each PCD parameter from a single sample and requires only detecting a single peak in the experimental PCH DFT (compare this to detecting many peaks with PCH). Since only a single peak is needed and the method is based on the DFT it is also easily automated and computationally inexpensive. Additionally, this method makes full use of the PCD model so that it can usually obtain lower uncertainty estimates of the conversion gain in comparison to the PT method. The major downside of this method is that the read noise needs to be small enough to guarantee the existence of the secondary peak. This ultimately limits the applicability of this method, like the PCH method, to the DSERN regime.
Nakamoto's Method
In response to the PCH method, Nakamoto and Hotaka introduced a characterization technique based on the principle of Maximum Likelihood Estimation (MLE) that also takes advantage of the full PCD model. 7 To understand how this method works, let y = {y 1 , . . . , y n2 } with y k ∼ PCD(0, g, µ, σ 2 ) d = N (µ, σ 2 ) be a sample of data captured under dark conditions with short enough integration time so that dark current is negligible (H = 0). Since the distribution of this data is normal, unbiased estimates for µ and σ 2 may be directly measured from the y-sample viaμ
= 1 n 2 n2 k=1 y k(20)andσ 2 = 1 n 2 − 1 n2 k=1 (y k −μ) 2 ,(21)
which are the sample mean and sample variance, respectively. Now consider capturing a second sample x = {x 1 , . . . , x n1 } with x k ∼ PCD(H, g, µ, σ 2 ) for some choice of H > 0. Since Ex k = µ + H/g we can estimate the sample meanx = 1 n1 n1 k=1 x k and then construct an estimator for the quanta exposure as a function of g in the formH (g) = g(x −μ).
Then, a constrained likelihood function is made from these three estimates via
L(g|x) = n1 k=1 f X (x k |H(g), g,μ,σ 2 )(23)
and an estimate for g is computed by maximizing this constrained likelihood function (or equivalently it's logarithm)g = arg max g L(g|x).
An estimate for the quanta exposure follows by evaluatingH =H(g). Since closed-form solutions for this maximization are intractable, any number of numerical methods can be employed.
Like the PCH and Fourier methods, Nakamoto's method incorporates the full description of the PCD model into the estimation procedure, which generally results in conversion gain estimates with less uncertainty than can be obtained with the traditional PT method. Furthermore, the H = 0 sample allows for direct measurement of µ and σ 2 , which generally helps stabilize the conversion gain estimates when the read noise is large. Because peak detection is not part of the estimation procedure, Nakamoto's method shows promise in being a viable method for both the sub-electron and multi-electron read noise regimes. The most significant disadvantage of this method is the requirement to obtain a sample at H = 0, which may not be possible depending on the available integration times of the sensor and the magnitude of dark current. This requirement is particularly challenging when trying to characterize an entire sensor array, which will inevitably contain hot pixels that cannot achieve a quanta exposure near zero. Lastly, we note that this method requires two samples but does not make full use of the information contained in both samples. This can be seen by the fact that the x-sample contains information about µ and σ; however, these parameters are estimated only from the y-sample.
PCH Expectation Maximization Algorithm
The Photon Counting Histogram Expectation Maximization (PCH-EM) algorithm (in review at the time of writing) is the latest iteration of methods for performing sensor characterization based on the PCD model. 8,9,17 It is the first technique devised to compute simultaneous maximum likelihood estimates for all four PCD parameters using only a single sample of data. This method was inspired from the fact that when the electron numbers associated with each observation are known, that is, we have the complete data (x, k) = {(x 1 , k 1 ), . . . , (x n1 , k n1 )}, closed-form maximum likelihood estimators for each PCD parameter are easily derived. 8 While the electron numbers cannot be directly observed, this fact motivates a latent (hidden) variables model of estimation, which is what the general expectation maximization algorithm provides. As such, PCH-EM is a specific implementation of the general EM algorithm with the PCD being the underlying distribution to be estimated.
To perform PCH-EM, a random sample x = {x 1 , . . . , x n1 } with x k iid ∼ PCD(H, g, µ, σ 2 ) is captured. Given an initial estimate of the parameters θ 0 = (H 0 , g 0 , µ 0 , σ 2 0 ) (obtained by one of the other methods, e.g. Fourier or PT method), the PCH-EM algorithm iteratively updates the parameter estimates via the update equations 8
H t+1 = A t (25a) g t+1 = B t − H 2 t+1 C t −xH t+1 (25b) µ t+1 =x − H t+1 g t+1 (25c) σ 2 t+1 =x − B t − H 2 t+1 g 2 t+1 ,(25d)wherex = 1 n1 n1 k=1
x k is the sample mean,x = 1 n1 n1 k=1 (x k −x) 2 is the sample variance, and
A t = 1 N N n=1 ∞ k=0 γ (t) nk k (26a) B t = 1 N N n=1 ∞ k=0 γ (t) nk k 2 (26b) C t = 1 N N n=1 x n ∞ k=0 γ (t) nk k,(26c)
where
γ (t) nk = e −H t H k t k! φ(x n ; µ t + k/g t , σ 2 t ) ∞ ℓ=0 e −H t H ℓ t ℓ! φ(x n ; µ t + ℓ/g t , σ 2 t )
.
The γ (t) nk are called membership probabilities because they represent the probability of x n belonging to the kth Gaussian component of the PCD given the current parameter estimates θ t . Since PCH-EM is just a specific implementation of the more general Expectation Maximization (EM) algorithm, each iteration of the algorithm guarantees an increase in the likelihood of the sample. The algorithm halts when a specified convergence criteria is achieved. In practical implementation, all of the series in the update equations can be truncated to finite sums by only considering the terms k ∈ {F −1 (ǫ), . . . , F −1 (1 − ǫ)}, where F −1 is the Poisson(H t ) quantile function and ǫ > 0 is a small positive number.
PCH-EM provides many positive characteristics in that it provides maximum likelihood estimates of all the PCD parameters using a single sample of data, incorporates the full PCD model, and does not require numerical optimization, e.g. Newton iteration, to maximize the sample likelihood. It is also easily automated and computationally inexpensive although not as inexpensive as traditional PT. Furthermore, because the general EM algorithm is so well studied, many extensions of PCH-EM are possible to improve the robustness of the algorithm and its estimates. The major downside of this method, which holds for all other methods excluding PT, is the requirement of starting points. Poor starting points can result in slow convergence or convergence to a local maxima while missing the global maxima of the sample likelihood function. While PCH-EM can suffer form the issue of local maxima, extensions of the algorithm using annealing are possible. 18-20
Two-Sample PCH-EM Algorithm
Currently in development, PCH-EM2 is the two-sample generalization of PCH-EM that incorporates two samples taken at different H into a single likelihood function, thus offering both advantages and disadvantages similar to the single-sample PCH-EM approach. One significant benefit of PCH-EM2 is that it enables one to obtain low-uncertainty estimates of all four parameters by combining samples taken at two different H-values, as the uncertainty of estimates for each PCD parameter varies differently with H. Therefore, PCH-EM2 is expected to be more accurate than PCH-EM. Further extensions of this method to an arbitrary number of samples is also possible. Additionally, the use of an extra sample stabilizes the algorithm in the presence of high read noise, making it a good candidate for general estimation procedure in the sub-electron and multi-electron read noise regimes. Unlike Nakamoto's method, PCH-EM2 can extract information about each parameter from both samples, making it theoretically more accurate. However, PCH-EM2, like other two-sample methods, is more sensitive to nonlinearity in comparison to single-sample methods because it requires the sensor to behave linearly over both samples instead of just one.
COMPARISON OF METHODS
Design of Experiment
To compare the uncertainty of each methods' conversion gain estimates, Monte Carlo experiments were performed. However, the four-dimensional parameter space of the PCD model made it challenging to fully explore in simulations. To reduce the area of exploration, the parameter µ was set to zero for all simulation runs since it only shifts the PCD without changing its shape. Moreover, to avoid over-quantization, the conversion gain was fixed at g = σ R /6 for all runs, leaving only two dimensions (σ R , H) ∈ R >0 × R ≥0 to consider. As the read noise surpasses about 0.5 e-, peak detection methods struggle, so the read noise interval was limited to σ R ∈ (0, 2). Similarly, the Poisson distribution's dynamic changes occur mainly when H is small, so the quanta exposure was limited to H ∈ (0, 10). Using these constraints, a grid of 64 σ R -values on (0.05, 2) and 32 H-values on (0.05, 10) was created, which paired with µ = 0 and g = σ R /6, resulted in a total of 2048 points in the PCD parameter space to simulate data on.
Once the desired parameters were selected, the following stage in the experimental design involved selecting the kinds of data to simulate along with their corresponding sample sizes. Six methods were available, three of which (PCH, Fourier, PCH-EM) necessitated just one sample, while the remaining three (PT, Nakamoto, PCH-EM2) required two samples. Specifically, Nakamoto's approach required two samples, with one of them being at H = 0. To accommodate all methods two types of data were generated including dark samples of the form y = {y 1 , . . . , y n2 } with y k ∼ PCD(0, g, µ, σ 2 ) and illuminated samples of the form x = {x 1 , . . . , x n1 } with x k ∼ PCD(H, g, µ, σ 2 ). Observations in each sample were generated according to the model (1). The uncertainty in any method's conversion gain estimates will be a function of the parameters. For this reason, the optimal sample size pairs in (8) for acv 0 = 0.015 and ζ = (1 + H/σ 2 R ) −1 were chosen to make the uncertainty in the PT conversion gain estimates mostly independent of the parameters. In this way, PT would be a reference to compare the uncertainty of the five other methods against. The specified value of acv 0 = 0.015 means that the PT conversion gain estimates should have an uncertainty of approximately 1.5% across all parameters. Figure 6 shows the total sample size n 1 + n 2 used as a function of the parameters. Parameters where n 1 > 10 5 (seen as the white region in Figure 6) were ignored to make sure the experiment did not take too long to complete.
To ensure a fair comparison of each method, both the dark and illuminated data were made available to all six methods, even if the method naturally used only one sample. This way, the two-sample methods did not have access to more information than the one-sample methods. The two-sample methods required no changes to their approach since they inherently incorporated both samples into their estimation procedure. However, for the one-sample methods, the information in the dark sample was integrated into the estimation procedure by providing the starting points g 0 = (x −ȳ)/(x −ŷ), µ 0 =ȳ, σ 2 0 =ŷ, and H 0 = g 0 (x − µ 0 ), wherex represents the sample mean andx represents the sample variance for the x-data (and likewise for the y-data). With this step, the experimental design phase was concluded.
Results
The experiment was executed on MATLAB code containing two nested loops. In the outer loop, each iteration consisted of selecting the next set of PCD parameters and associated sample sizes. For each iteration of the outer loop, the inner loop was repeated 512 times, where in each of the 512 iterations a x and y sample were generated and then supplied to each method so that the conversion gain could be estimated. This subsequently resulted in 512 conversion gain estimatesg k for each method, which were then used to compute the normalized Root Mean Squared Error (RMSE)
RMSE(θ) = 1 512 512 k=1 (1 −g k /g) 2 1/2 .(28)
As a result of the Monte Carlo experiment, a 64 × 32 array of normalized RMSE values for each method was generated. Figure 7 shows the Monte Carlo estimated RMSE for each method as a function of σ R and H. The first row is comprised of the one-sample methods with the second containing only the two-sample methods. The black region corresponds to parameters where data was not simulated due to the sample sizes becoming too large.
Several observations can be derived from the Figure 7. Initially, it should be acknowledged that the PT estimates' RMSE remained relatively stable at approximately RMSE ≈ 0.015. This value corresponds to the acv 0 = 0.015 uncertainty specification used for the optimal sample sizes. Therefore, the optimal sample sizes effectively controlled the PT estimate uncertainty for the parameters considered. Additionally, all five methods that utilized the full description of the PCD in the estimation process showed a region below σ R ≈ 0.42, e-where their conversion gain estimates' uncertainty was generally lower than PT (blue strip). This finding was due to the PCD-inclusive techniques utilizing the data's extra structure at low read noise values, which PT ignores by only incorporating the first two moments.
Regarding the one-sample methods, the PCH and Fourier techniques outperformed PT in terms of RMSE below approximately σ R ≈ 0.42, e-; however, their performance degraded above this read noise value due to their reliance on peak detection. Unlike the PCH and Fourier methods, PCH-EM did not necessitate observing peaks and could still estimate the conversion gain at higher read noise values, bridging the gap into the multielectron read noise regime for one-sample methods. For the two-sample methods, both Nakamoto's method and PCH-EM2 produced conversion gain uncertainties comparable to PT's in the σ R > 0.42, e-regime. This finding suggests that once the PCD's structure is lost due to increasing read noise, there is only a small advantage to using the full PCD model in the estimation process. Overall, PCH-EM performed the best for one-sample methods, while PCH-EM2 outperformed PCH-EM and showed potential as a general estimation technique in the sub-electron and multi-electron read noise regimes.
DISCUSSION AND FUTURE WORK
This study presented an overview of the PCD model as a universal framework for describing all currently available methods of conversion gain estimation in the sub-electron and multi-electron read noise regimes. By unifying the notation and model, it became possible to compare and contrast the differences between these methods.
Monte Carlo experiments revealed that utilizing the full PCD model in the estimation procedure produced conversion gain estimates with less uncertainty than the traditional PT method, especially when the read noise is below σ R ≈ 0.42 e-. Notably, the PCH-EM2 algorithm outperformed the time tested PT method below this threshold, while its performance merged with that of PT at higher read noise levels. This suggests that PCH-EM2 could potentially replace PT as a general estimation procedure. Future research will involve developing and implementing a multi-sample (≥ 2 sample) PCH-EM method and exploring the use of annealing to enhance the algorithm's robustness to poor starting points.
Figure 1 .
1Histograms produced by a scientific grade CCD pixel (left) and DSERN CMOS pixel (right).
Figure 2 .
2Plots of the PCD for various H and σ 2 with µ = 0 and g = 1 fixed.
Figure 3 .
3Simulated PCH showing ten detected peak locations.
Figure 4 .
4Simulated PCH showing two most prominent peaks with corresponding valley location used for quanta exposure and read noise estimation.
Figure 5 .
5Graph of |fX (ω)| (blue) and its asymptotic approximation near ω = g (purple) versus ω showing the two most dominant peaks at ω = 0 and ω = ω * .
Figure 6 .
6Total number of samples used for each set of PCD parameters. The black region corresponds to points where n1 > 10 5 , which were ignored in the simulation.
Figure 7 .
7Comparison of six characterization methods in terms of the RMSE achieved for conversion gain estimates. Single-sample methods are shown in first row with two-sample methods in the bottom row.
ACKNOWLEDGMENTSThe authors wish to express their gratitude to Nicholas Shade at Dartmouth University for his feedback on the implementation of the PCH method. The authors also would like to acknowledge and thank Katsuhiro Nakamoto from Hamamatsu Photonics for his help in implementing his method. Their contributions have been invaluable to the research, and the authors are appreciative of their assistance.
Determination of the conversion gain and the accuracy of its measurement for detector elements and arrays. B P Beecken, E R Fossum, Appl. Opt. 35Beecken, B. P. and Fossum, E. R., "Determination of the conversion gain and the accuracy of its measurement for detector elements and arrays," Appl. Opt. 35, 3471-3477 (Jul 1996).
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Determining conversion gain and read noise using a photon-counting histogram method for deep sub-electron read noise image sensors. D A Starkey, E R Fossum, IEEE Journal of the Electron Devices Society. 4Starkey, D. A. and Fossum, E. R., "Determining conversion gain and read noise using a photon-counting histogram method for deep sub-electron read noise image sensors," IEEE Journal of the Electron Devices Society 4, 129-135 (May 2016).
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| []
|
[
"Brownian Bees with Drift: Finding the Criticality",
"Brownian Bees with Drift: Finding the Criticality"
]
| [
"Donald Flynn \nUniversity of Oxford\n\n"
]
| [
"University of Oxford\n"
]
| []
| This dissertation examines the impact of a drift µ on Brownian Bees, which is a type of branching Brownian motion that retains only the N closest particles to the origin. The selection effect in the 0-drift system ensures that it remains recurrent and close to the origin. The study presents two novel findings that establish a threshold for µ: below this value, the system remains recurrent, and above it, the system becomes transient. Moreover, the paper proves convergence to a unique invariant distribution for the small drift case. The research also explores N-BBM, a variant of branching Brownian motion where the N leftmost particles are retained, and presents one new result and further discussion on this topic. | null | [
"https://export.arxiv.org/pdf/2304.14079v1.pdf"
]
| 258,352,479 | 2304.14079 | 5e85c16affcfdf5304ac25d76a4f0f4c377cf4c3 |
Brownian Bees with Drift: Finding the Criticality
27 Apr 2023
Donald Flynn
University of Oxford
Brownian Bees with Drift: Finding the Criticality
27 Apr 2023dissertation submitted for the degree of MMath Mathematics Trinity 2023
This dissertation examines the impact of a drift µ on Brownian Bees, which is a type of branching Brownian motion that retains only the N closest particles to the origin. The selection effect in the 0-drift system ensures that it remains recurrent and close to the origin. The study presents two novel findings that establish a threshold for µ: below this value, the system remains recurrent, and above it, the system becomes transient. Moreover, the paper proves convergence to a unique invariant distribution for the small drift case. The research also explores N-BBM, a variant of branching Brownian motion where the N leftmost particles are retained, and presents one new result and further discussion on this topic.
Introduction
Branching Brownian motion (BBM) is a stochastic process involving independent particles moving according to Brownian motion. Initially there is a single particle following a Brownian trajectory, until after an independent exponential time of rate 1, it branches into two new Brownian particles, each of which has its own exponential branching clock of rate 1. The study of BBM in its most basic form dates back to 1975; introduced by McKean [19].
As is common with many mathematical models, after the introduction of BBM different authors considered making small, physically motivated changes to the construction and considering what effects this has on its properties. One such alteration is the "N-BBM"; a model to which a large part of this dissertation is focused on. In this model we start with N particles performing branching Brownian motion in one dimension, and whenever a particle branches, such that there are N + 1 particles, we instantaneously remove the particle furthest to the left, i.e. the particle whose position is less than any other particles (and with arbitrary choice in a tie).
Another alteration is the "Brownian Bees" process 1 . Evolving similarly to N-BBM, such that whenever we have more than N particles one is removed -however this time we remove the particle furthest away from the origin. This process is a more well behaved object than the N-BBM due to the compacting effect of the selection rule, leading to the existence of limits for the cases N → ∞ and t → ∞. A pair of papers by J. Berestycki et al established a nice result of how in a certain sense these limits commute. [3,2].
In this dissertation we consider the effect of adding a drift µ to the Brownian Bee process, so that each particle evolves according to a Brownian motion with drift. The results of this paper can then be summarised as: There is a critical value v N such that if |µ| > v N then the system is transient, and if |µ| < v N the system is recurrent, and furthermore this v N is the asymptotic speed of the N-BBM. The precise statement of these will be given by Theorems 3.0.2 and 3.0.3.
We can construct a generalisation of the above processes (following [5] 2 ), where we describe the selection rule by a "score" function s : R d → R, and then we refer to the process as a "Brunet Derrida system with score function s". Furthermore we may add a drift µ ∈ R d to the Brownian motions driving the particles between branching times. The process Y (N,µ) = (Y 1 (t), . . . Y n (t)) is then defined informally by:
• Each particle moves according to an independent Brownian motion with drift µ.
• Every time an exponential clock of rate N fires, the particle with the lowest value of s(Y i (t)) jumps to the position of a randomly chosen particle Y k (t) where k is chosen uniformly from {1, . . . , N } Note in this definition that a particle can "jump to itself", where in this case no change will occur.
Following this notation, the N-BBM is recovered by the function s(x) = x, whereas the Brownian Bee process is recovered by s(x) = −|x|.
These models have received some attention in the literature since the initial conception of a one-dimensional such model, by Brunet, Derrida, Mueller and Munier 2006/7 in the papers [7,10].
Motivation
Some physical motivation for the study of Brunet-Derrida systems can be provided by the study of evolution [11,8] . Broadly speaking, the Brownian particles represent the fitness of different members of an asexually reproducing species. The population is taken to be fixed at size N due to external factors, and the members of the species that survive are those with the highest fitness function s. The Brownian motion aspect then represents some white noise caused by mutations upon reproduction. The different functions of s then allow for modelling of different types of evolutionary behaviour.
We can give additional motivation for the model of Brownian Motion with drift, although it should be prefaced with the fact that this author is not a biologist. In the standard Brownian Bees model, it could be viewed that the origin represents the "optimum fitness" for the environment that the species is in -giving the interpretation that the convergence in t of the Brownian Bee model represents the species eventually adapting to the environment they are in ([2] Theorem 1.2). Adding in a drift to the Brownian motion can be viewed as moving the origin to position µt at time t, this could represent this "optimum fitness" changing over time, for example due to climate change or other external factors. It is then useful to study whether the population is able to adapt fast enough to keep up with these external changes.
Notation and Formal Construction
In this section we will formally construct processes in enough generality to cover N-BBM, Brownian Bees and Brownian Bees with drift. We shall refer to such systems as "Brunet-Derrida systems with drift µ"
We will construct a process with a drift µ ∈ R d and a "score function" s : R d → R, where the score function determines which particle is removed at branching times. For example for standard N-BBM in one dimension, we take µ = 0, and s(x) = x.
We shall make the convention that in R d , d > 1 we label the particles such that
s(Y 1 (t)) ≤ . . . ≤ s(Y N (t))
however in R it is more convenient to use the ordering
Y 1 (t) ≤ . . . ≤ Y N (t)
We generally label the process by Y (N,µ) (where the score function used will be clear from context). Furthermore, we write Y (N,µ) = (Y 1 , . . . Y N ), suppressing the dependence on N and µ, which again will be clear from context. Now for the formal construction, we adapt from [5]: Let (J i ) i≥0 be the jump times of a Poisson process with rate N with J 0 = 0, and let (K i ) i≥1 be an independent sequence of i.i.d. uniform random variables on {1, . . . , N }. The process is started in some given initial condition (Y 1 (0), . . . Y N (0)). Then inductively, for each i ≥ 1, assuming that the system is defined up to time
J i−1 with Y 1 (J i−1 ) ≥ · · · ≥ Y N (J i−1 ), we define Y n (t) = Y n (J i−1 ) + B n (t) + µt, t ∈ [J i−1 , J i )
, where (B n (t), t ≥ 0) N n=1 are independent Brownian motions in R, independent from (K i ) and (J i ). At time J i , we duplicate particle Y K i J − i and remove the particle min 1≤n≤N s Y n J − i . Note that if the duplicated particle is the particle with minimum score, the net effect is that nothing happens. We now relabel the particles over this interval in so that they are increasing in R (or in the case R d , d > 1 we order such that they are increasing in their score function)
Y 1 (t) ≤ . . . ≤ Y N (t), t ∈ [J i−1 , J i ] The process Y (N,µ) = (Y 1 (t), . . . Y N (t))
is then a Brunet-Derrida system with score function s and drift µ.
We can now recover the processes mentioned at the beginning by taking:
• d = 1, s(x) = x and µ = 0 for standard N-BBM • s(x) = −||x||, µ = µ and d = d for d-dimensional Brownian bees with drift µ
Additionally, we introduce the concept of ancestors and children. The time s ancestor of a particle Y n (t) for s < t is a particle that Y n (t) is a direct descendent of through a duplication event. Additionally, the time t children of a particle Y k (s) are all the particles that are directly descended from Y k (s) through duplication events. We can see that each particle has precisely one ancestor at any given previous time, however a particle may have between 0 and N children at any later time.
We further remark that since the particles are ordered by position at any time t it is not necessarily the case that Y n (s) is an ancestor of Y n (t).
We will also refer to F t , the filtration generated by the whole process; so if N t := #{i ≥ 1 : J i ≤ t} is the counting process associated to the jump times J i then,
F t := σ (B 1 s ) s≤t , . . . , (B N s ) s≤t , (N s ) s≤t , (K i ) i≤Nt N-BBM
Heuristics and Current Results
One-dimensional Branching Bronwian motion with selection (N-BBM) first received attention, in Brunet, Derrida, Mueller and Munier's influential papers [7,10]. The original purpose of such a model was to study "noisy F-KPP equations" that is the Fisher-Kolmogorov-Petrovskii-Piskounov equation
∂ ∂t u = 1 2 ∂ ∂x 2 u + F (u) (2.1) Where the term F (u) = β(1 − u − f (1 − u)).
Since the models conception, however there have been several other applications of its study. For a more detailed discussion of these see the introduction of Pascal Malliads 2012 thesis [17].
A quick relation of the N-BBM to the F-KPP equation is from the free branching Brownian motion process, that is a particle started at the origin and moving according to a Brownian motion. The particle branches to form new independent branching particles at rate 1, with no killing occurring.
It can then be shown that the probability u(x, t), that there exists a particle to the right of x at time t satisfies Equation 2.1 (In fact this is the very reason BBM was first introduced [19]).
The precise reason for the relevance of the N-BBM model is a little more subtle, and comes from the study of the "cut-off" equation introduced by Brunet and Derrida in [12], where the forcing term in the F-KPP equation 2.1 is multiplied by 1 u≥N −1 , allowing for better modelling of real world phenomenon. For the purposes of this dissertation we simply assure the reader that N-BBM is a worthy object of study and direct them to [17] for more details on this.
Letting X (N ) = (X 1 (t), . . . X N (t)) be a standard N-BBM; some key conjectures made by Brunet and Derrida about this model in [9] are amongst others:
1. For every N , lim t→∞ X 1 (t) t = lim t→∞ X N (t) t = v N as N → ∞ 2. v N = √ 2 − π 2 √ 2(log N ) 2 1 − 6 log log N (log(N ) 3 (1 + o(1)) 3. diam t := X N (t) − X 1 (t) ≈ log(N ) √ 2
4. The genealogical timescale of the population is (log N ) 3 and converges to the Bolthausen-Sznitman coalescent
We won't delve into the 4 th point here too much, other than remarking that it is a very deep result and more discussion and definitions are given in [4]. Broadly speaking, understanding the genealogical time scale allows us to understand how far back in time we need to go before we find a common ancestor of the whole living population. It is easy to see then why this is a point of interest, due to the study of N-BBM being partially motivated by population genetics.
As for the other points on this list, A proof of (1) was first given for a very similar model by [13] by Bérard and Gouéré, and this was adapted by N.Berestycki and Zhao for the case of N-BBM. The authors extended this result to d dimensions (Theorem 1.1 in [5]), however we state their result for the simplest case d = 1 here:
Theorem 2.1.1. Let N > 1 and let (X 1 , . . . X N ) be an N-BBM process. Then |X N (t) − X 1 (t)| t → 0 (2.2) as t → ∞ almost surely. Moreover, X 1 (t) t → v N (2.3) almost surely, where v N are a deterministic constants
This result then tell us that for large times the N-BBM moves in a ballistic manor, and furthermore has a deterministic asymptotic velocity v N . Point (2) is harder to approach, though it has been partially settled by Bérard and Gouéré in the same paper. They showed that for a branching random walk with selection (a very similar but not quite identical process to N-BBM), the velocity of the system has a correction term of order (log N ) −2 + o ((log N ) −2 ). This result should in theory be easily adaptable to the case of N-BBM, however due to the technical nature of their proof no rigorous adaption to N-BBM has been given. Getting the "third order" 6 log log N (log(N ) 3 (1 + o(1)) correction term out is a harder and still open problem, with currently no known approaches for how to tackle this.
We state this conjecture for the second order correction to v N here:
v N = √ 2 − π 2 √ 2(log N ) 2 + o (log N ) −2 (2.4)
Although there is no formal proof for the asymptotic of v N , we do have a proof that v N are monotonically increasing in N , following from a useful coupling of the N-BBM between different values of N . This is given as Lemma 2.3 in [5] and we state this here also: Lemma 2.1.3. Let (X n (t), 1 ≤ n ≤ N ) t≥0 and (Y n (t), 1 ≤ n ≤ N ) t≥0 , N ≤ N be standard N-BBMs. Suppose they are initially ordered such that there is a coupling
Y 1 (0) ≥ X 1 (0); . . . ; Y N (0) ≥ X N (0)
Then we can couple X(t) and Y (t) for all times t such that X i (t) ≤ Y i (t) for all t ≥ 0 and 1 ≤ i ≤ N A further strong heuristic for why Conjecture 2.1.2 should be true is given by [17], where it is shown that at the timescale of (log N ) 3 the process converges as N → ∞ to a specified distribution around v N t (See Theorem 1.1 for exact setup). Where by timescale here, we mean taking the limits t → ∞ and N → ∞ simultaneously, such that t = t 0 (log N ) 3 . The speed here is then shown to be as conjectured, however this is not quite enough to settle the result, as we can't unpack the double limiting result as we would wish to.
Returning to the third conjecture of Brunet and Derrida about the diameter, it is not easy to even rigorously define what "≈" means in this context. A related result is proven by [5], where adapting the result for one dimension, the authors showed that provided the initial conditions satisfy some technical result there exists some constants a and c such that
lim inf N →∞ P (diam t ≤ a log N ) = 1, for t = c(log N ) 3 (2.5)
If we allow ourselves to interpret (log N ) 3 as some "large" time depending on N , this then gives us an upper bound for the diameter.
It is perhaps interesting now to remark on the results of Pain 2015 [20] who considered a model related to N-BBM. This model, titled "L-BBM" instead of having a fixed population size N has a fixed population diameter L, such that if any particle is further than L away from the leading particle, it is instantly killed. The particles otherwise move and branch as in a free branching Brownian motion.
Pain showed that the velocity of the L-BBM has asymptotic speed
v L := √ 2 − π 2 2 √ 2L + o 1 L 2 (2.6)
which if we allow ourselves to naively take the heuristic "L = log N 2 ", exactly recovers the second order conjectured asymptotic of the N-BBM. Though this gives a nice connection between L-BBM and N-BBM, it is far from rigorous, and in fact the proof of 2.6 does not make use of any coupling to an N-BBM type model.
Despite all this uncertainty surrounding the N-BBM model, some basic properties of the model are unambiguously and easily seen to be true, two of which are:
• The process is translation invariant. i.e. if X = (X 1 , . . . X N ) is an N-BBM started in position X 1 (0), . . . X N (0), and Y is an N-BBM started in position Y k (0) = X k (0)+χ for some χ ∈ R, then we may couple so that Y k (t) = X k (t)+χ
• The process is a strong Markov process. Since it is driven by a combination of Brownian motions and exponential distributions, both of which are strong Markov processes.
Theorem 2.2.1
We now prove a new result (to this author's knowledge) relating to the expectation of the hitting time of an N-BBM when we give the process a drift, −µ.
Theorem 2.2.1. Let X = (X 1 , . . . X N ) be a standard N-BBM with killing from the left started at X 1 (0) = . . . = X N (0) = 0. Then let µ be fixed with 0 ≤ µ < v N and When giving the proof of this theorem, we break it down into the cases N = 2 and N > 2. This is not because the case of N = 2 is more difficult but rather the opposite; the proof for N > 2 does work for N = 2 however N = 2 is much simpler. We present this therefore as an easier to follow "warm up" result. Additionally we break up the proof in this manor to record the progress of this dissertation, since before the proof of N > 2 was found, we coupled N > 2 to N = 2 to prove the above result for all N , but with the more restrictive assumption of µ < v 2 = 3 8 √ 2 . Instrumental in our proof will be the Lemma: Lemma 2.2.2 (First passage times of random sums). Let L 1 , L 2 , . . . be i.i.d random variables with E[L n ] > 0. Then let S n := n k=1 L k and for any R ∈ R + let τ R := inf{n : S n ≥ R} Then there exists constants α and β depending only on the distribution of L 1 such that:
τ R,µ := inf{t > 0 : X 1 (t) − µt ≥ R}E[τ R ] ≤ α + βR
Proof. This is given by [16] Equation (1.5). We have also found but omit an elementary proof under the assumption E[|L 1 | 3 ] < ∞. (Using a higher order Chebyshev-like inequality to bound P[S n ≤ R], and then using
E[τ ] = P[τ ≥ n] ≤ P[S n ≤ R])
In our proof we find a way to compare the N-BBM to a random variable that looks like n k=1 L k , which will allow us to use the above Lemma. Proof (N = 2): To prove the case N = 2 we make use of the fact that the 2-particle system has a regenerative structure that makes computation much easier, since at every time the rightmost particle branches, both particles return to the same point. This allows us to be much more explicit with our calculations, and in fact even allows us to calculate v 2 exactly.
We let T n be the branching times of the leftmost particle of X (since nothing happens when the rightmost particle branches). Then T n form a Poisson process of rate 1, and we may define a discrete time Markov chain Z n :
= X 1 (T n ) − µT n Hence if the i.i.d random variables L 1 , L 2 , . . . have the distribution of L k d = max B 1 (T ), B 2 (T ) − µT , where B 1 (t), B 2 (t) are i.i.d Brownian
Motions and T is an independent exponential time, then by the strong Markov property at T n we can see that Z n has the same distribution as n k=1 L k . Then by for example [14] 1.0.5 we may find that the distribution for B 1 (T ) is given by
P(B 1 (T ) ∈ dz) = 1 √ 2 e −|z| √ 2
We may use this to calculate E[L k ]. Giving:
E(L k ) = E max B 1 (T ), B 2 (T ) − µE(T ) = −µ + x=∞ x=−∞ y=∞ y=x y 2 e − √ 2(|x|+|y|) dydx = 3 8 √ 2 − µ
Where the constant 3 8 √ 2 comes from evaluating the integral. 1 Then
E(τ R,µ ) = ∞ 0 P(τ R,µ ≥ t)dt = E ∞ 0 1 τ R,µ ≥t dt
by Fubini. If N t := sup{n : T n ≤ t} is then the counting process corresponding to the Poisson process of branching times T n , then since T Nt ≤ t it follows that
E ∞ 0 1 τ R,µ ≥t dt ≤ E ∞ 0 1 τ R,µ ≥T N t dt = E ∞ k=1 (T k+1 − T k )1 τ R,µ ≥T k = ∞ k=1 E (T k+1 − T k )1 τ R,µ ≥T k = ∞ k=1 E 1 τ R,µ ≥T k
Where we have used Fubini again, and that since as τ R,µ and T k are stopping times for the strong Markov process X, then T k+1 − T k is independent from the F T k measurable random variable 1 τ R,µ ≥T k , and furthermore since T k+1 − T k is a rate 1 exponential distribution, its expectation is 1.
1 This constant is in fact equal to v 2 since, we can view the process Z n as a renewal reward process (see e.g. [21] Chapter 7) and so the Elementary Renewal Theorem for Renewal Reward processes gives that
v 2 − µ = lim n→∞ X 1 (T n ) − µT n T n = E[L k ] E[T 1 ] = 3 8 √ 2 − µ
Finally then, if we defineτ R,µ to be the first time that Z n exceeds R, i.e.τ R,µ = inf{n :
Z n ≥ R}, then ∞ k=1 E 1 τ R,µ ≥T k ≤ ∞ k=1 P (τ R,µ ≥ k) = E[τ R,µ ] ≤ α + βR
Where in the final step we have applied Lemma 2.2.2, since Z n = n k=1 L k
In order to modify this proof so that it works for N > 2 we have to make some changes. Firstly, we fail to get times T n such that the process renews at this point, since for N > 2 the probability that we have at least three particles in the same position at any time t > 0 is 0. The basic idea to get around this issue is to define times
T n ≈ inf t > T n−1 + 1 : sup 1≤n,m≤N |X n (t) − X m (t)| ≤
Which are times where the particles are almost at the same position. The precise definition of T n is slightly different as we also want to ensure that T n − T n−1 are i.i.d random variables. We then couple processes X ,+ and X ,− which correspond to moving all the particles at time T n to the position of X 1 (T n ) and X N (T n ) respectively. Since the process is very close together at time T n the result is that we "only lose " from the drift at each replacement. And then the bounding processes X ,+ and X ,− can be analysed in a similar manor to the proof of N = 2, since at the stopping times T n they are the sum of i.i.d random variables.
Proof (N ≥ 2): Let Y k (t) := X k (t) − µt
Then let > 0 be a constant to be fixed later and iteratively construct processes Y ,± = (Y ,± 1 , . . . , Y ,± N )and the stopping times T n . Firstly, let T 0 = 0 and Y ,−
i (0) = 0 = Y ,+ i (0) for 1 ≤ i ≤ N . Now, we inductively construct Y ,− and Y ,+ such that: • Y ,− i (t) ≤ Y i (t) ≤ Y ,+ i (t) • Y ,+ i (t) = Y ,− i (t) + n for t ∈ [T n , T n+1 )
• At time T n all the particles in Y ,− are in the same position, and similarly for Y ,+ .
Assume that we have constructed T n ; Y ,+ (t) and Y ,− (t) for all t ≤ T n . Then since T n is a stopping time, we may use the strong Markov property to note that for t ≥ T n , Y (t) behaves as an independent N-BBM. Then since at time
T n , Y ,− i (T n ) ≤ Y i (T n ) ≤ Y ,+ i (T n )
, we may use Lemma 2.1.3 (with the same values of N ) to couple the processes Y ,− (t), Y (t) and Y ,+ (t), where we will couple these process up to a stopping time T n+1 (about to be defined) Then in fact if we unpack this coupling process, we are just using the same Brownian Motions for each process between branching times, and immediately after branching times we choose the Brownian motions such that particles in the same relative position have the same Brownian motion (i.e. Y i (t) evolves by the same Brownian motion as say Y ,+ i (t)) between branching times). It follows then that whilst Y ,+ and Y ,− are evolving according to this coupling, they are exactly translated versions of each other, since at time T n both systems had all particles in one place.
Therefore if we define
T n+1 := inf t > T n + 1 : sup 1≤i,j≤N |Y ,− i (t) − Y ,− j (t)| ≤
Then this stopping time will also be the first time that the particles of Y ,+ are all within of each other.
Finally then, at time T n+1 we define Y ,− 1 (T n+1 ) to be the value given by the coupling, and then we redefine
Y ,− i (T n+1 ) := Y ,− 1 (T n+1 ) for all 1 ≤ i ≤ N
. Now let us check that we have satisfied the inductive properties we claimed:
• Y ,− i (T n+1 ) := Y ,− 1 (T n+1 ) ≤ Y 1 (T n+1 ) ≤ Y i (T n+1 ) Since Y ,− 1 (T n+1 )
is the value given by the coupling, and we know by construction that this is bounded above by
Y 1 (T n+1 ) • Y i (T n+1 ) ≤ Y ,+ i (T n+1 ) ≤ Y ,+ 1 (T n+1 ) + = Y ,− 1 (T n+1 ) + (n + 1)
Where the values of Y ,+ above are given by the coupling. These inequalities follow since we know at time T n+1 the particles of Y ,+ are at most apart, and furthermore, before we define the values at time T ,+ we have that Y ,
+ 1 (T n+1 ) = Y ,− 1 (T n+1 ) + n Hence, if we now redefine Y ,+ i (T n+1 ) := Y ,− 1 (T n+1 ) + (n + 1)
we complete the inductive definition and get the required properties.
Next, note that between times T n and T n+1 the processes Y , ± evolve as N-BBMs started with all particles at a single point, where here we are invoking the strong Markov property and that T n are stopping times adapted to the filtration on which Y , Y ,± are defined. Hence we can see from the definition of T n that the random variables S k := T k − T k−1 are i.i.d. And furthermore that the random variables
L k := Y ,− (T k ) − Y ,− (T k−1 ) are also i.i.d. Note also that from the coupling we have Y ,+ (T k ) − Y ,+ (T k−1 ) = L k +
The point now, is at the times T n = n k=1 S k , we have that Y ,− (T n ) = n k=1 L k and Y ,+ (T n ) = n k=1 (L k + ) It remains to be shown, however that E[S k ] < ∞. To this end let F t be the filtration to which the whole process is adapted, and we will show that S 1 = T 1 is bounded by a geometric random variable. Similar arguments to this will be used several times throughout this dissertation so will not go into too much detail (see proof of Proposition 3.0.4 and for a more detailed version of this argument) Consider deterministic times t = k for k = 1, 2, . . .. Then at each time let U k be the event that
• No particles branch during time t ∈ [k, k + 1/2) • sup 1≤n≤N −1 sup t∈[k,k+1/2) |Y i (t) − Y i (k)| ≤ 1/2 • The particle furthest to the right Y N (t) moves up by 1 during time t ∈ [k, k+1/2) so Y N (k + 1/2) − Y N (k) > 1
• During time t ∈ [k + 1/2, k + 1) None of the Brownian motions driving the particles move by more than /N
• The largest particle branches N times during time t ∈ [k + 1/2, k + 1)
By the Markov property of the system, it is then clear that U k are independent events of the same probability, and then by standard properties of Brownian motions and exponential distributions it follows that P(U k ) > 0. Additionally if the event U k happens then T 1 ≤ k + 1 since the largest particle branching N times without any particles moving very far ensure that all particles are contained within of each other.
Hence
P(T 1 ≤ k + 1 | F k ) ≥ P(U k ) = P(U 1 ) And hence by Lemma A.0.2 E[T 1 ] < ∞ Now we wish to show that E[L k ] > 0 for sufficiently small. to do this, note that as n → ∞ we have T n → ∞ since by construction S k ≥ 1. Hence by Theorem 2.1.1 lim n→∞ Y 1 (T n ) T n = v N − µ > 0
almost surely But then by the strong law of large numbers
lim n→∞ Y ,− 1 (T n ) T n = lim n→∞ 1 n n k=1 L k lim n→∞ 1 n n k=1 S k = E[L 1 ] E[S 1 ]
and similarly
lim n→∞ Y ,+ 1 (T n ) T n = E[L 1 ] + E[S 1 ] And so since Y 1 is sandwiched between Y ,− and Y ,+ it follows that E[L 1 ] E[S 1 ] ≤ v N − µ ≤ E[L 1 ] + E[S 1 ]
Then note that the map → E[S 1 ] is non-increasing, and hence we make take small enough so that
E[S 1 ] < v N −µ 2
Hence for some fixed small we have that
E[L 1 ] ≥ v N −µ 2 > 0. Then since t ≥ T N t , we calculate: E[τ R,µ ] = E ∞ 0 1 τ R,µ ≥t ≤ E ∞ 0 1 τ R,µ ≥T N t = E ∞ k=0 (T k+1 − T k )1 τ R,µ ≥T k = ∞ k=0 E (T k+1 − T k )1 τ R,µ ≥T k = E[S 1 ] ∞ k=0 E 1 τ R,µ ≥T k
Where this final step follows since τ R,µ and T n are stopping times, so the event {τ R,µ ≥ T k } ∈ F T k and hence by the strong Markov property,
S k+1 = T k+1 − T k is independent from F T k
Now proceeding a similar manor to the case of N = 2, define Z n := Y ,− 1 (T n ), and define N t := sup{n : T n ≤ t} the counting process associated with the renewal process T n . Then letτ R,µ := inf{n : Z n ≥ R} be the first time that Z n exceeds R.
Then
P(τ R,µ ≥ T k ) ≤ P(τ R,µ ≥ k) follows from the fact that Z n ≤ Y N (T n ). Hence E[τ R,µ ] ≤ E[S 1 ] ∞ k=0 E 1τ R,µ ≥T k ≤ E[S 1 ] (1 + E[τ R,µ ])
Then finally we deduce from Lemma 2.
2.2 that E[τ R,µ ] < α + β R for some (µ dependent) α and β since Z n is a random sum of the i.i.d variables n k=1 L k and E[L k ] > 0 And hence E[τ R,µ ] ≤ α + βR
We now move onto Brownian Bees with drift. To provide some (brief) background, the results in this section are inspired by the papers by J. Berestycki et al [2,3].
The results of these papers can be summarized in a commutative diagram, concerning Brownian Bees 1 in d dimensions with no drift (see [2] for more discussion and explanation of this diagram):
For this dissertation we focus on the left hand side of this diagram: the convergence in time of the Brownian Bees process. We state the corresponding result from [2] Theorem 3.0.1. Let X (N ) (t) be d dimensional Brownian Bees. Then for N sufficiently large, the process (X (N ) (t), t > 0) has a unique invariant measure π (N ) , a probability measure on R d N . For any χ ∈ R d N , under P χ , the law of (X (N ) (t) converges in total variation norm to π (N ) as t → ∞:
lim t→∞ sup C |P χ X (N ) (t) ∈ C − π (N ) (C)| = 0
This theorem tells us that the Brownian Bees are recurrent and have a unique stationary distribution, to which they converge as t → ∞. In this section we shall apply a drift µ to the Brownian Bees, and show that for µ smaller than the threshold v N a very similar result to 3.0.1 holds, whereas for µ larger than this threshold, we have transience, and the system behaves more like an N-BBM.
The main new results of this dissertation are then:
Theorem 3.0.2. Let Y (N,µ) = (Y 1 (t), . . . , Y N (t)) ∈ R N be a one dimensional Brown- ian Bee system with drift µ ∈ R, ordered such that at each time t, Y 1 (t) ≤ . . . ≤ Y N (t). Then if |µ| > v N ,
where v N is the asymptotic speed of an N-BBM (2.1.1), then it holds that:
lim t→∞ Y 1 (t) t = 1 − v N |µ| µ (3.1)
Theorem 3.0.2 tells us that if the drift µ is larger than the criticality v N then the Brownian Bee system is transient, and gives us an asymptotic of its escaping velocity. We contrast this with Theorem 3.0.3 which tells us the system is recurrent if |µ| < v N , and converges as t → ∞:
Theorem 3.0.3. Let Y (N,µ) = (Y 1 (t), . . . Y N (t)
) be a one dimensional Brownian Bee system with drift µ ∈ R such that |µ| < v N where v N is asymptotic speed of an N-BBM. Then there is a unique stationary measure π (N,µ) on R N so that for any χ ∈ R N , under P χ the law of Y (N,µ) converges in total variation norm to π (N,µ) as t → ∞: lim
t→∞ sup C P χ Y (N,µ) ∈ C − π (N,µ) (C) = 0
where the supremum is over all Borel measurable C ∈ R N
A key result in proving this theorem is being able to show that for any initial condition, the particle system will return to 0, and furthermore it will do so in a time of bounded expectation. The method of proof for this is by comparison with an N-BBM, and the use of Theorem 2.2.1, since if all particles are on one side of 0, the left/right most particles will always be the ones killed, allowing us to couple the Brownian Bees to a N-BBM.
Proposition 3.0.4. Let Y (N,µ) = (Y 1 (t), . . . Y N (t)) be a one dimensional Brownian Bee system with drift µ ∈ R such that |µ| < v N where v N is the asymptotic speed of an N-BBM.
Then for any initial condition, for
τ := inf {t > 0 : Y k (t) = 0, for some k ∈ {1, . . . , N }}
It holds that τ is almost surely finite. Moreover, if additionally Y (N,µ) (0) ∈ [−R 0 , R 0 ] N has all particles within R 0 from the origin, then there are deterministic constants α = α µ and β = β µ not depending on R 0 such that
E[τ ] ≤ α + βR 0 3.1 Proof of Theorem 3.0.2
Before stating the proof, we first give an overview of the idea. We first consider the systemX n = (Y n − µt). Which we can view as a Brunet-Derrida system where the fitness function is changing over time, and we remove the particle that is furthest away from the point −µt. We then use a lemma that allows us to dominate this process by a "kill from the right process"; which is intuitively saying that "always killing the rightmost particle results in the system as a whole moving further to the left than if we use any other rule". This allows us to show that in the limit the process X always stays to the right of the point −µt, and hence it becomes a "kill from the right" process.
Lemma 3.1.1. Let Y (N,µ) = Y 1 , .
. . Y N be a one dimensional Brownian Bee system with drift µ ∈ R and ordered such that Y 1 (t) ≤ . . . ≤ Y N (t) for all t. Let X (N ) = (X 1 , . . . , X N ) be an N-BBM with killing from the right. Then if X 1 , . . . , X N and Y 1 , . . . Y N have the same initial condition, there exists a coupling X (N ) to Y (N,µ) such that X n (t) ≤ Y n (t) − µt for 1 ≤ n ≤ N and all time t.
Proof. LetX n = Y n (t) − µt andX (N,µ) (t) = X 1 , . . . ,X N .
A trivial but key observation is that if σ, σ : {1, . . . , N } → {1, . . . N } are any two permutations, then it is sufficient to show that X σ(n) (t) ≤ Y σ (n) (t) for 1 ≤ n ≤ N , and from this we can deduce that X n (t) ≤ Y n (t). Now to construct the processes, let (J i ) i≥0 be the jump times of a Poisson process of rate N with J 0 = 0; Z i n be independent Brownian motions and (K i ) i≥1 be an independent sequence of uniform random variables on {1, . . . N }. Then letting i ≥ 1 we inductively assume that the processes have been constructed up to time J i−1 .
For
t ∈ [J i−1 , J i ), we define X n (t) = X n (J i−1 ) + Z i n (t − J i−1 ) andX n (t) =X n (J i−1 ) + Z i n (t − J i−1 )
We then relabel X (N ) andX (N,µ) so that they are ordered by position, and then by the remark above and the inductive assumption that X n (
J i−1 ) ≤X n (J i−1 ) we deduce that X n (t) ≤X n (t) for t ∈ [J i−1 , J i ).
Then at time J i we delete one particle and duplicate another, with the net result that we are moving one particle to the position of another. We can then try and keep track of which particle moved where, which is a slight shift in perspective as generally we only cared about the total process as opposed to the individual movements of the particles.
For the process X (N ) we want to move the particle X N (J i −) to X K i (J i −), since we move the furthest right particle. Then for the processX (N,µ) we move the particlẽ
X m (J i −) toX K i (J i −) where m maximises |X m (J i −) + µJ i |.
The idea is that we can then view this move as: . At time J 1 the leftmost particle branches. Then the rightmost particle of X (N ) moves to this particle. ForX (N ) , the second particle from the left moves to the leftmost particle, but we view this move as shown by the arrows, and then couple the corresponding Brownian motions so thatX (N ) lies to the right of X (N ) up to time J 2 By then assigning each particle at time t = J i an index according to their order before the move, in addition to the ordering X 1 (J i ), . . . , X n (J i ) then we get permutations σ (for X) and σ (forX). So σ(n) is the index of particle just before time J i before it jumped to X n (J i ) and similarly for σ andX n (and with arbitrary choice in case of a tie). We then see that X σ(n) (J i ) ≤X σ (n) (J i ), since before the jumps we have X n (J i −) ≤X n (J i −), for 1 ≤ n ≤ N and:
2. MoveX N −1 (J i −) toX N (J i −) 3. . . . 4. MoveX m (J i −) toX m+1 (J i −)
• The particles σ(N ), σ (N ) for X andX have moved to
X K i (J i −) andX K i (J i −) respectively, hence X σ(n) (J i ) = X K i (J i −) ≤X K i (J i −) =X σ (n) (J i )
• The particle σ(n) for X doesn't move for 1 ≤ n ≤ N − 1, and so is at position X n (J i ). And the particle σ (n) has either moved down one particle or stayed in the same place for 1
≤ n ≤ N − 1, i.e.X σ (n) (J i ) =X n (J i −) orX σ (n) (J i ) = X n+1 (J i −). However X σ(n) (J i ) = X n (J i −) ≤X n (J i −) ≤X n+1 (J i −)
We then see that we have constructed the processes X (N ) andX (N,µ) up to and including time J i , and hence proceeding by induction we have constructed the processes for all time t whilst retaining the monotone property. We can also check that the laws of the processes are correct.
Proof of Theorem 3.0.2: We assume without loss of generality that µ > 0 (since the whole system is symmetrical).
Then, let Y (N,µ) = (Y µ 1 (t), . . . , Y µ N (t)) ∈ R N be a Brownian Bee system with drift µ, ordered such that at each time t, Y µ 1 (t) ≤ . . . ≤ Y µ N (t). Then by Lemma 3.1.1, we may couple an N-BBM X (N ) with with killing from the right to Y (N,µ) such that X n (t) ≤ Y n (t) − µt. As in the proof above, we will denotẽ X n = Y n (t) − µt andX (N,µ) (t) = X 1 , . . . ,X N Then, by Theorem 2.1.1 we have that
lim t→∞ X 1 (t) t → −v N almost surely, Hence, lim inf t→∞X 1 (t) t ≥ lim t→∞ X 1 (t) t = −v N (3.2)
The point now, is that the particle that is killed at each reproduction in the processX (N,µ) (t) is determined by the largest value of |X m (t) + µt|, so if we let T be the random time such that for all t ≥ T ,
X 1 (t) t ≥ − µ + v N 2
Then by Equation 3.2, T is almost surely finite (since by assumption µ > v N ), and for t > T , the killing rule becomes "kill from the right", since for such t,X m (t) + µt is always a positive quantity. So if we fix any > 0, then we can take a deterministic number K ∈ R such that P(T ≤ K) ≥ 1 − . We then construct a process such that it follows the rule ofX up to time K, and then after that it follows the "kill from the right" rule. Let us call this process Z (N,µ,K) (t) = (Z 1 (t), . . . Z n (t)), and couple it toX such that we are using the same Brownian motions for both processes. Then for times t > K we can view the process Z (N,µ,K) (t) as an N-BBM started from the initial conditions of X 1 (K), . . . ,X N (K) , and hence by Theorem 2.1.1,
lim t→∞ Z 1 (t) t → −v N (3.3)
almost surely, But with probability at least 1 − we have that T ≤ K, and then in this case we have that the processesX (N,µ) and Z (N,µ,K) are identical. Hence,
P lim t→∞X 1 (t) t = lim t→∞ Z 1 (t) t = −v N ≥ 1 − (3.4)
And as is arbitrary, we conclude thatX 1 (t)
t → −v N a.s. Moreover, Y 1 (t) t → µ − v N a.s.
Proof of Theorem 3.0.3
The proof of this theorem is adapted from the proof of Proposition 6.5 and Theorem 1.2 in [2]. Throughout the entirety of this section we assume that Y (N,µ) is our Brownian Bees with drift process and that the drift µ is such that |µ| < v N . In an identical way to [2] Proposition 6.5 proving Theorem 1.2, Lemma 3.2.1 will be used to prove Theorem 3.0.3 in combination with Theorems 6.1 and 4.1 of [1]. These theorems say that a positive recurrent strongly aperiodic Harris chain admits a unique invariant probability measure, and that the distribution of the state of the Harris chain after n steps converges to that invariant probability measure as n → ∞.
Proof. Let Z n := (Y (N,µ) (t 1 n) Then to show that (Z n ) ∞ n=1 is a positive recurrent strongly aperiodic Harris Chain then by [1] we only need to show that there exists Λ ⊆ R N such that:
1. P χ (τ Λ < ∞) = 1 ∀χ ∈ R N , where τ Λ := inf{n ≥ 1 : Y n ∈ Λ} 2. There exists > 0 and a Probability measure q on Λ such that P χ (Z 1 ∈ C) ≥ q(C) for any χ ∈ Λ and C ⊆ Λ 3. sup χ∈Λ E(τ Λ ) < ∞
We will prove this with the somewhat arbitrary choice of taking Λ = [−1, 1] N .
It is now at this point that our proof will differ from that of Proposition 6.5 in [2]. The proof of item 2 above and the fact that this Lemma can be used to prove Theorem 3.0.3 are the same. However for items 1 and 3, [2] relied on a comparison for large N between the Brownian Bee system and a system where particles where killed upon reaching a deterministic radius. We however will rely on the results of Proposition 3.0.4 to prove these points, which has the advantage of also proving the results of [2] Theorem 1.2 for small N , however the disadvantage that our proof only works for the dimension d = 1, whereas the results of [2] hold for higher dimensions.
The rough strategy of proof for these three conditions is then 1. We use Proposition 3.0.4 to deduce that infinitely often Y (N,µ) (t) hits 0, and then at each time this happens there is some positive probability of the process branching, and then remaining inside [−1, 1] N for a time greater than t 1 , and hence τ Λ < ∞ 2. We may condition on no branching events occurring between time 0 and t 1 , and then the process moves as independent Brownian Motions, from which we can deduce the result explicitly 3. We use Proposition 3.0.4 to say that there are times T n such that the process hits 0 infinitely often and T n ≥ T n−1 +1. Then at each time T n there is a positive probability that the process duplicates so that all particles are in [−1, 1] N and remains there for time greater than t 1 . We also need to choose T n carefully in order to apply the expectation bound from Proposition 3.0.4, so that at some point between T n and T n+1 we are able to control the ball containing the process. To do this, after T n has occurred we wait until the particle that hit 0 has duplicated N times, and then argue by means of a coupling and Lemma A.0.5 that after this time we can control the ball containing the process. Then we choose T n+1 to be after this duplication -allowing us to get a bound uniform
in n on E[T n+1 − T n | F Tn ]
We shall now show the second part first: Conditional on no branching events taking place in time t ∈ [0, t 1 ] the particles move as N independent Brownian motions with drift. Hence,
P(Z 1 ∈ C) ≥ e −t 1 N C N i=1 1 √ 2πt 1 e − 1 2t 1 |y i −χ i −µt 1 | 2 dy 1 . . . dy n
And then since χ, y ∈ Λ = [−1, 1] N , we have that |y i − χ i − µt 1 | 2 ≤ (2 + |µ|t 1 ) 2 . Hence letting Leb (C) denote the Lebesgue measure of C in R N ,
P(Z 1 ∈ C) ≥ e −t 1 N 1 (2πt 1 ) N/2 exp − N (2 + |µ|t 1 ) 2 2t 1 Leb (C)
So taking q = Leb (C) /Leb (Λ) and
:= e −t 1 N 1 (2πt 1 ) N/2 exp − N (2 + |µ|t 1 ) 2 2t 1 Leb (Λ)
we have proven item 2 above. Now to show the first and third point: We will inductively define our times T n such that at each T n there is a positive probability that τ Λ < T n+1 , and also E[T n+1 − T n ] is bounded in n.
Let χ ∈ R N be the initial position of the particles, and let
R χ := inf{R ∈ R >0 : χ ∈ [−R, R] N } (3.5)
Then we define T 1 to be the first time that a particle hits 0, so
T 1 := inf{t > 0 : Y k (t) = 0 for some k ∈ {1, . . . N }}
Note in particular that by Proposition 3.0.4 we have
E[T 1 ] < α + βR χ (3.6)
Now to inductively define T n , we first define a random variable S n , so that S n > T n + 1 and the closest particle to 0 has duplicated N times.
S n := max {T n + 1, inf{t > T n : the particle closest to 0 has branched N times by time t}}
By particle closest to 0, we mean that at a branching time J i , the particle chosen to branch is the closest particle to 0 at time J i . Note since we have independence of the exponential random variables governing the branching rates from the Brownian motions, S n − T n has the distribution of the maximum 1 and the sum of N exponential distributions each of rate 1. i.e.
S n − T n d = max{Γ(N, 1), 1} (3.7)
where Γ(N, 1) is a gamma distribution of shape N and rate 1. The idea is that we can bound the tail of S n , and after time S n , any "far away" particles have been killed and replaced by particles in a controllable distance away from the origin. Next, we define T n := inf{t > S n : Y k (t) = 0 for some k ∈ {1, . . . N }} Now we claim that:
i. R Sn L n for L n i.i.d random variables independent of F Tn with E[L k ] < ∞, where R Sn := inf{R ∈ R >0 : Y (N,µ) (S n ) ∈ [−R, R] N }
and we mean in the sense of stochastic domination, i.e. there is a coupling so that R Sn ≤ L n holds for every n.
ii. P(Ñ ≤ n + 1 | F Tn ) ≥ c for some constant c whereÑ := inf{n > 0 : t 1 τ λ ≤ T n } has Suppose for a moment we have proven these two facts. Then by Lemma A.0.2 we have that E[Ñ ] < ∞.
Then consider, for n ≥ 1:
E[T n+1 − T n | F Tn ] = E[S n − T n | F Tn ] + E[T n+1 − S n | F Tn ]
Note then that S k − T k is independent from F Tn by the strong Markov property, since it only depends on the exponential distributions governing the process. Furthermore, we may then apply Proposition 3.0.4 and use the strong Markov property at time S n to conclude that E[S n − T n | F Sn ] ≤ α + βR Sn Hence, for n ≥ 1:
E[T n+1 − T n | F Tn ] ≤ E[S 1 − T 1 ] + α + βR Sn ≤ α + βE[R Sn | F Tn ]
Where this is using the fact that S k − T k is i.i.d with finite expectation (Since we identified the distribution in Equation 3.7)
≤ α + βE[L n | F Tn ] =: C (3.8)
Where we have used that R Sn ≤ L n almost surely, and that (L n ) n≥1 are i.i.d with each L n independent from F Tn Additionally, we have that
E[T 1 ] ≤ α + βR χ (3.9)
Where R χ is the radius of the ball initially containing all particles (defined in Equation
3.5) So if we set Z k = T k − T k−1 and define M n := n k=1 (Z k − C − (α + βR χ ))
Then equations 3.8 and 3.9 tell us that M n is a supermartingale adapted to the filtration G n := F Tn . We then have the properties:
• E[|M n+1 −M n | | F Tn ] is almost surely bounded in n (This follows from E[Z n | F Tn ] ≤ C, and Z n being positive)
•Ñ is a stopping time of the filtration G n
• E[Ñ ] < ∞
Which is enough to apply Optional Stopping for super-martingales to deduce that:
E[MÑ ] ≤ 0
And hence
E[t 1 τ ] ≤ E[TÑ ] = E Ñ k=1 Z k ≤ E Ñ k=1 (C + α + βR χ ) = (C + α + βR χ )E[Ñ ]
Which in particular after dividing through by t 1 proves items 1 and 3 from the start of the proof, since E[Ñ ] < ∞ and does not depend on χ; and R χ ≤ 1 for χ ∈ Λ Hence it suffices to prove items i and ii above. Let us start with i. We consider the positions of the N particles at time T n and we know that one particle is at the origin at this time. Then we couple the process from time T n to N free branching Brownian motion processes with drift started at the position of each particle Y i (T n ) for 1 ≤ i ≤ N . Since our model is then branching Brownian motion with selection, the model with a free branching Brownian motion with drift started at every particle will then contain our model, in the sense that the positions of N of the particles in the free BBM model will correspond to N particles in the Brownian Bees with drift model. Let us label these N BBM processes by X i u (t), u ∈ N i t , where 1 ≤ i ≤ N . (See Definition A.0.3 for the definition of free Branching Brownian motion), so that (X i u (t)) u∈N i t are the N i t particles that are the children of Y i (T n ), and they are at positions X i u (t). Then at time S n , let
R i Sn := inf R > 0 : X i u (t) ∈ [Y i (T n ) − R, Y i (T n ) + R], ∀ T n ≤ t ≤ S n
SoR i Sn gives us a radius surrounding the position of each particle at time T n that contains all the children of that particle until time S n .
Note then that in this free BBM model, since each particle and its children move independently, then by the strong Markov property we have thatR i Sn are i.i.d random variables whose law does not depend on n. Furthermore, we may use Lemma A.0.5 to deduce that:
E[R i Sn ] = C < ∞ (3.10)
for C independent of n. 2 We claim now that the radius of Brownian Bees at time S n , R Sn is dominated by 2 N i=1R i Sn . To see this, suppose for a contradiction that R Sn ≥ 2 N i=1R i Sn . Then there must be some particle k so that
Y k (T n ) +R k Sn ≥ 2 N i=1R i Sn , and hence Y k (T n ) −R k Sn ≥ 2 i =kR i
Sn . But then in the Brownian Bees model there is always at least one alive particle within distance 2 i =kR i Sn from the origin for t ∈ [T n , S n ]). Since if we denote
D i := [|Y i (T n )| −R i Sn , |Y i (T n )| +R i Sn ]
Then the distance to the origin of the children of particle i is always in the range D i for t ∈ [T n , S n ]. Then since there is initially one particle Y j (T n ) that is at the origin at time T n , then the only way that all the children of this particle can be killed, is if there's some other particle Y j (T n ) that is closer to the origin at some point, i.e. D j intersects D j . Then the only way that the particle j can be killed is if there's another particle that is closer to the origin than Y j at some point, i.e. D j intersects D j .
Then continuing this argument by induction, we see that there is always one particle alive within distance d = 2 i =kR i Sn from the origin, and furthermore whenever the particle that is closest to the origin branches, all its children remain within distance d from the origin.
Hence since we know that by time S n the particle closest to the origin will have branched at least N times, then by time S n , the particle Y k (T n ) and all its children must have been killed, as every time the particle closest to the origin branches, it adds one more particle which, along with its children, will stay within distance d from the origin. So since all the children of particle Y k are always further than d away from the origin, these will all be killed.
Hence in conclusion,
R Sn ≤ L n := 2 N i=1R
i Sn which gives the desired bound on R Sn since by the strong Markov property at time T n , we can see that L n are independent of F Tn , and so i.i.d. And furthermore by equation 3.10 we have that E[L n ] < ∞, proving item i above. Now to prove item ii, we will be a little brief with our argument, as similar arguments are used several times throughout this dissertation (see e.g. proof of Proposition 3.0.4 for a more detailed version of this argument)
We argue via the strong Markov property at time F Tn . Let A n be the event that:
• None of the Brownian motions with drift driving the particles will move by more than 1/(2N ) between time T n and T n + (1 − t 1 )
• The closest particle to the origin will branch N times between time T n and T n + (1 − t 1 )
• None of the Brownian motions with drift driving the particles will move by more than 1/2 between time T n + (1 − t 1 ) and T n + 1
• No branching events will occur between time T n + (1 − t 1 ) and T n + 1
Then by standard properties of Brownian motions and exponential distributions P(A k ) > 0. And furthermore since T n+1 ≥ S n ≥ T n + 1, by the strong Markov property the events A n are independent for different n, and of the same probability. Furthermore, if the event A n occurs then t 1 τ Λ will be less than T n+1 , since the particles will have stayed within Λ for a time at least t 1 , and hence there must be some discrete time-step kt 1 ∈ [T n + (1 − t 1 ), T n + 1] for which the particles lie in Λ. Hence P(Ñ ≤ n + 1 | F Tn ) ≥ P(A 1 ) -proving item ii.
The proof of Theorem 3.0.3 now follows by the exact same argument to the proof of Theorem 1.2 in [2] where we use Lemma 3.2.1 in place of Proposition 6.5. 3
Proof of Proposition 3.0.4
In order to prove Proposition 3.0.4, we start with stating and proving a weaker result: that if all particles are on one side of 0 initially, then the system will hit 0 in a time of bounded expectation:
Lemma 3.3.1. Let Y (N,µ) = (Y 1 (t), . . . Y N (t)
) be a one dimensional Brownian Bee system with drift µ ∈ R such that |µ| < v N where v N is the speed of a standard N-BBM. Then if we have initial conditions such that one of:
• Y 1 (0) ≤ Y N (0) ≤ 0 3
Due to limited space, we omit typing out this argument
• or 0 ≤ Y 1 (0) ≤ Y N (0)
Then the stopping time
τ := inf {t > 0 : Y k (t) = 0, for some k ∈ {1, . . . N }} is almost surely finite Moreover, if additionally Y (N,µ) (0) ∈ [−R 0 , R 0 ] N
has all particles within R 0 from the origin, then there are deterministic constants α = α µ and β = β µ not depending on R 0 such that
E[τ ] ≤ α + βR 0
Having this lemma will then allow us to prove the Proposition, that τ is finite for any initial condition. Furthermore we will also deduce that E(τ ) is finite.
Proof of Lemma 3.3.1. Without loss of generality we may assume that we start in some initial condition with 0 ≤ Y 1 (0) ≤ . . . ≤ Y N (0). Where here we are using the symmetry of the statement to assume we are above 0 as opposed to below 0; which we achieve by potentially reversing the direction of the drift µ to −µ. Now let X (N ) be a standard N-BBM with killing from the right, and couple X (N ) to Y (N,µ) so that we are using the same random variables to generate each process. Moreover, start X (N ) so that it is coupled to the same initial condition as Y (N,µ) . Then while Y (N,µ) is on the right of 0, it will behave identically tõ X (N,µ) := (X 1 (t) + µt, . . . , X N (t) + µt). In other words if τ = inf {t > 0 : Y k (t) = 0, for some k ∈ {1, . . . N }} Then for t < τ ,X n (t) = Y n (t). So τ is also the first time that the processX (N,µ) hits 0, and sinceX (N,µ) initially starts above 0, it must be the particleX 1 (t) that hits 0 first. However, by Lemma 3.1.1, we may further couple the process X (N ) so that it is dominated by a process X (N ) , where initially all particles in X (N ) are at the same position R 0 . If τ is the first time that this system hits 0, then it follows from this coupling that τ < τ almost surely. However by Theorem 2.2.1, we then deduce that for some α, β
E[τ ] ≤ E[τ ] < α + βR 0
In particular, τ is always finite Now, moving onto the main Proposition:
Proof of Proposition 3.0.4. Let ρ := inf {t > 0 : Y N (t) < 0 or Y 1 (t) > 0}, i.e. ρ is the first time that all particles are on one side of 0. Then conditional on ρ being finite, we may use the strong Markov property at time ρ, to get a Brownian Bees system started in position (Y 1 (ρ), . . . Y N (ρ)) -which is necessarily all on one side of 0. This allows us to apply Lemma 3.3.1 to conclude that τ is finite. Hence to show the finiteness of τ , it is sufficient to either show that either ρ < ∞ or τ < ∞. Now consider sampling the process Y (N,µ) at discrete time steps, t = k for k = 1, 2, . . .. Then consider the events A k and B k (which the reader may consider to stand for "Above" and "Below"), such that • A k is the event that at time k, if Y n (k) is the closest particle to 0, then Y n (k) > 0
• B k is the event that at time k, if Y n (k) is the closest particle to 0, then Y n (k) < 0 Then up to null sets, A k and B k are a partition of our probability space Ω. And clearly they are both also F k measurable events. Now we further define events U k and D k (which the reader may take as "Up" and "Down") such that U k = U 1 k ∩ U 2 k ∩ U 3 k ∩ U 4 k and denotingB n (t) = B n (t) + µt to be the Brownian motions with drift driving the process Y n (t) we have:
• U 1 k is the event that no branching events occur between time k and time k + 1/2 • U 2 k is the event that if Y n (k) is the closest particle to 0 at time k, andB n (t) is the drifting Brownian motion driving the child of Y n (k) between time k and k +1/2, thenB n (k+1/2)−B n (k) ≥ 1; and sup 1≤n≤N,n =n sup k≤t≤k+1/2 |B n (t) −B n (k)| ≤ 1/2
• U 3 k is the event that N branching events occur between time k + 1/2 and time k + 1 and the particles that duplicates is in position Y n (t) (i.e. there are n − 1 particles to its left) where Y n (k) is the closest particle to 0 at time k
• U 4 k is the event sup 1≤n≤N sup k+1/2≤t≤k+1 |B n (t) −B n (k)| ≤ 1/(2N )
We may read this event U k as the particle closest to 0 at time k moving up by 1 unit, then splitting many times in a row without any Brownian motions moving very far.
The event D k = D 1 k ∩ D 2 k ∩ D 3 k ∩ D 4 k is then defined very similarly, except instead of the closest particle moving up, it moves down instead. so D 1
k = U 1 k ; D 3 k = U 3 k ; D 4 k = U 4 k and D 2 k is the eventB n (k + 1/2) −B n (k) ≤ −1 and; sup 1≤n≤N,n =n sup k≤t≤k+1/2 |B n (t) −B n (k)| ≤ 1/2
The point then of constructing these complex events, is that if either A k ∩ D k or B k ∩ U k occur, then we have that either ρ ≤ k + 1 or τ ≤ k + 1. This is because if the closest particle to 0 at time k is above 0, and the event D k occurs, Then at time k +1/2, either this particle has crossed 0 and so τ ≤ k +1, or it still lies on the right of 0 and it or its children will be the closest particle to 0 during time t ∈ [k + 1/2, k + 1] -and hence when N branching events occur, all particles will now be on the right of 0 and ρ ≤ k + 1. The Event U k . Between t = k and t = k + 1/2 no particles move more than 1/2, except the closest particle to the origin which moves a distance 1 towards the origin. Then between time t = k + 1/2 and t = k + 1 no particles move more than 1/(2N ). Note due to a possible additive effect with the children of the closest particle branching this only allows us to say that the particles remain with in a radius 1/2 from the closest particle at t = k + 1/2. This control over the supremum ensures that no other particles can become closer to the origin between t = k + 1/2 and t = k + 1. Furthermore the closest particle or its children branch N times in t ∈ (k + 1/2, k + 1). The end result is that provided the closest particle at time k was below the origin, then at t = k + 1 all particles are now either on one side, or one particle has hit 0.
Note also that since we are controlling all the movements of the Brownian motions, then for the event U 3 k or D 3 k the particle in position n , will always be a child of the particle that moved by at least 1.
We now additionally claim that: (U k ) k≥1 are independent of each other, and similarly for (D k ) k≥1 . And furthermore, U k and D k are independent from F k .
It is then sufficient to show this last point, since U k is F k+1 measurable, and similarly for D k , so their mutual independence will follow from this.
It is clear then that the events U 1 k and U 4 k are independent from F k , since we have independence of increments for both the Brownian motions and the exponential distribution governing the next time a particle will branch. It is slightly less clear that U 2 k and U 3 k are independent from F k , since both events make explicit reference to the "closest" particle at time t = k: Y n (k), and n is explicitly F k measurable. Fortunately though since the random variables (B 1 (t) − B 1 (k)) , . . . , B N (t) − B N (k) are i.i.d and independent of F k for any t > k, we can relabel them using some order depending on F k and retain independence. This holds similarly for determining which random variable is branching, since we can view the particle in position n as each having their own Poisson process of rate 1 attached to them. And then since these Poisson processes are i.i.d, we can relabel them by choice using some F k dependent order. The details that this relabelling is valid are contained in the appendix Lemma A.0.1. Hence from this we deduce that U k and D k are independent from F k .
We now have all the ingredients to our argument, it finally remains to show that A k ∩ D k or B k ∩ U k occurs eventually for some k, as if this happens almost surely, then it tells us that at least one of ρ or τ is finite almost surely, from which we deduce the result.
From standard properties of Brownian motions and exponential distributions, we can see that U k and D k are both events with positive probability, and furthermore this probability does not depend on k; since each event depends only on the jump times between k and k +1 and the random variables B 1 (t) −B 1 (k) , . . . , B N (t) −B N (k) . Both of which are i.i.d for different values of k. Note however it is not the case that P(U k ) = P(D k ) since the drift µ gives a higher weight that the drifting Brownian motionsB n will move one way than the other. Let
c = min{P(U k ), P(D k )} (3.11)
so c > 0 Now define a stopping timẽ
K := 1 + inf{k ∈ N, ω ∈ (A k ∩ D k ) ∪ (B k ∩ U k )}
Then by the above discussion we know that almost surely at least one of ρ or τ is less thanK. It remains to show thatK is finite. we calculate:
P(K ≤ k + 1 | F k ) ≥ E 1 (A k ∩D k )∪(B k ∩U k ) | F k = E [1 A k 1 D k + 1 B k 1 U k | F k ] = 1 A k P(D k ) + 1 B k P(U k ) ≥ c
Where we have used the fact that A k and B k are a partition and F k measurable; and that U k and D k are independent of F k . Now we conclude by Lemma A.0.2 that E[K] < ∞ and henceK is finite almost surely. Therefore since either τ or ρ is less thanK almost surely, and conditional on ρ being finite we have that τ is finite, we conclude the first part of the lemma that τ is finite. Now for the moreover part we wish to get control over the ball containing the Brownian Bees process at timeK. Let R t denote the ball that has contained the process up to time t, so
R t := sup s≤t inf R>0 {R : |Y i (s)| ≤ R, 1 ≤ i ≤ N }
Clearly then R t is non-decreasing in t, and Y (N,µ) (t) ⊆ [−R t , R t ] N . The idea now is that we can make a crude approximation where we consider the process without selection allowing us to use Lemma A.0.5, which tells us that at timeK, RK has integrable tails. To make this arguement more precise, we know at time 0 the particles
Y (N,µ) (0) ⊆ [−R 0 , R 0 ] N .
Then to bound the upper tail and lower tail separately define the quantities:
• R + t := sup s≤t {inf R>0 {Y i (t) ≤ R, 1 ≤ i ≤ N }} • R − t := sup s≤t {inf R>0 {Y i (t) ≥ −R, 1 ≤ i ≤ N }}
Clearly then R t = max{R + t , R − t }. Then to bound R + t , we apply Lemma 3.1.1 to couple the process Y (N,µ) − µt − R 0 to an N-BBM with killing from the left, and all N particles starting at the origin 4 . This N-BBM can then be coupled to N free branching Brownian motion processes 5 (definition A.0.3) started at the origin, since these correspond to X (N ) but without deleting any particles. Then let R i,+ t be the smallest R such that the i th free branching Brownian motion process has not hit R by time t. It follows then that
R + K − µK − R 0 ≤ max{R 1,+ K , . . . R N,+ K }
Next,K is dominated by an exponential since for x large enough, (3.12) where c is from equation 3.11 and α is some positive constant. Then it follows by Lemma A.0.5 that P(R i,+ K ≥ x) ≤ e −c i √ x for x large enough. In particular E max{R 1,+ K , . . . R N,
P(K > x) = (1 − c) x ≤ e −αx+ K } ≤ N i=1 E[R i,+ K ] < ∞
And note that this quantity is independent of R 0 . Hence we conclude that
E[R + K ] ≤ R 0 + µE[K] + C = R 0 + C (3.13)
Where C, C are constants depending only on µ. The same argument works for bounding E[R − K ], so we deduce that E[RK] ≤ 2R 0 + C Now finally, we know that almost surely one of τ or ρ is less thanK, and we know that if ρ <K but τ >K, then at timeK all particles must be on one side of 0.
Hence
E(τ ) = E [E(τ |FK)] ≤ E K + E τ 1 τ >K |FK
Then we can use the strong Markov Property atK and Lemma 3.3.1 to deduce that E(τ ) ≤ E(K) + E(K + α + β RK )
And finally using equations 3.12 and 3.13
E(τ ) ≤ α + βR 0
Chapter 4 Conclusion
We have successfully found the critical case for the recurrence and transience of Brownian Bees. Perhaps the natural next question is what happens at the criticality µ = v N ? In order to answer this question more knowledge would be needed about N-BBMs; for example if we wanted to show that the case µ = v N is recurrent we would need a similar result to Theorem 2.2.1, which would require more understanding of the N-BBM than the fairly crude coupling used to prove the above theorem.
A different way to go further would be to consider the convergence in time of the N-BBM when viewed from the leftmost particle (i.e. the process (X i (t) − X 1 (t)) N i=2 for X an N-BBM). This has been conjectured in several papers and is widely considered to be true (see e.g. section 8 of [18]) -and in fact the method of proof of Theorem 3.0.3 should allow for the proof of a similar result for N-BBM viewed from the leftmost particle -though this has not been investigated in rigourous detail due to space and time constraints.
Appendix A Appendix
This appendix contains several technical lemmas which are stated here as providing the proofs or statements of them in the section above would detract from the ideas behind the relevant proofs. 1 ρ=σ P (X 1 ≤ x 1 ) . . . P (X N ≤ x n ) = P (X 1 ≤ x 1 ) . . . P (X N ≤ x n )
Where we are using the fact that X n is independent of G to remove the conditioning, and we are using the fact that X 1 , . . . X N are i.i.d to replace P X σ(n) ≤ x 1 with P (X n ≤ x 1 ) Lemma A.0.2. Let τ be a stopping time of some filtration (G n ) ∞ n=1 . Then suppose that there is some α ∈ N and > 0 such that P(τ ≤ n + α | G n ) ≥ Then it holds that P(τ > mα) ≤ (1 − ) m And in particular that E(τ ) < ∞ Proof. This is given as exercise E10.5 in [22] Definition A.0.3 (Free Branching Brownian Motion). A free branching Brownian motion is heuristically a system in which each particle moves independently according a Brownian motion, and at rate 1, it duplicates into two particles. We write N (t) for the number of particles alive at time t, and X u (t), u ∈ N (t) for the positions of the particles. See [6] for a more precise definition and discussion.
Lemma A.0.4 (Many-to-one Lemma). Let (X u (t), u ∈ N (t)) be a Branching Brownian Motion process started at 0. Let F : C [0,T ] → R be any measurable function. Then
E u∈N (t) F (X u (s), 0 ≤ s ≤ t) = e t E [F (B s , 0 ≤ s ≤ t)]
Proof. e.g. [15] Section 4.1
Lemma A.0.5 (Radius Bound). Let (X u (t), u ∈ N (t)) be a Branching Brownian Motion process started at 0. Let T be a random variable (possibly dependent) such that for some λ and t large enough We have that ∃c > 0 so that for x large enough P(R T ≥ x) ≤ e −c √ x Proof. We argue first by the many-to-one lemma. Let F (X s , s ≤ t) = 1 sup s≤t |Xs|≥x for x ∈ R Then,
P(T > t) ≤ e −λtP(R t ≥ x) = P sup u∈N (t) sup s≤t |X u (s)| ≥ x ≤ E u∈N (t) 1 sup s≤t Xu(s)≥x = e t P sup s≤t |B s | ≥ x
Then for each x ∈ R we choose a deterministic t x so that P(T > t x ) ≤ e −λ √
x . So for x large enough we can choose t x = √ x by the bound assumed for T . Then P (R T ≥ x) = P (R T ≥ x | T > t x ) P(T > t x ) + P (R T ≥ x | T ≤ t x ) P(T ≤ t x ) ≤ P(T > t x ) + P(R tx > x) where here we have used the fact that R t is non-decreasing in t.
≤ e −λ √
x + e tx P sup
s≤tx |B s | ≥ x ≤ e −λ √ x + 2e √ x e −( √ x) 3 2 = e −λ √ x + 2e −x/2
Using the standard bound that P(sup s≤t B s ≥ λt) ≤ e −λ 2 t 2 , and the fact that −B t is a Brownian motion. Hence we can take c so that for x large enough P(R T ≥ x) ≤ e −c √
x
Figure 3 . 1 :
31This diagram shows the coupling process. On the left we have the process X (N ) and on the rightX (N )
Lemma 3 .2. 1 .
31For any t 1 ∈ (0, 1), the Markov chain Y (N,µ) (t 1 n) ∞n=0 is a positive recurrent strongly aperiodic Harris chain.
Figure 3 . 2 :
32Figure 3.2: The Event U k . Between t = k and t = k + 1/2 no particles move more than 1/2, except the closest particle to the origin which moves a distance 1 towards the origin. Then between time t = k + 1/2 and t = k + 1 no particles move more than 1/(2N ). Note due to a possible additive effect with the children of the closest particle branching this only allows us to say that the particles remain with in a radius 1/2 from the closest particle at t = k + 1/2. This control over the supremum ensures that no other particles can become closer to the origin between t = k + 1/2 and t = k + 1. Furthermore the closest particle or its children branch N times in t ∈ (k + 1/2, k + 1). The end result is that provided the closest particle at time k was below the origin, then at t = k + 1 all particles are now either on one side, or one particle has hit 0.
Relabelling Lemma Let X 1 , . . . X N be i.i.d random variables independent of a sigma algebra G. Then if ρ : Ω → S(N ) is a random permutation of {1, . . . , N } and ρ is G measurable, Then X ρ(1) , . . . X ρ(N ) are i.i.d random variables independent of G. Proof. P X ρ(1) ≤ x 1 , . . . X ρ(N ) ≤ x n G) = σ∈S(N ) E 1 X σ(1) ≤x 1 . . . 1 X σ(N ) ≤xn 1 ρ=σ | G = σ∈S(N ) E 1 Xσ(1)≤x 1 . . . 1 X σ(N ) ≤xn | G 1 ρ=σ = σ∈S(N ) P X σ(1) ≤ x 1 . . . P X σ(N ) ≤ x
{R : |X u (t)| ≤ R, ∀u ∈ N (t)}
The name "Brownian Bees" was initially suggested by Jeremy Quastel[2] and it due to a rough analogy of the Brownian motions looking like a "swarm of bees".2 In this paper N.Berestycki and Zhao are exploring multidimensional Brunet-Derrida systems, however for this dissertation we shall stick primarily to the one dimensional process.
In these papers the authors use the term "N-BBM" to refer to the Brownian Bees, however in this dissertation we reserve this term for the process with killing from the left/right
To apply lemma A.0.5 to the drift case, use the fact that (the radius of the process without drift) + |µ|(S n − T n ) will dominate the radius of the process with drift. Then equation 3.7 tells us that S n − T n has the tail of a gamma distribution, which in particular has a tail dominated by an exponential, allowing us to both apply the Lemma, and deduce that E[S n − T n ] < C < ∞ for C independent of n
We are doing a two-step coupling here to dominate the Brownian Bees by a free branching Brownian motion with N particles started at the same position. There are other ways to achieve this but coupling to an N-BBM first to move the particles to the same position makes use of results already proven5 The topic of free branching Brownian motion is somewhat glossed over in this dissertation due to space constraints. The coupling is done in the intuitive way, since N-BBM is a branching Brownian motion with selection, we simply do not perform the selection to make the coupling.
AcknowledgementsWith thanks to Julien Berestycki for supervising me during this dissertation
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| []
|
[
"Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED",
"Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED"
]
| [
"Tom Blum ",
"Ran Zhou \nPresent Address\nDepartment of Physics\nGraduate School of Pure and Applied Science\nIndiana University\n47405BloomingtonIndianaUSA\n\nCenter for Accelerator-Based Science\nDepartment of Physics\nUniversity of Tsukuba\nTennodai 1-1-1305-8571, 351-0198Tsukuba, WakoIbaraki, SaitamaJapan and RIKEN Nishina, Japan\n\nand Theoretical Physics Laboratory, Nishina Center, RIKEN\nNagoya University\n464-8602, 351-0198Nagoya, WakoJapan, Japan\n\nDepartment of Physics\nand RIKEN-BNL Research Center, Brookhaven National Laboratory\nBrookhaven National Laboratory\nUpton, Upton11973, 11973NY, NYUSA, USA\n\nand RIKEN-BNL Research Center, Brookhaven National Laboratory\nNagoya University\nUpton464-8602, 11973NagoyaNYJapan, USA\n\nSchool of High Energy Accelerator Science\nKEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaJapan\n\nThe Graduate University for Advanced Studies (Sokendai)\n305-0801TsukubaJapan\n",
"Takumi Doi ",
"Masashi Hayakawa ",
"Taku Izubuchi ",
"Shunpei Uno ",
"Norikazu Yamada ",
"\nPhysics Department\nRIKEN-BNL Research Center\nUniversity of Connecticut\n06269-3046StorrsCTUSA\n",
"\nBrookhaven National Laboratory\nUpton11973NYUSA\n"
]
| [
"Present Address\nDepartment of Physics\nGraduate School of Pure and Applied Science\nIndiana University\n47405BloomingtonIndianaUSA",
"Center for Accelerator-Based Science\nDepartment of Physics\nUniversity of Tsukuba\nTennodai 1-1-1305-8571, 351-0198Tsukuba, WakoIbaraki, SaitamaJapan and RIKEN Nishina, Japan",
"and Theoretical Physics Laboratory, Nishina Center, RIKEN\nNagoya University\n464-8602, 351-0198Nagoya, WakoJapan, Japan",
"Department of Physics\nand RIKEN-BNL Research Center, Brookhaven National Laboratory\nBrookhaven National Laboratory\nUpton, Upton11973, 11973NY, NYUSA, USA",
"and RIKEN-BNL Research Center, Brookhaven National Laboratory\nNagoya University\nUpton464-8602, 11973NagoyaNYJapan, USA",
"School of High Energy Accelerator Science\nKEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)\n305-0801TsukubaJapan",
"The Graduate University for Advanced Studies (Sokendai)\n305-0801TsukubaJapan",
"Physics Department\nRIKEN-BNL Research Center\nUniversity of Connecticut\n06269-3046StorrsCTUSA",
"Brookhaven National Laboratory\nUpton11973NYUSA"
]
| []
| 2 Results computed in lattice QCD+QED are presented for the electromagnetic mass splittings of the low lying hadrons. These are used to determine the renormalized, non-degenerate, light quark masses. It is found that m M S u = 2.24 (10) (34), m M S d = 4.65 (15) (32), and m M S s = 97.6 (2.9) (5.5) MeV at the renormalization scale 2 GeV, where the first error is statistical and the second systematic.We find the lowest order electromagnetic splitting (m π + − m π 0 ) QED = 3.38(23)MeV, the splittings including next-to-leading order, (m π + − m π 0 ) QED = 4.50(23)MeV, (m K + − m K 0 ) QED = 1.87(10) MeV, and the m u = m d contribution to the(96)MeV. All errors are statistical only, and the next-to-leading order pion splitting is only approximate in that it does not contain all next-to-leading order contributions. We also computed the proton-neutron mass difference, including for the first time, QED interactions in a realistic 2+1 flavor calculation. We find (m p − m n ) QED = 0.383(68) MeV, (m p − m n ) (mu−m d ) = −2.51(14) MeV (statistical errors only), and the total m p − m n = −2.13(16)(70) MeV, where the first error is statistical, and the second, part of the systematic error. The calculations are carried out on QCD ensembles generated by the RBC and UKQCD collaborations, using domain wall fermions and the Iwasaki gauge action (gauge coupling β = 2.13 and lattice cutoff a −1 ≈ 1.78 GeV). We use two lattice sizes, 16 3 and 24 3 ( (1.8 fm) 3 and (2.7 fm) 3 ), to address finite volume effects. Non-compact QED is treated in the quenched approximation. The valence pseudo-scalar meson masses in our study cover a range of about 250 to 700 MeV, though we use only those up to about 400 MeV to quote final results.We present new results for the electromagnetic low energy constants in SU(3) and SU(2) partially-quenched chiral perturbation theory to the next-to-leading order, obtained from fits to our data. Detailed analysis of systematic errors in our results and methods for improving them are discussed. Finally, new analytic results for SU(2) L × SU(2) R -plus-kaon chiral perturbation theory, including the one-loop logs proportional to α em m, are given. 3 | 10.1103/physrevd.82.094508 | [
"https://arxiv.org/pdf/1006.1311v2.pdf"
]
| 118,650,042 | 1006.1311 | 66f9a230460afcdeeb8472925f114409f3159494 |
Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED
(Dated: May 26, 2010, revised: September 2, 2010) 3 Sep 2010
Tom Blum
Ran Zhou
Present Address
Department of Physics
Graduate School of Pure and Applied Science
Indiana University
47405BloomingtonIndianaUSA
Center for Accelerator-Based Science
Department of Physics
University of Tsukuba
Tennodai 1-1-1305-8571, 351-0198Tsukuba, WakoIbaraki, SaitamaJapan and RIKEN Nishina, Japan
and Theoretical Physics Laboratory, Nishina Center, RIKEN
Nagoya University
464-8602, 351-0198Nagoya, WakoJapan, Japan
Department of Physics
and RIKEN-BNL Research Center, Brookhaven National Laboratory
Brookhaven National Laboratory
Upton, Upton11973, 11973NY, NYUSA, USA
and RIKEN-BNL Research Center, Brookhaven National Laboratory
Nagoya University
Upton464-8602, 11973NagoyaNYJapan, USA
School of High Energy Accelerator Science
KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK)
305-0801TsukubaJapan
The Graduate University for Advanced Studies (Sokendai)
305-0801TsukubaJapan
Takumi Doi
Masashi Hayakawa
Taku Izubuchi
Shunpei Uno
Norikazu Yamada
Physics Department
RIKEN-BNL Research Center
University of Connecticut
06269-3046StorrsCTUSA
Brookhaven National Laboratory
Upton11973NYUSA
Electromagnetic mass splittings of the low lying hadrons and quark masses from 2+1 flavor lattice QCD+QED
(Dated: May 26, 2010, revised: September 2, 2010) 3 Sep 2010numbers: 1115Ha1130Rd1238Gc 1239Fe 4
2 Results computed in lattice QCD+QED are presented for the electromagnetic mass splittings of the low lying hadrons. These are used to determine the renormalized, non-degenerate, light quark masses. It is found that m M S u = 2.24 (10) (34), m M S d = 4.65 (15) (32), and m M S s = 97.6 (2.9) (5.5) MeV at the renormalization scale 2 GeV, where the first error is statistical and the second systematic.We find the lowest order electromagnetic splitting (m π + − m π 0 ) QED = 3.38(23)MeV, the splittings including next-to-leading order, (m π + − m π 0 ) QED = 4.50(23)MeV, (m K + − m K 0 ) QED = 1.87(10) MeV, and the m u = m d contribution to the(96)MeV. All errors are statistical only, and the next-to-leading order pion splitting is only approximate in that it does not contain all next-to-leading order contributions. We also computed the proton-neutron mass difference, including for the first time, QED interactions in a realistic 2+1 flavor calculation. We find (m p − m n ) QED = 0.383(68) MeV, (m p − m n ) (mu−m d ) = −2.51(14) MeV (statistical errors only), and the total m p − m n = −2.13(16)(70) MeV, where the first error is statistical, and the second, part of the systematic error. The calculations are carried out on QCD ensembles generated by the RBC and UKQCD collaborations, using domain wall fermions and the Iwasaki gauge action (gauge coupling β = 2.13 and lattice cutoff a −1 ≈ 1.78 GeV). We use two lattice sizes, 16 3 and 24 3 ( (1.8 fm) 3 and (2.7 fm) 3 ), to address finite volume effects. Non-compact QED is treated in the quenched approximation. The valence pseudo-scalar meson masses in our study cover a range of about 250 to 700 MeV, though we use only those up to about 400 MeV to quote final results.We present new results for the electromagnetic low energy constants in SU(3) and SU(2) partially-quenched chiral perturbation theory to the next-to-leading order, obtained from fits to our data. Detailed analysis of systematic errors in our results and methods for improving them are discussed. Finally, new analytic results for SU(2) L × SU(2) R -plus-kaon chiral perturbation theory, including the one-loop logs proportional to α em m, are given. 3
Results computed in lattice QCD+QED are presented for the electromagnetic mass splittings of the low lying hadrons. These are used to determine the renormalized, non-degenerate, light quark masses. It is found that m M S u = 2.24 (10) (34), (15) (32), and m M S s = 97.6 (2.9) (5.5) MeV at the renormalization scale 2 GeV, where the first error is statistical and the second systematic.
m M S d = 4.65
We find the lowest order electromagnetic splitting (m π + − m π 0 ) QED = 3.38 (23) MeV, the splittings including next-to-leading order, (m π + − m π 0 ) QED = 4.50 (23) MeV, (m K + − m K 0 ) QED = 1.87 (10) MeV, and the m u = m d contribution to the kaon mass difference, (m K + − m K 0 ) (mu−m d ) = −5.840(96) MeV. All errors are statistical only, and the next-to-leading order pion splitting is only approximate in that it does not contain all next-to-leading order contributions. We also com- We present new results for the electromagnetic low energy constants in SU(3) and SU(2) partially-quenched chiral perturbation theory to the next-to-leading order, obtained from fits to our data. Detailed analysis of systematic errors in our results and methods for improving them are discussed. Finally, new analytic results for SU(2) L × SU(2) R -plus-kaon chiral perturbation theory, including the one-loop logs proportional to α em m, are given.
I. INTRODUCTION
The mass splitting in the meson and baryon systems is an interesting topic in hadron spectroscopy. It is related to the quark masses which are fundamental parameters of the standard model. The mass splitting in the pseudo-scalar meson octet is a signature of the breaking of the strong isospin symmetry by the electromagnetic (EM) interaction and non-degenerate quark masses.
The hadron spectra are rich in diversity due to two origins: the nonperturbative quantum dynamics of the strong interaction and the presence of flavor symmetry breaking. In the standard model, the latter originates from non-degenerate quark masses as well as the difference between up-type quarks and down-type quarks. These sources of flavor symmetry breaking affect significantly the hadron spectra less than (1 ∼ 2) GeV. In the baryon octet, for instance, the mass difference of the proton and neutron is crucial to the phenomenological model of nuclei because it plays an important role in neutron β-decay, which is related to the stability of nuclei. If the up and down quark masses were degenerate, the proton would be heavier due to the EM interaction, and our Universe would not exist! Even though the mass differences in the baryon octet spectrum have already been measured in experiments to good accuracy, it is important to confirm that we can predict these splittings in the standard model from first-principles computation. Parameterizing the calculated splittings in terms of low energy constants (LEC) is also useful for effective theories like chiral perturbation theory.
The mass of a hadron is determined by both quantum chromodynamics (QCD) and quantum electrodynamics (QED), though the vast majority of the mass is due to QCD.
For the QCD interaction, since the coupling constant is large in the low energy regime (E 1 GeV), perturbation theory is not applicable, and we must turn to the techniques of lattice gauge theory to solve QCD for the hadron spectrum. On the other hand, the EM contributions to the masses, which break degeneracies due to flavor SU(3) (and isospin) symmetry since the up quark has charge +2/3 e and the down and strange quarks have charge −1/3 e, are expected to depend on the small QED coupling constant, α em ≡ e 2 /4π ≈ 1/137. However, since the hadrons are formed from bound-states of the quarks, there is no systematic way to treat the contributions in weak-coupling perturbation theory. Thus, calculations are done nonperturbatively in a combined lattice QCD+QED theory (indeed, even if the strong coupling constant were small, one is forced to a nonperturbative solution for QCD because of confinement). The state-of-the-art in lattice calculations is such that subpercent errors (statistical and systematic) on low-lying hadron masses and other observable quantities are becoming the norm(for a broad review, see [1] and references therein). As the precision of lattice calculations improve, the EM splittings become more and more relevant.
Indeed, the splittings themselves can be computed with sub-percent precision, at least for the statistical errors [2].
It is well known that the lowest order EM effect, the so-called Dashen term [3], which enters at O(α em ), is the dominant contribution to the charged-neutral pion mass difference.
In the chiral limit where the quarks are all massless, it is also true for the kaons. This theorem, known as Dashen's theorem, is broken by terms of order O(α em m) away from the chiral limit. Using an effective theory of QCD known as chiral perturbation theory, these corrections can be identified in the lattice calculation, and the non-degenerate quark masses determined by matching to the experimentally measured mass splittings [4],
m π ± − m π 0 = 4.5936(5) MeV,(1)m K ± − m K 0 = −3.937(28) MeV,(2)
m n − m p = 1.2933321(4) MeV.
In fact, any three hadron masses are enough to determine the three light quark masses, and we choose m π + , m K + and m K 0 for reasons explained later. The determination of the up quark mass, m u , is particularly interesting since one can check the simplest solution to the strong CP problem, m u = 0.
In the study presented here, we work with lattice domain wall fermions (DWF) [5,6] for the quarks and the Iwasaki gauge action for the gluons. We use an ensemble of gluon configurations with a single lattice spacing, generated by the RBC and UKQCD collaborations using (2+1)-flavors of dynamical quarks, a pair of degenerate quarks for the up and down, and a heavier strange quark [7]. The photons are simulated in non-compact, quenched QED, as was done in the pioneering quenched QCD+quenched QED calculations [8,9].
There are several differences between this work and our previous one [2]. The most obvious is that the dynamical strange sea quark has been included here. In Ref. [2] the QED gauge potential was fixed to the Coulomb gauge, and here we work in Feynman gauge in QED on finite volume, as described in Ref. [10]. Next, we perform fits to full next-to-leading order (NLO) partially-quenched chiral perturbation theory (PQχPT), including photons, for both SU(3) L × SU(3) R chiral symmetry [11] and SU(2) L × SU(2) R -plus-kaon [7,12,13], where the latter treats only the pion-triplet as pseudo-Nambu-Goldstone bosons. The NLO PQχPT for SU(2) L × SU(2) R -plus-kaon, including photons, is new, and is presented here for the first time. Calculations by the RBC and UKQCD collaborations [7,[14][15][16][17], and more recently PACS-CS [18,19], have shown that the physical strange quark mass is out of reach for NLO SU(3) chiral perturbation theory. Since we also wish to determine the strange quark mass in our calculation, we have developed the chiral perturbation theory for the SU(2) L × SU(2) Rplus-kaon system, including photons. In addition, since the photons are not confined, finite volume effects are expected to be large, so we work with two lattice sizes, 16 3 and 24 3 , with the same lattice spacing to investigate these effects. The leading EM finite volume effects have been computed in PQχPT [10], which we also use in our analysis.
This paper is organized as follows. In Sec. II we summarize the chiral perturbation theories used to fit our lattice calculations (details are given in the Appendix B). In Sec. III the basic framework and details of the lattice simulations are given. Section IV contains results and discussion of the calculation, including the fitted LEC's and the quark masses.
Section V discusses systematic errors, and in Secs. VI and VII we give final values for the quark masses and meson splittings, respectively. We examine the impact of (m u − m d ) on the decay constant ratio, f K /f π in Sec. VIII. The nucleon mass splitting is computed in Sec. IX. Finally, this work is summarized in Sec. X.
We reported the preliminary results from SU (3) PQχPT study in this work in Ref. [20,21]. The MILC collaboration also presented their first results on the EM splittings, using improved staggered fermions and non-compact, quenched QED configurations in Ref. [22].
II. CHIRAL PERTURBATION THEORY
We briefly review the framework and formulas of partially quenched chiral perturbation theory relevant for our 2+1 flavor calculation. The EM corrections in SU(2) chiral perturbation theory coupled to kaons are new. Details are given in Appendix B.
Recently it has been shown that SU(3) chiral perturbation theory is poorly convergent for quark masses near the physical strange quark mass, and that a straightforward and effective solution is to treat the strange quark mass m s as large compared to the light quark masses m l in an expansion in m l /m s [7,12,18,19]. We carry out fits to the data using both SU (3) and SU(2) chiral perturbation theory. We find the poor convergence extends to the EM sector as well, and use SU(2) chiral perturbation theory to quote our central results.
Before proceeding, it is important to discuss the order in chiral perturbation theory to which we work. For the SU (3) O(e 4 ) contributions have so far been ignored [2,11] since O(e 4 ) O(e 2 p 2 ) in practice, and we also follow this here.
In the SU(2) theory coupled to kaons the power counting becomes a bit more complicated for the kaon (for the pion it is the same as in the SU(3) case). Since the kaon is no longer a Nambu-Goldstone boson, LO for the mass-squared is now O(p 0 ), and NLO is O(p 2 ) and
O(e 2 ). The mass-squared EM splitting, however, remains the same order of magnitude as in (partially-quenched) SU(3) chiral perturbation theory. That is, to obtain the NLO contributions to the mass-squared splittings, we must work to NNLO for the masses. Since the aim here is to include all effects up to and including O(e 2 p 2 ) terms in the meson masssquared splittings as well as m d − m u , we include all O(e 2 p 2 ) contributions to the kaon mass. Because we compute m d − m u from the neutral-charged kaon mass-squared difference, the pure QCD effects at O(p 4 ), including the one-loop logarithms [12], cancel and are not included in our analysis.
Finally, to avoid confusion we emphasize that in this paper we only calculate correlation functions for "charged", or "off-diagonal" mesons. However, since we are free to change the charges and masses of the valence quarks making up these mesons, the total charge of the (unphysical) meson may happen to be zero. Sometimes we refer to these as "neutral"
mesons, but it must be kept in mind these never correspond to the π 0 meson which requires so-called disconnected quark diagrams in its correlation function as well as the full treatment of "diagonal" mesons in PQχPT.
A. SU(3) L × SU(3) R
The partially quenched chiral perturbation theory has been worked out in Ref. [11], and we adopt their notation. For three non-degenerate sea quarks and two non-degenerate valence quarks, labeled by "1" and "3", the meson mass-squared at NLO is (q 2 13 ) e 2 16π 2 χ 13 3 log
M 2 13 = χ 13 + 2Ce 2 F 2 0 q 2 13 + 48L r 6 − 24L r 4 F 2 0 χ 13χ1 + 16L r 8 − 8L r 5 F 2 0 χ 2 13 −48e 2 C F 4 0 L r 4 q 2 13χ 1 − 16e 2 C F 4 0 Lχ 13 µ 2 − 4 +e 2 δ mres (q 2 1 + q 2 3 ).(4)
Indices 1 − 3 always refer to valence quarks, 4 − 6 to sea quarks. The coefficients R m n13 and R p qπη are the residue functions written in terms of quark masses and are defined in Ref. [11]. The index p implies summation over valence indices 1 and 3, and if q is also present, then the sum is over pairs (1,3) and (3,1). The indices (m, n) signify a sum over pairs (π, η) and (η, π). χ ij = B 0 (m i + m j ) is the LO mass-squared for a meson made of quarks with masses m i and m j , q ij = q i − q j where q i is the electric charge of the ith quark in units of the fundamental charge e.χ 1 = 2B 0 (m 4 + m 5 + m 6 )/3 andq 2 = (q 2 4 + q 2 5 + q 2 6 )/3. χ π and χ η are given by the solution of
χ π + χ η = 2χ 1 ,(5)
χ π χ η = 4 3 B 2 0 (m 4 m 5 + m 5 m 6 + m 4 m 6 ). Following Ref. [11], the EM LEC's can be written in terms of five independent linear combinations of the K's, which is all that can be determined from lattice calculations,
Y 1 = K Er 1 + K Er 2 − K Er 7 − K Er 8 ,(7)Y 2 = K Er 9 + K Er 10 ,(8)Y 3 = −K Er 5 − K Er 6 + 2K Er 10 + 2K Er 11 ,(9)
Y 4 = 2K Er 5 + 2K Er 6 + 2K Er 18 + K Er 19 ,
Y 5 = K Er 8 .(10)
The EM mass-squared splitting of the pseudo-scalar meson is defined as ∆M 2 = M 2 (e = 0) − M 2 (e = 0). In terms of the Y i 's, it becomes
∆M 2 = 2Ce 2 F 2 0 q 2 13 −48e 2 C F 4 0 L r 4 q 2 13χ 1 − 16e 2 C F 4 0 L r 5 q 2 13 χ 13 −12e 2 Y 1q 2 χ 13 + 4e 2 Y 2 q 2 p χ p + 4e 2 Y 3 q 2 13 χ 13 − 4e 2 Y 4 q 1 q 3 χ 13 + 12e 2 Y 5 q 2 13χ 1 −2e 2 C F 4 0 1 16π 2 χ 14 log χ 14 µ 2 q 14 + χ 15 log χ 15 µ 2 q 15 + χ 16 log χ 16 µ 2 q 16 q 13 +2e 2 C F 4 0 1 16π 2 χ 34 log χ 34 µ 2 q 34 + χ 35 log χ 35 µ 2 q 35 + χ 36 log χ 36 µ 2 q 36 q 13 −
(q 13 ) 2 e 2 16π 2 χ 13 3 log
χ 13 µ 2 − 4 +e 2 δ mres (q 2 1 + q 2 3 ).(11)
Note that Y 1 is proportional to the sea quark charges. Since we work with quenched QED, this LEC can not be obtained from our calculation.
We carry out the fit in Section IV B 2 with the finite volume correction to the chiral logarithms taken into account. The finite volume correction to the leading-order chiral logarithms was computed in Ref. [10],
δM 2 13 (L) ≡ M 2 13 (L) − M 2 13 (∞) = 1 3 1 16π 2 F 2 0 R m n13 χ 13 M( √ χ m L) L 2 + 1 3 1 16π 2 F 2 0 R p qπη χ 13 M( √ χ p L) L 2 −2e 2 C F 4 0 1 16π 2 q 13 × q 14 M( √ χ 14 L) L 2 + q 15 M( √ χ 15 L) L 2 + q 16 M( √ χ 16 L) L 2 +2e 2 C F 4 0 1 16π 2 q 13 × q 34 M( √ χ 34 L) L 2 + q 35 M( √ χ 35 L) L 2 + q 36 M( √ χ 36 L) L 2 −3 (q 13 ) 2 e 2 4π κ L 2 + (q 13 ) 2 e 2 (4π) 2 K √ χ 13 L L 2 − 4 √ χ 13 H( √ χ 13 L) L .(12)
M(x) is the function appearing in the finite volume correction to the chiral logarithm induced by the tadpole diagram,
M(x) ≡ 4π ∞ 0 dλ λ 2 exp − x 2 4π λ T (λ) , T (λ) ≡ ϑ 3 0, i 1 λ 3 − 1 ,(13)
where ϑ 3 (v; τ ) is a Jacobi-theta function,
ϑ 3 (v, τ ) ≡ ∞ n=−∞ exp πτ in 2 + 2πvin .
The other functions and a constant κ are given by [10] κ
≡ ∞ 0 dλ λ 2 S(λ) = 2.837 · · · ,(14)S(λ) ≡ − ϑ 3 0, i 1 λ 3 − 1 − λ 3 2 ,(15)H(x) ≡ π ∞ 0 dλ λ 3 2 erf x λ 4π S(λ) ,(16)K(x) ≡ 4π ∞ 0 dλ λ 1 λ 1 − e − x 2 4π λ S(λ) ,(17)
where erf(x) is the error function,
erf(x) = 2 √ π x 0 ds e −s 2 . B. SU(2) L × SU(2) R -plus-kaon
Some time ago Roessl [12] worked out the low energy SU(2) Lagrangian of pions coupled to a kaon. Recently, the RBC and UKQCD collaborations showed that SU(3) chiral perturbation theory is poorly convergent for quark masses near the strange quark mass but that SU(2) chiral perturbation theory coupled to a kaon worked well for pions with masses less than about 400 MeV at NLO [7]. In Ref. [7], the unitary Lagrangian was extended to the partially quenched case. Here, we extend both works to include the EM interactions to the order α em m for the kaon mass, including the one-loop diagrams proportional to α em . For the pion mass, we begin with the partially-quenched SU(3) Lagrangian in Ref. [11] and expand in m l /m s .
pions
We derive the SU(2) L × SU(2) R result for the pion mass-squared splitting by expanding
∆M 2 = 2Ce 2 F 2 0 q 2 13 −48e 2 C F 4 0 L r 4 q 2 13 χ 4 + χ 5 3 − 16e 2 C F 4 0 L r 5 q 2 13 χ 13 −12e 2 Y 1q 2 χ 13 + 4e 2 Y 2 q 2 p χ p + 4e 2 Y 3 q 2 13 χ 13 − 4e 2 Y 4 q 1 q 3 χ 13 + 12e 2 Y 5 q 2 13 χ 4 + χ 5 3 −e 2 3 16π 2 χ 13 log χ 13 µ 2 q 2 13 + e 2 1 4π 2 χ 13 q 2 13 −e 2 C F 4 0 1 8π 2 q 13 q 14 χ 14 log χ 14 µ 2 + q 15 χ 15 log χ 15 µ 2 − q 34 χ 34 log χ 34 µ 2 − q 35 χ 35 log χ 35 µ 2 +e 2 δ mres (q 2 1 + q 2 3 ).(18)
In Eq. (18) all of the low energy constants now depend implicitly on the strange sea quark mass which is fixed (we rename them below to distinguish them from their SU(3) counterparts). In addition the Dashen term has absorbed contributions from the NLO SU(3) LEC's and the logs which do not depend on the charges or masses of the up and down quarks,
2Ce 2 F 2 0 q 2 13 + 12e 2 Y 5 q 2 13 χ 6 3 − 2e 2 16π 2 C F 4 0 q 2 13 χ 6 log χ 6 µ 2 − 48e 2 C F 4 0 L r 4 q 2 13 χ 6 3 .(19)
Including the contributions in pure QCD [7], the pion mass-squared to NLO becomes
M 2 = χ 13 1 + 24 F 2 (2L (2) 6 − L(2)
4 )
χ 4 + χ 5 3 + 8 F 2 (2L (2) 8 − L (2) 5 )χ 13 + 1 2 1 16π 2 F 2 R π 13 χ π log χ π µ 2 + R 1 π3 χ 1 log χ 1 µ 2 + R 3 π1 χ 3 log χ 3 µ 2 + 2C (2) e 2 F 2 q 2 13 − 12e 2 Y (2) 1q 2 χ 13 + 4e 2 Y (2) 2 q 2 p χ p + 4e 2 Y (2) 3 q 2 13 χ 13 − 4e 2 Y (2) 4 q 1 q 3 χ 13 + 12e 2 Y (2) 5 q 2 13 χ 4 + χ 5 3 − e 2 3 16π 2 χ 13 log χ 13 µ 2 q 2 13 + e 2 1 4π 2 χ 13 q 2 13 − e 2 C (2) F 4 1 8π 2 q 13 q 14 χ 14 log χ 14 µ 2 + q 15 χ 15 log χ 15 µ 2 − q 34 χ 34 log χ 34 µ 2 − q 35 χ 35 log χ 35 µ 2 + e 2 δ mres (q 2 1 + q 2 3 ) .(20)
The LO LEC's F and B are the counterparts of F 0 and B 0 from the SU(3) theory, and the other SU(2) LEC's are denoted by an explicit superscript "(2)". R i jk is given in SU (2) partially quenched case as
R i jk ≡ (χ i − χ 4 ) (χ i − χ 5 ) (χ i − χ j ) (χ i − χ k ) .(21)
The finite volume correction to Eq. (20) is given by
δM 2 (L) ≡ M 2 (L) − M 2 (∞) = 1 2 χ 13 16π 2 F 2 R π 13 M √ χ π L L 2 + R 1 π3 M √ χ 1 L L 2 + R 3 π1 M √ χ 3 L L 2 −2e 2 C (2) F 4 1 16π 2 q 13 q 14 M( √ χ 14 L) L 2 + q 15 M( √ χ 15 L) L 2 −q 34 M( √ χ 34 L) L 2 − q 35 M( √ χ 35 L) L 2 −3 (q 13 ) 2 e 2 4π κ L 2 + (q 13 ) 2 e 2 (4π) 2 K √ χ 13 L L 2 − 4 √ χ 13 H( √ χ 13 L) L .(22)
The constant κ and various functions are defined in Eqs. (13), (14), (16) and (17).
kaons
The kaon mass can be obtained from the tree-level Lagrangian, following Refs. [12,23], by constructing the kaon from one light and one "heavy" quark and writing down all operators with the desired symmetries in a non-relativistic theory where the power counting is straightforward. The needed relativistic Lagrangian is then constructed such that in the limit that the kaon is heavy, the non-relativistic theory is recovered. This has been done in the case of QCD to NNLO in Ref. [12] and to NLO in partially quenched QCD in Ref. [7].
Here we add the order e 2 p 2 terms induced by the EM interactions. Once the tree-level Lagrangian is known, the one-loop corrections can be computed. The O(e 2 ) Lagrangian and details of the one-loop calculation are given in the Appendix B.
The O(e 2 p 2 ) Lagrangian is quite complicated, with many operators appearing. While we have listed all possible operators in the Appendix B that contribute, we have not yet reduced them to a linearly independent set using relativistic invariance and the equations of motion. Still, this is enough to give the general quark mass and charge dependence. In the following, this is given by the generic LEC's x
(K) 3 ∼ x (K) 8 .
From Eqs. (B36), (B39) and (B42), the mass-squared of the kaon is
M 2 K = M 2 − 4B(A 3 m 1 + A 4 (m 4 + m 5 )) +e 2 2 A (1,1) K + A (2,1) K q 2 1 + A (s,1,1) K q 2 3 + 2A (s,2) K q 1 q 3 − e 2 (4π) 2 F 2 (A (1,1) K + 3A (2,1) K )q 2 1 + A (s,2) K q 1 q 3 s=4,5 χ 1s log χ 1s µ 2 +e 2 m 1 x (K) 3 (q 1 + q 3 ) 2 + x (K) 4 (q 1 − q 3 ) 2 + x (K) 5 (q 2 1 − q 2 3 ) +e 2 m 4 + m 5 2 x (K) 6 (q 1 + q 3 ) 2 + x (K) 7 (q 1 − q 3 ) 2 + x (K) 8 (q 2 1 − q 2 3 ) +e 2 δ mres (q 2 1 + q 2 3 ),(23)
where we have included the explicit chiral symmetry breaking LEC δ mres , the same as for the pion. Here the subscript "1" stands for a light valence quark, u or d, and "3" for the strange valence quark (charge). "4" and "5" refer to the u and d sea quarks, respectively.
To avoid confusion we note that LEC's without superscripts denote the pure QCD LEC's of Notice that the LO "Dashen" term is different than for the pion: the latter is a single LEC proportional to q 2 13 while the former consists of three LEC's and depends on the u, d, and s charges separately. This is a consequence of the different chiral symmetries assumed in the two cases.
We remind the reader that we do not keep terms of order p 4 and e 4 .
III. LATTICE FRAMEWORK
Following Ref. [8], the lattice calculation employs combined QCD+QED gauge configurations. A combined gluon-photon gauge link is simply the product of two independent links, a SU(3) color matrix for the gluons and a U(1) phase for the photons.
U x,µ = U (3) x,µ × U (1) x,µ Q i ,(24)
where Q i = eq i is the charge of the quark with flavor i. It is the combined link that appears in the lattice Dirac operator, in the usual gauge-invariant way. The gluon and photon links were generated independently in our calculation, so the sea quarks were not electrically charged. This quenched QED calculation suffers a systematic error that is expected to be O(α em α s ) from a simple vacuum polarization argument. In chiral perturbation theory, the charged sea quarks first contribute at O(α em m val ) for the valence quark mass m val , as we have seen in Sec. II. This drawback can be eliminated with the technique of re-weighting [24][25][26], which is becoming common in large scale dynamical calculations [15][16][17]27], and is under active investigation by us [28,29]. In a different context, combined dynamical simulations have also been performed for the first time [30], where the sea quarks are charged from the beginning.
For the QCD configurations, we use the 2 + 1 flavor QCD configurations generated with lattice sizes, respectively. The latter is slightly larger than the value 0.00315(2) determined in Ref. [7] on a smaller ensemble of configurations.
The ensembles and number of measurements on each are summarized in Table I. The stopping criterion in the conjugate-gradient algorithm used to compute quark propagators was 10 −8 , the same as in Ref. [7]. To increase our statistics on some of the ensembles, two or more different locations of the source are used on each configuration (see Table I).
The The quenched QED configurations were generated on the non-compact manifold [2,10].
Here we employ the Feynman gauge instead of the Coulomb gauge which was used previously [2] in our two-flavor calculation. Since the mass is a gauge invariant quantity, the result should be consistent within the statistical error, up to the effects of zero-modes. Further, removal of the modes also results in the satisfaction of Gauss' Law on the torus [10]. An advantage of the non-compact QED formalism is that the U (1) gauge potential A µ can be chosen randomly with the correct Gaussian distribution in momentum space, then Fourier transformed to coordinate space, so there are no autocorrelations in the ensemble. Finally, yet another advantage is that there is no lattice-artifact photon self-interactions in the action.
To couple A µ to the fermions, the non-compact field is exponentiated to produce the photon link, U
x,µ = exp (ieA x,µ ), where e = √ 4πα em ≈ 0.30286.
Since the QED interaction does not confine, it is possible that the finite volume may induce a significant systematic error. We thus do our simulation on both 16 3 ∆ is the separation between measurements in molecular dynamics time units. N meas denotes the total number of measurements, and t src is the Euclidean time-slice location of the source.
IV. RESULTS
In this section we present our results, focusing on the 24 3 ensembles, for the electromagnetic pseudo-scalar mass splittings (∆M 2 ), EM LEC's in SU(3) and SU(2) chiral perturbation theory describing the pseudo-scalar masses, and the quark masses. Results from the 16 3 ensemble are used for estimating finite volume effects which are discussed extensively in Sec. V. Before turning to the results for ∆M 2 , we first describe lattice-artifact electromagnetic effects induced by the finite size of the fifth dimension of the DWF used to simulate the four dimensional u, d, and s quarks.
In the following the notation uū (dd) denotes a meson whose two-point correlation function is made from just the connected quark diagram using degenerate light quarks with equal charges, q = 2/3 (−1/3). Such a meson is neutral, but should not be confused with the π 0 , which requires disconnected quark diagrams.
A. Electromagnetic effects in m res
We first calculate the residual mass m res [32][33][34] from the pure QCD configurations.
Then we consider the residual mass from the combined QCD+QED configurations so that the QED contribution to m res can be extracted. In the lattice DWF, m res is determined from the ratio
R(t) = x J a 5q ( x, t)π a (0) x J a 5 ( x, t)π a (0) ,(25)
where t is the Euclidean time, J a 5q is a pseudo-scalar density evaluated at the mid-point of the extra dimension, π a denotes the usual 4d pseudo-scalar density, and the superscript a The residual mass is an ultra-violet, additive shift of the input, bare quark mass. Because we are interested only in the EM meson mass-squared splittings, the leading order dependence of m res on the bare quark mass cancels, and we use a mass-independent residual mass in our later analysis that can be identified by extrapolating R(t) for the unitary quark masses to m f = 0 with a suitable t-average. Table II shows the numerical result of the residual mass computed from the QCD configurations alone. In each case, R(t) was averaged over the range 9 ≤ t ≤ N t /2 for the size N t of the lattice in the time direction, after folding the correlation function about N t /2. Next we consider the QED contribution to the residual quark mass. The QED contribution from quark flavor i can be expressed as
m res,i (QCD + QED) − m res (QCD) = e 2 C 2 q 2 i ,(26)
where m res,i (QCD + QED) means the residual mass computed on the combined QCD+QED configurations and m res (QCD), the residual mass computed on the pure QCD configurations.
Both are evaluated at m f = 0 and the former with physical quark charge q i . C 2 , which is of order O(m res ), parametrizes the QED contribution to the additive shift of the quark mass.
Although we compute this correction via the Ward-Takahashi identity for DWF [32], using a neutral meson made with degenerate, equally charged quarks, the form of Eq. (26) is completely consistent with a calculation in weak-coupling perturbation theory, say from the one-loop self-energy Feynman diagram for a quark with charge q i . In our chiral perturbation theory power counting, the QED contribution to the residual mass is O(α em m res ) and must therefore be included in our NLO analysis discussed in the next section.
To compute the residual mass and extract the EM contribution via Eq. (26), we use uū or dd correlation functions in Eq. (25) 3 . The total contribution to the meson mass-squared due to explicit chiral symmetry breaking is, as in the case of pure DWF QCD, just the sum of contributions from each quark in the meson, modulo higher order than O(α em m res )corrections. Table III shows the results for C 2 from uū and dd correlation functions. They agree well up to two digits, which implies that the O(α 2 em m res ) contribution is quite small. These differences are higher order in chiral perturbation theory relative to the one we work to in this paper, so we neglect them. We note that C 2 e 2 q 2 i is the expected size, O(α em m res ). The attained statistical precision on C 2 , which is impressive, of course stems from the fact that m res,i (QCD + QED) and m res (QCD) are computed on exactly the same set of gluon configurations, so they are highly correlated, and the QCD fluctuations cancel between them. In addition, C 2 appears to be insensitive to the volume (see Tab. III), presumably because the residual mass arises from the UV, short distance, regime.
B. Meson mass splittings
The electromagnetic mass splittings are determined from the pseudo-scalar masses computed with e = 0 and e = 0, using the same gluon configurations. We use the additional trick of averaging correlation functions over ±e, configuration-by-configuration [2,35]. 3 In an earlier paper [2] we mistakenly included an independent contribution, proportional to q i q j , to the residual mass for the charged mesons made of quarks with charges q i and q j . This is clearly inconsistent with flavor conservation and the definition of a renormalized quark mass defined in perturbation theory.
In Fig. 2, the improvement due to the ±e averaging is demonstrated for the meson masssquared splitting. The vertical axis shows the ratio of the statistical error without the trick to that with the trick, so that larger values indicate smaller statistical error for the ±e averaging trick. In most cases there is a large decrease (∼ 1/10) in the error over the naive factor of The pseudo-scalar meson masses are obtained from single state fits to wall source, point sink correlation functions with periodic boundary conditions in time with use of the fit
function C fit (t − t src ) = A[e −M (t−tsrc+Nt)%Nt + e −M (Nt−t+tsrc)%Nt ],(27)
where M is the ground state meson mass, and t src is the time slice where the source is placed.
To improve statistics in some cases, we average results from two sources (see Table I). The fitting procedure is done with the standard χ 2 minimization (maximum likelihood), and the error on the mass is obtained by the standard jackknife method. Since the meson correlation function is symmetric about the midpoint (from the source) in the time direction, we fold the data about this point and fit with a time range smaller than N t /2. Based on the obtained effective masses (a representative example is shown in Fig. 3), for all correlation functions we chose a fit range of 9 ≤ t − t s ≤ N t /2. The pseudo-scalar meson masses are tabulated in Tabs. XIII -XIV. We have extracted the masses in two ways, one being from the fits to the correlation functions using the full covariance matrix and the other being uncorrelated fits following [7,17]. The values of χ 2 /dof for the covariant fits are roughly one for the 24 3 ensembles, but somewhat higher in some cases for the 16 3 ones and for the heavier quark masses on both ensembles. Such behavior for the 16 3 ensembles was seen in the earlier, pure QCD, analysis using these configurations, and was attributed to an inferior gauge field evolution algorithm [31]. An We pause to compare the observed explicit chiral symmetry breaking effects to those expected from the discussion of the residual mass in the previous section. In the chiral limit, m f = −m res (QCD), and in the absence of EM induced explicit chiral symmetry breaking (L s → ∞), the neutral meson mass-squared should vanish (up to α 2 em corrections which we ignore), and so too should the splittings. But it is clear from Figs. 4 and 5 that thedd masssquared splitting does not (the same is true for the uū meson). Following the discussion in Sec. IV A and from the result of the pseudo-scalar mass-squared at the lowest-order in chiral perturbation theory, the shift in the splitting in the chiral limit should be 2B 0 C 2 e 2 q 2 d or 2BC 2 e 2 q 2 d , depending on whether we choose SU(3) or SU(2) chiral perturbation theory. A simple linear fit, also shown in Fig. 4, suggests this is true. Note that at NLO there are no logs in the splitting of neutral mesons made from only connected quark propagators, that is, a "charged" meson whose net charge happens to be zero. Further, by making L s larger, this lattice artifact should be (exponentially) reduced, which is also clear from the Fig. 4 where for L s = 32 the shift has been reduced by roughly a factor of ten, and the splitting nearly LEC to the fit, e 2 δ mres (q 2 i + q 2 j ). We conclude that the explicit chiral symmetry breaking artifacts induced by finite L s and QED interactions are precisely quantifiable at NLO in chiral perturbation theory and that higher order terms can be safely neglected, so these artifacts can be robustly eliminated, just as in the case for pure (DWF) QCD. Tables IV -VI. Before proceeding, we address a subtly in the kaon fits that was not recognized until after the correlation functions had already been computed. Our original plan was to use an SU (3) chiral perturbation theory analysis only, for quark masses in the range 0.005 − 0.03, and non-degenerate meson correlation functions were computed for these masses in all possible combinations. However, learning first from the pure QCD analysis [7,15,16], and later from our own, it became clear that 0.02 and 0.03 were too heavy, and that SU (2) , results in a stable fit with the same χ 2 , but with different values of the LEC's. While these fits all agree exactly when evaluated at the data points used in the fits, they differ elsewhere. There are two ways to fix this problem of an accidental flat direction in the χ 2 function at our disposal. First, keeping the same quark mass range, use the technique of singular-value-decomposition [36] (SVD) to determine all 10 LEC's. Second, increase the number of sea or valence quark mass points in the fit, so the parameter directions are all linearly independent. While treating the (next available) mass 0.02 quark simultaneously as light and strange contradicts our assumption that m l /m s 1, nevertheless it allows the LEC's to be linearly independent, and only slightly increases χ 2 which is still small. In practice, we only added the 0.02 valence quark mass to the kaon fit, keeping the light sea quark mass ≤ 0.01. As it happens, the quark masses determined from these two methods agree well, giving confidence that the SVD fit procedure, which we use for our central values, is reliable. Further, in the case where 0.02 data points were used, setting each of the LEC's to zero in turn resulted in much bigger χ 2 values except for x
(K) 6 − x (K)
8 , the ones related to sea quark masses which are not constrained as well. x (K) 8 = 0 gave the smallest χ 2 . In each of these cases the quark masses agreed within statistical errors to the full SVD fit. We use the difference in the central values of the quark masses from the two procedures as an estimate of one of the systematic errors due to fitting.
From Table V, we can see a large effect on C going from SU(3), where it is almost zero, to SU(2) where it is almost ten times larger. Recall that in the SU(2) theory the contributions of the strange quark terms in the SU(3) theory are absorbed into C (2) (see Eq. (19)). This situation is reminiscent of the pion decay constant in pure QCD computed on these lattices and for the same range of quark masses; the logs in that case also tend to significantly reduce the LO contribution over a simple analytic function, and the physical value [7,15,16]. Here, especially in the SU(3) case, the effect is even more dramatic. The other pion electromagnetic LEC's are roughly the same in both theories. In the SU (2) case, the size of the NLO EM correction turns out to be smaller than the LO one, showing compatibility with the chiral expansion. Finally, in Table V, we show LEC's corresponding to the phenomenological parameter set presented in Ref. [11]. The fact that the SU (3) NLO LEC's computed here (left-most column) do not agree is not surprising since the LO LEC, C, is clearly underestimated by a large degree. Note that to compare values of C, a factor of a −4 needs to be introduced, as well. We discuss the Dashen term further in Sec. V, after presenting the finite volume fits.
finite volume fits
Next we include in our fits the finite volume corrections to the chiral logarithms using Since the finite volume effects in QCD are very small compared to the QED ones, we ignore the former. Figure 9 shows the modified fits for the pions and kaons on the 24 3 lattice. The LEC's are given in Table V and VI. The largest change by far is in C, the LO Dashen term, which roughly doubles in the SU(2) case and increases by a factor of four in the SU(3) case.
Note, it is still much larger for the SU(2) fit. This is consistent with the observed large effect in the charged meson splitting compared to the neutral. Fortunately, this huge change does not greatly affect the values of the quark masses, as we shall see.
C. quark masses
Having determined the LEC's to NLO describing the pseudo-scalar masses in chiral perturbation theory, we now turn to fixing the physical quark masses at the (arbitrary) low M π ± = 139.57018 ± 0.00035 (28) that did not include logarithms. The strange quark mass is somewhat lower here, which may be a real flavor-dependent effect [1,2]. We also note that in the combined continuum limit analysis mentioned earlier, the RBC and UKQCD collaborations find that the strange quark mass is even smaller [15][16][17]. The average light quark mass is close to the value determined in pure QCD [7,15,16].
SU(3) SU(2) inf.v. f.v. inf.v. f.v. SU(3)+phenom.M K 0 = 497.614 ± 0.024(29)M K ± = 493.667 ± 0.016,(30)10 2 A 4 -1.89(45) -2.15(52) -2.21(56) - - - 10 3 A (1,1) K -9.1(1.1) -8.9(1.3) -8.8(1.4) -6.4(1.0) -5.8(1.2) -5.7(1.3) 10 3 A (2,1) K 8.SU(3) SU(2) inf.v f.v inf.v. f.v.
V. SYSTEMATIC ERRORS
In this section we examine the important systematic errors in our calculation: the chiral extrapolations, finite volume, non-zero lattice spacing, and QED quenching. In each case we estimate the size of the effect on the values of the quark masses and investigate the effect on the LO electromagnetic LEC's. Similar systematic uncertainties have been given for the pure QCD sector [7,15,16].
To estimate the systematic errors, the change in a quantity is computed under the influence of a change in how that quantity is computed, for example, by using a different fit formula. Since the data is the same, or there is significant overlap, in each case, we compute the change under the (super-)jackknife procedure in order to assess its significance. Central values of all quantities are quoted for the finite volume, SU(2) chiral perturbation theory fits which we believe give the most accurate results. The systematic errors computed in the following come from comparison to these central values.
A. chiral extrapolations
Previous studies have used the difference in analytic and chiral perturbation theory fits to estimate the chiral extrapolation error that stems from using unphysical heavy quarks [7,14,15,17,37,38]. One can also estimate the error in chiral perturbation theory alone by comparing the relative sizes of LO, NLO, or even NNLO corrections to a given quantity. For the latter to work, the estimates of the higher order contributions must be accurate.
It is perhaps not surprising to find that the meson mass-squared splittings show little trace of the chiral logarithms. For the mass range of pions in this study, it is well known that low energy observables like the meson mass-squared or decay constant exhibit more or less linear dependence on the quark mass. In Fig. 7, the charged pseudo-scalar splitting appears linear over the range of unitary points shown in the figure. Nevertheless, the fits to our data do show that NLO chiral perturbation theory (chiral logs) is consistent with the data. A similar conclusion was reached in the pure QCD case [7,[15][16][17]. To NLO in chiral perturbation theory, there are no logs for the neutral mesons made from connected quark propagators like those studied here, and indeed the neutral splittings, too, appear to be quite linear.
We do point out one aspect of the EM logs that leads one to expect a noticeable affect.
They behave like αm log m, not m 2 log m as the pure QCD logs do. A factor of α has replaced a factor of the quark mass. In fact they are like the quenched logs in pure QCD in this respect.
The first step in estimating the systematic error is to determine the fit range, or range of quark masses included in the fit. The available ranges are summarized in Table I. In
Refs. [7,[14][15][16][17] it was shown that for the same ensembles used in this work, SU (3)
B. finite volume
The effect of finite volume on the measured charged-meson splittings is large, as we have seen. In Fig. 11 the difference between the measured 16 3 and 24 3 EM splittings is about 15-20%. The LO LEC C changes dramatically, by about a factor of two, when the finite volume corrections at NLO are included in the chiral perturbation theory fits (see Table V and Fig. 10). Besides the usual special functions that replace the infinite volume logs, a large, negative constant appears in the finite volume formula, −3κq 2 13 /4πL 2 [10] with κ ≈ 2.837, which cancels against an enhanced value of C.
To estimate how reliable the NLO finite volume corrections are, one can use the LEC's from the 24 3 fits to predict the finite volume shift in the 16 3 splitting. The fit and prediction are shown in Fig. 11. First, the SU(2) fit agrees well with the 24 3 results for m f ≤ 0.01 which is the quark mass range used in the fit. For larger masses the fit deviates significantly from the data and suggests that NLO chiral perturbation theory is not reliable for these masses. Even for m f = 0.01, where we may trust NLO chiral perturbation theory, the theory over-predicts the shift on the 16 3 lattice by about a factor of two. The NLO LEC's Y 3 and Y 5 also have large finite volume shifts.
From Table VI From the pure QCD calculations, we know the finite volume effects in the 24 3 meson masses are at about the one percent level [15][16][17], and therefore the QED finite volume corrections dominate.
The finite volume errors on the quark masses are summarized in Table VIII. C. non-zero lattice spacing Since our calculation has only been done at a single lattice spacing, we can not estimate the non-zero lattice spacing errors directly. However, by now there is much evidence that these O(a 2 + m res a) discretization errors are small in pure DWF QCD, and they should largely cancel in the splittings. Even assuming they do not cancel, there is no reason to expect they are enhanced over the pure QCD case. In the first QCD calculation using the 24 3 ensemble, it was estimated that scaling errors were at about the four percent level for low energy quantities like the pion decay constant and the kaon [7]. Since then, a new calculation at the same physical volume but smaller lattice spacing has shown this estimate was about right, or perhaps a bit conservative [15][16][17]. Of course, here we are interested only in the mass splittings. The pion and kaon masses are fixed to their continuum values, so they have no scaling errors. Instead, the lattice spacing errors enter in the LEC's and the physical quark masses. Therefore we assign a robust four percent scaling error to the quark masses, which will be eliminated in up-coming calculations on the finer lattice spacing ensemble [15][16][17]. This error also encompasses the uncertainty in setting the lattice scale itself, which as mentioned earlier differs by about 2 ∼ 3 percent from the scale given in
Ref. [7].
The non-zero lattice spacing errors on the quark masses are summarized in Table VIII.
D. QED quenching
As mentioned our calculation is done in quenched QED where the sea quarks are neutral.
In chiral perturbation theory, we have neglected terms of order O(α em m sea ), including logs. (23)). However, we do note that sea quark charge effects from the logs can be included a posteriori in our determination of the quark masses.
Since the LEC's absorb changes of scale in the logs, one way to estimate the effect of the missing LEC's, or counter-terms, is to mark the change in the quark masses when these logs are included, or not. This leads to a negligible change in the quark masses. From Table V, the other EM LEC's have magnitudes roughly in the range 0.01 to 0.001. Setting Y 1 at the high end, Y 1 = ±0.01, the quark masses again change very little. Of course, the LEC's calculated with q sea = 0 will differ from those with q sea = 0, by O(α em ). This is higher order for all the LEC's determined here except C for the pions and A
(1,1)
K , A (2,1) K , A (s,1,1) K , and A (s,2) K
for the kaons. Taking all of the above into account, we quote a conservative two percent systematic error in our quark mass determination, stemming from the quenched approximation to QED.
Of course the above is only a rough estimate, so presently we are investigating the use of so-called re-weighting techniques to eliminate the quenching effects [24-26, 28, 29, 40].
Re-weighting is simply the use of ratio(s) of fermion determinants in observable averages in order to include the desired dynamical-quark effects. The calculation of a determinant which is non-local in the fields is quite expensive, so stochastic estimators must be used to make the calculation tractable. Re-weighting in the strange quark mass to the a posteriori determined physical value has proved quite useful and efficient in recent 2+1 flavor simulations [15][16][17]27].
VI. QUARK MASSES
We use finite volume SU (2)
which is obtained via nonperturbative renormalization using the RI/SMOM γµ scheme [17, [41][42][43][44][45]. The second error is systematic, including O((µa) 2 ), which will be removed when we take the continuum limit in future work (the O(α em ) QED correction to Z m is omitted).
where the first error is statistical, and the second is a total systematic error, derived by adding the individual errors summarized in Table VIII in quadrature. We remind the reader that these central values are obtained from our SU (2), finite volume fits on the 24 3 ensembles.
We note that the up quark mass obtained here is different from zero by more than six standard deviations, which seems to rule out the m u = 0 solution to the strong CP problem. However, there is an extensive literature concerning this scenario to which we refer the interested reader. For a discussion of extracting the up quark mass by using chiral perturbation theory, and its consequences, see [46][47][48]. The possibility of instanton effects additively shifting the up quark mass is discussed in many places [49][50][51][52]. In [53],
renormalization
VII. MESON MASS SPLITTINGS
In Tab. V we give the contribution to the charged pion mass splitting in the chiral limit, or Dashen's term. The physical splitting, given in Eq. (1) [54,55]. Our value for (m π ± − m π 0 ) QED is roughly consistent with, but two statistical standard deviations smaller than, the value from phenomenology and SU (3) chiral perturbation theory reported in [11], 3.7 MeV.
The above suggests that NLO contributions at the physical quark masses may be as large as 25% of the total pion mass difference, m π + − m π 0 . Away from the chiral limit, there are corrections to m π 0 that we have not computed in the lattice calculation (disconnected diagrams), nor in chiral perturbation theory (logs). However, we can estimate some of them by evaluating Eq. (20) for m u = m d = m ud , q 1 = q 3 = q u and averaging it with the case for
q 1 = q 3 = q d M 2 (q 1 , q 3 ; m 1 ) ≡ 1 2 M 2 (m 1 , q 1 , m 1 , q 1 ) + M 2 (m 1 , q 3 , m 1 , q 3 ) .(39)
This form can be inferred for the π 0 made with degenerate light valence quarks (m 3 = m 1 )
in our current study in which only the connected valence quark diagram is computed and QED is quenched. We focus on the one-particle irreducible two-point function Σ π 0 (p 2 ) of π 0 , and pick out the part depending on the valence EM charges induced from the connected diagram. Σ π 0 (p 2 ) can be divided into a pure QCD part Σ QCD π 0 (p 2 ) and a QED correction Σ QED π 0 (p 2 ) at order e 2 ,
Σ π 0 (p 2 ) = Σ QCD π (p 2 ) + Σ QED π 0 (p 2 ) ,(40)Σ QCD π (p 2 ) = 1 √ 2 2 tr (τ 3 ) 2 A QCD (p 2 ) = A QCD (p 2 ) ,(41)Σ QED π 0 (p 2 ) = 1 √ 2 2 tr (τ 3 Qτ 3 Q) D 1 (p 2 ) + 2 × 1 √ 2 2 tr (τ 3 ) 2 Q 2 D 2 (p 2 ) = 1 2 q 2 1 + q 2 3 D 1 (p 2 ) + q 2 1 + q 2 3 D 2 (p 2 ) .(42)
where τ a (a = 1, 2, 3) denote the Pauli matrices and Q = diag (q 1 , q 3 ). In Eq. (42), the first term originates from the Feynman diagram in which a virtual photon is exchanged between two valence quark lines, while a photon propagates on the same valence quark lines and induces the second term. Because the functions D 1, 2 (p 2 ) are given by QCD dynamics weighted by the photon propagator, the self-energy Σ π + (p 2 ) of the charged pion is also expressed in terms of these functions
Σ π + (p 2 ) = Σ QCD π (p 2 ) + Σ QED π + (p 2 ) ,(43)Σ QED π + (p 2 ) = tr (τ + Q τ − Q) D 1 (p 2 ) + tr (τ + τ − + τ − τ + ) Q 2 D 2 (p 2 ) = q 1 q 3 D 1 (p 2 ) + q 2 1 + q 2 3 D 2 (p 2 ) .(44)
From Eqs. (40)- (44), the charge dependence of m 2 π 0 and m 2 π + = M 2 (m 1 , q 1 , m 1 , q 3 ), to the order relevant to us, is found as
M 2 (m 1 , q 1 , m 1 , q 3 ) = K + q 1 q 3 F 1 + q 2 1 + q 2 3 F 2 , m 2 π 0 = K + 1 2 q 2 1 + q 2 3 F 1 + q 2 1 + q 2 3 F 2 = M 2 (q 1 , q 3 ; m 1 ) ,(45)
where K denotes the QCD part to NLO of chiral perturbation, and F 1, 2 the O(e 2 ) and
O(e 2 m l ) part. The chiral symmetry as well as QED gauge invariance should give F 1 | m 1 =0 = −2 F 2 | m 1 =0 to reproduce the EM charge dependence (q 1 − q 3 ) 2 of the LO EM correction to m 2 π + . Using Eq. (39) for m 2 π 0 , we find the LO + NLO EM pion mass difference at the physical point to be m π + − m π 0 = 4.50 (23) MeV. Phenomenology predicts that a small part of the total NLO correction is due to m u − m d = 0, 0.17(3) MeV [56] and 0.32 (20) MeV [57].
M 2 K (m u , 2/3, m s , −1/3) − M 2 K (m u , −1/3, m s , −1/3) = ∆ (EM) M 2 K + ∆ (mu−m d ) M 2 K +O(e 2 (m u − m d )) (46)
where the contributions to the mass-squared splitting are defined as
∆ (EM) M 2 K = M 2 K (m ud , 2/3, m ud , −1/3) − M 2 K (m ud , −1/3, m ud , −1/3),(47)∆ (mu−m d ) M 2 K = M 2 K (m u , 0, m s , 0) − M 2 K (m d , 0, m s , 0).(48)∆ (EM)/(mu−m d ) M 2 K /(M K 0 + M K ± ) are quoted above. So out of a physical mass-squared splitting (M K 0 ) 2 − (M K ± ) 2 = 3902.7 MeV 2 , about −47(2)% is ∆ (EM) M 2 K and +148(2)% is ∆ (mu−m d ) M 2
K . The breaking of Dashen's theorem can also be parametrized by ∆E [58],
∆E = M 2 K (m 1 , q 1 , m 3 , q 3 ) − M 2 K (m 1 , q 3 , m 3 , q 3 ) M 2 (m 1 , q 1 , m 1 , q 3 ) − M 2 (m 1 , q 3 , m 1 , q 3 ) − 1,(49)
where m 1 is the light quark mass and m 3 is the strange. M 2 (m 1 , q 3 , m 1 , q 3 ) is used here to represent m 2 π 0 ; no significant change of ∆E is observed in our numerical study even when the average (39) is adopted for m 2 π 0 in place of M 2 (m 1 , q 3 , m 1 , q 3 ). In the SU(3) chiral limit ∆E = 0 since the LO Dashen terms are the same in the numerator and denominator. If the strange quark mass is fixed to its physical value, then it does not vanish, and can be much larger than zero, even in the light quark chiral limit. Notice that ∆E vanishes trivially in both SU(2) and SU(3) theories when m 1 → m 3 .
We show ∆E for our data in Fig. 12 where the artifact δ mres (q 2 i + q 2 j ) has been subtracted for each value of the meson mass-squared. In the upper panel, fit results are shown for SU(3).
The fit evaluated at the simulated mass points does a reasonable job of reproducing the data, though as m 3 increases differences emerge. This is not surprising since only m 1 , m 3 ≤ 0.01 points were used in the fit, and including larger values yielded significantly poorer fits. More troublesome is the light quark extrapolation which yields a large value of ∆E at the physical point, which can be understood from two primary causes. First, the numerator is quite large since m 3 is evaluated at the physical strange quark mass, leading to a large O(αm) correction to the charged kaon mass-squared. Second, the denominator becomes quite small because the LO Dashen term is quite small in the SU(3) fit (compared to NLO terms). Both facts, of course, signal a breakdown in SU(3) chiral perturbation theory which renders the SU (3) ∆E unreliable. As noted in [11], the sea quark charge LEC's drop out of ∆E, and only known logarithms remain. Adding these to the (cyan) physical curve in Fig. 12 changes it only slightly.
In the lower panel of Fig. 12 we show analogous results for the SU(2) fits. While the SU(2) fits are more reliable since the LO contribution is larger compared to NLO, the latter corrections are still large (recall Fig. 10). As expected, the fit agrees better with the data points for larger values of m 1 , but the extrapolated value at the physical point and infinite volume is still much larger than the data points. We find in quenched QED that ∆E = 0.628 (59) where the error is statistical only. This is much larger than the value reported in our previous two flavor paper [2] and not much smaller than phenomenology and SU(3) chiral perturbation theory [11]. The main difference is that here we use full NLO chiral perturbation theory with finite volume corrections while in [2] only simple analytic fits were used. To properly address these large corrections, one needs to simulate with larger volumes and smaller quark masses, a project that is now underway.
The small statistical errors result because the physical pion and kaon meson masses were used to determine the physical quark masses from our fit.
Finally, based on the quark masses in Eqs. (32)- (34) and (36), we examine the ratio introduced in Ref. [59],
κ quark mass ≡ m d − m u m s − m ud 2m ud m s + m ud ,(52)
which is equal to
κ meson ≡ (M 2 K 0 − M 2 K ± ) QCD M 2 K − M 2 π M 2 π M 2 K (53) = M 2 (m d , 0, m s , 0) − M 2 K (m u , 0, m s , 0) M 2 K (m s , 0, m ud , 0) − M 2 (m ud, 0, m ud , 0) M 2 (m ud , 0, m ud , 0) M 2 K (m s , 0, m ud , 0) ,(54)
up to NNLO in SU (3) ChPT [56]. For SU(3) we obtain
κ quark mass = 0.00201(3),(55)κ meson = 0.00201(3),(56)κ meson = 0.00191(3),(57)
where the errors are statistical only. For SU(3) the values are quite consistent with each other, while for SU (2) there is a small difference. In [59], κ extracted from η → π 0 π + π − decays is 0.0019(3) while the O(p 6 ) analysis in [57] gives κ = 0.00260 at m s /m ud = 24.
VIII. ISOSPIN BREAKING EFFECTS ON THE KAON DECAY CONSTANT
In our results the up quark mass is about 35% smaller than average of the up and down quark masses, m ud . In principle, this isospin breaking effect may cause visible effects on phenomenologically important quantities when they are measured with sufficient accuracy.
As we saw in the previous section, a major part of the Kaon mass splitting comes from the quark mass difference, m u − m d .
Here we examine isospin breaking effects on the Kaon decay constant, f K . By combining the experimental decay widths, Γ(K → νµ(γ)) and Γ(π → νµ(γ)), and f π and f K , one can extract the corresponding ratio of CKM matrix elements [60]. In the latest global analysis by the FlaviaNet Working group on Kaon Decays [61], f K fπ Vus V ud is obtained from experimental results with an accuracy of 0.2%. The ratio of the decay constants used are from their world average of lattice QCD simulations, and is
f K f π = 1.193(5) [0.4%].(59)
We
IX. NUCLEON MASS SPLITTINGS
Isospin breaking also occurs in the nucleon system. The proton is slightly lighter than the neutron, which makes the proton a stable particle. In conjunction with baryon PQχPT, the lattice simulation helps us understand the relation between the baryon masses and their quark content [63].
In Nature, m p −m n =−1.293321(4) MeV as determined by experiments, and it is explained by two mechanisms. One is the EM interaction. The proton is a charged particle, but the neutron is neutral, so the QED interaction makes the proton heavier. The other is due to non-degenerate u, d quark masses. The valence quark content in the proton and neutron is uud and udd, respectively. So the proton is lighter than the neutron due to the fact that the d quark is heavier than the u. Combining these two effects in our lattice calculations, we can compute the p-n mass splitting.
For non-degenerate quark masses, we study the splitting using the pure QCD configurations. The nucleon mass in two flavor QCD is given by baryon PQχPT, to NLO [63],
m p = M 0 + 1 3 (5α + 2β)m u + 1 3 (α + 4β)m d + 1 2 σ(m j + m l )(60)m n = M 0 + 1 3 (α + 4β)m u + 1 3 (5α + 2β)m d + 1 2 σ(m j + m l )(61)
where m u , m d are the masses of the valence quarks and m j , m l are the masses of the sea quarks. The mass difference between the proton and the neutron is
(m p − m n ) (m d −mu) = − 1 3 (4α − 2β)(m d − m u ).(62)
We note only the sum of sea quark masses, m j + m l , appears in Eqs. (60) and (61) Next, we test the EM induced mass splitting on QCD+QED configurations with unitary (and therefore degenerate) mass points. The lowest order mass difference is parametrized as:
(m p − m n ) QED = α em (A 0 + A 1 m ud )(63)
where m ud = (m u + m d )/2, and the dependence on α em is made explicit to remind ourselves that the splitting vanishes in the absence of QED.
All of the above LEC's here can be extracted from fits to lattice data.
We first extract the nucleon masses from the two-point correlation function. The correlation function measured on the lattice with anti-periodic boundary condition in time has the form [64]:
G(t) = (1 + γ 4 )A B + e −M B + t − (1 − γ 4 )A B + e −M B + (Nt−t) +(1 + γ 4 )A B − e −M B − (Nt−t) − (1 − γ 4 )A B − e −M B − t ,(64)
where B + represents the nucleon state which has positive parity and B − represents the excited state of the nucleon which has negative parity. N t is the time-size of the lattice.
Since the mass of the excited state is much heavier than the ground state, we neglect its contribution. The nucleon and anti-nucleon terms left in the correlation function are picked up by multiplying G(t) by the projection operator 1 ± γ 4 and taking the trace. Then we average these two terms by taking t → N t −t for the anti-nucleon to improve the statistics of our measurements. The ±e trick is also used when QED configurations are included. Finally the nucleon masses are extracted from single state fits to point-sink correlation functions as
G(t) = Ae −M t ,(65)
where M is the ground state nucleon mass, and A measures the overlap between the nucleon state and the nucleon interpolation operator.
Initially, nucleon correlation functions were computed from the same wall source propagators used for the meson splitting analysis. However, on the 24 3 ensembles these exhibited poor plateaus and had poor signals for the EM neutron-proton mass difference. We then The configuration information of the additional measurements is listed in Tab. IX. Figure 14 shows representative plateaus for the sea quark mass 0.005 ensemble.
The nucleon masses are listed in Tabs. XVII and XVIII. They come from a standard χ 2 minimization with correlated fit, and the error on the mass is from the standard jackknife formula [9,65],
δm ele = 2παm 1 L 3 q =0 G E (q) 2 |q| · 2 q 2 + 4m 2 + 1 2m 2 1 + 4m 2 q 2 − 1 (66) δm mag = − πα 2m 3 1 L 3 q =0 |q|G M (q) 2 · 1 + 4m 2 q 2 − 1 − 1 2 1 1 + q 2 /4m 2 ,(67)
where δm ele (δm mag ) is the electric (magnetic) contribution to the nucleon mass m. We evaluate the above formulae at the physical point, using the dipole form for the nucleon elec-
tromagnetic form factors, G p E (Q 2 ) = G p M (Q 2 )/µ p = G n M (Q 2 )/µ n = G D (Q 2 )
, where µ p (µ n ), are proton(neutron) magnetic moment, and G D (
Q 2 ) = 1/(1 + Q 2 /Λ 2 ) 2 with Λ 2 = 0.71GeV 2 .
For G n E (Q 2 ), we use the Galster parametrization of G n E (Q 2 ) = AQ 2 /(4m 2 + BQ 2 ) · G D (Q 2 ) with A = 1.70, B = 3.30 [66]. We obtain (m p − m n ) Next we compute the mass splitting due to non-degenerate u and d quark masses, which is expected to switch the sign of the mass difference, in accord with Nature. Figure Ref. [63].
The quark mass dependence of m p −m n is simple in baryon chiral perturbation theory [63] to NLO in pure QCD, as seen in Eq. (60). The leading quark mass dependence for the EM splitting is unknown, so we assume that it is linear, and at this stage the measured values likely can not be used to discern a more complicated form anyway. In contrast, chiral perturbation theory for the nucleon mass itself predicts several non-analytic terms at NLO, and the careful extrapolation to the physical point is an important topic of current calculations. Because we have few data points, and our quark masses are relatively heavy, we do not attempt such an extrapolation here.
Combining the contributions from the EM interaction and non-degenerate u, d quark masses, we give the physical p-n mass splitting. We find m p −m n = −1.93 (12) and −2.13 (16) MeV, for 16 3 and 24 3 lattice sizes, respectively, which is larger than the experimental result (−1.293321 (4) MeV), but remarkable given that compared to the mass itself, the splitting is a 0.1% effect in Nature. The errors above are statistical only, and their small size is due to the facts that the difference is calculated on exactly the same configurations and with the ±e averaging trick.
To estimate the systematic error on the EM splitting from the chiral extrapolation we take the difference between the extrapolation using all of the data points (on the 24 3 lattice) and the lightest two mass points, or roughly 0.3 MeV. The finite volume effects, while quite noticeable at the simulated quark masses, are smaller in the quark mass extrapolated result.
To roughly estimate the finite volume effect, we consider the difference in the 16 3 to improve the extrapolation, with a different lattice spacing to take the continuum limit, and on a larger volume to improve the infinite volume extrapolation. determine the strange quark mass [7,[14][15][16][17][18][19]. When using the finite volume PQχPT formulas, we found that the NLO corrections relative to LO are about 25% for the physical pion masses, neglecting O(α 2 em )-terms in the π 0 mass that come from the axial anomaly (disconnected graphs) and are expected to be small [11]. Simple linear fits also work as well as the complicated NLO chiral perturbation theory ones, as has been seen in the case of pure QCD [7,[15][16][17]. Indeed, our data do not show significant curvature, so while they do not seem to require the presence of chiral logs from a theoretical point of view, they are consistent with them. The EM splittings and LEC's are significantly affected by the finite volume of the lattice, as expected since the long range interactions of the photons are not confined. For our final values, we used the finite volume formulas for the chiral logs computed in Ref. [10].
lattice size 10 2 A 0 A 1 χ 2 /dof (m p − m n ) QED (MeV)
The lattice-extracted, SU(3) L × SU(3) R LEC's were found to be somewhat inconsistent with the result of the phenomenological analysis in Ref. [11], although the latter were fit using an ad hoc set of choices for the LEC's. This may also be due to a lack of convergence of SU(3) chiral perturbation theory in the range of quark masses used here, or finite volume effects, or both.
The masses of the light quarks were also determined from our calculation. This is the first time EM interactions have been included directly in the quark masses determined from 2+1 flavor calculations. We employed the physical masses of the π ± , K 0 and K ± mesons as input to fix the quark masses in PQχPT. The SU(2) L × SU (2) Concerning the solution of the strong CP problem, it is of interest that our value for the up quark mass is different from zero by many (∼ 6 − 7) standard deviations. The Dashen term, or LO EM contribution to the pion mass difference is (m π ± −m π 0 ) QED = 3.38(23) MeV in our calculation, coming from the SU(2) chiral perturbation theory, finitevolume-corrected fit, which is our most reliable one. The error is statistical only. However, the value from the linear chiral fit agrees within errors. It is also consistent with the values of m 2 π ± in the chiral limit recently reported in [54,55], but somewhat smaller than the value from phenomenology and SU(3) chiral perturbation theory [11] and the value we reported for two flavor QCD in Ref. [2]. This suggests that NLO contributions at the physical quark masses may be as large as 25% of the total pion mass difference, and approximating the π 0 mass from the LEC's computed here, we find the LO+NLO EM contribution at the physical point is m π + − m π 0 = 4.50 (23) MeV. Phenomenology predicts that a small part of the NLO correction is due to m u −m d = 0, 0.17(3) MeV [56] and 0.32 (20) MeV [57]. Similarly, we find for the kaons that the pure EM mass difference is (m K ± − m In this work, quenched QED configurations were used to account for the EM interactions of the valence quarks, i.e., the sea quarks were neutral in our calculation. The systematic error due to this approximation can be removed by the re-weighing method [24,26]. We are now undertaking such a study. In similar spirit to the most recent RBC/UKQCD pure QCD calculation [15][16][17] on a finer lattice ensemble, a ≈ 0.086 fm, the analysis presented here is being replicated on those ensembles in order to take the continuum limit. Similarly, calculations on a third set of ensembles being generated by the RBC and UKQCD collaborations are on-going, with a new modified Iwasaki gauge action [67], to better explore the chiral regime. (12) The variables appearing in Eq. (B2),
m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(u µ ≡ i u † (∂ µ u − iR µ u) − u ∂ µ u † − iL µ u † , χ ± ≡ u † χu † ± uχ † u , χ ≡ 2B 0 M ,(B4)
are given in terms of the spurion field M in place of the ordinary quark mass matrix, and the external fields R µ , L µ , which transform under the local chiral rotation (g L (x), g R (x)) ∈ G as
R µ → R µ = g R R µ g † R + ig R ∂ µ g R , L µ → L µ = g L L µ g † L + ig L ∂ µ g L , M → M = g R Mg † L . (B5) For u[Π] → u[Π ] = g R u[Π] h((g L , g R ); Π) † = h((g L , g R ); Π) u[Π] g † L ,(B6)
with h((g L , g R ); Π) ∈ H, it turns out that u µ and χ ± transform covariantly with respect to
h A → A = hAh † ,(B7)
and that L QCD, 2 is invariant under the local chiral transformation.
The high frequency modes of photons coupled to quarks also generate local interactions in the low-energy effective Lagrangian of QCD. The coupling of quarks to photons preserves chiralities;
A µ (q L γ µ Qq L + q R γ µ Qq R ) ,(B8)
where Q represents the charge matrix and takes the form Q = e diag (q uV , q dV , q uS , q dS , q uV , q dV ) ,
for two-light flavors. The systematic dependence on these quark charges can hence be traced back once Q is promoted to a set of hermitian spurion fields, Q R, L , that transform under chiral rotations as
Q L → Q L = g L Q L g † L , Q R → Q R = g R Q R g † R .(B10)
On the other hand, to construct the effective Lagrangian, it is convenient to define quantities that transform covariantly by h((g L , g R ); Π), i.e., as in Eq. (B7)
Q L ≡ uQ L u † , Q R ≡ u † Q R u .(B11)
Since we will set Q L , Q R to the diagonal EM charge matrix Q after constructing the effective Lagrangian, we impose the chiral-invariant condition
Q R = Q L ≡ Q ,(B12)
which reduces just to the charge matrix in the end. The leading order (O(p 2 ) ∼ O(e 2 )) Lagrangian involving Nambu-Goldstone bosons is thus given by
L π, 2 = F 2 4 u µ u µ + χ + + C Q R Q L ,(B13)
The QED corrections from the low frequency photons can also be included by coupling the Nambu-Goldstone bosons to the U (1)-gauge potential A µ (x) and by setting the external fields L µ , R µ along the direction of Q in the end;
L µ = Q A µ = R µ .(B14)
The kaon sector
In SU(2) chiral perturbation theory, the strange quark is treated as being heavy, and hence the kaons are no longer treated as Nambu-Goldstone bosons. Since the EM charge of the sea strange quark, s S , differs from that of the valence strange quark, s V , in our simulation, these together with the ghost strange quark, s, are regarded as constituting the partially-quenched strange sector. Nevertheless, for the purpose of the analysis of our lattice data, it suffices to write down the effective Lagrangian with respect to the kaon multiplet including the valence anti-strange quark s V , keeping track explicitly of the dependence on the electric charges Q s, V , Q s, S (including e) of s V and s S , respectively, with the low energy constants having implicit dependence on the sea strange quark mass. The relativistic form of the kinetic and mass terms of the kaon multiplet K (U , D denote constituent quarks)
K ∼ [U V s V ] [D V s V ] [U S s V ] [D S s V ] [U G s V ] [D G s V ] ,(B15)
which is subject to the chiral rotation
K → h[Π(x), (g L , g R )] K ,(B16)
is given by
L K, kin = ∇ µ K † ∇ µ K − M 2 K † K ,(B17)
where M is the LO mass of the kaon and the covariant derivative ∇ µ K is with respect to the Maurer-Cartan form Γ µ ∇ µ K ≡ ∂ µ K − iΓ µ K ,
Γ µ ≡ − 1 2i u † (∂ µ u − iR µ u) + u ∂ µ u † − iL µ u † .(B18)
As is well-known [23], K is not suitable for the chiral order counting since that variable also carries the high frequency modes. The fluctuation is decomposed into the high frequency modes originating from M and the low frequency modes represented by k ≡ k v
K(x) = e iM v·x k(x) ,(B19)
where v is a light-like four-vector. In terms of k, Eq. (B17) becomes
L K, kin = −iM v µ k † ∇ µ k − ∇ µ k † k + ∇ µ k † ∇ µ k .(B20)
The field k carries the momentum of the order p 4πF π , M , and the above Lagrangian is O(p). In the succeeding sections the effective Lagrangian is constructed in terms of k and is converted to the relativistic form described by K. building block definition order P C To this end,
χ ± Eq. (B4) O(p 2 ) ±χ ± ( x) (χ ± ) T Q Q ± Q Q R ± Q L O(e 2 ) ± Q Q ± ( x) ± Q Q ± T Q 2 (±) Q R 2 ± Q L 2 O(e 2 ) ± Q 2 (±) ( x) ± Q 2 (±) T Q RL, ± Q R Q L ± Q L Q R O(e 2 ) ± Q RL, ± ( x) Q RL, ± Tx = (x 0 , x) transforms to x = (x 0 , −x). definition order P C kk † O(1) kk † ( x) kk † T k ±, µ Eq. (B21) O(p) k µ ± ( x) ± (k ±, µ ) T k (µν] Eq. (B22) O(p 2 ) k (µν] ( x) ± k (µν] T k ±, µν Eq. (B23) O(p 2 ) k µν ± ( x) ± (k ±, µν ) T k W, Q + ± Eq. (B24) O(e 2 ) k W, Q + ± ( x) ± k W, Q + ± T k W, Q − ± Eq. (B24) O(e 2 ) −k W, Q − ± ( x) ∓ k W, Q − ± Tk ±, µ ≡ i (∇ µ k)k † ± k (∇ µ k) † ,(B21)k (µν) ≡ ∇ (µ k∇ ν) k † = 1 2 ∇ µ k∇ ν k † + ∇ ν k∇ µ k † k [µν] ≡ ∇ [µ k∇ ν] k † = 1 2 ∇ µ k∇ ν k † − ∇ ν k∇ µ k † ,(B22)k ±, µν ≡ (∇ µν k) k † ± k (∇ µν k) † , ∇ µν ≡ ∇ µ ∇ ν + ∇ ν ∇ µ , (B23) k W, Q ± ± ≡ W kk † Q ± ± Q ± kk † .(B24)
In Table XIX, u µ , for instance, is omitted, as it will be not be used hereafter. Table XIX and XX include O(e 2 )-terms but not O(e) and O(ep), because EM charges are left in pairs in the low-energy effective theory after the high frequency photon modes are integrated out.
From
where W is Q s, V or Q s, S , and
K q 2 sV + 2e 2 A (s,2) K q uV q sV .(B36)
The O(e 2 )-correction to the neutral kaon mass-squared, (M e 2 K 0 ) 2 , is given by substituting q dV for q uV in Eq. (B35).
We next consider the one-loop contribution to kaon mass squared. The scalar QED Lagrangian density (B17) gives the correction from the diagrams, in each of which a photon propagates explicitly, but these contributions are absorbed by the redefinition of the coeffi-
where i = u or d, µ is the renormalization scale, and χ mn = χ m + χ n 2 , χ n = 2B 0 m n , Q = 1 N S n : sea q nS .
In our simulation, all sea quarks are neutral and the two light sea quarks are degenerate in mass m (S) . Hence (M log K, i ) 2 reduces to
(M log K, i ) 2 = −2 e 2 16π 2 1 F 2 0 × q 2 iV A (1, 1) K + 3 A (2, 1) K + q iV q sV A (s, 2) K χ i(S) ln χ i(S) µ 2 ,(B39)
where χ i(S) ≡ B 0 m i + m (S) .
There are two types of finite volume corrections induced at the one-loop level. The first type is given by the scalar QED diagrams
∆ (M K + ) 2 EM, photonic (L) = (q K ) 2 e 2 −3 κ 4π 1 L 2 + 1 (4π) 2 K(m K L) L 2 −4 1 (4π) 2 m K L H(m K L) ,(B40)
where κ and various functions are defined in Eqs. (14), (16) and (17). Another type is the finite volume correction to the terms (B37), ∆ M log
puted the proton-neutron mass difference, including for the first time, QED interactions in a realistic 2+1 flavor calculation. We find (m p − m n ) QED = 0.383(68) MeV, (m p − m n ) (mu−m d ) = −2.51(14) MeV (statistical errors only), and the total m p − m n = −2.13(16)(70) MeV, where the first error is statistical, and the second, part of the systematic error. The calculations are carried out on QCD ensembles generated by the RBC and UKQCD collaborations, using domain wall fermions and the Iwasaki gauge action (gauge coupling β = 2.13 and lattice cutoff a −1 ≈ 1.78 GeV). We use two lattice sizes, 16 3 and 24 3 ( (1.8 fm) 3 and (2.7 fm) 3 ), to address finite volume effects. Non-compact QED is treated in the quenched approximation. The valence pseudo-scalar meson masses in our study cover a range of about 250 to 700 MeV, though we use only those up to about 400 MeV to quote final results.
case where the kaon is a Nambu-Goldstone boson, the leading order (LO) includes all terms that are O(p 2 ) and O(e 2 ), and the next-to-leading order (NLO) includes all terms that are O(p 4 ), O(e 2 p 2 ), and O(e 4 ), where the conventional power counting is O(e) ∼ O(p). This counting is the same for the square of the masses and the mass-squared splittings.
µ 2 q 34 + χ 35 log χ 35 µ 2 q 35 + χ 36 log χ 36 µ 2 q 36 q 13 −
Eq.(11) in (m 1 , m 3 , m 4 , m 5 )/m 6 , where m 6 is the strange sea quark mass, m 1 and m 3 are taken as non-degenerate light valence quark masses, and m 4 and m 5 the light sea quark masses,
QCD configurations are separated by 20 or 40 Monte Carlo time units to suppress the autocorrelations in them. Our calculation is for the pseudo-scalar meson at the unitary point (m 1 = m 3 = m 4 = m 5 ) and the partially quenched point(arbitrary quark mass combination).
is a non-singlet flavor index. The correlation functions in Eq. (25) are computed from wall source, point sink, quark propagators.
Figure 1
1shows the chiral extrapolation of m res . The residual mass at the chiral limit is very close between the 16 3 and 24 3 lattices. m res is around 0.003, which is comparable to the lightest input sea quark mass, m l = 0.005, and larger than the smallest valence quark mass m f = 0.001, so the effect of the explicit violation of chiral symmetry from finite L s is not negligible in our calculation. Our measured values are roughly consistent with those found by the RBC and UKQCD collaborations[7,31].
FIG. 1 .
1The QCD residual mass for 16 3 (upper) and 24 3 (lower) lattice sizes. The data correspond to unitary mass points. The linear chiral extrapolation to the m f = 0 limit is also shown on the plot.
L
√ 2 that would result simply from doubling of the measurements (dashed line), while the few points with ratio exactly equal to one correspond to combinations that are trivially invariant under the change e → −e. This procedure corresponds to including the QED configuration −A x,µ for each A x,µ in the path integral and can exactly cancel unphysical O(e) noise with finite statistics which would have obscured the physical O(e 2 ) signal of interest, only the latter of which survives in the infinite statistics limit. Together, the complete procedure yields mass-squared splittings with sub-percent statistical precision.
FIG
. 2. A comparison of the statistical errors for the meson mass-squared splitting with and without the ±e averaging trick [2, 35]. The vertical axis shows the ratio of the error without the average to that with the average, so that larger values indicate smaller statistical error from the ±e averaging trick. In most cases there is a large decrease (∼ 1/10) in the error over the naive factor of √ 2 that would result simply from doubling of the measurements (dashed line). The few points with ratio exactly equal to one correspond to combinations that are trivially invariant under the change e → −e, i.e., m 1 = m 3 and q 1 = −q 3 .
FIG. 3 .
3Representative effective masses. Lattice size 24 3 . m sea =0.005, m 1 =m 3 =0.01, q 1 = 1/3 and q 3 = 0 (upper points) and m sea =m 1 =m 3 =0.005, q 1 = 1/3 and q 3 = −1/3 (lower points). The horizontal lines represent the fit result.
improved algorithm was used to generate the 24 3 ensemble. From Tabs. XIII -XIV the masses and errors determined with either fit method agree quite well. Our final analysis is based on the masses from the uncorrelated fits in order to be consistent with the analysis in Refs. [7, 17] from which we take the pure QCD LEC's. The typical statistical error on the mass is at the half of a percent level and smaller. The meson mass-squared splittings are given by ∆M 2 = M 2 (e = 0) − M 2 (e = 0), and the errors are again computed using a jackknife procedure. As an example, in Figs. 4 and 5, ∆M 2 for thedd meson is shown. Only the unitary points appear in the figure. A full summary of the mass-squared splittings is given in Tabs. XV and XVI. The promised statistical precision is observed. Even though the errors on the masses themselves are of the same order as the mass difference, the splitting is statistically well resolved under the jackknife analysis thanks to the strong statistical correlation between e = 0 and e = 0.
vanishes. Similar results hold for theūu mesons. The result based on the Ward-Takahashi
FIG. 4 .
4∆M 2 for the dd meson, with L s = 16 (upper set of line and plots) and 32, 16 3 lattice size from the SU(3) fit. The extrapolated values (box) are e 2 δ mres (q 2 1 + q 2 3 ). For comparison, we also show the values of B 0 C 2 e 2 (q 2 1 + q 2 3 ) obtained from the Ward-Takahashi Identity (values are slightly shifted horizontally to the left for clarity). δ mres is obtained from the fit range of 0.01-0.02 for L s = 16 and 0.01-0.03 for L s = 32. The error on B 0 C 2 e 2 (q 2 1 + q 2 3 ) comes mostly from the error on B 0 . Identity depends also on the value of B 0 or B, depending on whether we choose SU(3) or SU(2) chiral perturbation theory, which introduces some uncertainty. On the other hand, Figs. 4 and 5 clearly show this effect is due to finite L s chiral symmetry breaking, and that it can be precisely subtracted from the physical splitting by introducing a new lattice-artifact
FIG. 5 .
5Same as Fig. 4 but for lattice size 24 3 and L s = 16.1. infinite volume fits In Figs. 6 and 7, the meson mass-squared splittings are shown for the unitary quark mass points, for both 16 3 and 24 3 ensembles, respectively. For now, we concentrate on the 24 3 ensemble, and fit the mass-squared splittings to the infinite volume, NLO, chiral perturbation theory formulas described in Sec. II. The formulas require the values for the pure QCD LEC's, some of which we have not computed. The pure QCD LEC's, including F 0 , F , B 0 , B, and the Gasser-Leutwyler L's, have been calculated already by the RBC and UKQCD collaborations from a larger ensemble of which the present one is a subset. We use these values in our fits, in a combined super-jackknife analysis so that the statistical errors on the QCD parameters are fed into our analysis.
Figure 7
7shows the fit to the full SU(3) L × SU(3) R NLO formula, which is summarized in Tables IV and V. The quark mass range in the fit is m 1 , m 3 ≤ 0.01, and the χ 2 /dof for these uncorrelated fits is about two. χ 2 degrades significantly if larger quark mass points are used in the fits. Only unitary points are shown in the figure for clarity while all of the (allowed) quark mass and charge combinations for the mesons have been used in the analysis. For the 24 3 ensemble, this amounts to 52 data points for m 1 , m 3 ≤ 0.01. The charged meson splittings should not vanish in the chiral limit, m f = −m res ; this is just the LO Dashen term proportional to α em and the lattice-artifact chiral symmetry breaking. The neutral meson splittings do not vanish either due to the latter. The chiral logarithms reduce the LO Dashen term relative to the value given by a simple linear ansätz. Recall that the splittings of "neutral" mesons made from connected quark diagrams only do not contain logs at NLO, so their chiral behavior is particularly simple. Figures 7 and 8 show similar fits for SU(2) L × SU(2) R -plus-kaon chiral perturbation theory for the pions and kaons, respectively. Here we use the same range for the light quark masses, and for the kaons the valence strange quark is fixed to either 0.02 or 0.03. χ 2 /dof is similar to the SU(3) case for the pion and also significantly degrades when the quark mass range is extended upwards. For the kaon fits χ 2 is small. The total number of data points in the fits are 52 for the pions and 36 for the kaons. The SU(2) LEC's are also summarized in
. The QCD LEC's from RBC/UKQCD collaboration's infinite volume fits on 24 3 lattices with SU(3) and SU(2) PQχPT [17]. They were computed from a larger ensemble of lattices than used in [7]. All of the QCD LEC's are defined at the chiral scale Λ χ = 1 GeV. The labels in the first column correspond to SU(3) definitions; the analogous LEC for SU(2) is given in the third column.
Eq.(12) for the SU(3) fit, and Eq.(22) for the pion and the results in Appendix B 2 for the kaon in the SU(2) fit. We continue to use the pure QCD, infinite volume, LEC's from[17].
V. The SU(3) PQχPT and SU(2) pion PQχPT QED LEC's from fits of the mass-squared splittings measured on the 24 3 lattices. All of the LEC's are defined at chiral scale Λ χ = 1 GeV and are given in lattice units. The quark mass range in the fits is m 1,3 <= 0.01. "inf.v." and "f.v." means infinite and finite volume fits, respectively. "SU(3)+phenom." refers to a parameter set presented based on phenomenology and using SU(3) χPT [11]. Labels in the first column correspond to SU(3) definitions. "Dashen's term" is the LO result for the mass splitting in the chiral limit. energy scale of 2 GeV. First, the bare quark masses are determined by solving Eq. (4) or Eqs. (20) and (23) evaluated at the physical meson masses [4] (in MeV)
where only the central values are used in our analysis since the errors are negligible compared to the lattice results. Using a −1 = 1.784(44) GeV and the pure QCD nonperturbative renormalization constant Z m = 1.546 (2) (43) [17] computed by the RBC and UKQCD collaborations, M S light quark masses are given inTable VII, for infinite volume, finitevolume, SU(3), and SU(2) fits. We have not included the O(α em ) renormalization of the quark mass from QED interactions. These are similar to those found in our earlier two flavor work[2], also using DWF, but which used a more crude chiral perturbation theory analysis
. Kaon QCD and QED LEC's extracted from 24 3 lattice size data. LEC's are in lattice units. The kaon is composed by one light-(m 1 ) and one strange-(m 3 ) quark. We choose m 1 ≤ 0.01 and m 3 =0.02 or 0.03. The light sea quark is chosen as m sea ≤ 0.01. The mass of the strange sea quark is fixed at 0.04. The kaon QCD LEC's are quoted from RBC/UKQCD's work [17]. χ 2 /dof refers to the fit using the SVD method[36].
FIG. 6 .FIG. 7 .FIG. 8 .
678Meson mass-squared splittings. 16 3 lattice size. Infinite volume linear fit (upper panel) and infinite volume SU(3) chiral log fit (lower panel). The fit range of the linear fit is 0.01-0.03. Fit ranges of chiral log fits, 0.01-0.03 (solid line) and 0.01-0.02 (dashed line). Data points correspond to ud, uū and dd mesons, respectively, from top to bottom. Only unitary points are shown, although all of the partially quenched points were used in the fit. Meson mass-squared splittings. 24 3 lattice size. Infinite volume linear fit (upper panel), and infinite volume SU(3) and SU(2) chiral log fits (lower panel). The fit range of the linear fit is 0.005-0.03. Fit range of chiral log fits is 0.005-0.01. The solid (dashed) line in the lower panel represents the SU(3) (SU(2)) fit. Data points correspond to ud, uū and dd mesons, respectively, from top to bottom. Only unitary points are shown, although all of the partially quenched points were used in the fit. Kaon mass-squared splitting and infinite volume SU(2) kaon fit. The mass of the strange valence quark is fixed at 0.03, and m sea = 0.005. Different lines in the plot correspond to different charge combinations of the valence quarks.
FIG. 9 .
9VII. The u, d and s quark masses determined from QCD+QED interaction on 24 3 lattices. The values are given in MeV and the MS scheme at renormalization scale µ = 2 GeV. SU(3) or SU(2) mean quark masses from SU(3) PQχPT or SU(2) PQχPT + kaon theory. 24 3 SU(2) chiral log infinite volume and finite volume fits for pion (upper) and kaon (lower) mass-squared splittings. Lines correspond to fit results. The fit range is 0.005-0.01. The solid (dashed) line represents the infinite (finite) volume fit. In the upper panel, the fit curves are evaluated for degenerate unitary light quarks. For the lower panel, the curves are evaluated for m sea = 0.005 and m 3 = 0.03. Data points in the plot correspond to q 1 = 2/3e and q 3 = −1/3e, but all partially quenched points allowed by the fit range were used in the fit.
FIG. 10 .
10and SU(2) chiral perturbation theories give sensible fit results for pion masses less than about 400MeV, or for bare quark masses satisfying m f ≤ 0.01. It is possible that the range is different, perhaps larger, for the EM splittings. After all, most of the pure QCD contributions at LO and NLO completely cancel in the EM splittings (some of the pure QCD LEC's survive at O(α em m)). We work with uncorrelated fits, though our data are highly correlated, because there are too many mass and charge combinations to accurately determine the correlations on this finite statistical ensemble. The uncorrelated fits have been shown to agree with correlated ones when the covariance matrix is well determined, and when it is not, the correlated fits break down[17,39]. As already mentioned, when the quark mass range is extended upwards, for both SU(3) and SU(2) (pion) fits, χ 2 /dof increases noticeably, by more than a factor of two. Since we use uncorrelated fits, this χ 2 is not an absolute test ofgoodness of fit, though we expect changes do indicate relative goodness of fit. Thus we stick with the range m 1 , m 3 ≤ 0.01 for the light quarks to quote central values and to estimate systematic errors. One of the systematic errors is the difference in the central values for the quark masses determined from this restricted range and those values computed from the range m 1 , m 3 ≤ 0.02 for the light quarks. m 1 , m 3 ≤ 0.01 corresponds to valence pions in the range 250-420 MeV.For the mass range m 1 , m 3 ≤ 0.01, the most important change, which is anticipated inFig. 7, is that the Dashen term increases significantly when the logs are omitted from the SU(3) fit. C increases by about a factor of five, although it is still small compared to the value one would obtain from the physical splitting. For SU(2), the situation is much different; C changes very little, about two percent. Presumably, the large logs containing the strange quark mass contribute to the LO term in this case, and the remaining effect of the light logs is not as important. The higher order terms change more, but without logs to affect their running, there is not much sense in comparing the changes.InFig. 10the LO and NLO contributions in finite-volume SU(2) L × SU(2) R chiral perturbation theory for the charged pion mass splittings are shown. At the values of quark masses used in our calculation, after accounting for the δ mres contribution, the NLO contributions are about 50-100 % of the LO contribution. It is interesting to see how m d − m u is affected at the various orders in chiral perturbation theory. Using the LEC's determined in the full SU(2) L × SU(2) R -plus-kaon fits, we find that the NLO contributions increase m d − m u by a bit less than 2%. Taking all the above uncertainties due to fitting into account, we estimate systematic errors of about four and zero percent for the up and down, and strange quark masses, respectively. These are collected in Tab. VIII. The LO and NLO in finite volume SU(2) chiral perturbation theory contributions to the EM meson mass splitting. The dashed line corresponds to the finite volume fit result. The data points shown are for charged mesons with q 1 = 2/3 and q 3 = −1/3. The lower horizontal line gives the contribution of the lattice artifact e 2 δ mres (q 2 1 + q 2 3 ) while the upper horizontal line gives the sum of this contribution and Dashen's Term (in other words, their difference is just the LO contribution). The solid line corresponds to the total LO+NLO+e 2 δ mres (q 2 1 + q 2 3 ) contributions based on the fitted, finite volume LEC's, but evaluated with the infinite volume logarithms. The fit curves are evaluated for degenerate unitary light quarks.
the shifts in the kaon mass-squared LEC's are much smaller, especially the ones representing the LO Dashen term (A some of the LEC's show large finite volume shifts, the ultimate shifts in the quark masses are smaller. The largest, about 14%, occurs for the up quark mass. The down quark mass is affected much less, about 3%, and the shift in the strange quark mass is negligible.
FIG. 11 .
11Finite volume effect in the measured EM splittings. All of the data points have q 1 = 2/3 and q 3 = −1/3. Circles and squares correspond to 24 3 and 16 3 lattice sizes, respectively. The solid line is from the finite volume fit on 24 3 ensembles. The dashed line is the theoretical prediction for 16 3 lattices based on the LEC's extracted from 24 3 finite volume fit. The fit curves are evaluated for degenerate unitary light quarks.
chiral perturbation theory and the light quark mass range m ≤ 0.01 to obtain our central values of the physical quark masses.The physical strange quark mass is determined from the kaon mass-squared which is an implicit function of the bare sea and valence strange quark masses, through its LEC's which are calculated for fixed valence strange quark masses 0.02 and 0.03 and fixed sea strange quark mass 0.04. Assuming that the m s dependence is modest in this region, the physical kaon mass-squared is determined from a linear extrapolation in the valence strange quark mass. A similar procedure was carried out in[7] where three data points in the range 0.02-0.04 showed the kaon mass-squared is well approximated by a linear function (it turns out that the physical strange quark corresponds to about 0.035). Because we have only carried out calculations at a single strange sea quark mass value of 0.04, the kaon mass-squared can not be evaluated at the physical strange sea quark mass. This partial quenching leads to a small systematic error that was conservatively estimated in[7] to be 2% for m s which we adopt here. It is added in quadrature to the total systematic error for m s which appears below. The systematic error on the light quark masses is about 0.7% which is negligible compared to the other systematic errors, so we ignore it.The statistical errors come from fits underneath a standard jackknife analysis. The QCD LEC's come from an analysis of the extended RBC/UKQCD 24 3 ensembles; the results are consistent with those in Ref.[7]. All of the fits and corresponding LEC's are analyzed under a super-jackknife analysis so that statistical errors on all quantities, from all ensembles, are included. The systematic errors assigned have been discussed in this section. The massindependent quark mass renormalization factor is Z MS m (µ = 2GeV) = 1.546(2)(43),
scheme dependence of the renormalized quark mass was discussed in the context the isospin breaking. The effect vanishes in the (perturbative) M S scheme. At this point, there seem to be no common consensus on if there is any non-perturbative contribution, which is related to the aforementioned instanton effects, and how large it might be. Our results are potentially susceptible to this uncertainty, as are all other quark masses renormalized in a perturbative scheme.
, is 4 .5936( 5 )
45MeV. The SU(3) fit gives a very small value, about half an MeV. The finite volume fit dramatically increases the value, but it is less than half the physical value. The SU(2) fit gives a bigger value still, and after including finite volume corrections, it gives the LO EM correction to the pion mass difference (m π ± − m π 0 ) QED = 3.38(23) MeV. Coincidentally, this is about the same value obtained from the linear fit, 3.22(25)MeV. The value of m 2 π ± in the chiral limit, which comes from the LO EM correction is 929(64) MeV 2 and is similar to the values, using a sum rule and lattice-computed vector and axial-vector correlation functions in pure QCD, reported in
For the kaons, the pure EM mass difference is m K + − m K 0 = 1.87(10) MeV, while the contribution from m u − m d = 0 is −5.840(96) MeV. Here, of course, the result includes all NLO corrections, and LEC's from the finite volume fit are used. These values are obtained by taking the SU(2) formula for the kaon mass-squared M 2 K (m 1 , q 1 , m 3 , q 3 ), Eq. (23)
Perhaps more useful for other pure QCD simulations are the "physical" values of m π and m K in pure QCD deduced from our SU(2) fits with m u = m d = m ud :
FIG. 12 .
12Breaking of Dashen's theorem for the quenched QED case. The unphysical contribution δ mres (q 2 1 + q 2 3 ) has been subtracted from the data. Data for two values of the strange quark, 0.02 and 0.03, are shown. The curves correspond to the SU(3) fits (upper panel) and SU(2) fits (lower panel). The cyan bands denote the infinite volume extrapolations with one standard deviation statistical errors, using the LEC's extracted from the finite volume fits; the sea and strange quark masses are fixed at their physical values.
FIG. 13 .
13address a question: how far does the value of f K shift when the light quark mass in the Kaon is changed from from m ud to m u ? Some lattice determinations of f K use m ud as the light quark mass while the experiments measure decays of the charged Kaon to obtain f K ± , which is made of an up (and strange) quark. So it is relevant to know if the shiftf K (m ud ) − f K (m u )is comparable in size to the total error on the ratio, 0.4%. We note the analyses of V us /V ud in[60,61] (see also[62]) correct for the QED effects of the decay constants, and we only consider the decay constant for e = 0 but m u = m d in this section.InFig.13, f K (m x ) obtained by the RBC/UKQCD collaboration [7] is plotted as a function of valence light quark mass m x . The sea and valence strange quark masses are fixed. The square points are from light sea quark mass m l = 0.01(∼ 40MeV) while the circle data are for m l = 0.005(∼ 22 MeV). The curves are from the partially quenched SU(2) ChPT fits. The upper two curves denote f K (m x ) at fixed degenerate sea quark masses m l = 0.01 (upper) and 0.005 (lower), while the doted curve is evaluated for unitary quark mass, m x = m l . The lower three, almost degenerate, curves are for m l = 0.7 m ud , m ud , and 1.3 m ud . The inset magnifies the region close to the physical point. The filled square is f K for equal up and down quark masses, m l = m x = m ud . When the valence quark mass is decreased to a 30% smaller value, 0.7 × m ud , f K decreases by about 1%, if we simultaneously decrease the light sea quark masses to 0.7 × m ud (empty square). This setting of quark masses (empty square) underestimates the value of f K in Nature, since the down sea quark mass is also decreased to 0.7 × m ud for the empty square 4 . The more accurate estimation of f K for non-degenerate valence up and down quark masses is the empty circle, where the degenerate sea quark mass is fixed to m l = m ud , and only the valence quark mass is set to the lighter mass, m x = 0.7m ud . We note that the non-degenerate quark mass effect in the sea sector is suppressed by (m u − m d ) 2 , and setting degenerate sea quark mass to m l = m ud is a good approximation to estimate the f K shift due to the isospin breaking in the up and down quark masses. Because of the (accidental) decrease in the slope of f K (m x ) around the physical sea quark mass m l = m ud , the difference between f K (m ud ) and f K (0.7 × m ud ) is only about −0.2%, which is nevertheless sizable compared to the total error of 0.4% in the current world average of f K /f π . A similar analysis was done in[58], where f K + was properly estimated at m x = m u . An indirect error on f K + induced from (their) EM uncertainty in m u /m d (∼ 19%) was estimated to be ∼ 0.07%. So their shift of f K due to the quark mass difference between m ud and m u would be roughly 0.07/0.19 × (m ud /m u − 1) ∼ 0.25 % from their value of m u /m ud ≈ 0.6.This shift is slightly larger than our estimation, 0.2%, in part because m u /m ud in[58] is smaller than ours by about 10%. Kaon decay constant in pure QCD[7] computed from the same ensembles as used in this work. The valence strange quark mass is fixed to 0.03 and the sea strange quark mass is 0.04.
and the difference m j − m l appears first at NNLO in any observable due to the symmetry under switching sea up quark to sea down quark. So our degenerate sea up and down quark mass is enough to extract the isospin breaking to NLO (We will ignore possible contributions of O(e 2 (m u − m d ))).
switched to box sources (of size 16 3 ), which gave much better plateaus and signals, but only on the unitary points because of the additional computational cost. Thus, for the 16 3 and 24 3 QCD configurations the masses come from wall source correlation functions while for the 24 3 QCD+QED configurations, the masses are from box source correlation functions.
. Summary of additional configurations used for the box source nucleon calculation on the 24 3 lattices. QCD gauge configurations generated by the RBC and UKQCD collaborations[7, 17,31]. ∆ is the separation between measurements in molecular dynamics time units. The Iwasaki gauge coupling is β = 2.13.method. The results in Tab. XVII for the unitary masses and non-zero α em on the 24 3 ensembles are consistent with the pure QCD results obtained on the same ensembles reported in Ref.[7, 17], except for the m l = 0.005 case. The proton and neutron masses are about three standard deviations smaller than in the pure QCD case, or roughly three percent. It is of interest to further investigate how large the EM effect is on the nucleon masses themselves, as well as on the mass difference. Of course, in Nature there is no way to measure the nucleon mass due to QCD alone.
Figure 15 showsFIG. 14 .
1514the mass difference between the proton and neutron due to the QED interaction for the unitary points. If there is no QED interaction and m u = m d , then m n = m p , which is the result of isospin symmetry. When the QED interaction is included, the proton is heavier than the neutron, and the mass difference decreases with quark mass as observed inFig. 15. The 24 3 result is larger than the 16 3 result, once again signaling finite volume corrections. This simulation is on the unitary points, but m u = m d in nature. When we extrapolate (m p − m n ) QED to the physical point, we use the average light quark mass m ud , as determined in the previous section. Finally we find that (m p − m n ) QED is about 0.4 MeV (see Tab. X). From Fig. 15 there is a visible flattening of the splitting at the lightest quark mass for the 24 3 lattice size. Using only the lightest two quark masses in the extrapolation, we obtain (m p − m n ) QED = 0.63(23) MeV. The difference between the two results is used to estimate the systematic error in the chiral extrapolation.Since the photon is not confined, the EM proton-neutron mass difference could suffer from a large finite volume effect. In order to estimate this artifact, we use the Cottingham Proton effective masses, 24 3 lattice size, m l = 0.005. The upper panel is for the unitary point and box source. The lower panel is for a non-degenerate case and wall source.
for 16 3 volume, and (m p − m n ) (Cott.) QED = 0.16 MeV for 24 3 volume. Since the formula yields (m p − m n ) (Cott.) QED = 0.77 MeV for the infinite volume limit, the finite volume artifact corresponds to an underestimate of 0.73 MeV and 0.61 MeV for 16 3 and 24 3 , respectively. The tendency for the larger volume to correspond to larger (m p − m n ) QED is qualitatively consistent with the lattice results presented here.
16 shows the fit of the proton and neutron mass difference due to non-degenerate u, d quark masses computed on the QCD configurations. The LEC's and values of the splitting at the physical point are summarized in Tab. XI. Figure 16 confirms that (m p − m n ) (m d −mu) is proportional to m d − m u , which is predicted by baryon PQχPT (Eq. (62)). The slope is extracted and the physical (m p − m n ) (m d −mu) is estimated by setting m d − m u to its physical value, again as determined in the previous section. Our result is in good agreement with the one in
and 24 3 results which is about 0.05 MeV when all of the data are used in the fits, and roughly 0.3 MeV if only the lightest points on the 24 3 lattice are used. In light of the much larger artifact predicted by the Cottingham formula, we take the more conservative estimate of 0.3 MeV. The finite volume error on the pure QCD splitting appears to be under better control, and we simply take the difference of the two as an additional finite volume effect, or ∼ 0.25 MeV. The QCD splitting depends somewhat strongly on the value of m u −m d , and given the ∼ 20% uncertainty in this quantity, we estimate the systematic error due to the extrapolation by varying m u − m d over this range. This yields roughly a 0.5 MeV uncertainty. Adding all of these errors in quadrature, we find m p −m n = −2.13(16)(70) MeV. The result and errors are summarized in Tab. XII. Clearly further calculations are needed, at smaller quark masses
FIG. 15 .FIG. 16 .
1516The proton-neutron mass difference due to the QED interaction computed for unitary points. 24 3 (circle) and 16 3 (square) lattice sizes. The solid (dashed) line corresponds to a linear fit to the 24 3 (16 3 ) data points. The proton-neutron mass difference for m u − m d = 0 and e = 0. The solid (dashed) line corresponds to a linear fit to the 24 3 (16 3 ) data points, shown by circles (squares). lattice size m p − m n (MeV) fit error (MeV) finite vol. error (. Estimated result of the proton-neutron mass difference in Nature (systematic errors as described in the text).X. CONCLUSION In this paper we have investigated the EM mass splittings of the low-lying hadrons from first principles in the framework of lattice QCD+QED. Our simulations were based on the 2+1 flavor DWF QCD configurations generated by the RBC and UKQCD collaborations and quenched, non-compact, QED configurations generated by us. The mass splittings could be determined with very high statistical accuracy since the QCD part of the fluctuations in the hadron masses largely cancels in the splittings. The precision is further enhanced by applying our ±e trick [2, 35] to cancel O(e)-noise on each configuration, before averaging over the QCD ensemble. The statistical errors on the pseudo-scalar splittings are at an impressive sub-one-percent level, as are the errors on the masses themselves. The explicit chiral symmetry breaking induced by the finite extra 5th-dimension of DWF was studied in detail and shown to be under good control. This is important because the leading O(α em m res ) effect is comparable in size to the physical effects under investigation.We fit the pseudo-scalar meson mass-squared splittings to the theoretical predictions of partially-quenched chiral perturbation theory, including photons, to extract the EM low energy constants of the effective theory, up to NLO. We presented new analytic results for the kaon mass-squared in Sec. II and in the Appendix B. The fits were done to both SU(3) L × SU(3) R and SU(2) L × SU(2) R -plus-kaon theories, the latter being necessary to
R -plus-kaon theory was used to quote our final values since the physical strange quark mass is outside the range of convergence of SU(3) L × SU(3) R chiral perturbation theory. They are given in Eqs. (32)-(36), along with statistical and systematic errors. The down-up mass difference and quark mass ratios are given in Eqs. (35)-(38). These quark masses, up to EM effects, are consistent with the pure QCD values given in Ref. [7], which is not surprising since the pure QCD LEC's were taken from an identical analysis of extended ensembles of configurations [17] used there.
K 0 )
0QED = 1.87(10) MeV, while the contribution from m u − m d = 0 is −5.840(96) MeV. While these values are interesting, there is still systematic uncertainty in them which can only be removed by calculations with lighter quark masses and larger volumes.Finally, we also computed the proton-neutron mass difference, again for the first time in 2+1 flavor QCD+QED. Our result is somewhat bigger than the experimental one, but encouraging. We found m p −m n = 0.383(68) MeV for the EM mass splitting, and −2.51(14) MeV from m u = m d , both on the larger lattice (errors are statistical). Part of the systematic error, stemming mainly from finite volume and chiral extrapolations of the splittings, was estimated. The total splitting was found to be m p − m n = −2.13(16)(70) MeV, where the first error is statistical and the second, part of the systematic error. The central value is from the 24 3 lattice; we have not attempted either continuum limit or infinite volume extrapolations. The sign and relative size of the EM effect compared to the m d − m u mass difference effect is as expected.
fits to the pseudo-scalar two-point correlation functions on the 16 3 QCD+QED lattice configurations. Fit range is 9 ≤ t ≤ N t /2 in each case. m sea is the mass of the light quark in the sea sector. m 1 and m 3 are the masses of the valence quarks. The mass of the strange sea quark is fixed at 0.04. "cov" and "uncov" refer to covariant and uncovariant fits, respectively.m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (010 0.030 0.030 -2 -1 0.3823(10) 1.66(1.05) 0.3815(11) 0.04(6) 0.010 0.010 0.010 2 0 0.2437(14) 1.22(89) 0.2427(16) 0.02(4) 0.010 0.010 0.020 2 0 0.2843(13) 1.40(96) 0.2834(15) 0.03(5) 0.010 0.010 0.030 2 0 0.3201(12) 1.50(1.00) 0.3193(14) 0.04(6) 0.010 0.020 0.020 2 0 0.3201(12) 1.56(1.01) 0.3192(13) 0.04(6) 0.010 0.020 0.030 2 0 0.3524(11) 1.63(1.04) 0.3516(12) 0.04(6) TABLE XIII -continued from previous page m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (.3879(10) 0.96(80) 0.3880(10) 0.07(9) 0.020 0.010 0.020 -2 -1 0.2924(12) 1.04(83) 0.2928(13) 0.12(13) 0.020 0.020 0.010 2 1 0.2925(12) 1.03(83) 0.2929(13) 0.12(13) 0.020 0.010 0.030 -2 -1 0.3275(12) 0.95(80) 0.3278(12) 0.10(12) 0.020 0.030 0.010 2 1 0.3278(12) 0.95(79) 0.3281(12) 0.10(12) 0.020 0.020 0.020 -2 -1 0.3273(11) 1.02(82) 0.3276(11) 0.11(12)TABLE XIII -continued from previous page m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (.2510(15) 2.00(1.16) 0.2511(16) 0.16(12) 0.030 0.010 0.020 1 -1 0.2905(13) 2.00(1.16) 0.2905(14) 0.12(10) 0.030 0.010 0.030 1 -1 0.3259(12) 2.02(1.16) 0.3258(13) 0.10(9) 0.030 0.020 0.020 1 -1 0.3253(12) 2.06(1.17) 0.3251(12) 0.10(9) 0.030 0.020 0.030 1 -1 0.3574(11) 2.12(1.19) 0.3571(12) 0.09(8) 0.030 0.030 0.030 1 -1 0.3870(10) 2.20(1.21) 0.3866(11) 0.09(7)TABLE XIII -continued from previous page m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (
.2795(4) 0.94(41) 0.2796(5) 0.06(5) 0.005 0.005 0.010 -2 0 0.2172(4) 0.95(41) 0.2172(5) 0.10(9) 0.005 0.005 0.020 -2 0 0.2611(4) 0.92(40) 0.2612(5) 0.07(7) 0.005 0.005 0.030 -2 0 0.2990(4) 0.99(42) 0.2991(5) 0.05(5) 0.005 0.010 0.010 -2 0 0.2401(4) 0.97(41) 0.2402(5) 0.08(8) 0.005 0.001 0.001 -2 0 0.1410(6) 1.04(42) 0.1407(7) 0.15(11)TABLE XIV -continued from previous pagem sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (005 0.010 0.030 2 -2 0.3181(4) 0.96(41) 0.3183(5) 0.04(4) 0.005 0.020 0.020 2 -2 0.3179(3) 0.87(39) 0.3180(5) 0.04(3) 0.005 0.020 0.030 2 -2 0.3502(3) 0.89(40) 0.3504(4) 0.04(3) 0.005 0.030 0.030 2 -2 0.3800(3) 0.90(40) 0.3803(4) 0.04(4) 0.005 0.001 0.001 2 -2 0.1436(6) 1.03(42) 0.1433(7) 0.15(11) 0.005 0.005 0.005 2 2 0.1918(5) 0.97(41) 0.1917(6) 0.13(10) TABLE XIV -continued from previous page m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (.3173(3) 2.52(68) 0.3179(5) 0.20(15) 0.010 0.020 0.020 2 0 0.3173(3) 2.40(66) 0.3177(5) 0.20(15) 0.010 0.020 0.030 2 0 0.3495(3) 2.16(63) 0.3500(4) 0.18(15) 0.010 0.030 0.030 2 0 0.3794(3) 1.98(60) 0.3799(4) 0.17(15) 0.010 0.010 0.010 2 1 0.2416(3) 2.95(73) 0.2417(5) 0.27(18) 0.010 0.001 0.001 -2 1 0.1435(5) 4.09(86) 0.1440(8) 0.58(32) TABLE XIV -continued from previous page m sea m 1 m 3 q 1 ( 1 3 ) q 3 ( 1 3 ) M (cov) χ 2 /dof (cov) M (uncov) χ 2 /dof (
. Proton and neutron masses on unitary points for QCD+QED configurations. The masses are from size 16 3 box source, point sink correlation functions for both 16 3 and 24 3 lattices. The fit range for 16 3 lattices is 5-10. The fit range for 24 3 lattices is 6-11. The χ 2 /dof is from a covariant fit. lattice size m sea m u . Proton and neutron masses on QCD configurations with non-degenerate u, d quark masses. The p-n masses are from wall source, point sink correlation functions for 16 3 and 24 3 lattices. The fit range for 16 3 lattices is 5-10. The fit range for 24 3 lattices is 7-12. The χ 2 /dof is from a covariant fit.
3 .
3O(e 2 ) and O(e 2 p 2 ) Lagrangian for the kaon sector Having established the partially quenched framework and notation, we construct the electromagnetic part of the chiral Lagrangian by writing down all possible O(e 2 )-terms possessing the symmetries of massless (QCD + QED) in the non-relativistic theory and their relativistic counterparts, and O(e 2 p 2 )-terms that can induce the tree-level contribution to the kaon mass-squared.
XX. Parity (P ) and charge conjugation (C) transformation properties of operators at chiral order O(p 2 ), O(e 2 ) that are bilinear in kaon fields and transform as A → h A h † . W is either one of Q s, V or Q s, S . Under P , x = (x 0 , x) transforms to x = (x 0 , −x).
K
) , Q RL, + , Q Q + , Q 2 , Q RL, + , ( Q ) 2 , B ∈ Q + , Q ,(W 1 , W 2 ) ∈ {(Q s, V , Q s, V ) , (Q s, V , Q s, S ) , (Q s, S , Q s, S )} .(B26)The relativistic forms of the individual operators are read from the relation (B19), and4. EM correction to kaon mass-squaredIn this subsection, the explicit expression for the O(e 2 ) and the O(e 2 p 2 ) chiral-logarithmic correction to the kaon mass-squared is obtained by setting the charge matrices Q in light flavor partially quenched system as in Eq. (B9), Q s, V = e q sV and Q s, S = e q sS . The EM contribution to the K + mass-squared at order e 2 is (M e 2 K + ) 2 = 2e 2 (q uS + q dS )q sV + e 2 A
cients in Eq. (B27) and those of O(e 2 p)-operators in the infinite volume. The leading EM chiral-logarithmic correction comes only from the tadpole diagrams induced by Eq. iV (q iV − q nS ) + (q iV − q nS ) 2 χ in ln χ m : sea, n =m (q nS − q mS ) 2 χ mn ln χ iV − q nS ) χ in ln χ in µ 2 ,
(x) in Eq. (13). The finite size scaling effect on the O(e 2 ) wave function renormalization could induce O(e 2 p 2 )-correction to kaon mass squared after the renormalization of the kaon field. The explicit calculation, however, shows that such effects do not exist. There are as many LEC's as O(e 2 p 2 )-operators in Eqs. (B32) and (B34) participating in the O(e 2 p 2 )-contribution to kaon mass-squared, while our lattice study here can determine at best the linear combinations of LEC's of terms with the same charge and light quark mass dependence of order e 2 m, from the response of the data to the variation of these parameters in the (QCD + QED) action. The dependence on those parameters can be read off from Eqs. (B32) and (B34). In effect, Eq. (B32) alone leads to the following form of the charge
PACS numbers: 11.15.Ha, 11.30.Rd, 12.38.Gc 12.39.Fe
The LO Dashen term is proportional to the low energy constant (LEC) C and the fine structure constant α em . B 0 and F 0 1 are the LO QCD LEC's, the L's are the Gasser-Leutwyler LEC's at NLO, and the K's are the EM LEC's at O(α em m). δ mres is a pure lattice artifact LEC associated with the finite size of the extra dimension for DWF.We note from Eq. (4) that masses of mesons ∼ qq made from degenerate valence quarks
q, q with equal charges do not have logarithmic corrections at NLO. This happens for the
SU(2) case as well.
Refs.[12] and[7] while those with superscripts are EM LEC's defined in Appendix B. The finite volume correction to Eq. (23) is given in Appendix B, Eqs. (B40) and (B41).
, respectively. The domain wall height M 5 and the size of the extra dimension L s are 1.8 and 16, respectively. The residual quark mass in the chiral limit for pure QCD is found to be m res = 0.003148(46) and 0.003203(15), for the 16 3 and 243 DWF and the Iwasaki gauge action (β = 2.13) by the RBC and UKQCD collaborations [7,
17, 31]. The lattice sizes are 16 3 × 32 and 24 3 × 64. The lattice spacing 2 is a −1 = 1.784 (44)
GeV, as determined from the Ω baryon mass on the larger lattice, and which yields physical
volumes (1.76 fm) 3 and (2.65 fm) 3
and 24 3 lattice configurations with the same lattice spacing. This allows direct investigation of the finite volume effect in the mass spectrum.TABLE I. Ensembles of QCD gauge field configurations generated by the RBC and UKQCD collaborations[7, 17,31] for β = 2.13 with the Iwasaki gauge action that were used in this work.lat m sea
m val
Trajectories ∆ N meas
t src
16 3 0.01
0.01, 0.02, 0.03
500-4000 20 352
4,20
16 3 0.02
0.01, 0.02, 0.03
500-4000 20 352
4,20
16 3 0.02
0.01, 0.02, 0.03
500-4000 20 352
4,20
24 3 0.005 0.001, 0.005, 0.01, 0.02, 0.03 900-8660 40 195
0
24 3 0.01
0.001, 0.01, 0.02, 0.03
1460-5040 20 180
0
24 3 0.02
0.02
1800-3580 20 360 0,16,32,48
24 3 0.03
0.03
1260-3040 20 360 0,16,32,48
TABLE II .
IIThe QCD residual mass for 16 3 and 24 3 lattice sizes. The data correspond to unitary
mass points. The fit range for R(t) (defined in Eq. (25) is 9 ≤ t ≤ N t /2.
chiral perturbation theory would be needed to access the physical strange quark mass regime. We decided to include a lighter valence mass point, 0.001, to augment our fits, but since this was a new, separate calculation, only the mass-degenerate mesons could be computed. Thus, inour kaon fits, we have only two valence and two sea quark mass combinations available for
the region m u,d ≤ 0.01. Now the subtly: it turns out these combinations of quark masses
and charges are not enough to constrain all 10 LEC's appearing in Eq. (23). There is one
direction in the multi-dimensional parameter space that is not linearly-independent from the
rest. Fixing any one of the LEC's to zero, except A
(s,1,1)
5
or A
(s,2)
5
TABLE
TABLE
From a weak-coupling perturbation theory perspective in QCD+QED, we have neglected vacuum polarization effects at order O(α em α s ). For the pions, the consequence is that the single (linear combination) LEC Y 1 can not be determined. For the kaons there are several LEC's that can not be determined (see Eq.
TABLE VIII .
VIIISummary of quark mass systematic errors. Central values quoted from the finite volume, SU(2), chiral perturbation theory fit. Masses given in MeV. The quark mass renormalization error comes from the nonperturbative QCD result [17] plus a one percent error from QED, addedin quadrature. Systematic errors are given as a percent (%). The algebraic sign of each change
comes from the difference (quantity under change) − (central value).
value (stat. error) fit
fv lat. spacing QED quenching m s quenching renorm
m u
2.24(10)
+4.02 +13.50
4
2
-
2.8
m d
4.65(15)
+3.55 -2.48
4
2
-
2.8
m s
97.6(2.9)
+0.23 +0.07
4
2
2
2.8
m d − m u
2.411(65)
+7.77 -17.35
4
2
-
2.8
m ud
3.44(12)
+2.75 +2.71
4
2
-
2.8
m u /m d
0.4818(96)
+5.45 +16.40
4
-
-
m s /m ud
28.31(29)
+2.91 -2.56
4
2
2
-
TABLEX. Proton and neutron mass difference due to the QED interaction. The LEC's are extracted from the nucleon data on the unitary points. (m p − m n ) QED is given at the physical quark mass m ud determined in this work.16 3
2.42(95) 1.26(38) 0.002(96)
0.33(11)
24 3
2.72(55) 1.80(22) 0.7(1.2)
0.383(68)
lattice size − 1
3 (4α − 2β) χ 2 /dof (m p − m n ) (m d −mu) (MeV)
16 3
−1.452(45) 1.1(1.2)
−2.265(70)
24 3
−1.612(92) 0.06(24)
−2.51(14)
TABLE XI .
XIProton-neutron mass difference due to non-degenerate u, d quark masses, computed on QCD configurations only. (m p − m n ) (m d −mu) is calculated at the physical value of (m d − m u ) determined in this work.
TABLE XIV :
XIVSame as for Tab. XIII, except for lattice size
24 3 .
TABLE XV :
XVSummary of pseudo-scalar meson mass-squared
splittings (×10 3 ) obtained from the masses in Tab. XIII. Lat-
tice size 16 3 .
TABLE XV -
XVcontinued from previous page
m sea m 1 m 3 q 1 ( 1
3 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.010 0.030 0.030 -2
-2
0.984(10)
0.982(10)
0.010 0.010 0.010 -2
2
1.309(17)
1.310(19)
0.010 0.010 0.020 -2
2
1.480(17)
1.481(17)
0.010 0.010 0.030 -2
2
1.650(18)
1.648(17)
0.010 0.020 0.020 -2
2
1.659(17)
1.658(16)
0.010 0.020 0.030 -2
2
1.835(17)
1.832(16)
0.010 0.030 0.030 -2
2
2.019(17)
2.014(16)
0.020 0.010 0.010 0
1
0.1292(15)
0.1289(16)
0.020 0.010 0.020 0
1
0.1577(13)
0.1575(13)
0.020 0.010 0.030 0
1
0.1874(14)
0.1875(14)
0.020 0.020 0.020 0
1
0.1588(13)
0.1586(13)
0.020 0.020 0.030 0
1
0.1893(13)
0.1893(13)
0.020 0.030 0.030 0
1
0.1913(13)
0.1915(13)
0.020 0.010 0.020 1
0
0.1296(16)
0.1290(17)
0.020 0.010 0.030 1
0
0.1304(18)
0.1299(19)
0.020 0.020 0.030 1
0
0.1601(14)
0.1601(14)
0.020 0.010 0.010 1
1
0.1606(47)
0.1595(34)
0.020 0.010 0.020 1
1
0.1788(29)
0.1779(31)
0.020 0.010 0.030 1
1
0.1990(30)
0.1983(31)
0.020 0.020 0.020 1
1
0.1980(26)
0.1975(27)
0.020 0.020 0.030 1
1
0.2190(26)
0.2188(26)
0.020 0.030 0.030 1
1
0.2410(25)
0.2408(25)
0.020 0.010 0.010 1
-1
0.3563(43)
0.3558(45)
0.020 0.010 0.020 1
-1
0.3957(42)
0.3950(43)
0.020 0.010 0.030 1
-1
0.4366(43)
0.4364(44)
0.020 0.020 0.020 1
-1
0.4370(41)
0.4368(41)
0.020 0.020 0.030 1
-1
0.4797(41)
0.4800(42)
0.020 0.030 0.030 1
-1
0.5243(42)
0.5250(42)
0.020 0.010 0.020 -2
-1
0.4642(78)
0.4615(84)
0.020 0.010 0.020 1
2
0.5478(64)
0.5461(67)
0.020 0.010 0.030 -2
-1
0.4766(85)
0.4740(90)
0.020 0.010 0.030 1
2
0.6457(62)
0.6449(64)
0.020 0.020 0.020 -2
-1
0.5590(63)
0.5578(65)
0.020 0.020 0.030 -2
-1
0.5733(94)
0.5725(67)
0.020 0.020 0.030 1
2
0.6596(59)
0.6593(60)
0.020 0.030 0.030 -2
-1
0.6765(60)
0.6763(60)
0.020 0.010 0.010 2
0
0.5227(61)
0.5212(64)
0.020 0.010 0.020 2
0
0.5240(66)
0.5217(70)
0.020 0.010 0.030 2
0
0.5274(75)
0.5252(78)
0.020 0.020 0.020 2
0
0.6398(54)
0.6392(55)
0.020 0.020 0.030 2
0
0.6453(57)
0.6451(58)
0.020 0.030 0.030 2
0
0.7691(53)
0.7697(53)
0.020 0.010 0.010 2
1
0.4556(75)
0.4531(82)
0.020 0.010 0.010 2
-1
0.847(10)
0.846(10)
0.020 0.010 0.020 2
-1
0.898(10)
0.896(10)
0.020 0.010 0.020 -1
2
0.9822(96)
0.9811(99)
0.020 0.010 0.030 2
-1
0.952(10)
0.950(10)
0.020 0.010 0.030 -1
2
1.1218(99)
1.122(10)
0.020 0.020 0.020 2
-1
1.0376(94)
1.0372(96)
0.020 0.020 0.030 2
-1
1.0954(96)
1.0959(97)
0.020 0.020 0.030 -1
2
1.1819(96)
1.1828(97)
0.020 0.030 0.030 2
-1
1.2440(96)
1.2456(97)
0.020 0.010 0.020 0
-2
0.6357(54)
0.6349(56)
0.020 0.010 0.030 0
-2
0.7535(56)
0.7538(57)
0.020 0.020 0.030 0
-2
0.7609(53)
0.7612(53)
0.020 0.010 0.010 -2
-2
0.651(12)
0.647(13)
0.020 0.010 0.020 -2
-2
0.723(11)
0.719(12)
0.020 0.010 0.030 -2
-2
0.803(12)
0.800(12)
TABLE XV -
XVcontinued from previous page m sea m 1 m 3 q 1 ( 13 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.020 0.020 0.020 -2
-2
0.799(10)
0.797(11)
0.020 0.020 0.030 -2
-2
0.882(10)
0.880(10)
0.020 0.030 0.030 -2
-2
0.968(19)
0.968(10)
0.020 0.010 0.010 -2
2
1.436(17)
1.434(18)
0.020 0.010 0.020 -2
2
1.593(16)
1.591(17)
0.020 0.010 0.030 -2
2
1.756(17)
1.755(17)
0.020 0.020 0.020 -2
2
1.757(16)
1.757(16)
0.020 0.020 0.030 -2
2
1.928(16)
1.929(16)
0.020 0.030 0.030 -2
2
2.105(16)
2.108(17)
0.030 0.010 0.010 0
1
0.1292(15)
0.1289(16)
0.030 0.010 0.020 0
1
0.1588(13)
0.1584(14)
0.030 0.010 0.030 0
1
0.1886(13)
0.1882(14)
0.030 0.020 0.020 0
1
0.1599(13)
0.1594(14)
0.030 0.020 0.030 0
1
0.1907(13)
0.1901(13)
0.030 0.030 0.030 0
1
0.1932(13)
0.1924(13)
0.030 0.010 0.020 1
0
0.1291(16)
0.1289(17)
0.030 0.010 0.030 1
0
0.1297(18)
0.1295(19)
0.030 0.020 0.030 1
0
0.1615(14)
0.1610(14)
0.030 0.010 0.010 1
1
0.1590(31)
0.1587(33)
0.030 0.010 0.020 1
1
0.1790(28)
0.1790(30)
0.030 0.010 0.030 1
1
0.1995(28)
0.1998(30)
0.030 0.020 0.020 1
1
0.2008(25)
0.2010(27)
0.030 0.020 0.030 1
1
0.2228(24)
0.2232(26)
0.030 0.030 0.030 1
1
0.2462(24)
0.2465(25)
0.030 0.010 0.010 1
-1
0.3579(46)
0.3569(49)
0.030 0.010 0.020 1
-1
0.3968(44)
0.3955(47)
0.030 0.010 0.030 1
-1
0.4371(45)
0.4354(47)
0.030 0.020 0.020 1
-1
0.4387(42)
0.4367(44)
0.030 0.020 0.030 1
-1
0.4815(42)
0.4788(44)
0.030 0.030 0.030 1
-1
0.5265(42)
0.5232(44)
0.030 0.010 0.020 -2
-1
0.4629(75)
0.4627(80)
0.030 0.010 0.020 1
2
0.5510(71)
0.5504(65)
0.030 0.010 0.030 -2
-1
0.4751(80)
0.4759(86)
0.030 0.010 0.030 1
2
0.6500(59)
0.6499(63)
0.030 0.020 0.020 -2
-1
0.5659(59)
0.5659(63)
0.030 0.020 0.030 -2
-1
0.5824(62)
0.5828(66)
0.030 0.020 0.030 1
2
0.6690(56)
0.6690(60)
0.030 0.030 0.030 -2
-1
0.6889(57)
0.6889(61)
0.030 0.010 0.010 2
0
0.5229(63)
0.5215(68)
0.030 0.010 0.020 2
0
0.5224(66)
0.5213(71)
0.030 0.010 0.030 2
0
0.5247(73)
0.5238(77)
0.030 0.020 0.020 2
0
0.6446(53)
0.6427(56)
0.030 0.020 0.030 2
0
0.6509(56)
0.6489(58)
0.030 0.030 0.030 2
0
0.7767(53)
0.7738(55)
0.030 0.010 0.010 2
1
0.4527(74)
0.4517(80)
0.030 0.010 0.010 2
-1
0.850(10)
0.848(11)
0.030 0.010 0.020 2
-1
0.899(10)
0.896(11)
0.030 0.010 0.020 -1
2
0.987(10)
0.984(10)
0.030 0.010 0.030 2
-1
0.950(10)
0.947(11)
0.030 0.010 0.030 -1
2
1.126(10)
1.121(10)
0.030 0.020 0.020 2
-1
1.0424(96)
1.037(10)
0.030 0.020 0.030 2
-1
1.1003(98)
1.094(10)
0.030 0.020 0.030 -1
2
1.1871(97)
1.181(10)
0.030 0.030 0.030 2
-1
1.2503(97)
1.243(10)
0.030 0.010 0.020 0
-2
0.6403(55)
0.6386(57)
0.030 0.010 0.030 0
-2
0.7586(55)
0.7567(57)
0.030 0.020 0.030 0
-2
0.7669(52)
0.7643(54)
0.030 0.010 0.010 -2
-2
0.645(12)
0.644(13)
TABLE XV -
XVcontinued from previous page m sea m 1 m 3 q 1 ( 13 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.030 0.010 0.020 -2
-2
0.724(11)
0.724(12)
0.030 0.010 0.030 -2
-2
0.805(11)
0.806(12)
0.030 0.020 0.020 -2
-2
0.810(10)
0.811(10)
0.030 0.020 0.030 -2
-2
0.8978(98)
0.899(10)
0.030 0.030 0.030 -2
-2
0.9899(96)
0.991(10)
0.030 0.010 0.010 -2
2
1.442(18)
1.438(19)
0.030 0.010 0.020 -2
2
1.598(17)
1.592(18)
0.030 0.010 0.030 -2
2
1.758(18)
1.751(19)
0.030 0.020 0.020 -2
2
1.765(16)
1.756(17)
0.030 0.020 0.030 -2
2
1.935(17)
1.924(17)
0.030 0.030 0.030 -2
2
2.114(17)
2.101(17)
TABLE XVI :
XVISummary of pseudo-scalar meson mass-squared splittings (×10 3 ) obtained from the masses in Tab. XIV. Lattice size 24 3 .
TABLE XVI -
XVIcontinued from previous page m sea m 1 m 3 q 1 ( 13 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.005 0.005 0.020 1
-2
0.9837(62)
0.9850(62)
0.005 0.005 0.030 -2
1
0.9102(80)
0.9137(78)
0.005 0.005 0.030 1
-2
1.1305(73)
1.1342(71)
0.005 0.010 0.010 -2
1
0.8683(55)
0.8676(57)
0.005 0.001 0.001 -2
1
0.6851(55)
0.6869(62)
0.005 0.010 0.020 -2
1
0.9291(63)
0.9299(61)
0.005 0.010 0.020 1
-2
1.0165(63)
1.0183(60)
0.005 0.010 0.030 -2
1
0.9910(70)
0.9941(66)
0.005 0.010 0.030 1
-2
1.1670(70)
1.1705(66)
0.005 0.020 0.020 -2
1
1.0848(66)
1.0868(61)
0.005 0.020 0.030 -2
1
1.1539(68)
1.1565(64)
0.005 0.020 0.030 1
-2
1.2421(70)
1.2447(65)
0.005 0.030 0.030 -2
1
1.3172(70)
1.3196(67)
0.005 0.005 0.005 -2
-1
0.3861(35)
0.3846(38)
0.005 0.005 0.010 -2
-1
0.3878(38)
0.3870(41)
0.005 0.005 0.010 1
2
0.4322(33)
0.4321(34)
0.005 0.005 0.020 -2
-1
0.3967(48)
0.3970(49)
0.005 0.005 0.020 1
2
0.5282(38)
0.5299(34)
0.005 0.005 0.030 -2
-1
0.4112(61)
0.4122(63)
0.005 0.005 0.030 1
2
0.6312(46)
0.6322(39)
0.005 0.010 0.010 -2
-1
0.4367(34)
0.4370(34)
0.005 0.001 0.001 -2
-1
0.3500(42)
0.3489(48)
0.005 0.010 0.020 -2
-1
0.4492(39)
0.4499(36)
0.005 0.010 0.020 1
2
0.5365(36)
0.5381(31)
0.005 0.010 0.030 -2
-1
0.4658(45)
0.4664(42)
0.005 0.010 0.030 1
2
0.6417(40)
0.6425(33)
0.005 0.020 0.020 -2
-1
0.5546(35)
0.5556(29)
0.005 0.020 0.030 -2
-1
0.5758(37)
0.5755(31)
0.005 0.020 0.030 1
2
0.6639(36)
0.6635(29)
0.005 0.030 0.030 -2
-1
0.6878(36)
0.6864(30)
0.005 0.005 0.005 2
0
0.4626(30)
0.4614(32)
0.005 0.005 0.010 0
-2
0.5217(29)
0.5214(30)
0.005 0.005 0.020 0
-2
0.6413(37)
0.6428(34)
0.005 0.005 0.030 0
-2
0.7649(46)
0.7667(41)
0.005 0.010 0.020 0
-2
0.6458(34)
0.6474(31)
0.005 0.010 0.030 0
-2
0.7718(40)
0.7735(35)
0.005 0.020 0.030 0
-2
0.7870(36)
0.7878(32)
0.005 0.005 0.005 2
-2
1.3062(90)
1.3034(99)
0.005 0.005 0.010 2
-2
1.3910(94)
1.388(10)
0.005 0.005 0.020 2
-2
1.561(10)
1.563(11)
0.005 0.005 0.030 2
-2
1.734(12)
1.741(12)
0.005 0.010 0.010 2
-2
1.4780(98)
1.476(10)
0.005 0.010 0.020 2
-2
1.654(11)
1.657(10)
0.005 0.010 0.030 2
-2
1.833(12)
1.839(11)
0.005 0.020 0.020 2
-2
1.844(11)
1.847(10)
0.005 0.020 0.030 2
-2
2.034(12)
2.039(11)
0.005 0.030 0.030 2
-2
2.235(12)
2.240(12)
0.005 0.001 0.001 2
-2
1.1658(95)
1.169(10)
0.005 0.005 0.005 2
2
0.5427(59)
0.5406(64)
0.005 0.005 0.010 2
2
0.5757(59)
0.5753(62)
0.005 0.005 0.020 2
2
0.6498(68)
0.6516(67)
0.005 0.005 0.030 2
2
0.7349(80)
0.7361(79)
0.005 0.010 0.010 2
2
0.6135(57)
0.6142(57)
0.005 0.010 0.020 2
2
0.6934(61)
0.6953(55)
0.005 0.010 0.030 2
2
0.7816(68)
0.7823(60)
0.005 0.020 0.020 2
2
0.7820(59)
0.7834(50)
0.005 0.020 0.030 2
2
0.8764(62)
0.8754(52)
0.005 0.030 0.030 2
2
0.9747(62)
0.9718(52)
TABLE XVI -
XVIcontinued from previous page m sea m 1 m 3 q 1 ( 13 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.005 0.001 0.001 2
2
0.4943(70)
0.4920(80)
0.010 0.010 0.010 0
1
0.13426(85)
0.13325(75)
0.010 0.001 0.001 -1
0
0.1044(10)
0.1068(10)
0.010 0.010 0.020 0
1
0.16591(89)
0.16441(80)
0.010 0.001 0.001 -1
1
0.3012(32)
0.3008(35)
0.010 0.010 0.030 0
1
0.19850(99)
0.19621(91)
0.010 0.001 0.001 -1
-1
0.1162(17)
0.1262(20)
0.010 0.020 0.020 0
1
0.16935(82)
0.16667(80)
0.010 0.020 0.030 0
1
0.20233(88)
0.19932(88)
0.010 0.030 0.030 0
1
0.20593(92)
0.20282(90)
0.010 0.010 0.020 1
0
0.13724(92)
0.13465(81)
0.010 0.010 0.030 1
0
0.1403(10)
0.13637(95)
0.010 0.020 0.030 1
0
0.17274(88)
0.16931(86)
0.010 0.010 0.010 1
1
0.1581(15)
0.1562(15)
0.010 0.010 0.020 1
1
0.1806(15)
0.1762(15)
0.010 0.010 0.030 1
1
0.2032(17)
0.1975(16)
0.010 0.020 0.020 1
1
0.2022(15)
0.1975(14)
0.010 0.020 0.030 1
1
0.2247(17)
0.2201(15)
0.010 0.030 0.030 1
1
0.2474(19)
0.2439(15)
0.010 0.010 0.010 1
-1
0.3787(25)
0.3767(25)
0.010 0.010 0.020 1
-1
0.4257(27)
0.4218(27)
0.010 0.010 0.030 1
-1
0.4746(30)
0.4675(29)
0.010 0.020 0.020 1
-1
0.4751(27)
0.4691(28)
0.010 0.020 0.030 1
-1
0.5253(29)
0.5170(30)
0.010 0.030 0.030 1
-1
0.5763(30)
0.5672(31)
0.010 0.010 0.020 -2
-1
0.4746(40)
0.4624(38)
0.010 0.010 0.020 1
2
0.5599(37)
0.5508(32)
0.010 0.010 0.030 -2
-1
0.4936(45)
0.4767(44)
0.010 0.010 0.030 1
2
0.6663(38)
0.6543(35)
0.010 0.020 0.020 -2
-1
0.5779(36)
0.5658(33)
0.010 0.020 0.030 -2
-1
0.5968(40)
0.5837(36)
0.010 0.020 0.030 1
2
0.6846(38)
0.6727(34)
0.010 0.001 0.001 -2
0
0.4239(40)
0.4338(41)
0.010 0.030 0.030 -2
-1
0.7039(44)
0.6938(36)
0.010 0.010 0.010 2
0
0.5426(34)
0.5386(30)
0.010 0.010 0.020 2
0
0.5546(37)
0.5441(32)
0.010 0.010 0.030 2
0
0.5671(41)
0.5510(37)
0.010 0.020 0.020 2
0
0.6821(33)
0.6714(32)
0.010 0.020 0.030 2
0
0.6956(35)
0.6819(34)
0.010 0.030 0.030 2
0
0.8275(37)
0.8150(36)
0.010 0.010 0.010 2
1
0.4556(37)
0.4508(35)
0.010 0.001 0.001 -2
1
0.7125(73)
0.7150(80)
0.010 0.010 0.010 2
-1
0.8982(58)
0.8925(58)
0.010 0.010 0.020 2
-1
0.9665(62)
0.9543(60)
0.010 0.010 0.020 -1
2
1.0517(63)
1.0429(63)
0.010 0.010 0.030 2
-1
1.0380(69)
1.0173(66)
0.010 0.010 0.030 -1
2
1.2109(71)
1.1953(70)
0.010 0.020 0.020 2
-1
1.1251(62)
1.1100(64)
0.010 0.020 0.030 2
-1
1.1991(65)
1.1785(68)
0.010 0.020 0.030 -1
2
1.2871(66)
1.2677(70)
0.010 0.030 0.030 2
-1
1.3631(69)
1.3416(71)
0.010 0.001 0.001 -2
-1
0.3431(43)
0.3653(47)
0.010 0.010 0.020 0
-2
0.6685(35)
0.6624(32)
0.010 0.010 0.030 0
-2
0.7980(39)
0.7886(36)
0.010 0.020 0.030 0
-2
0.8131(35)
0.8010(35)
0.010 0.001 0.001 2
-2
1.214(12)
1.216(14)
0.010 0.001 0.001 2
2
0.4768(71)
0.5156(81)
0.010 0.010 0.010 -2
-2
0.6414(61)
0.6339(62)
TABLE XVI -
XVIcontinued from previous page m sea m 1 m 3 q 1 ( 13 ) q 3 ( 1
3 ) ∆M 2 (×10 3 )(cov) ∆M 2 (×10 3 )(uncov)
0.010 0.010 0.020 -2
-2
0.7300(63)
0.7129(61)
0.010 0.010 0.030 -2
-2
0.8197(68)
0.7972(66)
0.010 0.020 0.020 -2
-2
0.8159(61)
0.7970(57)
0.010 0.020 0.030 -2
-2
0.9051(68)
0.8864(60)
0.010 0.030 0.030 -2
-2
0.9947(75)
0.9806(63)
0.010 0.010 0.010 -2
2
1.529(10)
1.519(10)
0.010 0.010 0.020 -2
2
1.716(10)
1.698(10)
0.010 0.010 0.030 -2
2
1.911(12)
1.880(12)
0.010 0.020 0.020 -2
2
1.913(11)
1.887(11)
0.010 0.020 0.030 -2
2
2.112(11)
2.078(12)
0.010 0.030 0.030 -2
2
2.315(12)
2.278
TABLE XIX .
XIXParity (P ) and charge conjugation (C) transformation properties for operators at chiral order O(p 2 ), O(e 2 ) that do not contain kaon fields and transform as A → h A h † . Under P ,
TABLE
Table XIX and XX list the building blocks of chiral order O(p 2 ) and O(e 2 ) that transform as Eq. (B7). The definition of various variables appearing in Table XX are as follows;
Table XIX and XX, we find that there are 13 O(e 2 )-operators bilinear in kaon fields that are invariant under chiral, P and C transformations kk † A , W kk † B , W 1 W 2 kk † ,
F 0 is normalized such that the physical decay constant is roughly 92 MeV.
This result is slightly larger than the published one, 1.729(28) GeV, in Ref.[7] because it was determined on a larger ensemble of lattices. It is also larger than the result of a combined fit, including new 32 3 , β = 2.25, ensembles[15][16][17]. Later, we use the slight difference as a systematic error.
We thank Chris Sachrajda for pointing this out.
ACKNOWLEDGEMENTSWe thank Enno ScholzAppendix B: Partially quenched chiral perturbation theory frameworkThe aim of this appendix is twofold:(1)to obtain all possible terms in the chiral Lagrangian relevant to the kaon mass-squared at order O(e 2 ) and O(e 2 p 2 ) for the partially quenched SU(2) + kaon system, and(2)to derive the expression for the EM correction to the kaon mass-squared to order O(e 2 p 2 ). The appendix is compact, mainly summarizing results and defining notation. Much of the machinery, of course, has been worked out before, and we refer the interested reader to the literature. Here we follow closely the works in Refs.[10][11][12][13]. The new contributions in this work are O(e 2 )-terms and electromagnetic one-loop chiral logarithmic correction to the kaon mass-squared. We also list the O(e 2 p 2 )-operators relevant to the kaon-mass-squared, which serves as a check of the possible dependence of O(e 2 p 2 ) corrections on charges and masses.We begin by reminding the reader of the important details and notation, then construct the Lagrangian density, and finally compute the corrections to the kaon mass-squared to the order of our interest.SU(2) pion sectorIn the partially quenched system composed of N V valence quarks, N S sea quarks and N V ghost quarks, the field Π(x) representing the Nambu-Goldstone multiplet is the localWith the normalization of F such that F 92 MeV, the leading-order (LO) chiral Lagrangian readswhere , denotes the supertrace in the partially quenched light quark sector, whose flavors can be indexed as I = 1, · · · , N V : light valence quark flavors , I = N V + 1, · · · , N V + N S : light sea quark flavors ,O(e 2 )-Lagrangian density in the kaon sector is hence given byThere are O(e 2 p)-terms bilinear in kaon fields that are allowed from the symmetries. All possible terms are obtained from Eq. (B26) by the replacement kk † → v µ k µ − . These operators generate no O(e 2 )-contribution and chiral-logarithmic corrections to the kaon mass-squared.They, however, induce O(e 2 p 2 )-contribution to the kaon mass-squared after the renormalization of the kaon field. We shall come back to this point in Section B 4. Next we turn to listing up O(e 2 p 2 )-terms that induce the corrections to the kaon masssquared at the tree level. The building blocks are those inTable XIX, XX and XXI. The definition of the quantities inTable XXIis as follows;The O(e 2 p 2 )-terms with no derivatives arewhereThe O(e 2 p 2 )-terms with two derivatives that contribute to the kaon mass-squared at the tree level are (η µν is the metric in Minkowski space) η µν k (µν) A , η µν k +, µν A , W η µν k (µν) B , W η µν k −, µν Q − , W η µν k +, µν B , (q iV + q sV ) 2 + x (K) 7(q iV − q sV ) 2 + x (K) 8We note that m iV and m (S) are denoted by m 1 and m 4 = m 5 , respectively, in Eq.(23).
and SU by the JSPS Grant-in-Aid No. 227180 and Nagoya University Global COE program, Quest for Fundamental Principles in the Universe. This manuscript has been authored by an employee (TI) of Brookhaven Science Associates. Grant, Aid of the Japanese Ministry of Education. DE-AC02-98CH10886 with the U.S. Department of EnergyNo. 22224003, TI and NY by Grant-in-Aid of the Japanese Ministry of Education (Nos. 20105001, 20105002, 22740183), and SU by the JSPS Grant-in-Aid No. 227180 and Nagoya University Global COE program, Quest for Fundamental Principles in the Universe. This manuscript has been authored by an employee (TI) of Brookhaven Science Associates, LLC under Contract No. DE-AC02-98CH10886 with the U.S. Department of Energy.
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| []
|
[
"Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection",
"Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection",
"Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection",
"Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection"
]
| [
"Mohanad Sarhan ",
"Siamak Layeghy ",
"Marius Portmann ",
"Mohanad Sarhan ",
"Siamak Layeghy ",
"Marius Portmann "
]
| []
| []
| Machine Learning (ML)-based network intrusion detection systems bring many benefits for enhancing the cybersecurity posture of an organisation. Many systems have been designed and developed in the research community, often achieving a close to perfect detection rate when evaluated using synthetic datasets. However, the high number of academic research has not often translated into practical deployments. There are several causes contributing towards the wide gap between research and production, such as the limited ability of comprehensive evaluation of ML models and lack of understanding of internal ML operations. This paper tightens the gap by evaluating the generalisability of a common feature set to different network environments and attack scenarios. Therefore, two feature sets (NetFlow and CICFlowMeter) have been evaluated in terms of detection accuracy across three key datasets, i.e., CSE-CIC-IDS2018, BoT-IoT, and ToN-IoT. The results show the superiority of the NetFlow feature set in enhancing the ML models detection accuracy of various network attacks. In addition, due to the complexity of the learning models, SHapley Additive exPlanations (SHAP), an explainable AI methodology, has been adopted to explain and interpret the classification decisions of ML models. The Shapley values of two common feature sets have been analysed across multiple datasets to determine the influence contributed by each feature towards the final ML prediction. | 10.1016/j.bdr.2022.100359 | [
"https://export.arxiv.org/pdf/2104.07183v2.pdf"
]
| 237,355,092 | 2104.07183 | a7980aaab79acf74f39e7951367a1bbbe223a868 |
Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection
Mohanad Sarhan
Siamak Layeghy
Marius Portmann
Evaluating Standard Feature Sets Towards Increased Generalisability and Explainability of ML-based Network Intrusion Detection
Index terms-CICFlowMeterExplainableMachine LearningNetFlowNetwork Intrusion Detection SystemSHaP
Machine Learning (ML)-based network intrusion detection systems bring many benefits for enhancing the cybersecurity posture of an organisation. Many systems have been designed and developed in the research community, often achieving a close to perfect detection rate when evaluated using synthetic datasets. However, the high number of academic research has not often translated into practical deployments. There are several causes contributing towards the wide gap between research and production, such as the limited ability of comprehensive evaluation of ML models and lack of understanding of internal ML operations. This paper tightens the gap by evaluating the generalisability of a common feature set to different network environments and attack scenarios. Therefore, two feature sets (NetFlow and CICFlowMeter) have been evaluated in terms of detection accuracy across three key datasets, i.e., CSE-CIC-IDS2018, BoT-IoT, and ToN-IoT. The results show the superiority of the NetFlow feature set in enhancing the ML models detection accuracy of various network attacks. In addition, due to the complexity of the learning models, SHapley Additive exPlanations (SHAP), an explainable AI methodology, has been adopted to explain and interpret the classification decisions of ML models. The Shapley values of two common feature sets have been analysed across multiple datasets to determine the influence contributed by each feature towards the final ML prediction.
Introduction
Network Intrusion Detection Systems (NIDSs) aim to protect and preserve digital networks from cyber threats [1]. Traditional NIDSs aim to analyse incoming traffic signatures and match them to one of the known attack signatures [2]. This method provides high detection accuracy of known attacks, however, it fails to detect unseen threats to computer networks, known as zero-day attacks [3]. Therefore, researchers have applied emerging Machine Learning (ML) technologies to learn and detect the harmful patterns of network traffic to detect intrusions [4]. A large number of academic research has been applied in this domain, many ML-based NIDSs have been developed, mostly achieving high detection accuracy when applied to certain datasets. However, the number of practical deployments of such systems in networks is vastly low [5]. While several factors are contributing to the lack of translating academic research into operation, two main reasons seem to top the list; the lack of reliable comprehensive evaluation methodologies [6] and lack of understanding of internal ML-based NIDS operations [7]. In this paper, we investigate the possible solutions to both problems.
First, the issue of unreliable ML-based NIDS evaluation methodology is addressed. Current benchmark NIDS datasets used in the testing stage of such systems are often generated with a unique set of network data features. Therefore, the capability of evaluating a learning model's detection accuracy using a targeted feature set across multiple datasets is limited. This questions the model's generalisability across other network environments and attack scenarios. Most of the time, these systems and algorithms are evaluated via a single benchmark dataset, which can be attributed to the lack of availability of multiple benchmark datasets with a common feature set. A recent work [6] proposed a standard feature set, which enables the evaluation of ML-based NIDS across multiple datasets using a common NetFlow [8] feature set. NetFlow was chosen due to its prevalence in the networking industry, however, it has not been validated as the best option to be used in NIDS datasets. The performance of ML models using the proposed feature set has never been compared to another set of features across multiple datasets. Therefore, in this paper, we examine the feature set designed by the CICFlowMeter [9] to the NetFlow feature set proposed in [6] across three widely used datasets, i.e., CSE-CIC-IDS-2018, ToN-IoT and BoT-IoT.
The comparison of the two feature sets (NetFlow and CICFLowMeter) has been conducted across three widely used NIDS datasets. Each dataset contains network data flows that have been collected over a different network environment. Moreover, various attack scenarios are available in each dataset. Hence, a fair and comprehensive comparison of the feature sets is provided in this paper. The NetFlow-based formats of the three datasets and the CICFLowMeter-based format of the CSE-CIC-IDS-2018 [10] dataset are available in the community. Therefore, in this paper two additional datasets have been generated in the CICFlowMeter format, named CIC-ToN-IoT and CIC-BoT-IoT, extracted using the ToN-IoT [11] and BoT-IoT [12] datasets, respectively. Both datasets have been labelled with binary-and multi-class categories and made publicly available at [13]. This will accommodate for a reliable evaluation of proposed ML models within the research community using the NetFlow and CICFlowMeter feature sets across multiple datasets. We strongly believe that such comprehensive evaluation methods will lead to reliable conclusions on the detection performance and generalisability of ML-based NIDS.
Moreover, we also examine the lack of understanding of internal ML operations [14]. Currently, ML-based NIDSs are offered as a complex 'black-box' [15] that achieves promising detection results on datasets. The complexity of ML models makes it hard to explain the reasoning behind the predictions made. Consequently, organisations are reluctant to trust ML decisions in a highly sensitive area such as intrusion detection [7]. Therefore, eXplainable Artificial Intelligence (XAI) methods are being applied in modern systems [16] to explain and interpret the decisions made by ML models. Explainable ML outputs can help maintain and troubleshoot the deployment of an ML-based NIDS by understanding and modifying the factors contributing to the model's decision. The classification results have been explained by calculating the Shapley value of each feature using the SHapley Additive exPlanations (SHAP) methodology [17]. SHAP measures the contribution and impact of each data feature towards the influence of the model's decision [17]. This will aid in the identification of key features utilised in the model's prediction and assist in the explanation of ML decisions.
Overall, this paper aims to address two key limitations of ML-based NIDS causing the wide gap between research and production. Two common feature sets based on NetFlow and CICFlowMeter have been evaluated across three widely used NIDS datasets. As part of the evaluation, two new datasets have been generated and made available at [13] to be used within the research community. Finally, an explanation of the ML classification results is conducted via the analysis of feature importance using the SHAP technique. In Section 2, some of the key related works that focused on the generalisability and explainability of NIDS feature sets are discussed. In Section 3, the importance of maintaining common feature sets is highlighted and the utilised datasets are introduced. The evaluation results are provided and analysed in Section 4. Section 5 describes the SHAP technique used to explain the importance and influence of network data features across the datasets.
Related Work
In this section, some of the key works that focused on explaining and evaluating the generalisability of feature sets of ML-based NIDS are discussed. In [18], Moustafa et al. examined the attack detection performance of the UNSW-NB15 and KDD-99 datasets' feature sets. The experiment extracted the features of the UNSW-NB15 dataset from the KDD99 dataset. The Association Rule Mining (ARM) feature selection technique is applied to determine the most important features across the two datasets. The detection accuracy is measured using a Naive Bayes (NB) model and Expectation-Maximization (EM) clustering techniques. The results show that the original KDD-99 features are less efficient than the UNSW-NB15 features in terms of network attack detection. The NB and EM models achieved an accuracy of 62.02% and 52.54% using the original KDD-99 features and 78.06% and 58.88% using the UNSW-NB15 features, respectively. The authors did not replicate the KDD-99 features on the UNSW-NB15 dataset to reliably evaluate the two feature sets.
In [19], Sarhan et al. highlighted the limitations of existing NIDS datasets, which is a lack of a common feature set, as the current feature sets are unique and completely different from each other. Therefore, the evaluation methods of the proposed ML-based NIDS are often unreliable. The lack of a common ground feature set prohibits the evaluation of the ML model's performance ability to generalise across multiple datasets. As a solution, the paper generated and published four datasets; NF-UNSW-NB15, NF-BoT-IoT, NF-ToN-IoT, and NF-CSE-CIC-IDS2018, sharing the same 12 NetFlow-based features. The datasets are generated by converting existing NIDS datasets to NetFlow format. NetFlow features are practical as they are relatively easier to extract from network traffic due to their presence in packet headers compared to complex features requiring deep packet inspection. As a use case, an Extra Tree classifier is reliably evaluated across the four datasets using the common feature set. The detection results indicate a reliable performance following a binary-class detection scenario. However, inferior results are achieved following a multiclass detection scenario.
In [6], the authors extended the feature set proposed in [19], by extracting a total of 43 NetFlow-based features. The generated and labelled datasets named NF-UNSW-NB15-v2, NF-BoT-IoT-v2, NF-ToN-IoT-v2 and NF-CSE-CIC-IDS2018-v2 are generated. Their common feature set is proposed as a standard set to be used across future NIDS datasets. The authors argue the tremendous benefits of having a universal NetFlow-based standard feature set due to its wide usage in the networking industry, practical deployment, and scaling properties. An Extra Tree classifier is used to evaluate the proposed feature set and compare its detection accuracy with the basic NetFlow datasets generated in [19] and the original feature set of the datasets. The results indicate that the proposed NetFlow feature set vastly outperforms the other feature sets in terms of attack detection accuracy across all datasets. The additional extracted NetFlow-based features demonstrated an extra amount of security events that enhance ML model detection rates of network attacks.
While the previous papers aimed to address the common feature set limitations faced by NIDS datasets, papers such as [20] focused on explaining the internal operations of ML-based NIDS. The experimental methodology utilised the SHAP technique to interpret the decisions of the NIDS. To their knowledge, this is the first paper in the intrusion detection field to utilise the SHAP method in the understanding of the judgement and structure of the NIDS. The paper also explored the differences in interpretation between a binary-and multi-class classifier. The authors utilised a Deep Feed Forward (DFF) neural network with a ReLU activation function. The classifiers achieved an F1 score of 0.807 and 0.792 in binary-and multi-class experiments. However, the detection performance achieved by the model is lacking and the utilised single dataset, i.e., NSL-KDD dataset in the experiment, does not include or reflect the latest network threats and attacks which are commonly reported nowadays [21]. The paper used a single attack class by selecting a random 100 samples of 'Neptune' attacks for the local explanation phase. The classifiers utilise different features to each other in their decision-making process. Finally, the top 20 important features for each attack type used in the global explanation are listed and comprehensively analysed via related works in the paper.
In [22], the paper designed a DFF and proposed an XAI framework to add transparency in ML operations. The DFF model is made up of 3 hidden layers performing a ReLU activation function and the output layer performing the softmax activation function. One aspect of the paper is the exploration of multiple XAI techniques, i.e., LIME, SHAP, Boolean Decision Rules via Column Generation (BRCG), and Contrastive Explanation Method (CEM). However, the paper utilised the NSL-KDD dataset to validate their methodology, which has been criticised in the field due to the lack of complex current network attack scenarios in the KDD-based datasets [21]. The SHAP results deemed that a high value of the 'same srv rate' feature increases the probability of an attack prediction whereas a high value of the 'dst host serror rate' feature increases the chances of a benign prediction. The authors applied the BRCG method to the dataset to extract model rules and achieved an accuracy of 80%. The LIME method generated a local explanation on specific data samples indicating which features contributed towards an attack or a benign prediction. Finally, the CEM showed how the prediction can be changed by altering feature values on a single benign data sample.
Common NIDS Feature Set
NIDS datasets are made up of a number of network data features that reflect the information represented by datasets. These features are required to represent a sufficient amount of security events to aid the model's classification tasks. NIDS dataset features have a large impact on the final quality of the ML-based NIDS [23]. In the deployment of ML-based NIDS in production, two key aspects require to be designed; a particular feature set for extraction and an ML model for feature analysis. The development of an optimal combination of a feature set and ML model has been an ongoing research issue. A large number of experiments targeting various feature selection and attack detection have been conducted [24]. However, for a reliable evaluation of ML-based NIDS, multiple datasets are required to be utilised. As each dataset has been generated in a network environment where different attack types were conducted. Therefore, multiple datasets will aid in evaluating the generalisation ability of the ML-based NIDS detection capability in various attack scenarios and network environments.
The information represented by the datasets is determined by the choice of network features that make up the dataset. Currently, NIDS datasets have proprietary feature sets, which are unique in their design, and often completely different from each other. Dataset authors have applied their domain knowledge when selecting the network features. As a result, NIDS datasets are not similar in terms of network attributes represented to facilitate reliable experiments. It is important to note that it will be unfeasible to extract all of them, due to their large number, when the model is deployed over a practical network. The unique feature sets in datasets have restricted the evaluation of network data features across multiple datasets due to their absence in other datasets. It also prohibited the evaluation of an ML model across multiple datasets using a targeted feature set. Therefore, using a common feature set across multiple datasets is crucial in the design of the model, increasing the evaluation reliability and the chances of potential deployment.
Current Datasets
Common feature sets enable reliable experimental evaluation of ML models across various datasets and attack types. Currently, there are four NIDS datasets that share a common feature set sarhan2021toward. The NetFlow-based features are proposed as a standard set to be used across future NIDS datasets. However, the performance of ML models using the NetFlow feature set has never been evaluated with another common set of features. Therefore, in this paper, the features designed by the CICFlowMeter tool [9] are compared to the standard NetFlow feature set across three datasets. The key datasets (CSE-CIC-IDS2018, ToN-IoT, and BoT-IoT) discussed below have been widely used amongst the research community in the training and evaluation of ML-based NIDS. With the majority of the learning models achieving a reliable detection performance across the datasets. However, each dataset is made up of a unique, often completely different, set of network data features. Hence, the evaluation of such models has been conducted using different feature sets. This removes the conclusion of the model's generalisability to multiple datasets, each representing different network environments and attack types, using the proposed common feature set. The testbed used to emulate the network traffic was set up in an organisational network manner involving multiple departments. Attack types such as brute-force, bot, DoS, DDoS, infiltration, and web attacks were conducted from an external source. The dataset contains 75 features extracted using the CICFlowMeter-v3 tool [9]. There are 16,232,943 data flows in total, where 13,484,708 (83.07%) are benign and 2,748,235 (16.93%) are attack samples. • BoT-IoT [12]-The Cyber Range Lab of ACCS designed an IoT based network environment that consists of normal and botnet traffic. The non-IoT and IoT traffic were generated using the Ostinato and Node-red tools, respectively. A total of 69.3GB of pcap files were captured and the Argus tool was used to extract the dataset's original 42 features. The dataset contains 477 (0.01%) benign flows and 3,668,045 (99.99%) attack ones, that is, 3,668,522 flows in total.
• NF-CSE-CIC-IDS2018-v2 [6] -The CSE-CIC-IDS2018 dataset [10] have been converted into 43 NetFlow-based features using nProbe [25]
Generated Datasets
In order to compare the performance of the NetFlow-based feature set to the CICFLowMeter-based feature set, three NIDS datasets are required to be available in both formats. The CSE-CIC-IDS2018 dataset is available natively in the CICFlowMeter format and NetFlow format as NF-CSE-CIC-IDS2018-v2. However, the ToN-IoT and BoT-IoT are only available in a NetFlow format as NF-ToN-IoT-v2 and NF-BoT-IoT-v2, respectively. Therefore, in this paper, the CICFlowMeter tool [9] has been utilised to convert the ToN-IoT and BoT-IoT datasets into the CICFLowMeter format. The packet capture (pcap) files have been fed into the CICFlowMeter tool that extracts the required feature set and generated data flows. The flows have been labelled in a binary-and multi-class manner, using ground truth events to allow for supervised learning methods. The source/destination IPs and ports and protocol features have been utilised to locate each data sample in the ground truth events and identify the respective attack type. Figure 1 presents the overall process used to generate the new datasets. The generated datasets using the ToN-IoT and BoT-IoT datasets have been named CIC-ToN-IoT and CIC-BoT-IoT, respectively, and made publicly available for research purposes at [13].
Figure 1: Dataset Generation Process
• CIC-ToN-IoT-A dataset generated as part of this paper, where the feature set of the CICFlowMeter was extracted from the pcap files of the ToN-IoT dataset [11]. The CICFlowMeter-v4 tool [9] was utilised to extract 83 features. There are 5,351,760 data samples where 2,836,524 (53.00%) are attacks and 2,515,236 (47.00%) are benign samples. This dataset has been generated as a part of this paper and made available at [13].
• CIC-BoT-IoT-The CICFlowMeter-v4 [9] was used to extract 83 features from the BoT-IoT dataset [12] pcap files. The dataset contains 13,428,602 records in total, containing 13,339,356 (99.34%) attack samples and 89,246 (0.66%) benign samples. The attack samples are made up of four attack scenarios inherited from the parent dataset, i.e., DDoS, DoS, reconnaissance, and theft. This dataset has been published as a part of this paper at [13].
The generated datasets share the same feature set as the CSE-CIC-IDS2018 dataset, proposed by the CICFLowMeter. That provides a reliable comparison of the feature set across three different datasets. It also enables the evaluation of ML models across the datasets using a common feature set. The chosen datasets in this paper; NF-CSE-CIC-IDS2018-v2, NF-ToN-IoT-v2, and NF-BoT-IoT-v2 in a NetFlow-based format and CSE-CIC-IDS2018, CIC-ToN-IoT and CIC-BoT-IoT in a CICFlowMeter format will allow for an evaluation of ML-based NIDS on two common feature sets across multiple network environments. It will also comprehensively evaluate both feature sets and compare the performances of enabling ML models for the detection of a large variety of network scenarios. We believe that having two common feature sets across six different datasets will significantly assist the research community in the evaluation of the proposed system. Consequently, tightening the gap between academic research and practical deployments.
Evaluation
In this section, two common network data feature sets are being evaluated across six NIDS datasets. This includes three datasets in the CICFlowMeter format (CSE-CIC-IDS2018, CIC-BoT-IoT, and CIC-ToN-IoT) and their three respective datasets in the NetFlow format (NF-CSE-CIC-IDS2018-v2, NF-BoT-IoT-v2, and NF-ToN-IoT-v2). To the best of our knowledge, this is the first-ever evaluation of ML-based NIDS using two common feature sets across recent multiple NIDS datasets. The large variety of network traffic in each dataset includes different attack scenarios and benign applications that will extensively and comprehensively evaluate the combination of feature sets and ML models, which generally form the key components of an ML-based NIDS.
Methodology
During the experiments, Deep Feed Forward (DFF) and Random Forest (RF) classifiers are utilised to classify the network data flows present in the datasets. The DFF structure consists of an input layer and three hidden layers with 10 nodes each performing the ReLU activation function. The output layer is a single Sigmoidal node and the Adam optimisation algorithm is used. The RF model consists of 50 randomised decision trees, each performing the Gini function to measure the quality of a split. To avoid learning bias towards the attack and victim devices, the source/destination IPs and ports are dropped. In addition, the timestamp and flow ID features are removed as they are unique to each data sample. The min-max scaler is applied to normalise all values between zero and one. This is necessary to avoid increased attention to larger values. Several binary classification metrics are collected, including accuracy, F1 Score, Detection Rate (DR), False Alarm Rate (FAR), Area Under the Curve (AUC), and the Prediction Time required to predict a single data sample in microseconds (µs). The datasets have been split into 70%-30% for training and testing purposes. For a fair evaluation, five cross-validation splits are conducted and the mean results are measured. The DFF and RF classifiers are utilised to classify the dataset samples into attack and benign categories. Tables 1 and 2 list the attack detection results of the six datasets using the RF and DFF classifiers respectively. Both classifiers have achieved a higher detection accuracy on the NF-CSE-CIC-IDS2018-v2 dataset compared to the CSE-CIC-IDS2018 dataset by increasing the DR and lowering the FAR. Resulting in an increase of the F1 score from 0.93 to 0.98 and 0.90 to 0.97 using the RF and DFF classifiers respectively. These affirm the results published in [6]. The RF classifier achieved very similar detection results on the ToN-IoT and BoT-IoT datasets in both their NetFlow and CICFlowMeter feature sets. The F1 score increases from 0.99 to 1.00 in both datasets. The FAR drops from 1.22% in CIC-ToN-IoT to 0.58% in NF-ToN-IoT-v2 and from 1.53% in CIC-BoT-IoT to 0.25% in NF-BoT-IoT-v2. It is also noted that the NetFlow features require a significantly lower prediction time compared to the CICFlowMeter features in both datasets. The DFF model achieved a significantly increased DR in the NF-ToN-IoT-v2 dataset of 95.37% compared to 92.29% in the CIC-ToN-IoT dataset, which resulted in an F1 Score increase from 0.94 to 0.96. This is also the scenario in the NetFlow features of the NF-BoT-IoT-v2 dataset, in which a high DR of 99.54% is achieved compared to CIC-BoT-IoT of only 95.99% DR. The lower prediction time required by the NetFlow features compared to the CICFlowMeter features can be explained by the smaller total number of features that make up the dataset. The higher DR of attacks and lower FAR indicates that the NetFlow-based features contain more amount or better quality of security events that aid the ML models efficient network intrusion detection. Overall, the constantly higher achieved detection accuracies demonstrate that the proposed NetFlow feature set in [6] is in a better capacity to assist ML models in identifying the attacks present in all three datasets. Figure 2 visually presents the benefits of using the NetFlow feature set in attack detection scenarios compared to the CICFlowMeter feature set. The results are grouped by Figures 2a and 2b based on the DFF and RF classifiers respectively. The proprietary feature set of the ToN-IoT and BoT-IoT datasets has been added for comparison purposes, comparing both datasets in their original, NetFlow and CICFlowMeter formats. The CSE-CIC-IDS2018 dataset has been released in the CICFlowMeter format as its proprietary format. Therefore, the original and CICFlowMeter formats are identical. The F1 score is plotted on the y-axis and the datasets on the y-axis, the figures show that the NetFlow features constantly achieve a higher F1 score across the three datasets compared to the CICFlowMeter and original feature sets. The continuous improvement of the ML models' detection accuracy, when trained and evaluated using the NetFlow feature set over the CICFlowMeter features, demonstrates the advantages of standardising it in the NIDS community. The NetFlow feature set enables two ML models; DFF and RF, following deep and shallow learning structures, to achieve a reliable performance over three datasets designed over different network environments and include a large number of different attack types. The CICFlowMeter results are slightly less superior in the BoT-IoT and ToN-IoT datasets and less efficient to a certain extent in the detection of attacks in the CSE-CIC-IDS2018 dataset. Moreover, another advantage of the NetFlow feature set is it contains a lower number of features of 43 compared to the CICFlowMeter of 83 features. A lower number of features will aid in an enhanced operation of extraction, analysis, and storage of features over an operational network. Finally, the NetFlow features are naturally present in network packet headers that do not require additional tasks to collect, unlike the CICFlowMeter features that include statistically measured features based on sum, maximum, minimum, standard deviation and average calculations, which might be unfeasible to generate in a live high-speed network.
Results
Explainable ML-based NIDS
In this section, the classification results achieved above will be analysed and explained to identify the key features that contributed to the models' final predictions. While a great deal of effort has been put into the design of ML-based NIDSs to achieve great detection accuracy, the trust in their operation of securing computer networks has always been in question [7]. As a result, a very limited number of commercial NIDSs are equipped with ML capabilities. It is believed this is due to the nature of such tools where the cost of errors are significantly higher compared to other domains [5], which means the operation of NIDS requires to be accurate at all times to avoid significant disruption. ML is often known to be a 'black box' technology where there is no clear understanding of what patterns are learned or why predictions are made. In the context of ML-based NIDS, the way models train, predict, and classify network traffic into an attack category is often not transparent. Therefore, organisations are reluctant to implement an ML-based tool [7]. There are multiple reasons such as data source, data features, class imbalance, and inaccurate datasets that can have a vast impact on the predictions made by ML models [7]. This makes it critical to understand and gain insights into the ML internal operations and decisions made.
The above motivation led to the generation of an ongoing research area of eXplainable Artificial Intelligence (XAI), where the aim is to analyse and explain the internal operations of ML models. As such, implementing the technologies of XAI into the ML-based NIDS is important to tighten the gap between the extensive academic research conducted and the number of operational deployments by increasing trust in ML. One way to interpret the detection results of an ML model is to determine which network data features contribute to the decision of the classifier. Determining which features of the dataset were used by the classifier to differentiate between a benign and an attack sample is critical for many reasons. Firstly, it helps in identifying which features contain security events that should be used in the design of the model or a better feature set. On the other hand, invaluable features that contain a limited number of security events and should be omitted from datasets are also located. Moreover, the classification decision of the model will be justified based on the values of these features. This helps in troubleshooting the errors caused by the wrong predictions of the model, as it allows security experts to analyse the values of the influencing features that result in a miss-classification. Further tuning of the model parameters and utilised features can be conducted after such information.
Shapley Analysis
The interpretation of the model's decisions through the calculation of each feature contribution will help in uncovering the 'black box' of ML. In this paper, the Shapley values are used to explain the importance of network data features in the detection of network attacks. The Shapley value was invented by Lloyd Shapley in 1953 [26], which is the method of assigning players' payouts based on their contribution to the total game's payout. The theory behind the Shapley value has been adopted in ML, where the 'game' represents the prediction task of a single sample in a dataset. The 'payout' is the actual prediction for one sample minus
SHapley Additive exPlanations (SHAP) is a common XAI technique developed by Lundberg and Lee [17]. It is based on an additive feature importance method of calculating the Shapley values, known as KernelSHAP and TreeSHAP. Compared to other XAI methods, SHAP has a strong theoretical foundation and can be used to explain the output of any ML model. SHAP presents new methods that have shown enhanced performance compared to alternative techniques. KernelSHAP is a kernel-based calculation approach for Shapley values, inspired by local surrogate models. TreeSHAP is used to explain tree-based ML models such as decision trees, random forest, and extra trees by leveraging their internal 'tree' structure to speed up the explanation process. When implemented, SHAP explains the ML prediction of each data sample x by calculating the importance of each feature based on its contribution to the ML prediction process. Equation 2 defines the explanation as specified by SHAP,
g (z ) = φ 0 + M j=1 φ j z j(2)
where g is the explanation model, z ∈ {0, 1} M is the coalition vector of features used, M represents the maximum coalition size, and φ j ∈ R is the Shapley value of feature j.
Results
While evaluating the RF classifier using the CSE-CIC-IDS2018 dataset, network data features containing security events relating to the forward direction of the flow make up the top four features influencing the model's decision. In particular, the 'Fwd Seg Size Min' feature has almost double the influence of any other feature. For the DFF classifier, the top two features present the number of forward and backward packets per second, followed by the 'Fwd Seg Size Min' feature which is the most influencing feature in the RF classifier's decision. The Shapley analysis of the NF-CSE-CIC-IDS2018-v2 dataset using the RF and DFF classifiers has 13 common features in the top 20 influencing features for both classifiers. This indicates that they contain key security events that can be utilised Overall, a global interpretation has been conducted by analysing the Shapley values of six NIDS datasets. The datasets sharing two common feature sets (CICFlowMeter and NetFlow) have been utilised to analyse the impact of each feature across multiple datasets. The KernelSHAP and TreeSHAP methods have been utilised for the DFF and RF models, respectively. The results identify certain key features causing a major impact on the model detection. However, some features have been vastly insignificant in the classification's process. This relates to the amount and quality of security events that these features present relating to the detection of attack types in the dataset. Therefore, the amount of each feature's contribution is different based on the datasets. This could be explained by the different attack scenarios and the different benign application traffic that each dataset contain. Therefore, the influence and importance of each network feature vary in each dataset. However, there are some common findings across datasets, such as the RF classifier being heavily influenced by forward directional-based features in the CICFlowMeter feature set.
Conclusion
ML-based NIDSs have achieved outstanding attack detection performance in the research community. However, the number of operational deployments has been very small. The limited evaluation of a common feature set across multiple datasets and the missing explanation of classification results contribute to the failed translation of research into real-world deployments. In this paper, the proposed NetFlow-based feature set has been evaluated and compared with the feature set designed by the CICFlowMeter tool. The evaluation has been conducted over three datasets (CSE-CIC-IDS2018, ToN-IoT, and BoT-IoT) using two ML classifiers (RF and DFF). Two datasets, CIC-BoT-IoT and CIC-ToN-IoT, have been generated and published to conduct the experiments. This kind of reliable comparison demonstrates the importance and necessity of having common feature sets across NIDS datasets, such as evaluating the generalisability of ML models' performance across different network environments and attack scenarios. The classification results generated by the ML models indicate a constant superiority of the NetFlow feature across the three NIDS datasets. Where both the DFF and RF classifiers achieved a higher attack detection accuracy in a lower prediction time. In addition, the SHAP method is used to explain the prediction results of the ML models by measuring the feature importance. The key features influencing the predictions of the models have been identified for each dataset.
•
CSE-CIC-IDS2018 [10]-A well-known NIDS dataset was released in 2018 in a project involving the Communications Security Establishment (CSE) & Canadian Institute for Cybersecurity (CIC).
•
ToN-IoT [11] -An IoT-based heterogeneous dataset released in 2019 that includes telemetry data of IoT services, network traffic of IoT networks, and operating system logs. The data was generated using an industrial network testbed. The dataset contains several attack scenarios such as backdoor, DoS, Distributed DoS (DDoS), injection, Man In The Middle (MITM), password, ransomware, scanning and Cross-Site Scripting (XSS). Bro-IDS tool, now called Zeek, was utilised to extract 44 network traffic features.
Figure 2 :
2Classification performance of two feature sets across three NIDS datasets
the average predictions of all samples. The 'players' are the feature values of the samples that collaborate to obtain the 'payout'. Overall, the Shapley value is the weighted average of the respective contribution of a feature value. The Shapley value (φ j ) of feature j is defined via Equation 1 [27], where S is a subset of features (values) used by the model, X = {x j / j∈ [1, ...,p]} represents the vector of feature values of the dataset samples in which p is the number of total features and x j is a feature value, and val x (S) is the final prediction for feature values in a test set S. φ j (val) = S⊆{x1,...,xp}\{xj } |S|!(p − |S| − 1)! p! (val (S ∪ {x j }) − val(S))
in the detection of attacks present in the NF-CSE-CIC-IDS2018-v2 dataset. The common features mainly include Transmission Control Protocol (TCP)-and Time To Live (TTL)-based features. The 'TCP WIN MAX OUT' is the first and second most influencing feature in the DFF and RF classifiers respectively, where it presents the maximum TCP window from the destination to the source host. In the analysis of the CIC-ToN-IoT dataset, the RF classifier has determined that the 'Idle Mean, Min, and Max' as the key features, influencing more than 50% of the model's final predictions. The features present the average, minimum, and maximum time the flow was idle before becoming active. However, the DFF failed to fully utilise neither the idle-based features in the attack detection stage. Where there are only two idle-based features in the top 20 list. This can explain the lower performance of the DFF classifier compared to the RF classifier. Further analysis of the idle-based features is required. In the NF-ToN-IoT-v2 and NF-BoT-IoT-v2 datasets, there are 17 and 14 common features out of the top 20 features influencing the RF and DFF classifiers, respectively, this indicates that they withhold key security events to aid the ML model detection performance. In the CIC-BoT-IoT dataset, the forward directionalbased security events are presented in four out of the top five features that impact the RF classifier decisions. The Information Access Technology (IAT)-based features account for nine out of the top 20 features that aid the RF classifier to detect attacks present in the CIC-BoT-IoT dataset. In particular, the 'Fwd IAT Min and Mean' features are the top two, representing the minimum and average time between two packets sent in the forward direction. The DFF classifier likewise utilised the IAT-based features that made up eight features out of the top 20. Moreover, the average SHAP values of NetFlow and CICFlowMeter feature sets are calculated across the three NIDS datasets. The top 10 features of the NetFlow and CICFlowMeter feature sets are displayed in Figures 3 and 4 respectively. The features (y-axis) are ranked based on their mean Shapley values (x-axis) across the whole test data samples used to evaluate the ML models. Each figure presents the mean Shapley values as determined by the KernelSHAP and TreeSHAP methods for the DFF and RF models, respectively. For efficient comparison, the mean Shapley values have been normalised to a scale from 0 to 1. The error bars are set based on the maximum and minimum SHAP values across the three datasets. The large margin of error across both feature sets, particularly in the CICFlowMeter format, illustrates that there is no optimal choice of the most influencing features across the utilised datasets.However, there are some similarities inFigure 3, where the L7 PROTO, TCP-and TTL-based features tend to be the most influencing features on both ML classifiers across the NetFlow-based features. It is also noticeable that in the case of the NetFlow feature set, there are 5 common features in the top features of deep and shallow learning methods, while for the FlowMeter format it is only 2 features.
Figure 3 :
3Average SHAP values of three NetFlow-based datasets
Figure 4 :
4Average SHAP values of three CICFlowMeter-based datasets
to generate NF-CSE-CIC-IDS2018-v2. There are 18,893,708 total number of flows where 2,258,141 (11.95%) are attack samples and 16,635,567 (88.05%) are benign ones. There are six attack categories such as bruteforce, bot, DoS, DDoS, infilteration, and web attacks.• NF-ToN-IoT-v2 [6] -An IoT dataset generated based on 43 NetFlow features released in 2021. The
features are extracted using nProbe [25] from the pcaps of the original parent (ToN-IoT) dataset [11],
generated at the Cyber Range Lab by the Australian Centre for Cyber Security (ACCS). The total
number of attack data flows 10,841,027 (63.99%) and 6,099,469 (36.01%) are benign dataflows, adding
up to a total of 16,940,496 samples. There are nine attack categories known as backdoor, DoS, DDoS,
injection, MITM, password, ransomware, scanning, and XSS.
• NF-BoT-IoT-v2 [6] -A newly generated IoT dataset based on 43 NetFlow features was released in
2021. The features are extracted using nProbe [25] from the pcaps of the original dataset, known as
BoT-IoT, generated at the Cyber Lab of the ACCS [12]. It contains 37,763,497 labelled network data
flows, where the majority are attack samples; 37,628,460 (99.64%) and, 135,037 (0.36%) are benign.
There are four attack categories in the dataset, i.e., DDoS, DoS, reconnaissance, and theft.
Table 1 :
1RF classification resultsAccuracy
F1 Score
DR
FAR
AUC
Prediction Time
NF-CSE-CIC-IDS2018-v2
99.47%
0.98
96.82%
0.17%
0.9833
20.98µs
CSE-CIC-IDS2018
98.01%
0.93
94.75%
1.42%
0.9667
22.39µs
NF-ToN-IoT-v2
99.66%
1.00
99.80%
0.58%
0.9961
7.57µs
CIC-ToN-IoT
99.33%
0.99
99.80%
1.22%
0.9929
10.14µs
NF-BoT-IoT-v2
100.00%
1.00
100.00%
0.25%
0.9988
3.60µs
CIC-BoT-IoT
98.24%
0.99
98.24%
1.53%
0.9836
7.07µs
Table 2 :
2DFF classification resultsAccuracy
F1 Score
DR
FAR
AUC
Prediction Time
NF-CSE-CIC-IDS2018-v2
99.24%
0.97
94.67%
0.14%
0.9831
8.23µs
CSE-CIC-IDS2018
97.05%
0.90
85.71%
0.96%
0.9502
8.31µs
NF-ToN-IoT-v2
94.74%
0.96
95.27%
6.08%
0.9843
8.12µs
CIC-ToN-IoT
93.80%
0.94
92.29%
4.49%
0.9782
7.26µs
NF-BoT-IoT-v2
99.54%
1.00
99.54%
0.20%
0.9996
8.37µs
CIC-BoT-IoT
96.01%
0.98
95.99%
1.20%
0.9907
9.88µs
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| []
|
[
"ElasticHash: Semantic Image Similarity Search by Deep Hashing with Elasticsearch",
"ElasticHash: Semantic Image Similarity Search by Deep Hashing with Elasticsearch"
]
| [
"Nikolaus Korfhage \nDepartment of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany\n",
"Markus Mühling \nDepartment of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany\n",
"Bernd Freisleben \nDepartment of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany\n"
]
| [
"Department of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany",
"Department of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany",
"Department of Mathematics and Computer Science\nUniversity of Marburg\nMarburgGermany"
]
| []
| We present ElasticHash, a novel approach for high-quality, efficient, and large-scale semantic image similarity search. It is based on a deep hashing model to learn hash codes for fine-grained image similarity search in natural images and a two-stage method for efficiently searching binary hash codes using Elasticsearch (ES). In the first stage, a coarse search based on short hash codes is performed using multi-index hashing and ES terms lookup of neighboring hash codes. In the second stage, the list of results is re-ranked by computing the Hamming distance on long hash codes. We evaluate the retrieval performance of ElasticHash for more than 120,000 query images on about 6.9 million database images of the OpenImages data set. The results show that our approach achieves high-quality retrieval results and low search latencies. | 10.1007/978-3-030-89131-2_2 | [
"https://export.arxiv.org/pdf/2305.04710v1.pdf"
]
| 240,302,895 | 2305.04710 | 96d8343fa3156bbd0ed526a167e3d5652cb9b191 |
ElasticHash: Semantic Image Similarity Search by Deep Hashing with Elasticsearch
Nikolaus Korfhage
Department of Mathematics and Computer Science
University of Marburg
MarburgGermany
Markus Mühling
Department of Mathematics and Computer Science
University of Marburg
MarburgGermany
Bernd Freisleben
Department of Mathematics and Computer Science
University of Marburg
MarburgGermany
ElasticHash: Semantic Image Similarity Search by Deep Hashing with Elasticsearch
deep hashing · similarity search · Elasticsearch
We present ElasticHash, a novel approach for high-quality, efficient, and large-scale semantic image similarity search. It is based on a deep hashing model to learn hash codes for fine-grained image similarity search in natural images and a two-stage method for efficiently searching binary hash codes using Elasticsearch (ES). In the first stage, a coarse search based on short hash codes is performed using multi-index hashing and ES terms lookup of neighboring hash codes. In the second stage, the list of results is re-ranked by computing the Hamming distance on long hash codes. We evaluate the retrieval performance of ElasticHash for more than 120,000 query images on about 6.9 million database images of the OpenImages data set. The results show that our approach achieves high-quality retrieval results and low search latencies.
Introduction
Query-by-content approaches based on feature representations that are learned by deep convolutional neural networks (CNNs) have greatly increased the performance of content-based image retrieval systems. However, state-of-the-art methods in the field of semantic image similarity search suffer from shallow network architectures and small data sets with few image classes in the training as well as in the evaluation phases. Few image classes in the training phase lead to poor generalizability to query images with unknown content in the evaluation phase, i.e., a more fine-grained modeling of the image content is required. Thus, high accuracy for arbitrary search queries, fast response times, and scalability to millions of images are necessary to meet many users' needs both in scientific and commercial applications.
In this paper, we present ElasticHash, a high-quality, efficient, and scalable approach for semantic image similarity search based on the most popular enterprise full-text search and analytics engine Elasticsearch 1 (ES). ES processes queries very fast due to inverted indices based on Lucene 2 , scales to hundreds of servers, provides load balancing, and supports availability and reliability. Apparently, the properties of ES are not only desirable for full-text search, but also for semantic image similarity search. Furthermore, integrating image similarity search into ES allows multi-modal queries, e.g., combining text and images in a single query. The contributions of the paper are as follows:
-We present ElasticHash, a novel two-stage approach for semantic image similarity search based on multi-index hashing and integrate it via terms lookup queries into ES. -We present experimental results to show that ElasticHash achieves fast response times and high-quality retrieval results at the same time by leveraging the benefits of short hash codes (better search times) and long hash codes (higher retrieval quality). To the best of our knowledge, we provide the first evaluation of image similarity search for more than 120,000 query images on about 6.9 million database images of the OpenImages data set. -We make our deep image similarity search model, the corresponding ES indices, and a demo application available at http://github.com/umr-ds/ ElasticHash.
The paper is organized as follows. In Section 2, we discuss related work. Section 3 presents ElasticHash. In Section 4, we evaluate ElasticHash on the OpenImages data set in terms of search latency and retrieval quality. Section 5 concludes the paper and outlines areas for future work.
Related Work
Deep learning, in particular deep CNNs, led to strong improvements in contentbased image similarity search. With increasing sizes of the underlying image databases, the need for an efficient similarity search strategy arises. Since highdimensional CNN features are not suitable to efficiently search in very large databases, large-scale image similarity search systems focus on binary image codes for quantization or compact representations and fast comparisons rather than full CNN features.
Recently, several deep hashing methods were introduced [25,6,23,21,2,15,4]. Many of them employ pairwise or triplet losses. While these methods often achieve state-of-the-art performance on their test data sets, they are not necessarily suitable for very large data sets and fine-grained image similarity search based on thousands of classes. Existing deep hashing methods are often trained using small CNNs that usually cannot capture the granularity of very large image data sets. Often, CNN models like AlexNet [12] are used as their backbones, and they are usually evaluated on a small number of image classes [22,4,6,25] (e.g., a sample of 100 ImageNet categories [4], about 80 object categories in COCO [14], NUS-WIDE [5] with 81 concepts, or even only 10 classes as in MNIST or CIFAR). Additionally, the image dimensions in CIFAR and MNIST are very small (32x32 and 28x28, respectively) and thus not sufficient for image similarity search in real-world applications. Many approaches are trained on relatively small training data sets (e.g., 10,000 -50,000 images [3,4,15]). In addition, there are no standardized benchmark data sets, and each publication uses different splits of training, query, and database images, which further complicates a comparison of the methods. Furthermore, training from scratch can be prohibitively expensive for large data sets. We observed that for large data sets with a high number of image classes, a transfer learning approach that combines triplet loss and classification loss leads to good retrieval results. To the best of our knowledge, ElasticHash is the first work that presents a deep hashing model trained and evaluated on a sufficiently large number of image classes.
The currently best performing approaches for learning to hash image representations belong either to product quantization (PQ) methods [8,9] and methods based on deep hashing (DH) [23,6]. Amato et al. [1] present PQ approaches that transform neural network features into text formats suitable for being indexed in ES. However, this approach cannot match the retrieval performance of FAISS [9]. Therefore, we focus on deep hashing that in combination with multiindex hashing (MIH) [17] can circumvent exhaustive search in Hamming space and achieve low search latency while maintaining high retrieval quality.
ElasticHash is related to other image similarity search methods integrated into ES. For example, FENSHSES [16] integrates MIH into ES and has a search latency comparable to FAISS. The method works efficiently for small radii of the Hamming ball and relatively small data sets (500,000 images). Small hamming radii, however, often produce too few neighbors for a query [17]. MIH like FEN-SHSES is thus not suitable for our scenario of large-scale image retrieval in ES with long binary codes (256 bits), where we require sub-second search latency on a data set of about 7 million images. Furthermore, we solve the shortcomings of FENSHSES using only a subset of bits rather than the whole hash codes to perform our MIH-based coarse search. While other works extend ES for image similarity search by modifying the Lucene library [7], our approach is seamlessly integrated into ES without modifying its code base.
ElasticHash
ElasticHash consists of several components as shown in Figure 1: a deep hashing component, an ES cluster, and a retrieval component. The deep hashing component is realized as a web service using Tensorflow Serving where the integrated deep hashing model is applied to images and the corresponding binary codes are returned. In the first phase, the binary codes are extracted from the database images in the indexing phase using the deep hashing component and stored into the ES cluster. After initially building the index, the retrieval component handles incoming query images and visualizes the retrieval results. For this purpose, the binary codes are extracted from the query images using the web service, the corresponding ES queries are assembled and sent to the ES cluster that returns the final list of similar images. The entire similarity search system can be easily deployed for production via Docker. The deep hashing model is described in more detail in Section 3.1, including the training strategy and network architecture. In Section 3.2, the ES integration is presented.
Deep Hashing Model
We now describe our deep hashing model and how it is used to extract both short and long hash codes.
The model training consists of two phases that both use ADAM as the optimization method. First, an ImageNet-pretrained EfficientNetB3 [19] model is trained on a data set with a larger number of classes in order to obtain a more fine-grained embedding. In contrast to the original ImageNet dataset, it contains all ImageNet classes with more than 1000 training images and all classes of the Places2 [24] data set, which results in a total number of classes of 5,390. The model is trained with cross-entropy loss on a Softmax output. After two epochs of training the final layer with a learning rate of 0.01, all layers are trained for another 16 epochs with a learning rate of 0.0001.
In the second phase, the classification model's weights are used to initialize the deep hashing model. This model includes an additional 256-bit coding layer before the class output layer with tanh activation and 256 outputs. This model is trained for 5 epochs with a learning rate of 0.0001. It is trained on the same data set as before, however, by combining cross-entropy loss on the output and hard triplet loss [18] on the coding layer.
With the classification loss
L c = K i=1 y i log p i(1)
for K classes with labels y i and predictions p i , and the triplet loss
L t = max(d(a, p) − d(a, n) + γ, 0)(2)
for Euclidean distance d between the 256-dimensional output of the coding layer for anchor image a and positive example p and between a and a negative example n, respectively, the combined loss function is given by:
L = αL + βL ,(3)
where we set margin γ = 2 and weights α = 1 and β = 5. We first sample a batch of size b = 128 images from a uniform distribution of the classes. This batch is used for both computing the classification loss and generating b hard triplets.
To make the similarity search more robust, we used heavy data augmentation in both phases, which in addition to standard augmentation methods includes inducing JPEG compression artifacts. After training, the model generates 256-bit codes. These codes can be decomposed into four 64-bit codes for fast computation of Hamming distance on long integers. However, using codes of this length on a corpus of about 10 million images is too expensive, even when using multi-index hashing. We therefore extracted 64-bit codes from the original 256-bit codes to perform the filtering on shorter codes and thus smaller Hamming ball radii. To extract the 64 most important bits from the 256-bit codes, we first partition the 256-bit codes into four partitions by applying the Kernighan-Lin algorithm [10] on the bit correlations. From each of the four decorrelated partitions, we then take the first 16 bits to compose 64-bit codes.
Integration into ES
Before describing our image similarity search integration into ES, we will shortly review MIH in Hamming space [17]. The idea of MIH is based on the following observation: for two binary codes h = (h 1 , ..., h m ) and g = (g 1 , ..., g m ) where m is the number of partitions, h k and g k are the k th subcodes and H is the Hamming norm, the following proposition holds:
h − g H ≤ r ⇒ ∃k ∈ {1, ..., m} h k − g k H ≤ r m(4)
For the case of 64-bit codes that are decomposed into m = 4 subcodes, this means that a code is in a Hamming radius r < 12 if at least one of the subcodes has a distance of d ≤ r m = 2 from the query subcode. The performance of MIH can be increased if the subcodes are maximally independent of each other [20], especially for shorter codes [16]. Thus, after training a deep hashing model, the bit positions should be permutated accordingly.
The ES index used for retrieval contains four short codes (f 0 -f 3) and four long subcodes (r 0 -r 3) for each image. The short codes are used for MIH and efficiently utilize the reverse index structure of ES and are thus separated into four subcodes of type "keyword". The long codes are also separated into four subcodes in order to allow fast computation of Hamming distances for values of type long.
An additional index is used for fast lookup of neighboring subcodes within the retrieval query. The neighbors index does not change once it has been created and merely serves as an auxiliary index for term queries. It requires pre-computing all nearest neighbors for all possible 16-bit subcodes. Thus, the index of neighbors contains 2 16 documents. The document id corresponds to the unsigned integer representation of a 16-bit subcode and can therefore accessed within a term query. It contains a single field "nbs" that is assigned to a list of all neighboring 16-bit codes within a Hamming radius of d of the corresponding query subcode. Since this index basically works as a lookup table, it could also be realized somewhere else, i.e., not as an ES index. However, integrating the lookup table this way eliminates the need for external code and enables fast deployment of the whole system. All documents representing all possible 16-bit subcodes are inserted according to the query in Listing 1.1. In this stage, MIH is realized by querying the additional index of neighbors for fast neighbor lookup. Even with MIH, using the full code length of the deep hashing model trained for 256-bit codes is too expensive for larger databases. We therefore limit the code length for the filtering stage to 64-bit codes. To obtain a sufficiently large set of candidate hash codes in the first stage, we need to search within a Hamming ball with a correspondingly large radius. We set d = 2, which will return at least all codes within r = 11 of a 64-bit code. In our setting with d = 2, this results in 137 neighbors per subcode, i.e., 548 neighbors in total.
In ES, we realize MIH by using a terms lookup. It fetches the field values of an existing document and then uses these values as search terms (see Listing 1.1). In contrast to putting all neighbors into the query, using a dedicated index for subcode neighbors has the advantage that the retrieval of neighboring subcodes is carried out within ES. Thus, the query load is small, and no external handling of neighbor lookup is necessary.
In the second stage, all codes obtained by MIH are re-ranked according to their Hamming distance to the long code. To compute the Hamming distance of the 256-bit code, the Painless Script in Listing 1.2 is applied to each of the four subcodes. The query in Listing 1.3 combines the MIH step as a filter with a term query and the re-ranking step as an application of the painless script from Listing 1.2 on the filtered retrieval list.
Experimental Evaluation
To determine the search latency and retrieval quality of ElasticHash, we evaluate three settings for using the binary hash codes generated by our deep hashing model for large-scale image retrieval in ES: (1) short codes, i.e., 64 bits for both GET / es -retrieval / _search { " query " : { " fu nction_s core " : { " boost_mode " : " sum " , " score_mode " : " sum " , " functions " : [ ... , { " script_score " : { " script " : { " id " : " hd 6 4 " , " params " : { " field " : " r_ <i >" , " subcode " : <6 4 bit subcode for re -ranking > } } }, " weight " : 1 }, ... ] , " query " : { " co nstant_s core " : { " boost " : 0 . filtering and re-ranking, (2) long codes, i.e., 256 bits for both filtering and reranking, and (3) ElasticHash, i.e., 64 bits for filtering, 256 bits for re-ranking.
Settings (1) and (2) are similar to the MIH integration of Mu et al. [16].
To evaluate our approach, we use OpenImages [13], which is currently the largest annotated image data set publicly available. It contains multi-label annotations for 9.2 million Flickr images with 19,794 different labels and is partitioned into training, validation, and test data set. On the average, there are 2.4 positive labels for the training split, while the validation and test splits have 8.8. As our database images we use all training images being available when downloading the data set, i.e., 6,942,071 images in total. To evaluate the retrieval quality, we use all downloaded images from the OpenImages test and validation set as query images (121,588 images in total). From these images, we draw a sample of 10,000 images to measure the search latencies for the three different settings.
The quality of the retrieval lists is evaluated using the average precision (AP) score, which is the most commonly used quality measure in image retrieval. The AP score is calculated from the list of retrieved images as follows:
AP (ρ) = 1 |R ∩ ρ N | N k=1 R ∩ ρ k k ψ(i k ), with ψ(i k ) = 1 if i k ∈ R 0 otherwise (5)
where N is the length of the ranked image list, ρ k = {i 1 , i 2 , . . . , i k } is the ranked image list up to rank k, R is the set of relevant documents, R ∩ ρ k is the number of relevant images in the top-k of ρ and ψ(i k ) is the relevance function. We consider an image as relevant, if it has at least one label in common with the query image. To evaluate the overall performance, the mean AP score is calculated by taking the mean value of the AP scores over all queries.
Results
We first evaluate the search latency for the queries. Next, we compare the retrieval quality in terms of AP. The experiments were performed on a system with an Intel Core i7-4771 CPU @ 3.50GHz and 32 GB RAM. Table 1 shows that for a k up to 250 there is no notable decrease in retrieval quality when employing ElasticHash rather than using the long codes for both stages. Figure 2 shows examples of the top-10 retrieval results for the three settings. It is evident that the retrieval quality of ElasticHash is similar to using long codes, and both are superior to using short codes. On the other hand, Table 2 indicates that the average retrieval time only slightly increases compared to using short codes for both stages. This suggests that ElasticHash is a good trade-off between retrieval quality and search latency. Although our deep hashing model was trained on 5,390 classes, but almost 20,000 classes occur in the validation data set, high AP values are achieved for ElasticHash.
Conclusion
We presented ElasticHash, a novel two-stage approach for semantic image similarity search based on deep multi-index hashing and integrated via terms lookup queries into ES. Our experimental results on a large image data set demonstrated that we achieve low search latencies and high-quality retrieval results at the same time by leveraging the benefits of short hash codes (better search times) and long hash codes (higher retrieval quality).
There are several areas for future work. For example, it would be interesting to investigate how many classes are necessary to obtain a high degree of generalizability. Furthermore, our loss function could be adapted to multi-label image data. Finally, we plan to extend our approach to achieve intentional image similarity search [11] using ES.
Fig. 1 .
1Overview of the workflows for image similarity search in ES.
Query for adding an entry to neighbor lookup index.
" script " : { " lang " : " painless " , " source " : 6 4 -Long . bitCount ( params . subcode^doc [ params . field ] . value ) } } Listing 1.2. Query for adding a Painless Script.
" : [ ... , { " terms " : { " f_ <j >" : { " id " : " < 1 6 bit subcode for lookup >" , " index " : " nbs -d 2 " , " path " : " nbs " } } }, ... ] } } } }, } } } Listing 1.3. Query for performing two-stage similarity search.
Fig. 2 .
2Top-10 retrieval results for (a) short codes, (b) long codes, and (c) ElasticHash for the same query image (first on the left); green: relevant result; red: irrelevant result.
Table 1 .Table 2 .
12Retrieval quality in terms of mean AP for different thresholds of k on 121,588 query images. 87.94 86.08 84.44 82.54 79.41 76.44 72.86 long 95.35 94.72 94.23 93.71 92.90 92.09 90.95 ElasticHash 95.21 94.48 93.90 93.22 92.02 90.61 88.42 Search latencies for ES queries (ms) with standard deviation for different thresholds of k on 10,000 query images.top k
10
25
50 100 250 500 1000
short top k
10
25
50
100
250
500 1000
short
µ 23.09 23.98 24.45 25.58 28.38 33.09 42.20
σ 4.74 4.65 4.70 4.72 4.86 5.20 6.07
long
µ 111.83 111.58 111.99 113.05 116.77 121.98 132.60
σ 16.50 16.58 16.72 16.54 17.04 17.13 17.99
ElasticHash
µ 36.12 36.75 37.28 38.17 40.88 45.73 55.23
σ 7.80 7.96 7.81 7.89 7.93 8.12 8.64
https://www.elastic.co 2 https://lucene.apache.org
AcknowledgementsThis work is financially supported by the German Research Foundation (DFG project number 388420599) and HMWK (LOEWE research cluster Nature 4.0).
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"Anomalous Couplings in e + e − → W + W − γ at LEP2 and NLC Typeset using REVT E X 1"
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"S F Novaes \nInstituto de Física Teórica\nUniversidade Estadual Paulista\nRua Pamplona 14501405-900São PauloCEPBrazil\n",
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"Instituto de Física Teórica\nUniversidade Estadual Paulista\nRua Pamplona 14501405-900São PauloCEPBrazil"
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| We present sensitivity limits on the coefficients of a dimension-6 effective Lagrangian that parametrizes the possible effects of new physics beyond the Standard Model. Our results are based on the study of the process e + e − → W + W − γ at LEP2 and NLC energies. In our calculations, we include all the new anomalous interactions, involving vector and Higgs bosons, and take into account the Standard Model irreducible background. We analyse the impact of these new interactions on the total cross section, including the effects of the initial electron and final W polarizations. We then focus on the operators that will not be constrained by the e + e − → W + W − process, obtaining limits based on the photon energy distribution.14.80.CpTypeset using REVT E X | 10.1103/physrevd.56.4384 | [
"https://export.arxiv.org/pdf/hep-ph/9703388v2.pdf"
]
| 116,959,413 | hep-ph/9703388 | ea266b4d124dab19329130cdd9ed7789b5e643ce |
Anomalous Couplings in e + e − → W + W − γ at LEP2 and NLC Typeset using REVT E X 1
14 Aug 1997
F De Campos
Instituto de Física Teórica
Universidade Estadual Paulista
Rua Pamplona 14501405-900São PauloCEPBrazil
S M Lietti
Instituto de Física Teórica
Universidade Estadual Paulista
Rua Pamplona 14501405-900São PauloCEPBrazil
S F Novaes
Instituto de Física Teórica
Universidade Estadual Paulista
Rua Pamplona 14501405-900São PauloCEPBrazil
R Rosenfeld
Instituto de Física Teórica
Universidade Estadual Paulista
Rua Pamplona 14501405-900São PauloCEPBrazil
Anomalous Couplings in e + e − → W + W − γ at LEP2 and NLC Typeset using REVT E X 1
14 Aug 1997(March 26, 2022)arXiv:hep-ph/9703388v2
We present sensitivity limits on the coefficients of a dimension-6 effective Lagrangian that parametrizes the possible effects of new physics beyond the Standard Model. Our results are based on the study of the process e + e − → W + W − γ at LEP2 and NLC energies. In our calculations, we include all the new anomalous interactions, involving vector and Higgs bosons, and take into account the Standard Model irreducible background. We analyse the impact of these new interactions on the total cross section, including the effects of the initial electron and final W polarizations. We then focus on the operators that will not be constrained by the e + e − → W + W − process, obtaining limits based on the photon energy distribution.14.80.CpTypeset using REVT E X
I. INTRODUCTION
One of the main physics goals of LEP2 and future e + e − colliders is to directly test the gauge nature of couplings among the electroweak gauge bosons. The process with largest cross section at LEP2 involving these couplings is the W -pair production, e + e − → W + W − , which is sensitive to the trilinear W W γ and W W Z couplings. The measurement of these couplings and the sensitivity to possible deviations from the Standard Model (SM) predictions have been extensively studied in the recent years [1].
The most general phenomenological parametrization for these couplings [2] can be achieved by means of an effective Lagrangian [3] that involves operators with dimension higher than four, containing the relevant fields at low energies and respecting the symmetries of the Standard Model. The effective Lagrangian approach is a model-independent way to describe new physics that can occur at an energy scale Λ much larger than the scale where the experiments are performed.
The effective Lagrangian depends on the particle content at low energies and since the Higgs boson has not yet been found, there are two logical possibilities to describe the new physics effect at low energies. In one of them, the Higgs boson can be light, being present in the higher dimensional operators, in addition to the electroweak gauge bosons, and the SM symmetries are linearly realized [4,5]. Alternatively, the Higgs boson can be very heavy and it must be integrated out at low energies. In this case, the relevant fields at low energies are only electroweak gauge bosons and the SM symmetries are realized non-linearly [6]. Here we focus on a linearly realized SU L (2) × U Y (1) invariant effective Lagrangian to describe the bosonic sector of the Standard Model, keeping the fermionic couplings unchanged.
The same effective Lagrangian used to describe anomalous trilinear gauge couplings can, in general, lead to anomalous quartic interaction among gauge bosons and also to anomalous couplings of these particles with the Higgs field. All these interactions should also be investigated at LEP2 and at the Next Linear Colliders (NLC) in order to search for hints about the nature of the new physics described by these higher dimensional operators.
New quartic gauge boson couplings have been studied before in many different processes at future e + e − , eγ, γγ, e − e − and pp colliders [7]. However, most of these previous works have focused on the so-called genuinely quartic operators, i.e. operators that give rise only to quartic gauge boson interactions without altering the trilinear couplings [8]. Since these operators do not appear in a dimension-6 linearly realized SU(2) L ×U Y (1) invariant effective Lagrangian [9], they will not be considered here. Anomalous Higgs boson couplings have also been studied before in Higgs and Z boson decays [10], in e + e − [11,12] and γγ colliders [13].
The process with largest cross section in e + e − colliders that also involves quartic couplings, and possibly anomalous Higgs couplings, besides the trilinear couplings, is e + e − → W + W − γ. Therefore, it is the most promising channel to look for possible deviations from the Standard Model predictions. This process has been considered by Bélanger and Boudjema [8] and by Leil and Stirling [14] in the context of genuinely quartic operators, where the Higgs and trilinear couplings were set to the Standard Model values and 3 σ deviations in the total cross section were used to determine the reach of this reaction.
Grosse-Knetter and Schildknecht [15] have considered the effect of a single higher dimensional operator usually denoted by O W (see below) in the above process, taking into account modifications on both trilinear and quartic couplings. However, they assumed that the Higgs boson mass lies above the energy region to be investigated and therefore they disregarded its contribution.
The purpose of this work is to study the sensitivity to these anomalous couplings of the process e + e − → W + W − γ at LEP2 and the NLC. We consistently include in our calculations all new couplings introduced by the effective Lagrangian that has become widely adopted to describe new physics beyond the Standard Model. In particular, this process is sensitive to operators related to anomalous Higgs boson couplings that do not affect the self-coupling of gauge bosons and hence are not constrained by the LEP2 measurements of e + e − → W + W − . Therefore, the process e + e − → W + W − γ may provide important information about these operators at the NLC. This paper is organized as follows. In Section II, we review the framework of effective Lagrangians that we use to parametrize anomalous couplings and explain the methodology used to study the W + W − γ production. In Section III, we analyze the sensitivity at LEP2 based on the total cross section. In Section IV, we study the improvements arising from going to NLC energies, the effects of having a polarized electron beam, and the impact of being able to measure the W boson polarization. We then concentrate on the analysis of operators which will not be probed by the e + e − → W + W − process, obtaining limits based on the photon energy spectrum. We present our conclusions in Section V.
II. EFFECTIVE LAGRANGIAN AND THE PROCESS e
+ e − → W + W − γ
In order to write down the most general dimension-6 effective Lagrangian containing all SM bosonic fields, i.e. γ, W ± , Z 0 , and H, we adopt the notation of Hagiwara et al. [5]. This Lagrangian has eleven independent operators in the linear representation that are locally SU L (2) × U Y (1) invariant, C and P even. We discard the four operators which affect the gauge boson two-point functions at tree-level and therefore are strongly constrained by LEP1 measurements. We also do not consider the two operators that modify only the Higgs boson self-interactions, since they are not relevant for our calculations. We are then left with five independent operators, and the Lagrangian is written as,
L eff = L SM + 1 Λ 2 (f W W W O W W W + f W W O W W + f BB O BB + f W O W + f B O B ) ,(1)
with each operator O i defined as,
O W W W = Tr Ŵ µνŴ νρŴ µ ρ (2) O W W = Φ †Ŵ µνŴ µν Φ (3) O BB = Φ †B µνB µν Φ (4) O W = (D µ Φ) †Ŵ µν (D ν Φ)(5)O B = (D µ Φ) †Bµν (D ν Φ) ,(6)
where Φ is the Higgs field doublet, which in the unitary gauge assumes the form,
Φ = 0 (v + H)/ √ 2 , andB µν = i g ′ 2 B µν ,Ŵ µν = i g 2 σ a W a µν ,(7)
with B µν and W a µν being the field strength tensors of the U(1) and SU (2) rise to both types of new couplings. Therefore, the existence of anomalous trilinear gauge couplings could be related to the anomalous quartic gauge couplings and Higgs interaction, which are the subject of our investigation.
Studies of anomalous trilinear gauge boson couplings from W -pair production will significantly constrain combinations of the parameters f W W W , f W and f B . However they are "blind" with respect to f W W and f BB . We chose to study the reaction e + e − → W + W − γ since it is the process with the largest cross section involving triple, quartic gauge boson couplings and also anomalous Higgs-gauge boson couplings. Therefore, it is also sensitive to f W W and f BB , offering an excellent possibility for a detailed study of these couplings. The Standard Model cross section for the process e + e − → W + W − γ was evaluated in Ref. [16]. When we neglect the electron mass, Higgs contributions for this reaction do not appear at tree level since the couplings Hγγ and the HZγ are generated only at one loop [17,18]. Taking into account these contributions, there are 16 Feynman diagrams involved in the reaction e + e − → W + W − γ, which are represented in Fig. 1 (the crossed diagrams are not shown) which yields,
σ SM W W γ = 46 (418) fb, with E γ > 20 (5) GeV, at √ s = 190 GeV σ SM W W γ = 144 fb, with E γ > 20 GeV, at √ s = 500 GeV(8)
where we have required that the angle between any two particles is larger than 15 • . The cross section peaks at roughly √ s = 300 GeV and is typically two orders of magnitude smaller than the two-body process e + e − → W + W − , used to constrain anomalous trilinear couplings.
In order to compute the contribution from all possible anomalous couplings, we have developed a Mathematica code to automatically generate the Feynman rules for the Lagrangian (1) that were then incorporated in Helas-type [19] subroutines. These new subroutines were used to extend a Madgraph [20] generated code to include all the anomalous contributions and to numerically evaluate the helicity amplitudes and the squared matrix element. In our calculations, we have taken into account the standard loop Higgs contributions besides all the relevant anomalous couplings, which give rise to the 42 contributions shown in Fig. 1, 2, and 3. We have checked that our code passed the non-trivial test of electromagnetic gauge invariance. We employed Vegas [21] to perform the Monte Carlo phase space integration with the appropriate cuts to obtain the differential and total cross sections. Moreover, we have studied the angular variables in order to find optimal cuts to improve the anomalous contribution over the SM signal.
III. W W γ PRODUCTION AT LEP2
We studied the reaction e + e − → W + W − γ at LEP2 assuming a center-of-mass energy of √ s = 190 GeV and an integrated luminosity of L = 0.5 fb −1 . We applied a cut in the photon energy (E γ > 5 GeV), and we required the angle between any two particles to be
larger than θ ij > 15 • .
Our results for the sensitivity of LEP2 to the operators appearing in the effective Lagrangian (1), from an analyses of the total cross section, are summarized in Fig. 4 for a fixed value of the Higgs boson mass, M H = 170 GeV. We plot the contributions of the 5 different operators separately, assuming that only one operator contributes each time. We also show the result for an extension of the so-called HISZ scenario [5], where all the coefficient are the total cross section would be observed for the following ranges of the coefficients of these operators, for Λ = 1 TeV,
considered equal, i.e. f W W W = f W W = f BB = f W = f B = f ,− 75 < f W W W < 178 , −48 < f W < 192 , −188 < f B < 550 , −253 < f W W < 110 ;(9)
whereas for the extended HISZ scenario, we have,
− 33 < f < 119 .(10)
Of course, the operators that also give rise to changes in the triple vector boson couplings can also be constrained at LEP2 via the reaction e + e − → W + W − . A recent analyses of W -boson pair production based on a log-likelihood fit to a five-fold differential cross section obtained the 1σ limits [22], |f W W W | < 10, |f W | < 7.1, and |f B | < 46. However, one should keep in mind that this reaction is insensitive to f W W and f BB , and therefore the study of the process e + e − → W + W − γ can provide further information on these operators, as we show in this paper.
The contribution of the anomalous couplings involving only the Higgs boson, i.e. f W W and f BB (see Fig. 3), is dominated by on-mass-shell Higgs production with the subsequent
H → W + W − decay, σ(e + e − → W + W − γ) ∝ σ(e + e − → Hγ) × Γ(H → W + W − ) Γ(H → all) .(11)
For large values of the operator coefficients, the total Higgs width is dominated by the anomalous decay H → γγ [10], which is also proportional to f W W and f BB . On the other hand, the anomalous width Γ(H → W + W − ) depends only on f W W . Therefore, the contribution from the anomalous coupling f BB is much less sensitive than the contributions from the other operators since σ(e + e − → W + W − γ) becomes almost independent of this coefficient. Fortunately, this is not the case if one is sensitive to small values of the coefficients, as will occur at the NLC study in the next Section.
We have investigated various distributions to try to improve the LEP2 sensitivity. The most promising distribution is the angular distribution of the W bosons with respect to the beam direction (see Fig. 5). We computed the total cross section with the extra cut cos θ W + e + > 0, as suggested by this distribution, and found an increase in sensitivity from 2σ to 2.8σ. However, due to the small deviations in the shape of the kinematical distributions and small statistics, no further improvement seems to be possible.
IV. W W γ PRODUCTION AT NLC
The effect of the anomalous operators becomes more evident with the increase of energy, and we are able to put tighter constraints on the coefficients by studying their contribution to different processes at the Next Linear Collider. We studied the sensitivity of NLC to the process e + e − → W + W − γ assuming √ s = 500 GeV and an integrated luminosity L = 50 fb −1 . We adopted a cut in the photon energy of E γ > 20 GeV and required the angle between any two particles to be larger than 15 • . We have analyzed this process for different values of the Higgs boson mass.
In Fig. 6, we show the results for the total cross section, for M H = 170 GeV, including the effects of the anomalous operators. The values of the coefficients f 's for which a 2σ deviation is obtained are shown in Table I, being typically of the order of 1 − 10 TeV −2 .
As we could expect, the W -pair production at NLC is able to put a limit that is one order of magnitude better for the coefficients f B,W,W W W [22]. However this latter reaction is not able to constraint f BB,W W .
In an attempt to increase the sensitivity, we looked at the effects of a 90% polarized electron beam in order to reduce the SM background, mainly the one coming from diagrams of Fig. 1a and 1b were just left-handed electrons are present. We have considered both left-handed (LH) and right-handed (RH) polarizations, expecting a larger anomalous sensitivity for RH electrons.
In Fig. 7, we show the results for the total cross section, for a 90% right-handed (RH) polarized electron, for M H = 170 GeV. Comparing Fig. 6 and Fig. 7, we notice that the effect of the anomalous contributions in the total cross section are larger for the polarized case. However, the small absolute value of the cross section for the polarized case reduces the statistics and leads to no improvement in the established limits, as shown in Table II.
Since we expect the new interactions to involve mainly longitudinally polarized gauge bosons, we studied the sensitivity for different combinations of the polarizations of the W −pair. In Fig. 8, we show the analogous of Fig. 6 for the W L W L case. Again, the effect of the anomalous contributions to the total cross section is increased, but no further improvements are found due to the small statistics. The results for the bounds on the anomalous coefficients for the W L W L , W T W T , and (W L W T + W T W L ) cases can be seen in Table III. These bounds were obtained requiring a 2σ effect on the total cross section.
It is important to notice that the kinematical distributions of the longitudinally polarized W 's are quite different from the SM results. As we could expect, the new physics effects becomes more evident for longitudinal W 's since the decay H → W + W − is dominated by this state of polarization. In Fig. 9, we present the angular distribution of the longitudinal W + boson with the initial positron and with the final photon, the energy and the transverse momentum distributions. We can see, for instance, that the W energy distribution is very different from the SM prediction. Its characteristic behavior for 100 < E W < 175 GeV is due to the presence of the Higgs boson, which decays into the W pair giving rise, at the same time, to a monochromatic photon.
We present in Fig. 10 the percent deviation of the SM prediction in the photon transverse momentum distribution, i.e., ∆ = dσ ANO /dp Tγ dσ SM //dp Tγ − 1 × 100% ,
for the different polarization of the W 's. Once again the relevance of the W L W L case is evident: ∆ > 100% for p Tγ > 120 GeV. When a cut of p Tγ > 100 GeV is implemented, the background is drastically reduced and the ratio of anomalous over SM events per year goes from 576/442 to 424/74, for f all = 15 TeV −2 .
Using the reaction e + e − → W + W − γ, we are also able to establish bounds on the values of the coefficients f W W and f BB , for which the W -pair process is insensitive, since they only affect the Higgs boson couplings. In Fig. 11, we present the results of a combined sensitivity analysis in the form of a contour plot for the two free parameter, f BB and f W W ,
for M H = 170 GeV. These are the most relevant coefficients for the anomalous Higgs boson phenomenology and they are not constrained by the W −pair production. We should keep in mind that the W W γ production at LEP2 can put a 1σ bound on f W W (9) while it is not possible to impose a limit on f BB since the cross section is quite insensitive to this coefficient.
If the Higgs boson is found with a mass in the range from 170 to 300 GeV, one would have a large sensitivity for the anomalous Higgs couplings f W W and f BB in the photon energy distribution of the process e + e − → W + W − γ. This increased sensitivity comes about because the existence of a peak in the photon energy spectrum due to the 2-body nature of the dominant contribution, i.e. e + e − → Hγ followed by the subsequent decay H → W + W − (see Fig. 3). In Fig. 12, we illustrate this effect with a typical photon energy distribution, for f W W /Λ 2 = f BB /Λ 2 = 5 TeV −2 and M H = 170 GeV, where the Higgs peak appears very clearly in the photon spectrum of the anomalous contribution.
In order to analyse the significance of the signal based on the photon energy spectrum, we took different energy bins of 1, 3 and 5 GeV. The reason is to roughly mimic the effects of a realistic simulation including the finite energy resolution of the detector and the small spread in the real center-of-mass energy due to initial state radiation. We have not considered the experimental efficiency, ǫ eff , for W reconstruction. It can be easily incorporated by multiplying the obtained significances by √ ǫ eff . Table IV shows the improvement on the sensitivity compared to the total cross section analysis for the f W W /Λ 2 = f BB /Λ 2 = 5
TeV −2 and M H = 170 GeV case.
In Table V, we present our results for the sensitivity on f BB /Λ 2 and f W W /Λ 2 , assuming f BB = f W W , for the three energy bins above. We obtained a sensitivity of the order of a TeV −2 for M H = 170 GeV, decreasing by a factor of roughly four for M H = 300 GeV, which does not depend in a significant way of the bin size. For larger Higgs boson masses, the cross section is reduced due to phase space suppression. For smaller Higgs boson masses, the cross section is reduced since the Higgs boson is off-mass-shell, and in this case it would be better to study processes like e + e − → bbγ or e + e − → γγγ [12].
V. CONCLUSION
The search for the effect of higher dimensional operators that give rise to anomalous bosonic couplings should be pursued in all possible processes since the results may provide important information on physics beyond the Standard Model. We have studied here the production of a W -pair plus a photon in e + e − colliders in order to analyse the contributions of anomalous couplings arising from dimension-6 operators of a linearly realized SU L (2) ×
TABLES
FIG. 4 .
4Total cross section (SM + Anomalous) for the process e + e − → W + W − γ, at LEP2 as a function of different anomalous coefficients and also for the HISZ scenario (f all ). We assumed m H = 170 GeV, and L = 0.5 fb −1 . The results for the SM and for 1, 2, and 3σ deviations are displayed (see text for energy and angular cuts). FIG. 5. Normalized W + − e + angular distribution. The solid (dashed) line represents the SM (SM + Anomalous) contribution for f all /Λ 2 = 150 TeV −2 and m H = 170 GeV. FIG. 6. The same as Fig. 4 for NLC, with √ s = 500 GeV and L = 50 fb −1 . FIG. 7. The same as Fig. 6 for a 90% right-handed polarized electron , with √ s = 500 GeV and L = 50 fb −1 . FIG. 8. The same as Fig. 6 for longitudinal W bosons (W L W L ), with √ s = 500 GeV and L = 50 fb −1 . FIG. 9. Kinematical distributions of the longitudinally polarized W + vector boson for the SM (solid histogram) and for the anomalous contribution (dotted histogram). FIG. 10. Plot of deviation (∆) in the photon P Tγ distribution for the cases of W L W L (solid line), W L W T + W T W L (dashed line) and W T W T (dotted line).
IV. Number of standard deviations σ from the Standard Model from a sensitivity analysis based on the total cross section compared to a sensitivity analysis based on the peak of the photon energy distribution, considering a 1, 3 and 5GeV bin for different values of the Higgs mass. We fixed f W W /Λ 2 = f BB /Λ 2 = 5 TeV −2 . . The minimum and maximum values (min, max) of the coefficients f i /Λ 2 (for f BB = f W W = f ) in units of TeV −2 that generate a 95% C.L. signal for the total cross section analysis and for the photon energy spectrum analysis with 1, 3 and 5 GeV energy bins for different values of the Higgs mass.
The operator O W W W contributes only to anomalous gauge couplings, O W W and O BB contribute only to anomalous Higgs couplings, HZZ and HZγ, whereas O W and O B givegauge fields
respectively.
in order to reduce the number of free parameters to only one (f ). The Standard Model cross section and its value with 1, 2 and 3 σ deviations are depicted as horizontal lines. The most sensitive contribution comes from O W W W , O W , and O B . A 1σ deviation in
We present the limits attainable at LEP2 and at NLC, including the Standard Model irreducible background. Polarization of the electron beam and of the W -pair are found to be insufficient to improve the limits obtained from the total cross section.We also focused on the operators O W W and O BB , which cannot be tested in the W −pair production process. We showed, in particular, that for Higgs boson masses in the range M H = 170-300 GeV, the photon energy spectrum provides a sensitive signature for the anomalous Higgs couplings. Typical sensitivities of a few TeV −2 at the NLC are obtained for these coefficients, providing complementary information on different higher dimensional operators.This work was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), and by Fundação de Amparoà Pesquisa do Estado de São Paulo (FAPESP).FIG. 1. Feynmam diagrams for the Standard Model process e + e − → W + W − γ. Crossed diagrams are not shown. FIG. 2. The vector bosons anomalous contributions to e + e − → W + W − γ. Crossed diagrams are not shown. FIG. 3. The Higgs boson anomalous contributions to e + e − → W + W − γ.U Y (1) invariant effective Lagrangian. We have included all the anomalous trilinear and
quartic gauge couplings, as well as the anomalous Higgs couplings with gauge bosons.
ACKNOWLEDGMENTS
FIGURES
TABLE III .
IIIThe minimum and maximum values (min, max) of the coefficients f i /Λ 2 in units of TeV −2 for a 2σ deviation of the total cross section for different combinations of the final state W −pair polarization.M H (GeV)
Total Cross Section
1 GeV bin
3 GeV bin
5 GeV bin
170
4.2
52.2
43.1
35.8
200
2.8
17.8
20.9
17.8
250
1.8
8.3
10.7
9.9
300
1.0
2.5
3.8
4.3
TABLE
Anomalous CouplingsUnpolarized Anomalous CouplingsAnomalous Couplings
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The curves show the one, two, and three sigma deviations from the Standard Model value of the total cross section. FIG. 12. Photon energy distribution for the SM (solid line) and for the SM + Anomalous. Bb × F W W , For M H = 170 Gev, for f W W /Λ 2 = f BB /Λ 2 = 5 TeV −2 , and M H = 170 GeV. dashed line. with a 5 GeV binFIG. 11. Contour plot of f BB × f W W , for M H = 170 GeV. The curves show the one, two, and three sigma deviations from the Standard Model value of the total cross section. FIG. 12. Photon energy distribution for the SM (solid line) and for the SM + Anomalous (dashed line), for f W W /Λ 2 = f BB /Λ 2 = 5 TeV −2 , and M H = 170 GeV, with a 5 GeV bin.
| []
|
[
"The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS",
"The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS",
"The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS",
"The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS"
]
| [
"R K Cochrane \nInstitute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK\n",
"P N Best \nInstitute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK\n",
"D Sobral \nDepartment of Physics\nLancaster University\nLA1 4YBLancaster\n\nLeiden Observatory\nLeiden University\nP.O. Box 9513NL-2300 RALeidenThe Netherlands\n",
"I Smail \nDepartment of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"J E Geach \nCentre for Astrophysics Research\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK\n",
"J P Stott \nDepartment of Physics\nLancaster University\nLA1 4YBLancaster\n",
"D A Wake \nDepartment of Physics\nUniversity of North Carolina Asheville\nOne University Heights\n28804AshevilleNCUSA\n\nDepartment of Physical Sciences\nThe Open University\nMilton KeynesMK7 6AAUK\n",
"R K Cochrane \nInstitute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK\n",
"P N Best \nInstitute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK\n",
"D Sobral \nDepartment of Physics\nLancaster University\nLA1 4YBLancaster\n\nLeiden Observatory\nLeiden University\nP.O. Box 9513NL-2300 RALeidenThe Netherlands\n",
"I Smail \nDepartment of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"J E Geach \nCentre for Astrophysics Research\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK\n",
"J P Stott \nDepartment of Physics\nLancaster University\nLA1 4YBLancaster\n",
"D A Wake \nDepartment of Physics\nUniversity of North Carolina Asheville\nOne University Heights\n28804AshevilleNCUSA\n\nDepartment of Physical Sciences\nThe Open University\nMilton KeynesMK7 6AAUK\n"
]
| [
"Institute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK",
"Institute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK",
"Department of Physics\nLancaster University\nLA1 4YBLancaster",
"Leiden Observatory\nLeiden University\nP.O. Box 9513NL-2300 RALeidenThe Netherlands",
"Department of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"Centre for Astrophysics Research\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK",
"Department of Physics\nLancaster University\nLA1 4YBLancaster",
"Department of Physics\nUniversity of North Carolina Asheville\nOne University Heights\n28804AshevilleNCUSA",
"Department of Physical Sciences\nThe Open University\nMilton KeynesMK7 6AAUK",
"Institute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK",
"Institute for Astronomy\nSUPA\nRoyal Observatory Edinburgh\nEH9 3HJUK",
"Department of Physics\nLancaster University\nLA1 4YBLancaster",
"Leiden Observatory\nLeiden University\nP.O. Box 9513NL-2300 RALeidenThe Netherlands",
"Department of Physics\nCentre for Extragalactic Astronomy\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"Centre for Astrophysics Research\nScience & Technology Research Institute\nUniversity of Hertfordshire\nAL10 9ABHatfieldUK",
"Department of Physics\nLancaster University\nLA1 4YBLancaster",
"Department of Physics\nUniversity of North Carolina Asheville\nOne University Heights\n28804AshevilleNCUSA",
"Department of Physical Sciences\nThe Open University\nMilton KeynesMK7 6AAUK"
]
| [
"MNRAS in press",
"MNRAS in press"
]
| The deep, near-infrared narrow-band survey HiZELS has yielded robust samples of Hα-emitting star-forming galaxies within narrow redshift slices at z = 0.8, 1.47 and 2.23. In this paper, we distinguish the stellar mass and star-formation rate (SFR) dependence of the clustering of these galaxies. At high stellar masses (M * /M 2 × 10 10 ), where HiZELS selects galaxies close to the so-called star-forming main sequence, the clustering strength is observed to increase strongly with stellar mass (in line with the results of previous studies of mass-selected galaxy samples) and also with SFR. These two dependencies are shown to hold independently. At lower stellar masses, however, where HiZELS probes high specific SFR galaxies, there is little or no dependence of the clustering strength on stellar mass, but the dependence on SFR remains: high-SFR low-mass galaxies are found in more massive dark matter haloes than their lower SFR counterparts. We argue that this is due to environmentally driven star formation in these systems. We apply the same selection criteria to the EAGLE cosmological hydrodynamical simulations. We find that, in EAGLE, the high-SFR low-mass galaxies are central galaxies in more massive dark matter haloes, in which the high SFRs are driven by a (halo-driven) increased gas content. | 10.1093/mnras/stx3345 | [
"https://arxiv.org/pdf/1801.04933v1.pdf"
]
| 20,067,375 | 1801.04933 | f041d7f375d8aeae4586942078dffd6d7bbf5c83 |
The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS
2018
R K Cochrane
Institute for Astronomy
SUPA
Royal Observatory Edinburgh
EH9 3HJUK
P N Best
Institute for Astronomy
SUPA
Royal Observatory Edinburgh
EH9 3HJUK
D Sobral
Department of Physics
Lancaster University
LA1 4YBLancaster
Leiden Observatory
Leiden University
P.O. Box 9513NL-2300 RALeidenThe Netherlands
I Smail
Department of Physics
Centre for Extragalactic Astronomy
Durham University
South RoadDH1 3LEDurhamUK
J E Geach
Centre for Astrophysics Research
Science & Technology Research Institute
University of Hertfordshire
AL10 9ABHatfieldUK
J P Stott
Department of Physics
Lancaster University
LA1 4YBLancaster
D A Wake
Department of Physics
University of North Carolina Asheville
One University Heights
28804AshevilleNCUSA
Department of Physical Sciences
The Open University
Milton KeynesMK7 6AAUK
The dependence of galaxy clustering on stellar mass, star-formation rate and redshift at z = 0.8 − 2.2, with HiZELS
MNRAS in press
2018Accepted 2017 December 18. Received 2017 December 11; in original form 2017 September 21Preprint 17 January 2018 Compiled using MNRAS L A T E X style file v3.0galaxies: evolution -galaxies: high-redshift -galaxies: halo -cosmology: large-scale structure of Universe
The deep, near-infrared narrow-band survey HiZELS has yielded robust samples of Hα-emitting star-forming galaxies within narrow redshift slices at z = 0.8, 1.47 and 2.23. In this paper, we distinguish the stellar mass and star-formation rate (SFR) dependence of the clustering of these galaxies. At high stellar masses (M * /M 2 × 10 10 ), where HiZELS selects galaxies close to the so-called star-forming main sequence, the clustering strength is observed to increase strongly with stellar mass (in line with the results of previous studies of mass-selected galaxy samples) and also with SFR. These two dependencies are shown to hold independently. At lower stellar masses, however, where HiZELS probes high specific SFR galaxies, there is little or no dependence of the clustering strength on stellar mass, but the dependence on SFR remains: high-SFR low-mass galaxies are found in more massive dark matter haloes than their lower SFR counterparts. We argue that this is due to environmentally driven star formation in these systems. We apply the same selection criteria to the EAGLE cosmological hydrodynamical simulations. We find that, in EAGLE, the high-SFR low-mass galaxies are central galaxies in more massive dark matter haloes, in which the high SFRs are driven by a (halo-driven) increased gas content.
INTRODUCTION
A rich array of work reveals that key observable galaxy properties including stellar mass, colour, star-formation rate, and morphology correlate with galaxy environments (Butcher & Oemler 1978;Dressler 1980;Baldry et al. 2006;Peng et al. 2010;Koyama et al. 2013b;Scoville et al. 2013;Darvish et al. 2016), with massive, red, quiescent spheroids residing in the densest environments. Studies of galaxy environments can help constrain galaxy formation and evolu-E-mail: [email protected] tion processes (e.g. Peng et al. 2010). Yet quantifying galaxy environments on a galaxy-by-galaxy basis can be difficult, particularly at high redshifts, because the accuracy of such measurements is highly dependent on the depth and uniformity of the observations and the quality of the redshifts (e.g. Cooper et al. 2005).
The two-point correlation function, which quantifies the clustering strength of a population of galaxies, provides a fairly robust technique for identifying the typical dark matter halo environments of galaxy populations. On large scales, the two-point correlation function is dominated by the linear 'two-halo term', which depends on the clustering of galax-ies within different dark matter haloes. The two-halo term essentially measures the galaxy bias, a measure of the difference between the spatial distribution of galaxies and that of the underlying dark matter distribution. On small scales, the non-linear 'one-halo term', which quantifies the clustering of galaxies within the same dark matter halo, dominates. Given an understanding of the way in which haloes of different mass cluster (which is reasonably well understood from N-body simulations within the cosmological model, e.g. Bond et al. 1991;Lacey & Cole 1994;Jenkins et al. 2001), the observed (projected or angular) two-point correlation function enables us to derive the halo occupation of samples of galaxies from their observed clustering. This technique is known as Halo Occupation Distribution (HOD; Ma & Fry 2000;Peacock & Smith 2000;Berlind & Weinberg 2002;Cooray & Sheth 2002;Kravtsov et al. 2004) modelling. The HOD framework then provides typical host dark matter halo masses for galaxy samples. It is also possible to derive estimates of central and satellite galaxy fractions from the small-scale 'one-halo term' (e.g. Zheng et al. 2005;Tinker & Wetzel 2010).
Galaxy clustering measures provide a statistical description for a population of galaxies rather than quantifying environments on a galaxy-by-galaxy basis. Strong trends in clustering strength have been observed with galaxy morphological type (Davis & Geller 1976), colour (Zehavi et al. 2005;Coil et al. 2008;Simon et al. 2009;Hartley et al. 2010;Zehavi et al. 2011), star-formation rate (Williams et al. 2009;Dolley et al. 2014;Wilkinson et al. 2017) and stellar mass (Wake et al. 2011;McCracken et al. 2015;Coupon et al. 2015;Hatfield et al. 2016), with the more recent studies reaching back to z ∼ 2 − 3. A limited number of studies of Lyman break galaxies have probed even further, back to z ∼ 6 − 7 (e.g. Harikane et al. 2016Harikane et al. , 2017Hatfield et al. 2017). The largest samples have permitted the splitting of galaxy populations by more than one observed property. For example, Norberg et al. (2002), using low-redshift (z < 0.15) data from the 2dF survey (Cole et al. 2000), found that both early-and late-type galaxies display higher r 0 values and therefore stronger clustering at brighter B-band absolute magnitudes (M B ). Coil et al. (2008) found broadly consistent results at z ∼ 1 using the DEEP2 galaxy redshift survey (Newman et al. 2012), also confirming that at fixed M B , red galaxies are more strongly clustered than blue galaxies.
Splitting by multiple variables in this manner is important for galaxy evolution studies. A natural consequence of the apparent tight (∼ 0.4 dex scatter) correlation between stellar mass and star-formation rate of star-forming galaxies (the 'main sequence', e.g. Brinchmann et al. 2004;Daddi et al. 2007;Elbaz et al. 2007;Karim et al. 2011) is that fundamental trends in one of these properties manifest as trends in the other. Galaxies with star-formation rates below the main sequence can also complicate observed trends: the fraction of galaxies that are passive increases towards higher stellar masses (Peng et al. 2010;Sobral et al. 2011), and this can give rise to trends with stellar mass which might not exist for the star-forming population only (e.g. the bending of the main sequence, Lee et al. 2015). Therefore, in this work, we aim to investigate the dependence of galaxy clustering on galaxy stellar mass and star-formation rate separately.
The High-Redshift(Z) Emission Line Survey (HiZELS, Sobral et al. 2013a; see Section 2) identifies galaxies via their emission lines, yielding reliably-selected samples of Hα emitters within narrow redshift slices back to z = 2.2. Hα (rest-frame wavelength 6562.8) is the brightest of the hydrogen recombination lines, which trace the young massive stellar population. Given that Hα is sensitive to star formation on short time-scales (∼ 10 7 yr) and is also wellcalibrated and less strongly extincted by dust than ultraviolet light , it is often used as a tracer of starformation. The Hα line is red-shifted out of the optical and into the near-infrared at z ∼ 0.5, making it ideal for probing star-forming galaxies at high redshift using wide-field near-infrared ground-based telescopes (e.g. Moorwood et al. 2000;Geach et al. 2008;Koyama et al. 2010Koyama et al. , 2011Koyama et al. , 2013aLee et al. 2012). The well-defined redshift distributions of the HiZELS samples of Hα-selected star-forming galaxies are ideal for studies of galaxy clustering, and the large numbers of emitters allows for the study of the population divided into many subsamples. Sobral et al. (2010) presented the first study of Hα luminosity-binned HiZELS galaxies and found evidence of higher clustering strengths for the strongest emitters at z = 0.84. Geach et al. (2008) and Geach et al. (2012) performed the first clustering studies of L Hα -selected galaxies at z = 2.23, though the sample size was insufficient to split by luminosity. In our previous paper (Cochrane et al. 2017, hereafter referred to as C17), we confirmed that the trends found by Sobral et al. (2010) hold to higher redshifts, using larger HiZELS samples at z = 0.8, z = 1.47 and z = 2.23. Transforming clustering strengths to dark matter halo masses using HOD modelling, we found that halo mass increases broadly linearly with L Hα at all three redshifts. Scaling by the characteristic 'break' of the Hα luminosity function, L * Hα , transforms these relations to a single trend, revealing a broadly redshift-independent monotonic relationship between L Hα /L * Hα and halo mass (Sobral et al. 2010; see also Khostovan et al. 2017 for similar relations with other line emitters). For all of our samples, L * Hα galaxies reside in dark matter haloes of mass ∼ 10 12 M , the known peak of the stellar mass -halo mass relation (e.g. Behroozi et al. 2010). We also found low satellite fractions (∼ 5%) for these samples. This suggested that the starformation rates of central galaxies are being driven by the mass accretion rates of their dark matter haloes (see also Rodríguez-Puebla et al. 2016, for details of a stellar-halo accretion rate coevolution model that matches observational data well). Sobral et al. (2010) used the K-band luminosities of HiZELS galaxies as a proxy for their stellar mass, finding an increase in galaxy clustering with increasing K-band luminosity, though the trend was significantly shallower than was observed for Hα luminosities. Preliminary investigations in C17 involved splitting our larger sample of galaxies at z = 0.8 into two bins by observed K-band magnitude. Intriguingly, we found that the strong, roughly linear relationship between log 10 L Hα and r 0 held for our two samples, with any differences between the two K-band magnitude bins being much smaller than the trend with Hα luminosity. Khostovan et al. (2017) (Sobral et al. 2013a(Sobral et al. , 2015. Only emitters which exceed the limiting flux, f 50 , of their frames are included in this work.
sion line strength than with galaxy stellar mass.
In this paper, we extend our previous work to study the clustering of HiZELS star-forming galaxies as a function of both Hα luminosity and stellar mass in more detail. Rather than using K-band observed magnitude as a proxy for stellar mass, we use a full SED-fitting approach to estimate stellar masses. We then compare our observational results to the output of the state-of-the-art cosmological hydrodynamical simulation EAGLE (Crain et al. 2015;McAlpine et al. 2015;Schaye et al. 2015). The structure of this paper is as follows. In Section 2 we provide a brief overview of the HiZELS survey and discuss our stellar mass estimates in some depth. In Section 3 we review the clustering and HOD-fitting techniques presented in C17 that we adopt here. In Section 4 we present our results, and in Section 5 we compare these to the output of the EAGLE simulation. Conclusions are drawn in Section 6.
We use an H 0 = 70kms −1 Mpc −1 , Ω M = 0.3 and Ω Λ = 0.7 cosmology throughout this paper.
THE HIZELS SURVEY AND SAMPLE SELECTION
Samples of Hα emitters
Our sources are drawn from HiZELS, selected by their emission line strength as detailed in Sobral et al. (2013a) and Sobral et al. (2015). A combination of narrow-and broadband images are used to identify Hα emitters, yielding sources within narrow redshift ranges (∆z ∼ 0.02) centred on z = 0.81 & 0.84 (hereafter z = 0.8), z = 1.47, z = 2.23. The galaxies used in this paper are the same as those used by C17: we impose the criterion that sources exceed f 50 , the 50% completeness flux of their survey frames. Raw Hα narrow-band fluxes are corrected for dust extinction by 0.4 dex (A Hα = 1). An equivalent width-dependent [NII] line contamination correction is made to account for emission from the [NII]6548, 6584 lines that also fall into the narrowband filter (see Sobral et al. 2013a). Star-formation rates are derived directly from dust-corrected Hα luminosities, L Hα using SFR Hα (M year −1 ) = 4.6 × 10 −42 L Hα (ergs s −1 ),
adopting the calibration of Kennicutt (1998) and scaling by a factor 1.7 (Speagle et al. 2014) to convert from a Salpeter (1955) IMF to a Chabrier (2003) IMF.
Deriving stellar masses from deep broad-band imaging
In order to estimate stellar mass, we model each galaxy's stellar populations and dust content via spectral energy distribution (SED) fitting using a similar method to that described in Sobral et al. (2011) and Sobral et al. (2014). The observed photometry is first shifted into the rest-frame. Model galaxy SEDs are then convolved with the detector's spectral response function to compare modelled and observed flux, and fitted via χ 2 minimization. Our modelling draws upon the stellar population synthesis package of Bruzual & Charlot (2003), using the updated models commonly referred to as CB07. These models assume a Chabrier (2003) IMF and an exponentially declining star-formation history of the form e −t/τ , where τ is in the range 0.1 − 10Gyr. Although this is not a realistic description of the star-formation histories of individual galaxies, which are likely to be characterized by shorter bursts, triggered by stochastic accretion, τ is a reasonable estimate of the mean age of a galaxy (see also Sobral et al. 2014, who show that using single exponential star-formation models does not introduce any significant bias into the stellar mass estimates of HiZELS galaxies). We use a grid of ages from 30Myr to the age of the Universe at each redshift, with a grid of dust extinctions from Calzetti et al. (2000) up to E(B − V) = 0.5, and three metallicities (0.2 − 1.0Z ).
For the COSMOS field, up to 36 wide, medium and narrow bands are used, from GALEX's far-UV band to Spitzer's four IRAC bands. In the UDS field there are only 16 available bands, but J, H and K data from UKIRT/UKIDSS DR5 are very deep. Seven bands (ugrizJK) are used in SA22 (see Sobral et al. 2013b). All HiZELS sources are assumed to lie at the central wavelength of the redshift distribution, which is a reasonable approximation since the filter profile is extremely narrow (see Table 1). The resultant stellar masses are fairly well constrained, with typical statistical uncertainties of 0.23, 0.24 and 0.26 dex at z = 0.8, 1.47 and 2.23, which vary a little from source to source. SED masses are plotted against Hα luminosities for the HiZELS samples in Figure 1. At each redshift, our samples cover a very wide range in stellar mass (10 8 < M * /M < 10 11 ) and also around 1 dex in Hα luminosity, spanning the break of the luminosity function.
As a test of our stellar masses, especially in SA22, where fewer bands are available, we compare our stellar mass estimates to apparent K-band luminosities, which broadly trace the older stellar population (e.g. Kauffmann 1998;Longhetti & Saracco 2009). Figure 2 shows SEDderived stellar mass versus observed K-band magnitude for HiZELS galaxies in the SA22 field at z = 0.8. These galaxies occupy a clear locus in this plane, close to the line expected from direct proportionality between K-band flux (rest-frame 1.2µm) and stellar mass. At fixed K-band magnitude, redder galaxies (see colour coding) have higher SED masses than would be expected from a naive extrapolation of K-band flux, and bluer galaxies have lower derived SED masses. This is exactly as expected, since the red fraction is higher for higher luminosity sources. These galaxies are dominated by old stars and have high mass-to-light ratios. In contrast, the bluer (typically less luminous) galaxies in our HiZELS samples have younger stellar populations, and are thus particularly luminous for their mass. We conclude that our SED masses are reasonable, and fold in important colour information. Therefore, we use the SED-derived stellar masses for the remainder of this paper, with confidence. We note, nevertheless, that our results are qualitatively unchanged whether we use K-band-derived or SED-derived masses.
QUANTIFYING GALAXY CLUSTERING USING THE TWO-POINT CORRELATION FUNCTION
We quantify the clustering of subsamples of HiZELS galaxies using the same techniques as C17, and the interested reader should refer to that paper for more details. Here, we provide a brief overview of our methods.
Angular two-point clustering statistics
The angular two-point correlation function, w(θ), is defined as the excess probability of finding a pair of galaxies separated by a given angular distance, relative to that probability for a uniform (unclustered) distribution with the same areal coverage. The probability dP(θ) of finding galaxies in solid angles dΩ 1 and dΩ 2 is thus dP(θ) = N 2 (1 + w(θ)) dΩ 1 dΩ 2 , where N is the surface density of galaxies. w(θ) is generally calculated by comparing the distribution of sources to that of a randomly distributed population subject to the same sample selection criteria. We use random samples of galaxies as described in C17. Random galaxies have luminosities drawn from the luminosity function constructed from the same samples, not exceeding the limiting flux of their simulated detection frame, and taking into account the effects of incompleteness and flux boosting. Following C17, we use the minimum variance estimator proposed by Landy & Szalay (1993), which was shown to be minimally susceptible to bias from small sample sizes and . SED-derived stellar mass versus observed K-band magnitude for SA22 galaxies, colour-coded by r − J colour. The black line shows the direct proportionality between K-band flux (restframe 1.2µm) and stellar mass (i.e. gradient fixed at −0.4). The stellar mass is clearly well correlated with K-band flux, but at fixed K-band magnitude, redder galaxies have higher SED-derived stellar masses, as would be expected. This colour dependence appears to drive the scatter in the relation and the deviation of the points from the straight line shown.
fields:
w(θ) = 1 + N R N D 2 DD(θ) RR(θ) − 2 N R N D DR(θ) RR(θ) .(2)
N R and N D are the total number of random and data galaxies in the sample, and RR(θ), DD(θ), and DR(θ) correspond to the number of random-random, data-data, and data-random pairs separated by angle θ. w(θ) is normally fitted with a power-law, w(θ) = Aθ −0.8 . We estimate uncertainty using the bootstrap resampling method, with the HiZELS observed frames forming our resampled volumes. Each correlation function was constructed from 1000 bootstraps, taking the error on each Overplotted are the best-fitting relations log 10 M eff /M = 11.7 ± 0.7 + r 0 /(4.5 ± 0.3) and log 10 M min /M = 10.9 ± 0.7 + r 0 /(4.5 ± 0.3). We find excellent linear fits, so use r 0 as a proxy for halo mass in this paper.
w(θ) bin as the diagonal element of the bootstrap covariance matrix. These uncertainties are quite conservative (see Norberg et al. 2009), enhanced by variations between frames of different depths. As described in C17, we make a small correction, the integral constraint (Groth & Peebles 1977), to account for the underestimation of clustering strength due to the finite area surveyed.
Obtaining a real-space correlation length
In order to compare the clustering strengths of populations of star-forming galaxies at different redshifts quantitatively, we convert the angular correlation function to a spatial one. This conversion is often performed using Limber's approximation (Limber 1953), which assumes that spatial correlations that follow ξ = (r/r 0 ) γ are projected as angular correlation functions with slopes β = γ + 1. This simple power-law fit is not reliable for our samples of galaxies, which span fields with separations of degrees and use very narrow filters, meaning that on large scales, the angular separation directly traces the real-space separation (resulting in a slope β = γ on large scales). Therefore, we perform a numerical integration of the exact equation:
w model (θ) = ψ −1 ∫ +∞ 0 ∫ 2s s √ 2φ 2 f s (s − ∆) f s (s + ∆) R −γ−1 r γ 0 ∆ dRds.(3)
Here, ψ = 1 + cos θ, φ = 1 − cos θ, ∆ = (R 2 − 2s 2 φ)/2ψ, and f s is the profile of the filter, fitted as a Gaussian profile with µ and σ that depend on the filter being considered (see C17 for the parameters of our filters). We assume the standard value of γ = −1.8. χ 2 fitting of observed against modelled w(θ), generated using different r 0 values, allows us to estimate r 0 and its error (see Sobral et al. 2010).
Halo Occupation Distribution fitting to obtain halo masses
In C17, we used Halo Occupation Distribution (HOD) modelling to derive typical dark matter halo masses for Hα luminosity-binned samples of HiZELS galaxies. HOD modelling involves parametrizing the number of galaxies per halo as a function of dark matter halo mass, N |M . Given a set of HOD parameters, a halo mass function and halo bias (here both are adopted from ) and a halo profile (we use NFW; Navarro et al. 1996) we generate a real-space correlation function. For each parameter instance, we simulate the projection of this real-space correlation function and compare the result to our observed two-point correlation functions. We use Markov chain Monte Carlo (MCMC) techniques, implemented using the EMCEE package (Foreman-Mackey et al. 2013), to determine the bestfitting parameters. All fitting is performed using the HMF and HALOMOD packages provided by Murray et al. (2013). Satellite galaxies are parametrized to have a power-law occupancy above some halo mass, in line with most HOD models. The HOD parametrization of centrals differs from those formulated for mass-limited samples, because although all massive haloes will contain a central galaxy, this need not fall within a star-formation rate limited sample. Recent work by Gonzalez-Perez et al. (2018) supports adopting an alternative parametrization for star-forming galaxies, which includes a Gaussian peak for low-mass haloes. Thus, following Geach et al. (2012) and C17, we parametrize the number of central and satellite galaxies separately as:
N cen |M = F B c (1 − F A c )exp − log(M/M min ) 2 2(σ log M ) 2 + 1 2 F A c 1 + erf log(M/M min ) σ log M ,(4)N sat |M = F s 1 + erf log(M/M min ) σ log M M M min α .(5)
The key parameters are:
-M min : the minimum halo mass that hosts a galaxy. Note that our definition differs subtly to that used in work characterizing mass-limited samples, such as McCracken et al. (2015) and Hatfield et al. (2016), since in this work M min applies to both central and satellite galaxies. The total number of galaxies is given by:
N |M = N cen |M + N sat |M .(6)
When fitting the models to data, we use the observed number density of galaxies as an additional constraint. For a given N |M output from the halo model, the predicted number density of galaxies is:
n g = ∫ dMn(M) N |M ,(7)
where n(M) is the halo mass function, for which we use the determination of . The observed number density of galaxies used here is the integral of the luminosity function between the same limits used to select the real and random galaxy sample. For each set of HOD parameters, we may derive a number of parameters of interest for galaxy evolution. In this paper, we use the effective halo mass, the typical mass of galaxy host halo. This is given by:
M eff = 1 n g ∫ dM Mn(M) N |M .(8)
3.4 Calibrating r 0 to M halo using HOD models
For samples of galaxies with large satellite fractions, there will be a substantial one-halo term in the correlation function at small separations. In such cases, HOD modelling offers a better fit than a simple power-law. In C17, we found that HiZELS samples at z = 0.8, z = 1.47 and z = 2.23 have low satellite fractions (∼ 5%), and HOD fitting offers only marginal gains in goodness of fit at small scales (see Figure 3, left-hand panel). Instead, the main benefit of HOD fitting is to allow the conversion of clustering strengths into typical halo masses. Comparing measured r 0 to derived halo masses (Figure 3, right-hand panel), we find that these are tightly correlated, and can be reasonably approximated as simple linear fits. At z = 0.8, these are given by:
log 10 M eff /M = 11.7 ± 0.7 + r 0 /(4.5 ± 0.3)
log 10 M min /M = 10.9 ± 0.7 + r 0 /(4.5 ± 0.3).
Therefore, in some parts of this paper (Section 4.1 -4.4), we simply derive and quote r 0 values, as these are sufficient to indicate trends of clustering with stellar mass or starformation rate. When we require robust halo masses, as in Sections 4.5 and 5, we perform the full HOD fitting.
CLUSTERING OF HIZELS GALAXIES AS A FUNCTION OF STELLAR MASS AND SFR
Clustering as a function of Hα luminosity
In C17, we studied the clustering of HiZELS galaxies as a function of their Hα luminosity. We found strong relationships between L Hα and r 0 . The clustering strength increases monotonically with Hα luminosity at all redshifts, indicating that the most highly star-forming galaxies thrive in higher dark matter overdensities (see Figure 4). We speculated that this is where a plentiful gas supply fuels high star-formation rates. HOD fitting revealed that typical Hα-emitting galaxies are star-forming centrals, residing in host haloes with minimum mass increasing with Hα luminosity from ∼ 10 11.2 M to ∼ 10 12.6 M and corresponding effective halo masses ∼ 10 11.6 M − 10 13 M . At all three redshifts, L * Hα galaxies typically reside in haloes of effective mass ∼ 10 12 M . This coincides with the halo mass predicted by theory to be maximally efficient at converting baryons into stars. Samples selected within the same L Hα /L * Hα range inhabit similar populations of dark matter haloes. The relationship between scaled galaxy luminosity L Hα /L * Hα and dark matter halo mass is largely independent of redshift.
Clustering as a function of stellar mass
C17 briefly looked at K-band observed luminosities. We found that the trends in clustering strength with L Hα do not differ between two large K-band bins, concluding that they are unlikely to be driven by stellar mass. Here, we extend that study to provide a more definitive answer to the role stellar mass plays.
Initially we bin our sample of z ∼ 0.8 HiZELS galaxies by stellar mass, construct correlation functions and fit these as described in Section 3.1, obtaining a clustering strength r 0 for each subsample. We use the broad bins in Hα luminosity as defined by C17 (−0.4 < log 10 (L Hα /L * Hα ) < 0.3) for consistency, but find no significant differences when we re-run the analysis with no luminosity cuts except for the HiZELS selection. We find that the clustering strength is broadly constant with stellar mass at low galaxy masses. This is particularly clear at z = 0.8, where our samples are largest and probe lowest in stellar mass, but all three redshifts are consistent with this result. The clustering strength only increases when we reach stellar mass bins that contain a significant number of galaxies below the main sequence: at all three HiZELS redshifts, clustering strength increases significantly above a mass 2 − 3 × 10 10 M and the most massive galaxies are very strongly clustered (see Figure 4 and Table 2). For our Hα-selected samples, the M * − r 0 relationship appears substantially weaker than the L Hα − r 0 relation obtained by C17, and shown in Figure 4 for comparison, which continues to decrease at low Hα luminosities. At all three redshifts, the clustering strength is broadly flat at low stellar masses, with evidence for an increase for the most massive galaxies (above ∼ 2 − 3 × 10 10 M ). Bottom: r 0 versus L Hα from C17, replotted for comparison. Here, a strong monotonic trend is seen between r 0 and L Hα at z = 0.8 and z = 2.2; as shown in C17, the z = 1.47 data are consistent with the same trend (albeit noisier due to the smaller sample).
Whilst the gradient of the stellar mass -halo mass relation of mass-selected galaxies does decrease below M * ∼ 10 10 M (see Section 4.5; Moster et al. 2010Moster et al. , 2013Behroozi et al. 2013 and many others), the flattening we observe for these Hα-selected galaxies is very pronounced. This indicates that low-mass HiZELS galaxies reside in more massive dark matter haloes than would be expected for starforming central galaxies of these stellar masses. Although this might be surprising, given that C17 found low satellite fractions for these samples, it is important to remember that, at these masses, HiZELS Hα-selected galaxies lie well above the 'main sequence'. We explore the joint dependence of clustering on both stellar mass and L Hα in the following subsection.
Splitting by both stellar mass and Hα luminosity
Within the star-forming population, higher mass galaxies tend to have higher star-formation rates (and therefore higher Hα luminosities), so trends in mass can manifest as apparent trends in star-formation rate, and vice-versa. Here, r 0 increases significantly at both high L Hα and high stellar masses, and it is hard to tell the extent to which mass and luminosity are each independently correlated with halo mass. Our large samples of HiZELS galaxies allow us to break this degeneracy, and study trends in stellar mass and L Hα luminosity independently. At z = 0.8, where our sample is largest, we split the stellar mass -L Hα plane into ∼ 500 overlapping subsamples, constructing and fitting two-point correlation functions for each. In Figure 5, we present a 2D plot of stellar mass versus L Hα . Each region is colour-coded by its r 0 value, obtained via a smoothed grid using x and y values of each subsample's mean stellar mass and star-formation rate, respectively. Note that these r 0 measurements are not independent, due to the overlapping samples. With around 100 galaxies per bin, there are approximately 30 independent subsamples. We find that clustering strength increases broadly monotonically with L Hα at all stellar masses. At high stellar masses M * ≥ 10 10 M , r 0 also increases with stellar mass, as has been found by many mass-selected clustering studies. At low stellar masses, the stellar mass-r 0 relationship breaks down, as had been seen in stellar mass at fixed L Hα (if anything, r 0 increases slightly as we probe to lower stellar mass at higher L Hα , where we are probing star-formation rates well above the main sequence). Next, we show projections of this plot for the z = 0.8 data, and for the smaller samples at z = 1.47 and z = 2.23. We divide our galaxies at each redshift slice into two stellar mass bins, and bin further by L Hα . We construct twopoint correlation functions and obtain correlation strengths for these subsamples. The results are shown in Figure 6. We find that the increase in clustering strength with Hα luminosity holds for both stellar mass bins. The trends of the two stellar mass bins are almost indistinguishable. Only the most extremely luminous galaxies at z = 0.8 (L Hα > 10 42.2 ) show any departure from this, and, as found by Sobral et al. (2016), HiZELS samples at these luminosities suffer from significant AGN contamination.
We also divide our galaxies at each redshift slice into two L Hα bins, and bin further by stellar mass. The results are shown in Figure 7. Given the size of the sample, our results are clearest at z = 0.8. Here, we find that at all stellar masses, the higher luminosity galaxies are more strongly clustered than low luminosity galaxies at the same stellar mass, but this difference is most significant at low stellar masses. The data at z = 0.8 (top panel of Figure 4) clearly shows that below stellar masses of M * ∼ 10 10 M , . r 0 in the stellar mass -L Hα plane at z = 0.8, constructed using ∼ 500 overlapping (non-independent) subsamples and plotted using a smoothed linear interpolation. We overplot the main sequence derived by Speagle et al. (2014) at this redshift as a solid line, with the dashed lines showing the standard deviation. Clustering strength increases broadly monotonically with L Hα at all stellar masses. At high stellar masses M * 2 × 10 10 M , r 0 increases with stellar mass. We also find large r 0 values for highly star-forming low stellar mass galaxies that are located well above the main sequence.
HiZELS galaxies have a fairly flat r 0 -M * relation. At these stellar masses, the higher luminosity subsample displays much stronger clustering than the lower luminosity subsample, with r 0 ∼ 6 − 7h −1 Mpc (M eff ∼ 10 13 M ), compared to r 0 ∼ 3 − 4h −1 Mpc (M eff ∼ 10 12.4 M ). There is even a slight increase in clustering strength towards low masses for the higher luminosity subsample. We find similar trends for our second largest sample, at z = 2.23.
Together, our results present clear evidence for a dependence of star-formation activity of low-mass galaxies on environment. For these galaxies, Hα luminosity is a better predictor of clustering strength than stellar mass. As mentioned in the Introduction, the key difference between this work and many studies of galaxy clustering that use massselected samples is the clean, L Hα -selected sample of starforming galaxies yielded by our survey. In order to satisfy the HiZELS Hα flux limit, low stellar mass galaxies must lie significantly above the main sequence. One physical interpretation of this result is that these galaxies are highly star-forming centrals, which will soon form more stellar mass to put them on the main stellar mass -halo mass relation. Alternatively, we could be observing an increasing contribution of starbursting satellite galaxies (or galaxies that are infalling on to a massive halo and will soon become satellites) at low stellar masses.
Comparison of star-forming galaxies to mass-selected samples
Here, we compare the clustering of our Hα-selected samples to mass-limited samples. log 10 L Hα /ergs −1 z = 2. 23 8. 5 < log 10 M * /M < 10 10 < log 10 M * /M < 11. 5 Figure 6. Clustering strength as a function of L Hα for HiZELS galaxies split into two stellar mass bins at each redshift. The calculated r 0 values of the two mass-binned samples are consistent at fixed mass, with the possible exception of the very highest luminosities at z = 0.8. This implies that the Hα luminosity is the physical property most strongly correlated with clustering strength for our HiZELS galaxies. 42. 47 < log 10 L Hα /ergs −1 < 42. 7 42. 7 < log 10 L Hα /ergs −1 < 43. 17 Figure 7. Clustering strength as a function of stellar mass for HiZELS galaxies split into two Hα luminosity bins at each redshift. Both high-and low-luminosity massive galaxies are more strongly clustered than their less massive counterparts. Higher Hα luminosity galaxies tend to be more strongly clustered than less luminous galaxies at fixed mass. This is clear for the two largest samples, at z = 0.8 and z = 2.23. The offset in r 0 between the two luminosity bins is particularly large at low stellar masses, suggesting that low-mass galaxies with high luminosities have environmentally triggered star formation.
clustering of mass-limited galaxy samples from the VIDEO survey at a very similar redshift to our z = 0.8 sample, at 0.75 < z < 1.00 with median redshift z = 0.88. 1 Their selection is based on an apparent AB magnitude limit K S < 23.5. Our observations probe slightly deeper, reaching down to K ∼ 25, but the majority of our sources also satisfy K < 23.5.
1 Note that in Hatfield et al. (2016), r 0 is not derived from a power-law fit as in this work. Instead, r 0 is defined as the radius at which the best-fitting spatial correlation function equals unity.
The important difference between our samples is the Hα flux limit of our sample. Whereas we are probing mainly the star-forming population, a substantial proportion of the Hatfield et al. (2016) sample will comprise less highly starforming and passive galaxies. We characterize the clustering of HiZELS emitters down to the same stellar mass limits as Hatfield et al. (2016), using no luminosity cuts other than the source selection criteria described in Section 2.1. The results, shown in the left-hand panel of Figure . Left: r 0 as a function of stellar mass lower limit, for HiZELS Hα-selected galaxies and mass-selected galaxies from Hatfield et al. (2016). At fixed stellar mass limit, the star-forming galaxies display significantly lower r 0 values, with the difference only decreasing at the highest stellar mass limits. Right: Comparison of whole-sample r 0 values at different redshifts. There are clear differences in derived r 0 due to sample selection. In general, samples of passive galaxies (red points) and mass-selected samples (purple points) tend to be more highly clustered than samples of star-forming galaxies at the same redshift (blue points).
are approximately half of the VIDEO mass-selected sample r 0 values, with this difference only decreasing at the highest stellar masses. This shows that, at fixed stellar mass, starforming galaxies are markedly less strongly clustered than the galaxy population as a whole. Note that for the lowest two stellar mass bins of Hatfield et al. (2016), the K S < 23.5 selection may mean that only the reddest (and most passive, thus often most clustered) galaxies are included in the analysis, possibly biasing the points upwards relative to a fully mass-selected sample.
We now compare the clustering of our large samples of star-forming galaxies at the three HiZELS redshifts, z = 0.8, z = 1.47, z = 2.23, to other clustering measurements in the literature, to see whether these stark differences between differently selected samples persist at other redshifts. The right-hand panel of Figure 8 shows the results. We find that samples of passive galaxies and mass-selected samples tend to be more highly clustered than samples of star-forming galaxies at the same redshift, to at least z ∼ 2.
Those results form a parallel story to that already presented here. While we have studied the clustering of starforming galaxies and shown that more highly star-forming galaxies are more strongly clustered than their less starforming counterparts at fixed stellar mass, we show here that passive galaxies are more strongly clustered than starforming galaxies at fixed mass. How do these two apparently contradictory results fit together? Sobral et al. (2011) show that, at fixed stellar mass for M * < 10 10.6 M , the mean starformation rate of HiZELS galaxies increases strongly with environmental overdensity (Σ c ) across almost the full range of overdensities probed (2 < Σ c < 30), which included field galaxies and small groups. This is consistent with the main part of our study: the clustering strength of the most highly star-forming galaxies is largest. Janowiecki et al. (2017) study the atomic hydrogen gas fraction of field and small group galaxies, finding that low-mass (M * ≤ 10 10.2 M ) galaxies in the centres of groups have gas fractions ∼ 0.3 dex higher than those in the field at fixed stellar mass. They conclude that the higher star-formation activity of these galaxies is driven by their higher gas availability. Sobral et al. (2011) also use the underlying photometric sample to estimate the star-forming fraction for HiZELS galaxies as function of overdensity. Here, the trends are different. The starforming fraction increases slowly in the range 2 < Σ c < 10, but displays a sharp fall above these densities, falling to below 15% in the richest clusters. This is entirely consistent with our results: the mass-selected samples of Hatfield et al. (2016) display higher clustering strengths because they are dominated by passive galaxies in richer environments, which are not detected by the HiZELS survey due to its Hα flux selection. This interpretation, driven by the exclusion of environmentally quenched satellites from our HiZELS samples, is in line with both the low satellite fractions found in C17, and the low M eff values for HiZELS galaxies in general.
The stellar mass-halo mass relation
The stellar mass to halo mass ratio (SHMR) is defined as the total stellar mass within a halo divided by the dark matter halo mass. It reflects the relative star formation and satellite galaxy accretion of a halo, compared to its dark matter accretion history, and is effectively a measure of the efficiency of the conversion of baryons into stars. The least massive dark matter haloes build stellar mass inefficiently due to supernova feedback, resulting in low M * /M halo fractions. Efficiency appears to increase towards higher halo mass, up to M halo ∼ 10 12 M . A consensus has emerged that haloes of this mass are most efficient at forming stars, with substantial decrease in efficiency above this halo mass (e.g. Behroozi et al. 2013;Moster et al. 2013), which is associated with AGN Moster et al. (2013), the lowest mass Hα emitters lie significantly below it, which indicates that these galaxies are living in more massive haloes than would be expected for central galaxies of their stellar masses.
feedback. Birrer et al. (2014) find that the reduced stellarto-halo mass ratio can be accounted for at high halo masses by the quenching of massive galaxies at around M * , the knee of the stellar mass function. There is little evidence for redshift evolution in the peak of the SHMR. Here, we review one approach to modelling the SHMR, and compare our measurements to predictions. Moster et al. (2013) follow Moster et al. (2010) in adopting a double power-law parametrization for the SMHR. The four free parameters are fitted using populations of dark matter haloes and galaxies at redshifts from z = 0 to z = 4, specifically dark matter halo populations drawn from the Millennium and Millennium-II Simulations (Springel et al. 2005;Boylan-Kolchin et al. 2009) and galaxy populations from Li & White (2009) at low redshifts and Pérez-González et al. (2008) and Santini et al. (2012) at high redshifts. At each redshift, Moster et al. (2013) initiate an SMHR with a given set of parameters, and use this to simulate the stellar masses of galaxies within the dark matter haloes they draw from the N-body simulation at the same redshift. They then compare the stellar masses of their simulated galaxies to the observed stellar mass function, and assign the modelled SMHR a likelihood. They thus optimize the parameters of the SMHR at each redshift. By including observational errors on high-redshift stellar masses, they are able to derive models that agree well with observed stellar mass functions. Behroozi et al. (2010) show (using another stellar masslimited approach) that there is little difference between the SHMRs at low halo masses (M halo < 10 12 M ) derived when considering the total stellar mass within the halo or just that of the central galaxy. Given that we argued in C17 that the HiZELS samples are dominated by central galaxies, we use the stellar mass of HiZELS galaxies as a proxy for total stellar mass in the halo. We then compare our estimates of dark matter halo mass for HiZELS galaxies to the predictions of Moster et al. (2013). We take the same samples of galaxies within large L Hα /L * Hα bins at each of the three redshifts, as in C17. We estimate average SED masses as in Section 2.2, and use the effective halo masses derived from HOD fitting (see Section 3.3) to place these samples on to the SHMR. The left-hand panel of Figure 9 shows that our data are in excellent agreement with the predictions of Moster et al. (2013). At all three redshifts, HiZELS galaxies occupy a region at the peak of the SMHR. They reside in haloes that are able to support maximum conversion of baryons into stellar mass.
Nevertheless, these global averages include galaxies spread over > 2 dex in stellar mass, so are not necessarily representative of all HiZELS galaxies. To investigate this, in the right-hand panel of Figure 9 we place mass-selected subsamples of our z = 0.8 data on to the same relation. When we calculate the SMHR from the mean stellar mass and derived effective halo mass for each subsample, samples of galaxies with M * > 10 10 M lie approximately on the Moster et al. (2013) relation. However, at low stellar masses, our samples lie significantly below this modelled relation. As discussed in Section 4.3, our low-mass galaxies reside in particularly high-mass haloes for central galaxies of their stellar mass. One possible interpretation of this is that it could be indicative of a substantial amount of stellar mass contained in galaxies that are undetectable by HiZELS within the same halo (i.e. our assumption that the halo's total stellar mass is broadly given by the HiZELS stellar mass is wrong). This points towards some of our low-mass galaxies being satellites. In that case, our low-mass galaxies would be highly star-forming satellites of a (more massive) passive central. However, this would go against the conclusion of the HOD modelling in C17 that the majority of HiZELS galaxies are centrals. Alternatively, we could be picking out starbursting low-mass centrals that will soon gain sufficient stellar mass to place them on to the main SHMR. Given only the current HiZELS observational data, it is difficult to distinguish between these scenarios. We will return to this issue in Section 5.5, where we compare against the EAGLE simulations.
COMPARING OUR RESULTS TO SIMULATIONS
Overview of the EAGLE simulation
Historically, cosmological hydrodynamical simulations have struggled to reproduce observed properties of galaxy populations simultaneously with the same success as semi-analytic models. Observed statistics of galaxy populations such as stellar mass functions, luminosity functions and the detailed properties of individual galaxies such as sizes, bulge/disc masses and star-formation histories were poorly matched (see Somerville & Davé 2015, for a review). This is partly an issue of resolution: to maintain the broadest view of galaxies within the large-scale dark matter structure of the Universe, key processes that determine the detailed evolutionary path of individual galaxies such as star formation and feedback are left unresolved. The latest generation of hydrodynamical simulations has made notable strides by attempting to improve the calibration of sub-grid models to observed properties of galaxy populations. The Virgo Consortium's Evolution and Assembly of GaLaxies and their Environments project, EAGLE, comprises a suite of ΛCDM simulations based on SPH code GADGET 3 (Springel et al. 2005). EAGLE represents a significant improvement on previous hydrodynamical simulations due to its simple implementation of energy feedback from both massive stars and AGN. Subgrid models for these processes are calibrated using two main relations at z = 0.1: the galaxy stellar mass function, and the galaxy-black hole mass relation. EAGLE's success lies in its reproduction of various other observed relations (e.g. galaxy specific starformation rate distributions, passive fractions and the Tully-Fisher relation; Schaye et al. 2015) that are not explicitly used in the calibration. Artale et al. (2017) also find good agreement between the clustering of blue galaxies in EAGLE and those in the GAMA survey, concluding that these simulated and observed galaxies with similar properties occupy dark matter haloes of similar masses.
A number of EAGLE simulations are publicly available (McAlpine et al. 2015). Here, we use version Ref-L100N1504, due to its large volume (box of side length 100Mpc, comoving) and particle number (7 billion). We select galaxies at z = 0.87, close to the z = 0.8 HiZELS redshift slice.
Halo environments of EAGLE galaxies
Rather than calculating halo mass via the two-point correlation function as we have done for HiZELS galaxies, we identify the halo masses of EAGLE galaxies directly. We use the total friends-of-friends (FOF) mass of the galaxy's halo, labelled as GroupMass in the EAGLE FOF table, as opposed to the subhalo mass. We identify central galaxies as those galaxies for which SubGroupNumber = 0, and satellite galaxies as galaxies with SubGroupNumber > 0. In Figure 10, we show the typical halo masses of subsamples of EAGLE central and satellite galaxies at z = 0.87. The stellar mass and star-formation rates used are those within a 30pkpc (proper, as opposed to comoving, kpc) aperture, taken from the EA-GLE Aperture table. We see that the halo masses of central galaxies are strongly correlated with their positions on the SFR-stellar mass plane, with high-stellar mass galaxies residing in massive dark matter haloes. We also see hints of higher halo masses for higher luminosity low-mass central galaxies at fixed stellar mass. We quantify this in more detail in Section 5.3. For satellite galaxies, halo masses are less strongly correlated with stellar mass or star-formation rate. This reflects the fact that much of a satellite's mass is built up at earlier times, when it is the central of its own subhalo, before this subhalo is accreted on to the larger halo.
Mass and star-formation rate dependencies of halo mass from EAGLE
In Section 4.3, we showed that at fixed stellar mass, more highly star-forming low-mass galaxies appear more strongly clustered than their less highly star-forming counterparts.
Here, we mimic these stellar mass and star-formation rate selections and quantify the average halo masses of EAGLE central galaxies binned in the same way. We convert EAGLE star-formation rates to rough Hα luminosities, for comparison with HiZELS, using the Kennicutt (1998) L Hα −SFR conversion given in Section 2.1 and assuming the same Chabrier (2003) IMF as used by EAGLE.
Our results are presented in Figure 11. We see a strong M * − M halo correlation at high stellar masses, which flattens at low stellar masses, just like we found for the HiZELS samples. At low stellar masses (M * 10 10 M ), average halo mass increases with star-formation rate at fixed stellar mass. At high stellar masses (M * 10 10 M ), average halo mass is roughly independent of star-formation rate for central galaxies. This is broadly consistent with our HiZELS observational results. However, there appears to be a lack of very highly star-forming, low-mass galaxies in EAGLE (cf. Figure 10). EAGLE galaxies do not reach the high luminosities of HiZELS galaxies, perhaps because of insufficiently bursty star formation in the simulations, or the inability to resolve bursts on small time-scales. There are well-known tensions between EAGLE star-formation rates and observations. The specific star-formation rates of EAGLE star-forming galaxies are 0.2 − 0.5 dex below those inferred from observations, across all redshifts (Furlong et al. 2015). Despite the offset in global star-formation rate density, applying the required 0.3 dex star-formation rate offset to all star-formation rates would break the agreement between simulated and observed stellar mass densities. Nevertheless, the broad trends of our observational results are supported by EAGLE: for low stel- . z = 0.87 galaxies from EAGLE, plotted on the stellar mass -star-formation rate plane using a 30kpc (proper) aperture, colour-coded by their group halo mass. The halo masses of central galaxies (left-hand panel) are strongly correlated with their positions on this plane, with high stellar mass galaxies residing in massive dark matter haloes. The satellite galaxies (middle panel) have greater variance in halo mass at fixed stellar mass, due to the formation of their stellar mass in a smaller halo, before accretion on to more massive haloes. We also show the positions of z = 0.8 HiZELS galaxies (not colour-coded by halo mass) on the same plane (right-hand panel). HiZELS star-formation rates tend to be slightly higher than those of EAGLE galaxies at low stellar masses. Figure 11. Halo mass as a function of stellar mass for EAGLE central galaxies at z = 0.87, using moving average bins of size 0.15 dex. The errors plotted are the standard error on the mean. We select by EAGLE star-formation rate within an aperture of 30kpc (proper), and convert to a rough L Hα using the Kennicutt (1998) conversion, with correction to a Chabrier IMF. At low stellar masses, the most highly star-forming galaxies lie in more massive haloes than galaxies of the same mass but lower starformation rates, in line with our HiZELS observations. Low-mass HiZELS galaxies tend to reside in higher mass haloes than even the most highly star-forming EAGLE galaxies. As discussed in Section 5.2, this could be related to the known 0.2 − 0.5 dex global offset between the EAGLE star-formation rate density and observational measurements. lar mass central galaxies, galaxy dark matter halo mass is not a simple function of stellar mass, but also depends on the galaxy's star-formation rate.
Physical interpretation using EAGLE
Here, we use EAGLE to investigate why our most highly star-forming HiZELS galaxies tend to reside in the most massive dark matter haloes. We study the average gas content, M gas , star-formation rate, SFR, and star-formation efficiency, SFE = SFR M gas (the inverse of the gas depletion timescale), as a function of halo mass and stellar mass. We include only galaxies with SFR > 0 in this analysis. Figure 12 shows our results. The log 10 M halo − log 10 M gas relation for central galaxies is linear, and independent of galaxy stellar mass. At all stellar masses, the most massive haloes supply the most gas to their centrals. The same relation is strikingly different for satellite galaxies: the average gas mass of a satellite galaxy appears broadly independent of its halo mass, but varies significantly with stellar mass. At fixed halo mass, more massive satellite galaxies have larger gas reservoirs. This is likely due to the gas content being established earlier, prior to accretion on to a more massive halo, when the satellite galaxy's gas mass would have correlated with the mass of its subhalo (using the mass of the EA-GLE subhalo places centrals and satellites on to the same sequence), which in turn correlates more closely with stellar mass. Wetzel et al. (2013) argue that satellite galaxies retain their cold gas reservoirs upon infall and continue to form stars on long time-scales. This is broadly supported by EAGLE, where the gas mass of satellites of fixed stellar mass varies little with halo mass. The role of gas stripping in these galaxies' evolution appears to be sub-dominant.
The star-formation efficiencies of central and satellite galaxies are also markedly different. SFE falls with increasing halo mass for central galaxies at all stellar masses, with a particularly steep decrease above M halo ∼ 10 12 M . Higher stellar mass centrals also have slightly higher star-formation efficiencies, particularly in the lowest mass haloes. Satel- Mean gas mass, star-formation efficiency and star-formation rate as a function of halo mass for satellite and central EAGLE galaxies at z = 0.8, with 1σ error contours. For central galaxies at all stellar masses, galaxy gas mass correlates tightly with host halo mass. Although star-formation efficiency decreases with increasing halo mass, mean star-formation rate increases with halo mass, for central galaxies in haloes with M halo < 10 12 M . Dependencies on stellar mass are weak by comparison. In contrast, for satellites, star-formation rate does not depend strongly on M halo , but more on M * .
lite galaxies display a weak increase in SFE with halo mass (∼ 1 dex over ∼ 3 dex in M halo ), independently of stellar mass, perhaps due to increased intracluster medium pressure in higher mass haloes (e.g. Bekki 2014). The bottom row of Figure 12 shows the combination of the gas content and star-formation efficiency: the mean star-formation rate as a function of halo mass. Below M halo ∼ 10 12 M , mean SFR increases with M halo for central galaxies of all stellar masses. This increase appears to be driven by gas content: gas cooling from the halo fuels star formation in central galaxies, with higher cooling rates in more massive haloes and little variation in star-formation efficiency. At fixed halo mass, the more massive galaxies have higher SFRs due to increasing efficiency of gas conversion. Above M halo ∼ 10 12 M , the SFR − M halo relation appears to flatten due to decreasing star-formation efficiency; there are also few star-forming galaxies at these high halo masses. Satellite galaxies display a very weak increase in SFR with halo mass at the lowest halo masses, and subsequent flattening at high halo masses. This appears to be driven by a combination of increasing star-formation efficiency and decreasing gas content with increasing halo mass. At fixed halo mass, more massive satellites are more highly star-forming due to their higher gas content.
EAGLE thus provides insights into the drivers of the trends we observe with HiZELS. Simulated low-mass, highly star-forming galaxies also reside in higher mass haloes than their less highly star-forming counterparts. EAGLE shows that these trends are likely driven by gas supply rather than increased star-formation efficiencies in high-mass haloes. One remaining tension is the paucity of very highly starforming galaxies in EAGLE compared to those observed. Those EAGLE galaxies that are highly star-forming tend to be satellites (see Figure 10). Given the difficulties in an autocorrelation analysis of distinguishing star-forming satellites of passive centrals from star-forming centrals given only a star-formation rate-selected sample, there are significant uncertainties in our satellite fraction determination discussed in C17. Nevertheless, the scarcity of highly star-forming centrals in EAGLE may well be due to star formation in the high redshift Universe being more bursty and stochastic than is simulated or recorded in the timestep-smoothed EAGLE output.
Insights into the SHMR from EAGLE
In Section 4.5, we placed our HiZELS samples on to the SHMR, considering the typical halo mass derived from clustering measurements for galaxies in different stellar mass bins. We found that mass-selected subsamples of HiZELS galaxies tend to lie below the SHMR at the lowest stellar masses. We suggested that this could be due to significant additional stellar mass within the same haloes, indicating that some of our low-mass galaxies are satellites of central galaxies which lie below the HiZELS Hα detection limits. Alternatively, these galaxies could be very highly star-forming centrals which will soon gain enough mass to place them on to the main SHMR. Here, we investigate these scenarios, to ascertain whether either star formation at HiZELS observed rates or unaccounted stellar mass within the same halo (as estimated using the EAGLE simulations) can account for the additional stellar mass needed.
We begin by calculating the increase in stellar mass required to move our HiZELS measurements diagonally on to the Moster et al. (2013) SHMR, assuming little change in halo mass. For moderate to high-mass galaxies (M * = 10 10 − 10 11 M , the SHMR offsets are very small, but we find higher offsets (factors of tens) for galaxies at lower stellar masses. The required growth factors are shown as a function of stellar mass in Figure 13.
Next, we use the average L Hα within each stellar mass bin to calculate a typical stellar mass increase over 1Gyr of star formation if either the current star-formation rate or the current specific star-formation rate is maintained.
Finally, we select a sample of galaxies in EAGLE with comparable SFRs to those observed by HiZELS to evalu-10 9 10 10 10 11 M * , initial /M 1 10 100 M * , final /M * , initial mass needed to join SHMR mass from 1Gyr const. SFR mass from 1Gyr const. sSFR mass from satellites in EAGLE mass from satellites and centrals in EAGLE Figure 13. The growth factor required, as a function of stellar mass, to bring the z = 0.8 HiZELS galaxies on to the SHMR (thick blue line). Closed circles use the SHMR relation from Moster et al. (2013), and open circles use the SHMR constructed using EAGLE. This indicates the approximate uncertainty on the SHMR itself. We model corrections to the mass of the HiZELS galaxy obtained under the assumption of 1Gyr star formation at the measured star-formation rate and specific star-formation rate. For comparison, the other lines show the simulated corrections to the mass contained in the dark matter host haloes of HiZELS galaxies using EAGLE. High-mass HiZELS galaxies already lie on the SHMR. Low-mass EAGLE galaxies (M * < 10 10 M ) with comparable star-formation rates reside in dark matter haloes with significant stellar mass contributions from companion galaxies. A correction from these places HiZELS galaxies on or above the main SMHR.
ate the mass contribution of other galaxies in the halo. We do this in two ways. The first selects only star-forming central galaxies. This is motivated by C17, which estimated low satellite fractions for these samples. The second allows our star-forming EAGLE comparison galaxies to be either centrals or satellites. For each EAGLE comparison sample, we identify other EAGLE galaxies within the same dark matter haloes, and calculate a stellar mass correction, the difference between the stellar mass in the detected star-forming galaxy and the total stellar mass in the halo. These correction factors are shown in Figure 13. Figure 13 shows that for the high-mass galaxies, which already lie on the SHMR, stellar mass is little affected by ∼ 1Gyr of star formation at either fixed SFR or fixed sSFR, and that similarly accounting for satellite galaxies makes little difference to the stellar mass of the haloes. At lower stellar masses, ongoing star formation at fixed SFR over ∼ 1Gyr time scales can produce a significant increase in stellar mass (up to a factor of a few), but falls far short of that required to bring the galaxies on to the SHMR. Likewise, 1Gyr of star formation at fixed sSFR or considering the contribution of satellite galaxies in the same halo, both appear insufficient. Instead, it appears likely that some contribution from cen-trals within the same halo is required if our samples are going to move on to the SHMR, indicating that a proportion of our low-mass star-forming galaxies may be satellites of centrals with lower SFRs. Otherwise, we are detecting low-mass central galaxies that lie significantly below the SHMR, and will remain so for more than a Gyr, even if they maintain their current high specific star-formation rates.
CONCLUSIONS
We have studied the clustering of intermediate redshift starforming galaxies and its dependence on star-formation rate and stellar mass. Our samples comprise Hα-selected galaxies predominantly on and above the star-forming main sequence at three redshifts, z = 0.8, 1.47 and 2.23. We summarize the key results here.
• At all three redshifts, we find clear evidence for a monotonic increase in clustering strength, r 0 , with stellar mass above M * ∼ 2 − 3 × 10 10 M . At lower stellar masses, where star-forming galaxies selected by HiZELS lie significantly above the main sequence, this relation flattens. The M * − r 0 relation is very different from the log 10 L Hα − r 0 relation studied in C17, which shows a significant and monotonic increase of r 0 with increasing Hα luminosity, with no flattening at the lowest luminosities.
• At fixed stellar mass, higher Hα luminosity subsamples are more strongly clustered than their less luminous counterparts. This is particularly pronounced at the lowest stellar masses (M * < 10 10 M ). We find consistent results when we mimic our L Hα cuts using the EAGLE simulations. We deduce that these highly star-forming low-mass galaxies are undergoing environmentally driven star formation. Investigating the cause of this using EAGLE reveals that our trends are likely driven by enhanced gas supply in small groups compared to the field.
• We compare our mass-binned clustering measurements of L Hα -selected galaxies to those obtained from mass-selected samples, and show that measurements of galaxy clustering are strongly dependent on the galaxy selection criteria. We find that HiZELS star-forming galaxies are less strongly clustered than mass-selected galaxies at fixed stellar mass. Compilations of literature measurements confirm that passive and mass-selected samples tend to be more strongly clustered than star-forming samples back to at least z ∼ 2.
Mass-selected samples seem to be picking up many more quenched satellites in massive haloes. We argue that our results are in line with average star-formation rates increasing towards group densities but decreasing at the highest cluster densities, where environmentally driven quenching plays a stronger role.
• We place HiZELS samples on the SHMR obtained empirically using mass-selected galaxy samples by Moster et al. (2013). We find that, on average, these highly star-forming galaxies lie at its peak, where baryon to stellar mass conversion is most efficient. Extending this to mass-binned subsamples, we show that high-mass HiZELS galaxies (M * > 10 10 M ) lie on the SHMR, but that at lower stellar masses, our samples lie below the relation.
• Finally, we consider the effect of ongoing star formation and show that current star-formation rates are insufficient to return low-mass galaxies to the SHMR. Using EAGLE, we find that if a proportion of these are satellites, typical stellar mass corrections from HiZELS-undetected galaxies within the same haloes can easily bring low-mass galaxies up on to the main SHMR.
In conclusion, we use the clustering of carefully selected starforming galaxies with well-defined redshift distributions to determine their typical halo masses. We present evidence for environmentally driven star formation in low-mass galaxies, some of which lie well above the main sequence. We use the EAGLE simulation to strengthen the physical interpretation, and show that it is likely that these star-formation rates are driven by increased gas content in galaxies residing in higher mass haloes.
Figure 1 .
1Distributions of SED-estimated stellar masses and dust-corrected Hα luminosities for the three samples of HiZELS galaxies, at z = 0.8, z = 1.47 and z = 2.23. The dashed lines show L * Hα at each redshift, derived bySobral et al. (2013a) andCochrane et al. (2017). Overplotted are indicative regions of the 'main sequence' at each redshift with 2σ contours, derived bySpeagle et al. (2014).
Figure 2
2Figure 2. SED-derived stellar mass versus observed K-band magnitude for SA22 galaxies, colour-coded by r − J colour. The black line shows the direct proportionality between K-band flux (restframe 1.2µm) and stellar mass (i.e. gradient fixed at −0.4). The stellar mass is clearly well correlated with K-band flux, but at fixed K-band magnitude, redder galaxies have higher SED-derived stellar masses, as would be expected. This colour dependence appears to drive the scatter in the relation and the deviation of the points from the straight line shown.
Figure 3 .
3Left: The two-point angular correlation function constructed for the whole sample at z = 0.8, fitted with a power-law (r 0 = 2.58 +0.16 −0.14 h −1 Mpc) and HOD model (M eff = 12.13 +0.10 −0.09 M ). Right: r 0 − M halo calibration from Cochrane et al. (2017).
σ log M : characterises the width of the transition to N sat |M = F s M M min α around M min . -α: the slope of the power-law for N sat |M in haloes with M > M min . In line with the literature, we fix α = 1. Tests allowing α to vary confirm that this is an appropriate choice. -F A, B c : normalization factors, in range [0,1]. -F s : the mean number of satellite galaxies per halo, at M = M min
Figure 4 .
4Top: clustering strength, r 0 , as a function of stellar mass.
Figure 4 .zz
4There is little change in r 0 with = 1.47, 42.16 < log 10 (L Hα /erg s −1 ) = 2.23, 42.47 < log 10 (L Hα /erg s −1 )
Figure 5
5Figure 5. r 0 in the stellar mass -L Hα plane at z = 0.8, constructed using ∼ 500 overlapping (non-independent) subsamples and plotted using a smoothed linear interpolation. We overplot the main sequence derived by Speagle et al. (2014) at this redshift as a solid line, with the dashed lines showing the standard deviation. Clustering strength increases broadly monotonically with L Hα at all stellar masses. At high stellar masses M * 2 × 10 10 M , r 0 increases with stellar mass. We also find large r 0 values for highly star-forming low stellar mass galaxies that are located well above the main sequence.
Hatfield et al. (2016) measure the
72 < log 10 L Hα /ergs −1 < 42. 0 42. 0 < log 10 L Hα /ergs −1 < 16 < log 10 L Hα /ergs −1 < 42. 5 42. 5 < log 10 L Hα /ergs −1 <
Figure 8
88, are strikingly different. At identical stellar mass limits, HiZELS r 0 values
Figure 9 .
9Left: The stellar mass -halo mass relation fromMoster et al. (2013), with whole HiZELS samples at each redshift overplotted. We use the effective halo mass estimated via the HOD fitting to the whole HiZELS samples at each redshift (see C17). Error bars on the y-axis represent the 1σ uncertainty derived from the MCMC posterior distribution, combined in quadrature with the typical errors on the stellar mass measurements (0.23, 0.24, and 0.26 dex for z = 0.8, 1.47 and 2.23 respectively). At all three redshifts, HiZELS galaxies occupy a region at the peak of the SMHR, where conversion of baryons into stellar mass is at a maximum. Right: The stellar mass -halo mass relation fromMoster et al. (2013) as a function of stellar mass, with mass-binned HiZELS data from the z = 0.8 sample within the range 41.72 < L Hα < 42.42 overplotted. While high-mass emitters lie on the relation predicted by
Figure 10
10Figure 10. z = 0.87 galaxies from EAGLE, plotted on the stellar mass -star-formation rate plane using a 30kpc (proper) aperture, colour-coded by their group halo mass. The halo masses of central galaxies (left-hand panel) are strongly correlated with their positions on this plane, with high stellar mass galaxies residing in massive dark matter haloes. The satellite galaxies (middle panel) have greater variance in halo mass at fixed stellar mass, due to the formation of their stellar mass in a smaller halo, before accretion on to more massive haloes. We also show the positions of z = 0.8 HiZELS galaxies (not colour-coded by halo mass) on the same plane (right-hand panel). HiZELS star-formation rates tend to be slightly higher than those of EAGLE galaxies at low stellar masses.
Figure 12 .
12Figure 12. Mean gas mass, star-formation efficiency and star-formation rate as a function of halo mass for satellite and central EAGLE galaxies at z = 0.8, with 1σ error contours. For central galaxies at all stellar masses, galaxy gas mass correlates tightly with host halo mass. Although star-formation efficiency decreases with increasing halo mass, mean star-formation rate increases with halo mass, for central galaxies in haloes with M halo < 10 12 M . Dependencies on stellar mass are weak by comparison. In contrast, for satellites, star-formation rate does not depend strongly on M halo , but more on M * .
present consistent results in their study of Hβ + [OII] and [OIII] emitters from HiZELS: clustering strength increases more significantly with emis-Table 1. Numbers and mean redshifts of Hα emitters identified by the HiZELS survey and selected for this analysisFieldz Hα emitters
# Hα emitters
NBJ (COSMOS & UDS)
0.845 ± 0.011
503
NBJ (SA22)
0.81 ± 0.011
2332
NBH (COSMOS & UDS)
1.47 ± 0.016
451
NBK (COSMOS & UDS)
2.23 ± 0.016
727
Table 2 .
2Clusteringstrength, r 0 , for stellar mass-binned samples
of HiZELS galaxies at z = 0.8, 1.47, and 2.23.
R.K.Cochrane et al.
MNRAS in press, 1-17(2018)
ACKNOWLEDGEMENTSThis work is based on observations obtained using the Wide Field CAMera (WFCAM) on the 3.8-m United Kingdom Infrared Telescope (UKIRT), as part of the High-redshift(Z) Emission Line Survey (HiZELS; U/CMP/3 and U/10B/07). It also relies on observations conducted with HAWK-I on the ESO Very Large Telescope (VLT), programme 086.7878.A, and observations obtained with Suprime-Cam on the Subaru Telescope (S10B-144S). We acknowledge the Virgo Consortium for making their simulation data available. The EA-GLE simulations were performed using the DiRAC-2 facility at Durham, managed by the ICC, and the PRACE facility Curie based in France at TGCC, CEA, Bruyères-le-Châtel.We thank the anonymous reviewer for their detailed suggestions. We are grateful to Steven Murray for making the HALOMOD and HMF PYTHON packages available and for guidance on their use. We also thank Stuart McAlpine for help with the EAGLE simulations.RKC acknowledges funding from an STFC studentship. PNB is grateful for support from STFC via grant ST/M001229/1. DS acknowledges financial support from the Netherlands Organisation for Scientific research (NWO) through a Veni fellowship and from Lancaster University through an Early Career Internal Grant A100679. IRS acknowledges support from STFC (ST/P000541/1), the ERC Advanced Investigator programme DUSTYGAL 321334, and a Royal Society/Wolfson Merit Award.This paper has been typeset from a T E X/L A T E X file prepared by the author.
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| []
|
[
"Distance-Aware Occlusion Detection with Focused Attention",
"Distance-Aware Occlusion Detection with Focused Attention"
]
| [
"Journal Of L A T E X Class ",
"Files "
]
| []
| []
| For humans, understanding the relationships between objects using visual signals is intuitive. For artificial intelligence, however, this task remains challenging. Researchers have made significant progress studying semantic relationship detection, such as human-object interaction detection and visual relationship detection. We take the study of visual relationships a step further from semantic to geometric. In specific, we predict relative occlusion and relative distance relationships. However, detecting these relationships from a single image is challenging. Enforcing focused attention to task-specific regions plays a critical role in successfully detecting these relationships. In this work, (1) we propose a novel three-decoder architecture as the infrastructure for focused attention; 2) we use the generalized intersection box prediction task to effectively guide our model to focus on occlusion-specific regions; 3) our model achieves a new state-of-the-art performance on distance-aware relationship detection. Specifically, our model increases the distance F1-score from 33.8% to 38.6% and boosts the occlusion F1-score from 34.4% to 41.2%. Our code is publicly available.Index Terms-Focused attention, object pair detection, relative distance detection, relative occlusion detection, transformer model, visualizations of attention weights. | 10.1109/tip.2022.3197984 | [
"https://export.arxiv.org/pdf/2208.11122v1.pdf"
]
| 251,741,147 | 2208.11122 | 8908a10f6ae751b536fb6e8497e369445d24659d |
Distance-Aware Occlusion Detection with Focused Attention
AUGUST 2015 1
Journal Of L A T E X Class
Files
Distance-Aware Occlusion Detection with Focused Attention
148AUGUST 2015 1Index Terms-Focused attentionobject pair detectionrela- tive distance detectionrelative occlusion detectiontransformer modelvisualizations of attention weights
For humans, understanding the relationships between objects using visual signals is intuitive. For artificial intelligence, however, this task remains challenging. Researchers have made significant progress studying semantic relationship detection, such as human-object interaction detection and visual relationship detection. We take the study of visual relationships a step further from semantic to geometric. In specific, we predict relative occlusion and relative distance relationships. However, detecting these relationships from a single image is challenging. Enforcing focused attention to task-specific regions plays a critical role in successfully detecting these relationships. In this work, (1) we propose a novel three-decoder architecture as the infrastructure for focused attention; 2) we use the generalized intersection box prediction task to effectively guide our model to focus on occlusion-specific regions; 3) our model achieves a new state-of-the-art performance on distance-aware relationship detection. Specifically, our model increases the distance F1-score from 33.8% to 38.6% and boosts the occlusion F1-score from 34.4% to 41.2%. Our code is publicly available.Index Terms-Focused attention, object pair detection, relative distance detection, relative occlusion detection, transformer model, visualizations of attention weights.
I. INTRODUCTION
V Isual object detection has shown exciting progress [1] [2] in the last two decades thanks to the representation learning power of tailored deep neural network architectures in 2D [3] [4] and 3D [5] [6] settings. The community is now actively exploring the detection of higher-order semantic entities like human-object interactions (HOI) [7] [8].
In this paper, we go one step further from semantic to geometric. Specifically, we study two ubiquitous and essential relationships: relative distance and relative occlusion from the viewpoint.
Detecting these relationships from a 2D image is important because it will benefit other computer vision tasks, like HOI detection-the task of detecting human-object pairs and the relationships within each pair of them. Specifically, in HOI detection, models that explicitly possess the ability to understand The source code is available at https://github.com/Yang-Li-2000/Distance-Aware-Occlusion-Detection-with-Focused-Attention.git. This paper has supplementary demo videos available at https://youtu.be/gsfCxWO0xws and https://youtu.be/PfRZGlJXGOA provided by the authors. Understanding relative occlusion and relative distance may help HOI models rule out some unreasonable predictions such as (a) "a woman is holding an umbrella" when her hand is not occluding the handle of the umbrella at all, and (b) "a man (in the red box) is riding a skateboard (in the orange box)" that is much further to the viewpoint than the man. relative occlusion and relative depth may more easily rule out some unreasonable predictions. For example, in Fig. 1 (a), knowing that a woman's hand is not occluding the umbrella at all may help rule out "a woman is holding an umbrella." Additionally, in Fig. 1 (b), knowing that the skateboard in the orange box is much further than the man in the red box could help suppress "a man in the red box is riding a skateboard in the orange box."
Many other computer vision tasks, including embodied reference understanding [9] and scene de-occlusion [10], will also benefit from detecting relative distance and occlusion relationships. For example, in embodied reference understanding [9], if a person is referring to an object by its relative position to another object (e.g., behind and in front of ), being able to detect relative distance from the viewpoint can help the model understand which object that person is referring to. In scene de-occlusion, determining the relative occlusion relationship is the foundation of ordering recovery and subsequent deocclusion of the scene [10].
Determining relative distance relationships requires understanding the scene geometrically and deciding whether one object is closer or both are at the same distance.
Relative occlusion is more complicated. Its most straightforward form is object A occludes object B when object B does not occlude object A. For example, in Fig. 2 (a), the woman's hair, hip, and right leg occlude the man, but the man does not occlude the woman. Another form of relative occlusion is no occlusion. It can happen even when the bounding boxes of two objects overlap. In Fig. 2 (b), for example, a closer look at the space between vegetables and the knife helps conclude that no occlusion exists between them. A more complicated form of occlusion is mutual occlusion. To detect mutual occlusion, a arXiv:2208.11122v1 [cs.CV] 23 Aug 2022 person needs to refrain from concluding one object occludes another object and keep locating additional occlusion sites. For example, in Fig. 2 (c), the woman's arm occludes the man's wrist, but the relative occlusion is not the woman occludes the man. After noticing that the woman's arm occludes the man, additional attention needs to be placed near their legs to identify that the man's leg also occludes the woman's leg.
Successful detection of relative occlusion relationships between a pair of objects requires focusing attention on visual features from task-relevant regions and refraining from being distracted by irrelevant visual features. For example, when determining the relative occlusion relationship between the woman and the man in Fig. 2 (a), the woman's right leg and hair are more task-relevant than the window. Paying more attention to the woman's right leg, hip, and hair is beneficial to occlusion relationship detection. In contrast, detecting occlusion relationships between these two people by attending to the window will be futile. Attention, as a selection mechanism, enables us to give higher privileges to taskrelevant information and filter out distracting information [11]. Features from the window act as distractions and increase the complexity of identifying the relative occlusion relationship between the two people. Consequently, focused attention to task-relevant regions is critical for correctly detecting relative occlusion relationships.
Detecting the relative distance relationship similarly requires focused attention to task-relevant regions. However, different tasks may require focused attention to different regions. For instance, detecting relative distance requires attention largely to the background to gain a geometric understanding of the scene, while detecting object pairs requires focused attention primarily to both objects to accurately locate their positions and recognize their categories. Existing artificial intelligence models, as argued by authors of [12], perform unsatisfactorily when detecting relative distance and relative occlusion relationships.
Therefore, to design a model that performs well in detecting both relative occlusion and relative distance relationships, we carefully consider the importance of focused attention. Given that detecting relative occlusion and relative distance requires attention to different regions (mainly object parts v.s. objects and backgrounds), we use two separate decoders for occlusion and distance, respectively, to allow each decoder to focus on features relevant to its task without being distracted by features solely relevant to the other task. Additionally, to protect our relationship decoders from distractions from object pair detection, we use an extra decoder to propose object pairs before detecting relative relationships.
The above considerations motivated our three-decoder model architecture in Fig. 3. It prevents distractions between different tasks and functions as the infrastructure for focused attention. However, the architecture alone is not sufficient for accomplishing focused attention.
To achieve focused attention in our occlusion decoder, we use an extra task: predicting the bounding box for the region shared by (mostly when occlusions exist) or between two objects (mostly when no occlusion exists). We call this task the generalized intersection box prediction (GIT) (Fig. 4).
Our GIT, a novel way to guide the model to focus on taskrelevant regions, results in a more interpretable and robust system. Our experiments and attention weight visualizations demonstrate our model's ability to focus on task-specific regions when detecting relative occlusion relationships ( Fig. 9) and the concomitant improvements in correctness ( Fig. 10 and Table VII).
To recapitulate, 1) we propose to use the multi-decoder architecture as the infrastructure for focused attention; 2) our novel GIT effectively guides our model to focus on occlusionspecific regions; 3) our model achieves a new state-of-the-art performance.
II. PROBLEM FORMULATION
We study the detection of relative distance and relative occlusion relationships from the viewpoint. The input is an RGB image I. The required outputs are the bounding box of object A (b A ), the bounding box of object B (b B ), the relative distance relationship (d), and the relative occlusion relationship (o) between A and B.
A. Relative Distance
Relative distance is from the viewpoint and is object-centric [12]. "From the viewpoint" means the distance of an object is the length of the 3D line segment from the optical center of the input image to the object. At the same time, "objectcentric" means each object is assigned with only one distance, which is different from the "pixel-wise depth" in monocular depth estimation where each pixel has a distance [12].
There are four types of relative distance relationships: A is closer than B, B is closer than A, same distance, and not sure. The first two are straightforward: when the majority of human annotators in [12] believe one object is closer than the other, the relative distance is either A is closer than B or B is closer than A.
For "same distance", there is not a hard threshold. When the majority of human annotators in [12] perceive that a pair of objects are at the same distance, the relative distance relationship is "same distance". Similarly, when the majority of the annotators are not sure about the relative distance relationship for a pair of objects, the relationship is "not sure."
B. Relative Occlusion
The relative occlusion relationship is also from the viewpoint. Occlusion "from the viewpoint" means looking from the same direction as the camera.
There are four types of relative occlusion relationships: A occludes B, B occludes A, no occlusion, and mutual occlusion. The last one, mutual occlusion, means some parts of A occlude B while some parts of B also occlude A.
III. RELATED WORK
A. Models for Relative Occlusion and Distance Detection
To the best of our knowledge, no deep learning model was designed specifically to explicitly detect relative occlusion and relative distance relationships at the same time, except multilayer perceptrons (MLPs) in [12].
Models that detect relative occlusion relationships do exist [10] [13]. For example, in [10], the model can determine relative occlusion relationships and build an occlusion graph, but it does not support mutual occlusion.
Models that detect relationships between objects (visual relationship detection, or VRD) also exist [14][15] [16] [17]. Unlike relative distance and relative occlusion relationship, Su et al. [12] maintain that nearly all relationships are 2D in current VRD datasets and do not address relative occlusion and relative distance relationships using the viewpoint as reference.
To predict relative occlusion and distance relationships, Su et al. [12] designed several MLP models and modified multiple state-of-the-art VRD models to perform relative occlusion and relative distance detection. In [12], models designed for VRD did not achieve significantly better performance than the best MLP model when detecting relative occlusion and relative distance relationships.
In another very similar task-HOI detection-transformer models have been modified and applied by many works. Focused attention in transformer models for HOI detection has also been studied extensively in recent years.
B. Transformers for HOI detection
HOI detection resembles relative occlusion and relative distance detection, except that relationships in HOI detection are mostly semantic. In specific, while require outputs for relative distance and occlusion relationship are
(b a , b b , d, o) (as in Section II), those for HOI detection are (b h , b o , r, c o ). Here, b h , b o ,
r, and c o represents the bounding box of human, the bounding box of an object, the relationship between the human and the object, and the category of the object, respectively.
Both tasks require the detection of objects and relationships. In specific, HOI detection requires the prediction of the human, the object, and the interactions between the human-object pair. Relative occlusion and distance relationship detection requires the prediction of object A, object B, and relationships between the object pair.
The major difference lies in the relationships. In HOI, the relationships between humans and objects are mostly semantic (e.g. eat, cut, and hold). In relative occlusion and relative distance detection, however, the relationships are geometric (e.g. occludes and being closer). Additionally, semantic relationships in HOI detection are highly diverse and usually involve dozens or hundreds of relationship categories.
Recently, many one-stage transformer-based methods have been proposed to perform HOI detection [18] [19][20] [21]. These architectures first use a backbone to extract image features. The backbone is usually a convolutional neural network (CNN). Then these architectures use transformer encoders to extract global features from input images. After that, these architectures utilize transformer decoders and MLPs to produce the final predictions of bounding boxes, object classes, and relationship classes.
C. Focused Attention in Transformers
Sometimes, transformer models can naturally exhibit intense attention to task-relevant information. In object detection, decoders in DETR [22] can exhibit focused attention to the extremities of objects. In HOI detection, decoders in HOITR [18] can have focused attention on "the discriminative part" of object pairs.
More often, however, transformer models attend heavily to a huge amount of irrelevant information. As illustrated in Fig. 9, transformers decoder allocate a significant amount of attention to many regions that are irrelevant to relative occlusion even when the only task that decoder was asked to do was detecting relative occlusion relationship.
To let transformer models have more focused attention, many attempts have been made. Correia, Niculae, and Martins [23] let information with nearly zero weight have exactly zero weight. Zhao et al. [24] modified the self-attention module to explicitly select the elements that receive the highest attention scores and explicitly let their transformer model focus on these elements. Zhu et al. [25] designed their "deformable attention module," allowing their transformer model to have focused attention on a small number of pixels near a reference point.
It can be insufficient, however, to just focus on elements with the highest attention scores. In some complex tasks composed of multiple sub-tasks, relevant information to one sub-task can be distracting to another sub-task. Task-specific attention, in this case, is difficult to achieve due to the distinctions between different sub-tasks [19]. For example, in HOI detection, many locations within or near a human-object pair can be highly relevant for detecting the human and the object, but regions relevant to interaction detection can vary a lot based on the category of a specific interaction [18]. In other words, some information that is relevant to human-object pair detection can at the same time be distracting to interaction detection and vice versa.
To allow focused attention for each sub-task, researchers used separate decoders for different sub-tasks and connect them in parallel [21] or serial [19] manners. In specific, Chen et al. [21] proposed to use a parallel connection of two separate transformer decoders for two sub-tasks, and introduced an attention module between each layer of the two separate decoders to aggregate relevant information from one to the other. Additionally, Zhang et al. [26] proposed to use a series connection of separate transformer decoders for different sub-tasks, allowing each decoder to focus on only one task and attending to information that is relevant to that one task.
In the following section, we will introduce our method for achieving focused attention in the occlusion decoder. Specifically, we use a combination of series and parallel connections of transformer decoders to avoid distractions from different sub-tasks and use a generalized intersection box prediction task to further guide our occlusion decoders to attend to taskrelevant regions.
IV. METHOD
In this section, we describe the model architecture, matching strategy, and loss computation. Our model architecture has four components: feature extractor, object pair decoder, relationship decoders, MLP heads for classification and regression.
A. Overview
Our feature extractor utilizes a CNN backbone and a transformer encoder to extract context-aware visual features from input RGB images. Extracted features are fed into our object pair decoder for bounding box and class label predictions. Taking the outputs of our object pair decoder as queries, our occlusion and distance transformer decoders predict relative occlusion and relative distance relationships, respectively. Our occlusion decoder also predicts generalized intersection boxes. An illustration of our proposed model architecture is in Fig. 3.
We use transformer because it models long-range dependencies, which is critical for our task. Specifically, the transformer has a global receptive field, which helps detect the relative distance and occlusion relationships when two objects are very far away from each other. Additionally, when two objects are not very far from each other, the transformer's global receptive field will also help detect relative distance relationships because it allows a more comprehensive consideration of the spatial configurations of the scenes. CNNs, in contrast, according to Luo et al. [27], have a relatively small effective receptive field in which an input pixel can have a "nonnegligible impact" on the output. Therefore, we choose the transformer model instead of CNNs for its greater capacity to capture information from a larger area.
B. Feature Extractor
Our visual feature extractor consists of a CNN backbone and a transformer encoder. It models long-range dependencies and extracts context-aware visual features for downstream tasks.
1) CNN Backbone: Given an RGB image I of shape (3, H, W ), a CNN backbone (e.g., ResNet-101 [28]) produces a feature map F 0 of shape (C, h, w), where C is the number of output channels of CNN backbone, h = H 32 , and w = W 32 , where · is the ceiling function. We reduce the dimension of the feature map from C to d using a 1 × 1 convolutional layer with stride 1, producing reduced feature of shape (d, h, w). We then flatten the reduced feature and generate flattened feature F of shape (hw, d).
2) Transformer Encoder: Our transformer encoder takes the flattened feature F as input. It is composed of N l transformer encoder blocks. Each block is composed of a multihead self-attention layer and a feed-forward layer. Inside each self-attention layer, there are N h heads. The output A of each self-attention layer is formed by the concatenation of outputs A i 's from all N l heads.
Before feeding the flattened feature F into our transformer encoder, however, we generate position encoding [29] [30] PE to facilitate the use of relative position information in downstream modules. The shape of PE is (hw, d), where d is the hidden dimension of our transformer encoder.
After obtaining PE is (hw, d), we feed F and PE into the first transformer encoder block.
In each block, we denote the input feature as F in . The query Q, key K, and value V of each transformer encoder block are
(Q, K, V) = (F in + PE, F in + PE, F in )(1)
For each head i = 1, 2, · · · , N h inside a self-attention layer, the attention weights
W Q i , W K i and W V i all have shape (d, d N h ). For each head, the query Q i , key K i , and value V i are (Q i , K i , V i ) = (QW Q i , KW K i , VW V i )(2)
Each head produces an output A i
A i = Softmax Q i K T i √ d V i(3)
The outputs of all heads in a self-attention layer are concatenated to form A
A = Concat(A 1 , A 2 , · · · , A N h )(4)
After obtaining the concatenated A, the final output F attn of a self-attention layer is generated using a linear layer with dropout and a residual connection with input feature F in
F attn = Linear(Dropout(A) + F in ) ∈ R hw×d(5)
After obtaining F attn from the self-attention layer inside each transformer encoder block, F attn is fed into the feed-forward layer (FFN), which is composed of two fullyconnected linear layers with dropout and ReLU activation. FFN is applied to each feature vector in hw positions identically and individually and outputs a total feature F out of the same shape (hw, d), which will serve as input to the selfattention layer in the next next block.
Finally, the output F out of the last transformer encoder block is denoted as global memory M of shape (hw, d).
C. Object Pair Decoder
Our object pair decoder detects object pairs. It takes M, PE, and object pair queries QE as inputs. It is composed of N pair transformer decoder blocks. Each transformer decoder block is composed of a multi-head self-attention module and a multi-head cross-attention module followed by a feed-forward network. The attention mechanism is the same as that in encoder layers, so we only specify our choice of Q, K and V in the self-attention and cross-attention layers. Fig. 3. Model Architecture. Our model is comprised of a feature extractor, an object pair decoder, two relationship decoders, and multiple MLP heads. The features extractor uses a CNN backbone and a transformer encoder to extract global visual features M. Using M, the object pair decoder produces object pair embeddings F OP . Using F OP as queries, two relationship decoders parse relative occlusion and relative distance relationships, respectively. The outputs of relationship decoders are F d and Fo. Finally, taking the output embeddings of three decoders, MLP heads produce the final predictions of object boxes, object classes, relative distances, relative occlusions, and generalized intersection boxes.
To enable proposing a fixed number of object pairs, we choose a hyper-parameter N q as the number of object pairs queries to be outputted by the decoder, and we generate parameterized queries with N q learnable embeddings. 1) Self-attention Module: For k = 1, 2...N pair , we denote the input of the k-th self-attention module as F k in and the output of the k-th FFN as F k out . Both of them have shape (N q , d).
F k in = F k−1 out + QE (6)
where QE is a set of learnable parameters of shape (N pair ,d).
We set F 0 out = 0 for consistency of notation below. In the k-th self-attention module (k = 1, · · · , N pair ), the query Q, key K, and value V are
(Q, K, V) = (F k in , F k in , F k−1 out )(7)
2) Cross-attention Module: The k-th self-attention module outputs a feature F k attn with shape (N q , d) to be fed into the k-th cross-attention module. Together with the global memory M from the transformer encoder and the fixed positional encoding PE, we let
(Q, K, V) = (F k attn , M + PE, M)(8)
On top of the cross-attention layer, the FFN layer is the same as that in the encoder architecture. The k-th FNN layer outputs F k out , which is also the overall output embedding of the k-th decoder block. Finally, we stack F k out to obtain object pair output embeddings F op of shape (N pair , N q , d)
F op = Stack(F k out , k = 1, · · · , N pair )(9)
D. Relationship Decoders
The relationship decoders include the relative occlusion decoder and relative distance decoder. Both decoders share the same structure with the object pair decoder. That is, each decoder block contains self-attention, cross-attention, and feed-forward modules. For the self-attention module and the cross-attention module, as downstream decoders, they use F Npair out , the output of the last object pair decoder layer as queries. In particular, we choose
(Q, K, V) = (F Npair out , M + PE, M)(10)
The outputs of the relationship decoders are distance output embeddings F d and occlusion output embeddings F o . Both of them have shape (N q , d).
E. MLP Heads
The output embeddings F op , F d , and F o are fed into separate classification MLP heads and box regression MLP heads to produce the final predictions of instances in the input image. All MLP heads have the same structure except the number of nodes in the last layer, where each classification MLP head produces a probability vector via softmax over each class, while each regression MLP head produces a four-vector (c x , c y , w, h) ∈ [0, 1] 4 via sigmoid function. Here (c x , c y ) is the relative coordinate of the center of box and (w, h) denotes its width and height (see Table I).
Our model treats different types of relative distance relationships as different classes so there is not an explicit threshold between "same distance" and one object being closer than the other. In other words, our model predicts two objects are at the "same distance" when the logit for the "same distance" class is the greatest among those for all four.
F. Generalized Intersection Box Prediction Task
With GIT, our occlusion decoder produces an additional generalized intersection box for each pair of objects. First we denote the bounding box of objects A and B as
b A = [x A , X A ] × [y A , Y A ] b B = [x B , X B ] × [y B , Y B ](11)
where × denotes the Cartesian product,
(x i , y i ) is the coordi- nates of bottom left corner of b i , and (X i , Y i ) is the top rightb ∩ = [x ∩ , X ∩ ] × [y ∩ , Y ∩ ](12)
where x ∩ and X ∩ are the second and third smallest elements in {x A , X A , x B , X B } and y ∩ and Y ∩ are the second and third smallest elements in {y A , Y A , y B , Y B }.
As shown in Fig. 4 (a), the generalized intersection box is the overlapping region of the bounding boxes of two apes, and in (b), the bounding box of the tea table is contained in the bounding box of the sofa, so the overlapping region is the bounding box of the tea table itself. In (c) and (d), there is no intersection between object bounding boxes: the red car at the upper left corner and the wheel of the blue car at the lower right corner in (c), and two toys on the grassland in (d).
In these cases, we take the middle box that shares sides (or extended sides) with the bounding boxes of A and B. From a general perspective, the generalized intersection box is an important link between A and B, regardless of whether the bounding boxes of A and B actually intersect.
G. Matching Strategy
Since each image contains different numbers of ground truth instances, which is usually smaller than the number of predictions generated by the model N q (equals 100 in our implementation), we need to find a strategy to assign each ground truth to a prediction. Since each of the training images contains only labels for one pair of objects, whereas many other instances exist but are unlabelled, most of the predictions concerning the unlabelled instances should not be trained. Hence it is important that we only match the best prediction with the ground truth labels while viewing other predictions as background. We adapted the matching strategy in [18] as follows. For each training image, we assign the ground-truth targets, denoted as {g 1 , g 2 , · · · , g Nt }, where N t is the number of targets, to predictions {p 1 , p 2 , · · · , p Nq } using bipartite matching. That is, we seek for a one-to-one function σ : {1, 2, · · · , N t } → {1, 2, · · · , N q }, so that g i is matched to p σ(i) . Then we will minimize the cost among all possible matching functions σ ∈ S
σ 0 = arg min σ∈S Nt i=1 C(g i , p σ(i) )(13)
In order to get the most efficient training, we assign targets in the way that minimizes the matching cost C using the Hungarian algorithm [31]. The computation of the matching cost C(g, p) for a certain matched pair of target g and prediction p is as follows. C(g, p) is a weighted sum of classification cost C c (g c , p c ) and regression cost C r (g r , p r ):
C(g, p) = β c C c (g, p) + β r C r (g, p)(14)
Since the classification outputs are the confidence of each potential class (of objects and relationship predicates), C c measures the closeness of the model's confidence prediction of the ground-truth class to 1, that is:
C c (g i , p i ) = 1 − p i [k](15)
where k is the ground-truth class.
Using hyper-parameters α A , α B , α d and α o , where d and o refer to distance and occlusion classes respectively. We compute the total classification cost as a normalized weighted sum of C c (g i , p i ) for i ∈ {A, B, d, o}:
C c (g, p) = i α i C c (p i , g i ) i α i(16)
The regression cost of each predicted box is a weighted sum of l 1 cost and generalized IoU (GIoU) cost. For each j ∈ {b A , b B , int}, where int is the generalized intersection box, we let C r (g j , p j ) = α l1 · g j − p j 1 + α giou · GIoU(g j , p j ) (17) and if model predicts generalized intersection box, let
C r (g, p) = 1 3 j C r (g j , p j ), j ∈ {b A , b B , int}(18)
or if the generalized intersection box is not predicted, let
C r (g, p) = 1 2 j C r (g j , p j ), j ∈ {b A , b B }(19)
The generalized intersection box is always predicted when using GIT and always not predicted when not using GIT. One exception is the ablation study in Section V-E3, in which our model only predicts the generalized intersection box when two bounding boxes intersect with each other.
H. Loss Computation
The training loss is based on the result of matching, and its computation is slightly different from the cost function between matched ground truth-prediction pairs. Apart from the given classes of foreground objects in an image, we consider all objects not in the ground truth targets as a single class O bg called background object. The background object is encoded as a one-hot vector e Nc+1 without a bounding box, where N c is the number of instance classes.
1) Classification loss: The classification loss for objects and relationships are Negative Log Likelihood (NLL) loss. For i ∈ {A, B, d, o} and matched pairs (g, p), we use weight parameters α i :
L c (g i , p i ) = −α i · log(p i [k])(20)
where k is the ground truth class. For i ∈ {A, B} and unmatched prediction p, we use a small weight parameter α eos to control the background loss
L c (O bg , p i ) = −α eos · log(p i [N c + 1])(21)
2) Regression loss: The regression loss are only applied to matched pairs (g, p), because background object does not have a bounding box to be compared with. For j ∈ {A, B, int}, we use the same formula as Eq. (17):
L r (g j , p j ) = α l1 · g j − p j 1 + α giou · GIoU(g j , p j ) (22)
3) Total loss: Finally, the total loss is again weighted by β c and β r :
L total =β c matched (g,p) L c (g, p) + unmatched p L c (O bg , p) + β r matched (g,p) L r (g, p)(23)
I. Inference
To make a final inference of instances based on the model output, we first perform the non-maximum suppression (NMS) to filter out duplicate predictions. In particular, we first sort the list of pairs in decreasing order in overall confidence measured by the product of classification confidence of distance and occlusion:
Conf(p) = p d ∞ · p o ∞(24)
For two predictions p 1 and p 2 , we regard them as duplicate predictions if they satisfy the following similarity criterion:
IoU(p 1 b A , p 2 b A ) > 0.7 and IoU(p 1 b B , p 2 b B ) > 0.7(25)
and if the predicted object categories and relative relationships in p 1 and p 2 match. We will only keep the prediction with the highest overall confidence Conf(p) when there are duplications. This duplication removal strategy does not consider two predictions, "A is closer than B" and "B is further than A," as duplications because their relationships are different (closer v.s. further). Additionally, unless the bounding boxes of both objects significantly overlap, object A in the former prediction will not have greater than 0.7 IoU with object A in the latter one, not meeting the IoU threshed for being considered as a pair of duplications. Furthermore, unless both objects have the same predicted category, they will not meet the object category requirement for being duplications. Therefore, most of the time, our duplication removal strategy will not consider predictions such as "A is closer than B" and "B is further than A" for the same pair of objects as duplications. Our model is required to predict both of them because our ground truth contains both.
After filtering out duplicate predictions, we get a list of predictions as our result of inference: {p 1 , p 2 , · · · , p N }.
V. EXPERIMENTS
We conduct experiments to evaluate the performance of our model in detecting relative occlusion and relative distance relationships. We report F1-score (F1), precision (p), and recall (r); and discuss model performance boost by referring to F1-scores. To demonstrate the performance of our method, we also provide extensive qualitative results on images from the 2.5VRD validation and test set and video frames from in-the-wild videos. We also study the effects of GIT on occlusion decoder attentions through ablation experiments and provide qualitative comparisons. Additionally, we investigate the benefits of predicting the generalized intersection when no intersection exists and using separate decoders for different sub-tasks.
A. Dataset
The 2.5VRD dataset [12] contains annotated images for the within-image relative occlusion and relative distance detection task and image pairs for the across-image relative distance detection task. We conduct experiments using the within-image portion of the dataset. It contains 105660, 1196, and 3987 images in the training, validation, and test set, respectively. Annotations for the training set are very sparse. Only one pair of objects is annotated for each training image. Annotations for the validation and test sets, in contrast, are exhaustive.
B. Evaluation Metric
During the evaluation, we leak the number of annotated objects to our model and require our model to accurately detect object pairs and correctly predict corresponding relationships.
1) Leaking the Number of Objects: Using the same evaluation metric used in [12], we leak n, the number of annotated objects in an image, to our model. Knowing how many objects are in the annotations, our model then outputs at most n(n-1) predictions for evaluation. However, the maximum number of predictions our model can produce is limited by N q , and duplicated predictions are filtered out using non-maximum suppression. As a result, knowing the number of objects in annotations and limited by the number of predictions it can produce, our model would finally produce min{n(n − 1), N } predictions for evaluation, where N is the number of remaining predicted pairs after non-maximum suppression. 2) Correct Detection: As in [12], a prediction p is considered as a correct detection if both bounding boxes p b A , p b B have greater than 0.5 IoU with respect to ground truth bounding boxes.
3) Correct Prediction: A correct detection, if its predicates p d , p o for both relative distance and occlusion are correct, it is considered a correct prediction. Note that, as in [12], we do not take predicted object categories into consideration. To discourage duplicate detection, when counting true positive (TP) predictions, if a ground-truth instance is matched to multiple predictions, then all these predictions except the first one being matched are considered false positives (FP). Hence a prediction is considered as TP for a ground truth if and only if it is a correct prediction and it is the first one being matched with the corresponding ground truth.
C. Implementation Details
We apply data augmentation to images. For images in the training set, we apply random horizontal flip, random resize, and random adjustments of brightness and contrast. After applying data augmentation, we normalize all images.
We use the Res101 backbone in [18] pre-trained on a different dataset than 2.5VRD. The number of transformer encoder layers is 6. The number of queries in the object pair decoder is 100. We set the learning rate to 1e-4 and drop it once to 1e-5 at the 30th epoch. We use the Adam optimizer and train all models for 40 epochs. The dropout rate is 0.1. The weight for losses of background pairs is 0.02.
In Eq. (14), classification cost weight β c and regression cost βr is 1.2 and 1.0, respectively. In Eq. (16) class cost weight for object A (α A ), object B (α B ), distance (α d ) , and occlusion (α o ) is 1, 1, 2, and 2, respectively. In Eq. (17), the regression cost weight for l 1 cost (α l1 ) and GIoU cost is (α giou ) is 5 and 2, respectively.
During training, we sort predictions by the product of relationship confidences. When generating attention weight visualizations ( Fig. 9 and Fig. 10), however, we sort predictions by the product of relationship confidences and object class confidences. Photos and visualizations used in this paper may contain object pairs that are predicted by our models but are not in the ground truth annotations. Nonetheless, the occlusion relationships between these object pairs that are not in the ground truth are nonambiguous.
D. Comparisons with State-of-the-Art Methods
We compare the performance of our model with state-ofthe-art VRD models [14][15] [16][17] evaluated by [12]. On the 2.5VRD dataset, our model achieves 38.6% and 41.2% F1-score on relative distance and relative occlusion, respectively (Table II). Our model outperforms the previously best model-DRNet-14% and 20% on relative distance and relative occlusion, respectively.
E. Ablation Studies
We conduct ablation studies to investigate the effects of decoder layer numbers, the generalized intersection box prediction task, predicting the generalized intersection box when no intersection exists, and using separate decoders for different sub-tasks on model performance.
1) Number of Decoder Layers:
We study the effects of varying the number of transformer decoder layers in our object pair decoder, distance decoder, and occlusion decoder. Unless otherwise specified, we use 6, 3, and 3 decoder layers for object pair, distance, and occlusion, respectively.
We observe significant decreases in model performance when varying the number of object pair decoder layers (Table III). Specifically, decreasing the number of layers from 6 to 3 results in a 19% drop in distance F1-score and an 18% drop in occlusion F1-score. Increasing the number of layers from 6 to 9 leads to a 31% drop in distance F1-score and a 30% drop in occlusion F1-score. Using the object pair decoder, we aim to extract object pair features that serve as queries for downstream tasks. The significant performance decrease verifies the importance of these queries and indicates that the object pair decoder may also involve feature extractions for depth and occlusion. These results show that a 3-layer decoder cannot provide sufficient feature extraction for downstream tasks, and a 9-layer decoder may lead to an overfitting result. In the distance decoder and occlusion decoder, varying the number of transformer layers results in minor drops in model performance (Table IV and Table V). Specifically, decreasing the number of distance layers from 3 to 1 results in a 9% drop in distance F1-score and an 8% drop in occlusion F1-score. Increasing the number of distance layers from 3 to 6 results in a 3% drop in distance F1-score and a 3% drop in occlusion F1-score. Similarly, decreasing the number of occlusion layers from 3 to 1 causes a 5% drop in distance F1-score and a 6% drop in occlusion F1-score. Increasing the number of occlusion layers to 6 causes a 2% drop in distance F1-score and a 2% drop in occlusion F1-score. Likewise, simultaneously varying the number of distance decoder layers N d and occlusion decoder layers N o (Table VI) results in minor drops in model performance. These results show a 3-layer decoder can provide sufficiently fine features for classification and regression, and increasing or decreasing the number of layers leads to worse feature extraction results. At the same time, these smaller performance drops, compared to the significant ones when changing the number of object pair decoder layers, further indicates the more vital role the object pair decoder plays for downstream tasks.
2) Generalized Intersection Box Prediction Task: We study the effects of the generalized intersection box prediction task on model performance (Table VII). Since changing the number of object pair decoder layers would result in significant performance drops, we fix the number of object pair decoder layers to 6 and vary the number of decoder layers in distance and occlusion decoders to form three different settings. Under all these three settings, our proposed GIT leads to performance boosts on both distance and occlusion F1-scores.
3) Predict the Generalized Intersection Box when No Intersection Exists: We investigate the function of predicting GIT when no intersection exists for a pair of objects (Table VIII). For models trained with GIT, we use the best setting found in previous experiments: 6, 3, and 3 decoder layers in object pair decoder, distances decoder, and occlusion decoder, respectively. Additionally, since generalized intersection boxes are not always predicted, the matcher does not consider the costs of generalized intersection boxes. When trained to predict the intersection region if and only if an intersection exists, we observe a 2% and a 2% relative F1-score drop on distance and occlusion, respectively. Additionally, we observe no performance boost compared to the model trained without GIT if the model does not predict the generalized intersection boxes when no intersection exists.
Additionally, for the "no occlusion" class, we observe a 3% relative F1-score drop if the model does not predict the intersection box when no intersection exists.
Predicting the generalized intersection box even when no intersection exits is critical for our model's learning of relative occlusion relationship detection. It guides our model to identify the region where intersections may or may not happen. Specifically, when no intersection exists, it still guides our model to identify a region, which can be used to conclude no occlusion exists. In other words, to conclude no intersection exists, our model still needs to know where to look at. Not knowing where to look would degrade the performance when detecting "no occlusion". Our proposed GIT consistently helps the model to identify the region that is most critical for the conclusion of the relative occlusion relationships no matter intersection exists or not. 4) Separate Decoders: By performing experiments without a dedicated object pair decoder, we study how using separate decoders benefits the learning of relationships. Specifically, we use only one decoder with six layers to detect object pairs, relative distances, and relative occlusions. We observe a 4% and a 3% relative performance drop in relative distance and relative occlusion detection, respectively (Table IX).
The drops in performance indicate that using separate decoders is conducive to the learning of relative distance and relative occlusion relationships. Specifically, using a dedicated object pair decoder frees the two relationship decoders from detecting objects, allowing them to focus on task-specific regions. At the same time, using two relationship decoders for two different types of relationships helps each decoder to focus on its own task-specific without distracting each other. We provide attention weight visualizations and analysis to demonstrate the benefits of using separate decoders in Section V-G3 and Section VI-A2, respectively.
F. Qualitative Results
On the 2.5VRD dataset, qualitative results of our model are given in Fig. 5 (correct predictions), Fig. 6 (incorrect occlusion), and Fig. 7 (incorrect distance). Qualitative results on these images demonstrate that our model correctly predicts relative distance and occlusion relationships in diverse scenarios but fails under some situations. We hope these failure cases can inspire readers to further improve model performance on this fundamental task.
We also run our model (trained on the 2.5VRD dataset without any extra data) on videos that contain moving objects with changing relationships. These videos are provided by authors and are not from the 2.5VRD dataset. Links to the videos are given in the footnotes on the first page.
At the upper left corner of each figure, texts are printed in three different colors (black, white, and yellow). Except for the color, they are the same.
G. Attention Weight Visualizations
We provide visualizations of attention weights of our decoders. We compare and contrast the attention weights of occlusion decoders trained with and without GIT. The occlusion decoder trained with GIT demonstrates much more focused attention ( Fig. 9). In some images, the occlusion decoder trained with GIT exhibits improved prediction correctness (Fig. 10). Additionally, we compare visualizations of all three decoders in multi-decoder architecture with those of the decoder in the single-decoder architecture. We observe that using separate decoders for different sub-tasks allows each decoder to focus on the most relevant information.
1) GIT: More focused attention: With GIT, our occlusion decoder shows more focused attention to areas around occlusion sites (mostly when occlusions exist) or object extreme points close to another object (mostly when no occlusion exists) and focus much less on irrelevant areas such as regions outside the union of the object pair (Fig. 9).
For example, in Fig. 9 (b) left where no occlusion exists, our occlusion decoder trained with GIT pays heavy and focused attention to the bottom of the upper flower, which is the extreme point of the upper flower that is the closest to the lower flower. In contrast, in Fig. 9 (b) right, the one trained without GIT pays heavy attention to the upper left portion of the upper flower, which is very far away from the lower flower and is very unlikely to be a site of occlusion. When occlusion does exist (Fig. 9 (c)), the occlusion decoder trained with GIT attends heavily only to regions between the pair of flowers but the occlusion decoder trained without GIT attends heavily to multiple locations outside the pair of flowers. Similarly, in Fig. 9 (l), the occlusion decoder trained with GIT pays focused attention to the region between the car and the tree. In contrast, the occlusion decoder trained without GIT attends heavily to the rooftop and fence on both sides of the photo (Fig. 9 (l)). In the rest of the figures in Fig. 9, similar Fig. 11. Attention weights in each decoder when using separate decoders (left) or in the only decoder when not using separate decoders (right). Using separate decoders allows the model to focus on most-relevant information when addressing each sub-task. patterns can also be observed.
2) GIT: Improvements in correctness: In Fig. 10 (a), the occlusion decoder trained with GIT focuses much more on the areas that are mutually occluded and attends much less to areas that are outside the union of the object pair. In specific, in the upper two images of Fig. 10 (a), the occlusion decoder trained with GIT focuses much more on the white cat's front paw (which occludes the black cat) and hind paw (which is occluded by the black cat). In the lower two images of Fig. 10 (a), the occlusion decoder trained with GIT focuses much more on areas near the man's leg (which occludes the vehicle) and feet (which is partially occluded by the vehicle).
In addition to having more focused attention, model trained with GIT exhibits improved prediction correctness. Specifically, in Fig. 10 (a), the model trained with GIT correctly predicts that mutual occlusions exist between each of the two predicted pairs of objects. In contrast, the model trained without GIT only predicts that one object occludes the other object.
Similarly, in Fig. 10 (b), the occlusion decoder trained with GIT pays more focused attention to the front left leg of the dog on the left (in the upper image) and the bottom portion of the upper flower (in the lower image) and effectively suppresses attention to the tree (in the upper image) and the surrounding leaves (in the lower image). In these two pairs of images, the model trained with GIT successfully detects occlusion while the model trained without GIT incorrectly predicts that no occlusion exists.
Furthermore, in Fig. 10 (d), the occlusion decoder trained with GIT attends heavily to the areas between the objects which are very close to each other and successfully predicts that no occlusion exists. The occlusion decoder trained without GIT, in contrast, fails to conclude no occlusion exists between the two pairs of objects in Fig. 10 (d).
3) Separate Decoders: We provide the attention weight visualizations of occlusion decoder, distance decoder, and object pair decoder for our proposed multi-decoder model (left 3 columns of Fig. 11). We also provide the attention weights for the single-decoder model (the rightmost column of Fig. 11). Both model are trained with the GIT.
In the following section, We will provide analyses of attention weight visualizations.
VI. ANALYSES
In this section, we provide analyses for the observed more focused attention and improvements in prediction correctness in the model trained with the generalized intersection box prediction task (GIT). We also analyze the differences between decoder attention visualizations for the multi-decoder model and the single-decoder model. Additionally, we analyze the effects of object size and location. We also study the source of errors. Finally, we compare state-of-the-art VRD methods with our proposed method.
A. Attention Weight Analysis 1) GIT: Visualizations of attention weights in the experiment section demonstrate that GIT can lead to more focused attention and improvements in prediction correctness. We provide analyses for these observed results in this subsection.
GIT effectively guides attentions to locations near occlusions because the generalized intersection box proposed in Fig. 4 usually encloses the actual occlusion sites when there are occlusions. For example, in Fig. 8 (a), the train gradually moves to the right and occludes the building. Starting from the third frame in Fig. 8 (a), the train occludes the building, and the intersection happens inside the intersection of the red and blue bounding boxes. Since occlusions usually happen inside the intersection box, asking the occlusion decoder to predict intersection boxes effectively guides the occlusion decoder to attend to areas near occlusions.
Guiding the occlusion decoder's attention to the intersection box helps the decoder to focus on actual occlusions and protects the occlusion decoder from distractions. In Fig. 10 (a), the occlusion decoder trained with GIT demonstrated much more focused attention to the paws of the white cat (in the upper image) and the areas between the man's leg and the vehicle (in the lower image). At the same time, occlusion's attention to irrelevant regions was greatly reduced. In specific, in the upper two images of Fig. 10 (a), the occlusion decoder trained with GIT effectively suppressed its attention to the legs and belly of the black cat and the back of the white cat; in the lower two images of Fig. 10 (a), the occlusion decoder trained with GIT effectively ignored the tree far away from the object pair and focused on areas near the man's feet instead of the anterior and lateral parts of the vehicle.
Paying increased attention to regions near actual occlusion sites helped the decode to detect more occlusion sites inside the intersection box. At the same time, ignoring distractions from irrelevant regions helped the occlusion decoder to make decisions using information relevant to occlusion relationships. Much more focused attention to relevant regions and significantly decreased attention to distracting regions might contribute to the improvements in relative occlusion relationship predictions in these images.
When no occlusion exits, predicting the generalized intersection box is still very beneficial because attending to object parts near the intersection box is important for concluding that no occlusion exists, especially when two objects are very close to each other. For example, in Fig. 10 (d), the car is very close to the human and the two dears are very close to each other. To determine whether relative occlusions exist or not, paying close attention to regions near the tiny space between the car and the human and between two dears is required. Specifically, at first glance, the only potential occlusion site of the two dears in Fig. 10 (d) is the region between the front left leg of the left dear and the hind right leg of the right dear. Paying closer attention to this potential occlusion site would lead to the conclusion that two dears do not occlude each other.
Paying focused attention to potential occlusion sites was exactly what the occlusion decoder did when determining the relative occlusion relationships between the two dears. Specifically, in Fig. 10 (d), the occlusion decoder trained with GIT allocated heavy and focused attention to the regions near the front left leg of the left dear and correctly concluded no occlusion exists between two dears. The occlusion decoder trained without GIT, in contrast, paid much less attention to this potential occlusion site. Instead, it allocated heavy attention to the head of the left dear and the rocks above dears. Failing to pay focused attention to the potential occlusion site and being distracted by irrelevant information help to explain why the occlusion decoder trained without GIT failed to conclude that no occlusion exists between the two dears.
2) Separate Decoders:
a) The most relevant information: Using separate decoders for occlusion, distance, and the object pair allows each decoder to focus on the most relevant information.
For example, in Fig. 11 (a), the occlusion decoder attends heavily to the white cat's front paw (which occludes the black cat) and hind paw (which is occluded by the black cat). Attending to these paws is very helpful for concluding mutual relationships. The distance decoder mainly attends to the ground, wall, and the cats' contours. Attending to these regions helps the model to understand the spatial configurations. The object pair decoder attends heavily to the black cat's head, chest, back, and tail. Identifying these representative parts helps the object pair decoder recognize the cat.
b) Omission of important details: In contrast, when using a single decoder for all three sub-tasks, the decoder has to attend to much more information. The requirement to attend to too much information makes focusing on the most-relevant part more challenging. This increased difficulty can lead to the omission of some important details, making it more difficult to correctly detect occlusion relationships.
For instance, in the rightmost column of Fig. 11 (a), the single decoder attends extensively to the cats' bodies, with a greater emphasis near the black cat's ear, white cat's wrist, and white cat's chest. The decoder's attention to the white cat's chest and wrist could only indicate that the white cat is occluding the black cat. This single decoder, unfortunately, paid only a small amount of attention to the white cat's front and hind paws. Failing to pay enough attention to the white cat's hind paw makes it much harder to identify that the white cat is occluded by the black cat. This omission of this critical detail partly explains why the single decoder fails to detect mutual occlusion. Similarly, in Fig. 11 (b) the single decoder fails to pay sufficient attention to the left front paw of the dog on the left side (in the red box). This paw is the only occlusion site between these two dogs. Failing to attend to this paw makes it unlikely to detect the occlusion relationship between these two dogs. The omission of the occlusion of the paw helps to explain why the single decoder fails to detect occlusion between the two dogs. c) Distracting information: At the same time, using a single decoder could introduce distracting information for relationship detection.
For instance, in Fig. 11 (a), the single decoder's intense attention on the black cat's ear introduce information that is irrelevant to the detection of mutual occlusion. In specific, the black cat's ear (upper left corner of the black cat's head) does not occlude the white cat. This information could only help the detection of "no occlusion." Detecting that the black cat's ear does not occlude the white cat obviously does not contribute to the detection of mutual occlusion relationship. Therefore, this information (the black cat's ear) acts as distractions for relative occlusion relationship detection. Similarly, in Fig. 11 (c), the only occlusion is located at the intersection of the car d) Reduce distractions: Using a separate object pair decoder helps to partially eliminate these distracting information when detecting relative occlusion relationships.
For instance, the specialized occlusion decoder does not attend to the black cat's ear in Fig. 11 (a), the back of the car in Fig. 11 (c), the edge of the saucer in Fig. 11 (d), and the lower half of the toilet in Fig. 11 (f). Instead, the specialized occlusion decoder achieves highly focused attention to the regions where occlusions take place in these images.
B. Object Size and Location Analysis
We study the effects of object sizes, vertical positions, and horizontal positions to explore how our model predicts the relative relationships leveraging these priors. F1-score distribution maps are shown in Fig. 12.
An object looks larger in the photo as it gets closer to the viewpoint. Leveraging this prior distribution in natural images, the model can learn larger objects are more likely to be closer to the viewpoint while smaller objects are more likely to be further. As a result, correctly concluding that object A is closer to the viewpoint when object A is very small or object B is very large can be more challenging for the model. For the relationship "A is closer than B", F1-scores are low when object A size is small and when object B size is large (Fig. 12 (a) upper). For the same reason, F1-scores are relatively high when two objects are of similar sizes when determining "same distance" (Fig. 12 (b) upper figure).
Another prior in natural images is that, for objects on the ground, those whose base is further to the horizon are closer than those whose base is closer to the horizon. In other words, objects on the ground are more likely to be further away if their vertical location is high. Consequently, it is rare for object A to be closer than object B when the vertical position of object A in the image is very high while that of object B is very low. In Fig. 12 (a) second row, the lower F1 score part shows that it is more difficult for our model to predict the "A is closer than B" results correctly when A is very high or B is very low. And F1-score in the first column gets lower from left to right as the location of object B gets lower. This indicates that our model does consider the vertical location of objects when deciding the relative distance relationships. For relationships in which A and B are interchangeable (e.g., 'A and B have same distance' is equal to 'B and A have same distance'), there is no particular preference for either the size or position of A or B. Therefore, the size and position distributions of A and B are similar, which is reflected in Fig. 12 (b) (d) (e), that the F1-score distribution maps are relatively symmetric about their principal diagonals.
C. Error Source Analysis
As an end-to-end model of relationship detection, the overall performance of our model is determined by two major factors: object pair proposal and relationship classification. To study the performance of our relationship classifier, we attempt to bypass the effects of detection error. Since it is virtually impossible to obtain a perfect object detection (predicted bounding boxes are identical to ground truth), we need to use the notion of "good detection" to filter our predictions. An IoU threshold of 0.6 is used to select predictions and we computed the precision of occlusion/distance classification within this group, see Table X. From the table, we can see that the model with GIT outperforms the model without GIT when an occlusion exists ("A occludes B", "B occludes A", or "mutual occlusion"). In general, for both models, the relationship detection precision for "good predictions" is much higher than their overall F1-scores. This result shows that the main bottleneck is to devise a better object detection model so that most of the object pairs can be proposed.
D. Compare with VRD methods
We compare VRD methods, DRNet [17], ViP-CNN [14], VTransE [16], and PPR-FCN [15], with ours (Fig. 13).
a) DRNet: DRNet [17], as shown in the third row of Fig. 13), exploits statistical relation between objects and relations and enclosing boxes (a box that contains both object A and object B "with a small margin"). Specifically, DRNet models and refines the estimated probability between objects and relationships. For example, it aims to assign a higher probability to reasonable "(cat, eat, fish)" than impossible "(fish, eat, cat)" [17]. Additionally, it extracts features for the enclosing boxes to obtain contextual information that helps decide the relationships between a pair of objects [17]. While the above two types of information are useful for more general and high-level relationship detection tasks such as VRD and HOI detection, they are much less helpful when detecting relative occlusion and relative distance relationships. Take the cat-eat-fish example given by [17] as an example, cat is closer than fish and fish is closer than cat are both reasonable predictions. Similarly, a cat can occlude fish and a fish can occlude fish. In other words, when detecting relative distance and relative occlusion relationships, the statistical relationships between objects and relationships are less strong and less useful for relative distance and occlusion relationship detection.
Additionally, to detect the relative occlusion relationships, the model only needs to attend to the generalized intersection box instead of the larger enclosing box. Compared to our proposed generalized intersection box, the larger enclosing box used by DRNet contains too much information that is not germane to relative occlusion relationship detection. The information that is most relevant to relative occlusion relationship detection is located in the region where two bounding boxes intersect because occlusion cannot happen outside of this region. Thus, features extracted by DRNet for the enclosing box are too redundant. In contrast, our proposed GIT helps our model learn to focus on the potential occlusion sites (regions where two bounding boxes intersect) when detecting relative occlusion relationships. In contrast, our proposed generalized intersection box provides more relevant information for relative occlusion relationship detection. b) ViP-CNN: ViP-CNN [14], which predicts an enclosing box that "tightly covers both" objects similar to the one in DRNet (the first row of Fig. 13), similarly suffers from the redundancy problem as DRNet. c) VTransE: VTransE considers relationship detection "as a vector translation" such that, for the features in a lowdimensional space, object A + relation ≈ object B [16]. By learning the vector translation (the second row of Fig. 13), VTransE aims to make the learning of the triplet (object A, relation, B) less challenging "by avoiding learning the diverse appearances" of the triplet "with large variance" [16]. While its strategy makes easier the learning of more general relationships in which the detailed appearances of objects contribute less to the relationship, this strategy makes the learning of relative occlusion relationships very challenging because learning the relative occlusion requires learning the subtle details of object appearances, such as which parts are occluded. Therefore, VTransE's strategy to avoid learning diverse appearances partly contributes to its sub-optimal performance when detecting relative occlusion relationships. d) PPR-FCN: PPR-FCN [15] is designed for weakly supervised VRD (thus not inlucded in the comparisons of Fig. 13), where the ground truth is image level, i.e. no exact location of objects in the image is provided. In general, PPR-FCN aims to calculate a score for each specific predicate associated with a pair of subject-object. PPR-FCN consists of two modules: WSOD g(weakly supervised object detection) and WSPP (weakly supervised predicate prediction). Since no bounding box of objects is given, the model does not learn to find objects and their locations in the image but makes a prediction based on the class of objects and their relative spatial relation. This strategy works well with semantic relationships but not occlusion/distance relationships because the occlusion relationship can be flipped even when two objects do not change their positions. Thus, compared to PPR-FCN, our model focuses on the position of objects and tries to attend to relevant areas only, which is a more local approach.
VII. CONCLUSION
We propose a novel three-decoder architecture as infrastructure and use GIT to enforce focused attention. Visualizations of attention weights in our occlusion decoder confirm the effectiveness of GIT in creating focused attention. Our proposed occlusion decoder that exhibits focused attention to task-relevant regions is more interpretable than decoders that attend to a large portion of the image. Like humans, our proposed occlusion decoder pays intense attention to relevant information and suppresses distracting information. Moreover, visualizations of attention weights and error analyses indicate that our proposed occlusion decoder is more robust, especially when dealing with mutual occlusions and objects very close to each other, further highlighting the importance of focused attention.
This work was supported by Baidu and Anker. (Yang Li and Yucheng Tu contributed equally to this work.) (Corresponding author: Guyue Zhou) Yang Li, Xiaoxue Chen, and Guyue Zhou are with the Institute for AI Industry Research, Tsinghua University, Beijing, China. (e-mail: [email protected]; [email protected]; [email protected]) Yucheng Tu is with the Department of Mathematics, University of California San Diego, La Jolla, CA, 92093 USA (e-mail: [email protected]). Hao Zhao is with Intel Labs China and Peking University. (e-mail: [email protected]; [email protected])
Fig. 1 .
1Fig. 1. Understanding relative occlusion and relative distance may help HOI models rule out some unreasonable predictions such as (a) "a woman is holding an umbrella" when her hand is not occluding the handle of the umbrella at all, and (b) "a man (in the red box) is riding a skateboard (in the orange box)" that is much further to the viewpoint than the man.
Fig. 2 .
2Examples of relative occlusion and relative distance. (a) The woman's hair and leg occlude the man, but the man does not occlude the woman. The woman is closer to the viewpoint than the man. (b) No occlusion: although the two bounding boxes intersect, a closer look at the region between vegetables and the knife would lead to the conclusion that no occlusion exists between them. (c) Mutual occlusion: the woman's arm occludes the man's arm while the man's leg occludes the woman's leg.
Fig. 4 .
4Generalized Intersection Box (shown in red). (a) When two bounding boxes partially intersect, the generalized intersection box is the same as the intersection box. (b) When a smaller bounding box is completely inside a larger bounding box, the generalized intersection box is the same as the smaller bounding box. (c) and (d) When two bounding boxes do not intersect, the generalized intersection box is formed using the second and third smallest x coordinates and y coordinates of the two bounding boxes. corner of b i , for i ∈ {A, B}. Then the generalized intersection box b ∩ for a pair of objects A and B is defined as
Fig. 5 .
5Qualitative examples: correct predictions.
Fig. 6 .
6Qualitative examples: incorrect relative occlusion predictions.
Fig. 7 .
7Qualitative examples: incorrect relative distance predictions.
Fig. 8 .
8Model performance on videos. (a) Change in relative occlusion relationship: a moving train gradually occludes the building. (b) Change in both relative distance and relative occlusion relationships: a toy (on the left) moves further away from the viewpoint, moves right until being occluded by the other toy, and moves left and becomes not occluded. Links to videos are in the footnote on the first page.
Fig. 9 .
9Visualizations of attention weights in our occlusion decoders. The generalized intersection box prediction task (GIT) effectively guides our model to focus on occlusion-specific regions when detecting relative occlusion relationships. (Red indicates heavy attention. Blue indicates nearly no attention.) Fig. 10. Improvements in relative occlusion detection (from incorrect to correct) with the generalized intersection box prediction task (GIT). (Left: with intersection box prediction task. Right: without intersection box prediction task)
Fig. 12 .
12Model performance (F1-scores) w.r.t object size, vertical position, and horizontal position. and the tree. Information fron the back of the car is distracting and does not help the occlusion relationship detection.
Fig. 13 .
13Compare ViP-CNN, VTransE, and DRNet with our method.
TABLE I PREDICTION
IMLP HEADS: NOTATIONS (NOT) AND SHAPESMLP #
Prediction
NOT
Shape
Object Pair MLP-1 Object A Class
pA
(Npair, Nc)
Object Pair MLP-2
Object B Class
pB
(Npair, Nc)
Object Pair MLP-3
Object A Box
pb A
(Npair, 4)
Object pair MLP-4
Object B Box
pb B
(Npair, 4)
Distance MLP
Dist. Predicates
pd
(Npair, Nd)
Occlusion MLP-1
Occl. Predicates
po
(Npair, No)
Occlusion MLP-2
Intersection Box pint
(Npair, 4)
TABLE II COMPARISON
IITO STATE-OF-THE-ART METHODS. OUR MODEL OUTPERFORMS THE PREVIOUSLY BEST MODEL BY 4.8% AND 7.0% ON RELATIVE DISTANCE AND RELATIVE OCCLUSION, RESPECTIVELY.Dist F1
Occl F1
Dist p / r
Occl p / r
ViP-CNN [14]
33.6
34.2
33.3 / 33.9 34.1 / 34.3
PPR-FCN [15]
33.5
33.9
33.0 / 34.1 33.8 / 34.0
VTransE [16]
32.4
32.9
31.5 / 33.2 32.8 / 33.0
DRNet [17]
33.8
34.4
33.9 / 33.7 34.3 / 34.5
Ours
38.6 (+14%) 41.2 (+20%)
38.9 / 38.2 41.6 / 40.9
TABLE III EFFECTS
IIIOF VARYING THE NUMBER OF OBJECT PAIR DECODER LAYERS.Npair
Dist F1
Occl F1
Dist p / r
Occl p / r
3
31.3 (-19%)
33.9 (-18%) 31.7 / 30.9
34.3 / 33.5
6
38.6
41.2
38.9 / 38.2
41.6 / 40.9
9
26.7 (-31%) 28.9 (-30%)
27.0 / 26.5
29.1 / 28.6
TABLE IV EFFECTS
IVOF VARYING THE NUMBER OF DISTANCE DECODER LAYERS.N d
Dist F1
Occl F1
Dist p / r
Occl p / r
1
35.2 (-9%) 37.7 (-8%) 35.6 / 34.9 38.2 / 37.3
3
38.6
41.2
38.9 / 38.2 41.6 / 40.9
6
37.3 (-3%) 40.0 (-3%) 37.6 / 36.9 40.4 / 39.6
TABLE V EFFECTS
VOF VARYING THE NUMBER OF OCCLUSION DECODER LAYERS.No
Dist F1
Occl F1
Dist p / r
Occl p / r
1
36.6 (-5%)
38.6 (-6%)
37.0 / 36.2 39.0 / 38.2
3
38.6
41.2
38.9 / 38.2 41.6 / 40.9
6
37.7 (-2%)
40.4 (-2%)
38.0 / 37.4 40.7 / 40.0
TABLE VI EFFECTS
VIOF SIMULTANEOUSLY VARYING THE NUMBER OF DISTANCE AND OCCLUSION DECODER LAYERS.N d
No
Dist F1
Occl F1
Dist p / r
Occln p / r
1
1
36.5 (-5%) 39.0 (-5%) 36.9 / 36.1 39.4 / 38.6
3
3
38.6
41.2
38.9 / 38.2 41.6 / 40.9
6
6
38.1 (-1%) 40.8 (-1%) 38.4 / 37.8 41.2 / 40.5
TABLE VII
EFFECTS OF THE GENERALIZED INTERSECTION BOX PREDICTION TASK
(GIT). (INTER: INTERSECTION)
Nd
No
Inter Dist F1 Occl F1
Dist p / r
Occl p / r
1
1
F
36.3
38.3
36.6 / 36.0 38.6 / 38.0
1
1
T
36.5
39.0
36.9 / 36.1 39.4 / 38.6
3
3
F
38.1
40.5
38.5 / 37.7 41.0 / 40.1
3
3
T
38.6
41.2
38.9 / 38.2 41.6 / 40.9
6
6
F
38.0
40.3
38.5 / 37.6 40.8 / 39.9
6
6
T
38.1
40.8
38.4 / 37.8 41.2 / 40.5
TABLE VIII EFFECTS
VIIIOF PREDICTING THE GENERALIZED INTERSECTION BOX WHEN NO INTERSECTION EXISTS. (PINI: PREDICT THE GENERALIZED INTERSECTION BOX WHEN NO INTERSECTION EXISTS. OCCL: NO OCCL F1: THE F1 SCORE FOR THE CLASS "NO OCCLUSION")Inter
PINI
Dist F1
Occl F1
Occl: no occl F1
Dist p / r
Occl p / r
F
-
38.1
40.5
41.6
38.5 / 37.7 41.0 / 40.1
T
F
37.9 (-2%)
40.5 (-2%) 41.3 (-3%)
38.3 / 37.5
41.0 / 40.0
T
T
38.6
41.2
42.4
38.9 / 38.2 41.6 / 40.9
TABLE IX EFFECTS
IXOF NOT USING A DEDICATED DECODER FOR OBJECT PAIR DETECTION. (DOPD: DEDICATED OBJECT PAIR DECODER. IN OTHER WORDS, USING ONE DECODER FOR ALL THREE SUB-TASKS.)DOPD Dist F1
Occl F1
Dist p / r
Occl p / r
F
37.1 (-4%) 40.0 (-3%) 37.4 / 36.8 40.3 / 39.6
T
38.6
41.2
38.9 / 38.2 41.6 / 40.9
TABLE X PREDICTION
XPRECISION IN GOOD DETECTIONSRelationship Class
Without GIT With GIT
No occlusion
0.96
0.95
A occludes B
0.74
0.78 (+5%)
B occludes A
0.73
0.76 (+4%)
Mutual occlusion
0.68
0.69 (+1%)
Occlusion total
0.90
0.91
Distance not sure
0.90
0.92
A is closer
0.87
0.86
B is closer
0.87
0.86
Same distance
0.62
0.64
Distance total
0.85
0.85
ACKNOWLEDGMENTThe authors would like to express their sincere gratitude for Sijie Cheng's helpful discussions on attention heads. The authors would like to thank Jinjun Peng and Beiwen Tian for their help with GPU resources. The authors also thank Yupeng Zheng for his help with dataset processing.
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Yang Li received the B.S. degree in cognitive science with specializations in machine learning and neural computation from the University of California. 2021. He is currently a research intern at Tsinghua University. San Diego, La Jolla, CA, USA; Beijing, ChinaYang Li received the B.S. degree in cognitive sci- ence with specializations in machine learning and neural computation from the University of California San Diego, La Jolla, CA, USA, in 2021. He is currently a research intern at Tsinghua University, Beijing, China.
His specialization is differential geometry and geometric analysis. He is currently a teaching visitor at the University of California San Diego. Besides teaching, he is. 2015, and the Ph.D. degree in mathematics from the University of California. Beijing, China; San Diego, La Jolla, CA, USA, in 2021Yucheng Tu obtained the B.S. degree in mathematics from Tsinghua Universityinterested in research on deep learning and computer visionYucheng Tu obtained the B.S. degree in mathe- matics from Tsinghua University, Beijing, China, in 2015, and the Ph.D. degree in mathematics from the University of California San Diego, La Jolla, CA, USA, in 2021. His specialization is differential geometry and geometric analysis. He is currently a teaching visitor at the University of California San Diego. Besides teaching, he is interested in research on deep learning and computer vision.
She is currently a first-year Ph.D. student at the Institute for AI Industry Research. 2021Beijing, ChinaXiaoxue Chen obtained the B.S. degree in computer science and technology from Tsinghua University ; Tsinghua UniversityHer research interests are in the area of computer vision, especially 3D scene understandingXiaoxue Chen obtained the B.S. degree in computer science and technology from Tsinghua University, Beijing, China, in 2021. She is currently a first-year Ph.D. student at the Institute for AI Industry Re- search, Tsinghua University. Her research interests are in the area of computer vision, especially 3D scene understanding.
He is now a research scientist affiliated with Intel Labs China. He is also a joint postdoc affiliated with Peking University. His research interests cover various computer vision topics related to robotics, especially 3D scene understanding. Beijing, ChinaHao Zhao received the B.E. degree and the Ph.D. degree both from the EE department of Tsinghua UniversityPhotograph not available at the time of publicationHao Zhao received the B.E. degree and the Ph.D. degree both from the EE department of Tsinghua University, Beijing, China. He is now a research scientist affiliated with Intel Labs China. He is also a joint postdoc affiliated with Peking University. His research interests cover various computer vision topics related to robotics, especially 3D scene understanding. Photograph not available at the time of publication.
He is currently an Associate Professor with the Institute for AI Industry Research (AIR), Tsinghua University. His research interests include advanced manufacturing, robotics, computer vision, and human-machine interaction. 2010, and the Ph.D. degree from the Hong Kong University of Science and Technology. Harbin, China; Hong Kong, ChinaHarbin Institute of TechnologyGuyue Zhou received the B.E. degree from. Photograph not available at the time of publicationGuyue Zhou received the B.E. degree from Harbin Institute of Technology, Harbin, China, in 2010, and the Ph.D. degree from the Hong Kong University of Science and Technology, Hong Kong, China, in 2014. He is currently an Associate Professor with the Institute for AI Industry Research (AIR), Tsinghua University. His research interests include advanced manufacturing, robotics, computer vision, and human-machine interaction. Photograph not available at the time of publication.
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| Open RAN (ORAN, O-RAN) represents a novel industry-level standard for RAN (Radio Access Network), which defines interfaces that support inter-operation between vendors' equipment and offer network flexibility at a lower cost. Open RAN integrates the benefits and advancements of network softwarization and Artificial Intelligence to enhance the operation of RAN devices and operations. Open RAN offers new possibilities so that different stakeholders can develop the RAN solution in this open ecosystem. However, the benefits of Open RAN bring new security and privacy challenges. As Open RAN offers an entirely different RAN configuration than what exists today, it could lead to severe security and privacy issues if mismanaged, and stakeholders are understandably taking a cautious approach towards the security of Open RAN deployment. In particular, this paper provides a deep analysis of the security and privacy risks and challenges associated with Open RAN architecture. Then, it discusses possible security and privacy solutions to secure Open RAN architecture and presents relevant security standardization efforts relevant to Open RAN security. Finally, we discuss how Open RAN can be used to deploy more advanced security and privacy solutions in 5G and beyond RAN. | 10.1016/j.jnca.2023.103621 | [
"https://export.arxiv.org/pdf/2212.01510v1.pdf"
]
| 254,246,888 | 2212.01510 | d8c778ef67e53834fe637a1b65736c41bc1178dc |
Open RAN Security: Challenges and Opportunities
Madhusanka Liyanage
School of Computer Science
and Centre for Wireless Communications
University Collage Dublin
Ireland
University of Oulu
Finland
An Braeken
Department of engineering, technology (INDI)
Vrije Universiteit Brussel
Belgium
Shahriar Shahabuddin
Centre for Wireless Communications
University of Oulu
Mobile Networks, NokiaDallas, TXUSA, Finland
Pasika Ranaweera
School of Computer Science
University Collage Dublin
Ireland
Open RAN Security: Challenges and Opportunities
Open RANSecurityPrivacyO-RAN5G6GAIMachine LearningVirtualizationRadio Access Network
Open RAN (ORAN, O-RAN) represents a novel industry-level standard for RAN (Radio Access Network), which defines interfaces that support inter-operation between vendors' equipment and offer network flexibility at a lower cost. Open RAN integrates the benefits and advancements of network softwarization and Artificial Intelligence to enhance the operation of RAN devices and operations. Open RAN offers new possibilities so that different stakeholders can develop the RAN solution in this open ecosystem. However, the benefits of Open RAN bring new security and privacy challenges. As Open RAN offers an entirely different RAN configuration than what exists today, it could lead to severe security and privacy issues if mismanaged, and stakeholders are understandably taking a cautious approach towards the security of Open RAN deployment. In particular, this paper provides a deep analysis of the security and privacy risks and challenges associated with Open RAN architecture. Then, it discusses possible security and privacy solutions to secure Open RAN architecture and presents relevant security standardization efforts relevant to Open RAN security. Finally, we discuss how Open RAN can be used to deploy more advanced security and privacy solutions in 5G and beyond RAN.
Introduction
Mobile network communications are becoming one of the critical enablers of the current digital economy and interconnecting national critical infrastructure-based services [1]. The number of mobile subscribers and different mobile-based services is increasing in a rapid phase all across the globe [2]. However, the radio spectrum is still a scarce resource, and optimal utilization of radio resources is critical for developing a telecommunication network [3]. Thus, the orchestration and management of radio resources or the Radio Access Network (RAN) are also evolved with each mobile generation. The early mobile generation mobile networks architectures such as 2G and 3G had controllers responsible for orchestration and management of RAN and its resources [4]. The flat network architecture in 4G enables a new interface (i.e., X2) to support base station level communication to handle RAN resource allocation [5]. However, the RAN of existing mobile network generations is still based on monolithic building blocks. Thus, RAN functions of existing networks, including most of the 5G network, are still contained with the proprietary vendor-specific devices called Baseband Units (BBUs) at the base stations [6]. However, this approach leads to the proverbial vendor lock-in RAN since different RAN vendors can design their flavor of RAN Email addresses: [email protected] (Madhusanka Liyanage), [email protected] (An Braeken), [email protected] (Shahriar Shahabuddin), [email protected] (Pasika Ranaweera) equipment. This has eliminated the possibility for MNOs (Mobile Network Operators) to get mix-and-match services from other RAN vendors.
The introduction of the network softwarization concept in 5G [7,8,9] and added intelligence in beyond 5G networks have opened up a promising solution, called Open RAN, to mitigate this issue [9,10]. The Open RAN Alliance [11] went back to the controller concept to enable best-of-breed Open RAN. Open Radio Access Networks (Open RANs), also known as ORANs or O-RANs, have been considered one of the most exciting RAN concepts, designed for 5G and beyond wireless systems. Open RAN promotes openness and added intelligence for RAN network elements that could overcome the limitations of existing RAN technologies, [12,13]. The feature of openness allows smaller and new players in the RAN market to deploy their customized services, while the feature of intelligence is to increase automation and performance by optimizing the RAN elements and network resources. Moreover, Open RAN offers many RAN solutions and elements to the network operators to be more open and flexible. Further, the network operators can shorten the time-to-market of new applications and services to maximize the overall revenue because of the virtualization feature. Thus, the added intelligence in Open RAN could offer superior benefits even to existing network softwarization based virtual RAN and cloud RAN concepts.
There are two major Open RAN organizations, i.e., Telecom Infra Project (TIP) [14] and the O-RAN alliance [15] who are working on the advancement of Open RAN realization. The However, the benefits of Open RAN come at challenges, e.g., security, deterministic latency, and real-time control [16,10,17]. Among these factors, the security in Open RAN is quite essential. As 5G and beyond networks are responsible for inter-connecting many Internet protocol Telephony (IpT) based critical national infrastructure, attacks on future telecommunication networks will have a ripple effect [18,19]. Some devastating examples caused by such attacks are smart cities and factories shutting down, a complete black-out of the power grids and water supplies, a fall-out of the transportation infrastructure with crashes by autonomous vehicles, etc. [20,21]. All these challenges demand significant effort from the research and industry communities to standardize and implement security for all the sections of 5G and beyond networks, including Open RAN networks [22]. Specially, the decentralization of control functions with Open RAN increases the number of threat vectors and the surface area for attacks.
Open RAN has distinct features that bring intelligence to future networks. While AI helps overcome various challenges of 6G Open RANs via intelligent and data-driven solutions, it can hurt the security of RAN. Attackers can target the AI systems or even use AI-based attacks to jeopardize the operation of the Open RAN system. Thus, Open RAN will now be vulnerable to AI-related attacks such as denial-of-service (DoS) [23], spoofing [24] and malicious data injection [25] could affect the AI. For instance, AI training can be manipulated in an Open RAN spectrum access system by inserting fake signals. In addition, the integration of network softwarization will add a whole new set of security attacks related to virtualization. Similar to the 5G core and edge networks, now Open RAN needs to tackle softwarization associated attacks such as Virtual Network Function)/Cloud Network Function (VNF/CNF) manipulation [26], Virtual Machine (VM) misconfiguration [27], log leak attacks [28]. In addition, open interfaces defined in Open RAN will introduce another set of security and privacy vulnerability. Thus, it is necessary to develop correct security and privacy solutions to mitigate these new Open RAN-related security and privacy solutions at the radio network level. Existing security mechanisms, frameworks, and governance approaches will need to be upgraded to operate in open multi-vendor controlled Ecosystem.
On the other hand, added features of Open RAN can bring security and privacy advantages over traditional RAN. Open RAN can also build upon the security enhancements already enabled by 5G and allow the operator to control the network's security entirely, ultimately enhancing the operational security of their network. Less hardware dependency and support for complete software control in Open RAN allow isolating security breaches quickly and intelligently, reducing the impact of security risk. In addition, these features reduce the risks associated with security mechanism upgrades. Moreover, the modularity supported by the open interface in Open RAN allows the security and privacy deployments to support continuous integration/continuous delivery (CI/CD) operating model [40]. The CI/CD model supports seamless and effective security management against the security vulnerability in Open RAN.
Moreover, Open RAN enables the possibility for zero-touch and frequent software updates [41], which is more transparent, fast, secure, and low cost than the software upgrades in a traditional network. Finally, standardization of open interfaces can also reduce the security risks to a certain extent as it can help detect incongruencies and offer concrete steps to monitor the network. Thus, it is crucial to identify these new security benefits and rectify them correctly in future RAN deployments.
Motivation
The research on Open RAN security is still in its infancy. As Open RAN advocates open interfaces, it is imperative to analyze the security vulnerabilities and their mitigation of Open RAN in parallel to their system architecture development. The reason is that without a proper security framework in place, the idea of an open network might not be an attractive solution to the network operators. This is especially true in this era of complex geopolitics, where global powers are increasingly concerned about wireless infrastructure security. Table 2 summarizes existing research works about Open RAN security. The table highlights the lack of a comprehensive Open RAN security analysis in the literature. Most existing Open RANrelated publications focused on architecture, interfaces, and al-gorithms, while security was a secondary topic. A couple of technical specifications from Open RAN alliance present the security flaws and solutions of Open RAN in [34], and [36], respectively. However, they are not comprehensive because they lack a thorough discussion either on the solutions or the flaws. They also do not present the Open RAN security benefits and discuss any research directions. Similarly, other publications presented in Table 2 fail to provide a comprehensive analysis of Open RAN security.
Our Contribution
To the best of the authors' knowledge, this is the first attempt to provide a comprehensive security analysis of Open RAN. The main contributions of this article are presented below.
• Classification of security-related risks: A taxonomy distinguishing the risks present in Open RAN, is provided. Each of these risks is elaborated concerning a description of impact.
• Present Open RAN specific security solutions: Unique solutions for Open RAN security vulnerabilities based on blockchain, physical layer, and AI have been presented.
• Overview of general mistakes, consequences, and mitigation: A summary of the general design errors pertaining to Open RAN, their consequences, and potential mitigation are presented.
• Discussion on Open RAN security benefits: A list of security benefits specific to Open RAN, and already available in V-RAN and 5G networks are presented.
Outline
The rest of the paper is organized as follows. Section 2 presents the overview of Open RAN architecture and the difference from the conventional RAN architectures. The threat vectors and security risks associated with Open RAN are presented in Section 3. Several solutions for the security threats and vulnerabilities of Open RAN are elaborated in Section 4. Section 5 presents the security benefits of Open RAN implementation. Discussion and lessons learned towards realizing an Open RAN architecture are portrayed in Section 6. Finally, Section 7 concludes the paper.
3 [30] M L L L L This article discusses Open RAN deployment with a focus on 5G network device security 2021 [31] H M H L L A whitepaper by Altiostar on the security of Open RAN which presents a method to implement Open RAN with a zero-trust security framework 2021 [32] H L L L L An article that summarizes Open RAN specifications focusing on proposed architecture and building blocks 2021 [33] H L L L L This article presents an analysis of an Open RAN system with the aid of a traffic steering use case implemented in a modular way 2021 [34] H H L L L A technical specification by O-RAN alliance on Open RAN security threat modeling and remediation analysis 2021 [35] H H L M L A pre-print which identifies the limitations of current Open RAN architecture and the technologies and opportunities for research and development to overcome them 2021 [36] M M H L L A technical specification by O-RAN alliance on the security requirements and security controls per Open RAN defined interface and Open RAN defined network function 2021 [37] M L M L L Presented an analysis to demonstrate the urgent need to protect Open RAN fronthaul and proposed a security protocol as a potential solution 2020 [38] L M L L L A whitepaper by Ericsson on Open RAN security considerations that ensure an open and interoperable RAN is secure by design 2020 [10] H L L L L This article presents the basic functionalities and current research trends on C-RAN and its derivatives such as vRAN and Open RAN 2017 [39] L [9,12,42]. Figure 1 illustrated the key differences of traditional and Open RAN architectures. The Open RAN architecture is proposed to enable three main goals [32,10], i.e.;
• Cloudification: The goal is to support cloud-native RAN functions via disaggregated hardware and software components.
• Intelligence and automation: The goal is to utilize advanced AI/ML capabilities to enable automated management and orchestration in RAN
• Open internal RAN interfaces: The goal is to support various Open RAN interfaces, including interfaces defined by 3GPP.
As illustrated in Figure 1, the RAN in Open RAN architecture is disaggregated into four main building blocks, i.e., the Radio Unit (RU), the Distributed Unit (DU), the Centralised Unit (CU), and RAN Intelligence Controller (RIC). The RU is located with antennas, and it is responsible for transmitting, receiving, amplifying, and digitizing the radio frequency signals. The former BBU (Based Band Unit) is now disaggregated into DU and CU. They are the computation parts of the base station. Here, DU is physically located closer to RU, while CU can be located closer to the Core. RIC is possible for taking the intelligent and automated decisions related to RAN.
O-RAN appliance has proposed a more detailed architecture for Open RAN as represented in Figure 2 architecture, whose main responsibility is to manage the RAN domain, such as the provision of interfaces with network functions, near-real-time RIC for RAN optimization, and O-Cloud computing resource and workload management [43,29]. These SMO services can be performed through four interfaces, including A1, O1, O2, and open fronthaul M-plane.
• RAN Intelligence Control (RIC): This logical function enables Open RAN to perform real-time optimization of functions and resources through data collected from the network and end-users. It is the key element in Open RAN, which helps to realize disaggregation strategy, bringing multivendor interoperability, intelligence, agility, and programmability to RANs [44,30]. The RIC is divided into components as non-real-time RIC (Non-RT RIC) and nearreal-time RIC (Near-RT RIC). The Non-RT RIC is integrated with Open RAN SMO Framework. It handles the control request and RAN resources within the second range. To this task, Non-RT RIC utilizes specialized applications called rApps. Non-RT RIC can also collect network performance metrics and subscriber data to offer AIbased network optimization and policy guidance recommendations for Near-RT RIC. The Near-RT RIC resides within edge servers or regional cloud as it is responsible for performing network optimization actions within milliseconds range. Near-RT RIC uses the different xApps to support these tasks [45,33]. • O-DU: This logical node has functionalities of the physical and MAC layers. This element terminates the E2 with F1 interfaces.
• O-CU: This is a logical node in the Open RAN architecture and hosts all the functions of both the control plane and data plane. These two O-CU planes connect with the O-DU logical node via the F1-c interface and F1-u interface, respectively.
• O-RU: This logical node has a physical layer and radio signal processing capabilities to connect with the SMO framework via the open fronthaul M-plane interface and connects with end-users via radio interfaces.
One of the main goals of Open RAN is "opening" the protocols and interfaces between these RAN components, such as radios, hardware, and software. The O-RAN Alliance has defined eleven different interfaces, including A1, O1, E1, F1 open fronthaul M-plane, and O2. More specifically, the open fronthaul M-plane interface is to connect Service Management and Orchestration Framework (SMO) and Open RAN radio unit (O-RU), A1 is to connect non-real-time RAN intelligent controller (RIC) located in the SMO framework and near real-time RIC for RAN optimization, O1 is to support all Open RAN network functions when they are connected with SMO, and O2 is to connect SMO and O-Cloud for providing cloud computing resource and workflow management. According to [32], there are different deployment scenarios of the O1 interface, such as flat, hierarchical, and hybrid models, by which the SMO framework can provide numerous management services, for example, provisioning management services, trace management services, and performance management services.
Threat Vectors and Security Risks Associated with Open RAN
We start by explaining the taxonomy, used to distinguish the different types of risks. Next, each of the four identified domains are further elaborated.
Threat Taxonomy
We categorize the risks in three main domains: Process, Technology and Global. First, process risks are related to rules, regulations, oversight. Second, the technology risks correspond with the risks caused by the mechanisms for enforcing rules and procedures, as well as detecting threats. Fourth, global risks are broad risks related to the global communication instruction. Figure 3 provides an overview of the risk domains in Open RAN respectively.
Process
In the process risks, four categories are distinguished, corresponding to the preliminary assumptions or prerequisites, the general regulations, the privacy and human related aspects. In fact, all the process risks apply to any RAN implementation, but are in general more complex in Open RAN due to the modularity and the higher amount of stakeholders involved. Table 3 provides an overview of the key process risks associated with the Open RAN process. 6 The operational environment of the Open RAN system must provide reliable timestamps for e.g. the generation of audit records. In addition, the list of minimum prerequisites and assumptions, required to successfully operate the O-RAN system, needs to be defined for the operational environment. [47,16] This is applicable to any RAN implementation. However, it is more complex in Open RAN since some aspects (e.g. cloud services) are not under the control of the Open RAN system. Requirement of secure storage of stored logs, credentials and secrets Log files, secrets and credentials stored in external systems and related to Open RAN need to be protected, and access controlled should be enable to allow only privileged users. [47,31] This is applicable to any RAN implementation. However, Open RAN hardware should possess a hardware-based security module like TPM (Trusted Platform Module) to manage, generate, and securely store cryptographic keys, to offer secure boot, full disk encryption, and remote attestation. Requirement of Trusted certificate authorities (CAs) Trusted certificate authorities for identity provisioning are applied. [47,31] This is applicable to any RAN implementation. However, due to involvement of additional stakeholders, the CAs used in Open RAN for authenticating network elements should be properly audited by well established global organizations and SDOs General Requirement of Secure complete lifecycle process and assessment strategy Network operators should have an appropriate security process for the complete lifecycle of Open RAN deployment. [47,48] This is applicable to any RAN implementation. However, it is more complex in Open RAN due the involvement of additional stakeholders.
Requirement of Trusted assets/supply chain verification
There is a need to identify, locate, authenticate and verify the origin of the relevant assets in the Open RAN system. [47] This is applicable to any RAN implementation. However, it is more complex in Open RAN because an Open RAN system is built with components coming from different additional parties.
Increased complexity and inter-dependency Increased difficulty for identifying issues exists and accountability due to complexity is not evident. [49] This is specific to Open RAN. Due to the modularity of O-RAN and loss of total ownership. Multiple stakeholders need to collaborate to mitigate the threats Privacy
Violation of privacy policies such as GDPR Privacy issues arise due to new interfaces, shared environments and new players with different views and objectives on privacy. [47,50] Privacy issues arising from 5G C-RAN are already identified. However, the attack surface increases in case of Open RAN as components can be designed in different regions. People
Requirement of Trustworthy and qualified insiders
There is a need to provide sufficient security resources and sufficient security education and training for the users. [47,51,52] The availability of sufficient security educated people is a well-known problem. In the case of Open RAN additional expertise is required such as virtulaized component security Requirement of Trusted stakeholders All stakeholders involved with Open RAN System should be identified, authenticated and trusted. [47,51] This is applicable to any RAN implementation. However, it is more complex in Open RAN due to the increased and diversifed number of stakeholders.
Prerequisites
To operate a successful Open RAN system, a list of minimum prerequisites and assumptions of the the operational environment needs to be defined. However, The prerequisites are not under control of the RAN system, but should be carefully checked [53]. To start with, a reliable operational environment must be ensured, providing for instance reliable timestamps to be used in the audit records [54,47].
Next, secure storage of stored logs, credentials and secrets in external systems need to be guaranteed for instance by using hardware based security modules like trusted platform modules (TPMs) [55,47]. Cryptographic key management, remote attestation, disk image encryption, and secure booting are functions that are typically conducted by a TPM. O-RAN requires such an entity within its midst for managing hardware based security and a root of trust for facilitating signing and verification functions.
In addition, access to this sensitive data should only be allowed by privileged users [47]. The last prerequisite is that the certificate authorities (CAs), which authenticate the network elements, are fully trusted and audited by well established, worldwide recognized organizations [47,56]. In fact, all these prerequisites are essential for any RAN implementation. However, since there are more stakeholders involved in Open RAN, it is clear that these requirements are more challenging to enforce and verify, compared to other RAN implementations.
General regulations
The first step in the effective Open RAN launch that needs to be done is the standardization of critical processes like operation, administration and management, covering the complete lifecycle of the Open RAN deployment [57]. This includes a clear description of components used for secure establishment of mutual authentication, access control, key management, trusted communication, storage, boot and self-configuration, update, recoverability and backup, security management of risks in open source components, security assurance, privacy, continuous security development, testing, logging, monitoring and vulnerability handling, robust isolation, physical security, cloud computing and virtualization, and robustness.
Next, it is also important to identify, locate, authenticate and verify the origin of the relevant assets in the system. Furthermore, for each of the different assets, at rest and in transit and location, the type (data, component, etc.) and the security properties (Confidentiality, Integrity, Availability -CIA) should be carefully collected. In fact, a complete and efficient supply chain process is required [58]. In particular, this is more complex for Open RAN due to the decoupling of hardware and software and the modularity. For instance, there is a risk of firms from allied states purchasing relabeled products or components from adversarial states.
Finally, when an issue arises in the network, due to the complexity of the network it is not evident to identify and isolate the issues. Moreover, in case the issue is found, it is possible that the corresponding vendors do not take their responsibility as they can pass the blame to others because of the complexity and interdependence of the whole system.
Privacy
The privacy of end users encompasses privacy related to data, identity and personal information [59]. Privacy sensitive data for end users are mostly leaked via communication services that are gathering all types of personal information, which are often not needed for the functioning of the services. Adversaries can even further extract more personal information about end users, such as User Equipment (UE) priority, location information, trajectory, and preference. The protection of the user data is regulated by the law of the hosting country, where different jurisdictions can be applicable. There are, at least, three possible locations, the victim, the offender, or the service provider [50]. Therefore, clear guidelines should be developed in order to cope with these new interfaces, shared environments and new players available in Open RAN.
People
First of all, it is necessary to clearly identify and authenticate the stakeholders involved in the different processes like implementation, management, operation and maintenance of the Open RAN system. For each of the stakeholders, their roles and responsibilities should be clearly defined and assessed. Vendors should have well established and transparent security practices built into their engineering processes [47].
Moreover, adequate training and assessments need to be organized for the different stakeholders, going from administrators, integrators, operators and orchestrators in order to be capable to securely implement and manage the system according to the instructions provided by the Open RAN Alliance and the later to be developed standards [47].
Finally, strategies for security testing with published wellknown test plans at trusted lab facilities should be defined upfront and integrated in the regular operation [47] Moreover, adequate training and assessments need to be organized.
Technology
The largest class of risks is related to the different components and mechanisms in the network. Here, distinction is made based on [48], considering aspects related to open software, radio/open interface, intelligence, virtualization and general.
Open source software
The open source related risks are well known problems available in open source software code. An overview is provided in Table 4. Since, Open RAN is expected to be built (solely or partly) based on such open source codes, it is in particular vulnerable for this type of attacks.
A trusted developer can intentionally insert a backdoor by injecting a few lines of malicious code into an open source code component to be used within the Open RAN system [63]. It is then highly likely that a software project team picks it up and uses the infected open source code later, while the tools for vetting and testing of the development team do not detect the malicious code [47]. As a consequence, a vulnerability into Explicit legislative standards, guidance, requirements, or conditions to ensure trusted programming should become available. [47,61,60] This is a well-know problem in open source software code. Since Open RAN is expected to be built (solely or partly) based on such open source codes, it is in particular vulnerable for these attacks Dispute in patents
The actors involved in Open RAN development implement 5G functions at their discretion and under different copyright regimes. The establishment of a certain type of collaboration is required between those actors as the degree of their collaboration is not at the same level in many cases. [61,62] Due to the need of inherent agreements in the O-RAN alliance (With increased number of stakeholders), this is a threat specific for Open RAN.
the software code is included and can go undetected for a long period. The resulting effect on the Open RAN system can be diverse. It can either be simply annoying, but at the same time it can significantly decrease the system performance via for instance Denial of Service (DoS) attacks, or it can even lead to serious loss of sensitive data. Open source vulnerabilities are normally published on the National Vulnerability Database (NVD) [64]. This database is primarily intended for developers to disclose vulnerabilities. However, this source is also used by hackers to exploit those vulnerabilities enabling backdoors to attacks on e.g. the hypervisor, Operating System (OS), Virtual Machine (VM) or container. Moreover, vulnerabilities frequently propagate as developers often re-use free open source code. As a consequence, downloading open source libraries and their dependencies, as well as downloading open source code from untrusted repositories contain significant risks [47]. Open RAN vendors and operators should thus store at each moment up-to-date inventories containing the dependencies in their open source software used in the applications. In addition, this should be complemented by a process, which receives and manages all the notifications coming from the open source community that are related to newly discovered vulnerabilities, including newly developed patches to overcome them. This should enable a better supply chain traceability.
Existing legislation demonstrates implied security preferences but provides no explicit legislative standards, guidance, requirements, or conditions. These preferences should be explicit but transparent, reviewable, and auditable to ensure secure coding. Due to the fact a material amount of Open RAN code is being written by firms in different countries, security audits should be mandatory making code available to security researchers [61].
Finally, the last open source software risk is more linked to political and financial interests, instead of security interests. Both the 3GPP and Open RAN alliance operate a Fair, Reasonable and Non Discriminatory (FRANS) policy when it comes to patents that are held by contributors to those respective organizations. Patents are held on aspects of the 3GPP and Open RAN Alliance specifications, but the holders of those patents agree that it is mutually beneficial for everyone if the patents are licensed with a FRANS approach. The concern in this area is politically oriented. There might be a possibility that the patents held by competing manufacturers and service providers may be withdrawn from the FRANS licensing arrangement if trade relations between different countries dramatically deteriorate [61,65].
Radio/Open Interface
The different radio/open interface components include the Fronthaul, the central Unit/distributed unit (CU/DU) and the 5G radio network. with open interfaces allowing the software programmability of RAN. These components and interfaces may not be secured to industry best practices, for instance containing no proper authentication and authorization processes, ciphering and integrity checks, protection against replay attacks, prevention of key reuse, validation of inputs, response to error conditions, etc [47]. This follows often from the strict performance requirements (bandwidth, latency, fronthaul transport link length, etc.) that limit the use of some security features, enforced by the high bit rate fronthaul interface to increase the processing delay. As a consequence, different MITM, DoS, data tampering or even information disclosure attacks become possible.
-The first category of risks is due to attacks from the internet exploiting weak authentication and access control to penetrate the network boundary. There are several possibilities for this. First, it would allow the presence of a rogue Open RAN Radio Unit (O-RU) in order to fool the O-DU or UE into associating with it instead of the legitimate O-RUs [47]. This opens the door to subscriber's identity interception/disclosure and unauthorized user tracking attacks of user movements and activities by catching the SUPI/5G-GUTI of the subscriber's User Equipment (UE) and location of a device). Second, an adversary can inject DL/UL C-plane messages that falsely claim to be from the associated O-DU, which would impact the O-RU to process the corresponding U-Plane packets [67]. Also spoofing of DL/UL C-plane messages, leading to temporarily limited cell performance (or even DoS) on cells served by the O-RU and in addition, a consequential threat to all O-RUs parented to that O-DU might exist. Third, if in addition no trusted stakeholders are guaranteed, an attacker can attack a master clock by sending an excessive amount of time protocol packets or impersonate a legitimate clock, a slave, or an intermediate clock, by sending malicious messages to the master, thus degrading the victim's performance [66]. The attacker may be residing either within the attacked network (insider) or on an external network connected to the attacked network. This attack results in a situation where the clock service is interrupted completely or the timing protocol is operational but slaves are being provided inaccurate timing information due to the degraded performance of the master clock. This degradation in the accuracy of time may cause DoS to applications on all the RUs that rely on accurate time, potentially bringing down the cell. A cell outage caused by misaligned time may further impact performance in connected neighboring cells. Finally, when having two different vendors, the O-RU and the O-DU need to be managed as different The high bit rate Fronthaul interface imposes strict performance requirements which force to limit the use of some security features. [47] Yes, This interface is specific in O-RAN.
Attack on master clock
An attacker can attack a master clock by sending an enormous amount of time protocol packets. It can also impersonate a legitimate clock, a slave, or an intermediate clock, by sending malicious messages to the master, thus degrading the victim's performance. [47,66] No, however, there is an increased range of attacks in Open RAN due to the use of various xApps and rApps. Moreover, near-RT operation is expected in Open RAN for some of the functions. MITM random delay attack to desynchronize the clocks An attacker acting as MITM can introduce random packet delay on Precision Timing Protocol (PTP) sync messages and/or PTP delayreq/resp messages, which causes inaccurate PTP offset calculation, thus the clocks may not be synchronized properly. [47,66] No, however, there is an increased range of attack in O-RAN.
Spoofing of DL/UL C-plane messages
An adversary injects DL/UL C-plane messages that falsely claim to be from the associated O-DU which would impact the O-RU to process the corresponding U-Plane packets [47,67]. This will lead to temporarily limited cell performance (or even DoS) on cells served by the O-RU and in addition, a consequential threat to all O-RUs parented to that O-DU might exist.
No. However, there is an increased range of attacks in Open RAN. Moreover, this attack can be easier to perform in shared virtualized environment.
DoS attack against
O-DU C-plane DoS attacks against the O-DU C-plane are launched. [47]. Due to the cleartext nature of Enhanced Common Public Radio Interface (eCPRI) messages used for the Open Fronthaul C-Plane, an attacker can launch a volumetric DoS attack with bad or unauthenticated eCPRI Real-time control data messages (adopted for C-Plane communication) against the O-DU C-Plane, causing O-DU performance degradation and potentially its overall service interruption, which could further cascade to all its serving O-RUs.
No, however, there is an increased range of attacks in Open RAN. The openness in the Open RAN system will be a cause to lose explicit security due to a lack of know-how.
Intercept the Fronthaul (MITM) over U Plane
An attacker attempts to intercept the Fronthaul (MITM) over the User Plane due to the limited use of some security features at the Fronthaul interfaces. [47] No, This is a problem also in 3GPP RAN.
Attacks on user's data traffic
The integrity protection is enabled on the Control Plane messages, which still makes the data traffic of the user vulnerable because the Control Plane and User Plane are segregated. [47,38] No, This is a problem also in 3GPP RAN. However, Open RAN offers the computing resources (i.e. not available in other RANs) to implement 3GPP specified UP integrity protection algorithms without impacting on the user experience. Physical access to Fronthaul cable network An intruder into the exchange over the Fronthaul cable network attempts to gain electronic access to cause damage or access sensitive data. [47,68] The same type of attack can be applied in other RAN. However, the attack range and possibilities are increased in O-RAN. Insecure open Fronthaul interfaces An attacker penetrates and compromises the O-RAN system through the open O-RAN's Fronthaul due to the lack of industry level security best practices. [47,67,69] Yes, since this involves the new interfaces introduced specific in O-RAN. Although following [69], it is also inherently required for a secure 5G implementation. CU/DU Attacks via Shared Baseband Units An attacker exploits the lack of isolation on Shared Baseband Units and the edge platforms to perform attacks. [12,70] The same type of attack can be applied in V-RAN. However, the attack range and possibilities are increased in O-RAN. 5G Radio Network
Radio Jamming
An attacker could disrupt the communication by deliberate jamming, blocking or creating interference with the authorized wireless network. [47] No, this is a general attack that can be applied to any RAN.
Jam airlink via IoT devices
An attacker attempts to jam the airlink signal through IoT devices. [47] No, this is applicable to any RAN.
RAN Sniffing
An attacker could decode the essential network configuration details by sniffing the RAN. [47] No, this is a general attack that can be appied to any RAN.
RAN Spoofing
An attacker is spoofing the RAN signals by transmitting a fake signal meant to pretend as an actual signal. [47] No, this is a general attack that can be applied to any RAN. entities and may have heterogeneous security levels [38]. Instead, the O-DU will have to bridge the management traffic between the management system and the O-RU. Hence the possibility to reach the northbound systems beyond the O-DU through the Open Fronthaul interface becomes a possible attack vector in this split architecture.
-The second category of risks on the fronthaul is due to the ability of the attacker to compromise Open RAN data integrity, confidentiality and traceability in case the components are not secured to the industry best practices. This follows often from the strict performance requirements (bandwidth, latency, fronthaul transport link length, etc.) that limit the use of some security features, enforced by the high bit rate fronthaul interface to increase the processing delay. As a consequence, different MITM attacks become possible.
A MITM attacker over the fronthaul interface is able to intercept the data over the U-Plane and introduce random packet delay on the Precision Timing Protocol (PTP) sync messages and/or PTP delayrequest/response messages, which causes inaccurate PTP offset calculation, resulting in clocks which may not be synchronized properly [66]. Also, denial of service (DoS) attacks become possible. Moreover, after breaking the PDCP security, also access to content can be obtained. A Man-in-the-Middle (MITM) attack over the fronthaul interface or O1 is able to intercept the M plane, and thus also to do passive wiretapping and DoS, but needs to break M Plane Security prior to gain OAM access [47].
-The third category of risks on the fronthaul is if an attacker compromises the Open RAN monitoring mechanisms and integrity and availability of the log files [71].
-The fourth category of risks on the fronthaul is caused by a compromise of the integrity and availability of the Open RAN components in general. Insufficient assurance of Open RAN software package integrity could affect CIA of data, services, hardware and policies during installation or upgrade phases for Open RAN components [47]. An attacker could, in such a case, cause denial-of-service, data tampering, information disclosure, spoofing identity, etc.
-Finally, if an attacker is able to get physical access to the fronthaul components, it can result in a devastating impact on the confidentiality and integrity of the data [68]. Note that this is typically linked to the first type of process related threats dealing with trusted stakeholders.
• CU/DU. The shared units pool in the Open RAN cloud native deployment may suffer from insufficient isolation and impose the risk of breaking user privacy and accessing sensitive data [12]. The openness and exposure of the CU and DU entities in comparison to C-RAN are inviting intruders for gaining access of those entities through cyber hacking attempts. As the fronthaul of the O-RAN is expected to be deployed via enhanced Common Public Radio Interface (eCPRI), converged packet based network that contrives it is inviting cyber threats unlike the traditional frounthauls [72]. Although uncommon, intrusions can be perpetrated via the F interface in the Mid-haul that connect CU to its corresponding DUs. Such interventions are possible through the threat vectors such as service migration, offloading, or handover mechanisms that exist with edge computing base stations that are presumed to host CUs [70]. A compromised CU is capable of impregnating both fronthaul and the backhaul directions leveraging the open interfaces of the O-RAN.
• 5G Radio Network. These attacks are classical attacks, which can be applied to any RAN system and include radio jamming, jamming via IoT devices, RAN sniffing and spoofing [47].
Radio jamming can be impacting on the reference signals, the synchronization signal, the Physical Broadcast Channel (PBCH), the Physical Downlink Control Channel (PDCD), the Physical Uplink Control Channel, or the Physical Random-Access Channel [73]. This would enable an attacker to disrupt the communication by deliberate jamming, blocking or creating interference with the authorized wireless network. Additionally to blocking the communication flow, jamming the synchronization channels or the signaling flow is another method to disrupt the 5G services [74]. A capable adversary can target different entities of the 5G communication network simultaneously to impact an interference significant enough to subdue the communication. Thus, a jamming detection mechanism is mandatory to filter out the jamming frequencies in this era of 5G and beyond [75].
In addition, due to the millions of IoT devices in the network, jamming of the airlink signals through the IoT devices, can easily overload the Open RAN resources by means of Distributed DoS (DDoS) attempts carried via a botnet army of millions to billions of infected devices, on which a malware instructs to reboot all devices in a specific or targeted 5G coverage area at the same time [76]. Most IoT based services are Location Based Services (LBSs) and expect locational awareness with utmost availability. The attackers capable of jamming the GPS receiver will succeed in subduing the service to an inaccurate state [74]. Since the O-RAN interfaces should be open to a common standardization to avoid vendor-specific nature, adversaries have the ability to assimilate the firmware and software specifications and induce a race-like condition by exploiting its vulnerabilities.
RAN sniffing allows the attacker to decode essential network configuration details, assisting attackers to optimize and craft their attacks [77]. Vulnerabilities of the PBCH channel are allowing the attackers to sniff the 5G RAN network stats [78]. The open-source and low-cost natures of the software radio are inviting the attackers to exploit the existing vulnerabilities in software, protocol, and firmware layers. With RAN spoofing a fake signal pretending to be an actual signal is conveyed by targeting an RF receiver within the RAN [77]. Similar to sniffing, vulnerabilities of the PBCH channel and the software radio devices can be the main causes targeted through spoofing attempts, that embrace the masquerading signal as a legitimate one [79].
Intelligence
The different components and mechanisms that contribute to the intelligence in the Open RAN network are the Near Realtime Radio Access Network Intelligence Controller (Near-RT-RIC), Non Realtime RIC (Non-RT RIC), and machine learning (ML) algorithms. These risks are mostly specific to Open RAN as they operate on new components and new algorithms, which are currently not available. The threats related to intelligence are summarized in Table 6.
• Near-RT-RIC related Attacks: xApps have the capability to manipulate behavior of a certain cell, a group of UEs, and a specific UE. The related attacks are due to either malicious xApps, xApps with vulnerabilities, misconfigured xApps, compromised xApps or conflicting xApps [47].
As xApps are launched to perform intelligent functions for CU and DU entities in regards to radio resource management, a compromised xApp could attempt to take the control of a cell, a RU device, or a group of UE devices; and would be capable of tracking a certain consumer within its RIC domain. In addition, the same malicious xApp could gather priority information on the served UE devices through the A1 interface, where distinguishing and identifying serving UEs are possible. Such acts violate the location privacy of the important UEs and even the prioritization on the currently serving services can be manipulated. This will leads to compromise RAN performance as well as the privacy violation.
Malicious xApps can potentially be used as a sniffer for UE identification. In such a circumstance, RAN performance could be impacted negatively while the privacy of the subscribers may be violated. This follows from the fact that the A1 interface is able to point out a certain UE in the network (through its UE identifier), which creates correlations among the randomized and anonymized UE identities between the RAN nodes. As a consequence, UE location tracking and change in UE priority become possible. In particular, identification and tracking of a certain subscriber, for instance, a Very Important Person (VIP) becomes a real threat. The exposure of the UE identifier is most probable through E2 signaling channels in comparison to its counterpart A1, due to the Near-RT conditions of the E2. Further, such malicious xApps could change the Service Level Agreement (SLA) specifications of the assigned services similar to changing of the priority levels. Such acts could conflict with the Near-RT-RICs decision process as the program execution times might extend beyond the specified boundaries of a presumed Near-RT event or SLAs [35].
Vulnerabilities can potentially exist in any xApp, since it can come from either an untrusted or unmaintained source. Such vulnerabilities can then be exploited to take over another xApp or the whole near-RT RIC and often have the purpose to degrade the performance (e.g. a DoS). It may also be possible to alter data transmitted over A1 or E2 interfaces, or to extract sensitive information. Also, the xApp isolation can be exploited in order to break out of the xApp confinement and to deduce information from cohosted xApps. In addition, unauthorized access provides new opportunities to exploit vulnerabilities in other xApps or Open RAN components to intercept and spoof network traffic, and to degrade services (through DoS • Non-RT RIC related Attacks: rApps impact non-RT RIC functions such as AI/ML model training, A1 policy management, enrichment information management, network configuration optimization for the purpose of performance degradation, DoS, and enrichment data sniffing (UE location, trajectory, navigation information, and GPS data). rApps bearing many resemblances to xApps in their operational context, have the ability to manipulate the behavior of a certain cell, a group of UEs, and a specific UE. The related attacks are similar to xApps, due to either malicious rApps, rApps with vulnerabilities, misconfigured rApps, compromised rApps or conflicting rAPPs. Besides these similar ones, there are two more risks identified related to Non-RT RIC [47].
Untrusted or unmaintained sources can cause vulnerabil- Vulnerabilities and misconfiguration in rApps Vulnerabilities can potentially exist in any rApp, if it obtained from an untrusted or unmaintained source. An attacker exploits vulnerabilities and misconfiguration of such rAPPs to disrupt the offered network service and potentially take over another rApp or the whole non-RT RIC. [47,38,35] Yes, as these components are only defined in O-RAN.
Weak authentication and authorization in in rApps
If web front-end or REST API interfaces contain software vulnerabilities or implement authentication and authorization insufficiently, an attacker could bypasses authentication and authorization and able to gain access to the rApp and pose as a tenant. Such a way an attacker gains the ability manipulate configurations, access logs and implement back doors [47,67] Yes, as these components are only defined in O-RAN.
Compromising isolation in rApps An attacker compromises rApp isolation to break out of rApp confinement. Such a way, attacker can perform side channel attack to deduce information from co-hosted rApps in a shared resource pool [47] Yes, as these components are only defined in O-RAN.
Conflicts in rApps
Conflicting rApps (i.e. direct, indirect and implicit conflicts) unintentionally or maliciously impact non-realtime Open RAN system functions such as Carrier license scheduling, energy savings and subscription handling to degrade performance or trigger a DoS. [47] Yes, as these components are only defined in O-RAN.
ML Unanticipated results
If unexplainable AI is used, the results cannot be predicted and might have a fast impact [80]. Therefore, use of unexplainable AI/ML models in the Open-RAN can potentially lead to unanticipated consequences, which might have an impact on the security and privacy [67]. Such AI/ML models could unintentionally violate the security and privacy policies and offer bias results.
The same type of attack can be applied in V-RAN. However, the attack range and possibilities are larger in O-RAN.
Data poisoning attacks An attacker with access to the training set is able to poison the ML training data (Data poisoning attacks) and thus break the reliability of the training. This impacts the xApps/rApps managed Open RAN system functions such as mobility management, admission controls, bandwidth management, load balancing and results a performance degradation [47,81].
This attack only applies to O-RAN, where ML is explicitly included.
Evasion attacks/Adversarial examples An attacker uses an adversarial example (intentionally designed date) as an input to the ML models to make a mistake [82,83,84]. This impacts the xApps/rApps managed Open RAN system functions results a performance degradation.
This attack only applies to O-RAN, where ML is explicitly included.
Model poisoning Attacks
An attacker with access to the model can alter the ML model resulting in system manipulation and compromise of ML data confidentiality and privacy [47,84,85]. This impacts the xApps/rApps managed Open RAN system functions results a performance degradation. ities in any rApp. Exploitation of these vulnerabilities mostly leads to disruption of the offered network service and potentially taking over another rApp or the whole non-RT RIC. As a consequence, the attacker may gain the ability to alter data transmitted over A1 interface, or extract sensitive information. Also rApp isolation can be exploited to break out of rApp confinement and to deduce information from co-hosted rApps. Unauthorized access provides new opportunities to exploit vulnerabilities in other rApps or Open RAN components to intercept and spoof network traffic, to degrade services through DoS attempts; An attacker might also penetrate the non-RT RIC through A1/O1 interfaces or from external sources through SMO and attempts to trigger a DoS or degrade the performance of non-RT RIC [35].
In addition, rApps in the Non-RT RIC can cause conflicting decisions as they can be launched by different vendors targeting different purposes: Carrier license scheduling, or energy savings. Such conflicts could take the form of a direct, indirect, or implicit nature depending on the rApp parameters in question, and the effect that particular conflict is inducing. As in direct ones deals with the conflict of same parameter change requested by different rApps, indirect ones where the different parameter changes by different rApps would cause an opposite effect, and implicit ones that different parameter changes would lead to changing the network state. The effects can lead to an overall network performance degradation, or instabilities within the network entities. These conflicts are difficult to mitigate since dependencies are impossible to observe.
There is an additional vulnerability that can appear in the case the rApp management is exposed to a web front-end or REST API, whose software interfaces contain vulnerabilities or do not implement authentication and authorization in a proper way. This would allow an attacker to gain access to the rApp and pose as a tenant or to manipulate configurations, access logs, or to implement back doors.
• ML related Attacks: ML and AI play a vital role in the formation of the O-RAN concept. Thus, vulnerabilities or flows in existing ML models or algorithms can be envisaged as probable threats to the O-RAN system as they are deployed in the intelligence portion of the architecture. One of the most common threats is the data poisoning attacks, where the adversary is altering the data sets that are intended for training, testing, or validating [81,47]. The access to perform such modifications, however, can be gained via the penetration through fronthaul, midhaul, xApps, or rApps. Poisoning attempts could impact any stage of the ML process as in feature selection, prediction, decision making, model classification, or anomalous detection. The O-RANs openness and the Near-RT operations require the ML models to be formed online with continual updating during the operation. Though it would not impact in the long run, feeding bogus data to the online ML model is capable of impacting the RIC decision making negatively, especially in terms of radio resource allocation. Similarly, evasion/ adversarial attacks or model poisoning attacks represent two variants of the poisoning attack. In the evasion attempt, data is carefully tampered according to a perturbed design that would not detect as anomalous. In the model poisoning attempt, the entire model or the control parameters of the model are altered to impact the learning phases of the process [85].
Pre-trained and widely available ML models can be utilized by the attackers for gaining access or evading the system's anomalous detectors for launching transfer learning attacks. Model inversion and membership inference attacks are ensuing privacy leakages [82]. In model inversion attempts, adversary is reconstructing the training data set from the model parameters [86,87,89]. This is plausible as there are plenty of online repositories with training data that would aid the attacker in cross-validating the determined data. Membership inference threat would determine whether a particular data set was used in the training process of the ML model or not [88].
Finally, due to the complexity of the models in AI/ML, the results are not yet explainable in most of the cases [80]. Therefore, its use in the RAN can potentially lead to unanticipated consequences, which might have an impact on the security or performance [67].
The impact of data poisoning attempts would clearly target the allocation and management of radio resources within the O-RAN fronthaul, and could result in jeopardizing the accuracy of mobility management, load balancing, and QoS management functions that are administered under the Near-RT-RIC. In the long run, the entire RT intelligent framework could become compromised. Model poisoning, evasion and transfer learning attempts induce the same impact on the RT systems. For the Non-RT system, however, as the time is not a critical parameter, the impact would be less costly, as the decisions are made from the data gathered from an extended period in comparison to the RT instances.
Virtualization
The following components, Physical Network Function (PNF), Virtual Network Function (VNF), Cloud Network Function (CNF), SMO, hypervisor, Virtual Machine/Container (VM/CN), are involved in the virtualization process. Here, we discuss the security issues associated with each of these components. Some of these attacks can be applied also in V-RAN and C-RAN [97]. However, the attack range and impact of such attacks larger in Open RAN. The threats related to intelligence are summarized in Table 7.
• PNF related Attacks:. An attacker compromises a PNF to launch reverse attacks and other attacks against VNFs/CNFs. A lack of security policies to protect mixed PNF-VNF/CNF deployments, resulting to insecure interfaces, could be exploited to perform attacks against [47]. Spoofing on network traffic Intercept and spoof on network traffic via VMs/CNs [47].
Sharing hardware
Applications may share the same hardware resources in virtualization, which might be affected by vulnerabilities [38]. Compromised security services Compromises auxiliary/supporting network and security services [47].
VNFs/CNFs, potentially taking advantage of legacy security used by PNFs and not provided by the virtualization/containerization layer [47]. Apart from the security policies, service level agreements and service specifications are vital consensus for the PNF, CNF, and VNF entity deployment. As these entities are envisaged to launch security management entities as specified in [89], the original consensus should not be altered for all the service level guarantees.
• VNF/CNF related Attacks: Despite VNF/CNF images are effectively static archive modules including all components used to run a given Open RAN VNF/CNF, modules within an image may have vulnerabilities, introducing malware, missing critical security updates or are outdated. These images are only collections of files packaged together. Therefore, malicious files can be included intentionally or inadvertently within them. In addition, VNF/CNF images may also have configuration defects, e.g. configuring a specific user with greater privileges than needed. This could all be used to attack other VMs/CNs or hosts within the environment. An attacker can migrate a compromised VNF/CNF to a different location which has less security or privacy policies to gain additional access to the system. Since Open RAN uses different equipment with different vendors and different configurations, there can be less secure environments, which can lead to additional vulnerabilities if deployed in the same system [47].
Moreover, since many Open RAN VNFs/CNFs require secrets to enable authentication, access control and secure communication between components, these secrets are embedded directly into the image file system. In addition, the images often contain also sensitive components like an organization's proprietary software and administrator credentials. Anyone with access to the image (e.g. by means of insufficient authentication and authorization) can easily parse it to extract these secrets, resulting in the compromise, stealing or damage of the contents on the images. As a result, it can lead to Intellectual Property (IP) loss and expose significant technical details about an Open RAN VNF/CNF image to an attacker. Even more critically, because registries of images are typically trusted as a source of valid, approved software, compromise of a registry can potentially lead to compromise of downstream VMs/CNs and hosts [47].
There is an increased risk of MITM attacks by intercepting network traffic intended for registries in order to steal developer or administrator credentials within that traffic. This can result in fraudulent or outdated images to orchestrators [47]. Further, typical VNF/CNF based security threats exist, as in: location shift attack where the adversary is capable of displacing the VNF to a domain inheriting a lesser level of security policy assignment with the intention of gaining access, or interoperability issues between the VNF/CNF developers or service providers that can be exploited by an attacker [91,70].
• SMO related Attacks: As the SMO is the key entity behind the holistic autonomic environment of the O-RAN, its security is extremely vital for the O-RAN performance and the individual subscriber security and privacy. Improper or insufficient authentication or authorization of Open RAN external (e.g. AI/ML, Emotional Intelligence (EI), Human-Machine) or internal (e.g. over O1 or O2 interfaces, with Non-RT RIC) interfaces on SMO, allow access to the SMO and in particular the data stored on it.
Besides disclosing Open RAN sensitive information, the attacker may also alter the Open RAN components [47].
DoS attacks or increased traffic can cause overload situations and thus affects availability of the SMO data and functions. Further, an attacker may exploit weak orchestrator configuration, access control and isolation. A single orchestrator may run many different VMs/CNs, each managed by different teams, and with different sensitivity levels. If the access provided to users and groups is not conform their specific requirements, an attacker or careless user would be able to affect or subvert the operation of another VM/CN managed by the orchestrator. Malicious traffic from different VMs/CNs sharing the same virtual networks may be possible if VMs/CNs of different sensitivity levels are using the same virtual network with a poorly isolation of inter-VM/CN network traffic [47].
• Hypervisor related Attacks: An attacker can exploit the security weaknesses in the guest OS to attack the hypervisor of the hosting OS. Examples of guest OS vulnerabilities are OS command injection, SQL injection, buffer overflow or missing authentication for critical functions [92,90,93]. Privilege escalation is a common threat among hypervisor deployments that is also applicable in the context of the O-RAN. In this attack, any authorization violations are sought out by the perpetrator exploiting the infrastructure vulnerabilities formed through ill-maintenance or misconfigurations [70]. The administrative capabilities granted to the adversary through this threat is devastating as it could range from a simple excessive allocation of resources to a complete deletion of xApps or rApps [98,99].
An attacker may also change the configurations of compromised VNFs/CNFs to consume high amounts of CPU, hard disk, and memory resources in order to exhaust the hypervisor. Another way to compromise the hypervisor is by generating an excessive amount of log entries such that it is infeasible or very difficult to analyze the log files coming from other VNFs [92,90].
Finally, as the hypervisor provides its own security functions and Application Programming Interfaces (APIs) to the host system security functions, it is in full control of the security functionalities of the lower layers and thus needs to be fully trusted. When a malicious administrator has for instance root access to the hypervisor and by using a search operation, the user identity (ID), passwords and Secure Shell Protocol (SSH) keys from the memory dump can be extracted, which in turn violates the user privacy and data confidentiality [92,90].
• VM/CN related Attacks: VMs/CNs may be compromised due to flaws in the Open RAN VNFs/CNFs they run. For example, an Open RAN VNF/CNF may be vulnerable to cross-site scripting (SQL) injection [94] and buffer overflow vulnerabilities [95].
Insecure VM/CN runtime configuration by the administrator can lower the security of the [47].
An attacker hack into VM/CN is for instance possible if an attacker steals VMs/CNs private key from one VM/CN and so reveals the administrator privileges. Next, all tenant's tokens and the administrator rights of the whole Open RAN system can be obtained, [47].
From the side of the application, trust is required at all levels. In case the underlying host OS is malicious, access can be obtained to all data processed in the workloads, as in RAM memory and disk volumes. Techniques like secure enclaves [96] have the goal to provide a trusted environment. However, the application will be hardware-instance dependent [38].
If VM/CN migration is not secured or performed over a secure channel, a MITM attacker can modify arbitrary VM/CN OS or application states during the migration. An attacker may also use an older snapshot of VM/CN without the concern of the VM/CN owner to bypass the security system and obtain access to the system. This attack is possible after an already comprised hypervisor rollback to a previous snapshot. In the scheduler attack, the vulnerabilities in the hypervisor's scheduler are exploited to acquire system resources for the malicious VM at the expense of a victim VM [92,90].
Furthermore, due to virtualization and cloud computing, different applications might use the same hardware resources. Isolation between these applications are only at software level and not at the level of hardware. As a consequence, hardware related vulnerabilities like the recently discovered Meltdown and Spectre attacks (https://meltdownattack. com/) can have a larger attack range [38].
Finally, besides the main functionality of the VNF/CNF itself, the administrators may also decide to deploy additional network services on their VMs/CNs in order to do extra monitoring, remote configuration, remote access to other services such as SSH, etc. If these additional network services are directly accessible over the Internet or from another administrator, new entry points for attackers are created and if access is obtained to the VM/CN, more extra attacks become possible [47].
Global
Offering the highest level of security on the network is important for a nation. We here distinguish five major types of attacks or risks that need to be taken into account [100]. The related threats can be found in Table 8.
• Attack on digital economy: Since 5G is fully integrated in the digital economy, it can result in potential life or death consequences. For instance, currently a lot of data is sent from our mobile devices, smart homes, electrical cars via a network consisting of devices, which are remotely controlled and updated and thus present a potential attack vector. The possibility of a smart city shutting down, autonomous vehicles crashing, or factories going dark due to a cyber attack are frightening situations; that would eventually result in a major economic collapse. At this pivotal point in modern civilization where the global economic platforms are shifting to a holistic digital platform, a successful threat might endanger the entire growth of 5G and its predecessors through loss of trust from the subscribers. Thus, it is imperative to investigate the scope of such threat vectors that target economic platforms. It is evident the prescribed scope is reaching beyond the means of typical phishing, or identity thefts [102]. Moreover, the impending launching of Metaverse and its significance for O-RAN existence is further confirming the required focus on the robustness of digital economic platforms, as Metaverse is introducing a virtual serviceable platform built on top of monetary transactions [105].
• Espionage: There are currently no regulations for avoiding the collaboration between an Open RAN equipment manufacturer and an external party, like for instance a security agency of a certain country. Therefore, without no guarantee of good intentions of the equipment and software providers, possibilities for spying should be considered as viable. Such acts of espionage can be perpetrated by targeting corporate to government institutions. The flexibility offered through O-RAN standardization might be exploited, and privacy violations become the least of concerns for network operators. The AI-based decision These threats are very general and in particular related to attacks against the communication infrastructure. There are independent of the usage of Open RAN.
Violence against democracy
Besides espionage to dedicated people, also every other citizen can be envisaged. [100] This might be a real threat to the democracy or freedom of speech in the world and it should thus be avoided in any case that one actor receives full control.
These threats are very general and in particular related to attacks against the communication infrastructure. There are independent of the usage of Open RAN.
Majority attacks and supply chain concerns
It is a danger if there is a significant involvement in Open RAN development from one country or one region, facilitating possibilities for [104,61] any type of attack. Special attention should be given that no new secret alliances are formed, and therefore a well balanced spread among the suppliers of O-RAN equipment and software is required.
These threats are very general and in particular related to attacks against the communication infrastructure. There are independent of the usage of Open RAN. However, use of SW make Open RAN more vulnerable than RAN making the backbone of the O-RAN architecture is inviting instilling of botnet-type autonomous constructs that entail a sophisticated cyber intrusion; where prevention is quite arduous [103]. Therefore, proper ethical restraints should be drawn in a global scale to prevent such acts of espionage, while monitoring to detect such acts are equally pertinent.
• Attacks on critical infrastructure: Critical infrastructure typically consists of the management of power grids, water supplies, manufacturing, and transportation infrastructure. More and more, 5G is used as the backbone of communication in these infrastructures. Therefore, a dedicated cyber attack disrupting this critical infrastructure would have a devastating impact on the people dependent on this. Conversely, the control of the critical infrastructure is handed over to the 5G network operators. Therefore, their responsibility is ever so critical and honourable. Since the government level acts for blocking critical infrastructure to deliver threats in the geo-political arena are not rare occurrences, O-RANs dependence on the same network operators is raising concerns in the global shared resource market. Thus, the responsibilities of the network operators become extremely important within the O-RAN domain.
• Violence against democracy: If an actor receives the power to perform the role of big brother in all communication, there is a real threat for democracy and freedom of speech. As all the means of global economic infrastructure are envisaged to be shifted to a digital environment backed by the 5G enabled networks, democracy becomes merely a concept without any context or standing in case of a total takeover. The ideals that made O-RAN more efficient and flexible might lead to the downfall of modern democracy and its stance on the global scale.
• Majority attacks and supply chain concerns: As mentioned in [104,61], the Open RAN Alliance currently includes a wide range of high security risk companies. If the efforts in the development and standardization process for Open RAN is dominated by partners belonging to one country or even one region, it can cause an imbalance resulting in a new alliance that will still enable espionage possibilities and disrupt the intended openness. As supply chains formed through globalization are relying on online trading and financial platforms for international transactions, the responsibility of the O-RAN stakeholders is ever so vital in facilitating the required digital infrastructure. It is obvious that Blockchain serves as an appropriate solution to secure such a financial infrastructure. The majority of attacks or 51% attacks are however, proving to be realistic, where an attacker is capable of withdrawing the payment after the merchant has sent the product [106]. This threat is intimidating the credibility of the Blockchain networks by enabling plausible deniability -which is one of its foremost purposes for the emergence of Blockchain.
Open RAN Best Security Practices
We discuss the best security practices for Open RAN in this section. As Open RAN is a derivative of the conventional C-RAN, it will inherit many threats and vulnerabilities of C-RAN. Therefore, a number of C-RAN security solutions can be adopted by Open RAN without any significant modifications. For example, the existing security solutions to prevent primary user emulation attacks (PUEA) can be adopted for Open RAN [39]. In [107], the authors discussed cryptographic and wireless link signatures to distinguish between a legitimate user from an attacker. A helper node is proposed that is placed around a primary user. The helper node acts as a bridge between the primary and secondary users by sending authentic link signatures to the secondary nodes. The authors also proposed a corresponding physical layer authentication algorithm in [107]. There are other security mechanisms against PUEA based on the received signal strength. In [108], the authors proposed naive detection and variance detection methods against PUEA. The authors modeled advanced strategies of PUEA where both the legitimate user and the attacker can exploit estimation techniques and learning algorithms. The variance detection attack is effective against PUEA for a time-invariant channel.
The most widely researched security threat in the medium access control layer is the spectrum sensing data falsification (SSDF) attack where an active attacker transmits error observation to disrupt collaborative spectrum sensing and resource allocation [39]. A joint spectrum sensing and resource allocation scheme is proposed in [109] to combat the SSDF attack. The problem is formulated as a weighted-proportional-fairnessbased optimization problem with an additional constraint of the primary user being sufficiently protected. The authors decomposed the problem into two subproblems which are a resource allocation problem and a cooperative secondary user decision problem. The key idea of the scheme lies in improving the secondary users' sensing reliability and preventing the secondary user from acting maliciously. The computer simulations showed that the proposed scheme deals with the SSDF attack in the co-operative sensing process to improve system robustness.
As the Open RAN systems adopt cloud computing unlike conventional RAN systems, the security solutions for cloud computing are also relevant for Open RAN. In [110], the authors identified security and privacy vulnerabilities of cloud computing that can be exploited by an adversary for various attacks. The authors presented basic requirements to build a secure cloud system by addressing three main challenges, namely, outsourcing, multi-tenancy, massive data and intense computation. To address the outsourcing challenge, the cloud provider needs to provide secure and trustworthy data storage. The outsourced data also needs to be verifiable by the customer. For multi-tenancy, the cloud platform needs to securely perform resource allocation in the virtualized environment. Finally, massive data sets need to be broken down into small sets to accelerate the processing. The solutions to PUEA, SSDF and cloud computing vulnerabilities are applicable to both C-RAN and Open RAN. We invite interested readers to go through [39] for more discussion on the C-RAN security vulnerabilities and solutions.
We identify three key components to resolve security vulnerabilities that are exclusive to Open RAN. The first component to enhance Open RAN security is blockchain-based mu-tual authentication. As O-RAN promotes openness between a pool of untrustworthy O-RU and O-DUs unlike C-RAN, the blockchain can be a very important and unique tool to establish trust between them and enable a safe communication mechanism. The second key component is the physical layer itself. The difference from other RAN systems is the operators have the option to select O-RUs from different competing vendors in Open RAN technology. Thus, the O-RUs can be installed at any moment with the desired number of antennas, front-end processing, beamforming algorithms to enhance the security of the Open RAN. We also discuss RF-fingerprinting techniques which can be crucial to identifying rogue RUs trying to connect to the system. The third key component is AI algorithms. As Open RAN provides more interfaces to enable an intelligent RAN system, we discuss a few examples of AI-enabled enhanced security in Open RAN. Table 9 summarizes the key solutions to threats and vulnerabilities related to Open RAN. Finally, we present a subsection regarding the common mistakes in Open RAN design, their consequences and mitigation.
Blockchain-enabled Open RAN
Blockchain or distributed ledger technology (DLT) is a distributed database for exchanging identities of users and storing records of all user identities that are linked together using cryptography [119]. In other words, it is a chain of interconnected information blocks that creates a public ledger for recording a list of transactions. Blockchain is popular for its crucial role in modern cryptocurrency systems to provide a secure and decentralized record of transactions [120]. Blockchain or DLT has also emerged as a tool for designing a self-organized and secure radio access network (RAN). Blockchain-enabled identity management and authentication can lower the cost and aid the core network to provide more secure and user-oriented service in an era of open and distributed RAN deployments [121,122]. In [111], a RAN framework has been presented by leveraging the principles of blockchain. This framework proposes that the UEs and APs in the network agree about payments or spectrum assets based on a contract. The terms of this agreement are recorded by a smart contract, authorized by client signatures. Afterward, the contract is verified by the miners to determine whether the UEs have sufficient credit balance or the APs have sufficient spectrum assets. The verified contracts are aggregated to a block, which is added to the existing blockchain. In this process, a UE will be granted limited-time access to the spectrum assets while an AP can receive the payment automatically. As a result of enforcing the rights of relevant parties by means of a smart contract, trust has been established between initially untrustworthy UEs and APs. The application of blockchains and smart contracts for RAN can be further extended to cooperative communication, mobile ad-hoc networks and privacypreserving communication systems.
As the RAN technology is moving towards more open, intelligent, virtualized and interoperable networks in the form of Open RAN, it will be crucial to develop trust between a pool of O-RU and O-DU vendors. A blockchain-enabled smart contract establishes trust between these vendors and provides a mechanism to constantly monitor the development of the system by 20 A helper node can be used proposed which acts as a bridge between the primary and secondary user [107] or a variance detection method is adopted based on received signal strength [108] No, applies to both C-RAN and Open RAN Spectrum sensing data falsification A weighted-proportional-fairness-based optimization problem can be formulated with an additional constraint of primary user being sufficiently protected [109] No, applies to both C-RAN and Open RAN Cloud computing vulnerabilities A secure cloud system needs to address three main challenges, namely, outsourcing, multi-tenancy, massive data and intense computation [110] No, applies to both C- RAN Increasing the number of antennas and corresponding digital front-end processing with beamforming capabilities can diminish the threat of passive eavesdropping [114] and increase the probability of active attack detection [115] Yes, the solution is O-RAN specific because the operator has the freedom to choose suit-
able O-RUs and O-DUs from different vendors in O-RAN Conventional security framework
Open RAN enables intelligent zero-trust security framework upon which advanced AI algorithms can be developed to provide security in untrusted networks [116] Yes, the proposed framework in [116] adopts service based design by leveraging Open RAN architecture to ensure ease of integration DDoS attacks
Open RAN systems can employ machine learning algorithms that are trained to protect the network from DDoS attacks with very high accuracy [117,118] No, the machine learning solutions to detect DDoS attacks can be used by any RAN system independent third parties. One such example is a blockchainenabled privacy-preserving point-to-point (P2P) communication in Open RAN. Due to the advent of distributed and decentralized functionality of Open RAN, different P2P communication in mobile networks such as device-to-device (D2D) or machine-to-machine (M2M) will be benefited. However, the fundamental security flaws for P2P communication in a mobile network will remain in an Open RAN. The P2P has limitations in global peer discovery and routing without third party assistance and thus, the coverage is a bottleneck. In the centralized architecture, the UE is restricted to communicate directly with other users. Two users under the same BS cannot be directly connected to each other without the involvement of the core network. The reason is several key functionalities such as identity authentication, routing, etc, are only done at the core network in the state-of-the-art mobile networks. The distributed identity authentication issues in the current architecture can be addressed by blockchain due to its decentralized nature. The identity authentication can be performed locally at the RAN with a global identities record. In this way, two users under the same RAN unit can communicate with each other directly without accessing the core network.
In [112], a blockchain-enabled mutual authentication architecture is presented for identity management in Open RAN that does not require a third-party Certificate Authority (CA) or Public Key Infrastructure (PKI). The authors proposed a blockchain address (BC ADD) as a global identity for all UEs within a RAN, where all users generate their own address by hashing their public keys. These newly generated addresses are recorded by the ledger records and used as anonymous identities locally or globally. The relationship between the public key and BC ADD is strictly one-directional. As a result, it is difficult to fake a BC ADD when the pubic key is known, or vice versa. The authors compared the performance of their proposed mutual authentication method based on blockchain with Internet Key Exchange version 2 (IKEv2) and Transport Layer Security (TLS) from signaling, communication and computation perspectives. The authors noted blockchain based scheme only requires 2 signals while IKEv2 and TLS 1.3 require 4 and 9 signals, respectively. The authors also demonstrated that the blockchain method requires significantly less number of bytes for finite-field and elliptic curve cryptography (ECC) for communication and computations. For communication, the blockchain method requires 1060 and 356 bytes for finite-field and ECC, respectively. The IKEv2 requires 3820 and 3110 bytes for finite-field and ECC, respectively, for the same functionality. Similarly, the number of bytes required for both finitefield and ECC is significantly higher for computation in IKEv2.
In [123], the authors presented a privacy preserving framework of blockchain enabled RAN for increased efficiency and enhanced security. The authors simulated their system model in Hyperledger Fabric 1.2 based simulator. The simulation shows that the blockchain enabled RAN achieves higher throughput and lower resource consumption compared to conventional RANs. As Open RAN promotes open source software development for the base stations, it is imperative to develop a distributed security mechanism with many eyes to observe the changes in the Open RAN operation. Blockchain can provide an ideal framework to support RAN elements from different suppliers in a secured and organized manner. We believe blockchain can be a key element in future ORAN systems for authentication and identity management. However, several challenges remain to integrate blockchain technology into wireless networks. For power-limited node devices and cost-sensitive transmission networks, implementing blockchain based mutual authentication can be challenging. In addition, latency can be a critical issue of blockchains for delay-sensitive scenarios in a wireless network. Despite these challenges, blockchain can be an important component of the Open RAN systems as they suffer from more security challenges than traditional RANs due to their openness by design.
Leveraging physical layer to enhance Open RAN security
From an O-DU's perspective, it is crucial to differentiate a legitimate O-RU from a masquerading O-RU in the Open RAN systems. The transmitter of the O-RU consists of radio frequency (RF) modules such as digital-to-analog (DAC) converters, power amplifier (PA), analog band-pass filters, frequency mixers, etc. Despite decades of research and development effort by the microwave circuits community, longstanding imperfections still exist in the RF transmitter chain. These imperfections cannot be altered or corrected without significant effort and thus, can be exploited as radiometric signatures of different O-RUs. In addition to conscious design decisions, these imperfections can stem from uncontrollable factors in the manufacturing process such as differences in the semiconductor doping industry. As a result, different O-RUs can have very different flatness and ripples in the RF spectrum, differences in rejection and transition bands, mismatch in the I/Q phase, DC offset or gain imbalance, etc. The idea of using RF fingerprints to identify devices through such intrinsic features of the RF stages has been widely explored by the microwave circuit community. We believe such RF fingerprinting can also be used as the first line of defense to detect a masquerading O-RU.
A review of RF fingerprinting techniques has been presented in [113]. The authors described state-of-the-art techniques for RF fingerprinting based on transient response. These methods utilize the transition from the turn-off to the turn-on of a power amplifier that occurs before the start-up of a radio unit. The transient response of every power amplifier is unique and thus, can be used for wireless device identification. However, this method is effective when the transient is accurately known, i.e. the exact beginning and the exact end. The authors discussed several methods for detecting the start point of the transient, such as Bayesian step change detection, Bayesian rampup change detection, phase detection, mean change point detection, etc. In [124], the authors proposed an identification system based on the steady-state response of the hardware. The system is called passive radiometric device identification system (PAR-ADIS) and uses five features: frequency error, correlation, I/Q offset, magnitude errors and phase errors to identify a device. As detecting the transient response requires a very high sample rate, which is infeasible in many applications, the steady-state response is frequently used for RF fingerprinting.
In [125], a model based approach is presented for the identification of wireless users via power amplifier imperfections. The authors exploited the differences in non-linearities of I/O characteristics of a power amplifier modeled with the Volterra series. The authors proposed a generalized likelihood ratio test (GLRT) and a classical likelihood ratio test to identify the legitimate user. A symbol based statistical RF fingerprinting technique for fake base station identification is presented in [126]. The authors present a scheme to detect unique non-linearities based on hardware impairments of the transmitter. The proposed scheme is based on the assumption that a fake base station tends to violate the spectral mark and introduces large amplitude and phase errors compared to a legitimate base station. The RF fingerprinting can be an ideal mechanism to verify that an O-RU is secure enough to be connected to the O-DU of the O-RAN.
A massive multiple-input multiple-output (MIMO) O-RU can also improve the security in an Open RAN system. A massive MIMO system equips the base station with a large number of antenna elements which can serve a large number of user terminals in the same frequency band [127]. It should be noted that the antennas reside in the O-RU of Open RAN while the baseband layer processing is performed in the O-DU. The number of layers in baseband is typically 16 or less in a 5G base station. For an N-layer O-DU, the O-RU must support at least a number of N antennas and RF-front end circuitry. If the number of antennas in O-RU is significantly higher (e.g. 8-10 times) than N, the Open RAN system can be considered a massive MIMO system. Every layer of baseband data in the O-DU can exploit the higher number of antennas in O-RU with beamforming techniques. The base station can direct its baseband data in a specific direction by constructively adding multiple antenna streams and improving the signal quality. Due to their beamforming capability, the massive MIMO systems are more secure than the small-scale MIMO systems. It is possible to direct a narrow beam toward a legitimate user in a massive MIMO beamforming system. If an eavesdropper is not in the vicinity of the legitimate user, the received signal power of the eavesdropper is significantly diminished while the received power of the legitimate user increases manifold.
In [114], the authors presented analytical results that showed a passive eavesdropper has a negligible effect on the secrecy capacity in a massive MIMO system. Their simulation shows that a passive eavesdropper's capacity remains the same with an increasing number of antennas. However, the legitimate user's capacity increases greatly for a large number of antennas. For a small-scale MIMO system with 2-8 antennas, the legitimate user's secrecy capacity is about half of the channel capacity. When the number of antennas is 100, the secrecy capacity reaches about 85 percent of the channel capacity. The primary reason for the resilience of a massive MIMO system against a passive eavesdropper is based on the assumption that the uplink channel estimation is independent of the eavesdropper's channel. However, an active eavesdropper can transmit pilot signals to the base station to influence the base station's transmit beamforming design. In such a scenario, the physical layer security of a massive MIMO system is compromised and the achievable secrecy rate vanishes with the increasing power of the eavesdropper's pilot signal. However, the probability of detecting an attack increases with increasing eavesdropper's signal power [115]. Two active eavesdropper detection methods have been proposed in [114]. The first scheme is based on random quadrature phase-shift keying (QPSK) pilot transmission by the legitimate user. The idea is that the phase of two legitimate pilot signals converges to valid PSK symbols as the number of antennas is large. In the second scheme, the beamformer is constructed in such a way that the received signal at the legitimate user is equal to an agreed value. These two detection schemes are only effective due to the large number of antennas in a massive MIMO system. Due to their centralized structure, the current base stations typically employ a fixed number of antennas and baseband layers. In the Open RAN design paradigm, the operators can select an O-RU with a higher number of antenna chains and thus, with a capability to beamform and enhance security. We believe the ability to select O-RUs with the desired configuration will be crucial to improving the overall security of an Open RAN system. The most popular physical layer candidates for future wireless standards such as cell-free MIMO or reflective intelligent surface (RIS) can utilize a high number of antennas. Thus, applying specific physical layer configurations is a viable solution to combat security threats such as eavesdropping in an Open RAN system. Identifying rogue O-RU will be crucial for Open RANs to succeed and replace the conventional RANs. We believe RF fingerprinting could be the first line of defense against a rogue O-RU that is trying to connect to the network.
AI enabled Open RAN Security
Since the introduction of deep learning by Hinton et al., there has been a reinvigorating interest in AI applications in the wireless communication research community. The ML based solutions have also been popular in the network security research community. Despite some security concerns associated with AI based solutions as mentioned before, the AI automated security solutions will represent an essential key element of future wireless networks. The application of AI is so crucial that entire security frameworks have been proposed to utilize the AI algorithms. In [116], an architectural concept design of an intelligent zero trust architecture upon which advanced AI algorithms can be developed is proposed in order to provide security in untrusted networks. This framework adopts a service-based design by leveraging Open RAN architecture to ensure ease of integration. The three main components of zero-trust architecture in Open RAN are intelligent agent or portal (IGP), intelligent network security state analysis (INSSA), and intelligent policy engine (IPE).
The IGP employs a reinforcement learning approach to analyze the incoming traffic, provides an initial risk assessment, and a model for their security posture. The reinforcement learning model used by multiple IGPs can be a common model that is trained in the federated learning approach. By utilizing federated learning, a more comprehensive model of the local envi-ronment is trained by different subjects. The second component INSSA provides a dynamic risk assessment for every access request. The authors proposed a graph neural network to model the state of Open RAN. The neural network models the communication patterns of the Open RAN with the goal to assign risk scores in such a way that the overall security metric is maximized while granting access. The final component of the zero trust architecture in Open RAN is the IPE which takes the final decision to grant access. The IPE is based on a neural network called long short-term memory (LSTM) to evaluate the risk of granting access based on reports from IGPs and INSSA. After making a decision, the IPE monitors the security state of the session. The IGP, the INSSA, and the IPE work together to provide a cohesive framework for zero trust in Open RAN.
Conventional hardware dependent security such as firewalls or deep hardware inspections might not be the ideal solution for a dynamic and open environment of Open RAN. Therefore, it would be crucial to develop automated mechanisms for intrusion detection, attack response and mitigation. An Open RAN system can employ an ML mechanism that is trained to protect the network from DDoS attacks. A plethora of ML based mechanisms for DDoS detection can be found in the literature. Five classification methods, including K-nearest neighbors (KNN), Decision Tree (DT), Random Forest (RF), Support Vector Machine with linear kernel (L-SVM), and Neural Network (NN) have been studied for intrusion detection in [117]. The authors used a limited set of features to enable real-time classification and middlebox deployment. The authors found that all five methods were able to detect DDoS attacks with a high level of accuracy. However, the authors considered only three types of DDoS attacks. A total of 13 different DDoS attacks were considered in [118]. The accuracy of the ML algorithms decreased significantly for this scenario. In addition, both works of [117] and [118] used supervised learning which requires labeled data. Such labeled data can be challenging to obtain and thus, the application of supervised learning is not always realistic.
In [128], the authors discussed network intrusion detection systems using different autoencoder architectures. Autoencoders are a type of artificial neural network based on unsupervised learning that aims to reconstruct its original input vectors. The proposed intrusion detection autoencoder develops a threshold heuristic of the reconstruction error which represents the proportion of abnormality in training data. The authors considered four types of autoencoders namely basic autoencoder, stacked autoencoder, denoising autoencoder, and variational autoencoder. The results showed that stacked and variational autoencoder perform better than the rest. A timebased anomaly detection system, named Chronos, is presented in [129]. Chronos is an autoencoder that utilizes time-based features to detect anomalous DDoS traffic. This method extracts statistical information from time-based features for each small set of packets collected during a time window. The efficacy of Chronos was evaluated by performing extensive evaluations on the CICDDoS2019 dataset. The authors also evaluated the impact of different window sizes to detect DDoS attacks. Chronos achieves an accuracy of over 99% for most attacks and greater than 95.86% for all attacks.
The application of AI for Open RAN security is not limited to intrusion detection. AI is an effective tool to identify devices based on RF fingerprinting. Contrary to the hand-engineered approaches, the ML approaches are able to rapidly identify a rogue O-RU before sharing any network information. In [130], the authors presented a convolutional neural network with a triple loss for RF fingerprinting. The authors demonstrated the feasibility of the proposed scheme over the experimental POWDER platform in Salt Lake City, Utah, USA. The proposed method achieves a 99.86% detection accuracy for different training and testing days on real-world datasets.
In [131], the authors studied four ML techniques to identify RF devices in the time domain. These four schemes are deep neural networks, convolutional neural networks, support vector machines, and multi-stage training using accelerated Levenberg-Marquardt. The authors examined data originating from 12 different transmitters. The accelerated Levenberg-Marquardt based training method achieved 100% accuracy and outperformed state-of-the-art ML methods. A massive experimental study of deep learning for RF fingerprinting has been presented in [132]. The authors analyzed 400 GB of I/Q data transmitted by 10,000 radios. The authors chose convolutional neural networks because of their ability to interpret features better than conventional ML techniques. This work demonstrated that the proposed solution can handle different channel conditions and signal-to-noise ratios, and is scalable to very large populations. Its almost certain that AI will play an integral role in the security of Open RAN systems. However, ML techniques themselves are vulnerable to security threats. The effectiveness of ML algorithms greatly depends on the quality of training data sets. An adversary can also send false data during the training process of the system. Therefore, robust ML algorithms that can tolerate malicious inputs need to be adopted. In addition, stability training can be adopted so that the ML schemes do not deteriorate for different and independent data sets.
General mistakes, consequences and mitigation
Almost all of the technology related attacks mentioned in Section 3.3 are caused because of some general design errors. Table 10 summarizes these major errors, together with their consequences and potential mitigation measures 1 .
The following six general mistakes are identified.
• The hardware-software Open RAN system suffers from insecure design:
The open fronthaul and its interfaces, xApps based radio resource management, Decoupled hardware, open management interfaces, and open source deployments are some insecure design strategies that require more investigation to determine remedial possibilities [133]. This includes misconfigured or poorly configured Open RAN interfaces due to the outdated components or improperly configured permissions, insufficient/improper mechanisms for authentication, encryption and authorization in This type of weakness would allow attackers to inject malware in order to manipulate and harm the Open RAN components, which may result in a variety of consequences going from launching DoS attacks to retrieval of sensitive information including unprotected private keys. As a consequence, the attacker gets unauthenticated/unauthorized access to Open RAN components via the different Open RAN interfaces.
In order to offer protection against this, security standards (e.g. Media Access Control Security -MACsec) for Open RAN devices and interfaces should be clearly defined and people should be trained in order to apply these standards and processes [66]. In particular, special attention should be given to all access control mechanisms at every access point. A Security by Design (SbD) approach might be suitable for automating the authorization framework [135]. Security certifications should be issued via trusted security standardization bodies. It is also essential to monitor network traffic for suspicious activity on all levels and to add firewalls and rate limiters to act appropriately.
• Software flaws: The tendency to utilize open source solutions for virtualization deployments as well as for the RU, DU, and CU based deployments is a way to assert the required interoperability within a multi-vendor O-RAN; At the same time exposes the software specifications and configurations to the general public [133]. This opportunity allowing the assimilation of the software standards would benefit either negatively or positively depending on the capability of the attacker or the defender. Software flaws are present in the different components of the Open RAN system, which are not notified and mitigated in time.
In particular, software providing network functionalities play an important role in firewall protection. If vulnerabilities like buffer overflows are exploited, arbitrary commands can be executed with devastating consequences.
As Open RAN is built with open source software, it is important to keep the software always up-to-date and to make sure that they are developed by trusted suppliers, which use third party certificate authorities. Security training for employees is required such that software is developed with the highest standards. Further, Software Defined Security (SDS) concept can be extended to the application layer from the network layer for automating the security functions and detecting software based flaws through machine learning means [137,136].
• Security event log files, generated by the different Open RAN components, are not sufficiently or improperly protected: Security has not been recognized as a function by the O-RAN setup to maintain proper and organized logs for event recording.For instance, they lack information on host name, IP or MAC address, correct timing of incidents, etc. [133]. Malware injection resulting in DoS attacks to retrieval of sensitive information via unauthenticated/unauthorized access [134].
Define security standards and protocols, as in Media Access Control Security (MACsec) for Open RAN devices and interfaces [66]. Issue security certificates via standardized bodies. Train people to apply the defined standards and processes. Monitor network traffic for suspicious activity on all levels. Add firewalls and rate limiters to act appropriately. Proper Access control mechanism should be deployed as SbD [135]. Software flaws Software for firewall protection can result to failures. Exploitation of buffer overflows results in the execution of arbitrary commands with devastating consequences.
Be careful with open source software and keep software always up-to-date. Invest in security training for employees. Purchase software from trusted suppliers and use third party certificate authorities. Follow SDS approach for automating flaw detection [136,137]. Insufficient protection of security event log files.
Security restoration delays, wrong audits and threats persistence.
Automate the log monitoring process and add rate limits. Define log management clearly in the standard. SbD approach for automating log maintenance while introducing anomalous log detection using ML [138,135]. Insufficient protection of data storage Attacks against privacy, including data tampering, information disclosure, elevation of privilege, etc.
MACsec protocol suite can be followed for the fronthaul, while SDS based approach can be deployed for the network and application layer network automation [139,135]. Follow data poisoning prevention methods [140]. Compromise of integrity and availability DoS attacks. See also consequences of first error in case of improper authorization and authentication A proper autonomous authentication and authorization mechanism is required to protect integrity and ensure availability [116].
Physical access
Retrieval of stored private keys, certificates, user plane data, control plane data and management data in cleartext. Modification of Open RAN components settings and configurations in order to disable security features and allow eavesdropping or wiretapping on various planes, creates performance issues [38].
Define security standards for physical security. Ensure that all stakeholders in the Open RAN system are identified, authenticated and trusted [67].
25
Compromise or incomplete logs can result in security restoration delays, wrong audits and threats persistence.
The standard should clearly define how to manage the log monitoring process and rate limits should be added. SbD approaches can be employed for centralized log auditing, while an automated anomalous detection method for security logs can be an efficient directive to overcome this pitfall [138,135].
• Sensitive data, stored, processed and transferred among the different Open RAN components, are not secured according to industry best practices: For instance, appropriate encryption and integrity protection mechanisms are missing, inappropriate access control, lack of traceability of the access in the audits, etc.
This weakness will mostly result in attacks against privacy, including data tampering, information disclosure, elevation of privilege, etc.
Security standards for authentication and end-to-end encryption (MACsec) following the latest developments should be defined [66]. Security certifications should be issued via security standardization bodies. People should also be trained to apply the defined standards and processes. Finally, data poisoning prevention algorithms should be deployed [140].
• Integrity and availability of Open RAN components can be compromised: Integrity and availability of Open RAN components can be compromised due to overload situations caused by DoS attacks or increased traffic and where the Open RAN components do not possess the required functionalities to deal with it.
A direct consequence is of course a DoS attack. However, other possibilities may appear in case of insufficient or improper configuration (see first error). For instance, an attacker might be able to boot Open RAN components from unauthorized memory devices and thus instill a selected malware to a xApp or any other operational entity [47].
Similar counter measures as in the previous mistake should be taken.
• Several possibilities allow physical access to different components in the Open RAN system: First, it can appear via ports and consoles (such as JTAG, serial consoles or dedicated management ports) which are insufficiently secured. Second, credentials of the administrator may be insufficiently protected. Another possibility is that the configuration module of the hardware and software might be insufficiently protected against malware injection and manipulation.
Physical attacks on the Open RAN deployment enables the retrieval of stored private keys, certificates, user plane data, control plane data and management data in cleartext. Moreover, attackers can try to modify the Open RAN components settings and configurations via local access in order to disable security features and allow eavesdropping or wiretapping on various planes, create performance issues [38].
Also for physical access, the required security standards should be developed. In addition, all stakeholders in the Open RAN system need to be identified, authenticated and trusted [67].
Security Benefits of Open RAN
Besides all these risks, Open RAN brings of course a whole series of benefits. Several benefits are typical for Open RAN, others are also available in V-RAN and some of them are common for all 5G networks. Table 11 provides an overview of the main benefits.
Open RAN specific
Full visibility
Due to virtualization and the disaggregated components connected through open interfaces, operators have direct access to all network performance data and operational telemetry data representing activities between/within the different network functions. The integrity of this data is more ensured as this data is created isolated from the functions' executing environment. Combining this data with security log data results in an earlier detection of security problems and easier detection of the root cause [34,141,142].
Note that full visibility can also be a risk. Due to the complexity, the root cause cannot always be easily detected and there is a danger that different vendors will not take accountability for potential issues. Following [144], it is claimed that the time and cost to perform a complete security review would seem to be multiplied by the number of vendors the operators take on board.
Selection of best modules
Operators can more easily select the vendors offering the best products, meeting the required industry security standards and certifications [34,141,142]. Examples of industry best practices are for instance "secure by design" DevSecOps in which information security operations are integrated into De-vOps workflows and automated testing in development of containerized applications [145]. The operator can also collaborate with the vendor to determine and influence Continuous Integration/Continuous Deployment (CI/CD) processes with continuous regression testing and software security auditing used by the supplier. Other good practices are the adoption of Supplier Relationship Management (SRM) with an inbound development process and strict security controls for Free and Open Source Software (FOSS), trust stack management with software coming from reliable supply chains and trusted, well-defined operations, intelligent vulnerability management, and multivendor System Integration (SI) with continuous verification on vendors sharing the same interpretation and implementation of functions [34,31]. The zero trust security principle can be implemented. [34] There are a range of industry best practices that can be adopted including Groupe Speciale Mobile Association (GSMA), National Institute of Standards and Technology (NIST), European Union Agency for Cybersecurity (ENISA), National Telecommunications and Information Administration (NTIA), Center for Internet Security (CIS), Open Web Application Security Project (OWASP), Open Standards, Open Source (OASIS), national cyber security organisations, Building Security In Maturity Model (BSIMM), Cloud Native Computing Foundation (CNCF), the Linux Foundation, SAFECode and CNTT [141,142].
Diversity
Integrating independent and individual modules decreases the risk that common coding errors or practices of one single entity impact large parts of the network and thus decrease the attack range. Consequently, diversity helps to balance the security risks. Open RAN enables an expanded pool of vendors on the market, reducing a nation's dependence on any sole vendor for wireless services [141,48,143]. Despite the O-RAN entities being contrived following common and established standards, multiple vendors might embed diverse mechanisms, technologies to meet the standards or guarantees. This will eventually create competition among the vendors for better market returns. This competition can be considered healthy from the security perspective, as the standards might have to be improved from the security front to convince the consumers.
Modularity
Due to the modularity of the network, operators can switch to a CI/CD operating model, enabling seamless and effective patch management for fixing any detected security vulnerability. As a consequence, the vulnerabilities in the network are faster removed. In addition, updates become more transparent and have less impact on the overall network. Moreover, also operational agility is obtained making it possible to replace functional elements by new versions or capabilities [34,142]. The CI/CD method combined with the DevSecOps principles can insure individual modules carry out the updates or patches separately, eliminating any opportunity for complete compromise of the system in case of a malicious agent was conveyed via the updating process [146].
Enforcement of security controls
Due to the choice among different vendors, modularity and open interfaces, the operator is in the position to demand strong security capabilities and control of its suppliers. For instance, in the case of a cloud architecture, the operator and the cloud infrastructure supplier have a common agreement in which this last one is responsible for the deployment of the latest security tools for detection and prevention [31,141].
Open interfaces
Open interfaces at the different levels give a higher exposure, resulting in more scrutiny and thus higher overall security. Thanks to the open interfaces, operators are not dependent anymore on the supplier in case of (security) issues and can do upgrades themselves, being able to react faster. It also gives the possibility to experiment with new functions and new vendors, exploring new ways to secure the network and its operation [30,48,141]. This is at the same time a risk as in order to explore new possibilities by the operator, sufficient qualified people are required, which is not evident due to the complexity of the overall system.
Open source software
Open-source software presents security challenges regarding its open nature but has the advantage of being verified by multiple independent parties, being rigorously and varied tested, and customized against threats [30].
As mentioned before, the use of open source also includes many risks. It was concluded in the Github 2020 State of the Octoverse Report that vulnerabilities remain undetected in many cases for more than four years, before being disclosed [147]. Therefore, one cannot simply state that open source software is faster patched than proprietary software.
Automation
The introduced intelligence in Open RAN can be used to automate the management and control via big data analysis, AI and ML. As a consequence, closed loop responses to changes in the network can be automatically performed. This has the advantage that no human interactions are required anymore, which inherently includes threats like humans accidentally altering the security posture of a network function or maliciously harvesting credentials, changing configurations, or implanting malware within the network [34,30,48].
Again, automation can bring risks as previously identified at the ML algorithms.
Open standards
Open RAN will be developed based on open standards, defined by the Open RAN consortium. Such standards enable to align on a common approach approved by leading members in the field and coordinate all information regarding security threats, vulnerabilities and exploits [34,141].
A prerequisite is of course the presence of these standards, which are not fully available at the moment. In addition, these standards should be correctly implemented.
V-RAN specific
Isolation
Isolation is obtained via the defined interfaces between functional elements in an Open RAN. It offers on the one hand the possibility to insert controls for monitoring and on the other hand allows software updates and patches to be installed with less risk that version dependencies will create issues [141,10].
Increased scalability for security management
Often, there are trade-offs between application, performance and security requirements. Due to the modularity, operators can tailor their deployments and shift more easily the resources for monitoring and control to meet better to these requirements and improve scalability [148]. Also vRAN functional elements can be shifted to provide better isolation [141].
Control trust
Since operators control the platforms on which virtualized functions run in Open RAN, they have also complete control on the trust infrastructure [149]. The identity and provenance of each functional element is known and managed by using strong cryptographic mechanisms like signature operations. Each new version is validated by the operators and therefore they have control on what is running where on their networks [141].
However, it must be taken into account that the situation becomes more complex as there are more assets and stakeholders involved.
Less dependency between HW and SW
In an Open RAN, there is less dependency between the network software and hardware. This makes it in the first place easier to perform the required upgrades in a faster way. Second, it also avoids risks associated to isolated security breaches [142,150].
Private network
Private 5G networks will soon become the general trend as they enable companies the possibility to fully customize the network according to their specific needs with respect to speed, bandwidth, security requirements, on their locations and own timetable. It will enable companies to offer their customers a dedicated 5G experience, with applications in a large range of domains from healthcare, manufacturing, transportation, education, etc. Companies will have the option to build out and run their own private 5G network, or they can also outsource it to a mobile network operator or systems integrator [151]. One such option is via network slicing, where each slice can be seen as a complete end-to-end network and includes the security capabilities according to the needs [142].
There will be soon many players on the market to launch this innovative network as a service concept, replacing in many cases their existing Wi-Fi and fixed wireless/wired infrastructure.
More secure storage of key material
In traditional network architectures, sensitive cryptographic key material such as for instance Access Stratum keys are more vulnerable to various threats as they are stored at the cell site [152]. In Open vRAN, this key material can be stored deep inside the network in a secure vCU, hosted in a data center [142].
5G networks Related
Edge oriented
Due to the open interfaces, the operator is able to spread the security analysis throughout the network and include monitoring at the edge. These edge-focused analytics will facilitate the detection and prevention of attacks at the lower part in the network in order to avoid DDoS and to block malicious data from reaching the core network. This is in particular important to support mobility services like services offered by IoT [34,153].
Simpler security model
In zero trust [154], nothing is trusted unless it is verified, regardless of the location. The O-RAN Alliance completely embraces this principle. Therefore, everything needs to be verified and results need to be communicated [34]. Zero trust networking can enhance security in different domains relying on robust standards. First it enables to secure the technology and application stack including all interfaces and APIs. Second, it allows the leverage of the cloud-based nature of 5G and the deployment of cloud security functionality and telemetry. Third, it ensures the tailoring and customization of the security control via network slicing. Finally, it makes it possible to deploy multiple layers of authentication [155].
Lessons Learned and Discussion
Lessons Learned
In this section we summarize the key lessons learned for previous sections of this survey.
Threat Vectors and Security Risks Associated with Open
RAN We have provided a clear taxonomy and an extensive overview of the different types of risks in Open RAN. There are basically three main domains of risks: process, technology and global. The global risks are general and apply for any type of RAN. In particular, we have shown that most of the technical risks follow from basic errors like insufficient mechanisms for encryption, authentication and authorization, improper configuration, software flaws, inappropriate event log management, lack of integrity and availability protection and unprotected physical access. Corresponding risk mitigation measures are provided, which are also related to the identified process risks. Basically, it all falls or stands with well defined standards and policies on all different processes covering the complete lifecycle, which can be clearly implemented, verified and audited in an automatic way. This is currently an ongoing work, but progress is on the way.
Open RAN Best Security Practices
As a derivative of C-RAN, Open RAN can inherit many security solutions and practices straight from C-RAN [156,10]. However, due to its open architecture, it also requires a significant number of unique security solutions. Most of such solutions are required due to their lack of restrictions on O-RUs from different vendors. Open RAN enables blockchain based mutual authentication and privacy preserving P2P communication. Unlike conventional RAN technologies, Open RAN platforms can be upgraded with beamforming functionality and provide a countermeasure against eavesdroppers. Open RAN provides a platform to utilize the full benefit of AI algorithms for security solutions. In addition, many technology related attacks are caused by general design errors and can be mitigated by defining security standards and automation.
Security Benefits of Open RAN
A significant amount of additional security benefits have been identified for Open RAN, compared to v-RAN and even 5G networks. However, most of them are closely related to the security risks. For instance, full visibility, selection of best modules, diversity, modularity and open interfaces also bring increased complexity and interdependency, requiring sufficiently trained people and trusted stakeholders. The same holds for open source software, which clearly has several advantages, but also brings several risks as identified here. Another example is the possibility to create an advanced level of automation in the network, but at the same time can lead to vulnerabilities from potential AI/ML attacks. Finally, for the enforcement of security controls and the adoption of open standards, the required standards, processes and policies still need to be fully defined.
Discussion
Cost Of Security in O-RAN Deployments
The RAN section of the telecommunication domain typically costs around 70% to 80% of the entire cost of the network; and it represent the best opportunity to reduce the cost of the network. In comparison, O-RAN is economically beneficial than PHY RAN or vRAN. The openness of the O-RAN is inviting the vendors to be more competitive with their apparatus, where 30% less amount can be expected on Open Radios and O-RAN software. As proprietary BBUs are replaced by the typical servers, and their cost become less in comparison. According to [157], 32.5% Capital Expenditure (CapEx) savings and 21% Operational Expenditure (OpEx) savings can be expected regardless of whether the O-RAN is configured D or C setting. O-D-RAN in contrast to O-C-RAN has higher CapEx and OpEx values.
The major security repercussions of the O-RAN deployments are forecasted due to its openness of interfaces and modules. Ramifications for such flaws are typically automated security solutions or standards that define secure protocols for communication channels that cover confidentiality, integrity, and accountability; and an automated access control scheme. In addition, an AI or ML driven firewall and IDS facilities are imperative for securing the sub domains. Even with highest level of cryptographic primitives, computation power required for the secure communication protocols can be managed within the available O-RAN resources. The security overhead applicable to the channels are aggregated to the OpEx. The service oriented 5G networks are offering most of its services as cloud based services. Security can also be offered as a service where access control framework, firewall, and IDS facilities can be different flavors of the Security as a Service (SECaaS) use case [158]. These cloud oriented edge leveraged services eliminate the requirement for dedicated hardware; and nullify the required CapEx for launching SECaaS. Such a service can be purchased as a subscription based service that covers the entire O-RAN fronthaul domain, and it can be shared among the other O-RAN services. Hence, OpEx can be manageable. This service outsourcing will allow dynamic scaling ability. For security management and orchestration however, an agent of the SECaaS service should be deployed within the O-RAN for monitoring the security related actions and responses. Substantial amount of CapEx should be allocated for this agent as it requires to be high performing. Considering all these aspects where CapEx and OpEx are allocated for, the cost does justify the benefits granted through the SECaaS based services that would mitigate any disruptions to the O-RAN system.
Impact of Quantum Computing
The Quantum Computing (QC) represents a superior computing power that exceeds almost 158 million times faster than a state-of-the-art supercomputer [159]. The QC manifests an amalgamation of quantum mechanics of superposition, interference, and entanglement. The models that form the QC computations are often based on quantum bits (qbits), where the qbit value scopes higher than a typical computing bit, with its states. With this enormous computing power, QC is an obvious candidate for O-RAN processing core components, specifically for RIC-based computations. A QC enabled core RIC can deliver the real time outcomes of the Near-RT services. In spite of its power, QC based computing requires unique models for performing computing operations. Typical radio resource management and allocation computations might have to adhere to QC models or algorithms for solving the problems.
From the perspective of security, QC is a technology that challenges the complexity of modern applied cryptographic algorithms [160]. The RSA algorithm, which is believed to be unbreakable in the current context, can be broken with a QC employing Shor's algorithm factoring discrete logarithms. Thus, QC poses a threat to the security of future networks and systems despite its rare existence. As a solution, Quantum Resistance (QR) cryptography was introduced and bares a significant interest among the research community. QR algorithms can be formulated from lattice-based, multivariate, hash-based, or elliptic curve based methods [70]. Thus, O-RAN can leverage the capabilities offered through QC for achieving the guaranteed service level requirements. A Quantum Key Distribution (QKD) infrastructure can be adopted as the PKI for O-RAN internal entities including the xApps and rApps. On the contrary, QR based defense mechanisms should be adopted for signaling and critical channels for improving both the security and efficiency of the O-RAN.
Role of Securing B5G/6G Applications
Metaverse is a concept introduced to transcend physical space into a digital reality where all the possible actions in the real-world are enabled within the digitized world. Though the technologies Virtual Reality (VR), Augmented Reality (AR), Mixed Reality (MR), and Extended Reality (XR) are available, a holistic solution that incorporates all these digital visualization technologies is lacking [161]. Metaverse satisfies that void and envisages possibilities beyond measure. The network bandwidth, latency, jitter, availability, and reliability aspects of the RAN should be at its highest capacity to deliver a successful Metaverse [162]. In addition, a significant level of interoperability and flexibility should be maintained within the network to guarantee performance with haptic sensory feedback. The O-RAN is capable of delivering the required flexibility and interoperability within its network. The security concerns of the Metaverse mostly exist in the virtual domain. Therefore, O-RAN cannot guarantee the internal security of the Metaverse; but can secure the network domain via typical security mechanisms.
Digital Twin (DT) technology contrives an exact replica of an object in the digital space. The replica or the twin, however, is formed in a simulated environment where all the actions committed by the actual object and its reactions can be mimicked on its digital counterpart [163]. The main purpose of a DT application is to enable remote controlling and monitoring of apparatus or equipment situated in a factory environment align with Industrial IoT (IIoT) deployments [164]. The DT concept can be visualized as a miniature version of the Metaverse where an interactive interface is formed through AR-based collaborative tools. The O-RAN dynamic and flexible launching of xApps, with their Near-RT standards can facilitate the DT applications successfully. It requires lesser network based requirements than Metaverse. Though, the service rendered by the DT can be mostly critical, and require dedicated service channels with priority. Thus, fronthaul communication channels should embed better security credentials for DT deployments.
Conclusion
In order to cope with the continuous growth of mobile subscribers, mobile data and mobile services, a drastically new approach is needed in order to ensure that the network resources are used in the most optimal way. In addition, this solution should particularly take into account a thorough security protection as also the amount and impact of cybersecurity attacks are continuously increasing.
Open RAN offers all the possibilities to enable a great breakthrough in the network technology landscape and is able to address most of the current shortcomings in RANs thanks to the added openness and intelligence. However, due to this totally new approach where multiple vendors can now simultaneously integrate their technology, a more complex ecosystem exists resulting in a multitude of new risks and opportunities. We have provided a comprehensive overview of these different risks and benefits. We also discussed the best security practices to be applied. As an important conclusion, in order to fully benefit from the most essential opportunities and to avoid the most important risks, the existence of an extended standard describing in detail the different processes in Open RAN is an essential step.
Figure 1 :
1High level Comparison of Open RAN with Traditional RAN
Figure 2 :
2. The main elements of the Open RAN architecture include Service Management and Orchestration (SMO), RAN Intelligence Control (RIC), O-Cloud, Open RAN central unit (O-CU), Open RAN distributed unit (O-DU), and Open RAN radio unit (O-RU). • Service Management and Orchestration (SMO): The SMO framework is a core component of the Open RAN The high-level architecture of Open RAN proposed by the O-RAN alliance.
Figure 3 :
3The Threat Taxonomy of Open RAN Systems • O-Cloud: This is a physical computing platform. It creates and hosts the various virtual network functions (VNFs) and cloud network functions (CNFs) which are used by near-real-time RIC, O-CU control plane, O-CU user plane, and O-DU[46].
Figure 4 provides an overview of the main technology related risks in Open RAN.
Figure 4 :
4A Summery of Related Attacks on Open RAN Architecture and It's Components
Table 1 :
1Summary of Important AcronymsAcronym
Definition
Acronym
Definition
3GPP
3rd Generation Partnership Project
5G
Fifth Generation
AI
Artificial Intelligence
AICPA
American Institute of Certified Public Accountants
API
Application Programming Interface
BBU
Baseband Unit
CA
Certificate Authority
CIA
Confidentiality, Integrity, Availability
CICA
Canadian Institute of Chartered Accountants
CI/CD
Continuous Integration / Continuous Delivery
CN
Container
CNF
Containerized or Cloud-Native Network Function
COTS
Commercial Off-The-Shelf
CPU
Central Processing Unit
(D)DoS
(Distributed) Denial of Service
DL
Downlink
DU
Distributed Unit
E2E
End-to-end
eCPRI
Enhanced Common Public Radio Interface
EI
Election Infrastructure
ETSI
European Telecommunications Standards Institute
FH
Fronthaul
FRANS
Fair, Reasonable and Non Discriminatory
FTP
File Transfer Protocol
GDPR
General Data Protection Regulation
gNB
Next generation NodeB
GPS
Global Positioning System
GUTI
Global Unique Temporary Identifier
HTTP
Hypertext Transfer Protocol
HW
Hardware
IoT
Internet of Things
IP
Intellectual Property
IPSec
Internet Protocol Security
JTAG
Joint Test Action Group
LTE-M
Long Term Evolution for Machines
MI
Model Inversion
MIMO
Multiple Input Multiple Output
MITM
Man In The Middle
ML
Machine Learning
M-Plane
Management Plane
MTBF
Mean Time Between Failures
Near-RT
Near-Real-Time
NF
Network Function
Non-RT
Non-Real-Time
NVD
National Vulnerability Database
OAM
Operations, Administration and Maintenance
O-Cloud
Open Cloud
O-CU
Open Centralized Unit
O-DU
Open Distributed Unit
OFH
Open Fronthaul
Open RAN
Open Radio Access Network
O-RU
Open RAN Radio Unit
OS
Operating System
OSS
Operations Support Systems
PBCH
Physical Broadcast Channel
PDCCH
Physical Downlink Control Channel
PNF
Physical Network Function
PTP
Precision Time Protocol
RAM
Random Access Memory
RAN
Radio Access Network
rApp
non-real-time intelligence Application
RIC
Radio Access Network Intelligent Controller
RRM
Radio Resource Management
RRU
Remote Radio Unit
RU
Radio Unit
SSH
Secure Shell
SI
System Integration
SMO
Service Management and Orchestration
S Plane
Synchronization Plane
SRM
Supplier Relationship Management
SUPI
Subscription Permanent Identifier
SW
Software
SQL
Structured Query Language
TCP
Transmission Control Protocol
TPM
Trusted Platform Module
UDP
User Datagram Protocol
UE
User Equipment
UL
Uplink
U Plane, UP
User plane
vCU
Virtual Computational Unit
VIP
Very Important Person
VM
Virtual Machine
VNF
Virtual Network Function
vRAN
virtualized Radio Access Network
xApp
near-real-time intelligence Application
TIP's OpenRAN program is an initiative that focuses on de-
veloping solutions for future RANs based on disaggregation of
multi-vendor hardware, open interfaces, and software. O-RAN
alliance is another Open RAN organization that mainly focuses
on defining and enforcing new standards for Open RAN to en-
sure interoperability among the different vendors. At the be-
ginning of 2020, a liaison agreement between TIP and O-RAN
was made to ensure their alignment in developing interopera-
ble Open RAN solutions. OpenRAN development of TIP has
similar original goal similar to O-RAN. Thus, we use the term
"Open RAN" throughout the paper with refer to both Open-
RAN and O-RAN development efforts.
O-RAN refers to the O-RAN Alliance or designated specifi-
cation.
Table 2 :
2Summary of publications relevant to Open RAN securityYear & Ref.Open RAN Architecture
Open RAN Security Flaws
Open RAN Security Solutions
Open RAN Security Benefits
Research Directions
Remarks
2022 [29]
H L
L
L
L
A review article on RAN evolution towards open models and potential Open RAN
benefits and market trends
2022
Did not Consider the factor or only very briefly discussed it through mentioning it in passing M Medium Coverage: Partially considers the factor (leaves out vital aspects or discusses it in relation to other factors) Unlike traditional RAN technology, Open RAN decouples hardware and software bonds in proprietary RAN equipment. This feature offers more flexibility for mobile operators to deploy and upgrade their RAN segmentM M L
L
A survey of C-RAN security flaws and solutions where many threats and solutions
are relevant for Open RAN.
This Paper
H H H H H A comprehensive security analysis of Open RAN which througly discusses the
Open RAN security architecture, security flaws and solutions and security benefits
of Open RAN
L
Low Coverage: H High Coverage: Consider the factor in reasonable or high detail
4
2. Brief Overview of Open RAN Architecture
Table 3 :
3Overview of process related Open RAN risksRisk category
Threat
Description
Specific to Open-RAN
Prerequisite
Requirement
of
reliable
operational
environment
Table 4 :
4Overview of Open source software related Open RAN risksImpacting Open
RAN Component
Threat
Description
Specific to Open-RAN
All
Known vulnerabilities
Attention should be paid to developers using SW
components with known vulnerabilities or untrusted
libraries and without proper management of interde-
pendencies and patch management. [47, 60]
This is a well-know problem in open source
software code. Since Open RAN is expected
to be built (solely or partly) based on such
open source codes, it is in particular vulner-
able for these attacks
Backdoors
Attention should be paid to a trusted developer inten-
tionally inserting a backdoor into an open source code
Open RAN component.[47, 61]
This is a well-know problem in open source
software code. Since Open RAN is expected
to be built (solely or partly) based on such
open source codes, it is in particular vulner-
able for these attacks
No standards for trusted
coding available
Table 5
5summarizes the threats related to this radio/open interface.• Fronthaul. The Fronthaul of Open RAN, consisting of
O1, O2, A1, and E2 are the new components, all available
Table 5 :
5Overview of Radio/Open Interface related Open RAN risksAn attacker penetrates O-DU and beyond through O-RU or the Fronthaul interface due to heterogeneous security levels in the split architecture.[47,38] Yes, This is happen only in O-RAN as different vendor equipment is possible for O-RU and O-DU.Impacting
Open RAN
Component
Threat
Description
Specific to Open-RAN
Fronthaul
Rogue O-RU
The idea is to fool O-DU or UE into associating it to a rogue O-RU
over the legitimate O-RUs. [47]
It is possible to set up a rogue RU in other
RAN systems as well. However, it will be
more easy to develop a rogue O-RU in O-
RAN due to the open nature. No implicit se-
curity due to lack of knowhow.
Security
level
mismatch
be-
tween O-DU and
O-RU
Intercept
the
Fronthaul
(MITM)
over
M Plane
). The open source nature of the xApps are advertising the vulnerabilities to the adversaries while misconfigurations and incompatibilities are inevitable with the open nature of the O-RAN.Finally, since there is no clear functional split between the Near-RT RIC and the Open RAN Next Generation Node B (O-gNB), possible conflicts, including conflicts in xApps, between the decisions taken by the Near-RT RIC and the O-gNB regarding the radio resource management can appear, both unintentionally or maliciously. This can have an impact on the Open RAN system functions such as mobility management, admission controls, bandwidth management and load balancing, potentially resulting in performance degradation. Moreover, isolation of xApps is critical for the independent operation of O-RAN services and for accurate Near-RT-RIC decision making. Such an isolation or confinement can be penetrated through underlying system vulnerabilities, deducing access information via shared resource applications, or masqueraded authentication attempts. A compromised isolation could subdue the xApp operations to the attacker.
Table 6 :
6Overview of Intelligence related Open RAN risks xApps have the capability to manipulate behavior of a certain cell, a group of UEs, and a specific UE to track a certain subscriber or change priority level of an UE.[47,38] Yes since xApps and E2, A1 interfaces are only defined in O-RAN. UE identification due to malicious xApps Malicious xApps can exploit UE identification and track UE location. For example, a xApp can potentially be used as a "sniffer" for UE identification.[47,38,67] Yes since xApps and E2, A1 interfaces are only defined in O-RAN. Vulnerabilities and misconfiguration in xApps Vulnerabilities can potentially exist in any xApp, if it obtained from an untrusted or unmaintained source. An attacker exploits vulnerabilities and misconfiguration of such xAPPs to disrupt the offered network service and potentially take over another xApp or the whole near-RT RIC.[47,38,35] Conflicts in xApps Conflicting xApps unintentionally or maliciously impact O-RAN system functions such as mobility management, admission controls, bandwidth management and load balancing in the purpose of performance degradation. Moreover, a threat actor can utilize a malicious xApp that intentionally triggers RRM (Radio Resource Management) decisions conflicting with the O-gNB internal decisions to create DoS.[47,38] An attacker compromises xApp isolation to break out of xApp confinement. Such a way, attacker can perform side channel attack to deduce information from co-hosted xApps in a shared resource pool.[47] An attacker performs UE sniffing in the Non-RT RIC via A1 interface or via R1 interface via rApps in order to identify UE. For example, a rApp can potentially be used as a "sniffer" for UE identification.[47] Impacting
Open
RAN
Component
Threat
Description
Transfer learning attack A transfer learning attack becomes possible[47]. This impacts the xApps/rApps managed Open RAN system functions results a performance degradation. An attacker can have the aim to reconstruct training data from model parameters[82,86,87]. This impacts the xApps/rApps managed Open RAN system functions results a performance degradation.Membership Inference AttacksAn attacker tries to identify the data samples used for the model training[82,88]. This impacts the xApps/rApps managed Open RAN system functions results a performance degradation.This attack only applies
to O-RAN, where ML is
explicitly included.
This attack only applies
to O-RAN, where ML is
explicitly included.
Model inversion
This attack only applies
to O-RAN, where ML is
explicitly included.
This attack only applies
to O-RAN, where ML is
explicitly included.
Table 7 :
7Overview of Virtualization related Open RAN risksCan exploit the improper/missing authorization on SMO functions[47].However, the attack range and possibilities are larger in Open RAN.Exploits weak orchestrator configuration, access control and isolation[47].Can be applied in V-RAN. However, the attack possibilities are higher in Open RAN.Eavesdrop on network traffic via a malicious VM/CN or hypervisor/container engineImpacting
Component
Threat
Description
Specific to Open-RAN
PNF-
VNF/CNF
Lack of security policies
to protect mixed PNF-
VNF/CNF
Compromises a PNF to launch reverse attacks and other at-
tacks against VNFs/CNFs due to the lack of security policies
to protect mixed PNF-VNF/CNF[47, 89].
Can be applied in V-RAN. How-
ever, the attack possibilities are in-
creased in Open RAN.
VNF/CNF
Attacks via compromised
or outdated images
Compromises VNF/CNF images or used outdated images[47]
The same type of attack can be ap-
plied in V-RAN.
Configuration defects
Utilizes the configuration defect of VNF/CNF to attack[47].
However, the attack range and pos-
sibilities are increased in Open
RAN.
Stealing security creden-
tials
Steals embedded security credentials from VNF/CNF
images[47].
Insufficient authentication
and authorization
Insufficient authentication and authorization can lead to IP loss
and expose significant technical details about an Open RAN
VNF/CNF image to an attacker. [47]
Stealing or damage of em-
bedded information from
VNF/CNF images
Steal or damage sensitive information from/in VNF/CNF
images[47].
MITM on VNF/CNF mi-
gration
Performs MITM to intercept network traffic and jeopardize the
VNF/CNF image migration. [47, 70]
Interoperability issues
Exploits the security level mismatches of different
VNF/CNFs[90, 50, 70, 91].
Location shift attack
A compromised VNF/CNF changes the run-time environments
to perform an attack[90, 91, 70]
SMO
Improper/missing authen-
tication
Can exploit the improper/missing authentication on Service
Management and Orchestrator (SMO) functions to illegally ac-
cess the SMO and its functions[47].
The same type of attack can be ap-
plied against the Management and
Orchestration (EMM) in C-RAN.
Improper/missing autho-
rization
DoS attacks
Performs overload or flooding DoS attacks at SMO[47].
Orchestration associated
security issues
Hypervisor
VM/guest OS manipula-
tion
Exploits the security weaknesses in the guest OS to attack the
hypervisor [92, 90, 93].
The same type of attack can be ap-
plied in V-RAN.
Exhausting the hypervisor
Changes the configurations of compromised VNFs/CNFs to
consume more resources and exhaust the hypervisor [92, 90,
70].
However, the attack range and pos-
sibilities are larger in Open RAN.
Exceeding logs trou-
bleshooting failure
Change the configurations of compromised VNFs/CNFs
to generate excessive amounts of logs and exhaust the
hypervisor[92, 90].
Insider attacks
An insider who has access to the hypervisor misuses his privi-
leges to perform an attack[92, 90].
VM/CN
Misuse to attack others
A VM/CN can be misused to attack another VM/CN, hypervi-
sor/container engine, other hosts (memory, network, storage),
etc. [47]
The same type of attack can be ap-
plied in V-RAN.
Insecure run time configu-
ration
Insecure VM/CN run time configuration by the administrator
can lower the security[47].
However, the attack range and pos-
sibities are larger in Open RAN.
Compromise due to flaws
VMs/CNs may be compromised due to flaws in the
VNFs/CNFs they run[47, 94, 95].
Attacker hack
Hack into VM/CN retrieves the administrator privileges, re-
sulting in obtaining all tenant's tokens and the administrator
rights of the whole Open RAN system[47].
Malicious host
The host OS has access to all data[38, 96].
Migration attack
During the VM/CN migration, a MITM attacker can modify
arbitrary VM/CN OS or application states[92, 90].
VM rollback attack
An attacker uses and older snapshot of VM/CN to obtain ac-
cess to the system[92, 90].
Scheduler Attacks
Misconfigures the hypervisor scheduler to allocate more re-
sources to malicious VMs[92, 90].
Eavesdropping on network
traffic
the virtual machines and containers. This means that a malicious VM/CN or hypervisor/container engine can get access to all Open RAN network data processed in the workloads. An attacker can launch a noisy neighbor attack against the shared O-Cloud infrastructure to cause the Open RAN system performance degradation and/or the services disruption by depriving the resources required by various Open RAN running functionsOpen RAN system. It may
expose VMs/CNs and the hypervisor/container engine to
increased risk from a compromised VM/CN. For example,
it could be used to elevate privileges and attack VMs/CNs,
the O-Cloud infrastructure/services, etc [47].
A compromised VM/CN will be able to alter that VM/CN
in order to access other VMs/CNs, monitor VM/CN to
VM/CN communications, attack the O-Cloud infrastruc-
ture/services, scan the network to which it is connected to
in order to find other weaknesses to be exploited, etc. The
container engine (in case of CN) or hypervisor (in case
of VM) has access to all RAM memory, disk volumes
mounted on
Table 8 :
8Overview of Global Open RAN risksNetwork communications play an important role in the digital economy of a country and can cause huge damage in case of failure[100,101] , e.g. shutting down of smart cities, crashing of autonomous electrical vehicles or going dark of factories. In particular, special attention should be given to avoid loss of trust with the users in case of such attacks as they might endanger the entire growth of the network[102].Threat
Description
Table 9 :
9Security Solutions to Open RAN risksThreats and vulnerabilities
Solutions
Table 10 :
10Overview of most common errors, consequences and mitigation measurementsGeneral error
Consequences
Risk mitigation
Insecure design of
Open RAN interfaces
Novel design strategies can be left with critical flaws
due to the open and flexible approach of O-RAN
Table 11 :
11Overview of Open RAN benefitsFull visibilityThe operator has increased visibility, allowing better security control and response to incidents.[34,141,142] Selection of best modules Operators will be able to integrate best-in-class security platforms.[34,31,141,142, 48, 38] Diversity Diversity and independency among the diverse modules will decrease the attack range. [141, 48, 143] Modularity Enabling more efficient, seamless patch management and SW updates to remove vulnerabilities[]. [34, 142] Enforcement of security controls security controls can be better enforced. [31] Open interfaces Operators are independent of the supplier to react on security issues. [30, 48, 141] Open source software Open source software has been verified by multiple parties. [30] Automation The complete automation of network management can be speed up. [34, 30, 48] Open standards Better coordination of security measures becomes possible. [34, 141] Also V-RAN Isolation Isolation enables more control and less issues during updates for security management. [141] Increased scalabilty It enables better trade-offs between performance and security. [141] Control trust Operators are able to control full trust in their network. [141] Less dependency between HW and network SW There are less risks for SW upgrades. [142] Private networks There is easier migration to private networks. [142, 49] More secure storage of key material More secure storage of key material. [142] Also 5G Edge oriented Security is dealt closer to the edge of the network in order to stop attacks closer to the source. [34] Simpler security modelRAN type
Benefit
Short description
Open RAN
specific
Note that these mitigation measures are closely related to the process related risks, which can in fact also be considered as guidelines (seeTable 3)different hardware-software components of the Open RAN system.
AcknowledgmentThis work is supported by 6Genesis Flagship (grant 318927) project. The research leading to these results partly received funding from European Union's Horizon 2020 research and innovation programme under grant agreement no 101021808 (H2020 SPATIAL project). The paper reflects only the authors' views. The Commission is not responsible for any use that may be made of the information it contains.
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| []
|
[
"Predicted Trajectory Guidance Control Framework of Teleoperated Ground Vehicles Compensating for Delays",
"Predicted Trajectory Guidance Control Framework of Teleoperated Ground Vehicles Compensating for Delays"
]
| [
"Qiang Zhang ",
"Zhouli Xu ",
"Yihang Wang ",
"Lingfang Yang ",
"Xiaolin Song ",
"Zhi Huang "
]
| []
| []
| Maneuverability and drivability of the teleoperated ground vehicle could be seriously degraded by large communication delays if the delays are not properly compensated. This paper proposes a predicted trajectory guidance control (PTGC) framework to compensate for such delays, thereby improving the performance of the teleoperation system. The novelty of this PTGC framework is that teleoperators' intended trajectory is predicted at the vehicle side with their delayed historical control commands and the LiDAR 3D point cloud of the environment, and then the vehicle is guided by the predicted trajectory. By removing the teleoperator from the direct control loop, the presented method is less sensitive to delays, and delays are compensated as long as the prediction horizon exceeds the delays. Human-in-the-loop simulation experiments are designed to evaluate the teleoperation performance with the proposed method under five delay levels. Based on the repeated measurement analysis of variance, it is concluded that the PTGC method can significantly improve the performance of the teleoperated ground vehicles under large delays(>200ms), such as the task completion time (TCT), deviation to centerline (D2C) and steering effort (SE). In addition, the results also show that teleoperators can adapt to smaller delays (≤ ), and the presented method is ineffective in such cases. | 10.1109/tvt.2023.3269517 | [
"https://export.arxiv.org/pdf/2212.02706v1.pdf"
]
| 254,274,899 | 2212.02706 | 8bea193e4a57371d82f7d6e653d748cb8ffa3f2a |
Predicted Trajectory Guidance Control Framework of Teleoperated Ground Vehicles Compensating for Delays
Qiang Zhang
Zhouli Xu
Yihang Wang
Lingfang Yang
Xiaolin Song
Zhi Huang
Predicted Trajectory Guidance Control Framework of Teleoperated Ground Vehicles Compensating for Delays
1Index Terms-Delay compensationTrajectory predictionGuidance controlTeleoperation
Maneuverability and drivability of the teleoperated ground vehicle could be seriously degraded by large communication delays if the delays are not properly compensated. This paper proposes a predicted trajectory guidance control (PTGC) framework to compensate for such delays, thereby improving the performance of the teleoperation system. The novelty of this PTGC framework is that teleoperators' intended trajectory is predicted at the vehicle side with their delayed historical control commands and the LiDAR 3D point cloud of the environment, and then the vehicle is guided by the predicted trajectory. By removing the teleoperator from the direct control loop, the presented method is less sensitive to delays, and delays are compensated as long as the prediction horizon exceeds the delays. Human-in-the-loop simulation experiments are designed to evaluate the teleoperation performance with the proposed method under five delay levels. Based on the repeated measurement analysis of variance, it is concluded that the PTGC method can significantly improve the performance of the teleoperated ground vehicles under large delays(>200ms), such as the task completion time (TCT), deviation to centerline (D2C) and steering effort (SE). In addition, the results also show that teleoperators can adapt to smaller delays (≤ ), and the presented method is ineffective in such cases.
I. INTRODUCTION
A. Motivation
Although significant achievements have been made in recent years, fully autonomous driving remains challenging [1]. Humans still surpass machine intelligence in cognition. Benefiting from the merits of safety and cost, teleoperated vehicles are widely applied in dangerous areas and occasions unreachable by humans or too complex for an autonomous system, such as reconnaissance, route clearing, surveillance, and rescue [2]. In an unmanned ground vehicle (UGV) teleoperation system, the teleoperator watches the video feedback that is captured by onboard cameras and transmitted via wireless communication, and takes proper steering, braking and throttle operations like driving on a simulator. A typical teleoperation framework is shown in Fig. 1, which is a Direct Control (DC) framework. The driver station and vehicle are spatially separated, and the remote vehicle is controlled directly by the teleoperator. Connections between vehicle and operator are realized by transmitting the teleoperator's commands ( )(including steering, throttle and brake commands) and the feedback ( ) (including vehicle states, video feedback, etc.) through the wireless network. There exist control delays d1 and feedback delays d2 during transmission. This round-trip delay of d1 + d2 leads to a temporal desynchronization between the operator's control actions and the observation of its corresponding vehicle response. When the delay is slight, human operators can adapt to delays by predicting the outcome of operations. However, when this delay is large, the human's adaptability to delay decreases due to high cognitive workloads resulting from a lack of clear correspondence between input and output [3], [4]. Our tests on a long-distance low-latency graph transmission system in an industrial district show that the typical latency ranges from 0.18s to 0.55s depending on the radio interference and terrain. When the input, i.e., video feedback, suffers from a considerable time delay (>200 ms) [5], the performance and stability of the teleoperation system could be degraded. To cope with the detrimental effects of large delay, a simple, effective method is to slow down the speed or adopt a moveand-wait strategy. Nevertheless, low efficiency is not acceptable in most cases.
Fig. 1. Delays in a teleoperation system
This paper proposes a novel supervisory control framework for the ground vehicle teleoperation system to compensate for delays. The guidance trajectory is predicted based on the 3D point cloud in a bird's eye view (BEV) and driving commands. Then the teleoperated vehicle is directly controlled by a tracking controller to follow the predicted trajectory. The delay is compensated as long as the prediction horizon is greater than the delay.
B. Related work
Since the latency increases the operator's cognitive workload [6], the principle of reducing cognitive workload is to maintain a correlation between commands issued by the operator and the expected result of those commands [7]. The predictive display aims to compensate for delays by predicting the vehicle and operator motion. In a predictive display solution, the vehicle response that is likely to result from the current operation of the operator is predicted and displayed immediately to help the operator receive feedback regarding their control actions. Brudnak [5] adopted a feed-forward vehicle model as a high-fidelity state estimator to predict the vehicle response. Graf [8] presented a curvature model using both the teleoperator's inputs and the vehicle states to calculate the trajectory curvature. The above vehicle model-based methods' performance depends on the models' accuracy. Dybvik [9] studied the effect of using a simple predictive display on performance and the operator's workload. Results from 57 participants showed a significant 20% improvement with the help of the predictive display. However, accurate acquisition of vehicle dynamics is challenging. Solutions that do not require the knowledge of vehicle dynamics have been considered in pursuit of robustness. Zheng [10], [11]proposed a model-free predictor to compensate for communication delays. In Zheng's study, the predictor is a first-order time delay system whose parameters are designed based on the stability analysis and the frequency domain performance analysis of coupling errors, so the presented method is sensitive to the delay and the frequency characteristic of coupling errors. Zheng's experiments show that human operators are affected more by the asynchrony between the generating steering commands and monitoring the subsequent vehicle heading than by the asynchrony between controlling and monitoring the vehicle's longitudinal speed. Zheng [12] further proposed a blended architecture for the vehicle heading prediction by combining the performance benefits of a modelbased method with the robustness benefits of a model-free prediction scheme. To improve the situational awareness of the teleoperation system, Jung [13] developed head-mounted displays combined with a predictive display compensating for bidirectional network and operation delays to afford immersive 3D visual feedback.
Delay in the control loop results in deteriorated performance and instability. The robust control strategies against time delay have been intensively studied for teleoperation systems. Methods including Lyapunov function-based approaches [14], delay estimation techniques [15] and heuristic algorithms [16]- [18] are proven to be effective. Most studies focus on the bilateral teleoperation system to maintain stability and transparency. While for a ground vehicle teleoperation system, the main objective is to follow the operator's intention stably and avoid collisions. To deal with time-varying internet delay, Thomas [19] designed an adaptive Smith Predictor, which combined a delay estimation technique based on characteristic roots of delay differential equations to measure the delay with an adaptive Smith Predictor. In Thomas' approach, the teleoperator is modeled as a part of the control loop. Therefore, the variation of the teleoperator's response characteristics would affect the control system performance.
Adding autonomy capabilities to the teleoperated vehicle is an alternative approach handling delays. Studies verified that cooperative control could improve the performance and safety of unmanned ground vehicles [20], [21]. Cooperative control is classified into two categories: shared control and supervisory control. The key to shared control is the distribution of control right between the human operator and machine intelligence [22], [23]. Storms [22] presented an MPC-based shared control method. They found that communication delay's effect on safety has been improved considerably, while the control stability and the operator's workload are not discussed.
The supervisory control mitigates the sensitivity to delays by removing the operator from control loop. In a supervisory control system, the operator makes decisions based on environmental information and sets the global [24] or local guidance points/path to vehicles. The vehicle completes the maneuvering with its autonomous system. For some teleoperation applications, the environment is unknown or dynamic, so the global path guidance may not be applicable. Researchers paid more attention to the local path guidance model, also known as the point-to-go mode [25]. The operator needs to actively determine guidance points without decision support, which results in a significant cognitive workload. The vehicle speed fluctuates if the teleoperator cannot pick the guidance point timely. Zhu [26] proposed a method to generate candidate guidance points with the local perception information, decreasing the workload of picking the adequate guidance point. The main problem of the local guidance point/path-based approach is that teleoperators can hardly pick collision-free waypoints or paths due to the complexity of the driving environment and insufficient field feedback. Schitz [27] proposed an interactive corridor-based path planning framework. In Schitz's research, the human operator manually specified a corridor towards the destination in advance, and the vehicle planned a collision-free path in the specified corridor.
The above studies have addressed delay issues effectively. From the human perspective, experienced drivers get used to the normal driving manner, i.e., gazing at the area of interest, turning with a steering wheel, accelerating/decelerating with a pedal. Therefore, human operators using the normal driving manner could achieve better performance, such as stability and efficiency, than others due to lower cognitive workload. Predictive display mode seems preferable at this point. However, uncertain vehicle dynamics decrease the prediction accuracy and add to the workload of handling a vehicle. Although supervisory control is less insensitive to delays and vehicle dynamics since the autonomous system, instead of the teleoperator, is responsible for vehicle dynamics control, the pick-and-go mode could increase the cognitive burden. This study aims to develop a novel supervisory control framework to combine the merits of supervisory control and normal driving manner, in which the human teleoperates the vehicle as usual as normal driving, and the vehicle automatically follows the intended path compensating for delays.
C. Contribution
This paper proposes a predicted trajectory guidance control (PTGC) framework for teleoperated ground vehicles, aiming to improve the maneuverability and drivability of teleoperated vehicles under large delays. The proposed method uses a deep learning model to predict the operator's intended future trajectory and a tracking controller to follow this predicted trajectory. By removing the teleoperator from the control loop, the presented method is insensitive to delays while retaining the teleoperator's main authority over the vehicle by allowing the vehicle to drive as the driver intended.
The main contributions of this paper are summarized as follows:
1) A novel predicted trajectory guidance control (PTGC) framework is proposed to compensate for time delays, which reduces the teleoperator's cognitive workload and improves system performance by using a normal driving manner.
2) A deep learning-based multimodal prediction model using the operator's history operations and LiDAR 3D point cloud to predict the operator's intended trajectory is designed. The model achieves accurate trajectory prediction within one second. Thus, a delay of less than one second is compensated effectively.
D. Paper Organization
The remainder of the paper is organized as follows. Section II summarizes the system structure of the PTGC framework. The deep learning-based multimodal trajectory prediction model is presented in Section III. Section IV describes the details of the trajectory tracking controller. The design of human-in-the-loop experiments, the experimental results and discussions are given in Section V. Finally, Section VI makes the conclusions.
II. PREDICTED TRAJECTORY GUIDANCE CONTROL FRAMEWORK
We propose a predicted trajectory guidance control (PTGC) framework, as shown in Fig. 2, to compensate for delays and improve the control stability. The PTGC framework comprises two modules: trajectory prediction and trajectory tracking.
Fig. 2. Predicted trajectory guidance control framework
As discussed above, if the teleoperator controls the vehicle directly, the teleoperation system is sensitive to latency. In the PTGC framework, the teleoperator's commands are sent to the trajectory prediction module instead of directly to the vehicle and combined with environment information, i.e., 3D point cloud, to predict the teleoperator's intended trajectory. Since the teleoperator's control command ( ) is the response to the vehicle-road system states ( − d2 ). Therefore, the received control command ( − d1 ) at the vehicle side is aligned with ( − d ) , here d = d1 + d2 , and fed into the intended trajectory prediction model. A deep learning-based multimodal trajectory prediction model generates the intended future trajectories with the prediction horizon greater than the total time delay. Compared to predictive display, where only driving commands are fed into a vehicle model to generate the future trajectory, the 3D LiDAR point cloud is incorporated here. The reasons are as follows: (1) Constant Turn Rate and Acceleration (CTRA) model that takes driving operations as input can only predict an accurate trajectory in a short period;
(2) drivable area implied in the point cloud help generate a collision-free and feasible trajectory.
The predicted trajectory acts as the guidance trajectory and is fed into the tracking module, which controls the vehicle directly. The predicted trajectory is the outcome of input at time − d , so the first d of the predicted trajectory is truncated and then aligned with ( ). A tracking controller outputs steering commands ( ) to vehicle according to the error between the actual and predicted trajectory. The trajectory tracking performance only depends on the tracking controller and is irrelevant to the time delay and the teleoperator's proficiency in driving skills.
Advantages of this PTGC framework are that, by predicting the intended trajectories, deploying the tracking controller at the vehicle side and removing the teleoperator from the control loop, the presented method is less sensitive to delays and vehicle dynamics, and the stability of trajectory following is improved.
III. INTENDED TRAJECTORY PREDICTION USING LIDAR POINT CLOUD AND OPERATOR'S COMMANDS
In the proposed method, the vehicle is guided by the teleoperator's intended trajectory that is not picked directly by the operator but predicted with the operator's control command and environmental 3D point cloud. Our goal is to predict a future guidance trajectory over the next T time steps.
Intended future trajectory prediction can be considered a sequence-to-sequence problem. LSTM is with the ability to handle time sequence issues, so we construct trajectory prediction models based on LSTM networks. However, the general LSTM network can only predict one trajectory sequence and cannot perform multimodal prediction for the uncertainty of the operator's intention, which is prone to degradation of prediction accuracy. We proposed a method combining LSTM and multimodal prediction methods to address these problems, in which the Resnet is used to encode the context feature and LSTM is used to encode the motion feature.
A. Trajectory prediction model The proposed prediction model is illustrated in Fig. 3. This model consists of three modules, i.e., motion encoding module, context encoding module and decoding module.
The motion encoding module is an LSTM network encoding control commands and vehicle states that implies the operator's intention. We predict the trajectory of the future T time steps with information in the past ℎ time steps. The historical vehicle states are denoted as The historical control commands are denoted as
= − h ,⋯, − ,⋯,(1)=[ − h ,⋯, − ,⋯, ] (2) where − = ( − , ℎ − , − ) . − , ℎ − , and −
are the steering, throttle, and brake commands at the time step − , respectively. LSTM encoding module is to obtain the motion feature vector:
= ( • ( , ) + ) (3) where the function (•)
represents the input-output function of the LSTM, and are the weights and bias of LSTM, respectively.
The Context encoding module uses a Resnet network [28] to encode contextual constraints. The surrounding environment is described by the 3D point cloud. The point cloud in the range of 32 m × 32 m × 5 m ( length × width × height ) is converted into a binary image on a BEV grid [29]. The grid resolution is 0.125 m. Therefore, the size of the binary image is 256 × 256 pixels. The binary image is then divided into two channels, one for ground points and the other for non-ground points, and produces the pseudo-image denoted as . The vector of context features is obtained by encoding :
= ( ) (4) The dimensions of and are 128 and 512, respectively. We concatenate the context and motion features to generate a 640-dimensional feature that is fed into the decoding module to obtain the predicted output Pout: out = ( ( , )) (5) Note that we are not to generate one trajectory but N candidate trajectories and their corresponding probabilities, so the dimension of out is (2 + 1) • .
out =[ 1 , 2 ,⋯, ,⋯ ](6)
where = [ +1 , +1 ,⋯, + , + ,⋯, + , + , ]. + , + is the predicted waypoint of the j-th trajectory at the time step + , is the probability of j-th trajectory, ∈ (1,⋯, ), ∈(1,⋯, ) and =1 =1. The candidate with the highest probability is chosen as the predicted trajectory.
B. Loss function
The loss function [30] is constructed for the multimodal prediction. Specifically, we use the binary cross-entropy loss of classification and ℎ L 2 loss for the trajectory regression tasks to calculate the total loss ℒ .
The trajectory regression loss ℒ re is defined based on --loss:
ℒ re = ∑ = * ∑ − 2 =1 =1(7)
where is the predicted waypoint of the j-th candidate trajectory at time step + , and is the ground-truth position. = * is a selection function setting to 1 if = * is true and 0 otherwise. * is the number of the closest trajectory to the ground-truth trajectory according to the trajectory distance function.
* = argmin ∈{1,⋯, } ∑ − 2 =1 (8) ℒ
is the classification cross-entropy loss defined as ℒ = −∑ = * =1 log (9) The total loss ℒ is the sum of the trajectory regression loss and classification loss.
ℒ = ℒ + ℒ (10) where
is the classification loss weight to balance classification and regression performance.
C. Ablation Experiments and Error Analysis
Ablation experiments were conducted to analyze the significance of input components in the proposed model. The driving simulation was conducted on a CARLA-based simulator to collect the dataset for model training. We recruited six volunteers to drive vehicles on the simulator for data collection. The sample rate is 20Hz, and the time step st is 0.1 seconds. The trajectory in the past 20 steps (2s) was observed, and motion in the next 5, 10, and 20 steps (0.5 s, 1 s, and 2 s) was predicted. The collected data were split by a sliding window, and 63897 records were acquired. The ratio of records for training, validation and testing was 3:1:1.
We use the CTRA model [31] as the baseline. Variants of the proposed model, i.e., M-model using motion features, Cmodel using context features and MC-model using both motion and context features, are devised for comparison. The performance was evaluated using the average deviation error (ADE) and the final deviation error (FDE). Results of ablation experiments are shown in Fig. 4 and Table I. 86 We can find that the history command sequence and vehicle states are more significant for trajectory prediction than the current ones since M-model is better than the CTRA model. The motion feature and context feature are almost of equal importance. Combining motion and context features further benefits the trajectory prediction and is with the minimum ADE and FDE.
The distribution of large FDE errors for the prediction horizon of one second was further analyzed and presented in Table II. It shows that MC-model is with a smaller percentage of large error than the M-model and C-model. Especially, the error distribution of the MC-model is less than 1% in the range of error greater than 2m, which is much smaller than the other two models. It further proves that combining motion and context features is beneficial for trajectory prediction with a smaller error range. Therefore, the ablation experiments verified the feasibility of the proposed method. If not otherwise stated, the MC-model was adopted as the prediction model of the PTGC method.
IV. TRAJECTORY FOLLOWING CONTROL
As mentioned in section II, the predicted trajectory is the outcome of input at time − d , so the first d of the predicted trajectory is truncated. As shown in Fig. 5, the predicted trajectory consists of two segments, i.e., the history segment to be truncated (dashed line) and the future segment (solid line) to be followed. The split point is the predicted waypoint at the predicted time step + d . The Stanley control method [32] based on distance and heading angle errors at the point is adopted for trajectory tracking control. A simplified bicycle model with infinite tire stiffness is adopted in the design of the controller.
As shown in Fig. 5, the point is set as the preview point, θ v is the vehicle heading angle, θ p is the tangential angle of at , and e is the lateral deviation. The heading deviation is defined as = − (11) The steering control variable ( ) can be obtained intuitively from the relative deviation of the vehicle position to the predicted trajectory, which contains the lateral deviation e and the heading deviation θ e .
( ) = ( ) + ( ) (12) If ignoring the lateral error, the direction of the front wheel is aligned with the tangential direction of its corresponding preview point and the heading component
( ) is defined as ( ) = ( )(13)
where we define d(t) relating to the vehicle speed v(t), i.e., d(t)= ( )/ , and k is a gain parameter greater than zero. The function arcsin(•) produces a front-wheel deflection angle pointing directly to the trajectory to be tracked and is limited by the vehicle speed v(t).
Considering the above two control components together in the steering angle control law ( ), we have ( ) = ( ) + arcsin
( ) is further expressed aṡ ( ) = − ( ) (18) We can get ( ) = (0)e −(17)
Thus, the lateral error converges exponentially to zero, and the parameter k determines the convergence rate.
V. EXPERIMENTS AND ANALYSIS
A. Human-in-the-loop simulation platform
As shown in Fig. 6, a real-time driver-in-the-loop simulation platform was developed for teleoperation experiments and algorithm evaluation. The simulation platform consists of four parts, i.e., the drive station, the vehicle-road system, the controller and the communication network.
At the drive station, the operator uses a Logitech ○ R G27 joystick for steering, braking and acceleration control, and a monitor for visual feedback. As shown in Fig. 7, the front view, vehicle speed and the predicted trajectory are shown on the monitor. Operators drive the vehicle based on this feedback. The vehicle-road system is simulated with a CARLA-based simulator. The CARLA simulator runs vehicle dynamics and physical world simulation, and outputs vehicle states and environment information, including the driver's view of the environment and the 3D LiDAR point cloud. The resolution of RGB camera is 800×600 pixels at a frame rate of 20 Hz. The LiDAR has 32 lines, scans 128,000 points per second, and outputs a laser point cloud at 20Hz.
The communication network is simulated by a ROS node. The control command and visual feedback are sent to the ROS node and queued in a first-in-first-out (FIFO) pipeline. The communication delay is realized by setting the depth of the FIFO pipeline greater than zero. Compared to the real teleoperation system, only the vehicle-road system dynamics and communication system are simulated, while the humanmachine interface is almost identical. Therefore, the simulation platform can ensure the fidelity of the operator's response to delays.
B. Test Road
The test road is designed to evaluate the performance of the presented method under various delays. Due to the restriction of CARLA, the structural road scene was applied. As shown in Fig. 8, the 622m long closed-loop road features two-way four lanes and six turns with the radii ranging from 17 m to 45 m.
Fig. 8. Overview of test road
C. Teleoperation tasks and performance metrics
The study aims to improve the maneuverability and drivability of the teleoperated vehicle under large delays. We expect the proposed method to enable the vehicle to complete driving tasks as fast and safely as possible under different delay conditions. Therefore, the participants are required to operate the vehicle to complete the driving task as fast as possible while staying as close to the road centerline as possible to minimize tracking errors.
Three independent parameters, i.e., task completion time (TCT), deviation to centerline (D2C) and steering effort (SE), are used as performance metrics. TCT is defined as the time it takes for a participant to complete one loop of the driving task as "fast" and "smooth" as possible. The D2C is defined as the area between the actual vehicle track and the road centerline, indicating the magnitude of deviation to the centerline. These two metrics reflect the teleoperation system's longitudinal and lateral maneuverability performance, respectively. The lower value of metrics indicates higher performance. The SE is denoted by the average absolute steering angle, which characterizes the controllability of the teleoperated vehicle. Due to the detrimental effects of delay on teleoperated driving tasks, driving operations without timely visual feedback could result in oversteering and repetitive correction in the form of overdriving behavior. Less steering effort means more manageable and more comfortable control of the vehicle.
Teleoperated driving tasks under five delay levels ranging from 200ms to 1000ms with the interval of 200ms and applying two control frameworks, i.e., direct control (DC) and predicted trajectory guidance control (PTGC), are studied. The experiments follow a 5×2 within-subject factorial design and aim to determine how the delay magnitude and control method affect mobility and drivability. Including the zero-delay case as the baseline, eleven driving tasks are tested, and each task is repeated three times. Therefore, each participant needs to complete 33 runs. The task sequence is randomly scheduled to reduce the learning impact on individual scenes or one trajectory.
D. Experimental procedure
Nine people with an average age of 22±3years were recruited to participate in the experiments. They had a driver's license and at least one year's driving experience. All participants had a normal or corrected-to-normal vision and some experience driving in a virtual environment with a steering wheel and pedals (e.g., playing a virtual racing game) but no teleoperation driving experience in a delayed condition.
The whole testing process was divided into two sessions: the training session and the testing session. The training session served to help participants adapt to teleoperation under large latency based on the simulation platform. Participants were verbally informed of the test details, including the driving task and performance goals, i.e., completion time, deviation error and steering effort. Participants were asked to complete the driving task as quickly as possible but were not told which metric had a higher priority. Instead, it is up to them to adapt and adjust to the driving task. Once the training session had been completed, the test session began. Participants were asked to run 11 tasks in a randomized order during the testing session, and each task was repeated three times. A run was valid if the following events did not occur.
1) The vehicle ran off the road for 5 s.
2) Vehicle rollover.
3) The average speed is less than 18 km/h.
E. Analysis methods
A two-way RM-ANOVA was used to study the effects of two independent variables, i.e., delay level and control method, on TCT, D2C and SE. Here, two-way refers to two factors: delay level and control method.
The two null hypotheses for each metric were tested using an F-test based on the type III sum of squares and 95% confidence level. These null hypotheses are as follows:
(1) There is no significant difference in performance metrics when different control methods are used for teleoperation.
(2) No significant differences in performance measures exist between the different delay levels.
If the F-test indicated that at least one mean is different from the others (i.e., P < 0.05), Fisher's least significant difference method was used to identify the groups with pairwise significant differences in the means.
F. Experimental results and discussion
A total of 297 records were obtained, and one participant's result was discarded as the average speed was lower than 18km/h. Therefore, 264 valid records were used for experimental analysis.
A case (800 ms delay) study is shown in Fig. 9. Compared with the DC, the actual path is closer to the destined path when using the PTGC, and the steering intention of the driver can be recognized in advance before entering the intersection, thus controlling the vehicle steering as early as possible. Especially in the adjusting phase after the turning, the DC case has a longer overshoot, while using the PTGC reaches stability more quickly after turn, which means that using the PTGC has better maneuverability and stability. Two-way RM-ANOVA on each performance metric was conducted individually as a general linear model. The details of RM-ANOVA are shown in Tables III-V. With a significance level of 0.05, the P values of the F-test with respect to the factor of delay level are close to 0 and much smaller than 0.05 in all three RM-ANOVA tables, which indicates that the hypothesis of no significant difference between the different delay levels tested can be rejected with 95% confidence level. In terms of the effect of the control method, the P values for the metrics of TCT, D2C, and SE are 0.000, 0.000 and 0.001, respectively. All P values are smaller than 0.05. Thus, there is a significant difference in performance metrics when different control methods are used with a 95% confidence level. However, the P values of the factor Control*Delay for three metrics are 0.002, 0.013, and 0.004, respectively, indicating a significant interaction effect between the Control Method and the Delay Level. Therefore, we conducted a pairwise ANOVA comparison to determine whether the control method significantly affects the teleoperation performance at different delay levels. Results are shown in Fig. 10, where the '*' denotes the pairs with a statistically significant difference. For delays greater than 200 ms, the performance improvements in D2C and TCT metrics are significantly different using the PTGC relative to the DC, as shown in Fig. 10(a) and 10(b). The performance improvements in the SE metric are significant only when delays are greater than 400 ms, as shown in Fig. 10(c). All three metrics slightly worsen when the delay is not greater than 200 ms, indicating that humans can adapt to slight delay without assistance. The performance even decreases in the existence of assistance due to the prediction errors. To find out why there exist performance differences under different delay levels using the PTGC, a one-way ANOVA was applied in DC cases to explore the performance differences to the baseline. The results are shown in Table VI. All three performance metrics under 200ms delay are not significantly different from the zero-delay cases (P < 0.05), which indicates that human drivers can adapt to the low delays (≤ 200 ms) without compromising driving performance. Under 400 ms delay, the SE of DC cases is also not significantly different from the baseline (P<0.05). The reason could be that human drivers try to control the vehicle with minimal effort at relative low delay levels. The significance results at different delay levels in Table VI are consistent with those shown in Fig. 10, which indicates that the PTGC framework can result in remarkable performance improvements only when the DC framework is significantly affected by the delay. The reason is that the PTGC framework is based on the driver's intention prediction, and due to errors in trajectory prediction, its performance is always inferior to the baseline. To analyze the improvement quantitatively, results are further normalized using the average performance metrics of the zero-delay case as the benchmark [33]. The D2C, TCT and SE performance improvement of PTGC cases relative to DC cases at different delay levels is denoted as P D2C , P TCT , and P SE , respectively. Referring to [34] and taking P D2C for example, the improvement is calculated by
2 = | c − d | | d − 0 |(20)
where , 0 are the means of D2C of PTGC, DC and zero-delay cases, respectively.
Assuming each metric contributes equally to the overall performance ove , we get:
ove = 2 3 + 3 + 3(21)
Note that the overall performance is based on the significance analysis. If the performance improvement at a certain delay level is not statistically significant, the value is set to zero. The normalized improvement in the three metrics and the pairwise ANOVA comparison of the normalized overall performance metrics at different delay levels are shown in Table VII and Fig.11, respectively. It shows that under large delay levels, e.g., 400ms, 600ms, 800ms and 1000ms, the overall performance improvement of PTGC cases over the DC cases is 30%, 49%, 41% and 27%, respectively, and the overall performance improvement with the PTGC framework is statistically significant at these delay levels. The performance improvement decreases as the time delay increases when the delay is greater than 600ms. The reason is that the prediction error increases as the prediction horizon increases, and the prediction error is insignificant when the prediction horizon is less than 600ms. So, the overall improvement reaches its maximum at the delay level of 600ms.
VI. CONCLUSION
This paper proposes a predicted trajectory guidance control framework for teleoperation of ground vehicles, aiming to improve the maneuverability and drivability of teleoperated vehicles under delays. The control method is novel in that it uses a deep learning model to predict the teleoperator's driving intentions and intended trajectories at the vehicle side, and the vehicle is guided by the predicted trajectory using a closedloop tracking controller. The advantage of this approach is that it removes the teleoperator from the closed-loop control system and reduces the sensitivity of the human driver to time delays. The performance of the proposed method is verified with a human-in-loop driving simulation at delay levels ranging from 200ms to 1000ms. Three performance metrics, i.e., D2C, TCT and SE, are used to evaluate the performance improvement. The results show that the proposed method improves maneuverability and drivability under delays>200ms. Under 600ms delay, the overall improvement is about 49%. However, there is no improvement for cases of delay ≤ 200ms due to the human teleoperator's adaptability to small delays.
Fig. 3 .
3Framework of the prediction model
where − =[ − , − , − , − ]. ( − , − ) , − , and − are the position, velocity, and heading at the time step − , ∈ (1,⋯, h ), respectively.
Fig. 4 .
4Predicted trajectory using different model
Fig. 5 .
5Trajectory tracking control algorithm Similarly, if not consider the heading deviation, the larger the lateral error is, the larger the front wheel steering angle is. Assuming that the expected vehicle trajectory intersects the tangent line of the preview point at a distance d(t) from the front wheel, the lateral component ( ) can be derived approximately from the geometric relationship if ( ) is not large.
linear bicycle kinematic model, the change rate of the lateral error( ) is given bẏ ( ) = − ( )sin ( ) (16) where sin ( ) is known from the geometric relationship.
Fig. 6 .Fig. 7 .
67Human-in-the-loop simulation platform for teleoperated ground vehicle system Visual feedback displayed on the monitor
Fig. 9 .
9Actual path under 800 ms delay
Fig. 10 .
10Pairwise ANOVA comparison on D2C, TCT and SE at different delay levels
TABLE I RESULTS
IOF ABLATION EXPERIMENTS
Time
0.5 s
1 s
2 s
Metric (m) ADE FDE ADE FDE ADE FDE
CTRA
0.17 0.33 0.66 1.04 1.09 1.49
M-model
0.13 0.21 0.43 0.76 0.66 1.27
C-model
0.15 0.20 0.39 0.72 0.71 1.22
MC-model 0.13 0.18 0.24 0.54 0.56 0.
TABLE III
IIIRM-ANOVA RESULT FOR THE METRIC D2CFactor
DF
F
P
Control
1 30.63 0.000
Delay
4 28.61 0.000
Control * Delay 4
4.39 0.002
TABLE IV
RM-ANOVA RESULT FOR THE METRIC TCT
Factor
DF
F
P
Control
1 30.63 0.000
Delay
4 28.61 0.000
Control * Delay 4
4.39 0.013
TABLE V
RM-ANOVA RESULT FOR THE METRIC SE
Factor
DF
F
P
Control
1 30.63 0.001
Delay
4 28.61 0.000
Control * Delay 4
4.39 0.005
TABLE VI ONE
VI-WAY ANOVA RESULTS FOR DIFFERENCES TO THE BASELINE USING DC METHODMetric
Delay (ms)
Mean
Difference
P
D2C
0
200
-133.464
0.433
400
-222.104*
0.008
600
-326.271*
0.000
800
-528.021*
0.000
1000
-616.021*
0.000
TCT
0
200
-1.929
1.000
400
-15.785*
0.000
600
-22.554*
0.000
800
-29.098*
0.000
1000
-40.285*
0.000
SE
0
200
-0.195
1.000
400
-1.078
0.095
600
-3.023*
0.000
800
-3.847*
0.000
1000
-4.168*
0.000
* indicates a significant difference
TABLE VII PERFORMANCE
VIIIMPROVEMENT VS. DELAY LEVELS Delay 400 ms 600 ms 800 ms 1000 ms Fig. 11. Pairwise ANOVA comparison on the normalized overall performance metrics2
41%
59%
60%
35%
48%
49%
33%
21%
0
38%
28%
23%
ove
30%
49%
41%
27%
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| []
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[
"M 2 KD: Multi-model and Multi-level Knowledge Distillation for Incremental Learning",
"M 2 KD: Multi-model and Multi-level Knowledge Distillation for Incremental Learning"
]
| [
"Peng Zhou \nUniversity of Maryland\nCollege Park\n",
"Long Mai \nAdobe Research\nSan Jose\n",
"Jianming Zhang \nAdobe Research\nSan Jose\n",
"Ning Xu \nAdobe Research\nSan Jose\n",
"Zuxuan Wu \nUniversity of Maryland\nCollege Park\n",
"Larry S Davis \nUniversity of Maryland\nCollege Park\n"
]
| [
"University of Maryland\nCollege Park",
"Adobe Research\nSan Jose",
"Adobe Research\nSan Jose",
"Adobe Research\nSan Jose",
"University of Maryland\nCollege Park",
"University of Maryland\nCollege Park"
]
| []
| Incremental learning targets at achieving good performance on new categories without forgetting old ones. Knowledge distillation has been shown critical in preserving the performance on old classes. Conventional methods, however, sequentially distill knowledge only from the last model, leading to performance degradation on the old classes in later incremental learning steps. In this paper, we propose a multi-model and multi-level knowledge distillation strategy. Instead of sequentially distilling knowledge only from the last model, we directly leverage all previous model snapshots. In addition, we incorporate an auxiliary distillation to further preserve knowledge encoded at the intermediate feature levels. To make the model more memory efficient, we adapt mask based pruning to reconstruct all previous models with a small memory footprint. Experiments on standard incremental learning benchmarks show that our method preserves the knowledge on old classes better and improves the overall performance over standard distillation techniques. | null | [
"https://arxiv.org/pdf/1904.01769v1.pdf"
]
| 102,486,799 | 1904.01769 | 1ffb4eb0cc564673cc8908e0887c2fb740dd3efd |
M 2 KD: Multi-model and Multi-level Knowledge Distillation for Incremental Learning
Peng Zhou
University of Maryland
College Park
Long Mai
Adobe Research
San Jose
Jianming Zhang
Adobe Research
San Jose
Ning Xu
Adobe Research
San Jose
Zuxuan Wu
University of Maryland
College Park
Larry S Davis
University of Maryland
College Park
M 2 KD: Multi-model and Multi-level Knowledge Distillation for Incremental Learning
Incremental learning targets at achieving good performance on new categories without forgetting old ones. Knowledge distillation has been shown critical in preserving the performance on old classes. Conventional methods, however, sequentially distill knowledge only from the last model, leading to performance degradation on the old classes in later incremental learning steps. In this paper, we propose a multi-model and multi-level knowledge distillation strategy. Instead of sequentially distilling knowledge only from the last model, we directly leverage all previous model snapshots. In addition, we incorporate an auxiliary distillation to further preserve knowledge encoded at the intermediate feature levels. To make the model more memory efficient, we adapt mask based pruning to reconstruct all previous models with a small memory footprint. Experiments on standard incremental learning benchmarks show that our method preserves the knowledge on old classes better and improves the overall performance over standard distillation techniques.
Introduction
Deep neural networks perform well on many visual recognition tasks [5,19,15] given specific training data. However, problem arises when adapting networks to unseen categories while remembering seen ones, which is known as catastrophic forgetting [22,7,13]. To tackle this issue, there is a growing research attention on incremental learning where the new training data is not provided upfront but added incrementally. The target of incremental learning is to achieve good performance on new data without sacrificing the performance on old and it has been widely explored across different tasks such as classification [17,25] and detection [28].
To alleviate catastrophic forgetting in incremental learning, one possibility is to maintain a subset of old data to Figure 1. Concept overview. We propose to distill knowledge from all previous models efficiently to preserve old data information rather than sequentially applying distillation only to the last model. (e.g. using both S1 and S2 in S3 for distillation instead of sequentially using S1 for S2 and then S2 for S3). The confusion matrix is LWF-MC [25] on the left and our method on the right for the exemplar-free incremental setting. avoid over fitting on new data [25,2,16]. However, an issue in practice is that when models embedded in a product are delivered to customers, they no longer have access to trained data for privacy purposes. To tackle the situation, a stricter exemplar-free setting was introduced in [17], which requires no exemplar set for previous categories and only distills previous knowledge from the current categories.
Prior methods typically apply knowledge distillation sequentially during the incremental procedure to preserve previous knowledge. Since they apply distillation only to the last model, it is difficult to maintain all past knowledge completely (the left side of Figure 1). From that observation, we propose using all the model snapshots. Prior knowledge is preserved better through our approach (the right side of Figure 1). In addition, we enhance the distillation process from different feature levels to further improve the performance. However, saving all previous models may incur a great penalty in memory storage and without somehow compressing this historical information would not be practical. To address this, we reconstruct previous outputs using only "necessary" parameters during training. These parameters are determined by applying a compression algorithm to the series of models. With this pipeline, our approach only requires storing necessary parameters on-thefly and avoids saving all previous model snapshots.
To this end, we propose an end-to-end Multi-model and Multi-level Knowledge Distillation (M 2 KD) framework as depicted in Figure 2 for incremental learning. We introduce a multi-model distillation loss which leverages the snapshots of all previous models to serves as teacher models during distillation, and then directly matches the outputs of a network with those from the corresponding teacher models. To make the pipeline more efficient, we adapt mask based pruning methods to reconstruct the previous models. We prune the network after each incremental training step and identify significant weights to reconstruct the model. This allows us to reconstruct previous models and utilize them as teacher models in our multi-model distillation. To further enhance the distillation process, we also include an auxiliary distillation loss to preserve more intermediate features of previous models. Additionally, our approach addresses catastrophic forgetting in sequential distillation, and thus generalizes well for both exemplar based and exemplar-free settings. (See Section 4)
To show the effectiveness of our approach, we evaluate our model on Cifar-100 [14] and a subset of ImageNet [15]. We achieve state-of-the-art performance for all the datasets in the exemplar-free setting. We also show improvement when adapting to exemplar-based incremental learning and our exemplar-free setting outperforms [25] with a 200 exemplar budget.
In summary, our contributions are three fold. First, we propose a multi-model distillation loss, which directly matches logits of the current model with those from the corresponding teacher models. Secondly, for efficiency, we reconstruct historical models via mask based pruning such that model snapshots can be reconstructed with low memory footprint. Experiments on standard incremental learning benchmarks show that our method achieves state-of-theart performance in exemplar-free incremental setting.
Related Work
The ultimate goal of incremental learning is to achieve good performance on new data while preserving the knowledge about old data. Generally, two types of evaluation settings [3] have been considered. One is multi-head incremental learning which utilizes multiple classifiers at inference, and the other is single-head incremental learning which only utilizes one classifier at inference. Multi-head incremental learning. The evaluation setting in this stream is that a specific classifier is selected during testing according to the tasks or categories. With this prior information, no confusion exists across different classifiers, and thus the target becomes how to adapt the old model for new tasks or categories. Kirkpatrick et al. [13] explore Elastic Weight Consolidation (EWC), which constrains the important weights on the old tasks when adapting to the new ones. Mallya et al. [21,20] learn a mask for pruning to further constrain the weights on the old tasks. Hou et al. [11] rely on a subset of old data and distill the knowledge of the old model when adapting to new tasks. Different from this setting, we do not assume the task or category information is known during inference and follow the setting of single-head incremental learning. Also, even though we apply pruning in our approach, our goal is different from Mallya et al. [21,20] as the masks are utilized to reconstruct previous models for single-head setting. Single-head incremental learning. Single-head evaluation uses only one classifier to predict both the old and the new classes. This setting is more challenging [3] compared to the multi-head counterpart because of the confusion between old and new categories. Knowledge distillation is frequently utilized to preserve information. Li et al. [17] Learn Without Forgetting (LWF) by distilling the knowledge from the last model. Dhar et al. [4] introduce Grad-CAM [27] in the loss function. Rebuffi et al. [25] introduced exemplar set for the old data and match previous logits through distillation. Castro et al. [2] explore the balance between old and new data during training. Li et al. [16] focus on constructing exemplar set and Caselles et al. [1] replay the seen categories with GANs [6]. Instead of saving exemplars, we save the parameters of previous models for reconstruction. With that, this paper can be considered a complement research direction. In fact, as knowledge distillation is an important component in these methods, they can potentially benefit from our approach as well. Network pruning. Considerable research has explored this area to reduce network redundancy. Han et al. [8,9] propose to compress network through quantization and Huffman coding. Yu et al. [29] compress the weights according to their scores. Other methods [24,12,18] explore compression for fast inference. In contrast to these methods, we leverage network redundancy and use pruning to reconstruct all previous models in incremental learning with low memory footprint.
Approach
We propose novel distillation losses to preserve previous information without introducing too much memory overhead (See Figure 2). The model is agnostic to the backbone architecture and generalizes well to both exemplar based and exemplar-free methods. Given images from the current training data, we preserve previous knowledge directly from the reconstructed output through matching the logits with the corresponding model and classifying the current data with its ground truth. As an example, each layer contains a mask matrix Mt i at the ti-th incremental step recording significant weights for previous data. The gray dots represents the weights to be trained on the current data. Red and green dots denote the weights retained from the first and second incremental step respectively-they are fixed during training. The gray dots are fine-tuned for the current data before pruning. After pruning, a subset of the gray dots will be marked as important weights and become blue dots, and remaining weights will be fine-tuned during the next incremental step. Accordingly, Mt 2 is updated and used as Mt 3 at the end of this round. In multi-model distillation, the red and green output logits of the current model are matched with the model 1 and 2 respectively while the blue logits are matched with its ground truth.
Multi-model Distillation
The incremental learning process consists of a sequence of incremental class inclusion steps, referred to as incremental steps. Samples from a batch of new classes C k are added at the k-th incremental step. For instance, 20 classes will be added per incremental step in a 20-class batch setting. Accordingly, the network assigns new logits (output nodes) for the incremental classes. At inference, the maximum logit score in the output is treated as the final decision.
The knowledge distillation used in incremental learning [17,25] mainly aims to match the output of the current model to a concatenation of the last model logits and ground truth labels. Formally, it optimizes the cross entropy for both the old and new logits,
L D = − 1 N N i=1 Co j=1 s ij log(s ij ) − 1 N N i=1 C j=Co+1 y ij log(s ij ),(1)
where N and C denotes the number of samples and the total class number so far respectively, and C o denotes the old classes. s ij is the output score of the network obtained by applying Sigmoid function to the output logits for sample i at logit j. s ij denotes the old score obtained by the most previous model. y ij denotes the ground truth.
Treating the most previous model as the teacher and applying this distillation sequentially helps preserve historical information, especially when no previous exemplar set is stored, which is the protocol for prior methods [25,2,17,4]. However, the historical information will be gradually lost in this sequential pipeline as the current model must reconstruct all the prior information from the penultimate model alone. To address this limitation, we propose multi-model distillation, which directly leverages all previous models as our teacher model set. Since we mainly have current training data and labels for both settings, the network is more confident on current classes than old ones. Therefore, matching the previous logits of the current model directly with their corresponding old models preserve information better than always using the last model. Formally, we minimize the cross entropy for the logits between the current model and corresponding teacher models from previous in-
L M M D = − 1 N N i=1 P −1 k=1 C k j=C k−1 +1 s ijk log(s ijk ) − 1 N N i=1 C j=C P −1 +1 y ij log(s ij ),(2)
where classes from C k−1 + 1 to C k belong to the k-th incremental step and P denotes the number of incremental steps. Classes from C P −1 + 1 to C belong to the current categories. s ijk is the output score of the current model for sample i at logit j in the k-th incremental step. s ijk denotes the output score of the k-th previous model.
Multi-model distillation matches the logits in the current model with the corresponding teacher model directly, reducing the information loss between incremental steps. At inference, we directly choose the maximum among the output logits, which acts as an ensemble of all the previous teacher models and the current model.
Auxiliary Distillation
Previous incremental learning methods preserve old class information through matching the final output. However, the features from intermediate layers also contain useful information. Inspired by the auxiliary loss in segmentation task [30], we propose an auxiliary distillation loss to preserve the intermediate statistics of previous models. Similar to using the final output to represent network statistics, the prediction made by lower level features also represents intermediate feature statistics. Following the main branch classification, we extract lower level features and use an auxiliary classifier to conduct classification based on intermediate features (See Figure 3).
Also, a multi-model distillation loss is added on this auxiliary classifier for the purpose of preserving prior lower level features, and a standard cross entropy loss is also included for classifying the current data. Formally, the loss function becomes Algorithm 1 Pruning Algorithm input X 1 , . . . , X k // input image sets of incremental step 1, . . . , k require Θ // current model parameters // store pre-update parameters and masks m for y = 1, . . . , k do Grad(Θ y (m < y)) = 0 // apply mask update optimizer through Back-Propagation Θ y ← min(L M M D (Θ y ) + λL AD (Θ y )) adjust threshold by pruning ratio //update threshold Θ y (Θ y < threshold) = 0 // prune and update Θ y m(Θ y >= threshold) = y //update masks end for
L AD = − 1 N N i=1 P −1 k=1 C k C k−1 +1 a ijk log(a ijk ) − α N N i=1 C j=1 y ij log(a ij ),(3)
where a ijk denotes the output score from previous auxiliary classifiers, a ijk or a ij is the output score of the auxiliary branch, α is the ratio between the distillation and cross entropy loss. Notice that all the logits in ground truth labels are utilized in the classification cross entropy to enforce the correct prediction of current data.
The total loss function of the network becomes,
L total = L M M D + λL AD ,(4)
where λ is the ratio between the main classification multimodel distillation and the auxiliary classification distillation. This auxiliary classification branch is only used during training. At inference time, we only use the main branch classifier for prediction.
Model Reconstruction
One drawback of multi-model distillation in its original form is that it utilizes all previous models, requiring additional memory storage for the models. However, we observe that distillation aims to match logits. Therefore it is only necessary to preserve the outputs of previous networks, not the entire networks themselves. Our key idea is to save only a small set of the most important parameters of the networks from which we can approximate the output. By that way, all the models can be recovered on-the-fly without large memory penalty.
To determine the necessary parameters, we adapt mask based pruning [21] for model reconstruction. Specifically, after training each incremental step we sort the magnitude of weights in each layer, freeze the important ones to reach a specified pruning ratio, and use the residual weights to train the next incremental class set. We repeat this procedure for all future incremental steps until all the incremental classes are included. (See Algorithm 1)
We use a mask M to identify the important weights of each layer for all previous incremental steps. After each pruning procedure, we update the mask for the current incremental step. With the saved biases, batch normalization and classifier parameters, we can reconstruct all previous models from the last model (pre-updated model). Specifically, the output of a network with n convolutional layers is obtained from its classifier (the last layer) and features,
s = Ψ(f (n) ),(5)
where Ψ denotes the classifier and f (n) denotes the features in the n-th layer and can be generally written as
f (n) = σ(w (n) f (n−1) + b (n) ),(6)
where w and b are weights and biases respectively, σ denotes the activation function and f (0) is the input.
With the mask M k for the k-th incremental step, we reconstruct the corresponding feature by:
f (n) k = σ(w (n) k δ(M (n) k <= k)f (n−1) k + b (n) k ),(7)
where M (n) k denotes the mask in the n-th layer at incremental step k, f (n) k denotes the feature in the n-th layer in k-th incremental step, and δ denotes delta function.
Thus the output of the k-th model is reconstructed by
s k = Ψ k (f (n) k ),(8)
where s k and Ψ k denotes the output of the network and the classifier for the k-th incremental step respectively. We update masks and save parameters including biases, batch normalization and classifier on-the-fly.
Experiments
We evaluate our method in the exemplar-free setting in subsection 4.3. Then we extend our method to the exemplar-based setting in subsection 4.6. We also compare our memory cost with other methods in subsection 4.7.
Datasets and Evaluation Metrics
The dataset and training strategy for incremental learning are as followed: iILSVRC-small [26]: A small subset of 100 classes out of the 1000-class ImageNet. The evaluation metric is the 5 iteration average accuracy of the 100 classes. The number of classes in every incremental step is determined by the number of steps. We train from scratch and the performance is evaluated on the validation set of ImageNet. Cifar-100 [14]: All 100 classes are evaluated. The average result out of 5 random selections of classes for each incremental step is reported. We train from scratch and the evaluation results are based on the test set. Evaluation Metrics. Following the same metrics in prior methods [17,25], the top-1 classification accuracy is reported for Cifar-100 and top-5 classification accuracy is reported for iILSVRC-small.
Implementation Details
We use PyTorch for implementation. The network architecture follows prior works [25,17,2]: we use ResNet-32 [10] with input size 32 × 32 for Cifar-100 and ResNet-18 with input size 224 × 224 for iILSVRC-small. We extract the output of the second residual block for auxiliary distillation and empirically set α to 0.5. We train 80 epochs for Cifar-100 and 60 epochs for iILSVRC-small. Following the setting in [25], we use the training batch size of 128 and the initial learning rate of 2.0 to train the model. The learning rate decays by a factor of 5 every 40 epochs for Cifar-100 and 20 epochs for iILSVRC-small. Weight decay with a factor of 1e-5 is applied for the first incremental step and 0 for the rest in our full model to ensure weights from previous models remain the same. We optimize the network using standard Stochastic Gradient Descent (SGD) with a momentum 0.9. The pruning ratio is 0.75 for class batch less than 20 and 0.8 for 20 groups. After cutting off insignificant weights in the pruning stage, we fine tune the network for another half number of the epochs in normal training. λ is set to be 1.0 to balance the losses. Only random horizontal flipping is applied as data augmentation for all experiments.
Exemplar-free setting
We evaluate our methods in exemplar-free single-head setting. For evaluation, we also compare with the following baselines and state-of-the-art approaches. FT: A baseline approach that fine-tunes the whole model on new coming incremental classes without applying knowledge distillation. LWF-MC [25]: A multi-class classification version of [17] as described in [25], applying distillation to the logits from the last previous model sequentially. outperforms LWF-MC and FT from 5-classes to 20-classes per incremental step. The margin becomes larger as more incremental steps are added. This demonstrates the advantage of multi-model distillation as it avoids accumulating loss of historical information. Similar observation can be made when evaluating on iILSVRC-small. It is interesting to note that our model with pruning achieves comparable performance with the no-pruning version. This indicates the effectiveness of the pruning procedure in terms of saving memory while maintaining performance. Even though the residual active weights decrease gradually due to pruning, we still preserve the performance up to 20 incremental steps. See Section 4.4 for more detailed discussions of each component of our approach.
Ablation Studies
We investigate the effectiveness of each component of our method in this section. In particular, we compare our full model with the following baselines. LWF-MC aux: Add auxiliary distillation to LWF-MC. LWF-MC MMD: Change the original loss to our multimodel distillation. No auxiliary distillation is applied. Ours skip1: Instead of using all previous models, we study the case when skipping some snapshots. Starting from the last previous model, we skip the first model in multi-model distillation. The skipped model is replaced by the next model for multi-model distillation. Ours skip2: Skip the first two models instead of one compared to Ours skip1. Figure 5 shows the comparison for each of the component in our approach. LWF-MC aux improves our baseline model LWF-MC on all the datasets after adding auxiliary distillation, indicating that the intermediate level information also contributes to preserving previous knowledge. With only multi-model distillation (LWF-MC MMD), the performance gradually improves for both datasets as more incremental steps are involved, which demonstrates that directly distilling knowledge from the corresponding model helps to reduce the lost in sequential distillation. Note that our multi-model distillation reduces to the standard distillation used in [17] if only one or two incremental steps are added. By incorporating the auxiliary distillation, however, our method still shows improved performance. Lastly, our model achieves nearly the same performance as our upper bound which saves all previous snapshots, showing the effectiveness of our pruning based approach. See Section 4.5 for more experiments about the pruning ratio. Figure 7 compares how multi-model distillation is affected by the number of models. LWF-MC can be regarded as a special case which skips 3 models in the last round. The performance trend from LWF-MC to Ours shows that the performance improves as the number of model preserved increases, confirming the values of multi-model distillation.
To further analyze the performance behind our model, we show the resulting confusion matrices in Figure 6. It can be observed from the confusion matrix that LWF-MC has a strong bias to the data from the newly added classes while the performance on the old classes degrades dramatically. With the knowledge from the intermediate level, the confusion of previous data gets reduced. More clearly, with the favor of multi-model distillation, the knowledge from all the previous data preserves better and cause less confusion. Also, if we skip some previous models in the distillation and use other models to guide the network, the skipped logits become less confident than directly using the corresponding model for distillation. In short, the comparison from confusion matrices confirm the advantage on preserving previous knowledge via multi-model distillation.
Analysis on pruning ratio
We compare the results corresponding to different pruning ratios to investigate the robustness of our approach. Table 1 summarizes the results. Marginal performance variation (around 3%) is observed for different pruning ratios. Even though a higher (0.9) pruning ratio affects the performance as the active weights decrease in the current incremental step and a lower (0.6) ratio affects the performance as available weights decrease in the future steps, the relatively trivial influence indicates that a large redundancy exists in the network architecture. Benefitting from it, our approach shows robustness to different pruning ratios.
Exemplar Based Setting
Our approach also works for exemplar based incremental learning which use distillation sequentially on the output of networks [25,2,16]. To evaluate our model in this setting, we add exemplar selection to our approach and compare with exemplar based methods. iCaRL [25]: A prominent exemplar based incremental learning approach which constructs exemplar set for the old data according to the feature means and do distillation on the last previous model. An external nearest class mean classifier [23] is applied at inference. iCaRL aux: Adding auxiliary distillation to iCaRL. iCaRL M 2 KD: Change the original distillation function which only match logits from the last previous model to our multi-model distillation. Auxiliary distillation is also appended for a better performance.
The results are shown in Figure 8. With the introduction of multi-model and auxiliary distillation, the performance of iCaRL improves. It indicates that with direct access to all the previous models for distillation, the knowledge preserves better even with exemplar set.
Memory Comparison
Starting from the memory footprint of LWF as our baseline, we compare the extra memory storage between exemplar based method such as iCaRL [25] and our approach. The memory is calculated in the 10-class incremental step setting for both iILSVRC-small and Cifar-100. For our approach, we directly calculate the storage difference between the last and the initial step as stored parameters are accumulating along the training procedure. For iCaRL, the memory is approximately calculated by the average size of image for 2000 samples (i.e. the default exemplar size), and the compensation for saving the record of exemplar set. To optimize the memory consumption of iCaRL, we resize the images in iILSVRC-small to 256 × 256 and compress to JPG with quality 95 before calculate the image size to match their image size during training. Table 2 shows the memory compensation for different methods. It indicates that our approach has approximately 7× smaller memory compensation on iILSVRC-small and 10× smaller on Cifar-100 than iCaRL. On average, for each incremental step, our approach only take 0.98 MB and 0.08 MB for iILSVRC-small and Cifar-100 respectively. The memory advantage to exemplar based methods might become larger in real scenario as higher resolution images take more storage.
We perform further memory analysis in Figure 9. We compare our approach with iCaRL given the constraint of the same memory on Cifar-100. For fair comparison, we reduce the exemplar set as a penalty of the additional memory we use for network parameters to match with the memory size used for iCaRL. The performance is evaluated by av- Dataset iILSVRC-small Cifar-100 LWF-MC [17] 0 0 iCaRL [25] 68.0 9.4 M 2 KD (ours) 9.80 0.84 iCaRL Figure 9. Analysis on performance and memory compared to iCaRL on Cifar-100 (10-class batch). We increase memory budget for exemplar set from 200 to 4000 images and report the average accuracy of all the 10 incremental steps.
eraging the top-1 accuracy across all the incremental steps. When memory budget limit equals to 200 images, we do not use any exemplar set but still performs better than iCaRL. The reason for this is that the sequential distillation pipeline tends to lose information even when exemplars from old classes are available. Moreover, increasing memory budget makes the performance gap between our approach and iCaRL larger, showing our strength to memorize what has been learned.
Conclusion and Discussion
This paper presents a novel distillation strategy that mitigate catastrophic forgetting in single-head incremental learning setting. We introduce multi-model distillation which directly guides the model to distill knowledge from the corresponding teacher models. To further improve our performance, we incorporate auxiliary distillation to preserve intermediate features. More efficiently, we avoid to save all the model snapshots through reconstructing all previous models using mask based pruning algorithm. Extensive experiments on standard incremental learning benchmarks demonstrate the effectiveness of our approach. Incremental learning is still far from solved. There's still a significant gap between one-step training versus incremental training. It remains to be a open question how to reduce the confusion between different incremental steps especially without access to previous data, which might be a future exploration for our research.
Figure 2 .
2Framework overview.
Figure 3 .
3Illustration of auxiliary distillation. We extract the intermediate features and connect directly with an auxiliary classifier to preserve middle level knowledge. cremental steps,
M 2
2KD (ours): Our full model applying multi-model, auxiliary distillation along with pruning procedure to save memory storage. M 2 KD (no pruning): The upper bound of our model which directly loads all the previous snapshots for multimodel distillation.
Figure 4
4highlights our performance compared to stateof-the-art methods. For Cifar-100, our methods consistently
Figure 4 .
4Performance on iILSVRC-small and Cifar-100 dataset in exemplar-free setting. (a) Top-1 accuracy on Cifar-100 (5-class batch). (b) Top-1 accuracy on Cifar-100 (10-class batch). (c) Top-1 accuracy on Cifar-100 (20-class batch). (d) Top-5 accuracy on iILSVRC-small (10-class batch). (e) Top-5 accuracy on iILSVRC-small (20-class batch).
Figure 5 .
5Ablation Studies for our approach. a) Top-1 accuracy comparison on Cifar-100 (20-class batch). b) Top-5 accuracy performance on iILSVRC-small (20-class batch)
Figure 6 .
6Confusion matrix comparison for Cifar-100 in exemplarfree setting. (20-class batch) a)LWF-MC. b)LWF-MC with auxiliary distillation. c)Ours skip2. d)M 2 KD (ours).
Figure 7 .
7Comparison between different number of models used in multi-model distillation on Cifar-100 20-class batch.
Figure 8 .
8Performance comparison in exemplar based setting. a) Top-1 accuracy performance on Cifar-100 (10-class batch). b) Top-5 accuracy performance on iILSVRC-small(10-class batch).
Table 1 .
1Top1 accuracy comparison among different pruning ratios
on Cifar-100 (20 classes per incremental step).
Table 2 .
2Memory compensation comparison (MB). Each entry is the additional memory requirement for methods across different datasets based on the memory footprint of LWF.1000
2000
3000
4000
Memory budget K
45
55
65
Acurracy (%)
M 2 KD(ours)
AcknowledgementThis work was partly funded by Adobe. The authors acknowledge the Maryland Advanced Research Computing Center (MARCC) for providing computing resources.
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[
"The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates",
"The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates"
]
| [
"B H Margolius "
]
| []
| []
| We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series. | 10.1007/s11134-022-09851-x | [
"https://arxiv.org/pdf/2107.13244v1.pdf"
]
| 236,469,047 | 2107.13244 | 14b5e72712fa8b6a6376dcdeff8de6974348685a |
The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates
B H Margolius
The periodic steady-state solution for queues with Erlang arrivals and service and time-varying periodic transition rates
Received: date / Accepted: datearXiv:2107.13244v1 [math.PR] 28 Jul 2021 Noname manuscript No. (will be inserted by the editor)Erlang queues · Time-varying · Waiting time · Matrix analytic methods · Asymptotic periodic solution Mathematics Subject Classification (2020) 60K25 · 05A15 · 65C40 · 60J27
We study a queueing system with Erlang arrivals with k phases and Erlang service with m phases. Transition rates among phases vary periodically with time. For these systems, we derive the asymptotic periodic distribution of the level and phase as a function of time within the period. The asymptotic periodic distribution is analogous to a steady-state distribution for a system with constant rates. If the time within the period is considered part of the state, then it is a steady-state distribution. We also obtain waiting time and busy period distributions. These solutions are expressed as infinite series. We provide bounds for the error of the estimate obtained by truncating the series. Examples are provided comparing the solution of the system of ordinary differential equation with a truncated state space to these asymptotic solutions involving remarkably few terms of the infinite series.
Introduction
In this paper, we explore several quantities related to the Erlang arrival, Erlang service queue with time-varying periodic transition rates. The E k /E m /1 queue is a single server queue. Arrivals occur in k phases visited sequentially with transitions among phases occurring at rate λ (t). The transition rate is a periodic function of time. Throughout this paper, we take the length of the period to be one. The service process is also Erlang. It consists of m phases. Transitions among phases occur at rate µ(t) (also periodic), with each phase completed in sequence.
When service is exponential, the standard deviation of the service time is equal to its expectation. For Erlang-m service (when rates are constant), with parameter µ, the mean is m µ and the variance is m µ 2 . Of course, similar facts hold for Erlang−k arivals. This is an advantage when modeling processes for which the variance and standard deviation of the service distribution are not equal. Erlang-m service or Erlang-k arrivals also lets us track the stage of service or arrival, respectively, of the customer. These are two advantages cited by Gayon, et al, in choosing Erlang service for modeling a single-item-make-to-stock production system in which items have Erlang production times [7]. Foh and Zukerman [6] used Erlang service to model random access protocols. Jayasuriya, et al [11] use generalized Erlang service to model channel holding times in a mobile environment. Kuo and Wang [29] use an M/E m /1 queue to model a machine repair problem. Maritas and Xirokostas [19] also study a machine repair problem using Erlang service. Their model allows for more than one server. Grassmann [9] provides additional examples of applications of E k /E m /1 queues.
Many researchers have studied the E k /E m /1 queue, or the simpler E k /M/1 or M/E m /1 queues with constant transtion rates. A traditional approach using generating functions can be found, for example, in Saaty [23], Kleinrock [12] and Medhi [20]. This is the approach that we use in this paper, extending it to queues with timevarying periodic transition rates. This paper extends related work applying this approach to other queues with time-varying periodic transition rates. See [17] when the generating functions for the queue-length process are scalar, and [18] for quasi-birthdeath processes (QBD) when the generating functions for the queue-length process are vectors with a component for each phase. In the 2021 paper, we use a two priority queue with finite waiting room for priority 2 customers as an extended example.
Queues with Erlang arrivals, Erlang service or both, have been analyzed by Smith [25], Syski [26] and Takács [27] using Laplace transform techniques. Takács studies the waiting time, queue length and busy period for a queue with Erlang arrivals and general service. Truslove [28] considers this queue with finite waiting room. Leonenko [15] studies the transient solution to the M/E k /1 queue following an approach due to Parthasarathy, [21]. A paper by Griffiths, Leonenko and Williams [10] also provides an exact solution to the transient distribution of the M/E k /1 queue. Arizono, et al [2] use generating functions for the number of minimal lattice paths to find the equilibrium distribution for the E k /E m /1 queue length distribution. E k /E m /1 queues may also be analyzed using matrix analytic methods [13]. Grassmann [9], in his 2011 paper, derives an effective method for finding the characteristic roots of the k + m degree polynomial related to the waiting time distribution that arises from these methods. He builds on the approach due to Syski [26] and Smith [25]. Ivo Adan and Yiqiang Zhao [1] study a GI/E m /1 system and show that for arbitrarily distributed inter-arrival times and Erlang service, the waiting time distribution can be expressed as the finite sum of exponentials which depend on the roots of an equation. They also develop a method for finding these roots. Luh and Liu [16] study the E k /E m /1 queue and show that the roots of the characteristic polynomial associated with the process are simple if the arrival and service rates are real. They use this result to construct a general solution space of vectors for the stationary solution of the queue length distribution. Poyntz and Jackson [22] find the steady-state solution for the E k /E m /r queue, illustrating the method with the E k /E m /2 queue due to the "tediousness of the algebra".
In this paper, we are studying the E k /E m /1 queue when the transition rates vary with time. For two fairly recent surveys of research on queueing systems with timevarying parameters, the reader is referred to the papers by Schwarz, et al [24] and Whitt [33].
The remainder of this paper is divided into several sections. In section 2 we provide a brief review of results for QBDs with time-varying periodic transition rates. Section 3 provides the set up and analysis of the E k /E m /1 queue. We explore the singularities of the generating function for the level distribution in section 4 to find an exact formula for the level and phase probabilities. In section 5, we provide error bounds for the level probabilities and show that for smooth functions, the truncated series for our exact formulas can be made arbitrarily close. Section 6 gives formulas for the waiting time distribution and section 7 derives the busy period distribution as the solution of a Volterra equation of the second kind.
2 Review of results for quasi-birth-death (QBDs) processes with time-varying periodic rates
We study the asymptotic periodic solution of ergodic queues with Erlang arrivals and service and time-varying periodic transition rates. The solutions are expressed in terms of an integral over a single period. The integrals involve the idle probabilities for the system. These idle probabilities may be estimated using Tikhonov regularization. We do this within the framework developed in [17] and [18]. The solutions are exact, but involve an infinite series. The solutions may be estimated to arbitrary precision using finitely many terms.
We begin by recapping one of the main results from [18]. The infinitesimal generator for a (QBD) with time-varying periodic transition rates:
Q(t) = B(t) A 1 (t) A −1 (t) A 0 (t) A 1 (t) A −1 (t) A 0 (t) A 1 (t) . . . . . . .
This leads to the system of differential equations:
p 0 (t) = p 0 (t)B(t) + p 1 (t)A −1 (t) p n (t) = p n−1 (t)A 1 (t) + p n (t)A 0 (t) + p n+1 (t)A −1 (t), n ∈ N\{0},(1)
where p n (t) is a K-element row vector whose jth component reflects the probability of being in phase j and level n at time t. The A m (t), m = −1, 0, 1 are K × K matrices reflecting transitions among phases and within the current level or to an adjacent level. We can use the system of ordinary differential equations given in (1) to solve for the generating function for the asymptotic periodic distribution (see Breuer [4] for more details). The asymptotic periodic distribution is the limiting distribution at time t within the period as the number of periods tends to infinity. Such a limit will exist if the process is ergodic. To obtain equation (2), we have assumed that p n (t) = p n (t −1), so P(z,t) = ∑ ∞ n=0 p n (t)z n = P(z,t − 1). Note that P(z,t) is a row vector of generating functions. The coefficient of z n of the jth component gives the asymptotic periodic probability of being in level n and phase j at time t within the period.
The key equation for the generating function is given by:
P(z,t) = ∞ ∑ j=0 p j (t)z j = t t−1 p 0 (u) B(u) − A 0 (u) − z −1 A −1 (u) Φ(z, u,t)du × (I − Φ(z,t − 1,t)) −1 ,(2)
where Φ(z, u,t) is the generating function for the unbounded process, that is, the process that permits negative levels. Φ(z, u,t) is an evolution operator that satisfies
d dt Φ(z, u,t) = Φ(z, u,t)A z (t),(3)d du Φ(z, u,t) = −A z (u)Φ(z, u,t),(4)
and Φ(z,t,t) = I,
where
A z (t) = A 1 (t)z + A 0 (t) + A −1 (t)z −1 .(6)
For further details, see [18].
3 Erlang arrivals and service, the E k /E m /1 queue
The queue with Erlang arrivals and service, the E k /E m /1 queue, can be modeled with a three-dimensional state space {X(t), K(t), J(t)} in which X(t) represents the level at time t, K(t) is the arrival phase, and J(t) is the service phase. Arrivals are k-Erlang with time-varying periodic transition rate λ (t) among arrival phases. Service is m-Erlang with time-varying periodic transition rate µ(t) among service phases. This process can be modeled as a QBD. Define the following transition rate matrices:
D 0 (t) = −λ (t) λ (t) −λ (t) λ (t) . . . . . . −λ (t) , D 1 (t) = λ (t) , C 0 (t) = −µ(t) µ(t) −µ(t) µ(t) . . . . . . −µ(t) , C 1 (t) = µ(t) .
The matrices D 0 (t) and D 1 (t) are k × k, and the matrices C 0 (t) and C 1 (t) are m × m reflecting transitions among arrival and service phases, respectively. Let e 1 represent an appropriately dimensioned row vector with a one in the first position, and zeros elsewhere. The inter-arrival arrival distribution in the constant rate case is given by
F T a (t) = 1 − e 1 exp(D 1 t)1 k = 1 − e −λ t k−1 ∑ j=0 λ j t j j!
where 1 k is a k column vector of ones. In the time-varying case, we have
F T a,u (t) = 1 − e 1 Λ (u, u + t)1 k = 1 − e − t+u u λ (ν)dν k−1 ∑ j=0 e t+u u λ (ν)dν j j! where Λ (u,t) is an evolution operator satisfying d dt Λ (u,t) = Λ (u,t)D 1 (t), d du Λ (u,t) = −D 1 (u)Λ (u,t),
and Λ (t,t) = I.
An explicit formula for Λ (u, u + t) is
Λ (u, u + t) = e − t+u u λ (ν)dν 1 t+u u λ (ν)dν · · · ( t+u u λ (ν)dν) k−1 (k−1)! 0 . . . ( t+u u λ (ν)dν) k−2 (k−2)! . . . . . . . . . 0 · · · 0 1 .
The departure process is similarly defined with
F T d (t) = 1 − e 1 exp(C 1 t)1 k = 1 − e −µt k−1 ∑ j=0 µ j t j j!
in the constant rate case and
F T d,u (t) = 1 − e 1 M(u, u + t)1 k = 1 − e − t+u u µ(ν)dν m−1 ∑ j=0 e t+u u µ(ν)dν j j!
when rates are time-varying. The matrix function M(u, u + t) is given by
M(u, u + t) = e − t+u u µ(ν)dν 1 t+u u µ(ν)dν · · · ( t+u u µ(ν)dν) m−1 (m−1)! 0 . . . ( t+u u µ(ν)dν) m−2 (m−2)! . . . . . . . . . 0 · · · 0 1 .
The infinitesimal generator for the E k /E m /1 queue, Q(t), is given below. We arrange states in lexicographic order, i.e.
. . Then the infinitesimal generator for this process is given by
Q(t) = D 0 Q 0,1 Q 1,0 D 0 ⊕ C 0 D 1 ⊗ I m I k ⊗ C 1 D 0 ⊕ C 0 D 1 ⊗ I m I k ⊗ C 1 D 0 ⊕ C 0 D 1 ⊗ I m . . . . . . . . . . . . . . . ,(7)
where
Q 0,1 (t) = λ (t) is a k × mk matrix, Q 1,0 (t) = I k ⊗ 0 . . . 0 µ(t)
, I k is a k × k identity matrix, ⊗ and ⊕ represent the Kronecker product and Kronecker sum, respectively. For definitions of the Kronecker product and Kronecker sum see, for example, the textbook by Alan Laub [14], or MathWorld [31], [32]. The dependence of Q(t) on t is suppressed in the notation in equation (7).
For the E k /E m /1 QBD,
A −1 (t) = I k ⊗ C 1 (t), A 0 (t) = D 0 (t) ⊗ I m + I k ⊗ C 0 (t)
and (6), then the function Φ(z, u,t) is an evolution operator which satisfies equations (3), (4) and (5). For each of these km × km matrices, we reference the components of the matrix as ((a 1 ,
A 1 (t) = D 1 (t) ⊗ I m . With A(z,t) = z −1 A −1 (t)+A 0 (t)+zA 1 (t) as in equations 1 )(a 2 , s 2 ))
where (a 1 , s 1 ) refer to the arrival and service phases of the row and (a 2 , s 2 ) give the arrival and service phases of the column. The function Φ(z, u,t) is a Laurent series in z with km × km matrix coefficients where the ((a 1 , s 1 ), (a 2 , s 2 )) entry of the coefficient on z ℓ represents the probability of a net change of ℓ levels during the time interval [u,t) and a sequence of transitions that begin in arrival phase a 1 and service phase s 1 at time u and end in arrival phase a 2 and service phase s 2 at time t.
The key equation gives the generating function for this QBD in terms of an integral over a single time period. See [18] for the general case for QBDs. For the E k /E m /1 system, the key equation is given by
P(z,t) = ∞ ∑ j=1 p j (t)z j = t t−1 (p 0 (u)zQ 0,1 (u) − p 1 (u)A −1 (u)) Φ(z, u,t)du × (I − Φ(z,t − 1,t)) −1 . (8)
We can write an explicit formula for the evolution operator Φ(z, u,t). Note that A(z,t) may be expressed in terms of a Kronecker sum as
A(z,t) = (D 0 (t) + zD 1 (t)) ⊕ C 0 (t) + z −1 C 1 (t) .
The eigenvalues of A(z,t) are the sum of eigenvalues of the matrices (D 0 (t) + zD 1 (t)) and C 0 (t) + z −1 C 1 (t) and the eigenvectors are the Kronecker product of the corresponding eigenvectors.
Define
ω K = e 2πi K = cos 2π K + i sin 2π K ,
a Kth primitive root of unity. The matrix D 0 (t) + zD 1 (t) has eigenvalues
ξ ℓ (z,t) = λ (t)(ω ℓ k z 1/k − 1), ℓ = 0, . . . , k − 1,(9)
and corresponding eigenvectors
v ℓ = 1 √ k z (1−k)/k ω 0 k z (2−k)/k ω ℓ k z (3−k)/k ω 2ℓ k . . . z 0 ω (k−1)ℓ k .(10)
The matrix C 0 (t)
+ 1 z C 1 (t) has eigenvalues ε j (z,t) = µ(t)(ω j m z −1/m − 1), j = 0, . . . , m − 1,(11)v ℓ ⊗ u j . Matrix Eigenvalue A(z,t) ξ ℓ (z,t) + ε j (z,t) Φ(z,u,t) exp{ t u (ξ ℓ (z,ν) + ε j (z,ν))dν} (I − Φ(z,t − 1,t)) −1 (1 − exp{(ξ ℓ (z) +ε j (z)}) −1 Φ(z,u,t)(I − Φ(z,t − 1,t)) −1 exp{ t u (ξ ℓ (z,ν) + ε j (z,ν))dν} ×(1 − exp{(ξ ℓ (z) +ε j (z)}) −1
and corresponding eigenvectors
u j = 1 √ m z (m−1)/m ω 0 m z (m−2)/m ω j m z (m−3)/m ω 2 j m . . . z 0 ω (m−1) j m .(12)
We can now compute the eigenvalues and eigenvectors of
A(z,t) = (D 0 (t)+zD 1 (t))⊕ (C 0 (t) + 1 z C 1 (t)). The eigenvalues are ξ ℓ (z,t) + ε j (z,t), j = 0, . . . , m − 1, ℓ = 0, . . . , k − 1. The eigenvector for (D 0 (t) + zD 1 (t)) ⊕ (C 0 (t) + 1 z C 1 (t)) corresponding to the eigen- value ξ ℓ (z,t) + ε j (z,t) is v ℓ ⊗ u j j = 0, . . . , m − 1, ℓ = 0, . . . , k − 1.(13)
Note that while the eigenvalues depend on t, the eigenvectors do not. This enables us to easily compute the eigenvalues for the matrices: Φ(z, u,t),
(I − Φ(z,t − 1,t)) −1 and Φ(z, u,t) (I − Φ(z,t − 1,t)) −1 .
The eigenvectors for each of these matrices are those given in equation (13), the same as for A(z,t). Let
ξ ℓ (z) = 1 0 λ (u)(ω ℓ k z 1/k − 1)du =λ (ω ℓ k z 1/k − 1) andε j (z) = 1 0 µ(u)(ω j m z −1/m − 1)du =μ(ω j m z −1/m − 1)
give the average value of the eigenvalues for the arrival and departure processes, respectively, over a single time-period. We have definedλ = t t−1 λ (u)du, the average value of λ (t) over a single time period, andμ = t t−1 µ(u)du, the average value of µ(t) over a single time period. Then the eigenvalues for the four matrices with common eigenvectors are as given in table 1.
Define the matrices
Ω K = ω 0 K ω 0 K · · · ω 0 K ω 0 K ω 1 K · · · ω K−1 K . . . . . . . . . . . . ω 0 K ω K−1 K · · · ω (K−1) 2 K ,
and Ω K , its complex conjugate. Let
H C D C H −1 C = C 0 + z −1 C 1
where D C is a diagonal matrix of the eigenvalues, ε j (z,t), of C 0 + z −1 C 1 and H C is a matrix whose columns are the eigenvectors, u j , of C 0 + z −1 C 1 . The matrix
H C = 1 √ m diag[ z (m−1)/m z (m−2)/m · · · 1 ]Ω m . Similarly, let H D D D H −1 D = D 0 + zD 1 where D D is a diagonal matrix of the eigenvalues, ξ ℓ (z,t), of D 0 + zD 1 and H D is a matrix whose columns are the eigenvectors, v ℓ , of D 0 + zD 1 . The matrix H D = 1 √ k diag[ z (1−k)/k z (2−k)/k · · · 1 ]Ω k .
Then
H D ⊗ H C = 1 √ km diag[ z (1−k)/k z (2−k)/k · · · 1 ] ⊗ diag[ z (m−1)/m z (m−2)/m · · · 1 ] × (Ω k ⊗ Ω m ) .
Similarly, Generating functions and roots of unity. Recall that
H −1 D ⊗ H −1 C = 1 √ km Ω k ⊗ Ω m diag[ z (k−1)/k z (k−2)/k · · · 1 ] ⊗ diag[ z (1−m)/m z (2−m)/m · · · 1 ] . Then Ξ (a 1 ,s 1 )(a 2 ,s 2 ) = 1 km z s 2 −s 1 m + a 1 −a 2 k m−1 ∑ i=0 ω i(s 1 −s 2 ) m k−1 ∑ ℓ=0 d ℓ,i ω ℓ(a 1 −a 2 ) k where Ξ is one of the matrix functions A(z,t), Φ(z, u,t), (I − Φ(z,t − 1,t)) −1 or Φ(z, u,t) (I − Φ(z,t − 1,t)) −1 ,F (K, j) (x) = 1 K x − j K K−1 ∑ ℓ=0 F(ω ℓ K x 1 K )ω − jℓ K(14)
where F(x) = ∑ ∞ n=0 a n x n and F (K, j) (x) = ∑ ∞ n=0 a Kn+ j x n . See Herbert Wilf's text generatingfunctionology [34] for more details on generating functions and the role of roots of unity.
We apply this formula twice to
1 km z s 2 −s 1 m + a 1 −a 2 k m−1 ∑ i=0 ω i(s 1 −s 2 ) m k−1 ∑ ℓ=0 d ℓ,i ω ℓ(a 1 −a 2 ) k
to obtain explicit formulas for [z n ]Ξ (a 1 ,s 1 )(a 2 ,s 2 ) , the components of the coefficient matrices.
The generating function for a Poisson random variable appears several times in these expressions. Recall that the generating function for a Poisson random variable is
e λ (z−1) = e −λ ∞ ∑ n=0 λ n z n n!
and for steps to the left, the generating function is
e µ(z −1 −1) = e −µ ∞ ∑ n=0 µ n z −n n!
with the product of these forming the generating function for a random walk:
e λ (z−1)+µ(z −1 −1) = e −λ −µ ∞ ∑ n=−∞ z n ∞ ∑ ℓ=0∨(−n) µ ℓ λ n+ℓ ℓ!(n + ℓ)! .
Note that these Poisson generating functions appear in three of our Ξ matrices. We work out in detail, the simplest of these.
When Ξ = Φ(z, u,t) with s = s 2 − s 1 , a = a 2 − a 1 , λ = t u λ (ν)dν and µ = t u µ(ν)dν, we have [Φ(z, u,t)] (a 1 ,s 1 ),(a 2 ,s 2 ) (15) = 1 km z s m − a k m−1 ∑ i=0 ω −is m k−1 ∑ ℓ=0 e λ (ω ℓ k z 1/k −1)+µ(ω i m z −1/m −1) ω −ℓa k = z s m m m−1 ∑ i=0 e µ(ω i m z −1/m −1) ω −is m z − a k k k−1 ∑ ℓ=0 e λ (ω ℓ k z 1/k −1) ω −ℓa k (factoring exponentials) = z s m m m−1 ∑ i=0 e µ(ω i m z −1/m −1) ω −is m e −λ ∞ ∑ n=0
λ nk+a z n (nk + a)! (applying equation (14))
= e −λ −µ ∞ ∑ ℓ=0 µ mℓ+s z −ℓ (mℓ + s)! ∞ ∑ n=0
λ nk+a z n (nk + a)! (applying equation (14))
= e −λ −µ ∞ ∑ n=−∞ z n ∞ ∑ ℓ=0∨(−n) µ ℓm+s λ (n+ℓ)k+a (ℓm + s)!((n + ℓ)k + a)! (coefficient on z n ).
The coefficient on z n in this Laurent series reflects the probability of n more arrivals (which require completion of k phases at rate λ ) than service completions (which require completion of m service phases at rate µ) and a transition from arrival phase a 1 to arrival phase a 2 (net change a = a 2 − a 1 ) and from service phase s 1 to s 2 (net change s = s 2 − s 1 ) occurring during the time interval from u to t. The ((a 1 , s 1 ), (a 2 , s 2 )) component of the matrix coefficient on z n of the function Φ(z, u,t) (I − Φ(z,t − 1,t)) −1 gives the expected number of times t within the period that the process has made a net change of n levels and is in arrival phase a 2 and service phase s 2 having started at phases (a 1 , s 1 ) at time u within an earlier period. This coefficient does not count the expected number of visits, but rather the expected number of periods that the process is in a given state at time t within the period.
The components of the matrix function Φ(z, u,t) (I − Φ(z,t − 1,t)) −1 are linear combinations of generating functions of the form
e t u (λ(ν)(z−1)+µ(ν)(z −1 −1))dν 1 − eλ (z−1)+μ(z −1 −1) −1 = ∞ ∑ n=−∞ z n ∞ ∑ ℓ=0∨(−n) ∞ ∑ j=0 e − t+ j u (λ (ν)+µ(ν))dν t+ j u λ (ν)dν ℓ+n t+ j u µ(ν)dν ℓ (ℓ + n)!ℓ!
evaluated at kth and mth roots of the indeterminate z times a root of unity. An exact formula for the coefficient on z n of the ((a 1 , s 1 )(a 2 , s 2 )) component is given by
[z n ] Φ(z, u,t) (I − Φ(z,t − 1,t)) −1 (a 1 ,s 1 )(a 2 ,s 2 ) = ∞ ∑ ℓ=0∨(−n) ∞ ∑ j=0 e − t+ j u (λ (ν)+µ(ν))dν t+ j u λ (ν)dν (ℓ+n)k+a t+ j u µ(ν)dν ℓm+s ((ℓ + n)k + a)!(ℓm + s)! .
For the ((a 1 , s 1 )(a 2 , s 2 )) component of (I − Φ(z,t − 1,t)) −1 , we have
[z n ] (I − Φ(z,t − 1,t)) −1 (a 1 ,s 1 )(a 2 ,s 2 ) = ∞ ∑ ℓ=0∨(−n) ∞ ∑ j=0 e −(λ +μ) j ( jλ ) (ℓ+n)k+a ( jμ) ℓm+s ((ℓ + n)k + a)!(ℓm + s)! .
These formulas, while exact, are not conducive to computation.
Singularity analysis
Following the approach of Sedgewick and Flajolet [5], we note that the singularities of the generating function are reflected in the coefficients. In this section, we explore the zeros of the denominator of the generating function, P(z,t). Note that the generating function P(z,t) has singularities wherever
1 − exp λ ω ℓ k z 1/k − 1) +μ(ω j m z −1/m − 1) = 0.
(a) Zeros of equation (18). The k = 7 petaled rose is inside the unit circle.
(b) Zeros of the real part of equation (18) This occurs for z such that
λ (ω ℓ k z 1/k − 1) +μ(ω j m z −1/m − 1) = 2πin, n ∈ Z.(16)
Let y = z 1 km when ℓ = j = 0, then equation (16) becomes λ y m+k − (λ +μ + 2πin)y k +μ = 0, n ∈ Z.
(17) Figure 1 shows the roots of 1 − eλ (y m −1)+μ(y −k −1) .
Using Rouchè's theorem, we can show that the polynomial given in equation (17) has k roots on or inside the unit circle and m roots outside of the unit circle.
Provided that the m solutions to y m = k(λ +μ+2πin) λ (k+m)
are not solutions to (17), the roots are distinct. We can substitute
y * = k(λ +μ + 2πin) λ (k + m) 1 m
(a) Zeros of the real part of equation (18) are asymptotic to solutions to r k =μ λ +μ cos(kθ ) (shown in blue inside the unit circle) as r → 0 and asymptotic to r m =λ +μ λ sec(mθ ) (shown in red outside the unit circle) as r → ∞. Zeros of the real part of equation (18) by Young's inequality. Equality holds only ifλ k =μ m , or if k or m is zero. We have assumed ergodicity, soλ k <μ m ; that is, the mean arrival rate must be less than the mean service rate. neither k nor m equals zero since the arrival and service processes must have at least one phase. When n = 0, it is clear that the two sides of the equation are not equal because the real and imaginary parts are not equal.
Therefore, the roots of equation (17) are distinct. In fact, we can use the following contraction mappings to find the k + m roots for each fixed n. To find the m roots outside the unit circle, we may use the iteration: Because P(z,t) is a generating function for an ergodic process, it must converge for all complex |z| < 1. This means that zeros of the denominator inside the unit circle are also zeros of the numerator of the generating function. We focus our attention on the m roots of equation (17) outside of the unit circle. We label these roots, χ j,n , j = 1, . . . , m and n ∈ Z.
We consider examples where k and m are relatively prime. Suppose α is a root of the polynomial (17), then ω x km α is a root of λ ω ℓ k y k+m − (λ +μ + 2πin)y k +μω j m = 0, n ∈ Z, where x is the minimum non-negative integer such that m j + mx ≡ 0 mod mk and kq − kx ≡ 0 mod mk. Note that α is the kmth root of χ j,n for some j ∈ {1, . . . , m}.
More generally, we have
independent of the indices j and q. (Note that if k and m are not relatively prime, the approach in this paper can still be used, but the limit given in equation (19) would not be independent of j and q. We would need to find the roots of more than one equation for each n.) So, we approximate e t u εq(y km ,ν)+ξ j (y km ,ν)dν 1−eε q(y km )+ξ j (y km ) with the series e t u (λ (ν)(α m −1)+µ(ν)(α −k −1))dν
mλ α m − kμα −k ∞ ∑ n=0 y n ω nx km α n . Define D χ = 1 χ −1/k · · · χ (1−k)/k χ 1/k 1 · · · χ (2−k)/k . . . . . . . . . . . . χ (k−1)/k χ (k−2)/k · · · 1 and C χ = 1 χ 1/m · · · χ (m−1)/m χ −1/m 1 · · · χ (m−2)/m . . . . . . . . . . . . χ (1−m)/m χ (2−m)/m · · · 1 .
Then we may express the generating function given in equation (8), as
P(z,t) = ∞ ∑ j=1 p j (t)z j = ∞ ∑ j=1 t t−1 ∞ ∑ n=−∞ m ∑ ℓ=1 e t u (λ (ν)(χ 1/k ℓ,n −1)+µ(ν)(χ −1/m ℓ,n −1))dν mλ χ 1/k ℓ,n − kμχ −1/m ℓ,n × p 0 (u)χ ℓ,n Q 0,1 (u) − p 1 (u)A −1 (u) du × D χ ℓ,n ⊗ C χ ℓ,n z j χ j ℓ,n = ∞ ∑ j=1 t t−1 ∞ ∑ n=−∞ m ∑ ℓ=1 e t u (λ (ν)(χ 1/k ℓ,n −1)+µ(ν)(χ −1/m ℓ,n −1))dν mλ χ 1/k ℓ,n − kμ χ −1/m ℓ,n × p 0,k−1 (u)χ ℓ,n λ (u) − µ(u) k−1 ∑ q=0 p 1,q,m−1 (u)χ q/k ℓ,n du × 1 χ −1/k ℓ,n · · · χ (1−k)/k ℓ,n ⊗ 1 χ 1/m ℓ,n · · · χ (m−1)/m ℓ,n z j χ j ℓ,n(20)
Define
f (x,t) = t t−1 e t u (λ (ν)(x 1/k −1)+µ(ν)(x −1/m −1))dν mλ x 1/k − kμx −1/m × p 0,k−1 (u)xλ (u) − µ(u) k−1 ∑ q=0 p 1,q,m−1 (u)x q/k du,(21)
then the probability vector for level j is [z n ]P(z,t) from equation (20),
p j (t) = ∞ ∑ n=−∞ m ∑ ℓ=1 f (χ ℓ,n ,t)χ − j ℓ,n 1 χ −1/k ℓ,n · · · χ (1−k)/k ℓ,n ⊗ 1 χ 1/m ℓ,n · · · χ (m−1)/m ℓ,n .(22)
This expression is exact. See [18] for more details.
To illustrate the method, we consider an example of an E 7 /E 4 /1 queue with
λ (t) = 3 − 2 sin(2πt)
and µ(t) = 5 + 4 sin(2πt).
We approximate the distribution with p (q)
j (t) = q ∑ n=−q m ∑ ℓ=1 f (χ ℓ,n ,t)χ − j ℓ,n 1 χ −1/k ℓ,n · · · χ (1−k)/k ℓ,j (t)
to the level probabilities p j (t) as the number of terms in the estimate increases. In section 5 we provide error bounds for the asymptotic estimates of the level probabilities.
Error Bound
Our goal is to estimate the error (23). Our first bound applies for j ≥ 3. We do this by finding bounds for (a) the modulus of the roots |χ ℓ,n |, (b) f (χ ℓ,n ,t), defined in equation (21), and on
p j (t) − p (q) j (t) ∞ where p (q) j (t) is defined in equation(c) 1 χ −1/k ℓ,n · · · χ (1−k)/k ℓ,n ⊗ 1 χ 1/m ℓ,n · · · χ (m−1)/m ℓ,n ∞ .
Our asymptotic estimates for the p j (t) are governed by the singularities of the generating function P(z,t) and the function f (x,t) given in equation (21). The kmth roots of these are the zeros of
1 − eλ (y m −1)+μ(y −k −1) .
We examine the asymptotic behavior of the roots which are outside the unit circle. Return again to equation (16). Write z = re iθ in polar form and consider the limit of z 1/k n as n → ∞. Assume r > 1 and that r 1/k is also positive. From equation (16), the roots of the singularities of the generating function satisfȳ
λ z 1/k ω ℓ k =λ −μ(ω j m z −1/m − 1) + 2πin, n ∈ Z.(24)
Dividing both sides byλ ω ℓ k e iθ /k n,
lim n→±∞ r 1/k n = lim n→±∞ λ λ n −μ (ω j m r −1/m e −iθ /m − 1) λ n + 2πin λ n e −iθ /k ω −ℓ k = lim n±→∞ 2πin λ n e −iθ /k ω −ℓ k = 2π λ
where the last equality follows from the fact that r 1/k is real and positive. This, in turn, implies that the limiting value of θ , θ * , as n → ±∞ is such that e −iθ * /k−2πiℓ/k+πi/2 = 1 1 (t) for the probability of being in level 1 and the specified arrival and service phases for the E 7 /E 4 /1 system. See equation (23). Fig. 4: These graphs compare the asymptotic estimate given by p 1,1,0 (t) for the probability of being in level 1, arrival phase 1 and service phase 0 for the E 7 /E 4 /1 system. The asymptotic estimate is shown with the blue dashed line and the solution from a system of ordinary differential equations, truncated at 50 levels is shown in red. The graphs are for three different values of q, with estimate given by p (q)
1,1,0 (t) as defined in equation (23).
or θ * = kπ 2 .
Hence, for r > 1,
r ∼ 2πn λ k .
Similarly, if r < 1, then lim n→±∞ r −1/m n = 2π µ and the limiting value of θ is − mπ 2 . From the preceding analysis, we see that the modulus of the kth root of the singularity is bounded by
2π|n| λ −λ + 2μ √ 2λ < χ 1/k ℓ,n < 2π|n| λ +λ + 2μ √ 2λ .(25)
These bounds are independent of ℓ = 0, . . . , m − 1.
Next we consider the function f (x,t) given in equation (21). We consider three expressions separately:
f 1 (x, u,t) = e t u (λ (ν)(x 1/k −1)+µ(ν)(x −1/m −1))dν , f 2 (x) = mλ x 1/k − kμx −1/m and f 3 (x, u) = p 0,k−1 (u)xλ (u) − µ(u) k−1 ∑ q=0 p 1,q,m−1 (u)x q/k so that f (x,t) = t t−1 f 1 (x, u,t) f 2 (x) f 3 (x, u)du.
We can compute the following bound for | f 3 (χ, u)|, with |χ| ≥ 1:
p 0,k−1 (u)χλ (u) − µ(u) k−1 ∑ q=0 p 1,q,m−1 (u)χ q/k ≤ p 0,k−1 (u) |χ| λ (u) + µ(u) k−1 ∑ q=0 p 1,q,m−1 (u) χ q/k ≤ λ (u) |χ| + µ(u) k−1 ∑ q=0 p 1,q,m−1 (u) χ q/k ≤ (λ (u) + µ(u)) |χ| ,
where we have used the fact that the phase transition rates are real and non-negative, as are probabilities. We can find a lower bound for | f 2 (χ)|, |χ| > 1. χ ℓ,n is a root of equation (24). Because of the limit (19), we may take the exponents j and ℓ equal to zero, so
χ 1/k ℓ,n = 1 λ λ +μ −μχ −1/m ℓ,n + 2πin .≥ m λ +μ + 2πin − (k + m)μ χ −1/m ℓ,n < 1 = m (λ +μ) 2 + 4π 2 n 2 − (k + m)μ m λ +μ + 2πin ≥ (k + m)μ
for m µ > k λ (our ergodicity condition).
Now consider | f 1 (χ ℓ,n , u,t)|.
| f 1 (χ ℓ,n , u,t)| = e t u (λ (ν)(χ 1/k ℓ,n −1)+µ(ν)(χ −1/m ℓ,n −1))dν = e t u λ (ν) 1 λ λ +μ 1−χ −1/m ℓ,n +2πin −1 +µ(ν)(χ −1/m ℓ,n −1) dν substitution for χ 1/k ℓ,n = e t u λ (ν) λ [μ+2πin]+ µ(ν)− λ (ν)μ λ χ −1/m ℓ,n −µ(ν) dν simplification = e t u λ (ν)μ λ + µ(ν)− λ (ν)μ λ χ −1/m ℓ,n −µ(ν) dν e t u λ (ν) λ 2πindν = 1 = e t u λ (ν)μ λ −µ(ν)+µ(ν)ℜ{χ −1/m ℓ,n }dν ≤ e t u λ (ν)μ λ dν . |χ −1/m ℓ,n | < 1
Then, putting these inequalities all together,
| f (χ ℓ,n ,t)| ≤ |χ ℓ,n | m (λ +μ) 2 + 4π 2 n 2 − (k + m)μ t t−1 (λ (u) + µ(u))e¯µλ t u λ (ν)dν du = |χ ℓ,n |C n , where C n = t t−1 (λ (u) + µ(u))e¯µλ t u λ (ν)dν du m (λ +μ) 2 + 4π 2 n 2 − (k + m)μ .
The C n form a decreasing sequence. A bound on 1 χ
−1/k ℓ,n · · · χ (1−k)/k ℓ,n ⊗ 1 χ 1/m ℓ,n · · · χ (m−1)/m ℓ,n ∞ is |χ ℓ,n |.
Applying the lower bound (because the exponent is negative) for χ 1/k ℓ,n given in inequality (25), we have
p j (t) − p (q) j (t) ∞ ≤ 2mC q ∞ ∑ n=q+1 2πn λ −λ + 2μ λ √ 2 −k j+2k
The leading coefficient 2m is for m roots for each fixed n, and two tails of the sum over n. We also employ a bound on |χ ℓ,n | in this step.
≤ 2mC q ∞ q 2πx λ −λ + 2μ λ √ 2 −k j+2k dx
For a monotone decreasing function, the given integral is greater than the sum.
= mC q π 2πq − 1 √ 2 (λ + 2μ) k(2− j)+1 (k( j − 2) − 1)λ k(2− j)
This bound goes to zero as q → ∞ for j ≥ 3.
The plots in figures 3 and 4 show rapid convergence even for level one. We explore why this is so in subsection 5.1.
A Riemann-Lebesgue type lemma
The functions f (χ ℓ,n ,t) defined in equation (21), for fixed t, are not Fourier coefficients, but they behave somewhat similarly. We have
f (χ ℓ,n ,t) = t t−1 e t u λ (ν) λ [μ+2πin]+ µ(ν)− λ (ν)μ λ χ −1/m ℓ,n −µ(ν) dν m(λ +μ + 2πin) − (k + m)μχ −1/m ℓ,n × p 0,k−1 (u)χ ℓ,n λ (u) − µ(u) k−1 ∑ q=0 p 1,q,m−1 (u)χ q/k ℓ,n du.
We perform a change of variables. For fixed t, let
x = t u λ (ν) λ dν = g(u). g(u)
is a decreasing function. Hence it has an inverse. The differential
dx = − λ (u) λ du, so du = −λ λ (g −1 (x)) dx. f (χ ℓ,n ,t) = 1 0 e 2πinx+μx−μxχ −1/m ℓ,n + t g −1 (x) µ(ν) χ −1/m ℓ,n −1 dν m(λ +μ + 2πin) − (k + m)μχ −1/m ℓ,n × p 0,k−1 (g −1 (x))χ ℓ,n λ (g −1 (x)) − µ(g −1 (x)) k−1 ∑ q=0 p 1,q,m−1 (g −1 (x))χ q/k ℓ,n ×λ λ (g −1 (x)) dx
As |n| → ∞, χ −1/m ℓ,n ≈λ k/m λ +μ + 2πin −k/m .
Define h ℓ,n (x) = eμ
x−μxχ −1/m ℓ,n + t g −1 (x) µ(ν) χ −1/m ℓ,n −1 dν m(λ +μ + 2πin) − (k + m)μχ −1/m ℓ,n × p 0,k−1 (g −1 (x))χ ℓ,n λ (g −1 (x)) − µ(g −1 (x)) k−1 ∑ q=0 p 1,q,m−1 (g −1 (x))χ q/k ℓ,n ×λ λ (g −1 (x))
so that f (χ ℓ,n ,t) = 1 0 e 2πinx h ℓ,n (x)dx.
Let T = [0, 1). If h ℓ,n (x) is a continuous N times differentiable periodic function (h n (x) ∈ C N (T) with h (k) ℓ,n (0) = h (k)
ℓ,n (1) for 0 ≤ k ≤ N), then repeated applications of integration by parts will yield f (χ ℓ,n ,t) = −1 2πin
N 1 0 h (N) ℓ,n (x)e 2πinx dx where h (N) ℓ,n (x) is the Nth derivative of h ℓ,n (x). Then f (χ ℓ,n ,t) ≤ 1 2π|n| N 1 0 h (N) ℓ,n (x) dx.
Note that
χ ℓ,n = 1 λ λ +μ 1 − χ −1/m ℓ,n + 2πin
k so that h ℓ,n (x) ∼ C(x)n k−1 for some function C(x) that does not depend on n. The contribution from χ −1/m ℓ,n → 0 as |n| increases. However, so long as h ℓ,n (x) is sufficiently smooth, the integral f (χ ℓ,n ,t) = 1 0 e 2πinx h ℓ,n (x)dx → 0 as n → ∞. This happens because the rapid oscillations introduced by the factor e 2πinx cause the integral to go to zero. Figure 5 shows a graph of e 2πinx h ℓ,25 (x) for t = 0.25 to illustrate this idea. See Loukas Grafakos text Classical Fourier Analysis [8], theorem 3.3.9, p. 196 for a similar result for Fourier coefficients.
Waiting time distribution
If a customer enters the system when there are already j customers ahead of him and the customer being served is in service phase s, then at least m − s + m( j − 1) additional service phases must be completed before he begins service and m − s + m j must be completed before his service is finished and he leaves the queue. Let W q (u) represent the waiting time until a customer arriving at time u reaches the front of the queue and W (u) represent the waiting time including service for that customer. The waiting time distributions, given that X(u) = j, J(u) = s and j ≥ 1 is so the waiting time distribution for a customer entering at time u is given by
P{W q (u) ≤ t|X(u) = j, J(u) = s} = ∞ ∑ q=m j−s u+t u µ(ν)dν q q! e − u+tP{W q (u) ≤ t} = ∞ ∑ n=−∞ m ∑ ℓ=1 f (χ ℓ,n , u) k−1 ∑ a=0 χ −a/k ℓ,n χ −1/m ℓ,n 1 − χ −1/m ℓ,n × 1 − e u+t u µ(ν)(χ −1/m ℓ,n −1)dν and P{W (u) ≤ t} = ∞ ∑ n=−∞ m ∑ ℓ=1 f (χ ℓ,n , u) k−1 ∑ a=0 χ −a/k ℓ,n χ −1/m ℓ,n 1 − χ −1/m ℓ,n × 1 − e − u+t u µ(ν)dν m ∑ q=0 u+t u µ(ν)dν q q! −χ ℓ,n e − u+t u µ(ν)dν e χ −1/m ℓ,n u+t u µ(ν)dν − m ∑ q=0 χ −1/m ℓ,n u+t u µ(ν)dν q q! .
We estimate the waiting time distribution with the expression
P{W (q) q (u) ≤ t} = q ∑ n=−q m ∑ ℓ=1 f (χ ℓ,n , u) k−1 ∑ a=0 χ −a/k ℓ,n χ −1/m ℓ,n 1 − χ −1/m ℓ,n × 1 − e u+t u µ(ν)(χ −1/m ℓ,n −1)dν . (26)
An example appears in figure 6.
Busy period distribution
In this section, we follow the approach of Baek, Moon and Lee [3] and apply it to the case of time-varying periodic parameters to find the busy period in terms of a Volterra equation of the second kind. Let us define the first passage time
τ j = inf{t > u, N(t) = 0|N(u) = j, J(u) = q}.
We note that τ j is the length of a busy period that starts with j customers in the system and with an arriving customer in phase q. Let us define the following probabilities:
Q ( j)
n (t) = P{N(t) = n, τ j > t|N(u) = j, J(u) = q}, Q ( j) 0 (t) = P{τ j < t|N(u) = j, J(u) = q}. We find the busy time distribution. {N(t), J(t)} is a continuous time Markov chain with absorbing boundary at N(τ j ) = 0. We set up the following system of ordinary differential equations:
d dt Q ( j) 0 (t) = Q ( j) 1 (t)A −1 (t) d dt Q ( j) 1 (t) = Q ( j) 1 (t)A 0 (t) + Q ( j) 2 (t)A −1 (t)(27)
d dt Q Fig. 6: These graphs compare the asymptotic estimate for the waiting time distribution to the ODE estimate for the distribution for the specified arrival and service phases for the E 7 /E 4 /1 system. See equation (26).
To solve the system of differential equations (27), we define the generating function
G ( j) (z, u,t) = ∞ ∑ n=0 Q ( j)
n (t)z n .
The differential equation for the generating function is
d dt G ( j) (z, u,t) = G ( j) (z, u,t)A z (t) − Q ( j) 0 (u,t)A z (t)
with solution G ( j) (z, u,t) = G ( j) (z, u, u)Φ(z, u,t) − t u Q ( j) 0 (ν)A z (ν)Φ(z, ν,t)dν
where G(z, u, u) = z j e q and e q is a row vector with a one at component q and zeros elsewhere. Since Q
0 (t) = e q [z − j ]Φ(z, u,t) − t u Q ( j) 0 (ν) A −1 (ν)[z 1 ]Φ(z, ν,t) +A 0 (ν)[z 0 ]Φ(z, ν,t) + A 1 (ν)[z −1 ]Φ(z, ν,t) dν. (28)
The matrix coefficient on z n in the generating function for the unbounded process that appears (several times) in equation (28) is given in equation (15). For example, with a = a 2 − a 1 , and s = s 2 − s 1 and a ≥ 0, s ≥ 0.
Conclusion
In this paper, we developed a method for computing the asymptotic periodic distribution of the level and phase probabilities for a queue with k Erlang arrival phases and m Erlang service phases. We also showed how to compute the waiting time distribution seen by a customer arriving at any time within the period assuming that the system is in its asymptotic periodic "steady-state". This calculation requires computing an integral over a single time-period. We provide exact Fourier like expansions, but require only finitely many of these terms to compute the level probabilities to arbitrary accuracy. We compare our results to those obtained by solving a truncated version of the infinite system of differential equations and letting the system run until steady-state is achieved. The computations require the asymptotic periodic solution for the queue being idle or having a single customer. These probabilities can be computed using Tikhonov regularization. We also express the busy period as a solution of a Volterra equation of the second kind.
Declarations
Conflict of interest
The authors declare that they have no conflicts of interest.
Funding
Not applicable.
Conflicts of interest/Competing interests
Not applicable.
Code availability Not applicable.
are shown in white; zeros of the imaginary part for n = 3 are shown in cyan. The m = 4 intersections of the cyan and white curves outside of the unit circle show the m roots for n = 3 outside the unit circle. The intersection of the white and cyan seven petal roses show the k = 7 roots corresponding to n = 3 inside the unit circle.
Fig. 1 :
1These figures were produced using Matlab code written by Elias Wegert[30]. Both plots show the region from −2 − 2i to 2 + 2i. The expression given in equation(18) is plotted in the complex plane. Shading shows contour lines. The colors represent the argument, so points where multiple colors come together are zeros of the function. In this example, k = 7, m = 4,λ = 3 andμ = 5. Note that there are k = 7 petals in the rose inside the unit circle and m = 4 inverted petals outside the unit circle.
are shown as dashed black lines. (b) Zeros of the real part of equation (18) are shown in white for an E 3 /E 5 /1 system; zeros of the imaginary part for n = −7 are shown in cyan. The m = 5 intersections of the cyan and white curves outside of the unit circle show the m roots for n = −7 outside the unit circle. The intersection of the white and cyan three petal roses show the k = 3 roots corresponding to n = −7 inside the unit circle.
Fig. 2 :
2Zeros of the real part of the denominator of the generating function for an E 7 /E 4 /1 queueing system are shown on the left, and zeros of the denominator of the generating function for an E 3 /E 5 /1 queueing system are shown on the right. Note that the petals inside the unit circle correspond to the number of arrival phases.into equation(17) to show that if y * solves(17)
,
roots command from Matlab, for example, works perfectly well. To find the k roots on or in the unit circle, we may use the iteration: q = 0, . . . , k − 1.
q (y km ,ν)+ξ j (y km ,ν)dν1 − eε q (y km )+ξ j (y km ) = e t u (λ (ν)(α m −1)+µ(ν)(α −k −1))dν mλ α m − kμα −k
in figures 3 and 4 show convergence of the asymptotic estimates p
Fig. 3 :
3These graphs compare the asymptotic estimate given by p
m andλ are positive and |a − b| ≥ ||a| − |b||
P{WFig. 5 :
5(u) ≤ t|X(u) = j, J(u) = s} = Graph of e 2πinx h ℓ,25 (x) for t = 0.25 for E 7 /E 4 /1 example From equation (22), P{X(u) = j, J(u) = s} =
( j) 0
0(t) = [z 0 ]G ( j) (z, u,t), Q
+ s)!((− j + ℓ)k + a)!
Table 1 :
1Four matrix functions which share the eigenvectors
given in table 1 and the d ℓ,i are the corresponding eigenvalues.
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"FERMILAB-PUB-22-319-AD (accetpted) Cascade Models in Simulation of Extended Heavy Targets Irradiated by Accelerated Proton and Deuteron Beams a",
"FERMILAB-PUB-22-319-AD (accetpted) Cascade Models in Simulation of Extended Heavy Targets Irradiated by Accelerated Proton and Deuteron Beams a"
]
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"M Baznat \nJoint Institute for Nuclear Research\nDubnaRussia\n\nInstitute of Applied Physics\nChisinauMoldova\n",
"A Baldin \nJoint Institute for Nuclear Research\nDubnaRussia\n\nInstitute for Advanced Studies \"OMEGA\"\nDubnaRussia\n",
"E Baldina \nJoint Institute for Nuclear Research\nDubnaRussia\n\nInstitute for Advanced Studies \"OMEGA\"\nDubnaRussia\n",
"M Paraipan \nJoint Institute for Nuclear Research\nDubnaRussia\n\nInstitute of Space Science\nMagureleIlfovRomania\n",
"V Pronskikh \nFermi National Accelerator Laboratory\n60510BataviaILUSA\n",
"P Zhivkov \nInstitute of Nuclear Research and Nuclear Investigations\nSofiaBulgaria AS\n"
]
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"Joint Institute for Nuclear Research\nDubnaRussia",
"Institute of Applied Physics\nChisinauMoldova",
"Joint Institute for Nuclear Research\nDubnaRussia",
"Institute for Advanced Studies \"OMEGA\"\nDubnaRussia",
"Joint Institute for Nuclear Research\nDubnaRussia",
"Institute for Advanced Studies \"OMEGA\"\nDubnaRussia",
"Joint Institute for Nuclear Research\nDubnaRussia",
"Institute of Space Science\nMagureleIlfovRomania",
"Fermi National Accelerator Laboratory\n60510BataviaILUSA",
"Institute of Nuclear Research and Nuclear Investigations\nSofiaBulgaria AS"
]
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| The paper presents a survey of the main numerical models used for simulation of interaction of accelerated particle beams with target nuclei. These models form the core of the software for simulation of various experiments and experimental facilities both for scientific and applied purposes. The beam and target parameters considered in detail in this study (protons and deuterons with energies from 0.66 to 4 AGeV and bulk U targets) cover the range of interest in development of new concepts of nuclear power production aided by accelerated particle beams. | 10.1134/s1063779622050021 | [
"https://arxiv.org/pdf/2204.11940v1.pdf"
]
| 248,392,077 | 2204.11940 | a778422909394827f054d67755f09ebb35d1444c |
FERMILAB-PUB-22-319-AD (accetpted) Cascade Models in Simulation of Extended Heavy Targets Irradiated by Accelerated Proton and Deuteron Beams a
April 2022
M Baznat
Joint Institute for Nuclear Research
DubnaRussia
Institute of Applied Physics
ChisinauMoldova
A Baldin
Joint Institute for Nuclear Research
DubnaRussia
Institute for Advanced Studies "OMEGA"
DubnaRussia
E Baldina
Joint Institute for Nuclear Research
DubnaRussia
Institute for Advanced Studies "OMEGA"
DubnaRussia
M Paraipan
Joint Institute for Nuclear Research
DubnaRussia
Institute of Space Science
MagureleIlfovRomania
V Pronskikh
Fermi National Accelerator Laboratory
60510BataviaILUSA
P Zhivkov
Institute of Nuclear Research and Nuclear Investigations
SofiaBulgaria AS
FERMILAB-PUB-22-319-AD (accetpted) Cascade Models in Simulation of Extended Heavy Targets Irradiated by Accelerated Proton and Deuteron Beams a
April 2022
The paper presents a survey of the main numerical models used for simulation of interaction of accelerated particle beams with target nuclei. These models form the core of the software for simulation of various experiments and experimental facilities both for scientific and applied purposes. The beam and target parameters considered in detail in this study (protons and deuterons with energies from 0.66 to 4 AGeV and bulk U targets) cover the range of interest in development of new concepts of nuclear power production aided by accelerated particle beams.
INTRODUCTION
Numerical study of ADS systems, namely, interaction of accelerated proton and light ion beams with extended targets, encounters a well known problem of description of neutron breeding in such systems. Various approaches and numerical models differ in their predictions of neutron yield and spectrum up to a factor of two. That is why the comparative examination of the widely used models and codes for description of interaction of accelerated ion beams with heavy bulk targets with the idea of further experimental verification is important.
The intranuclear cascade(INC) approach was apparently first developed by Goldberger [1], who in turn based his work on the ideas of Heisenberg and Serber [2], who regarded intranuclear cascades as a series of independent collisions using on-mass-shell free particle-nucleon cross sections. The colliding particles are treated as classical point-like objects moving between collisions on well defined trajectories in the target potential well.
Many intra-nuclear cascade models have been proposed and developed in the past by several groups. In many cases the motivation was to provide a satisfactory level of description of finalstate hadron spectra in the problem of few MeV to few GeV reactions of hadrons with nuclei. They find application in low-energy calorimetry, studies of nucleon shielding, accelerator based nuclear-waste degradation, neutrino beams, or studies of design and application of spallation neutron sources.
Let's remind the main basic assumptions of the INC. The main condition for the applicability of the intranuclear cascade model is that the DeBroglie wavelength λ of the particles participating in the interaction be sufficiently small: It is necessary that for most of these particles λ be less than the average distance between the intranuclear nucleons Δ ∼ 10 −13 cm. Only in this case does the particle acquire quasi-classical features and can we speak approximately of particle trajectory and two-particle collisions inside the nucleus. It is clear that for this to be the case the primary particle kinetic energy T must be greater than several tens of MeV.
Another important condition for applicability of the INC is the requirement that the time in which an individual two-particle intranuclear collision occurs on the average, τ ∼ 10 −23 s, be less than the time interval between two such consecutive interactions = / ≳ 4 , 3 ≳ 3 • 10 344 ( ) , > where l is the mean range of the cascade particle before the interaction, c is the velocity of light, R= ? @ , ⁄ is the mean radius of the nucleus, and σ is the cross section for interaction with an intranuclear nucleon. This permits the interaction of the incident particle with the nucleus to be reduced to a set of individual statistically independent intranuclear collisions.
Since the energy of the particles participating in the cascade is greater than the binding energy of the intranuclear nucleons − the same characteristics can be used for interaction of cascade particles inside the nucleus as for the interaction of free particles. The effect of other intranuclear nucleons is taken into account by introduction of some average potential V , and also by the action of the Pauli principle. The nucleus is considered to be a degenerate Fermi gas of nucleons enclosed in the nuclear volume. Both projectile and nucleus have to be initialized. Regarding the projectile, type and energy are known, but its impact parameter is taken randomly and coulomb deviation considered. If the projectile is a composite particle, its structure must be given in the same way as for the target. The target nucleus is defined by its mass, its charge, the potentials felt by the particles, the momentum of each nucleon (most of the time a Fermi gas distribution is used), and the spatial distribution of the nucleons. Two ways exist to define this distribution. The distribution is either continuous, often several concentric density regions, or discrete, i.e. positions are sampled in a Wood-Saxon distribution, for example. According to the Pauli principle, the nucleons, after an intranuclear collision, must have energy above the Fermi energy; otherwise such an interaction is forbidden. The effect of Pauli principle is very important. The action of the Pauli principle leads to an increase of the mean free path of fast particles inside the nucleus. It is especially pronounced at Einc< 40 MeV causing to rise even though the nucleonnucleon cross section is strongly increasing. Understanding of the limitations of INC at low energies is important for evaluation of reliability of transport calculations used in wide variety of applications. In collisions of high energy particle with the Fermi sea, the momentum transfer is small, and Pauli principle limits the interaction to small fraction of the Fermi sea close to its surface, thus increasing the mean free path. Most of the collisions are not central. Calculations show that in the energies of few tens to few hundreds MeV about 60% of the collisions leading to inelastic reactions occur at impact parameters at which the nuclear density is less than a half of the central density. The target periphery is modelled in all the INC implementations, but each has a different way to deal with the low energy participants chosen considering agreement with the experimental data rather than from basic physical considerations. The incident particle and target constituents are moving on classical trajectories in the potential well and scatter whenever their relative distance is less than B ( DE )/ , σ(Ecm) being the free space cross section, and Ecm their center of mass energy.
Two different methods are applied to move and follow the particles participating to the cascade. With the time-like transport all particles are followed at the same time, while with the space-like transport the particles are followed one after another.
Three events exist during the intranuclear cascade: collision, resonance decay and reflection/transmission at the nucleus surface. Collisions can be elastic or inelastic. Most of the time experimental data (cross sections) are used to define interaction probabilities first and second what are the output products and their characteristics (types, energies, momenta), each selection being done randomly. When necessary, Pauli blocking is taken into account. Crosssection parameterizations, number and type of collisions and of resonances taken into account, and the way to apply Pauli blocking different for different models.
When the particle reaches the surface; it can be emitted or reflected. To be emitted the particle must be energetic enough, i.e. be able to overcome nuclear and Coulomb potential. However, the basics of INC are clearly the treatment of the transport of nucleons with their two-body interactions, i.e. without clusterization. So, up to now,, the only way to produce composite particles during the cascade is to add a coalescence model. Before leaving the nucleus, a nucleon can drag and aggregate one or more nucleons, close enough to it in space and momentum. This procedure extends the INC applicability to a satisfactory level.
Finally, different criteria are used to stop the cascade and to start the de-excitation phase of the remnant nucleus. We can mention three of them: cutoff energy, stopping time and deviation from an optical absorptive potential. Different scenarios for intranuclear cascade are possible depending on the energy and the impact parameter of the incident particle going from ejection of a single nucleon, taking with it all of the incident energy, to the capture of the projectile leaving the nucleus in a state of strong excitation. Once excited, the nucleus enters a second and slower phase, the de-excitation. Here again, different scenarios compete according to the mass, excitation energy and angular momentum of the remnant nucleus. The first and rapid phase is of about 10 −22 s and the second in the order of 10 −18 s. In addition to these two phases sometimes included is a third one named pre-equilibrium. This step is actually an intermediate step since it deals with the transition between cascade and de-excitation and more precisely how the cascade is stopped. The need of this additional phase is then strongly connected with the cascade modeling.
INC reproduces successfully a wide variety of experimental data of hadron and pion induced reactions, using a small number of adjustable parameters, most with clear physical meaning.
I. LIEGE INTRANUCLEAR-CASCADE MODEL INCL4.6
The original Liege INC model for nucleon-induced reactions is described in [3,4]. The standard INCL4.2 model is described in detail in [5,6] and in references cited therein. The INCL4 model is a time-like intranuclear cascade model. In the initial state, all nucleons are prepared in phase space. Target nucleons are given position and momentum at random in agreement with Saxon-Woods and Fermi sphere distributions, respectively. They are moving in a constant potential well describing the nuclear mean field. The incident particle (nucleon or pion) is given the appropriate energy and an impact parameter at random. In this version, incident light particles (up to alphas) are viewed as a collection of on-shell nucleons, with a Fermi motion inside their reference frame, and with a total energy equal to the nominal total incident energy. The collision mechanism is assumed to proceed from a succession of binary collisions (and decays) well separated in space and time. The fate of all particles is followed as time evolves. The particles travel along a straight-line trajectories until two of them reach their minimum distance of approach, in which case they can be scattered provided the value of this distance is small enough, or until they hit the border of the potential well, supposed to describe the nuclear target mean field. Initial positions of target nucleons are taken at random in the spherical nuclear target volume with a sharp surface, initial momenta are generated stochastically in a Fermi sphere. Inelastic collisions, pion production, and absorption are supposed to appear and disappear through the ⇌ ∆ and ∆⇌ reactions. For πN interaction, experimental cross sections are uses, including nonresonant scattering, but the latter is treated as proceeding through the formation of a Δ with a very short lifetime; inelastic πN scattering is neglected for convenience. In the NN → NΔ process, the Δ particle is given the mass mΔ , taken at random from the distribution
( ∆ ) = K L M L M NL O M @ @NPQ R ∆ SR ∆ O T O U V , (1) 4 = XE Y V 3(E Z 3E [ ) V \XE Y V 3(E Z NE [ ) V \ PE Y V ,(2)
with mΔ lying in the interval [ K + _ , √ − K ], √ being the center of mass (c.m.) energy of the collision, and consistent with energy-momentum conservation. The quantity FN in Eq. (1) is the normalization constant. The parameters are q0 = 0.18 GeV, ∆ ? = 1.215 GeV, and Γ ? = 0.13
GeV. The introduction of the q-dependent factor is required by the fit of ⇌ ∆ data and can be justified as follows: a Δ resonance is a correlated pion-nucleon system and the phase space of the latter system is considerably reduced when its c.m. energy is low. The average intrinsic lifetime τΔ was taken as follows:
@ de = L M L M NL O M @ d O ,(3)where proper time ? = ħ h O
. This is justified as follows: if the Δ resonance is going to decay into a πN pair with low energy (which is the case for small mΔ in our classical picture), the decay width is considerably diminished due the reduction of phase space. The stopping time of the cascade is determined self-consistently by the model itself. It can simply be parameterized (in fm/c) by ijkl = 29.8 q ?.@r , (4) for incident nucleons(ZT and AT are the charge and mass numbers of the target, respectively). At the beginning of the cascade process, the incident nucleon or pion is located with its own impact parameter on the surface of the working sphere, which is centered on the target with the radius
Est = ? + 8 ,(5)
where R0 and a are respectively the radius and the diffuseness of the target nucleus density. Particles are moving along straight-line trajectories between collisions inside the working sphere. They are divided into participants and spectators in the usual sense. When participants leave the working sphere, they are considered as ejectiles and do not interact anymore. The potential radius for particles with energy larger than the Fermi energy is also taken to be equal to Rmax. Pions do not experience any potential. The depth of the potential well felt by the nucleons is dependent on the energy of the nucleons and is not the same for protons and neutrons. The energy dependence is taken from the phenomenology of the real part of the optical-model potential.
( ) = v w O @Nxtly (zS{ O ) | } , < Est 0, > Est .(6)
The values of R0 and a are taken from electron scattering measurements and parameterized, for convenience, from Al to U, as follows: R0 = (2.745x10 −4 AT + 1.063)(AT) 1/3 fm, = 0.510 + 1.63x10 −4 AT fm (in the numerical code, other values, as well as another shape for ρ(r), can optionally be introduced). The quantity ρ0 is such that the distribution is normalized to AT , the target mass number. The momentum distribution is kept as a uniform Fermi sphere with Fermi momentum pF. Nucleons with high momentum in the central part of the nucleus are expected to travel farther out than those with low momentum.
Therefore it is considered that a nucleon with the momentum p is to reach the maximum radial distance R(p). Because of these r-p correlations, a nucleon with the momentum between p and p + dp should be located, with a constant uniform probability, in a sphere of the radius R(p). This radius can be deduced by assuming that the number of nucleons populating the layer of density profile ρ(R(p)) and ρ(R(p + dp)) is the same as the number of nucleons with the momentum between p and p + dp:
from which R(p) can be deduced. The initial position and momentum of any target nucleon are generated as follows: ⃗ is taken at random in a sphere with the radius pF , R(p) is calculated using Eq. (8), and ⃗ is chosen at random in a sphere with the radius R(p). This is equivalent to taking ⃗, ⃗ at random according to the joint probability distribution
where θ(x) is the Heaviside function. In practice, the value of p can be generated from the uniform Fermi sphere distribution and the position is generated uniformly in a sphere with the radius R(p). After integration over the relevant variables, the joint distribution in Eq. (9) corresponds to the spatial density ρ( ⃗) and to the sharp Fermi sphere momentum distribution:
∫ ( ⃗, ⃗) , = ( ⃗),(10)∫ ( ⃗, ⃗) , = q ç(l É 3l) Ç[ M l É M . (11)
The procedure outlined above is at variance with the one used in many transport models, where nucleons are placed in a potential with a Saxon-Woods or similar shape. The dynamical Pauli blocking in INCL4.2 operates in phase space and is implemented as follows: if two nucleons i and j are going to suffer a collision at positions é(è) êêêêêêê⃗ leading to the final state with momenta é(è) êêêêêêê⃗, the phase space occupation probabilities fi are evaluated by counting nearby nucleons in a small phase-space volume,
ë = @ 4 4_ħ M Ç[ M â íì M l íì M × ∑ ( óò − ô ö êêê⃗ − è ê ê⃗ô) × ( óò − ô ö êêêê⃗ − è êêê⃗ô) öõú ,(12)
where the sum is limited to particles k with the same isospin component as particle i (or j). The factor ½ is introduced because spin components are ignored. The parameters rPB and pPB were taken to have the following values: rPB = 3.18 fm and pPB = 200 MeV/c. The collision between i and j is allowed or forbidden following the comparison of a random number with the product (1 −fi)(1 − fj ). Pauli blocking is not applied to Δ particles because their density is always very small. On the other hand, it is enforced for nucleons resulting from Δ decays. At the end of the cascade, surviving Δ resonances from inelastic collisions are forced to decay and the conservation of baryon number, charge, energy, momentum and angular momentum is verified,
AP+AT=Aej+Arem,(13)
ZP+ZT=Zej+Zπ+Zrem, (14) Tlab=Kej+Wπ+Erec+E*+S, (15) ùsû êêêêêêê⃗ = xè êêêêê⃗ + _ êêêê⃗ + âxE êêêêêêêêê⃗, (16) ⃗ = xè êêêê⃗ + _ êêê⃗ + âxE êêêêêêêê⃗ + * êê⃗ , (17) for baryon number, charge, energy, momentum, and angular momentum, respectively. The projectile P colliding with the target T and generating (baryonic) ejectiles, pions, and a remnant (which is the remaining part of the target up to the end of the cascade stage) are considered. In Eq. (15), Kej is the kinetic energy of the ejectiles, Wp is the total energy of the pions, Erec is the recoil energy of the remnant, E * is the remnant excitation energy, and S is the separation energy, i.e., the minimum energy needed to remove all ejectiles and pions from the target nucleus ground state. In the other equations, the indices have the similar meaning. In Eq. (17), ⃗ is the angular momentum of the incident particle, âxE êêêêêêêê⃗ is the angular momentum corresponding to the c.m. motion of the remnant, and * êê⃗ is the intrinsic angular momentum of the remnant. The INCL4.2 version was tested successfully, in the 200 MeV -2 GeV range, against a large data base, but some phenomenological aspects of nuclear physics were neglected. The model cannot describe production of clusters in the cascade, i.e. with a kinetic energy definitely larger than the typical evaporation energies, as it can be seen experimentally. Concerning the predictive power of the model, several deficiencies can be noted. Pion production is generally overestimated. Quasielastic peaks in (p, n) reactions are generally too narrow and sometimes underestimated. Finally, the reaction cross sections are severely underestimated below ~100 MeV. Residue production cross sections are sometimes unsatisfactorily reproduced, especially for residues close to the target.
In 2009, an upgraded version (INCL4.5) was released [6] where the following model improvements were implemented.
An average isospin-dependent potential well, of the Lane type, is introduced for pions, as well as reflection and/or transmission at the border of this potential. The depth of the potential was taken as far as possible from the phenomenology of the real part of the pion-nucleus optical potential (dispersive effects due to the strong imaginary part have to be removed). This depth amounts to 22 MeV for π+ and 38 MeV for π− on the Pb target. The radius of the potential is taken as R0+4 , in rough accordance with phenomenology. This modification reduces the pion production cross section, thus compensating the overestimation by INCL4.
2.
An improvement of the INCL4.2 model is that outgoing nucleon crossing the nuclear periphery is supposed to be able to carry along other nucleons to form a cluster, provided the involved nucleons are lying sufficiently near each other in phase space. To limitations in computing time, clusters up to Dù Est = 8 are considered. The emitted cluster should have sufficient energy to escape, i.e. Dù = ∑( ë − ë ) − Dù > 0, where the Ti are the kinetic energies of the nucleons and Vi are the depths of their potential wells and the cluster has also to succeed the test for penetration through the Coulomb barrier. At the end of the cascade process, short-lived clusters with a lifetime of less than 1 ms (e.g. 5 Li) are forced to decay, isotropically in their c.m. frame. Clusters with a lifetime larger than 1 ms are considered detectable, prior to decay.
In this way, the following features are introduced. First, the deflection of charged particles in the Coulomb field was taken into account, both for incident and outgoing particle at the nuclear periphery. Second, light charged particles can be emitted by the cascade owing to a dynamical coalescence model: nucleons leaving the target may carry other nucleons provided they are sufficiently close by in phase space. Third, the behavior of the model at low incident energy is improved, mainly by a better account of soft collisions, especially in the first instances of the reaction process.
Finally, the last version of the model, INCL4.6, was published in 2013. A detailed account can be found in [7]. The main new development involves the treatment of cluster-induced reactions.
Treatment of cluster-induced reactions is as follows. In INCL4.2, an incident cluster (up to an alpha particle) is considered as a collection of independent nucleons with internal Fermi motion superimposed to the motion of the incident cluster as a whole (see [5]), adjusted in such a way that the sum of the total energies of the constituting nucleons is equal to the nominal energy of the physical cluster. In other words, the cluster is replaced by independent on-shell nucleons with the correct nominal total energy, but with an incorrect (smaller than nominal) total momentum. This approximation is justified at high energy, but it is not really appropriate for reactions at low incident energy.
Initialization of the incident cluster is as follows. Nucleon momenta é £ êêê⃗ and positions é £ êêê⃗ i inside the cluster are generated as before [5] (note, however, that a special method is applied to ensure ∑ é £ êêê⃗ = 0 and ∑ é £ êêê⃗ = 0). At the beginning of the event, the cluster center of mass is positioned on the classical Coulomb trajectory in such a way that one of the nucleons is touching a sphere of radius RCoul. The latter represents the Coulomb barrier. The value of RCoul is taken from the phenomenology of the Coulomb barrier heights and has been tabulated as function of the target mass for p, d, t, 3 He and 4 He projectiles.
Collisions are, of course, governed by Pauli blocking, treated in a different way in the first and in the subsequent collisions. The nucleons involved in the first collision are subject to a strict blocking: after the collision, both of them should lie outside the Fermi sphere. In subsequent collisions, the blocking is applied stochastically, with a probability given by the product of final state blocking factors. A careful definition of the latter allows one to account for surface effects and for the depletion of the Fermi sphere during the evolution of the cascade.
An important novelty of recent versions of the code is the introduction of a coalescence model based on phase space, which permits the emission of light clusters, with mass A ≤ 8, during the cascade stage, in keeping with experimental evidence.
The INCL4.6 version uses modified value for Rmax
Rmax=R0+ 8 +rint = R1+ 8 ,(18)
and the model separation energy Si is replaced by the physical separation energy ë l• ¶i , taken from mass tables, for the emission from the actual nucleus. The modification of Rmax increased the maximum time of the cascade, which now corresponds to the time of passage of the incident particle through the "working sphere" along a diameter, when this time exceeds the usual stopping time, given by Eq. (4).
An important characteristic of the model is the self-consistent determination of the stopping time of the cascade, which can be simply parameterized as ijkl = 29.8 q ?.@r fm/s, where AT is the mass of the target nucleus. At = ijkl many physical quantities, such as the excitation energy of the target nucleus and the average kinetic energy of the ejectiles, switch from a fast time evolution, dominated by intranuclear cascade, to a much slower evolution, which is taken as a signature of equilibration. Thanks to this choice of the stopping time, it is not necessary to introduce a pre-equilibrium model describing the intermediate stage between the fast cascade and the evaporation-fission decay.
Intranuclear-cascade models in general (and INCL in particular) only describe the fast, dynamical stage of a spallation reaction, leading to the formation of excited nuclei which subsequently de-excite by emitting particles and/or fissioning. It is therefore necessary to follow the de-excitation of this cascade remnant if one requires a complete description of the nuclear reaction. Since the time scale for de-excitation is much longer than for cascade, a different physical description is usually employed. This may include an optional pre-equilibrium stage, which then handles the thermalization of the remnant; if pre-equilibrium is used, the intranuclear-cascade stage is stopped earlier. Either way, thermalization is attained and subsequent de-excitation of the remnant is described by statistical de-excitation models.
Such a pre-equilibrium model is sometimes used between the cascade and the de-excitation phases. Several versions exist, but almost all are based on the exciton model developed by Griffin [8]. According to the use or rejection of this intermediate phase, the duration of the cascade is obviously different or, maybe more correctly, mass, charge and excitation of the remnant nucleus are larger, if this phase is called. While some intranuclear cascade models need such pre-equilibrium models to improve their capability, this is not the case of some others.
Boudard, co-developer of INCL, initiated the translation of the Fortran77 version of INCL in C++. The work, started by P. Kaitaniemi [11], was continued and finalized by D. Mancusi [12]. This provided the opportunity to re-consider the INCL code and made its maintenance easier.
The main transport code implemented the INCL4.6 [7] version as the default intranuclear cascade model.
For this purpose, the Liege intranuclear cascade model (INCL) [12] is used; this model has been recently extended towards high energies (≈ 15 GeV) including multipion production [13,14], strange particles, such as kaons and hyperons [15,16], and the production of η and ω mesons [17]. This new version of the INCL allows us to predict the formation of hyperremnants and their characterization in atomic ,mass, and strangeness numbers together with their excitation energies and angular momenta. These improvements in INCL also require de-excitation models considering the emission of hyperons, in particular, the evaporation of particles. Currently, there are a few numbers of de-excitation models that treat the evaporation of hyperons and the formation of hypernuclei, such as the evaporation model ABLA07 developed at GSI by Kelic and collaborators [18] and recently extended to hypernuclei by us including the evaporation of particles on the basis of Weisskopfs approach according to [19].
II. BINARY CASCADE
Binary Cascade is a hybrid between a classical intranuclear cascade and a QMD [20] model, for the simulation of inelastic scattering of pions, protons and neutrons, and light ions of intermediate energies off nuclei [21]. The nucleus is modeled by individual nucleons bound in the nuclear potential. Binary collisions of projectiles or projectile constituents and secondaries with single nucleons, resonance production, and decay are simulated according to measured, parameterized or calculated cross sections. The Pauli exclusion principle, i.e. blocking of interactions due to Fermi statistics, reduces the free cross section to an effective intra-nuclear cross section. Secondary particles are allowed to further interact with remaining nucleons. The Binary Cascade models interactions of nucleons, pions, and light ions with nuclei for incident particle energies in the energy range starting from few MeV up to few GeV.
The Binary Cascade introduces a new approach to cascade calculations. It is based on a detailed 3-dimensional model of the nucleus, and exclusively based on binary scattering between reaction participants and nucleons within this nuclear model. This feature makes it a hybrid between a classical cascade code, and a quantum molecular dynamics model (QMD) [20]. In Binary Cascade, like in QMD, each participating nucleon is seen as a Gaussian wave package, (19) propagating in time and space, undergoing collisions with nucleons in the nuclear medium in the process.
( ; ë ; ë ; ) = 2 ( ) , P ⁄ ⁄ exp (− 2 X − ( )\ 4 ⁄ + ë ( ) ),
Here, x and t are space and time coordinates, and qi and pi describe the nucleon position in the configuration and momentum space. The total wave function is assumed to be the direct product of the wave functions of the participating nucleons and hadrons. Participating means that they are either primary particles, or were generated or scattered in the process of the cascade.
Binary Cascade is an intra-nuclear cascade propagating primary nucleons and all secondary particles within a nucleus. Interactions take place between a primary or secondary particle and an individual nucleon of the nucleus. The nucleus is modeled by explicitly positioning nucleons in space, and assigning momenta to these nucleons. This is done in a way consistent with the nuclear density distributions, Pauli's exclusion principle, and the total nu-clear mass.
Propagating particles in the nuclear field is done by numerically solving the equations of motion, using time-independent fields derived from optical potentials. The cascade begins with a projectile and the nuclear description, and terminates when the average energy of all participants within the nuclear boundaries are below a given threshold. The remaining pre-fragment will be treated by pre-equilibrium decay and de-excitation models.
For the primary particle an impact parameter is chosen randomly on a disk outside the nucleus, perpendicular to a vector passing through the center of the nucleus. The initial direction of the primary is perpendicular to this disk. Using straight-line transport, the distance of closest approach ë Eëå to each nucleon i in the target nucleus, and the corresponding time-of-flight ë Å is calculated. The interaction cross-section σi with target nucleons is calculated based on the momenta of the nucleons in the nucleus, and the projectile momentum. The target nucleons for which the distance of the closest approach ë Eëå is smaller than Ø ∞ ± _ i are candidate collision partners for the primary. All candidate collisions are ordered by increasing ë Å . If no collision is found, a new impact parameter is chosen. This way transparency effects at the nuclear boundaries are taken into account. The primary particle is then transported in the nuclear field by the time step given by the time to closest approach for the earliest collision candidate. Outside the nucleus, particles travel along straight-line trajectories. Particles entering the nucleus have their energy corrected for Coulomb effects. Inside the nucleus particles are propagated in the nuclear field. The equation of motion in the field is solved for a given time step using a Runge-Kutta integration method. At the end of each step, the interaction of the collision partners is simulated using the scattering term described below, resulting in a set of candidate particles for further transport. The secondaries from a binary collision are accepted subject to Pauli's exclusion principle. If the momentum of any of the particles is below the Fermi momentum, the interaction is suppressed, and the original primary continues to the time of its next collision. In case an interaction is Pauli allowed, the tracking of the primary ends, and the secondaries are treated like the primary. All their possible binary collisions with the residual nucleus are calculated, with the addition of decay in case of strong resonances. For resonance decay, the collision time is the time to the decay of the particle, sampled from the resonance's lifetime.
Herein the stochastic masses and decay widths are taken into account. All secondaries are tracked until they react, decay or leave the nucleus, or until the cascade stops due to the cut-off condition described above.
A 3-dimensional model of the nucleus is constructed from A nucleons and Z protons with coordinates ri and momenta pi, with i = 1, 2, . . . ,A. Nucleon radii ri are selected randomly in the nucleus rest frame according to the nuclear density ρ(ri). For nuclei with A > 16, the Woods-Saxon form of the nucleon density [22] is used,
( ë ) = w O @N≤≥¥ [(â µ 3ä) s ⁄ ] ,(20)
where ρ0 is approximated as To take into account the repulsive core of the nucleonnucleon potential, a minimum inter-nucleon distance of 0.8 fm is taken. The nucleus is assumed to be spherical and isotropic, i.e. each nucleon is placed using a random direction and the previously determined radius ri.
? = , P_ä M y1 + s V _ V ä V } 3@ ,(21)
The momenta pi of the nucleons are chosen randomly between 0 and the Fermi momentum π Est ( ë ). The Fermi momentum, in the local Thomas-Fermi approximation as a function of the nuclear density ρ, is π Est ( ë ) = ħ (3 4 ( )) @/, .
The total vector sum of the nucleon momenta has to be zero, i.e. the nucleus must be constructed at rest. To achieve this, one nucleon is chosen to compensate the vector sum of the remaining nucleon momenta âxij = − ∑ ë ë∫Ü3@ ë∫@ . If this sum is larger than the maximum allowable momentum π Est ( ë ), the direction of the momenta of the nucleons with the largest contribution to the net nucleus momentum is iteratively flipped, until the residual sum is an allowed momentum value for a nucleon.
The effect of collective nuclear interaction upon participants is approximated by a time-invariant scalar optical potential, based on the properties of the target nucleus. For protons and neutrons the potential used is determined by the local Fermi momentum π ( ) as
( ) = l É V 4E ,(24)
where m is the mass of the neutron or the mass of the proton, respectively. For pions the potential used is a simple approximation given by the lowest-order optical potential as derived in [24]:
( ) = − 34_(ℏD) V Ü E [ y1 + E [ º } ? ( ).(25)
Here, A is the nuclear mass number, and mπ and M are the pion and nucleon masses, respectively; _ is the reduced pion mass, _ =
E [ NE Z E [ º ,
where K is the mass of the nucleus; ρ(r) is the nucleon density distribution. The parameter b0 is the effective s-wave scattering length. The value used was obtained from the analysis to pion atomic data and resulted in b0 to be about -0.042 fm. It is assumed that the nucleus is in its ground state and all states below Fermi energy are occupied. Thus, collisions and decays for which any secondary nucleon has a momentum pi below the local Fermi momentum, i.e. ë < π Est ( ë ), (26) are suppressed.
The basis of the description of the reactive part of the scattering amplitude are two-particle binary collisions, also with associated or direct resonance production, and decay. Based on the cross section described later, collisions will occur when the transverse distance dt of any participant target pair becomes smaller than the black-disk radius corresponding to the total cross-section σt
∞ Ω l µ > j 4 .(27)
Experimental data and parameterizations thereof are used in the calculation of the total, inelastic and elastic cross-section wherever available.
For the case of proton-proton (pp) and proton-neutron (pn) collisions, as well as π + -and π −nucleon collisions, experimental data and parameterizations are readily available as collected by the Particle Data Group (PDG) [25] for both elastic and inelastic collisions. The tabulation based on a sub-set of these data for √ below 3 GeV, and the PDG parameterization at higher energies, are applied.
It also defines an upper limit of applicability of the model. Below 10 GeV kinetic energy, the resonance contributions considered are wholly sufficient to describe the total cross-section.
Most of the cross-sections of individual channels involving meson-nucleon scattering can be modeled as resonance excitation in the s-channel.
The initial states included in the model at present include all pion-nucleon scattering channels. The product resonances taken into account are the Delta-resonances with masses of 1232, 1600, 1620, 1700, 1900, 1905, 1910, 1920, 1930, and 1950 MeV, and the excited nucleons with masses of 1440, 1520, 1535, 1650, 1675, 1680, 1700, 1710, 1720, 1900, 1990, 2090, 2190, 2220, and 2250 MeV.
In the resonance production in the t-channel, single and double resonance excitations in nucleonnucleon collisions are taken into account.
The resonance production cross-sections are as much as possible based on parameterizations of experimental data for proton-proton scattering. The formula used for parameterizing the crosssections is motivated from the form of the exclusive production cross-section of the Δ1232 in proton-proton collisions:
Üò = 2 Üò Üò √i3Bi O (√i3Bi O ) V N¿ ¡ì V ¬ BiON¿¡ì √i √ Üò ƒ .(28)
For all other channels, the parameterizations were derived from these by adjusting the threshold behavior accordingly. Cross-sections for the reminder of the channels are derived from those described above, by applying detailed balance. Isospin invariance is assumed. The formalism used to apply detailed balance is
( → ) = ∑ ∆ D D Å Å ô V ∆ s s û û ô V À,º (4Ã | N@)(4Ã Õ N@) (4Ã OE N@)(4Ã oe N@) 〈l ¡ì V 〉 〈l -- V 〉 ( → ).(29)
Angular distributions for elastic scattering of nucleons are taken as closely as possible from experimental data, i.e. from the result of a phase shift analysis. They are derived from differential cross-sections obtained from the SAID database, R. Arndt et al. [26].
Angular distributions for final states other than nucleon-nucleon elastic scattering are calculated analytically, derived from the collision term of the in-medium relativistic Boltzmann-Uehling-Uhlenbeck equation [27] via scaling of the center-of-mass energy. The modular structure of GEANT4 allows the generation of single events with a known incident particle energy and any explicitly defined hadronic final-state generator. The kinematics of secondaries produced in the interaction are then analyzed and the resulting angular, momentum, energy, and baryon number spectra are stored in histograms. The energy-momentum balance can be controlled as well. The histograms are compared to published measurements of the differential and double differential dσ/dE cross sections, dσ/dE, dσ/dΩ, d 2 σ/dEdΩ, and the invariant cross-sections, Ed 3 σ/d 3 p.
The range of Binary Cascade model applicability in nucleon nuclear reactions stretches from < 100 MeV to about 10 GeV, allowing for a consistent calculation of the secondary hadron spectra in the low and intermediate energy domains.
III. BERTINI CASCADE
The INC model developed by Bertini [28][29][30][31] solves on the average the Boltzmann equation of this particle interaction problem. The Bertini nuclear model consists of a three-region approximation to the continuously changing density distribution of nuclear matter within nuclei. Relativistic kinematics is applied throughout the cascade and the cascade is stopped when all the particles which can escape the nucleus, have done so. The Pauli exclusion principle is taken into account and conformity with the energy conservation law is checked. Path lengths of nucleons in the nucleus are sampled according to the local density and free nucleon-nucleon cross-sections. Angles after collisions are sampled from experimental differential cross-sections. Intermediate energy nuclear reactions up to 10 GeV energy are treated for proton, neutron, pions, photon and nuclear isotopes.
The necessary condition of validity of the INC model is ò ≪ D ≪ Δ ⁄ , where ò is the de Broglie wavelength of the nucleons, is the average relative nucleon-nucleon velocity and Δt is the time interval between collisions. The physical foundation becomes approximate at energies less than about 200 MeV , and there needs to be supplemented with a pre-equilibrium model. Also, at energies higher than 5-10 GeV the INC picture breaks down. The basic steps of the INC model are summarized below.
The nucleons are assumed to have a Fermi gas momentum distribution. The Fermi energy is calculated in a local density approximation i.e. it is made radius dependent with the Fermi momentum π ( ) = y ,_ V w(â) 4 } @/, . The initialization phase fixes the nucleus radius and momentum according to the Fermi gas model.
If the target is Hydrogen (A = 1), a direct particle-particle collision is performed, and no nuclear modeling is used. If A > 11, nuclei are modeled with three concentric spheres as well. The sphere radii are then defined as:
ë ( ë ) = 4 log ⁄ @Nx S -± -V € µ − 1‹ + @ ,
where C2 = 1.7234.
The potential energy for nucleon N is
K = π 4 2 K + K ( , )
where pF is the Fermi momentum and BE the binding energy.
The momentum distribution in each region follows the Fermi distribution with zero temperature.
( ) = 4 ,(30)
where
∫ ( ) l É ? = l å .(31)
Here np and nn are the numbers of protons and neutrons in the region and pF is momentum corresponding the Fermi energy
π = l É V 4E Z = ℏ V 4E Z y ,_ V fl } V M ,(32)
which depends on the density n/v of particles, and which is different for each particle and each region. The path lengths of nucleons in the nucleus are sampled according to the local density and free nucleon-nucleon cross-sections. The angles after collisions are sampled from experimental differential cross-sections. Thus, the free particle-particle cross-sections and region-dependent nucleon densities are used to select the path length for the projectile particle. The tabulated total reaction cross-sections are calculated by Letaw's formulation [32][33][34]. For nucleon-nucleon cross-sections, parameterizations based on the experimental energy and isospin dependent data are used.
For pions the INC cross-sections are provided to treat elastic collisions, and inelastic channels: π − n → π 0 n, π 0 p → π + n and π 0 n → π − p. Multiple particle production is also implemented.
The S-wave pion absorption channels π + nn → pn, π + pn → pp, π 0 nn → X , π 0 pn → pn, π 0 pp → pp, π − nn→ X , π − pn → nn , and π − pp → pn are implemented.
The Pauli exclusion principle forbids interactions where the products would be in occupied states. Following the assumption of a completely degenerate Fermi gas, the levels are filled from the lowest level. The minimum energy allowed for a collision product corresponds to the lowest unfilled level of system, which is the Fermi energy in the region. So, in practice, the Pauli exclusion principle is taken into account by accepting only secondary nucleons which have EN > EF .
After INC, the residual excitation energy of the resulting nucleus is used as input for a nonequilibrium model. The Geant4 cascade model implements the exciton model proposed by Griffin [8]. In this model nuclear states are characterized by the number of exited particles and holes (the exitons). INC collisions give rise to a sequence of states characterized by increasing exciton number, eventually leading to an equilibrated nucleus. For practical implementation of the exciton model we use level density parameters from [35] and the matrix elements from [36]. In the exciton model the possible selection rules for particle-hole configurations in the course of the cascade are: Δp = 0,±1; Δh = 0,±1; Δn = 0,±2, where p is the number of particles, h is number of holes, and n = p + h is the number of exitons. The cascade pre-equilibrium model uses target excitation data, and exciton configurations for neutrons and protons to produce the nonequilibrium evaporation. The angular distribution is isotropic in the frame of rest of the exciton system. The parameterizations of the level density used are tabulated both with their A and Z dependence and including high temperature behavior. The smooth liquid high energy formula is used for the nuclear binding energy.
Fermi break-up is allowed only in some extreme cases, i.e. for light nuclei (A < 12 and 3(A − Z) < Z < 6 ) and if Eexcitation > 3Ebinding. A simple explosion model decays the nucleus into neutrons and protons and decreases exotic evaporation processes. The fission model is a phenomenological model using potential minimization. The binding energy parametrization is used and some features of the fission statistical model are incorporated as in [37]. The statistical theory for particle emission from exited nuclei remaining after INC was originally developed by Weisskopf [38]. This model assumes complete energy equilibration before particle emission, and re-equilibration of excitation energies between successive evaporation emissions. As a result, the angular distribution of emitted particles is isotropic.
The emission of particles is computed until the excitation energy falls below the cutoff value. If a light nucleus is highly exited, the Fermi break-up model is executed. In addition, fission is performed when the fission channel is open. The main chain of evaporation is followed until Eexcitation falls below Ecutoff = 0.1 MeV. The evaporation model ends with a γ emission chain, which is followed until Eexcitation < E γ cutoff = 10 −15 MeV.
Extensive benchmarking of the INC physics provided by Bertini cascade sub-models, exitons, pre-equilibrium state, nucleus explosion, fission, and evaporation has been made. The Geant4 evaporation model for cascade implementation adapts the widely used computational method developed by Dostrowski [39,40]. The model is validated up to 10 GeV incident energy and users from various fields have been using it successfully.
To validate Bertini isotope production physics performance, extensive simulations on protoninduced reactions in Pb and Au targets were performed with Geant4 [41]. The Bertini cascade model in Geant4 simulates the hadronic interactions of protons, neutrons and pions with surrounding materials.
IV. CEM AND LAQGSM MODELS
The Los Alamos National Laboratory (LANL) Monte-Carlo N-particle transport code MCNP6 [42] uses by default the latest version of the cascade-exciton model (CEM), CEM03.03 [43][44][45], as its event generator to simulate reactions induced by nucleons, pions, and photons with energies up to 4.5 GeV and the Los Alamos version of the quark-gluon string model (LAQGSM), LAQGSM03.03 [45][46][47], to simulate such reactions at higher energies, as well as reactions induced by other elementary particles and by nuclei with energies up to ∼ 1 TeV/nucleon. Details, examples of results, and useful references to different versions of CEM and LAQGSM can be found in [45].
The cascade-exciton model (CEM) [44] of nuclear reactions is based on the standard Dubna intranuclear cascade model [48,49] and the modified exciton model (MEM) [50,51]. The CEM code calculates nuclear reactions induced only by nucleons, pions, and photons. A detailed description of the initial version of the CEM can be found in [44], therefore we outline here only its basic assumptions. The CEM assumes that reactions occur in three stages. The first stage is the INC, in which primary particles can be re-scattered and produce secondary particles several times prior to absorption by, or escape from, the nucleus. All the cascade calculations are carried out in the three-dimensional geometry. The nuclear matter density ρ(r) is described by the Fermi distribution with two parameters taken from the analysis of electron-nucleus scattering, namely
( ) = l ( ) + å ( ) = ? {1 + [( − )⁄ ]},(33)
where c = 1.07A 1/3 fm, A is the mass number of the target, and = 0.545 fm. For simplicity, the target nucleus is divided by concentric spheres into seven zones in which the nuclear density is considered to be constant. The energy spectrum of the target nucleons is estimated in the perfect Fermi-gas approximation with the local Fermi energy π ( ) = ħ 4 [3 4 ( )] 4 , ⁄ (2 ⁄ K ), where mN is the nucleon mass. The influence of intranuclear nucleons on the incoming projectile is taken into account by adding to its laboratory kinetic energy the effective real potential V, as well as by considering the Pauli principle which forbids a number of intranuclear collisions and effectively increases the mean free path of cascade particles inside the target. For incident nucleons V≡ VN(r)=TF(r)+ϵ, where TF(r) is the corresponding Fermi energy and ϵ is the binding energy of the nucleons. For pions, CEM03.01 uses a square-well nuclear potential with the depth Vπ ≃ 25 MeV, independently of the nucleus and pion energy, as was done in the initial Dubna INC [48,49].
The Pauli exclusion principle at the cascade stage of the reaction is handled by assuming that nucleons of the target occupy all the energy levels up to the Fermi energy. Each simulated elastic or inelastic interaction of the projectile (or of a cascade particle) with a nucleon of the target is considered forbidden if the "secondary" nucleons have energies smaller than the Fermi energy. If they do, the trajectory of the particle is traced further from the forbidden point and a new interaction point, a new partner and a new interaction mode are simulated for the traced particle, etc., until the Pauli principle is satisfied or the particle leaves the nucleus.
If the residual nuclei after the INC have atomic numbers with A≤AFermi=12, CEM uses the Fermi break-up model to calculate their further disintegration instead of using the pre-equilibrium and evaporation models. Fermi break-up, which estimates the probabilities of various final states by calculating the approximate phase space available for each configuration, is much faster to calculate and gives results very similar to those from using the continuation of the more detailed models for lighter nuclei.
An important ingredient of the CEM is the criterion for transition from the intranuclear cascade to the pre-equilibrium model. The cascade model uses a different criterion to decide when a primary particle is considered to have left the cascade (cutoff energy Tcut or cutoff time tcut). In CEM the effective local optical absorptive potential Wopt.mod.(r) is defined from the local interaction cross section of the particle, including Pauli blocking effects. This imaginary potential is compared to the one defined by the phenomenological global optical model Wopt.exp.(r). The degree of similarity or difference of these imaginary potentials is characterized by the parameter = ôX klj.EkÅ. − klj.xtl. \ klj.xtl. > ô.
When increases above an empirically chosen value, the particle leaves the cascade, and is then considered to be an exciton. CEM uses the fixed value = 0.3.
When the cascade stage of a reaction is completed, CEM uses the coalescence model to create high-energy d, t, 3 He, and 4 He fragments by final-state interactions among emitted cascade nucleons outside of the target nucleus. The value of the momentum p of each cascade nucleon is calculated relativistically from its kinetic energy T. It is assumed that all the cascade nucleons having differences in their momenta smaller than pc and with the correct isotopic content form an appropriate composite particle.
The coalescence model first checks all nucleons to form 2-nucleon pairs, if their momenta permit it. It then takes these 2-nucleon pairs and the single nucleons left and forms 4 He, 3 He, and/or tritium, if their momenta permit it. The extended coalescence model further takes these two-nucleon pairs, tritium, 3 He, and 4 He to see if they can coalesce to form heavier clusters: 6 He, 6 Li, 7 Li or 7 Be. All coalesced nucleons are removed from the distributions of nucleons so that atomic and mass numbers are conserved.
The results show significant improvement in the production of heavy clusters in the whole energy range. However, too many alpha particles were lost (coalesced into heavy clusters); so pc ( The emission of the cascade particles determines the particle-hole configuration, Z, A, and the excitation energy that is the starting point for the pre-equilibrium stage of the reaction. The subsequent relaxation of the nuclear excitation is treated in terms of an improved Modified Exciton Model (MEM) [50,51] of pre-equilibrium decay, followed by the equilibrium evaporation/fission stage described using a modification of the generalized evaporation model (GEM) code GEM2 by Furihata [52]. The transition from the pre-equilibrium stage of a reaction to the third (evaporation) stage occurs when the probability of nuclear transitions changing the number of excitons n with ∆n = +2 becomes equal to the probability of transitions in the opposite direction, with ∆n = −2, i.e., when the exciton model predicts that equilibration has been established in the nucleus.
Generally, all three components can contribute to experimentally measured particle spectra and other distributions.
The Los Alamos version of the Quark-Gluon String Model (LAQGSM) [46,47] is a further development of the Quark-Gluon String Model (QGSM) by Amelin, Gudima, and Toneev (see [55] and references therein) and is intended to describe both particle-and nucleus-induced reactions at energies up to about 1 TeV/nucleon. The basis of QGSM is the time-dependent version of the intranuclear-cascade model developed at Dubna, often referred in literature simply as the Dubna intranuclear Cascade Model (DCM) (see [53] and references therein). LAQGSM also describes nuclear reactions as three-stage processes: an INC, followed by pre-equilibrium emission of particles during the equilibration of the excited residual nuclei formed after the INC, followed by evaporation of particles from and/or fission of the compound nuclei.
The DCM models interactions of fast cascade particles ("participants") with nucleon spectators of both the target and projectile nuclei and includes as well interactions of two participants (cascade particles). It uses experimental particle+particle cross sections at energies below 4.5 GeV/nucleon, or those calculated by the quark-gluon string model (QGSM) at higher energies (see, e.g., [54] and references therein) to simulate angular and energy distributions of cascade particles, and also considers the Pauli exclusion principle.
When the cascade stage of a reaction is completed, QGSM uses the coalescence model described in [53] to "create" high-energy d, t, 3 He, and 4 He by final-state interactions among emitted cascade nucleons outside of the colliding nuclei. After calculating the coalescence stage of a reaction, QGSM moves to the description of the last slow stages of the interaction, namely to pre-equilibrium decay and evaporation, with a possible competition of fission using the standard version of CEM [44]. If the residual nuclei have atomic numbers A ≤ 12, QGSM uses the Fermi break-up model to calculate their further disintegration instead of using the pre-equilibrium and evaporation models. LAQGSM differs from QGSM by replacing the pre-equilibrium and evaporation parts of QGSM described according to the standard CEM [44] with the new physics from CEM2k [56,57] and has a number of improvements and refinements in the cascade and Fermi break-up models. A detailed description of LAQGSM and further references can be found in [46,47]. The coalescence model was extended to be able to produce light fragments up to 7 Be in CEM and up to 12 C in LAQGSM.
The pre-equilibrium interaction stage of nuclear reactions is considered by the current CEM and LAQGSM in the framework of the latest version of MEM [50,51]. At the pre-equilibrium stage of a reaction, CEM03.03 and LAQGSM03.03 take into account all possible nuclear transitions changing the number of excitons n with ∆n = +2, -2, and 0, as well as all possible multiple subsequent emissions of n, p, d, t, 3 He, and 4 He. The corresponding system of master equations describing the behavior of a nucleus at the pre-equilibrium stage is solved by the Monte-Carlo method [44]. In [58], the modified exciton model MEM was extended to include the possibility of emitting heavy clusters, with A > 4, up to 28 Mg (66 types of particles and LF). For incident energies below about 200 MeV, Kalbach has developed a phenomenological systematics for preequilibrium particle angular distributions by fitting available measured spectra of nucleons and complex particles [59]. As the Kalbach systematics are based on measured spectra, they describe very well the double-differential spectra of pre-equilibrium particles and generally provide a better agreement of calculated pre-equilibrium complex-particle spectra with experimental data.
The inverse cross sections used by these models at the pre-equilibrium stage (and at the evaporation/fission-stage) have a significant impact on the calculated particle width, and affect greatly the final results and the-accuracy of the MCNP6, MCNPX [60] and MARS15 [61][62][63]transport codes, which use these models as their event-generators. This is why it is necessary to use as good as possible approximations for the inverse cross sections in the extended models.
The unmodified codes use the inverse cross sections σinv from Dostrovsky's formulas [39,40] for all emitted nucleons and complex particles (d, t, 3 He, and 4 He) is not very suitable for emission of fragments heavier than 4 He. Better total-reaction-cross-section models that can be used as an estimate for inverse cross sections are available today, especially such as the NASA model [64], the approximations by Barashenkov and Polanski [65], and those by Kalbach [66]. A quite complete list of references on modern total-reaction-cross-section models, as well as on recent publications where these models are compared with each other and with available experimental data can be found in [67].
An extensive comparison of the systematics for total reaction (inverse) cross sections showed that the NASA approach is better, in general, than the other available models. This is why we implemented the NASA inverse cross sections into the MEM to be used at the pre-equilibrium stage of reactions.
The NASA approximation, as described by Eq. (36,) attempts to simulate several quantummechanical effects, such as the optical potential for neutrons (with the parameter Xm) and collective effects like Pauli blocking (through the quantity δT).
KÜÃÜ = ? 4 X ó @ , ⁄ + q @ , ⁄ + q \ 4 y1 − D ò á q OER } E ,(36)
where r0, AP , AT , δT , Rc, BT , Tcm, and Xm are, respectively, the constant used to calculate the radii of nuclei, the mass number of the projectile nucleus, the mass number of the target nucleus, the energy-dependent parameter, the system-dependent Coulomb multiplier, the energydependent Coulomb barrier, the colliding system center of-momentum energy, and the optical model multiplier used for neutron-induced reactions. The calculation of inverse cross sections at the pre-equilibrium stage of reactions was improved with a new hybrid NASA-Kalbach approach, instead of the old Dostrovsky model used previously. This extended version of the MEM is implemented into the upgraded CEM, labeled CEM03.03F, as well as into the new LAQGSM03.03F.
After the INC, LAQGSM uses the same pre-equilibrium, coalescence, Fermi break-up, and evaporation/fission models as described above for CEM.
The improved CEM, LAQGSM, was implemented as event generator into MCNP6 and allow one to describe particle-and nucleus-induced reactions and provide a good agreement with available experimental data. They have a good predictive power for various reactions and can be used as reliable tools in scientific and applied research.
Emission of energetic heavy clusters heavier than 4 He from nuclear reactions play a critical role in several applications, including electronics performance in space, human radiation dosages in space or other extreme radiation environments, proton and hadron therapy in medical physics, accelerator and shielding applications, and so on. The CEM and LAQGSM event generators in MCNP6 describe quite well the spectra of fragments with sizes up to 4 He in a broad range of target masses and incident energies (up to ∼ 5 GeV for CEM and up to ∼ 1 TeV/A for LAQGSM).
V. Numerical simulation of extended heavy targets irradiated by proton and deuteron beams.
Comparison of models and codes
Thorough understanding of the mechanisms and approaches used in different simulation models is important for obtaining reliable numerical data on irradiation of big heavy targets by accelerated proton and ion beams. Such experiments performed at the JINR accelerator facilities with Quinta and BURAN targets contribute to the research aimed at advanced schemes of nuclear power production with accelerated particle beams. Of course, experimental studies of such scale and complexity should be preceded by comprehensive numerical study.
Below we give the results of simulation of extended heavy targets irradiated by proton and deuteron beams. The focus of comparison of different applied models is neutron production and absorption in the target material depending on the target dimensions.
We consider targets from uranium-238 with the following dimensions: a radius of 15 cm, a length of 40 cm; a radius of 30 cm, a length of 80 cm; and a radius of 60 cm, a length of 160 cm.
The following accelerated beams are considered: protons with an energy of 0.66 GeV, 1 GeV, 2 GeV, 4 GeV, and deuterons with an energy of 0.66 GeV/nucleon, 1 GeV/nucleon, 2 GeV/nucleon, 4 GeV/nucleon.
Among the most important parameters characterizing beam interaction with fissionable materials is neutron production rate per beam particle. Neutron flux leaving the target is also important, as it provides information on the energy accumulated in the target due to ion-target interaction.
Tables 1-12 below summarize the results of calculations via several models for the beam and target parameters given above. Table 9. Neutron production/loss in interaction of d beam with U-238 target with a radius of 30 cm and a length of 80 cm obtained using the following models: a -SHIELD [68,69]; b -GEANT4 (BC-Bertini [28][29][30][31]); c -MCNP6 (CEM03 [43][44][45] and INCL [5][6][7]); d -MARS15 [61][62][63].
Ed, Neutron production/part. Neutron leakage/part.
It can be seen that the difference between the predictions by different codes and models is, on average, within 30%, although, in certain cases it may be almost twice as high. This demonstrates the difficulties encountered in simulation of beam-matter interaction, especially in bulk targets and proves that in order to obtain reliable numerical picture of interaction, codes should be verified to experimental data for particular experimental conditions.
An important and least studied part of the neutron spectrum from large extended targets is that of fast neutrons. Below we give the numerical results obtained with different models on production and escape of fast neutrons for the target parameters corresponding to those of BURAN target: a radius of 60 cm and a length of 160 cm. Table 13. Production and leakage of fast neutrons (En > 1 MeV) for U-238 target with a radius of 60 cm and a length of 160 cm irradiated by the proton beam obtained using the following models: a -SHIELD [68,69]; b -GEANT4 (BC-Bertini [28][29][30][31] and INCL [5][6][7]), c -MCNP6 (CEM03 [43][44][45] and INCL [5][6][7] It can be seen from Table 13 that, on average, the agreement of the considered models is not bad. However, the discrepancy increases with increasing energy of the incident beam. Already for a proton energy of 1 GeV the model predictions may differ by as much as 40%, this discrepancy increasing to 43% for a 2 GeV proton beam and 50% for a 4 GeV proton beam. It should be noted that considered beam energies are of interest from the point of view of development of new concepts of nuclear power production with the aid of accelerated ion beams, which explains utmost importance of both theoretical and experimental study of these processes. Of course, neutron production and escape depends strongly on the target parameters: material and dimensions. Below we illustrate the effect of the target dimensions on these processes.
VI. Effect of target dimensions on neutron production and capture in a heavy bulk target irradiated by accelerated proton and deuteron beams Irradiation of heavy extended targets by light ion beams is of substantial interest for development of the new concept of power production aided by an accelerator [70,71] and for research toward transmutation of radioactive waste. Note that the neutron spectrum, especially the hard part, is of extreme importance in ADS nuclear power production. Although the fraction of fast neutrons is rather small, they carry a substantial part of the energy. Therefore, it is important that the geometry of the target is such that the produced fast neutrons do not leave its volume, carrying away a noticeable fraction of energy. Below, we estimate the appropriate target length and radius.
The numerical experiment on irradiation of bulk heavy targets by proton and deuteron beams is illustrated in Figs. 1 and 2. Experimental studies in this field are carried out at JINR. The comparison of calculations and measured data will be the topic of another paper.
The integrated number of fissions and captures per projectile, for the targets with a radius of 60 cm and different lengths is shown in Fig. 1, for 0.66 GeV proton, and 0.66, 1, 2 and 4 AGeV deuteron beams. It can be seen from Fig. 1 that both the number of fissions and the number of captures per one beam particle reach a plateau rather promptly, already for a target length of about 40 cm, for all considered beam types and energies.
The integrated number of fissions and captures per projectile, for the target with a length of 160 cm and different radii is shown in Fig. 2, for 0.66 GeV proton, and 0.66, 1, 2 and 4 AGeV deuteron beams. It can be seen from Fig. 2 that, similar to the case of the target length variation, the number of fissions and neutron captures promptly reach a plateau with increasing target radius. The radius equal to 30 cm is roughly sufficient for it.
This testifies that a bulk target, being irradiated by proton and deuteron beams in a wide energy range, demonstrates a "saturation" mode when the number of fissions and captures per beam
CONCLUSIONS
The paper presented the comparative analysis of the basic cascade models of nuclear interactions applied in software for simulation of beam-target collisions: Liege intranuclear cascade, binary cascade, Bertini cascade, cascade exciton model and quark-gluon string model. The main physics approximations underlying these models were discussed.
The simulation of bulk heavy targets irradiated by accelerated proton and deuteron beams of energies from 0.66 GeV/nucleon to 4 GeV/nucleon was performed using five different software packages: SHIELD, GEANT4, MCNP6 and MARS15. The neutron production and escape from 238 U targets of three sizes: a radius of 15 cm and a length of 40 cm, a radius of 30 cm and a length of 80 cm, and a radius of 60 cm and a length of 160 cm were analyzed. The beam and target parameters were chosen close to those studied experimentally at the accelerator complex of the Joint Institute for Nuclear Research.
It was shown that, on the whole, the agreement between the models and codes is within 30% and better, worsening in some cases to 50% discrepancy. The highest discrepancy was observed for the high energy part of the neutron spectrum, which is of importance in design of novel accelerator-aided nuclear power production facilities and nuclear waste transmutation issues. This indicates the need in further theoretical and experimental studies of inelastic interactions with production of fast neutrons and interaction of these neutrons with bulk heavy materials. The comparison of calculations and experimental data obtained at JINR will be the subject of another paper.
conditions are naturally set to R(0) = 0 and R(pF ) = Rmax , and the integration of Eq.
with the correction ? = 1.16(1 − 1.16 3 V M ) fm. For light nuclei, the harmonic-oscillator shell model for the nuclear density[23] is used,
If 1 <
1A < 4, a nuclei model consisting of one layer with a radius of 8.0 fm is created.
For 4 <
4A < 11, a nuclei model is composed of three concentric spheres i 6.4B−log ( ë ), where ë = {0.01, 0.3, 0.7} and @ = 3.3836 @/, .
4 He) was increased to compensate it. The new values for pc for the extended coalescence model are:
Fig. 1 .
1Fissions and captures in the bulk 238 U target depending on the target length.
Fig. 2 .
2Fissions and captures in the long irradiated target depending on the target radius.
Table 1 .
1Neutron production/loss in interaction of p beam with U-238 target with a radius of 15 cm and a length of 40 cm obtained using the following models: a -SHIELD[68,69]; b -Table 2. Number of fissions and neutron captures per particle in interaction of p beam with U-238 target with a radius of 15 cm and a length of 40 cm obtained using the following models: a -SHIELD[68,69]; b -GEANT4 (BC-Bertini[28][29][30][31]), c -MCNP6 (CEM03[43][44][45] and INCL[5][6][7]); d -MARS15[61][62][63].Table 3. Neutron production/loss in interaction of p beam with U-238 target with a radius of 30 cm and a length of 80 cm obtained using the following models: a -SHIELD [68Table 4. Number of fissions and neutron captures per particle in interaction of p beam with U-238 target with a radius of 30 cm and a length of 80 cm obtained using the following models: a -SHIELD[68,69]; b -GEANT4 (BC-Bertini[28][29][30][31]), c -MCNP6 (CEM03[43][44][45] and INCL[5][6][7]); d -MARS15[61][62][63].GeVNumber of fissions/part.Number of captures/part.Table 5. Neutron production/loss in interaction of p beam with U-238 target with a radius of 60 cm and a length of 160 cm obtained using the following models: a -SHIELD [68Table 6. Number of fissions and neutron captures per particle in interaction of p beam with U-238 target with a radius of 60 cm and a length of 160 cm obtained using the following models: a -SHIELD[68,69]; b -GEANT4 (BC-Bertini[28][29][30][31] and INCL[5][6][7]), c -MCNP6 (CEM03[43][44][45] and INCL[5][6][7]); d -MARS15[61][62][63].Number of fissions/part. Number of captures/part.Table 7. Neutron production/loss in interaction of d beam with U-238 target with a radius of 15 cm and a length of 40 cm obtained using the following models: a -SHIELD [68GEANT4 (BC-Bertini [28-31]), c -MCNP6 (CEM03 [43-45] and INCL [5-7]); d -MARS15
[61-63].
Ep,
GeV
Neutron production/part.
Neutron leakage/part.
a
b
c
d
a
b
c
d
CEM
03
INCL
CEM0
3
INCL
4
272
306
345
270
267
168
172
203
159
164
2
148
164
170
151
161
92
93
102
90
86
1
72
78
80
72
78
45
45
49
44
42
0.66 42
42
46
40
44
27
25
28
25
24
Ep,
GeV
Number of fissions/part.
Number of captures/part.
a
b
c
d
a
b
c
d
CEM
03
INCL
CEM0
3
INCL
4
43.0
42.5
42
30
48
56
53.6
64
49
50,5
2
23.0
22.8
20
17
25
31
28.4
31
27
30
1
11.4
11.0
9
8
12
14.9
13.7
14
13
14.4
0.66 6.7
6.1
5
4
7
8.3
7.2
8
7
7.9
, 69]; b -
GEANT4 (BC-Bertini [28-31]), c -MCNP6 (CEM03 [43-45] and INCL [5-7]); d -MARS15
[61-63].
).Ep,
GeV
/part
Neutron production
(En>1MeV)/part
Neutron leakage(En>1MeV)/part
Neutron leakage(En>1MeV)/
Neutron production
(En>1MeV)( %)
a
b
a
b
c
a
b
BC
INCL
BC
INCL CEM03 INCL
BC
INCL
4
272.6 300.4 286.1 16.3 16.9
16.1 18
12
6
5.6
5.6
2
143.2 157.2 131.7 10.2 10.1
8.4
12
9
7.1
6.4
6.4
1
65.1 69.6 63.7 5.6
5.4
5.2
7
5
8.6
7.8
8.2
0.66 37.1 36.4 32.1 3.8
3.6
3.3
4
4
10.2
10
10.3
Table 11. Neutron production/loss in interaction of d beam with U-238 target with a radius of 60 cm and a length of 160 cm obtained using the following models: a -SHIELD[68,69]; b -GEANT4 (BC-Bertini[28][29][30][31]and INCL[5][6][7]); c -MCNP6 (CEM03[43][44][45]and INCL[5][6][7]); d -MARS15[61][62][63].
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| []
|
[
"Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system",
"Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system"
]
| [
"Eduard Feireisl [email protected] ",
"Mária Lukáčová -Medvid'ová ♠ ",
"Bangwei She [email protected] \nAcademy for Multidisciplinary studies\nCapital Normal University\nWest 3rd Ring North Road 105100048BeijingP. R. China\n",
"\nInstitute of Mathematics\nInstitute of Mathematics\nAcademy of Sciences of the Czech Republič\nZitná 25CZ-115 67Praha 1Czech Republic\n",
"\nJohannes Gutenberg-University Mainz\nStaudingerweg 955 128MainzGermany\n"
]
| [
"Academy for Multidisciplinary studies\nCapital Normal University\nWest 3rd Ring North Road 105100048BeijingP. R. China",
"Institute of Mathematics\nInstitute of Mathematics\nAcademy of Sciences of the Czech Republič\nZitná 25CZ-115 67Praha 1Czech Republic",
"Johannes Gutenberg-University Mainz\nStaudingerweg 955 128MainzGermany"
]
| []
| We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient. | 10.1007/s00211-023-01346-y | [
"https://arxiv.org/pdf/2205.04076v1.pdf"
]
| 248,571,819 | 2205.04076 | 684b947374f4fd48e389ff79d797816dacf211d5 |
Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system
9 May 2022
Eduard Feireisl [email protected]
Mária Lukáčová -Medvid'ová ♠
Bangwei She [email protected]
Academy for Multidisciplinary studies
Capital Normal University
West 3rd Ring North Road 105100048BeijingP. R. China
Institute of Mathematics
Institute of Mathematics
Academy of Sciences of the Czech Republič
Zitná 25CZ-115 67Praha 1Czech Republic
Johannes Gutenberg-University Mainz
Staudingerweg 955 128MainzGermany
Improved error estimates for the finite volume and the MAC schemes for the compressible Navier-Stokes system
9 May 2022compressible Navier-Stokes systemerror estimatesrelative energystrong solutionupwind finite volume methodMarker-and-Cell finite difference method
We present new error estimates for the finite volume and finite difference methods applied to the compressible Navier-Stokes equations. The main innovative ingredients of the improved error estimates are a refined consistency analysis combined with a continuous version of the relative energy inequality. Consequently, we obtain better convergence rates than those available in the literature so far. Moreover, the error estimates hold in the whole physically relevant range of the adiabatic coefficient.
Introduction
The Navier-Stokes equations governing the motion of viscous compressible fluids have numerous applications in engineering, physics, meteorology or biomedicine. In this paper we consider the viscous barotropic fluid endowed, for simplicity, with the isentropic pressure-density state equation p = a̺ γ , where a > 0 is a positive constant, and γ > 1 denotes the adiabatic coefficient. The global-in-time existence of weak solutions is known for any γ > d 2 in the d-dimensional setting, see Lions [20] and [7]. More recently, Plotnikov and Vaigant [26] extended the existence theory for any γ ≥ 1 if d = 2. Unfortunately, the multilevel approach used in the existence proof is rather difficult to adapt directly to a numerical scheme; whence the numerical analysis of the problem remains rather incomplete.
In the last few decades, many efficient and robust numerical methods have been proposed to simulate the motion of viscous compressible fluid flows. We refer the reader to the monographs by Dolejší and Feistauer [2], Eymard, Gallouët and Herbin [5], Feistauer [3], Feistauer, Felcman and Straškraba [4], Toro [27], and the references therein. Despite a good agreement of the obtained results with experiments, a rigorous convergence analysis with the associated error estimates have been performed only in a few particular cases.
In his truly pioneering work, Karper [18], see also [9], showed convergence (up to a subsequence) of a mixed finite element-finite volume (or discontinuous Galerkin) approximation to a weak solution of the compressible multidimensional Navier-Stokes system under the technical restriction γ > 3. His proofs basically follows step by step the existence theory developed in [7] and as such is difficult to adapt to other numerical methods. Moreover, as the weak solutions are not known to be unique, the result holds up to a subsequence and no convergence rate is available.
Recently, see [10,11,12], we have developed a new approach based on the concept of more general dissipative weak (dissipative measure-valued) solution, which, combined with the weak-strong uniqueness and conditional regularity results, yields a rigorous proof of convergence for the mixed finite elementfinite volume, finite volume and finite difference Marker-and-Cell (MAC) methods for any γ > 1 as long as the sequence of numerical solution remains uniformly bounded and/or if the strong solution exists. The aim of the present paper is to derive error estimates for the finite volume and the MAC methods for full range of the adiabatic coefficient γ > 1.
There are several results concerning error estimates for the compressible Navier-Stokes equations. Under the assumption of the L 2 -bounds of the discrete derivatives of the numerical solutions, Jovanović [17] studied the convergence rate of a finite volume-finite difference method to the barotropic Navier-Stokes system. In [21,22] Liu analyzed the errors for P k conforming finite element method, k ≥ 2, assuming the existence of a suitably regular smooth solution. However, the stability of the method with respect to the discrete energy was not investigated.
Furthermore, Gallouët et al. [14,15] analyzed the unconditional convergence rates of the mixed finite volume-finite element method [9] and the MAC scheme for γ > 3/2 in the dimension d = 3. Similar results have been obtained by Mizerová and She [23]. All the above mentioned convergence results are based on a discrete version of the relative energy inequality estimating the error between the numerical and the strong solution. The obtained convergence error is O(h A ), where h > 0 is a mesh parameter and A = min 2γ−3 γ , 1 2 , cf. [14,15,23]. In particular, the convergence order tends to zero when γ → 3 2 and remains positive only if γ > 3/2. Moreover, if γ ≥ 2, the convergence rate is only 1 2 in the energy norm, though the numerical experiments indicate the second order convergence rate.
In view of the existing results, the main novelty of the present paper is two-fold:
• Extending the error analysis to the full range γ > 1.
• Improving the convergence rate via a detailed consistency and error analysis.
Following the strategy proposed in the monograph [12,Chapter 9], we combine the standard consistency errors with the "continuous" form of the relative energy inequality. In contrast with the existing methods based on ad hoc construction of an approximate relative energy inequality, the new approach is rather versatile and free of additional discretization errors. In particular, we can handle any consistent energy stable numerical method in the same fashion. We focus on the finite volume method proposed in [12] and the MAC method from [23]. The application to the mixed finite element-finite volume method of Karper [18] was studied independently and presented in the recent work by Novotný and Kwon [19]. Compared to the previous results of Gallouët et al. [14,15], we employ the consistency formulation of the numerical solution where the test function is smooth. This new approach avoids the complicated integration by parts formulae on the discrete level and improves the convergence rates of the MAC method presented in [14,23].
The paper is organized in the following way. After presenting the continuous model and the corresponding relative energy, we formulate the numerical schemes: the finite volume and the MAC method, see Section 2. Next, we discuss their energy stability and consistency. The main results on the error estimates are formulated and proved in Section 3.
Compressible Navier-Stokes system
We begin with formulating the compressible Navier-Stokes system
∂ t ̺ + div x (̺u) = 0, ∂ t (̺u) + div x (̺u ⊗ u) + ∇ x p(̺) = div x S (1.1) in the time-space cylinder [0, T ] × Ω, Ω ⊂ R d , d = 2, 3,
where ̺ is the density, u is the velocity field, and S is the viscous stress tensor given by
S = µ(∇ x u + ∇ T x u − 2 d div x uI) + λdiv x uI, µ > 0, λ ≥ 0.
The pressure is assumed to satisfy the isentropic law
p = a̺ γ , a > 0, γ > 1. (1.2)
To avoid technical problems related to a proper numerical approximation of the physical boundary, we impose the periodic boundary conditions and identify the computational domain with the flat torus
Ω = T d ≡ [0, 1]| {0,1} d .
The system (1.1) is supplemented with finite energy initial data (̺ 0 , u 0 ) :
T d → R + × R d , ̺(0, x) = ̺ 0 > 0, (̺u)(0, x) = ̺ 0 u 0 , and E 0 = T d 1 2 ̺ 0 |u 0 | 2 + P (̺ 0 ) dx < ∞, (1.3)
where P is the so-called pressure potential, P (̺) = a̺ γ γ−1 for the isentropic gas law (1.2).
Relative energy
The main tool to evaluate the distance between numerical and strong solutions is the relative energy functional, cf. [8]:
E(̺, u|r, U ) = T d 1 2 ̺ |u − U | 2 + E(̺|r) dx, with E(̺|r) = P (̺) − P ′ (r)(̺ − r) − P (r).
As pointed out, relative energy functionals are often used to estimate the distance between a suitable weak solution and the strong solution; whence yielding the weak-strong uniqueness property. Recently, a discrete version of the relative energy has been applied in the error analysis of numerical schemes, see [14,15,23].
Classical solutions
It will be useful to identify the regularity class of smooth (classical) solutions to the Navier-
̺ 0 ∈ C 3 (T d ), ̺ 0 > 0 in T d , u 0 ∈ C 3 (T d ; R d ).
Let (̺, u) be a weak solution to problem (1.1) originating from the initial data (1.3) such that 0 ≤ ̺ ≤r and |u| ≤ū a.e. in (0,
T ) × T d . (1.4) Then (̺, u) is a classical solution of (1.1)-(1.3) in [0, T ] × T d .
If, in addition, ̺ 0 , u 0 belong to the class
̺ 0 ∈ W k,2 (T d ), u 0 ∈ W k,2 (T d ; R d ), k ≥ 6, (1.5) then ̺ ∈ C([0, T ]; W k,2 (T d )), u ∈ C([0, T ]; W k,2 (T d ; R d ))
, and the following estimate hold
∂ ℓ t ̺ C([0,T ]×T d ) + ̺ C 1 ([0,T ]×T d ) + 1/̺ C([0,T ]×T d ) + ̺ C([0,T ];W k,2 (T d )) ≤ D, ℓ = 1, 2, ∂ ℓ t u C([0,T ]×T d ;R d ) + u C 1 ([0,T ]×T d ;R d ) + u C([0,T ];W k,2 (T d ;R d )) ≤ D, ℓ = 1, 2, (1.6)
where D depends solely on T,r,ū and the initial data
(̺ 0 , u 0 ) via the norm (̺ 0 , u 0 ) W k,2 (T d ;R d+1 ) and min x∈T d ̺ 0 (x).
Proof. The first part was proved in [6, Proposition 2.2] via the local existence theory by Valli and Zajaczkowski [25] combined the weak-strong uniqueness principle and the conditional regularity result by Sun, Wang and Zhang [24]. In particular, the bounds (1.6) were established for k = 3, ℓ = 1.
Next, as shown in [1,Theorem 3.3], the solution inherit higher Sobolev regularity from the data as long as the norm u C([0,T ];W 2,∞ (T d ;R d )) is controlled. In particular, the estimates (1.6) can be established. Similarly to Gallagher [13], the proof in [1] is based on the particular isentropic form of the pressure that enables to transform the problem to a parabolic perturbation of a symmetric hyperbolic system.
Numerical methods
First, we introduce suitable notation. By c we denote a positive constant independent of the discretization parameters ∆t and h. We shall frequently write A B if A ≤ cB and A ≈ B if A B and B A. We also write c ∈ co{a, b} if min(a, b) ≤ c ≤ max(a, b). Moreover, we denote by · L p , · L p L q , and · L p W q,s the norms · L p (T d ) , · L p (0,T ;L q (T d )) , and · L p (0,T ;W q,s (T d )) , respectively.
Time discretization
We divide the time interval [0, T ] into N t equidistant parts with a fixed time increment ∆t (= T /N t ).
For a function f n given at the discrete time instances t n = n∆t, n = 0, 1, · · · , N t , we define a piecewise constant approximation f (t) in the following way f (t, ·) = f 0 for t < ∆t and f (t) = f n for t ∈ [n∆t, (n + 1)∆t), n ∈ {1, . . . , N t }.
The time derivative is approximated by the backward Euler method
D t f = f (t, ·) − f (t − ∆t, ·) ∆t for all t ∈ [0, T ].
Space discretization
To begin, we introduce a uniform structured mesh including primary, dual and bidual grids.
Primary grid
We call T the primary grid with the following properties and notations:
• The domain T d is divided into compact uniform quadrilaterals T d = K∈T K, where T is the set of all elements that forms the primary grid.
• E denotes the set of all faces of the primary grid T . Given an element K ∈ T , E(K) is the set of its faces; E i is the set of all faces that are orthogonal to the unit basis vector e i ; E i (K) = E(K) ∩ E i for any i ∈ {1, . . . , d}.
• h denotes the uniform size of the grid, meaning |x K − x L | = h for any neighbouring elements K and L, where x K and x L are the centers of K and L, respectively.
• σ K,i− and σ K,i+ denote the left and right face of an element K in the i th -direction, respectively.
• N (K) denotes the set of all neighbouring elements of K ∈ T .
• σ = K|L denotes the face σ that separates the elements K and L. Moreover, σ = − − → K|L means σ = K|L and x L − x K = he i for some i ∈ {1, . . . , d}.
• n denotes the outer normal of a generic face σ and n σ,K denotes the outer normal vector to a face σ ∈ E(K).
Dual grid
The dual of the primary grid is determined as follows.
• For any face σ = K|L ∈ E i , a dual cell is defined as Figure 1(a) for a two dimensional graphic illustration.
D σ = D σ,K ∪ D σ,L , where D σ,K = {x ∈ K, x i ∈ co{(x K ) i , (x σ ) i }}, see• D i = {D σ | σ ∈ E i }, i ∈ {1, .
. . , d}, represents the i th dual grid of T . Note that for each fixed i ∈ {1, . . . , d} it holds
T d = σ∈E i D σ , int(D σ ) ∩ int(D σ ′ ) = ∅ for σ, σ ′ ∈ E i , σ = σ ′ .
• E i is the set of all faces of the i th dual grid D i and E i,j = {ǫ ∈ E i |ǫ is orthogonal to e j }.
• A generic face of a dual cell D σ is denoted as ǫ ∈ E(D σ ), where E(D σ ) denotes the set of all faces of D σ .
• ǫ = D σ |D σ ′ denotes a dual face that separates the dual cells D σ and D σ ′ . Moreover,
ǫ = −−−−→ D σ |D σ ′ means ǫ = D σ |D σ ′ and x σ ′ − x σ = he i for some i ∈ {1, . . . , d}.
• N ⋆ (σ) denotes the set of all faces whose associated dual elements are the neighbours of D σ , i.e.,
N ⋆ (σ) = {σ ′ | D σ ′ is a neighbour of D σ }.
Bidual grid
• Similarly to the definition of the dual cell, a bidual cell D ǫ := D ǫ,σ ∩ D ǫ,σ ′ associated to ǫ = D σ |D σ ′ ∈ E i,j is defined as the union of adjacent halves of D σ and D σ ′ , where D ǫ,σ = {x ∈ D σ |x j ∈ co{(x σ ) j , (x ǫ ) j }} see Figure 1(b) for a two dimensional graphic illustration.
• B i,j denotes the j th dual grid of D i , that is set of all bidual cells associated to the bidual faces of E i,j . Note that B i,j = T in the case of i = j. Discrete function spaces. We introduce the following spaces of piecewise constant functions:
K L σ = K|L D σ,K D σ,L D σ = D σ,K ∪ D σ,L σ ′ = K|M M N (a) Dual grid in two dimensions K L D ǫ D σ ′ D σ σ ′ = − − → M|N σ = − − → K|L M N ǫ = D σ |D σ ′ (b) Bidual grid in two dimensionsQ h = {φ | φ h | K = constant for all K ∈ T } , Q h = Q d h , W h = (W 1,h , . . . W d,h ) , W i,h = {φ | φ h | Dσ = constant for all σ ∈ E i } , i ∈ {1, . . . , d}.
The corresponding projections read
Π Q :L 1 (T d ) → Q h , Π Q φ = K∈T (Π Q φ) K 1 K , (Π Q φ) K = 1 |K| K φ dx, Π (i) E :W 1,1 (T d ) → W i,h , Π (i) E φ = σ∈E (Π (i) E φ) σ 1 Dσ , (Π (i) E φ) σ = 1 |σ| σ φ dS(x),
where 1 K and 1 Dσ are the characteristic functions. Further, for any
φ = (φ 1 , . . . , φ d ) we denote Π E φ = Π (1) E φ 1 , . . . , Π (d) E φ d .
Moreover, for any bidual grid D ǫ we define
Π ǫ φ| Dǫ = 1 |ǫ| ǫ φ dS(x). (2.1)
Discrete operators
Average and jump. First, for an piecewise smooth function f h , we define its trace
f out h (x) = lim δ→0+ f h (x + δn) and f in h (x) = lim δ→0+ f h (x − δn).
Then for any r h ∈ Q h we define the average operator
{ {r h } } σ (x) = r in h (x) + r out h (x) 2
for any x ∈ σ ∈ E.
If in addition, σ ∈ E i for an i ∈ {1, . . . , d}, we write { {r h } } σ as { {r h } } (i) σ and denote { {r h } } (i) = σ∈E i 1 Dσ { {r h } } (i) σ ∀x ∈ σ ∈ E.
Analogously to the average operator, we define the jump operator for r h ∈ Q h as
r h σ (x) = r out h (x) − r in h (x). Further, for vector-valued functions v h = (v 1,h , . . . , v d,h ) ∈ Q d h and u h = (u 1,h , . . . , u d,h ) ∈ W h , we define { {v h } } = { {v 1,h } } (1) , . . . , { {v d,h } } (d) , u i,h | K = u i,h | σ K,i+ + u i,h | σ K,i− 2 , u i,h = K∈T 1 K u i,h | K , and u h = (u 1,h , . . . , u d,h ) .
Note that for any
u h ∈ W h we have u h = Π Q u h .
Gradient operator. For any r h ∈ Q h and u h ∈ W h we introduce the following gradient operators.
∇ D r h (x) = (ð D 1 r h , . . . , ð D d r h ) (x), ∇ B u h (x) = ∇ B u 1,h (x), . . . , ∇ B u d,h (x) with ∇ B u i,h (x) = ð B i1 u i,h (x), . . . , ð B id u i,h (x) , where ð D i r h (x) = σ∈E i 1 Dσ (ð D i r h ) σ , (ð D i r h ) σ = r L − r K h , σ = − − → K|L ∈ E i , ð B i,j u i,h (x) = ǫ∈ E i,j (ð B i,j u i,h ) Dǫ 1 Dǫ , (ð B i,j u i,h ) Dǫ = u σ ′ − u σ h , for ǫ = −−−−→ D σ |D σ ′ ∈ E i,j . Furthermore, for any v h ∈ Q h and φ ∈ W 1,2 (T d ) we set ∇ Q v h = K∈T 1 K ∇ Q v h | K with ∇ Q v h | K = σ∈E(K) |σ| |K| { {v h } } ⊗ n, ∇ Π E T φ = ∂ (1) T Π (1) E φ, · · · , ∂ (d) T Π (d) E φ . Here, ∂ (i) T is defined for any u i,h ∈ W i,h , i ∈ {1, . . . , d} as ∂ (i) T u i,h (x) = K∈T 1 K (∂ (i) T u i,h ) K , ∂ (i) T u i,h K = u i,h | σ K,i+ − u i,h | σ K,i− h , K ∈ T .
Note that for any r h ∈ Q h and u i,h ∈ W i,h , there hold
ð D i r h = ∂ (i) T { {r h } } (i) and ð B i,i u i,h = ∂ (i) T u i,h .
Divergence operator. For u h ∈ W h and v h ∈ Q h we define the following discrete divergence operators adjoint to the above discrete gradient operators
div W T u h (x) = d i=1 ∂ (i) T u i,h (x) and div Q T v h (x) = d i=1 ∂ (i) T { {v i,h } } (i) (x) = d i=1 ð D i v i,h (x).
It is easy to observe for any
v h ∈ Q h that div W T { {v h } } = div Q T v h . (2.2)
Upwind flux. Given a velocity field u h ∈ Q h ∩ W h , the upwind flux function for r h ∈ Q h is given by
Up[r h , u h ] σ = r in h (u σ ) + + r out h (u σ ) − , where r ± = 1 2 (r ± |r|), u σ = { {u h } } · n, if u h ∈ Q h , u h · n, if u h ∈ W h .
To approximate nonlinear convective terms we apply the following diffusive upwind flux
F ε h [r h , u h ] σ = Up[r h , u h ] σ − h ε r h σ , ε > −1. For φ h ∈ Q h we define a vector-valued upwind flux componentwise Up[φ h , u h ] = (Up[φ 1,h , u h ], · · · , Up[φ d,h , u h ]) , F ε h [φ h , u h ] = (F ε h [φ 1,h , u h ], · · · , F ε h [φ d,h , u h ]) .
Preliminary estimates and inequalities
In this section we present a preliminary material. First, it is easy to check that the following integration by parts formulae hold, see e.g. [16, Lemma 2.1].
Lemma 2.1. Let r h , φ h ∈ Q h , and u h , φ h ∈ W h . Then T d r h div W T u h dx = − T d u h · ∇ D r h dx, T d r h ∂ (i) T u i,h dx = − T d u i,h ð D i r h dx. (2.3a)
Next, we report the following useful lemmas whose proofs are presented in Appendix A.
Lemma 2.2. For any r h ∈ Q h , v h ∈ Q h , u h ∈ W h , ψ ∈ W 1,2 (T d ) and U ∈ W 1,2 (T d ; R d ), there hold T d r h div x U dx = T d r h div W T Π E U dx, (2.4) T d v h · ∇ x ψ dx = T d v h · ∇ Π E T ψ dx. (2.5) Lemma 2.3. For any u h ∈ W h , v h ∈ Q h and ψ ∈ W 1,2 (T d ) there hold T d u h · ∇ x ψ dx = − T d Π ǫ ψ div W T u h dx, (2.6) T d v h · ∇ x ψ dx = − d i=1 T d Π (i) E ψ ð D i v i,h dx. (2.7) Lemma 2.4. For any u h ∈ W h , v h ∈ Q h and U ∈ W 2,2 (T d ; R d ), we have T d Π Q u h · ∆ x U dx = − d i=1 d j=1 ǫ=Dσ|D σ ′ ∈ E j,i Dǫ ð B j,i u j,h (Π (i) E ∂ i U j ) Dσ + (Π (i) E ∂ i U j ) D σ ′ 2 dx, (2.8a) T d u h · ∇ x div x U dx = − T d div W T u h Π ǫ (div x U ) dx, (2.8b) T d v h · ∆ x U dx = − T d ∇ D v h : Π E ∇ x U dx, (2.8c) T d { {v h } } · ∇ x div x U dx = − T d Π ǫ div x U div Q T v h dx. (2.8d) Lemma 2.5. Let v h ∈ Q h , u h ∈ W h , U ∈ W 2,2 (T d ; R d ), and Φ ∈ W 3,2 (T d ; R d ). Then for any i, j ∈ {1, . . . , d}, we have Π Q u h − u h L 2 ≤ h 2 ∇ B u h L 2 , { {v h } } − v h L 2 ≤ h 2 ∇ D v h L 2 , (2.9a) Π (i) E ∂ i U j − ∂ i U j L 2 ≤ h U W 2,2 (2.9b) div x U − Π ǫ div x U L 2 ≤ h U W 2,2 , Π ǫ div x U − Π (i) E div x U L 2 ≤ h U W 2,2 . (2.9c) ∇ x div x Φ − ∇ Q div h Π Q Φ L 2 ≤ h Φ W 3,2 , ∆ x Φ − div W T ∇ D Π Q Φ L 2 ≤ h Φ W 3,2 . (2.9d)
Finite volume and finite difference methods
We proceed by presenting a finite volume and a finite difference numerical method that will be used to approximate the Navier-Stokes system (1.1)-(1.3). Both methods have been already successfully applied in numerical simulations, see, e.g., [12]. In our recent work [11,12,23], the convergence was shown for γ > 1 via the concept of dissipative measure-valued solutions. However, the error analysis was missing for the finite volume method and suboptimal for the finite difference method.
Finite volume method
We introduce the finite volume (FV) method approximating the Navier-Stokes system (1.1)-(1.3).
Definition 2.6 (FV scheme). Given the initial data (1.3), we set (̺ 0 h , ̺ 0 h u 0 h ) = (Π Q ̺ 0 , Π Q [̺ 0 u 0 ]). The FV approximation (̺ n h , u n h ) ∈ Q h × Q h , n = 1, . . . , N, of the Navier-Stokes system (1.1)-(1.3)
is a solution of the following system of algebraic equations:
T d D t ̺ n h φ h dx − E F ε h [̺ n h , u n h ] φ h dS(x) = 0 for all φ h ∈ Q h , (2.10a) T d D t (̺ n h u n h ) · φ h dx − E F ε h [̺ n h u n h , u n h ] · φ h dS(x) − T d p n h div h φ h dx = −µ T d ∇ D u n h : ∇ D φ h dx − ν T d div Q T u n h div Q T φ h dx for all φ h ∈ Q h , (2.10b) where ν = d−2 d µ + λ.
Finite difference MAC method
We proceed by presenting the finite difference MAC scheme that is based on a staggered grid approach. On the one hand, the discrete density ̺ h and pressure p h = p(̺ h ) are approximated on the primary grid T . On the other hand, the i th component of the velocity field u i,h is approximated on the i th dual grid D i . The MAC scheme reads as follows.
Definition 2.7 (MAC scheme). Given the initial data (1.3), we consider (̺ 0 h , ̺ 0 h Π Q u 0 h ) = (Π Q ̺ 0 , Π Q [̺ 0 u 0 ]). The MAC approximation of the Navier-Stokes system (1.1)-(1.3) is a sequence (̺ n h , u n h ) ∈ Q h × W h , n = 1, 2, .
. . , N, which solves the following system of algebraic equations:
T d D t ̺ n h φ h dx − E F ε h [̺ n h , u n h ] φ h dS(x) = 0 for all φ h ∈ Q h , (2.11a) T d D t (̺ n h Π Q u n h ) · φ h dx − E Up[̺ n h Π Q u n h , u n h ] · φ h dS(x) + µ T d ∇ B u n h : ∇ B φ h dx + ν T d div W T u n h div W T φ h dx − T d p n h div W T φ h dx = −h ε+1 d i=1 d j=1 T d { {u i,h n } } (j) (ð D j ̺ h )ð D j φ i,h dx, for all φ h = (φ 1,h , . . . , φ d,h ) ∈ W h , (2.11b) where ν = d−2 d µ + λ.
In what follows, we will denote by ̺ h (t), u h (t) the piecewise constant approximations of ̺ n h , u n h , n = 0, 1, . . . , N on the time interval [0, T ], see Section 2.1. We note that both methods, the FV method (2.10) as well as the MAC method (2.11), preserve the positivity of density and conserve the mass
̺ h (t) > 0 and T d ̺ h (t) dx = M for all t ∈ (0, T ),(2.12)
where M := T d ̺ 0 dx denotes the fluid mass, see e.g. [12, Lemma 11.2].
Energy stability
The essential feature of any numerical scheme is its stability. We now recall the energy stability of both numerical methods introduced above, see [12,
τ ∈ (0, T ), it holds T d 1 2 ̺ h |Π Q u h | 2 + P (̺ h ) (τ ) dx + µ τ 0 T d |∇ h u h | 2 dxdt + ν τ 0 T d |div h u h | 2 dxdt ≤ E 0 , (2.13) where E 0 = T d 1 2 ̺ 0 |u 0 | 2 + P (̺ 0 ) dx is the initial energy and (∇ h u h , div h u h ) = (∇ D u h , div Q T u h ) for u h ∈ Q h in the FV scheme; (∇ B u h , div W T u h ) for u h ∈ W h in the MAC scheme.
Moreover, there exists c > 0 which may depend on the fluid mass M and the initial energy E 0 but is independent of the parameters h and ∆t such that
̺ h |Π Q u h | 2 L ∞ L 1 ≤ c, ̺ h L ∞ L γ ≤ c, ̺ h Π Q u h L ∞ L 2γ γ+1 ≤ c, (2.14a) div h u h L 2 L 2 ≤ c, ∇ h u h L 2 L 2 ≤ c, u h L 2 L 6 ≤ c. (2.14b)
Consistency formulation
The next important ingredient of our approach is the consistency formulation of the numerical scheme.
Lemma 2.9 (Consistency formulation). Let (̺ h , u h ) be either a solution of the FV scheme (2.10) or the MAC scheme (2.11) with ∆t ≈ h ∈ (0, 1), γ > 1 and ε > −1.
Then for all τ ∈ (0, T ), φ ∈ L ∞ (0, T ; W 2,∞ (T d )), ∂ 2 t φ ∈ L ∞ ((0, T )×T d ) and φ ∈ L ∞ (0, T ; W 2,∞ (T d ; R d )), ∂ 2 t φ ∈ L ∞ ((0, T ) × T d ; R d ) there holds T d ̺ h φ dx τ t=0 = τ 0 T d (̺ h ∂ t φ + ̺ h Π Q u h · ∇ x φ) dxdt + e ̺ (τ, ∆t, h, φ), (2.15a) T d ̺ h Π Q u h · φ dx τ t=0 = τ 0 T d (̺ h Π Q u h · ∂ t φ + ̺ h Π Q u h ⊗ Π Q u h : ∇ x φ + p h div x φ) dxdt − µ τ 0 T d ∇ h u h : ∇ x φ dxdt − ν τ 0 T d div h u h div x φ dxdt + e m (τ, ∆t, h, φ), (2.15b)
where the consistency errors are bounded as follows:
|e ̺ (τ, ∆t, h, φ)| ≤ C ̺ ∆t + h + h 1+ε + h 1+β D for the FV method C ̺ ∆t + h 1+ε + h 1+β D for the MAC method |e m (τ, ∆t, h, φ)| ≤ C m √ ∆t + h + h 1+ε + h 1+β M for the FV method C m √ ∆t + h + h 1+ε + h 1+β M + h 1+ε+β D for the MAC method.
(2.15c)
Here, the constant C ̺ depends on the initial energy E 0 , T, and φ L ∞ (0,T ;
W 2,∞ (T d )) , ∂ 2 t φ L ∞ ((0,T )×T d ) ,
and C m depends on
E 0 , T, φ L ∞ (0,T ;W 2,∞ (T d ;R d ) , ∂ 2 t φ L ∞ ((0,T )×T d ;R d )
. Further, the exponents β D and β M are given by
β D = max − 3ε+3+d 6γ , γ−2 2γ d , if γ ∈ (1, 2), 0, if γ ≥ 2, β M = − 3ε+3+d 6γ , if γ ∈ (1, 2), γ−3 3γ d, if γ ∈ [2, 3), 0, if γ ≥ 3 for d = 3, 0, if γ > 2 for d = 2.
(2. provides an explicit bound in terms of the numerical step and regularity of the associated test function. Moreover, we improve the result of [12] by requiring less regularity of the test functions.
Proof of Lemma 2.9. The consistency errors arising from the time derivative term can be evaluated in the following way. First, by a direct calculation, we obtain
t n+1 0 T d D t r h (t)Π Q ϕ(t) dxdt = t n+1 0 T d r h (t) − r h (t − ∆t) ∆t ϕ(t) dxdt = 1 ∆t t n+1 0 T d r h (t)ϕ(t) dxdt − 1 ∆t t n −∆t T d r h (t)ϕ(t + ∆t) dxdt = − t n+1 0 T d r h (t)D t ϕ(t + ∆t) dxdt + 1 ∆t t n+1 t n T d r h (t)ϕ(t + ∆t) dxdt − 1 ∆t 0 −∆t T d r h (t)ϕ(t + ∆t) dxdt = − t n+1 0 T d r h (t)D t ϕ(t + ∆t) dxdt + 1 ∆t t n+1 t n T d r h (t)ϕ(t) dxdt − 1 ∆t ∆t 0 T d r 0 h ϕ(t) dxdt = − t n+1 0 T d r h (t)∂ t ϕ(t) dxdt + T d r h (τ ) =r n h ∀τ ∈[t n ,t n+1 ) ϕ(τ ) dx − T d r 0 h ϕ(0) dx + I 1 + I 2 + I 3 ,(2.16)
for any τ ∈ [t n , t n+1 ), n = 1, . . . , N T , where
I 1 = T d r 0 h 1 ∆t ∆t 0 (ϕ(0) − ϕ(t)) dt dx ∆t ∂ t ϕ L ∞ L ∞ r 0 h L 1 , I 2 = T d 1 ∆t r n h ϕ(t + ∆t) − ϕ(τ ) dt dx ∆t r n h L 1 ∂ t ϕ L ∞ L ∞ , I 3 = t n+1 0 T d r h (t) (∂ t ϕ(t) − D t ϕ(t + ∆t)) dxdt = T d n k=0 t k+1 t k r h (t) ∂ t ϕ(t) − D t ϕ(t + ∆t) dt dx ≤ ∆t ∂ 2 t ϕ L ∞ L ∞ r h L ∞ L 1 .
Collecting the above estimates we obtain from (2.16) that
T d r h ϕ dx τ 0 − t n+1 0 T d (D t r h (t)Π Q ϕ(t) + r h (t)∂ t ϕ(t)) dxdt ≤ ∆t ∂ 2 t ϕ L ∞ L ∞ r h L ∞ L 1 ,(2.T d ̺ h φ dx τ t=0 = t n+1 0 T d (̺ h ∂ t φ + ̺ h Π Q u h · ∇ x φ) dxdt + e ̺ (τ, ∆t, h, φ), (2.18a) T d ̺ h Π Q u h · φ dx τ t=0 = t n+1 0 T d (̺ h Π Q u h · ∂ t φ + ̺ h Π Q u h ⊗ Π Q u h : ∇ x φ) dxdt + t n+1 0 T d p h I − µ∇ h u h − νdiv h u h : ∇ x φ dxdt + e * m (τ, ∆t, h, φ), (2.18b)
where e * m is controlled by
|e * m (τ, ∆t, h, φ)| ≤ C m ∆t + h + h 1+ε + h 1+β M for the FV method , C m ∆t + h + h 1+ε + h 1+β M + h 1+ε+β D
for the MAC method.
In order to derive (2.15a) it suffices to realize that the time integral from τ to t n+1 at the right hand side of (2.18a) is of order O(∆t) . Indeed
t n+1 τ T d ̺ h ∂ t φ + ̺ h Π Q u h · ∇ x φ dxdt ≤ ∂ t φ L ∞ L ∞ ̺ n h L ∞ L 1 + ∇ x φ L ∞ L ∞ ̺ n h Π Q u n h L ∞ L 1 t n+1 τ 1dtt n+1 τ T d ̺ h Π Q u h · ∂ t φ + ̺ h Π Q u h ⊗ Π Q u h + p h I − µ∇ h u h − νdiv h u h : ∇ x φ dxdt ≤ t n+1 τ ̺ n h Π Q u n h L 1 (T d ) ∂ t φ L ∞ (T d ) dt + t n+1 τ ̺ n h Π Q u n h ⊗ Π Q u n h + p n h I L 1 (T d ) ∇ x φ L ∞ (T d ) dt + t n+1 τ µ∇ h u n h + νdiv h u n h L 1 (T d ) ∇ x φ L ∞ (T d ) dt ≤ ∆t ̺ n h Π Q u n h L ∞ L 1 ∂ t φ L ∞ L ∞ + ∆t ̺ n h Π Q u n h ⊗ Π Q u n h + p n h L ∞ L 1 ∇ x φ L ∞ L ∞ + ∇ x φ L ∞ L ∞ µ∇ h u n h + νdiv h u n h L 2 L 1 t n+1 τ 1 2 dt 1/2 √ ∆t.
Substituting the above estimate into (2.18b) we obtain (2.15b), which completes the proof. Then for all τ ∈ (0,
T ), φ ∈ L ∞ (0, T ; W 2,∞ (T d )), ∂ 2 t φ ∈ L ∞ ((0, T )×T d ) and φ ∈ L ∞ (0, T ; W 2,∞ (T d ; R d ))∩ L 2 (0, T ; W 3,2 (T d ; R d )), ∂ 2 t φ ∈ L ∞ ((0, T ) × T d ; R d ), there holds T d ̺ h φ dx τ t=0 = τ 0 T d (̺ h ∂ t φ + ̺ h Π Q u h · ∇ x φ) dxdt + e ̺ (τ, ∆t, h, φ), (2.21a) T d ̺ h Π Q u h · φ dx τ t=0 = τ 0 T d (̺ h Π Q u h · ∂ t φ + ̺ h Π Q u h ⊗ Π Q u h : ∇ x φ + p h div x φ) dxdt + τ 0 T d u h · (µ∆ x φ + ν∇ x div x φ) dxdt + e m (τ, ∆t, h, φ), (2.21b)
where the consistency errors can be bounded as follows
|e ̺ (τ, ∆t, h, φ)| ≤ C ̺ (∆t + h), |e m (τ, ∆t, h, φ)| ≤ C m (∆t + h) (2.21c)
Here, the constant C ̺ depends on
̺, u, E 0 , T, φ L ∞ (0,T ;W 2,∞ (T d )) , ∂ 2 t φ L ∞ ((0,T )×T d ) , and C m depends on ̺, u, E 0 , T, φ L ∞ (0,T ;W 2,∞ (T d ;R d )) , φ L 2 (0,T ;W 3,2 (T d ;R d )) , ∂ 2 t φ L ∞ ((0,T )×T d ;R d ) .
Proof. We will present the proof for the FV method, the proof for the MAC method is analogous. First, we denote the errors of the inviscid fluxes as
e 1 = t n+1 0 T d ̺ h Π Q u h · ∇ x φ dxdt − t n+1 0 E F ε h [̺ h , u h ] Π Q φ dS(x)dt, (2.22) e 2 = t n+1 0 T d ̺ h Π Q u h ⊗ Π Q u h : ∇ x φ dxdt − t n+1 0 E F ε h [̺ h u h , u h ] · Π Q φ dS(x)dt + t n+1 0 T d p h div x φ − p h div h Π Q φ dxdt.
(2.23)
Analogously as in the proof of [12, Theorem 11.3] we get
|e 1 | ≤ c( φ L ∞ W 2,∞ )h ̺ h L 2 L 2 and |e 2 | ≤ c( φ L ∞ W 2,∞ )h ̺ h u h L 2 L 2 .
In view of assumption (2.20) the errors e 1 and e 2 are controlled by
|e 1 | ≤ c( φ L ∞ W 2,∞ )h ̺ h L 2 L 2 ≤ c( φ L ∞ W 2,∞ , ̺)h, |e 2 | ≤ c( φ L ∞ W 2,∞ )h ̺ h u h L 2 L 2 ≤ c( φ L ∞ W 2,∞ , ̺, u)h.(2.= ̺ h u h we get T d ̺ h u h · φ dx τ 0 = τ 0 T d ̺ h u h · ∂ t φ + ̺ h u h ⊗ u h + p h I : ∇ x φ + u h · (µ∆ x φ + ν∇ x div x φ) dxdt + e 2 + e 3 + e 4 ,(2.
25) where e 2 is given in (2.23). The error terms e 3 and e 4 can be estimated in the following way
|e 3 | = − t n+1 0 T d u h · (µ∆ x φ + ν∇ x div x φ) + µ∇ D u h : ∇ D Π Q φ + νdiv h u h div h Π Q φ dx = t n+1 0 T d µu h · (div W T ∇ D Π Q φ − ∆ x φ) + νu h · (∇ Q div h Π Q φ − ∇ x div x φ) dx ≤ c( φ L 2 W 3,2 , u)h, |e 4 | = t n+1 τ T d ̺ h Π Q u h · ∂ t φ + ̺ h Π Q u h ⊗ Π Q u h + p h I : ∇ x φ + u h · (µ∆ x φ + ν∇ x div x φ) dxdt ≤ ∆t φ C 1 ̺ h Π Q u h L ∞ L 1 + ̺ h |Π Q u h | 2 L ∞ L 1 + p h L ∞ L 1 + u φ L ∞ W 2,∞ t n+1 τ dt ≤ c( φ L ∞ W 2,∞ , φ C 1 , ̺, u)∆t.
Consequently, collecting the estimates of e 2 , e 3 and e 4 we observe that (2.21b) follows from (2.25), which completes the proof.
Error estimates
This section is the heart of the paper. We prove the main result -the convergence rates for the FV (2.10) and MAC (2.11) schemes. If, in addition, the numerical solutions are uniformly bounded, the convergence rates can be improved to the first order.
Theorem 3.1 (Convergence rates). Let γ > 1 and the initial data (̺ 0 , u 0 ) satisfy
̺ 0 ∈ W k,2 (T d ), ̺ 0 > 0 in T d , u 0 ∈ W k,2 (T d ; R d ), k ≥ 6.
Suppose that the Navier-Stokes system (1.1) admits a classical solution (̺, u) defined on [0, T ] × T d , with the initial data (̺ 0 , u 0 ). Further, let (̺ h , u h ) be a numerical solution obtained either by the FV scheme (2.10) or by the MAC scheme (2.11) emanating from the projected initial data (̺ 0 h , u 0 h ). Then there exists a positive number
c = c(T, (̺ 0 , u 0 ) W k,2 (T d ;R d+1 ) , inf ̺ 0 , (̺, u) C([0,T ]×T d ;R d+1 ) ) such that sup 0≤t≤T E(̺ h , u h |̺, u) + µ T 0 T d |∇ h u h − ∇ x u| 2 dxdt + ν T 0 T d |div h u h − div x u| 2 dxdt ≤ c(h A + √ ∆t), (3.1) ̺ h − ̺ L ∞ L γ + ̺ h u h − ̺u L ∞ L 2γ γ+1 c( √ ∆t + h) 1/2 + c( √ ∆t + h A ) 1/γ for γ ≤ 2, ̺ h − ̺ L ∞ L 2 + ̺ h u h − ̺u L ∞ L 2γ γ+1 c( √ ∆t + h A ) 1/2 for γ > 2,(3.
2)
and
u h − u L 2 L 2 c( √ ∆t + h A ) 1/2 . (3.3)
The convergence rate A reads
A = A F V := min {1, 1 + ε, 1 + β D , 1 + β M } for the FV method, A M AC := min {1, 1 + ε, 1 + β D , 1 + β M , 1 + ε + β D } for the MAC method. (3.4)
Here the constants β D and β M are given in (2.15d). • For the case d = 2, we obtain the following convergence rate A:
Remark 2. Let us discuss the obtained convergence rate O(h
-Let γ ≥ 2. Then for any ε ≥ 0 both numerical methods have the first order convergence rate, i.e. A = 1.
-Let γ ∈ (1, 2). The convergence rates are different for the FV and MAC schemes. * A F V = min 1 − 5+3ε 6γ , 1, 1 + ε . Choosing the optimal value of ε, ε = − 5 3+6γ ∈ (− 5 9 , − 1 3 ), the convergence rate A F V = 1 + ε varies between 4 9 for γ ց 1 and 2 3 for γ ր 2. * A M AC = min 1 − 5+3ε 6γ , 1, 1 + ε, 1 + ε − 5+3ε 6γ reaches its maximum value 6γ−5 6γ > 0 at ε = 0. Thus, the convergence rate varies between 1 6 for γ ց 1 and 7 12 for γ ր 2.
• For the case d = 3, we obtain the following convergence rate A:
-Let γ ≥ 3. Then for any ε ≥ 0 both methods have first order convergence rates, i.e. A = 1.
-Let γ ∈ [2, 3). Then for any ε ≥ 2γ−3 γ we have A = 2γ−3 γ and the convergence rate varies between 1 2 for γ = 2 and 1 for γ ր 3. -Let γ ∈ (1, 2). * A F V = min 1 − 2+ε 2γ , 1, 1 + ε . Choosing an optimal value of ε, ε = − 2 1+2γ ∈ (− 2 3 , − 2 5 ), A F V = 1 + ε and varies between 1 3 for γ ց 1 and 3 5 for γ ր 2.
* A M AC = min 1 − 2+ε 2γ , 1, 1 + ε, 1 + ε − 2+ε 2γ
reaches its maximum value γ−1 γ > 0 at ε = 0. Note that A M AC varies between 0 when γ ց 1 and 1 2 when γ ր 2. Remark 3. In view of the above results, the convergence rates available in the literature, see e.g. [14,15,23], are not optimal. Indeed, for d = 3 and γ = 3 2 , they degenerate to 0. Moreover, no error analysis is available for γ < 3 2 . Our approach yields error estimates also for γ ∈ (1, 3 2 ]. In addition, we have better convergence rates, e.g., for d = 3 and γ = Proof of Theorem 3.1. First, by a straightforward but lengthy calculation, see Appendix D, we observe the following relative energy inequality
[E(̺ h , u h |̺, u)] τ 0 + τ 0 T d µ |∇ h u h | 2 + ν |div h u h | 2 dxdt ≤ τ 0 T d ̺ h ∂ t |u| 2 2 + ̺ h Π Q u h · ∇ x |u| 2 2 dxdt + e ̺ τ, ∆t, h, |u| 2 /2 − τ 0 T d (̺ h ∂ t P ′ (̺) + ̺ h Π Q u h · ∇ x P ′ (̺)) dx − e ̺ (τ, ∆t, h, P ′ (̺)) − τ 0 T d (̺ h Π Q u h · ∂ t u + ̺ h Π Q u h ⊗ Π Q u h : ∇ x u + p h div x u) dxdt + τ 0 T d (µ∇ h u h : ∇ x u + νdiv h u h div x u) dxdt + e m (τ, ∆t, h, −u) + τ 0 T d ∂ t ̺P ′ (̺) − P (̺) dxdt. (3.5)
Next, we observe the following identities
̺ h Π Q u h · ∇ x |u| 2 2 − ̺ h Π Q u h ⊗ Π Q u h : ∇ x u = −̺ h (Π Q u h − u) ⊗ (Π Q u h − u) : ∇ x u − ̺ h (Π Q u h − u) · (u · ∇ x u), P ′′ (̺) = 1 ̺ p ′ (̺), ̺P ′ (̺) − P (̺) = p(̺), ∂ t (̺P ′ (̺) − P (̺)) = ∂ t p(̺).
Then by substituting the above equalities into (3.5) and denoting
e S = e ̺ τ, ∆t, h, |u| 2 /2 − e ̺ (τ, ∆t, h, P ′ (̺)) + e m (τ, ∆t, h, −u), we obtain [E(̺ h , u h |̺, u)] τ 0 + τ 0 T d µ |∇ h u h − ∇ x u| 2 + ν |div h u h − div x u| 2 dxdt ≤ e S + τ 0 T d ̺ h (u − Π Q u h ) · (∂ t u + u · ∇ x u) dxdt − τ 0 T d ̺ h (Π Q u h − u) ⊗ (Π Q u h − u) : ∇ x u dxdt + µ τ 0 T d |∇ x u| 2 − ∇ h u h : ∇ x u dxdt + ν τ 0 T d |div x u| 2 − div h u h div x u dxdt + τ 0 T d ∂ t p(̺) − ̺ h ∂ t p(̺) ̺ − ̺ h Π Q u h · ∇ x p(̺) ̺ − p h div x U dxdt. (3.6)
As (̺, u) satisfies the Navier-Stokes system (1.1), we know that
̺(∂ t u + u · ∇ x u) = µ∆ x u + ν∇ x div x u − ∇ x p(̺).
Substituting this equality into (3.6) we get
[E(̺ h , u h |̺, u)] τ 0 + τ 0 T d µ |∇ h u h − ∇ x u| 2 + ν |div h u h − div x u| 2 dxdt ≤ e S + τ 0 T d (̺ h − ̺)(u − Π Q u h ) · (∂ t u + u · ∇ x u) dxdt, − τ 0 T d ̺ h (Π Q u h − u) ⊗ (Π Q u h − u) : ∇ x u dxdt, + µ τ 0 T d |∇ x u| 2 − ∇ h u h : ∇ x u + (u − Π Q u h ) · ∆ x u dxdt + ν τ 0 T d |div x u| 2 − div h u h div x u + (u − Π Q u h ) · ∇ x div x u dxdt + τ 0 T d ̺ − ̺ h ̺ ∂ t p(̺) − ̺ h ̺ Π Q u h · ∇ x p(̺) − p h div x u dxdt − τ 0 T d (u − Π Q u h ) · ∇ x p(̺) dxdt.
Rearranging the terms on the right hand side, we arrive at
[E(̺ h , u h |̺, u)] τ 0 + τ 0 T d µ |∇ h u h − ∇ x u| 2 + ν |div h u h − div x u| 2 dxdt ≤ e S + 5 i=1 R E i ,
where the integrals R E i , i = 1, · · · , 5, read
R E 1 = τ 0 T d (̺ h − ̺)(u − Π Q u h ) · (∂ t u + u · ∇ x u + ∇ x p(̺) ̺ ) dxdt = τ 0 T d (̺ h − ̺)(u − Π Q u h ) · div x S(∇ x u) ̺ dxdt R E 2 = − τ 0 T d ̺ h (Π Q u h − u) ⊗ (Π Q u h − u) : ∇ x u dxdt, R E 3 = −µ τ 0 T d (∇ h u h : ∇ x u + Π Q u h · ∆ x u) dxdt, R E 4 = −ν τ 0 T d (div h u h div x u + Π Q u h · ∇ x div x u) dxdt, R E 5 = − τ 0 T d p h − p ′ (̺)(̺ h − ̺) − p(̺) div x u dxdt.
Next, for i = 1, · · · , 5 we analyze R E i such that it can be controlled either by the relative energy or the mesh parameter h.
Term R E 1 . Applying Hölder's inequality and Lemma B.4 we obtain
R E 1 ≤ 1 r div x S(∇ x u) L ∞ ((0,T )×T d ) C 0 τ 0 E(̺ h , u h |̺, u)dt + C 1 δ ∇ h u h − ∇ x u 2 L 2 + C 2 δh 2 = C * 0 τ 0 E(̺ h , u h |̺, u)dt + C * 1 δ ∇ h u h − ∇ x u 2 L 2 + C * 2 δh 2 , where C * 0 > 0 depends on u L ∞ W 2,∞ , ̺ C([0,T ]×T d ) , M, E 0 , γ, δ, and r = min [0,T ]×T d ̺; C * 1 > 0 depends on u L ∞ W 2,∞ , M, E 0 , and γ; C * 2 > 0 depends on u L ∞ W 2,∞ , M, E 0 , γ, and ∇ x u L ∞ ((0,T )×T d ) .
Term R E 2 . Thanks to Hölder's inequality we observe the following estimate.
R E 2 ≤ C τ 0 E(̺ h , u h |̺, u)dt, where C depends on ∇ x u L ∞ ((0,T )×T d ) .
Term R E 3 . We analyze the third term R E 3 in two cases. First, we consider the case of the FV scheme. In this case u h ∈ Q h , ∇ h u h = ∇ D u h and Π Q u h = u h . Thus,
R E 3 = µ τ 0 T d (∇ h u h : ∇ x u + Π Q u h · ∆ x u) dxdt = µ τ 0 T d ∇ D u h : ∇ x u + u h · div W T Π E ∇ x u dxdt = µ τ 0 T d ∇ D u h : (∇ x u − Π E ∇ x u) dxdt ≤ µh ∇ h u h L 2 L 2 u L 2 W 2,2 ,
where we have used the equality (2.4), the integration by parts formula (2.3a), and the estimate (2.9b). Second, we consider the case of u h ∈ W h obtained by the MAC scheme. In this case, ∇ h u h = ∇ B u h and the term R E 3 can be estimated in the following way
R E 3 = µ τ 0 T d (∇ h u h : ∇ x u + Π Q u h · ∆ x u) dxdt = µ τ 0 d i=1 d j=1 ǫ=Dσ|D σ ′ ∈ E j,i Dǫ ð B j,i u j,h ∂ i U j − (Π (i) E ∂ i u j ) Dσ + (Π (i) E ∂ i u j ) D σ ′ 2 dxdt ≤ µh ∇ h u h L 2 L 2 u L 2 W 2,2 ,
where we have applied (2.8a), Hölder's inequality and the estimate (2.9b). Consequently, we have for both cases R E 3 ≤ Ch, where the constant C depends on µ, the initial energy E 0 and U L 2 W 2,2 .
Term R E 4 . We analyze the term R E 4 also in two cases. First, for u h ∈ Q h obtained by the FV method, we have div h u h = div Q T u h , Π Q u h = u h and thus
R E 4 = ν τ 0 T d (div h u h div x u + Π Q u h · ∇ x div x u) dxdt = ν τ 0 T d div Q T u h div x u + u h · ∇ x div x u − Π ǫ div x u div Q T v h + { {v h } } · ∇ x div x u dxdt = ν τ 0 T d div Q T u h (div x u − Π ǫ div x u) + (u h − { {u h } }) · ∇ x div x u dxdt ≤ h ( div h u h L 2 L 2 + ∇ h u h L 2 L 2 ) u L 2 W 2,2 ,
where we have used the identity (2.8d), Hölder's inequality, the estimates (2.9a) and (2.9c). Second, for the case of u h ∈ W h we have
R E 4 = ν τ 0 T d (div h u h div x u + Π Q u h · ∇ x div x u) dxdt = ν τ 0 T d div W T u h div x u − div W T u h Π ǫ div x u + u h · ∇ x div x u + Π Q u h · ∇ x div x u dxdt = ν τ 0 T d div W T u h (div x u − Π ǫ div x u) + (Π Q u h − u h ) · ∇ x div x u dxdt ≤ νh ( div h u h L 2 L 2 + ∇ h u h L 2 L 2 ) u L 2 W 2,2 ,
where (2.8b), Hölder's inequality, the estimates (2.9a) and (2.9c) were applied. Consequently, we have for both cases R E 4 ≤ Ch, where the constant C depends on ν, initial energy E 0 , and u L 2 W 2,2 .
Term R E 5 . The estimate of R E 5 is straightforward by applying Hölder's inequality, i.e.,
R E 5 ≤ C τ 0 E(̺ h , u h |̺, u)dt,
where C depends on div x u L ∞ ((0,T )×T d ) .
Consequently, collecting the above estimates of R E i for i = 1, · · · , 5, we find
E(̺ h , u h |̺, u)(τ ) + τ 0 T d (µ − C * 1 δ) |∇ h u h − ∇ x u| 2 + ν |div h u h − div x u| 2 dxdt ≤ e S + E(̺ h , u h |r, u)(0) + C * 0 τ 0 E(̺ h , u h |r, u)dt + C * 2 δh 2 .
(3.7)
Applying the standard projection error estimates we get
E(̺ h , u h |̺, u)(0) ≤ Ch 2 ,(3.8)
where C depends on ̺ 0 C and u 0 L 2 W 1,2 . Consequently, by choosing δ < µ C * 1 , substituting (3.8) into (3.7), using Gronwall's lemma and recalling the consistency error (2.15c), we may infer that
E(̺ h , u h |̺, u)(τ ) + τ 0 T d |∇ h u h − ∇ x u| 2 + |div h u h − div x u| 2 dxdt ≤ Ce τ C * 0 1−∆tC * 0 ( √ ∆t + h A ) for ∆t < 1 C * 0 .
Here, the constant C depends on ̺ L ∞ W 2,∞ , u L ∞ W 2,∞ and the exponent A is given by (3.4).
Finally, we combine the above estimate with Lemma C.1 and Lemma B.2 in order to obtain (3.2) and (3.3), respectively. Note that E 0 and M are bounded by the norm (̺ 0 , u 0 ) W k,2 (T d ;R d+1 ) . Due to Proposition 1.1 all terms depending on the norms of the exact solution (̺, u) as well as r are bounded by a constant c = c(T,
(̺ 0 , u 0 ) W k,2 (T d ;R d+1 ) , (̺, u) C([0,T ]×T d ;R d+1 ) ) which finishes the proof.
Finally, we observe that under the assumption that the numerical solutions (̺ h , u h ) are uniformly bounded, the above error estimates can be improved. Indeed, applying Lemma 2.10, Lemma C.1 and Lemma B.2 we derive the first order error rate.
Theorem 3.2 (Error rates for bounded numerical solutions). In addition to the hypotheses of Theorem 3.1, let the numerical solution (̺ h , u h ) be uniformly bounded,
̺ h L ∞ ((0,T )×T d ) ≤ ̺ and u h L ∞ ((0,T )×T d ;R d ) ≤ u. (3.9)
Then there exists a positive number
c = c T, (̺ 0 , u 0 ) W k,2 (T d ;R d+1 ) , inf ̺ 0 , ̺, u, such that sup 0≤t≤τ E(̺ h , u h |̺, u) + µ τ 0 T d |∇ h u h − ∇ x u| 2 dxdt + ν τ 0 T d |div h u h − div x u| 2 dxdt ≤ c(h + ∆t)
for all τ ∈ [0, T ], and
̺ h − ̺ L ∞ L 2 + ̺ h u h − ̺u L ∞ L 2 + u h − u L 2 L 2 c(∆t 1 2 + h 1 2 ).
Conclusion
In this paper we have presented improved error estimates for two well-known numerical methods applied to compressible Navier-Stokes equations. Specifically, we consider the upwind finite volume method and the Marker-and-Cell (MAC) method with implicit time discretization and piecewise constant approximation in space. However, the approach presented in the paper can be applied also to other well-known numerical methods for compressible Navier-Stokes equations. The novelty of our approach lies in the use of continuous form of the relative energy inequality combined with a refined consistency analysis. Thus, following the framework of the Lax equivalence theorem it suffices to show the (energy) stability, cf. Lemma 2.8, and the consistency of a numerical scheme, cf. Lemma 2.9, in order to obtain the convergence rates for the scheme. Indeed, the consistency errors directly yield global errors in the relative energy. To obtain the corresponding error estimates we only assume that the initial data are sufficiently regular and a strong solution exists globaly in time. The error estimates presented in Theorem 3.1 improves the results already presented in the literature [15,14,23], see Remarks 2,3 for a detailed discussion. In particular, our error estimates hold for the full range of the adiabatic coefficient γ > 1.
Moreover, we have considered a natural hypothesis on uniformly bounded numerical solutions and proved that the error estimates can be further improved, cf. Theorem 3.2. Indeed, we prove that both numerical methods converge with the first order in time and mesh parameter in terms of the relative energy and with the half order in the L ∞ (0, T ; L 2 (T d ))-norm for the density and momentum, as well as in the L 2 ((0, T ) × T d )-norm for the velocity.
Proof of Lemma 2.4. First, we recall (2.4) and (2.3a) to derive the first equality
T d Π Q u h · ∆ x U dx = T d Π Q u h · (div x ∇ x U ) dx = T d Π Q u h · (div W T Π E ∇ x U ) dx = − T d ∇ D Π Q u h : Π E ∇ x U dx = − d i=1 d j=1 σ∈E i Dσ ð D i u j,h Π (i) E ∂ i U j dx = − d i=1 d j=1 σ∈E i Dσ 1 2 ǫ∈ E j,i (Dσ) (ð B j,i u j,h ) Dǫ Π (i) E ∂ i U j dx = − d i=1 d j=1 ǫ=Dσ|D σ ′ ∈ E j,i Dǫ ð B j,i u j,h (Π (i) E ∂ i U j ) Dσ + (Π (i) E ∂ i U j ) D σ ′ 2 dx
Next, it is easy to check (2.8b) by setting ψ = div x U in (2.6), i.e.,
T d u h · ∇ x div x U dx = − T d div W T u h Π ǫ (div x U ) dx.
Further, thanks to (2.4) and (2.3a), we observe (2.8c), i.e.,
T d v h · ∆ x U dx = T d v h · (div x ∇ x U ) dx = T d v h · (div W T Π E ∇ x U ) dx = − T d ∇ D v h : Π E ∇ x U dx. Finally, by setting ({ {v h } } , div x U ) as (u h , ψ) into (2.6) we get (2.8d), i.e. T d { {v h } } · ∇ x div x U dx = − T d Π ǫ div x U div W T ({ {v h } }) dx = − T d Π ǫ div x U div Q T v h dx,
where we have used the identity (2.2).
Proof of Lemma 2.5. Note that the estimates stated in (2.9b) -(2.9d) hold due to the standard interpolation error; whence we omit the proof. Now we prove (2.9a). First, by a direct calculation, we have
Π Q u h − u h 2 L 2 = K∈T d i=1 σ∈E i (K) |D σ,K | u i,σ K,i+ + u i,σ K,i− 2 − u i,σ 2 = 1 4 K∈T d i=1 u i,σ K,i+ − u i,σ K,i− 2 2 σ∈E i (K) |D σ,K | = h 2 4 K∈T |K| d i=1 ∂ (i) T u i,h 2 ≤ h 2 4 ∇ B u h 2 L 2 ,
where we have used the fact that
ð B i,i = ∂ (i)
T in the last inequality, which proves the first estimate of (2.9a). Analogously, we compute
{ {v h } } − v h 2 L 2 = K∈T d i=1 σ∈E i (K) |D σ,K | v in i,h + v out i,h 2 − v in i,h 2 = h 2 4 K∈T d i=1 σ∈E i (K) |D σ,K |(ð D i v i,h ) 2 = h 2 4 d i=1 σ∈E i |D σ |(ð D i v i,h ) 2 ≤ h 2 4 ∇ D v h 2 L 2 ,
which proves the second estimate of (2.9a). This concludes the proof of Lemma 2.5.
B Sobolev-Poincaré type inequality
First, we recall [12,Theorem 17] for a generalized Sobolev-Poincaré inequality.
Lemma B.1 ( [12]). For a structure mesh let γ > 1 and ̺ h ≥ 0 satisfy
0 < c M ≤ T d ̺ h dx and T d ̺ γ h dx ≤ c E ,
where γ > 1, c M and c E are positive constants. Then there exists c = c(c M , c E , γ) independent of h such that
f h 2 L q (T d ) ≤ c ∇ h f h 2 L 2 (T d ) + T d ̺ h |f h | 2 dx .1 = C 1 (M, E 0 , γ) > 0 and C 2 = C 2 (M, E 0 , γ, ∇ x U L ∞ , U W 2,∞ ) > 0 such that u h − U 2 L 2 ≤ C 1 ∇ h u h − ∇ x U ) 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 2 h 2 , (B.1) Π Q u h − U 2 L 2 ≤ C 1 ∇ h u h − ∇ x U h 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 2 h 2 , (B.2)
where M and E 0 are the fluid mass and initial energy.
Proof. Firstly, by setting f h = u h − U h for some U h belonging to the same discrete space as u h in Lemma B.1 we know that
u h − U h 2 L 2 (T d ) ≤ C 1 ∇ h (u h − U h ) 2 L 2 (T d ) + T d ̺ h |u h − U h | 2 dx ,
where the constant C 1 depends on c M ≡ M, c E ≡ E 0 and γ. Note that the choices of c M and c E are owing to the mass conservation (2.12) and energy stability (2.13). Next, for u h ∈ Q h and u h ∈ W h we set U h = Π Q U ∈ Q h and U h = Π E U ∈ W h , respectively. Then by the triangular inequality and projection error we derive
u h − U 2 L 2 ≤ u h − U h 2 L 2 + U h − U 2 L 2 ≤ C 1 ∇ h (u h − U h ) 2 L 2 (T d ) + T d ̺ h |u h − U h | 2 dx + (h ∇ x U L 2 ) 2 ≤ C 1 ∇ h u h − ∇ x U 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 1 ∇ x U − ∇ h U h 2 L 2 (T d ) + T d ̺ h |U h − U | 2 dx + h 2 ∇ x U 2 L 2 ≤ C 1 ∇ h u h − ∇ x U 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 1 h 2 U 2 W 2,∞ + h 2 ∇ x U 2 L ∞ T d ̺ h dx + h 2 ∇ x U 2 L 2 = C 1 ∇ h u h − ∇ x U 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 2 h 2 ,
where C 2 depends on C 1 , U W 2,∞ , ∇ x U L ∞ , M, and ∇ x U L 2 , which proves (B.1). Finally, we proceed with the proof of (B.2). On the one hand, for the case of u h ∈ Q h we have Π Q u h = u h , meaning (B.2) automatically holds as it is the same as (B.1). On the other hand, for the case of u h ∈ W h we employ (B.1) and the triangular inequality to derive
Π Q u h − U 2 L 2 ≤ Π Q u h − u h 2 L 2 + u h − U 2 L 2 ≤ h 2 div h u h 2 L 2 + C 1 ∇ h u h − ∇ x U 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 2 h 2 C 1 ∇ h u h − ∇ x U 2 L 2 (T d ) + T d ̺ h |u h − U | 2 dx + C 2 h 2 ,
where we have used the fact that div h u h 2 L 2 E 0 in view of (2.14b), which completes the proof.
Next, we recall [12,Lemma 14.3] in order to show the following statement formulated in Lemma B.4. Now we are ready to show the following lemma.
Lemma B.4. Let (̺ h , u h ) be a solution obtained either by the FV method (2.10) or the MAC method (2.11), and let U ∈ L ∞ (0, T ; W 2,∞ (T d ; R d )). Then there holds
τ 0 T d |(̺ h − r)(Π Q u h − U )| dxdt ≤ C 0 τ 0 E(̺ h , u h |r, U )dt + C 1 δ ∇ h u h − ∇ x U 2 L 2 + C 2 δh 2 ,
where C 1 , C 2 are the same as in Lemma B.2, and C 0 depends on r, r, δ, M, E 0 , γ.
Proof. First, thanks to Lemma B.3 we observe
τ 0 T d 1 res (̺ h ) ̺ h dxdt = τ 0 T d 1 ̺ h <r ̺ h dxdt + τ 0 T d 1 ̺ h >r ̺ h dxdt ≤ r τ 0 T d 1 ̺ h <r 1 dxdt + τ 0 T d 1 ̺ h >r ̺ γ h dxdt ≤ C τ 0 E(̺ h , u h |r, U )dt,
where C = C(r, r) is given in Lemma B.3. Next, using the triangular inequality, Young's inequality, the above estimate, Lemma B.2 and Lemma B.3 we find
τ 0 T d |(̺ h − r)(Π Q u h − U )| dxdt ≤ τ 0 T d 1 ess (̺ h )|(̺ h − r)(Π Q u h − U )| dxdt + τ 0 T d 1 ̺ h <r r|Π Q u h − U | dxdt + τ 0 T d 1 ̺ h >r ̺ h |Π Q u h − U | dxdt ≤ τ 0 T d 1 ess (̺ h ) 1 2 (̺ h − r) 2 + ̺ h |Π Q u h − U | 2 /r dxdt + τ 0 T d 1 ̺ h <r 1 2 1 δ r 2 + δ|Π Q u h − U | 2 dxdt + τ 0 T d 1 ̺ h >r 1 2 ̺ h + ̺ h |Π Q u h − U | 2 dxdt C 0 τ 0 E(̺ h , u h |r, U )dt + C 1 δ ∇ h u h − ∇ x U 2 L 2 + C 2 δh 2 ,
where C 0 depends on r, C(r, r), δ, and C 1 . We have completed the proof.
C Relative energy norm
In this section we show how to control the errors in the conservative variables by the relative energy.
√ ̺ L 2γ √ ̺(u − U ) L 2 + ̺ − r L γ U L 2γ γ−1 ̺ 1/2
L γ ̺|u − U | 2 1/2 L 1 + ̺ − r L γ U L ∞ E(̺, u|r, U ) 1/2 + E(̺, u|r, U ) 1/γ which proves (C.1a). Next, again by the triangular inequality and Lemma B.3 we observe for γ ≥ 2 that ̺ − r L 2 ≤ (̺ − r)1 ess (̺) L 2 + (̺ − r)1 res (̺) L 2
E(̺|r) 1/2 + T d ̺ 2 1 ̺>r dx 1/2 + T d 1 res (̺) dx 1/2 E(̺|r) 1/2 + T d ̺ γ 1 ̺>r dx 1/2 E(̺|r) 1/2 ,
where we have used the fact that ̺ 2 ≤ ̺ γ for large ̺ with γ ≥ 2. Further, it is easy to check that
m − M L 2γ γ+1 ≤ ̺(u − U ) L 2γ γ+1 + (̺ − r)U L 2γ γ+1 √ ̺ L 2γ √ ̺(u − U ) L 2 + ̺ − r L 2 U L 2γ ̺ 1/2 L γ ̺|u − U | 2 1/2 L 1 + ̺ − r L 2 U L ∞ E(̺, u|r, U ) 1/2
which proves (C.1b).
When assuming an upper bound on ̺, we derive via Lemma B.3 that
̺ − r L 2 ≤ (̺ − r)1 ess (̺) L 2 + (̺ − r)1 res (̺) L 2 E(̺|r) 1/2 + 1 res (̺) L 2 E(̺|r) 1/2 which implies m − M L 2 ≤ ̺(u − U ) L 2 + (̺ − r)U L 2 √ ̺ L ∞ √ ̺(u − U ) L 2 + ̺ − r L 2 U L ∞ E(̺, u|r, U ) 1/2 .
Combining the above two estimates we get (C.2) and complete the proof.
D Derivation of the relative energy
In this section we show the relative energy inequality (3.5). We start with the reformulation of the relative energy.
This research was initiated during our "Research in Pairs" stay at the Mathematisches Forschungsinstitut Oberwolfach in 2021. * The research of E.F. and B.S. leading to these results has received funding from the Czech Sciences Foundation (GAČR), Grant Agreement 21-02411S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840. ♠ M.L. has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) -Project number 233630050 -TRR 146 as well as by TRR 165 Waves to Weather. She is grateful to the Gutenberg Research College and Mainz Institute of Multiscale Modelling for supporting her research.
Figure 1 :
1MAC grid in two dimensions
Lemma 2 . 10 (
210Consistency formulation for a bounded numerical solution). Let the assumptions of Lemma 2.9 hold. Moreover, let ̺ h and u h be uniformly bounded, i.e., there exist positive constants ̺ and u such that ̺ h ≤ ̺ and |u h | ≤ u.(2.20)
A ) for the choice ∆t = h and different values of γ > 1, d = 2, 3.
for the FV and MAC schemes, respectively.
Then there exists C = C(r, r) > 0 such that (̺ − r) 2 1 ess (̺) + (1 + ̺ γ )1 res (̺) ≤ CE(̺|r), where E(̺|r) = P (̺) − P ′ (r)(̺ − r) − P (r) and(1 ess (̺), 1 res (̺)) = (1, 0) if ̺ ∈ [r, r], (0, 1) if ̺ ∈ R + \[r, r]. (B.3)
Lemma C. 1 .•
1Let γ > 1 and (r, U ) If ̺ > 0 and T d ̺ γ dx ≤ E 0 hold, then ̺ − r L γ + m − M m = ̺u and M = rU .• In addition, let ̺ < ̺. Then ̺ − r L 2 + m − M L 2 (E(̺, u|r, U )) First, by the triangular inequality and Lemma B.3 we obtain for γ ≤ 2 that̺ − r L γ ≤ (̺ − r)1 ess (̺) L γ + (̺ − r)1 res (̺) L γ (̺ − r)1 ess (̺) L 2 + (̺ − r)1 res (̺) L γ E(̺|r) 1/2 + ( ̺ L γ + r L γ ) 1 res (̺) + E(̺|r) 1/γ ≤ E(̺, u|r, U ) 1/2 + E(̺, u|r, U ) 1/γ ,where 1 ess (̺) and 1 res (̺) are given in Lemma B.3. Further, utilizing the above estimate with the triangular inequality, Hölder's inequality, and the L γ bound on ̺, we findm − M L 2γ γ+1 ≤ ̺(u − U )
Proposition 1.1. Let the initial data belong to the classStokes
system (1.1) inherited from the initial data (1.3). The following result can be the deduced from [1,
Theorem 3.3] and [6, Proposition 2.2].
Theorem 11.1 and 14.1] Lemma 2.8 (Energy estimates). Let (̺ h , u h ) be a numerical solution obtained either by the FV scheme (2.10) or by the MAC scheme (2.11) with γ > 1. Then for all
17 )
17whenever τ ∈ [t n , t n+1 ), where r h stands for ̺ h or ̺ h u h .Analogously as in the proofs of [12, Theorem 11.2] and [12, Theorem 14.2], we obtain
Now we are ready to show the following lemma.Lemma B.2. Under the assumption of Lemma B.1 let (̺ h , u h ) be a solution obtained either by the FV method (2.10) or the MAC method (2.11). Let U ∈ W 2,∞ (T d ; R d ), then there exists C
t n+1 t n r h (t)ϕ(t + ∆t) − r h (τ )ϕ(τ ) dt dx = T d 1 ∆t t n+1 t n
T d v h · ∇ x ψ dx = T d v h · ∇ Π E T ψ dx = d i=1 T d v i,h ∂ (i) T Π (i) E ψ dx = − d i=1 K∈T K ð D i v i,h Π (i) E ψ dx.
Appendix A Proof of the preliminary lemmasIn this section we present the proofs of Lemmas 2.2 -2.5.Proof of Lemma 2.2. First, we calculateAnalogously, we findwhich completes the proof.Proof of Lemma 2.3. First, we calculatewhere ǫ − and ǫ + are the left and right edges of D σ in the i th -direction of the canonical system for σ ∈ E i . Note that D ǫ ± ⊂ T are elements of the primary grid T . Then we can rewrite the above relation aswhere we have used (2.3a). This proves (2.6). The proof of (2.7) follows from (2.5) and (2.3a), specifically,Moreover, the term T 4 readsSumming up the above terms we get (3.5)
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"Generalized Chaplygin gas model: constraints from Hubble parameter versus Redshift Data",
"Generalized Chaplygin gas model: constraints from Hubble parameter versus Redshift Data"
]
| [
"Puxun Wu \nDepartment of Physics\nInstitute of Physics\nHunan Normal University\n410081ChangshaHunanChina\n\nSchool of Sciences and Institute of Physics\nCentral South University of Forestry and Technology\n410004ChangshaHunanChina\n",
"Hongwei Yu \nDepartment of Physics\nInstitute of Physics\nHunan Normal University\n410081ChangshaHunanChina\n"
]
| [
"Department of Physics\nInstitute of Physics\nHunan Normal University\n410081ChangshaHunanChina",
"School of Sciences and Institute of Physics\nCentral South University of Forestry and Technology\n410004ChangshaHunanChina",
"Department of Physics\nInstitute of Physics\nHunan Normal University\n410081ChangshaHunanChina"
]
| []
| We examine observational constraints on the generalized Chaplygin gas (GCG) model for dark energy from the 9 Hubble parameter data points, the 115 SNLS Sne Ia data and the size of baryonic acoustic oscillation peak at redshift, z = 0.35. At a 95.4% confidence level, a combination of three data sets gives 0.67 ≤ A s ≤ 0.83 and −0.21 ≤ α ≤ 0.42, which is within the allowed parameters ranges of the GCG as a candidate of the unified dark matter and dark energy. It is found that the standard Chaplygin gas model (α = 1) is ruled out by these data at the 99.7% confidence level. | 10.1016/j.physletb.2006.11.028 | [
"https://export.arxiv.org/pdf/gr-qc/0612055v1.pdf"
]
| 119,409,651 | gr-qc/0612055 | f7df8bf57c32b5bbe3cf960a7ae7dd0ec53fa90e |
Generalized Chaplygin gas model: constraints from Hubble parameter versus Redshift Data
arXiv:gr-qc/0612055v1 9 Dec 2006
Puxun Wu
Department of Physics
Institute of Physics
Hunan Normal University
410081ChangshaHunanChina
School of Sciences and Institute of Physics
Central South University of Forestry and Technology
410004ChangshaHunanChina
Hongwei Yu
Department of Physics
Institute of Physics
Hunan Normal University
410081ChangshaHunanChina
Generalized Chaplygin gas model: constraints from Hubble parameter versus Redshift Data
arXiv:gr-qc/0612055v1 9 Dec 2006PACS numbers: 9880-k, 9880Es * corresponding author
We examine observational constraints on the generalized Chaplygin gas (GCG) model for dark energy from the 9 Hubble parameter data points, the 115 SNLS Sne Ia data and the size of baryonic acoustic oscillation peak at redshift, z = 0.35. At a 95.4% confidence level, a combination of three data sets gives 0.67 ≤ A s ≤ 0.83 and −0.21 ≤ α ≤ 0.42, which is within the allowed parameters ranges of the GCG as a candidate of the unified dark matter and dark energy. It is found that the standard Chaplygin gas model (α = 1) is ruled out by these data at the 99.7% confidence level.
I. INTRODUCTION
Many astrophysical and cosmological observations, including Type Ia Supernovae (Sne Ia) [1] and cosmic microwave background radiation (CMBR) [2,3] etc, indicated that the universe is undergoing an accelerating expansion. Many works have being done in order to explain this discovery. Some people attribute the observed acceleration to a possible breakdown of our understanding of the laws of gravitation, thus they attempted to modify the Friedmann equation [4,5]. However, many more think that the cosmic acceleration is driven by an exotic energy component with the negative pressure in the universe, named dark energy, which at late times dominates the total energy density of our universe and accelerates the cosmic expansion. Up to now there are many candidates of dark energy, such as the cosmological constant Λ [6], quintessence [7], phantom [8] and quintom [9] etc.
Recently an interesting model of dark energy, named the Chaplygin gas, was proposed by Kamenshchik et al [10]. This model is characterized by an exotic equation of state
p ch = − A ρ α ch(1)
with a positive constant A and α = 1. Progress has been made toward generalizing these model parameters. In this regard, Bento et al. generalized parameter α from 1 to an arbitrary constant in Ref. [11], and this generalized model was called the generalized Chaplygin gas (GCG) model and can be obtained from a generalized version of the Born-Infeld action. For α = 0 the GCG model behaves like the scenario with cold dark matter plus a cosmological constant.
Inserting the above equation of state of the GCG into the energy conservation equation, it is easy to obtain
ρ ch = ρ ch0 A s + 1 − A s a 3(1+α) 1 1+α ,(2)
where ρ ch0 is the present energy density of the GCG and A s ≡ A/ρ 1+α ch0 . It is worth noting that, when 0 < A s < 1, the GCG model smoothly interpolates between a non-relativistic matter phase (ρ ch ∝ a −3 ) in the past and at late times a negative pressure dark energy regime (ρ ch = −p ch ). As a result of this interesting feature, the GCG model has been proposed as a model of the unified dark matter and dark energy (UDME). Meanwhile, for A s = 0 the GCG behaves always like matter while for A s = 1 it behaves always like a cosmological constant.
The GCG model, thus, has been the subject of great interest and many authors have attempted to constrain this UDME model by using various observational data, such as the Sne Ia [12,13,14,15,16,17,18], the CMBR [18,19,20], the gamma-ray bursts [21], the gravitational lensing [14,17,22], the X-ray gas mass fraction of clusters [13,14,15], the large scale structure [18,23], and the age of high-redshift objects [24].
In this paper we shall consider the new observational constraints on the parameter space of the GCG for a flat universe by using a measurement of the Hubble parameter as a function of redshift [25], the new 115 Sne Ia data released by the Supernova Legacy Survey (SNLS) collaboration recently [27] and the baryonic acoustic oscillation (BAO) peak detected in the large-scale correlation function of luminous red galaxies from Sloan Digital Sky Survey (SDSS) [28]. We perform a combined analysis of three databases and find that the degeneracy between A s and α is broken. At a 95.4% confidence level we obtain a strong constraint on the GCG model parameters: 0.67 ≤ A s ≤ 0.83 and −0.21 ≤ α ≤ 0.42, a parameter range within which the GCG model could be taken as a candidate of UDME and the pure Chaplygin gas model could be ruled out.
II. CONSTRAINT FROM THE HUBBLE PARAMETER AS A FUNCTION OF
REDSHIFT
Last year, based on differential ages of passively evolving galaxies determined from the Gemini Deep Deep Survey [29] and archival data [30], Simon et al. [31] gave an estimate for the Hubble parameter as a function of the redshift z,
H(z) = − 1 1 + z dz dt(3)
where t is the time. They obtained 9 data points of H(z) at redshift z i and used the estimated H(z) to constrain the dark energy potential. Later these 9 data points were used to constrain parameters of holographic dark energy model [32], parameters of the ΛCDM,
XCDM and φCDM models [34] and the interacting dark energy models [33]. Here we will use this data to constrain the GCG model.
For a flat universe containing only the baryonic matter and the GCG, the Friedmann equation can be expressed as
H 2 (H 0 , A s , α, z) = H 2 0 E 2 (A s , α, z) ,(4)
where
E(A s , α, z) = [Ω b (1 + z) 3 + (1 − Ω b )(A s + (1 − A s )(1 + z) 3(1+α) ) 1 1+α ] 1/2 ,(5)
Ω b is the present dimensionless density parameter of baryonic matter and H 0 = 100hKms −1 Mpc −1 is present Hubble constant. The Hubble Space Telescope key projects
give h = 0.72 ± 0.08 [35] and the WMAP observations give Ω b h 2 = 0.0233 ± 0.0008 [3]. The
best fit values for model parameters A s , α and constant H 0 can be determined by minimizing
χ 2 (H 0 , A s , α) = Σ 9 i=1 [H(H 0 , A s , α, z i ) − H obs (z i )] 2 σ 2 (z i ) .(6)
Since we are interested in the model parameters, H 0 becomes a nuisance parameter. We marginalize over H 0 to get the probability distribution function of A s and α: L(A s , α) = dH 0 P (H 0 )e −χ 2 (H 0 ,As,α)/2 , where P (H 0 ) is the prior distribution function for the present Hubble constant. In this paper a Gaussian priors H 0 = 72 ± 8kmS −1 Mpc −1 is considered.
In Fig. (1), we show the data of the Hubble parameter plotted as a function of redshift for the case H 0 = 72kms −1 Mpc −1 . Fig. (
III. JOINT STATISTICS WITH SDSS BAO AND SNLS SNE IA
Using a large spectroscopic sample of 46,748 luminous red galaxy from the SDSS, last
year Eisenstein et al [28] successfully found the size of baryonic acoustic oscillation (BAO) peak and obtained a parameter A, which is independent of cosmological models and for a flat universe can be expressed as
A = √ Ω m E(z 1 ) 1/3 1 z 1 z 1 0 dz E(z) 2/3 ,(7)
where z 1 = 0.35, A is measured to be A = 0.469 ± 0.017 and Ω m is the effective matter density parameter given by Ω [14,15,26]. Using parameter A we can obtain the constraint on dark energy models from the BAO. In Fig. (3) we show the constraints from this measurement on the parameter space A s − α. The best fit happens at A s = 0.76 and α = 0.01. Although the BAO data constrains efficiently the parameter plane into a narrow strip, parameters A s and α are also degenerate.
m = Ω b + (1 − Ω b )(1 − A s ) 1/(1+α)
However, from Fig. (2, 3) it is interesting to see that possible degeneracies between these parameters may be broken by combining these two kinds of observational data. In Fig. (4) we show the results of such an analysis. The best fit happens at A s = 0.61 and α = −0.28.
At the 95.4% confidence level we obtain 0.46 ≤ A s ≤ 0.79 and −0.53 ≤ α ≤ 0.2, a stringent constraint on the GCG. Apparently at the 68% confidence level the scenario of standard dark energy plus dark matter scenario (i.e. the case of α = 0) is excluded.
If further adding the new 115 SNLS Sne Ia data [27], which contains 44 previously published nearby Sne Ia (0.015 < z < 0.125) plus 71 distant Sne Ia (0.15 < z < 1) discovered by SNLS and gives the best fit values, A s = 0.78 and α = 0.16, for the GCG model, we find that a more stringent constraint is obtained, namely, at the 95.4% confidence level a combination of three databases gives 0.67 ≤ A s ≤ 0.83 and −0.21 ≤ α ≤ 0.42 with the best fits A s = 0.75 and α = 0.05. In Fig.(5) we show the 68.3%, 95.4% and 99.7% confidence level contours from these three data sets. It is easy to see that our results are consistent with the standard dark energy plus dark matter scenario at a 68% confidence level.
IV. CONCLUSION AND DISCUSSION
The constraints on the generalized Chaplygin gas (GCG) model, proposed as a candidate of the unified dark matter-dark energy scenario (UDME), has been studied in this paper.
The Hubble parameter as a function of redshift has been used to constrain the parameter space of the GCG model. We find, although the Hubble parameter gives a degeneracy between model parameters A s and α, the complementary and interesting constraints on the parameters of the model could be obtained. Combining the new SNLS Sne Ia data and the recent measurements of the baryon acoustic oscillations found in the SDSS, we obtained a very stringent constraint on model parameters of GCG. At the 95.4% confidence level, we found 0.67 ≤ A s ≤ 0.83 and −0.21 ≤ α ≤ 0.42. At addition we find at a 68% confidence level the combination of these three databases allows the scenario of standard dark energy plus dark matter, although the Hubble parameter plus the SDSS BAO exclude it.
Using the X-ray gas mass fractions of galaxy clusters and the dimensionless coordinate distance of Sne Ia and FRIIb radio galaxies, Zhu [15] obtained, at a 95.4% confidence level, A s = 0.70 +0.17 −0.17 and α = −0.09 +0.54 −0.33 . Using the CMBR power spectrum measurements from BOOMERANG and Archeops, together with the Sne Ia constraints, Bento et al. [20] found that 0.74 < A s < 0.85, and α < 0.6. Apparently these results are comparable with our results in this paper, which are within the allowed parameters ranges of the GCG as a candidate of UDME. However the standard Chaplygin gas model (α = 1) is ruled out by these data at the 99.7% confidence level. Meanwhile it is easy to see that at a 68.3% confidence level our result is consistent with the standard dark energy plus dark matter scenario (i.e. the case of α = 0), which is also in agreement with what obtained in Ref. [15,20].
FIG. 1 :FIG. 2 :FIG. 3 :FIG. 4 :FIG. 5 :
12345The Hubble parameters H(z) as a function of z for the case H 0 = 72kms −1 M pc −1 . The solid curve corresponds to our best fit to 9 Hubble parameter data plus SNLS SNe Ia data and SDSS baryonic acoustic oscillation peak with A s = 0.75, α = 0.05. The dotted line and dashed line correspond to A s = 1.0 and A s = 0.0 respectively. The 68.3%, 95.4% and 99.7% confidence level contours for A s versus α from the measurement of Hubble parameter with a Gaussian priors H 0 = 72 ± 8kmS −1 M pc −1 . The best fit happens at A s = 0.82 and α = 0.71. The 68.3%, 95.4% and 99.7% confidence level contours for A s versus α from the SDSS baryonic acoustic oscillations. The best fit happens at A s = 0.76 and α = 0.01. The 68.3%, 95.4% and 99.7% confidence level contours for A s versus α from the Hubble parameter data plus the SDSS baryonic acoustic oscillations peak. The best fit happens at A s = 0.61 and α = −0.28. The 68.3%, 95.4% and 99.7% confidence level contours for A s versus α from the Hubble parameter data plus the SDSS baryonic acoustic oscillations peak and the SNLS Sne Ia data. The best fit happens at A s = 0.75 and α = 0.05.
2) shows the results of our statistical analysis for the Hubble parameter data. Confidence contours (68.3%, 95.4% and 99.7%) in the A s -α plan are displayed by considering the Hubble parameter measurements discussed above. The best fit happens at A s = 0.82 and α = 0.71. It is very clear that two model parameters, A s and α, are degenerate.
AcknowledgmentsWe would like to thank Z. Zhu for his valuable discussions and help. This work was supported in part by the National Natural Science Foundation of China under Grants No.
10575035, the Program for NCET under Grant No. 04-0784 and the Key Project of Chinese Ministry of Education. and No. 10575035, the Program for NCET under Grant No. 04-0784 and the Key Project of Chinese Ministry of Education (No. 205110).
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| []
|
[
"RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE",
"RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE",
"RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE",
"RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE"
]
| [
"Gunther Uhlmann ",
"Yiran Wang ",
"Gunther Uhlmann ",
"Yiran Wang "
]
| []
| []
| We study the determination of the mass of a de Sitter-Schwarzschild black hole from one quasinormal mode. We prove a local uniqueness result with a Hölder type stability estimate. | 10.1007/s00220-023-04666-0 | [
"https://export.arxiv.org/pdf/2203.13749v1.pdf"
]
| 247,748,850 | 2203.13749 | be54793e901ac0ceca956d5efaac8d79752d804d |
RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE
Gunther Uhlmann
Yiran Wang
RECOVERY OF BLACK HOLE MASS FROM A SINGLE QUASINORMAL MODE
We study the determination of the mass of a de Sitter-Schwarzschild black hole from one quasinormal mode. We prove a local uniqueness result with a Hölder type stability estimate.
Introduction
The possibility of inferring black hole parameters from quasinormal modes (QNMs) has been explored in the physics literature, see Section 9 of the review paper [2]. For example, for slowly rotating black holes, Detweiler showed by numerical calculation in [7] that the wave parameters for the most damped mode are unique functions of the black hole parameters. Later, Echeverria in [10] investigated the stability issue. Since the success of gravitational wave interferometers, the topic has gained increasing attention, see e.g. [3]. One particular motivation for the study is to verify the black hole no hair theorem for which two QNMs are needed: one QNM is used to recover the black hole parameter and another QNM is used to test the theorem. We refer to [2, Section 9.7] for a review and [15] for the state of the art. Despite some convincing evidence, it seems that the theoretical justification is not complete. For example, most of the analysis in the literature is done for the fundamental modes corresponding to low angular momentum. However, it is generally not known which modes are excited and are extractable from the actual black hole ring down signals, see [3,2]. In this short note, we aim to provide a mathematical justification of the recovery of black hole parameters from a single QNM. We consider the model of a non-rotating de Sitter-Schwarzchild black hole (M, g dS ):
(1) M = R t × X • , X = (r bH , r sI ) × S 2 g dS = α 2 dt 2 − α −2 dr 2 − r 2 dw 2
where dw 2 denotes the standard metric on S 2 and
(2) α = (1 − 2m r − 1 3 Λr 2 ) 1 2
Here, m > 0 is the mass of the black hole and Λ > 0 is the cosmological constant. They satisfy 0 < 9m 2 Λ < 1. r bH , r sI are the two positive roots of Date: March 28, 2022. α(r) = 0 which corresponds to horizons. Throughout the note, we assume that Λ is known. Consider the d'Alembertian on (M, g dS ):
(3) M = α −2 (D 2 t − α 2 r −2 D r (r 2 α 2 )D r − α 2 r −2 ∆ S 2 )
where D r = −i∂ r and ∆ S 2 the positive laplacian on S 2 . The stationary scattering is governed by the operator [19]. On L 2 (X; Ω) with measure Ω = α −2 r 2 drdw, ∆ X is an essentially self-adjoint, non-negative operator, see [17]. Consider the resolvent
(4) ∆ X = α 2 r −2 D r (r 2 α 2 )D r + α 2 r −2 ∆ S 2 see(5) R X (λ) = (∆ X − λ 2 ) −1
Here, we use λ 2 as the spectral parameter and take Im λ ≥ 0 to be the physical plane such that R X (λ) is bounded on L 2 (X; Ω) for Im λ >> 0, according to the spectral theorem. Sá Barreto and Zworski demonstrated in [19, Proposition 2.1] that R X (λ) has a meromorphic continuation as operators from C ∞ 0 (X) to C ∞ (X) from Im λ ≥ 0 to C with poles of finite rank. The poles of R X (λ) are called resonances. The fact that they are equivalent to the quasinormal modes defined by using Zerilli's equation (see e.g. [6,5]) are discussed in [19], see also [4,17].
We denote the set of resonances by R(m) and set O = (0, 1/(3 √ Λ)). We call λ a trivial resonance if λ ∈ R(m) for all m ∈ O. For example, it is known that 0 is a trivial resonance, see [17]. Trivial resonances can not be used to determine black hole parameters. Our main result is Theorem 1.1. Let Λ > 0 and m ∈ O. For any λ ∈ R(m)\i(−∞, 0] not a trivial resonance, there exists δ > 0 (depending on λ) such that for any
m ∈ O with | m − m| < δ, if λ ∈ R( m) then m = m. Moreover, if λ ∈ R( m)
is sufficiently close to λ, then | m − m| ≤ C| λ − λ| 1/N for some C > 0 and N ∈ N depending on λ.
We point out that resonances on i(−∞, 0] are excluded in the theorem. There is a set of resonances on i(−∞, 0] described by (22) which cannot be treated with our method, although their dependency on m can be found explicitly. We do not investigate it further because such resonances are purely imaginary and they seem to be less relevant in practical cases, see for instance [15].
We also remark that for recovering black hole parameters, it is common to use only one or a few QNMs. This is very different from the usual inverse spectral/resonance problem for which the whole set R(m) is used to determine the parameters. In fact, there is a large literature on distribution of resonances for large angular momentum. For example, Theorem in [19] states that there exists K > 0, θ > 0 such that for any C > 0 there is an injective map b from the set of pseudo-poles recovers m. Similar results exist for rotating black holes, see for example [8]. Note that resonances sufficiently close to the lattice points cannot be trivial resonances. It is desirable to identify the trivial resonances, if there is any except 0. There is some interesting recent result in [13] which shows the convergence of resonances to a set of i(−∞, 0] for small masses. The numerical study [13,Fig. 6(b)] seems to indicate that such points are not trivial resonances.
(±l ± 1 2 − i 2 (k + 1 2 )) (1 − 9Λm 2 )
Our proof of the theorem is based on analytic perturbation argument, by observing that the coefficients of the operator ∆ X are analytic functions in m. There are some resonance perturbation theories, see for instance Agmon [1], Howland [14], which are developed upon perturbation theory for eigenvalues, see for example [18]. Here, we use that ∆ X has asymptotically hyperbolic structure near the two horizons to construct a parametrix modulo a trace-class error term, following Mazzeo and Melrose [16]. We then use the Fredholm determinant and its analyticity in m to finish the proof. This approach has the benefit of not relying on the spherical symmetry of the black hole metric. It is clear from the proof that one can add metric or potential perturbations with suitable decay at the horizons to obtain a similar result to Theorem 1.1.
The note is organized as follows. We begin in Section 2 with a scattering problem to demonstrate the possibility of recovering parameters from a single resonance. In Section 3, we discuss the asymptotic hyperbolic structure and the analyticity. We construct the resolvent in Section 4 and finish the proof in Section 5.
An example: the potential barrier
In this section, we give an example of a scattering system depending on one parameter, for which a single resonance recovers the parameter. The example was actually used by Chandrasekhar and Detweiler in [6] to illustrate the concept of quasinormal mode.
Consider
u (x) − V (x)u(x) + σ 2 u(x) = 0, x ∈ R
where σ is constant and V is the rectangular barrier Figure 2. Note that the potential is characterized by L. In this case, the scattering resonances can be defined as the poles of the scattering matrix.
V (x) = 1, x ∈ [−L, L] 0, otherwise See
It is a standard exercise in scattering theory to find the scattering matrix. Let's look at a wave traveling to the right, hit the potential and gets reflected and transmitted. In this case, the solution looks like
u R (x) = e iσx + re −iσx , x < −L te iσx , x > L
Here, r is the refection coefficient and t is the transmission coefficient. Similarly, we can consider a wave traveling to the left of the form
u L (x) = t e −iσx , x < −L e −iσx + r e iσx , x > L
with r , t the reflection, transmission coefficient respectively. The scattering matrix is
S = t r r t
By matching the solution and its derivatives at x = L, −L, we can find the coefficients as
r = r = e −2iσL+2iqL − e −2iσL−2iqL K , t = t = e −2iσL−2iqL + r q + σ q − σ e −2iqL where K = q + σ q − σ e −2iqL − q − σ q + σ e 2iqL
Thus, the resonances are solutions of K = 0 or equivalently
(6) ( q + σ q − σ ) 2 = e 4iqL
where q 2 = σ 2 − 1. Suppose we have σ such that Im q = 0. Then we can take modulus of (6) to find L as
(7) L = − 1 2 Im q ln | q + σ q − σ |.
This shows that one can recover L from one resonance. Now we provide a numerical verification. We compute resonances using a Matlab code from [20] and identify L using (7). It is important to note that the code from [20] does not calculate resonances by solving (7). Take L = 1.3. The potential and resonances are plotted in Figure 2. The numerical values of the four resonances nearest to the origin with positive real parts are
λ 1 = 1.2127 − 0.4432i, λ 2 = 2.2120 − 1.1135i, λ 3 = 3.4242 − 1.4810i, λ 4 = 4.6501 − 1.7230i
Using any of these resonances in (7), we find L = 1.3 with a 10 −4 error.
The asymptoically hyperbolic structure and analyticity
It is known that ∆ X in (4) can be essentially viewed as perturbed Laplacians associated with some asymptotically hyperbolic metrics near ∂X. We follow the presentations in [17]. Let X be a compact manifold of dimension n + 1 with boundary ∂X. Let ρ be a boundary defining function such that ρ > 0 in X, ρ = 0 at ∂X, dρ = 0 at ∂X. A metric g on X is called conformally compact if G = ρ 2 g is a non-degenerate Riemannian metric on the closure X. If in addition |dρ| 2 G | ∂X = K a constant, the metric g is called asymptotically hyperbolic. In this case, the sectional curvature approaches −K along any curve towards ∂X, see [16,Lemma (2.5)]. There is a normal form of the metric near ∂X, see e.g. Graham [11]. In particular, there is a choice of boundary defining function x such that in a neighborhood U = [0, ) x × Y, Y ⊂ ∂X of p ∈ ∂X, we can use local coordinate (x, y), y ∈ Y and get
(8) g = dx 2 + h(x, y, dy) x 2 Now we consider ∆ X in (12) on X. We define β = 1 2 dα 2 dr = m r 2 − Λ 3 r
We see that β is a smooth function of r on [r bH , r sI ] and analytic in m ∈ O.
We set β bH = β(r bH ) > 0, β sI = β(r sI ) < 0. Here, we recall that
(9) r bH = Im( 1 − (3m √ Λ) 2 + i3m √ Λ) 1/3 / √ Λ, r sI = Im(− 1 − (3m √ Λ) 2 + i3m √ Λ) 1/3 / √ Λ,
see page 6 of [19]. Thus β sI , β bH are both analytic functions of m ∈ O. Now we write (4) as
(10) ∆ X = βr −2 αD α (βr 2 αD α ) + α 2 r −2 ∆ S 2
For convenience, we denote ∂X = ∂X sI ∪ ∂X bH with
∂X sI = {r sI } × S 2 , ∂X bH = {r bH } × S 2 .
Note that α only vanishes at ∂X. We let ρ be a boundary defining function defined through (11) α = 2r bH β bH ρ near r = r bH and α = 2r sI β sI ρ near r = r sI .
Here, the smooth structure on X is changed. Before, r − r bH is a smooth boundary defining function near ∂X bH but now we think of (r − r bH ) 1 2 as a smooth boundary defining function, see [19,Section 2]. By using ρ, (10) becomes (12) ∆ X = βr −2 ρD ρ (βr 2 ρD ρ ) + 4ρ 2 β 2 bH r 2 bH r −2 ∆ S 2 near ∂X bH ∆ X = βr −2 ρD ρ (βr 2 ρD ρ ) + 4ρ 2 β 2 sI r 2 sI r −2 ∆ S 2 near ∂X sI Let g bH be the metric defined in a neighborhood of ∂X bH given by (13) g bH = dρ 2 β 2 ρ 2 + r 2 (2β bH r bH ) 2 dw 2 ρ 2 and let g sI be the metric defined in a neighborhood of ∂X sI given by (14) g sI = dρ 2 β 2 ρ 2 + r 2 (2β sI r sI ) 2 dw 2 ρ 2 These can be viewed as metric perturbations of the hyperbolic metrics
g bH,0 = 4dz 2 β 2 bH (1 − |z| 2 ) g sI,0 = 4dz 2 β 2 sI (1 − |z| 2 ) on B 3 = {z ∈ R 3 : |z| ≤ 1} with constant negative sectional curvature −β 2 bH
and −β 2 sI respectively. Here, (1 − |z| 2 ) 1 2 is the boundary defining function. Also, g bH , g sI are even asymptotically hyperbolic metrics as defined in Guillarmou [12].
After some calculation see [17,Proposition 8.1], we conclude that there are two smooth functions W bH , W sI such that (15) α∆
X α −1 = ρ∆ X ρ −1 = ∆ g bH + ρ 2 W bH − β 2 bH , near ∂X bH α∆ X α −1 = ρ∆ X ρ −1 = ∆ g sI + ρ 2 W sI − β 2 sI , near ∂X sI
This shows the asymptotically hyperbolic structure of α∆ X α −1 near ∂X.
Consider the dependency of the operator α∆ X α −1 on m. Note that the manifold X varies when varying m. We change the notation from X to X(m). The dependency can be fixed by transforming X(m) to a fixed reference manifold. Let X = (1, 2) × S 2 and let Ψ : X → X(m) be a diffeomorphism defined by (s, w) = Ψ(s, w) = (ψ(s), w) with ψ(s) = (s − 1)r sI + (2 − s)r bH Note that Ψ extends smoothly to X → X(m). Since r bH , r sI are analytic functions of m ∈ O, ψ and Ψ are also analytic in m ∈ O. Now, the pull back ρ * = ψ * (ρ) is a family of smooth boundary defining functions for ∂X. To see their dependency on m, we write (2) as
α = Λ/3(r − r bH ) 1 2 (r − r sI ) 1 2 (r − r 0 ) 1 2
where r bH , r sI are two positive roots of α = 0 and r 0 is the third negative root. Using (11) and on X near ∂X bH , we have (16)
ρ * = Λ/3(r − r bH ) 1 2 (r − r sI ) 1 2 (r − r 0 ) 1 2 2β bH r bH = Λ/3(r sI − r bH )(s − 1) 1 2 (2 − s) 1 2 ((s − 1)r sI + (2 − s)r bH − r 0 ) 1 2 2β bH r bH = (s − 1) 1 2 A bH (s, m)
where A bH (s, m) is defined through the last two lines. It is clear that A bH is smooth in s. Since (s − 1)r sI + (2 − s)r bH − r 0 > 0 for s ∈ [1, 2], we see that A bH , hence ρ * , is analytic in m ∈ O. Near ∂X sI , we have a similar form
(17) ρ * = (2 − s) 1 2 A sI (s, m).
From (13), (14), we see that the pull-back of the metrics are (18)
Ψ * g bH = (dρ * ) 2 (r sI − r bH ) 2 (β * ) 2 (ρ * ) 2 + [(s − 1)r sI + (2 − s)r bH ] 2 (2β bH r bH ) 2 dw 2 (ρ * ) 2 Ψ * g sI = (dρ * ) 2 (r sI − r bH ) 2 (β * ) 2 (ρ * ) 2 + [(s − 1)r sI + (2 − s)r bH ] 2 (2β sI r sI ) 2 dw 2 (ρ * ) 2
near ∂X bH , ∂X sI respectively. Here, β * = ψ * (β). Let V 0 (X) be the Lie algebra of smooth vector fields on X vanishing at ∂X. In local coordinate (x, y) near ∂X with x being the boundary defining function, V 0 (X) is generated by x∂ x , x∂ y . The space of zero-differential operators of order m on X, denoted by Diff m 0 (X) is generated by m fold products of vector fields in V 0 (X). From the analyticity of the diffeomorphism Ψ on m, we see that the pull-back of α∆ X α −1 to X is a differential operator on X with coefficients analytic in m. To see it belongs to Diff 2 0 (X) with coefficients analytic in m ∈ O, we consider α∆ X α −1 near ∂X for example near s = 1. We change the boundary defining function from ρ * to γ = (s − 1) 1 2 so ρ * = γA bH (γ 2 , m). Then the metric in (18) becomes
Ψ * g bH = (1 + ∂ γ A bH )(dγ) 2 (r sI − r bH ) 2 (β * ) 2 A 2 bH γ 2 + [(s − 1)r sI + (2 − s)r bH ] 2 (2β bH r bH ) 2 A 2 bH dw 2 γ 2
which is a family of Riemannian metrics on X analytic in m.
The resolvent construction
We obtain an approximation of R X (λ) in (5) following Mazzeo-Melrose [16]. In fact, we will find the resolvent of α∆ X α −1 on X. We will be using operators acting on half densities on X. For convenience, we introduce an auxiliary Riemannian metric g X on X which equals g bH , g sI near ∂X bH , ∂X sI respectively. Such a metric can be obtained by gluing g bH , g sI near ∂X and some Riemannian metric in the interior of X. The choice is clearly not unique and its dependency on m is not important. We use g X to trivialize the (zero) one-density bundle Ω 0 , that is we take Ω 0 to be the volume form |dg X |. The half-density bundle is Ω 1 2 0 . Let x be a boundary defining function such that in local coordinates (x, y), x ≥ 0, y ∈ S 2 near ∂X, g X is expressed in form of (8). In this coordinate,
Ω 1 2 0 = H(x, y)| dx x dy x | 1 2
for some smooth function H. Now we consider α∆ X α −1 acting on smooth sections C ∞ (X; Ω 1 2 0 ) in the following way α∆ X α −1 (uΩ
1 2 0 ) = (α∆ X α −1 u)Ω 1 2 0
The resolvent R α (λ) = (α∆ X α −1 − λ 2 ) −1 is acting on Ω 1 2 0 in the same way. The parametrix is constructed on the 0-blown-up space of X × X as in [16]. Let Diag = {(z, z) ∈ X × X} be the diagonal of X × X. Let ∂Diag = Diag ∩ (∂X × ∂X) which has two (disjoint) connected components. As a set, the 0-blown-up space is
X × 0 X = (X × X)\∂Diag S ++ (∂Diag),
where S ++ (∂Diag) denotes the inward pointing spherical bundle of T * ∂Diag (X× X). Let
β 0 : X × 0 X → X × X(19)
be the blow-down map. Then X× 0 X is equipped with a topology and smooth structure of a manifold with corners for which β 0 is smooth. The manifold X × 0 X has the following boundary hyper-surfaces: the left and right faces
L = β −1 0 (∂X × X), R = β −1 0 (X × ∂X)
, and the front face ff = β −1 0 (∂Diag). Since ∂X = ∂X bH ∪ ∂X sI where the asymptotic behavior of the resolvent is different at each connected component, it is convenient to introduce Figure 3. The lifted diagonal is denoted by Diag 0 = β −1 0 (Diag \ ∂Diag). X × 0 X has co-dimension two corners at the intersection of two of the boundary faces L, R, ff and co-dimension three corners given by the intersection of all the three faces. See Figure 3. to Diag 0 and vanishing to infinite orders at L, R. Here, it is understood that Ω 1 2 0 denotes the half-density bundle lifted from the one on X × X by β 0 . The corresponding class of pseudo-differential operators is denoted by Ψ m 0 (X, Ω 1 2 0 ). Next, let V b be the space of smooth vector fields on X × 0 X which are tangent to each of the boundary faces L, R, ff. Let ρ • , • = L bH , L sI , R bH , R sI , ff be boundary defining functions. We set
L bH = β −1 0 (∂X bH × X), L sI = β −1 0 (∂X sI × X), R bH = β −1 0 (X × ∂X bH ), R sI = β −1 0 (X × ∂X sI ) so L = L bH ∪L sI , R = R bH ∪R sI , see(20) A a,b,c,d (X × 0 X) = {u ∈ D (X × 0 X) : V 1 · · · V k u ∈ ρ a L bH ρ b R bH ρ c L sI ρ d R sI L ∞ (X × 0 X), V i ∈ V b , i = 1, 2, · · · , k, ∀k ≥ 0} Then define K −∞,a,b,c,d 0 (X × 0 X) = A a,b,c,d (X × 0 X) ⊗ C ∞ (X × 0 X; Ω 1 2 0 ). Finally, define K m,a,b,c,d 0 (X) = K −∞,a,b,c,d 0 (X × 0 X) + K m 0 (X)
. Then we let Ψ m,a,b,c,d 0 (X) be the space of operators on X whose Schwartz kernel when lifted to X × 0 X belongs to K m,a,b,c,d 0 (X). We have the following result.
(X) with (21) a = 1 + λ β bH i, b = 1 + λ |β sI | i, analytic in m ∈ O and holomorphic in λ ∈ C\Q with (22) Q . = −i β bH N ∪ −i |β sI | N ∪ {0} such that (23) (α∆ X α −1 − λ 2 )M (λ, m) = Id +E(λ, m)
Here, E(λ, m) ∈ ρ ∞ ff Ψ −∞,∞,a,∞,b 0 (X) is trace class on x l L 2 (X) for any l. Moreover, its Schwarz kernel is holomorphic in λ ∈ C\Q, and analytic in m ∈ O.
Proof. For fixed m, the construction of M (λ, m) and E(λ, m) and their holomorphy in λ is essentially contained in Proposition (7.4) of [16], which applies to the laplacian of asymptotically hyperbolic metrics. As argued in Proposition 2.2 of [19], the result applies to α∆ X α −1 − λ 2 as the normal operator is elliptic. Because we argued in Section 3 that α∆ X α −1 ∈ Diff 2 0 (X) with coefficients analytic in m ∈ O, the construction in [16] produces M (λ, m), E(λ, m) analytic in m. The set Q comes from the poles of the resolvent of β −2 bH ∆ 0 − λ 2 and β −2 sI ∆ 0 − λ 2 see Lemma (6.15) of [16]. Here, ∆ 0 denotes the positive laplacian of the standard hyperbolic metric. By rescaling the operator, we find the set Q.
To find a, b, it is convenient to work on X. It suffices to consider the operators near ∂X bH , ∂X sI respectively. Write g bH = ρ −2 h bH . From [16,Theorem (7.1)], see also [12, Theorem 1.1], we know that the resolvent of (24) |dρ| −2 h bH ∆ g bH + ζ(ζ − 2) belongs to Ψ −2,ζ,ζ 0 (X). Here, we followed [16] and used a different spectral parameter ζ. Near ∂X bH , −|dρ| 2 h bH approaches −β 2 bH . Comparing (24) with (12), we get β 2 bH ζ(ζ − 2) = −λ 2 − β 2 bH which gives ζ = 1 + iλ/β bH . This gives a, and b can be found in the same way near ∂X sI . To see the trace class property, we recall a result [16,Lemma (5.24)]. The push forward of the space ρ ∞ ff K −∞,a,b,c,d
0 (X × 0 X) is A a,b,c,d 0 (X × X; Ω 1 2 0 ⊗ Ω 1 2 0 ) = p (|y − y | 2 + ρ 2 + (ρ ) 2 ) p A a,b,c,d (X × X)
with the latter defined similarly to (20). Let K E (z, z ) be the Schwarz kernel of E. As E ∈ ρ ∞ ff Ψ −∞,∞,a,∞,b 0 (X), we conclude that for N ∈ N we can write
K E (z, z, m) = ρ N F N (z, z, m)
where F N ∈ C ∞ (X) is analytic in m ∈ O. Near ∂X bH , using (13) and z = (x, y) as local coordinate, we see that dg(z) = H(x, y)x −3 |dρdy|. We get a similar expression near ∂X sI . Then we see that the integral X |K E (z, z)|dg X (z) is finite so E is of trace class.
Proof of Theorem 1.1
We apply the resolvent to (23) to get
For the stability, we write K(λ, m) = (m − m 0 ) N f (λ, m) for some N ≥ 0 and f analytic in m with f (λ 0 , m 0 ) = 0. Now we set t = (m − m 0 ) N and get K(λ, t) = K(λ, m) = tf (λ, m 0 + t 1/N ). We see that ∂ t K(λ, t)| t=0 = f (λ, m 0 ) = 0 Using the implicit function theorem, we get that t = g(λ) is differentiable in a neighborhood of λ 0 . Thus, |t| ≤ C|λ − λ| which implies |m − m| ≤ C|λ − λ| 1/N This completes the proof of the Theorem 1.1.
in to R(m, Λ) such that all the poles in Ω C = {λ : Im λ > −C, |λ| > K, Im λ > −θ| Re λ|} are in the image of b and for b(µ) ∈ Ω C , we have b(µ) − µ → 0 as |µ| → ∞. See Figure 1. By looking at the sequence of resonances for large l, k, one
Figure 1 .
1Resonances for de Sitter-Schwarzshild black holes. The black dots are resonances captured by Theorem in[19]. The hollow dots and resonances in the shaded region are not.
Figure 2 .
2A rectangular barrier and its resonances.
Figure 3 .
3The 0-blown up space. The blown-up at the two components of ∂Diag 0 are shown. Now we introduce spaces of operator on X × 0 X. First, let K m 0 (X) ⊂ D (X × 0 X;
Proposition 4 . 1 .
41There is a family of operators M (λ, m) ∈ Ψ −2,a,a,b,b 0
AcknowledgmentThe authors thank Peter Hintz for his thorough reading of a previous version of the manuscript and for making very useful comments. GU was partly supported by NSF, a Simons Fellowship, a Walker Family Endowed Professorship at UW and a Si-Yuan Professorship at IAS, HKUST.
M (λ, m) = R α (λ)(Id +E(λ, m). M (λ, m) = R α (λ)(Id +E(λ, m))
using analytic Fredholm theorem, we see that for any m ∈ O, (Id +E(λ, m)) −1 is a family of bounded operators, meromorphic in λ ∈ C\Q. The poles (at least away from Q) are the resonances. Since E(λ, m) is compact on x l L 2 (X). In fact, the resolvent R α (λ) is also meromorphic at Q, as clarified in [12Since E(λ, m) is compact on x l L 2 (X), using analytic Fredholm theorem, we see that for any m ∈ O, (Id +E(λ, m)) −1 is a family of bounded operators, meromorphic in λ ∈ C\Q. The poles (at least away from Q) are the reso- nances. In fact, the resolvent R α (λ) is also meromorphic at Q, as clarified in [12].
We recall that if A is a trace class operator on a Hilbert space H with eigenvalues λ k , k = 1, 2, · · · with |λ 1 | ≥ |λ 2 | ≥ · · · ≥ 0. Then the Fredholm determinant det(Id +A) = Π ∞ k=1 (1 + λ k ). Now we use the determinant of Id +E to analyze the poles. See [9, Appendix BNow we use the determinant of Id +E to analyze the poles. We recall that if A is a trace class operator on a Hilbert space H with eigenvalues λ k , k = 1, 2, · · · with |λ 1 | ≥ |λ 2 | ≥ · · · ≥ 0. Then the Fredholm determinant det(Id +A) = Π ∞ k=1 (1 + λ k ). See [9, Appendix B].
Id +A is invertible if and only if det(Id +A) is non-zero, see. Also, Proposition B.28Also, Id +A is invertible if and only if det(Id +A) is non-zero, see [9, Proposition B.28].
Therefore, the resonances of R α (λ) is contained in the zero set of K(λ, m) = det. Id +E(λ, m)Therefore, the resonances of R α (λ) is contained in the zero set of K(λ, m) = det(Id +E(λ, m))
we conclude that K(λ, m) is a function holomorphic in λ ∈ C\Q, and analytic in m ∈ O. Now we suppose λ 0 is a resonance so K(λ 0 , m) = 0. By the analyticity in m, either K(λ 0 , m) is identically zero for all m which means λ 0 is a resonance for all m ∈ O. Using the argument in the end of [9, Section B.5. or m is the only (discrete) zeroUsing the argument in the end of [9, Section B.5], we conclude that K(λ, m) is a function holomorphic in λ ∈ C\Q, and analytic in m ∈ O. Now we suppose λ 0 is a resonance so K(λ 0 , m) = 0. By the analyticity in m, either K(λ 0 , m) is identically zero for all m which means λ 0 is a resonance for all m ∈ O, or m is the only (discrete) zero
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Gravitational-wave measurements of the mass and angular momentum of a black hole. F Echeverria, Physical Review D. 403194F. Echeverria. Gravitational-wave measurements of the mass and angular momentum of a black hole. Physical Review D 40.10 (1989): 3194.
Volume and area renormalizations for conformally compact Einstein metrics. R Graham, Proceedings of the 19th Winter School. Circolo Matematico di Palermo. the 19th Winter School. Circolo Matematico di PalermoR. Graham. Volume and area renormalizations for conformally compact Einstein met- rics. Proceedings of the 19th Winter School. Circolo Matematico di Palermo, 2000.
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Quasinormal modes of small Schwarzschild-de Sitter black holes. P Hintz, Y Xie, Journal of Mathematical Physics. 6311509P. Hintz, Y. Xie. Quasinormal modes of small Schwarzschild-de Sitter black holes. Journal of Mathematical Physics 63.1 (2022): 011509.
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Testing the no-hair theorem with GW150914. M Isi, M Giesler, W Farr, M Scheel, S Teukolsky, Physical Review Letters. 12311111102M. Isi, M. Giesler, W. Farr, M. Scheel, S. Teukolsky (2019). Testing the no-hair theorem with GW150914. Physical Review Letters, 123(11), 111102.
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| []
|
[
"ARTICLE Controlled spatial separation of spins and coherent dynamics in spin-orbit-coupled nanostructures",
"ARTICLE Controlled spatial separation of spins and coherent dynamics in spin-orbit-coupled nanostructures"
]
| [
"Shun-Tsung Lo \nDepartment of Physics\nNational Cheng Kung University\n701TainanTaiwan\n",
"Chin-Hung Chen \nDepartment of Physics\nNational Cheng Kung University\n701TainanTaiwan\n",
"Ju-Chun Fan \nDepartment of Physics\nNational Cheng Kung University\n701TainanTaiwan\n",
"L W Smith \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"G L Creeth \nDepartment of Electronic and Electrical Engineering\nDepartment of Physics\nUniversity College London\nWC1E 7JELondonUK\n\nDepartment of Electronic & Electrical Engineering\nNational Cheng Kung University\n701TainanTaiwan\n\nUniversity of Sheffield\nMappin StreetS1 3JDSheffieldUK\n",
"Che-Wei Chang \nDepartment of Physics\nNational Cheng Kung University\n701TainanTaiwan\n",
"M Pepper \nDepartment of Electronic and Electrical Engineering\nDepartment of Physics\nUniversity College London\nWC1E 7JELondonUK\n\nDepartment of Electronic & Electrical Engineering\nNational Cheng Kung University\n701TainanTaiwan\n\nUniversity of Sheffield\nMappin StreetS1 3JDSheffieldUK\n",
"J P Griffiths \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"I Farrer \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"H E Beere \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"G A C Jones \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"D A Ritchie \nCavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK\n",
"Tse-Ming Chen [email protected]. \nDepartment of Physics\nNational Cheng Kung University\n701TainanTaiwan\n"
]
| [
"Department of Physics\nNational Cheng Kung University\n701TainanTaiwan",
"Department of Physics\nNational Cheng Kung University\n701TainanTaiwan",
"Department of Physics\nNational Cheng Kung University\n701TainanTaiwan",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Department of Electronic and Electrical Engineering\nDepartment of Physics\nUniversity College London\nWC1E 7JELondonUK",
"Department of Electronic & Electrical Engineering\nNational Cheng Kung University\n701TainanTaiwan",
"University of Sheffield\nMappin StreetS1 3JDSheffieldUK",
"Department of Physics\nNational Cheng Kung University\n701TainanTaiwan",
"Department of Electronic and Electrical Engineering\nDepartment of Physics\nUniversity College London\nWC1E 7JELondonUK",
"Department of Electronic & Electrical Engineering\nNational Cheng Kung University\n701TainanTaiwan",
"University of Sheffield\nMappin StreetS1 3JDSheffieldUK",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Cavendish Laboratory\nJ J Thomson AvenueCB3 0HECambridgeUK",
"Department of Physics\nNational Cheng Kung University\n701TainanTaiwan"
]
| []
| The spatial separation of electron spins followed by the control of their individual spin dynamics has recently emerged as an essential ingredient in many proposals for spin-based technologies because it would enable both of the two spin species to be simultaneously utilized, distinct from most of the current spintronic studies and technologies wherein only one spin species could be handled at a time. Here we demonstrate that the spatial spin splitting of a coherent beam of electrons can be achieved and controlled using the interplay between an external magnetic field and Rashba spin-orbit interaction in semiconductor nanostructures. The technique of transverse magnetic focusing is used to detect this spin separation. More notably, our ability to engineer the spin-orbit interactions enables us to simultaneously manipulate and probe the coherent spin dynamics of both spin species and hence their correlation, which could open a route towards spintronics and spin-based quantum information processing. | 10.1038/ncomms15997 | null | 9,433,734 | 1801.10148 | 72808404fcac7abcec8b31da1b2b22f6f1c88518 |
ARTICLE Controlled spatial separation of spins and coherent dynamics in spin-orbit-coupled nanostructures
Published 10 Jul 2017
Shun-Tsung Lo
Department of Physics
National Cheng Kung University
701TainanTaiwan
Chin-Hung Chen
Department of Physics
National Cheng Kung University
701TainanTaiwan
Ju-Chun Fan
Department of Physics
National Cheng Kung University
701TainanTaiwan
L W Smith
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
G L Creeth
Department of Electronic and Electrical Engineering
Department of Physics
University College London
WC1E 7JELondonUK
Department of Electronic & Electrical Engineering
National Cheng Kung University
701TainanTaiwan
University of Sheffield
Mappin StreetS1 3JDSheffieldUK
Che-Wei Chang
Department of Physics
National Cheng Kung University
701TainanTaiwan
M Pepper
Department of Electronic and Electrical Engineering
Department of Physics
University College London
WC1E 7JELondonUK
Department of Electronic & Electrical Engineering
National Cheng Kung University
701TainanTaiwan
University of Sheffield
Mappin StreetS1 3JDSheffieldUK
J P Griffiths
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
I Farrer
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
H E Beere
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
G A C Jones
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
D A Ritchie
Cavendish Laboratory
J J Thomson AvenueCB3 0HECambridgeUK
Tse-Ming Chen [email protected].
Department of Physics
National Cheng Kung University
701TainanTaiwan
ARTICLE Controlled spatial separation of spins and coherent dynamics in spin-orbit-coupled nanostructures
Published 10 Jul 201710.1038/ncomms15997Received 10 Jan 2017 | Accepted 17 May 2017 |OPEN * These authors contributed equally to this work. w Present addresses: (I.F.). Correspondence and requests for materials should be addressed to T.-M.C. (email:
The spatial separation of electron spins followed by the control of their individual spin dynamics has recently emerged as an essential ingredient in many proposals for spin-based technologies because it would enable both of the two spin species to be simultaneously utilized, distinct from most of the current spintronic studies and technologies wherein only one spin species could be handled at a time. Here we demonstrate that the spatial spin splitting of a coherent beam of electrons can be achieved and controlled using the interplay between an external magnetic field and Rashba spin-orbit interaction in semiconductor nanostructures. The technique of transverse magnetic focusing is used to detect this spin separation. More notably, our ability to engineer the spin-orbit interactions enables us to simultaneously manipulate and probe the coherent spin dynamics of both spin species and hence their correlation, which could open a route towards spintronics and spin-based quantum information processing.
T he spin-orbit interaction in materials gives rise to a separation of different spin species in momentum space, creating many interesting phenomena such as the spin Hall 1-3 , the quantum spin Hall 4,5 effects and the spin-momentum locking 6,7 . However, it does not separate the spin-up and spin-down electrons in real space. In other words, even though different spins behave very differently they cannot be resolved and tracked in real space, similar to spin-degenerate systems where the spin-orbit interaction is negligible. So far most of the spintronic technologies which require spin to be resolved before subsequent operations have to rely on the creation of a spin imbalance with, for example, ferromagnets or optical injection. However, these methods are limited in both fundamental and practical aspects since only one spin type (that is, the majority spin) can be utilized. For example, the correlation between different spin types remains experimentally unexplored unless one can resolve and track both spin types simultaneously, for which it is necessary to spatially split electron spins rather than polarize them. Developing a simple way to spatially separate the opposite spin types, then manipulate and track the coherent spin dynamics of both of the two spin types and, more importantly, their phase correlation is therefore essential and a frontier in current research.
The Stern-Gerlach magnet is well-known for separating spins but is limited to uncharged particles, and modified proposals for electron spins using inhomogeneous spin-orbit effective fields [8][9][10] have yet to be realized. The spin Hall effect geometry [1][2][3] can also produce spin separation, where the diffusive electrons that are scattered to opposite edges of a conductor are coupled to spins of opposite orientations; however, no control can be exercised in such a random scattering system. A promising way to achieve spatial separation of electron spins in a spin-orbit coupled system is to apply a transverse magnetic field. Spin-up and spin-down electrons have different momenta and thus, when moving through a magnetic field, will experience different Lorentz forces and consequently undergo different cyclotron motions. This concept has been successfully demonstrated, using a hole gas in which the spin-orbit interaction was not tunable [11][12][13] , but to manipulate and study the behaviour of the spatially separated spins remains an outstanding challenge.
Here we combine this simple concept of spatial spin separation with techniques to coherently manipulate and detect spins, and thereby demonstrate a spatial spin splitting of a coherent electron beam together with full control of the dynamics of these spatially separated spins. The spatial separation, coherent spin dynamics and phase correlation between the up-and down-spin electrons can all be-electrically and on-chip-controlled and probed. This allows both of two spin types (instead of just the majority one as in most previous studies) to be simultaneously probed and manipulated, which promises to advance spintronic technologies that require both spin types to be operated together.
Results
Spatial separation of spins. Figure 1a captures the operation of our devices. A quantum point contact (QPC)-a one-dimensional (1D) constriction created by applying voltages to split gates patterned on the surface of an InGaAs heterostructure-is used to inject an unpolarized electron beam into a two-dimensional electron gas (2DEG). The 2DEG is formed in the InGaAs quantum well (Methods section), wherein the structural inversion asymmetry of the well generates a momentum-dependent magnetic field B SO R on the spin of every moving electron, the so-called Rashba spin-orbit interaction. This Rashba spin-orbit effective magnetic field B SO R lies in the plane of the 2DEG (that is, the x-y plane in Fig. 1a) and is orientated perpendicular to the electron's momentum. It lifts the spin degeneracy in momentum space and leads to two spin-polarized Fermi circles, parallel and antiparallel to B SO R (Fig. 1b). Electrons in the parallel and antiparallel spin states (hereafter, we refer to these as the up and down spins, respectively), though moving in the same direction and spatially unresolved when injected from a QPC into the 2DEG, have different Fermi wavevectors and thus will be deflected along different cyclotron trajectories in the presence of a transverse magnetic field. Spin-selective spatial separation of an electron beam is therefore achieved.
To study the spatial separation of the two spin species, another QPC is placed at a distance L from the QPC emitter to act as a charge collector, forming a geometry (Fig. 1a,c and the inset of 1d) known as transverse magnetic focusing [11][12][13][14][15][16][17][18] . Magnetic focusing occurs when the electrons that leave the QPC emitter are focused into the QPC collector, giving peaks in collector voltage (that is, focusing peaks) at magnetic fields where an integer multiple of cyclotron diameter is equal to L. The two spatially separated spin species travel with different cyclotron radii and thus will require two different magnetic fields
B "# ¼ 2' k "# eL ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 2m à E F p Ç m à a=' À Á eL ;ð1Þ
to focus themselves directly into the collector (inset of Fig. 1d), where ' is Planck's constant divided by 2p, e is the elementary charge, m à is the electron effective mass, E F is the Fermi energy, k m (k k ) refers to the Fermi wavevector of spin-up (-down) state, and a parameterizes the strength of Rashba spin-orbit interaction. A spatial splitting of electron spins is therefore visible as a peak splitting in the magnetic focusing spectrum, allowing us to easily track and investigate the spatial spin separation. Figure 1d shows the magnetic focusing spectrum, with the emitter (G E ) and collector conductance (G C ) both set to 100 mS (above the quantized plateau at 2e 2 /h) to allow both spin species to propagate through the 1D channels (Supplementary Notes 1 and 2). For Bo0 focusing peaks appear periodically at integer multiples of BE0.19 T, corresponding to when electrons are focused into the collector. This value is consistent with the cyclotron motion B ¼ 2 ffiffiffiffiffiffiffiffiffiffiffiffiffi 2m à E F p =eL calculated using the 2D electron density. For B40 electrons are directed in the opposite direction, therefore no peaks in collector voltage are observed. The splitting of the focusing peak (hereafter referred to as the focusing peak doublet) is observed on the first and the third focusing peaks as evidence of spatial spin splitting. The low-field B m and high-field B k peak within the doublet corresponds to the spin-up and spin-down electrons, respectively. The Rashba parameter a estimated from the peak splitting using equation (1) is 3.1  10 À 11 eVm, close to the value estimated from the beating pattern in the Shubnikov-de Haas oscillations (Supplementary Note 1). There is additional structure around the focusing peaks which is likely due to the quantum interference effects 19 . We note that the focusing peak doublet is not visible on the second focusing peak. This is consistent with the model 20 that the electrons are subject to spin flip with respect to the momentum when they are reflected from the edge of the 2DEG and hence the two spatially separated spin branches reunite with each other at the collector (Supplementary Note 3).
Control of charge and spin dynamics. So far the magnetic focusing spectrum can only show that the electrons leaving from a QPC emitter are spatially spin-split, without being able to shed any light on the spin dynamics afterwards. An important open question remains on how each spin species evolves due to the influence of a rotating B SO R (in the reference frame of the spin)-which rotates along the cyclotron trajectories as the momentum rotates-in such a spin-orbit coupled 2DEG. For example, it is desirable to understand whether electron spins can maintain their coherence before reaching the collector, and also whether these spins adiabatically follow B SO R . To study the binary spin dynamics, we now force the collector to act as a spin analyzer by introducing the lateral spin-orbit interaction 21,22 and manipulating the energy and population of the 1D subbands in the collector. A voltage difference between the two sides of the split gate is used to create a lateral inversion asymmetry and consequently a lateral spin-orbit effective magnetic field B SO L pointing along the z axis (Fig. 1a). The electrically tunable B SO L þ B SO R within the emitter and collector QPC allows us to respectively prepare and analyse the electron spins along any specific direction in the y-z plane. In addition, a top gate (gate T in Fig. 1c) covers the entire focusing path and is used to vary B SO R (and equivalently a) in the 2DEG region.
The electron spins transmitted through the QPC emitter stabilize at the state determined by B SO L þ B SO R and consequently their orientations are initialized out of the 2DEG plane. In other words, the spin-up (spin-down) electrons are tilted toward negative (positive) z-direction by B SO L owing to being in the 1D B SO L þ B SO R parallel (antiparallel) spin states. After leaving the emitter, the electrons experience only the in-plane B SO R (since the focusing transverse magnetic field is small compared to B SO R ) and therefore can precess about it as depicted in Fig. 1a if they propagate coherently. We can alter the spin orientation by controlling the spin precession frequency using top gate voltage V T , which determines B SO R . Here we first demonstrate an electrically tunable spatial spin separation in Fig. 2a, where the evolution of focusing spectrum of the first doublet is measured as a function of V T at G E ¼ 160 mS and G C ¼ 100 mS. The two superimposed dashed lines are the calculated B m and B k focusing fields using the model of spin precession described below. The spatial separation between the two spin species, manifested as the peak splitting
|B k À B m | ¼ 4m à a/' eL, increases with increasing V T .
We now move on to study the spin dynamics of the two spin species and the phase correlation between them. This is achieved by lowering the collector conductance to G C ¼ 20 mS such that the QPC acts as a spin analyzer, as described in Supplementary Note 2. The orientation of incident electron spins is indicated by the magnitude of the collector voltage (that is, the focusing peak height). Electrons can propagate through the collector if their spin is parallel to the polarization direction, and cannot pass if their spin is antiparallel. Figure 2b shows that both the B m and B k focusing peaks in collector voltage oscillate with V T . These oscillations are p out-of-phase with each other, that is, each local maximum (minimum) in collector voltage along the B m focusing accordingly, the direction of electron spins with respect to B SO R remains conserved (that is, the spinors can be described as a superposition of the adiabatic B SO R eigenstates with conserved probabilities; see Supplementary Note 4 for more details). Hence, the electron spins precess about B SO R with a Larmor frequency of o s "# ¼ 2a j k "# j =' . The spin precessional angle accumulated by electrons travelling along a semiclassical cyclotron orbit to the collector is therefore given as 23
z y x B SO R B SO R B SO L B SO R d a c b V E M C T Vexc Iexc E1 E2 C1 C2 T k x |k ↑ | |k ↓ | Δk Device A Device B B ↑ B ↓ (iii) B > 0 (i) B ↑ (ii) B ↓ x y 1.0 0.5 0.0 B (T) -0.5 -1.fðV T Þ ¼ o s "# t "# ¼ pm à aðV T ÞL=' 2ð2Þ
where t mk is the time interval for the focusing process. This angle is irrespective of spin orientation and depends only on the strength of spin-orbit interaction a(V T ) for a fixed L, consistent with the observation of antiphase oscillations in the collector voltage for B m and B k in Fig. 2b. Moreover, the oscillations enable us to calculate the gate-voltage-dependent variation of a using equation (2), which is consistent with the value obtained from the splitting of focusing peaks using equation (1). Later in this paper we will compare these a values derived independently using these two methods. Such a consistency is here evident from the excellent quantitative agreement between the position of the focusing peaks measured experimentally and the values calculated using equation (1) in accordance with the a value derived from the spin precessional motion (dashed lines in Fig. 2a,b). The fact that the two antiphase oscillations are quantitatively described by considering spin precession in the adiabatic limit indicates that both of the spatially separated up and down spin adiabatically follow and precess about the rotating B SO R as illustrated in Fig. 1a. This also suggests that the phase correlation between the two separate, neighbouring spin types can be electrically controlled via tuning a. It is worth noting that the phase correlation observed in focusing spectra is equal to p regardless of the strength of spin-orbit interaction because both spin types are focused into the same collector by different magnetic fields, while in reality opposite spins travel along different trajectories and gain different phase shifts determined by equation (2).
Similar results are observed in other devices, as shown in Fig. 3 where the data are obtained using device B after illumination (Methods section). Figure 3a,b compare the magnitude of the B m and B k focusing peaks as a function of V T , for G C ¼ 20 and 100 mS, respectively. For G C ¼ 20 mS (Fig. 3a) the collector acts as a spin analyzer. The B m and B k collector voltages oscillate with V T , and are p out-of-phase with each other. In contrast, no oscillations are observed when G C is raised to 100 mS, where the QPC collector acts only as a charge detector (Fig. 3b). The oscillations also disappear when either the emitter or the collector QPC is biased symmetrically (Supplementary Note 5), which is consistent with our spin precession model. When the emitter is biased symmetrically (that is, B SO L ¼ 0), the electron spins which are emitted are aligned along the axis of B SO R and hence no spin precession shall occur. Also, when B SO L is removed from the collector, the spin polarization is analysed along the stationary B SO R spin states, and hence no spin precession can be probed. Note that as with device A, there is a quantitative agreement between the a(V T ) obtained with equations (1) and (2), which use the peak splitting and oscillatory collector voltage data, respectively.
One advantage of device B is that the QPC emitter and collector can be independently controlled since they do not share a common middle gate. This enables us to reverse the polarity of the lateral inversion asymmetry of the QPC and hence B SO L , simply by reversing the polarity of the voltage difference between the two sides of the split gate. Figure 3c,d presents a comparison of the focusing spectra for DV E ¼ þ 1.5 V and À 1 V. Here the focusing spectra are plotted as a function of magnetic field and a(V T ) Â L, instead of V T (as in other figures), since when DV E changes the distance L between the emitter and collector also changes and thus needs to be taken into account (Supplementary Note 6). A phase inversion in the oscillations for both the B m and B k focusing peaks is apparent as the lateral bias DV E is changed from þ 1.5V to À 1 V. Such an inversion can be easily understood using a schematic in Fig. 3e which illustrates the phase evolution, depicted using Bloch spheres, of the spin-up (red arrows) and spin-down (blue arrows) electrons travelling along the cyclotron trajectory at positive and negative DV E , respectively. Since the initial phase correlation between spin-up and spin-down electrons is inverted as the direction of B SO L is reversed, the observed phase correlation for the arrivals that undergo the same phase evolution (and equivalently aL) must also be inverted. Figure 4 summarizes values obtained for the Rashba coefficient a. The values obtained via the focusing peak splitting using equation (1) (open symbols) and via the oscillatory collector voltage using equation (2) (solid symbols) are both shown and are in excellent quantitative agreement with each other. To directly compare data before (red symbols) and after (blue symbols) illumination, we plot the Rashba coefficient a as a function of carrier density. The value of a follows the same trend line both before and after illumination. For comparison, we also plot the values of a(V T ) published in recent work 22 using a spin field-effect transistor (fabricated on the same wafer used here), where a is estimated from spin precession measurements in a steady-instead of rotating-B SO R . There is excellent quantitative agreement between the values of a obtained from these different devices and methods. The spin focusing technique appears more informative than the conventional SdH beating analysis 24 which is sometimes difficult to observe (Supplementary Note 1), and provides a reliable means for the determination of a value in the ballistic transport regime.
Discussion
The ability to manipulate and probe coherent spin dynamics in materials with high spin-orbit interaction is important for understanding the physics of emerging materials, as well as to having implications for spintronics and (topological) quantum computing [25][26][27] . Distinct from most previous studies 22, 28 -which rely on the introduction of polarized electrons to break the spin symmetry and are limited in that only the majority spin type can be resolved and used-our spin focusing technique provides a route to probe and manipulate the coherent spin dynamics of both spin species and their phase correlation in semiconductor nanostructures, and can be readily extended to materials with unusual band structures such as topological insulators 6,7,29 , graphene and its hybrid structures 30 . A recent study 18 that used the conventional magnetic focusing technique to probe the properties of graphene is a successful example. From a technological viewpoint, our ability to spatially bifurcate the two electron spin types and coherently manipulate them to any specific orientation (through spin precession and the fast manipulation of B SO R and B SO L using surface gates) make it possible to prepare two separate, neighbouring spins with an electrically controllable phase correlation, which has implications for interferometer and quantum logic operations.
Methods
Devices. A gated modulation-doped In 0.75 Ga 0.25 As/In 0.75 Al 0.25 As heterostructure is used in this work. The layer sequence is grown by molecular beam epitaxy as follows: 250 nm In 0.75 Al 0.25 As; 30 nm In 0.75 Ga 0.25 As (quantum well); 60 nm In 0.75 Al 0.25 As (spacer); 15 nm In 0.75 Al 0.25 As (Si-doped); 45 nm In 0.75 Al 0.25 As; and 2 nm In 0.75 Ga 0.25 As (cap). A dielectric layer (27 nm and 40 nm for device A and B, respectively) of SiO 2 is deposited on the wafer surface by plasma-enhanced chemical vapour deposition. Subsequently, surface gates are defined using electron-beam lithography and thermal evaporation of Ti/Au. There are two device designs, denoted device A and device B, as shown in Fig. 1c. In device A the lateral biases of the emitter and collector QPCs are defined as DV E ¼ V E À V M and DV C ¼ V C À V M , respectively, whereas in device B DV E ¼ V E1 À V E2 and DV C ¼ V C1 À V C2 . Note that the emitter is covered by the top gate, such that the Fermi wavevector of the focusing electrons that transit from the emitter to the bulk can be reliably controlled with the top gate. The collector is not covered by the top gate so that the spin polarization can be analysed along a fixed axis, independent of the top gate voltage. Data from device A and B are taken before and after illumination, respectively, which give very different characteristics of the 2DEG.
Measurements. Experiments are performed at a base temperature of 25 mK in a dilution refrigerator equipped with a superconducting magnet. The carrier density and mobility of the 2DEG are measured to be 2.1 Â 10 11 cm À 2 and 1.7 Â 10 5 cm 2 V À 1 s À 1 , respectively, using four-terminal magnetotransport measurements (Supplementary Note 1). This gives a mean free path of 1.3 mm for momentum relaxation. After illumination, they increased to 3.9 Â 10 11 cm À 2 , 2.6 Â 10 5 cm 2 V À 1 s À 1 and 2.7 mm, respectively. For transverse magnetic focusing experiments, simultaneous lock-in measurements of emitter and collector QPC conductances are carried out by supplying two-independent excitation sources of a 77 Hz a.c. voltage V exc ¼ 100 mV to the emitter and a 37 Hz a.c. current I exc ¼ 1 nA to the collector. The magnetic field is applied normal to the 2DEG plane to focus electrons into the collector. The focusing signal is measured as a voltage drop developed across the QPC collector in linear response to the 77 Hz a.c. current from the QPC emitter.
Data availability. The data that support the findings of this study are available from the corresponding author upon reasonable request.
Figure 1 |Figure 2 |Figure 3 |
123Scheme for spatial spin separation and control of spin dynamics. (a) Schematic view of a spin focusing device. The structural inversion asymmetry gives rise to an in-plane Rashba spin-orbit field B SO R on the spin of every moving electron, illustrated by the inset. We define the spin-up, m (spin-down, k), as parallel (antiparallel) to B SO R . Spin-up and spin-down electrons have different Fermi wavevectors and thus will be deflected along different cyclotron trajectories in a transverse magnetic field, resulting in spatial spin separation. Within the QPC constriction, an additional lateral spin-orbit field B SO L can be created via laterally biasing the gates to tilt spins toward either positive or negative z-direction. The two spatially separated spin species thus precess about B SO R in the 2DEG region. The spin-orbit fields B SO R and B SO L are represented by green arrows, while the red and blue arrows represent up and down spins, respectively. (b) The Fermi surface (red and blue circle of radius k m and k k for spin-up and spin-down) is spin-split with a wavevector separation Dk( ¼ k k À k m ) in the presence of Rashba spin-orbit interaction. The arrows are coloured following the same convention as in a. (c) Scanning electron microscope images of device A and B, with scale bar of 1 mm. Devices A and B are measured before and after illumination, respectively, which gives markedly different electron densities and mobilities (Methods section). Device B contains two pairs of split gates to allow independent control of the QPC emitter (using E1 and E2) and collector (using C1 and C2). (d) Transverse magnetic focusing spectrum measured from device B. The inset shows representative trajectories for spin-up (red trace) and spin-down (blue trace) electrons at different magnetic fields. which corresponds to the incident spins being parallel (antiparallel) to the polarization direction of the collectorcoincides with the local minimum (maximum) along the B k peak. Evidently, both the up and down spin coherently precess and maintain their initial p out-of-phase correlation after undergoing the action of the rotating B SO R . Within the adiabatic approximation in which B SO R changes its direction slowly such that the system adapts its configuration Magnetic spin focusing spectra.(a) Collector voltage as a function of magnetic field B and top gate voltage V T for device A with emitter conductance G E ¼ 160 mS and collector conductance G C ¼ 100 mS. The lateral bias DV E is fixed at 1.33 V (see Methods section for the quantification of DV E ) whereas DV C ranges from 2.15 V to 2.41 V as V T increases to keep both QPCs at fixed conductance values. The solid line illustrates the average B between the spin-up and spin-down focusing peaks (B m þ B k )/2, which can be used to determine the carrier density n 2D . The dashed lines show the focusing peak positions calculated using the spin precessional motion. (b) As in a but with G C reduced to 20 mS to turn the collector into a spin analyzer. DV E is fixed at 1.23 V whereas DV C ranges from 2.02 V to 2.30 V. The subsequent maxima (minima) of the oscillating collector voltage along the B m and B k focusing peaks correspond to rotations of the incident spins by np, where n is an integer, such that the spin is parallel (antiparallel) to the polarization direction of the collector. Spin precession in a rotating B SO R . (a) Collector voltage of the B m (red) and B k (blue) focusing peaks as a function of V T , with G E ¼ 100 mS and G C ¼ 20 mS. Data for all panels in this figure are from device B. The lateral biases of the QPC emitter and collector are set at DV E ¼ 0.25 V and DV C ¼ 0.5 V, respectively. (b) As in a except with G C increased to 100 mS for comparison. (c) Magnetic spin focusing spectrum as a function of aL and magnetic field for DV E ¼ 1.5 V and DV C ¼ 0.5 V. (d) As in c but with DV E changed to À 1 V to invert the direction of B SO L . This gives rise to an inverse p out-of-phase oscillation in the B m and B k focusing peaks with respect to that in c. Only the data with the collector voltage above 5 mV are shown to highlight the varying focusing peak height. Data for c,d are obtained in a different cooldown to a,b. The dashed lines indicate the focusing peak positions calculated using the same method as in Fig. 2. (e) A sequence of Bloch spheres illustrate the phase evolution of the spin-up (red arrows) and spin-down (blue arrows) electrons moving along the focusing trajectory. The top (bottom) row of spheres represents the phase evolution for DV E 40 (DV E o0); the vertical (horizontal) axis represents B SO R (B SO L ). Starting with electrons within the QPC emitter, the combination of Rashba and lateral spin-orbit interactions prepares the B SO L þ B SO R parallel and antiparallel spin states. After leaving the QPC and entering the 2DEG both spin types experience only the Rashba effective field B SO R and precess about it.
Figure 4 |
4Comparison of the measured Rashba coefficients. The Rashba coefficient a is plotted as a function of carrier density n 2D . Red and blue data points correspond to data obtained using the magnetic spin focusing technique (illustrated in inset a) before (device A) and after (device B) illumination, respectively. Two methods are used to extract a. Open symbols show the values given by equation(1)in the main article which considers the spatial spin separation of electrons. Solid symbols show values obtained using equation(2) which considers the precessional motion of the spin. The dashed line shows a polynomial fit to the data from spin focusing. For comparison, the Rasha coefficient obtained in recent measurements of a spin field-effect transistor22 (illustrated in inset b) are shown by the black solid line.
NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15997
NATURE COMMUNICATIONS | 8:15997 | DOI: 10.1038/ncomms15997 | www.nature.com/naturecommunications
AcknowledgementsWe thank S.-C. Ho and C.-T. Liang for discussions. This work was supported by the Ministry of Science and Technology (Taiwan), the Headquarters of University Advancement at the National Cheng Kung University and the Engineering and Physical Sciences Research Council (UK).Author contributionsPublisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/ licenses/by/4.0/ r The Author(s) 2017 NATURE COMMUNICATIONS | DOI: 10.1038/ncomms15997
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| []
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[
"The weak lensing radial acceleration relation: Constraining modified gravity and cold dark matter theories with KiDS-1000",
"The weak lensing radial acceleration relation: Constraining modified gravity and cold dark matter theories with KiDS-1000"
]
| [
"Margot M Brouwer [email protected] \nKapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands\n\nInstitute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n",
"Kyle A Oman \nKapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands\n\nInstitute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK\n",
"Edwin A Valentijn \nKapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands\n",
"Maciej Bilicki \nCenter for Theoretical Physics\nPolish Academy of Sciences\nLotników 32/4602-668WarsawPoland\n",
"Catherine Heymans \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n\nFaculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany\n",
"Henk Hoekstra \nLeiden Observatory\nLeiden University\nP.O.Box 95132300RALeidenThe Netherlands\n",
"Nicola R Napolitano \nSchool of Physics and Astronomy\nSun Yat-sen University\n519082Guangzhou, Zhuhai CampusP.R. China\n",
"Nivya Roy \nSchool of Physics and Astronomy\nSun Yat-sen University\n519082Guangzhou, Zhuhai CampusP.R. China\n",
"Crescenzo Tortora \nINAF -Osservatorio Astronomico di Capodimonte\nSalita Moiariello 1680131NapoliItaly\n",
"Angus H Wright \nFaculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany\n",
"Marika Asgari \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n",
"Jan Luca Van Den Busch \nFaculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany\n",
"Andrej Dvornik \nFaculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany\n",
"Thomas Erben \nArgelander-Institut für Astronomie\nAuf dem Hügel 7153121BonnGermany\n",
"Benjamin Giblin \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n",
"Alister W Graham \nCentre for Astrophysics and Supercomputing\nSwinburne University of Technology\n3122HawthornVICAustralia\n",
"Hendrik Hildebrandt \nFaculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany\n",
"Andrew M Hopkins \nAustralian Astronomical Optics\nMacquarie University\n105 Delhi Road\n\nNorth Ryde\n2113NSWAustralia\n",
"Arun Kannawadi \nDepartment of Astrophysical Sciences\nPrinceton University\n4 Ivy Lane08544PrincetonNJUSA\n",
"Konrad Kuijken \nLeiden Observatory\nLeiden University\nP.O.Box 95132300RALeidenThe Netherlands\n",
"Jochen Liske \nHamburger Sternwarte\nUniversity of Hamburg\nGojenbergsweg 11221029HamburgGermany\n",
"Huanyuan Shan \nShanghai Astronomical Observatory (SHAO)\nNandan Road 80200030ShanghaiChina\n\nUniversity of the Chinese Academy of Sciences\nYuquanlu 19A100049BeijingChina\n",
"Tilman Tröster \nInstitute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK\n",
"Erik Verlinde \nInstitute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands\n",
"Manus Visser \nDepartment of Theoretical Physics\nUniversity of Geneva\n24 quai Ernest-Ansermet, 1211 Genève 4Switzerland\n"
]
| [
"Kapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands",
"Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands",
"Institute for Computational Cosmology\nDepartment of Physics\nDurham University\nSouth RoadDH1 3LEDurhamUK",
"Kapteyn Astronomical Institute\nUniversity of Groningen\nPO Box 800NL-9700 AVGroningenthe Netherlands",
"Center for Theoretical Physics\nPolish Academy of Sciences\nLotników 32/4602-668WarsawPoland",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Faculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany",
"Leiden Observatory\nLeiden University\nP.O.Box 95132300RALeidenThe Netherlands",
"School of Physics and Astronomy\nSun Yat-sen University\n519082Guangzhou, Zhuhai CampusP.R. China",
"School of Physics and Astronomy\nSun Yat-sen University\n519082Guangzhou, Zhuhai CampusP.R. China",
"INAF -Osservatorio Astronomico di Capodimonte\nSalita Moiariello 1680131NapoliItaly",
"Faculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Faculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany",
"Faculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany",
"Argelander-Institut für Astronomie\nAuf dem Hügel 7153121BonnGermany",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Centre for Astrophysics and Supercomputing\nSwinburne University of Technology\n3122HawthornVICAustralia",
"Faculty of Physics and Astronomy\nRuhr University Bochum\nAstronomical Institute (AIRUB)\nGerman Centre for Cosmological Lensing\n44780BochumGermany",
"Australian Astronomical Optics\nMacquarie University\n105 Delhi Road",
"North Ryde\n2113NSWAustralia",
"Department of Astrophysical Sciences\nPrinceton University\n4 Ivy Lane08544PrincetonNJUSA",
"Leiden Observatory\nLeiden University\nP.O.Box 95132300RALeidenThe Netherlands",
"Hamburger Sternwarte\nUniversity of Hamburg\nGojenbergsweg 11221029HamburgGermany",
"Shanghai Astronomical Observatory (SHAO)\nNandan Road 80200030ShanghaiChina",
"University of the Chinese Academy of Sciences\nYuquanlu 19A100049BeijingChina",
"Institute for Astronomy\nUniversity of Edinburgh\nRoyal Observatory\nBlackford HillEH9 3HJEdinburghUK",
"Institute for Theoretical Physics\nUniversity of Amsterdam\nScience Park 9041098 XHAmsterdamThe Netherlands",
"Department of Theoretical Physics\nUniversity of Geneva\n24 quai Ernest-Ansermet, 1211 Genève 4Switzerland"
]
| []
| We present measurements of the radial gravitational acceleration around isolated galaxies, comparing the expected gravitational acceleration given the baryonic matter (g bar ) with the observed gravitational acceleration (g obs ), using weak lensing measurements from the fourth data release of the Kilo-Degree Survey (KiDS-1000). These measurements extend the radial acceleration relation (RAR), traditionally measured using galaxy rotation curves, by 2 decades in g obs into the low-acceleration regime beyond the outskirts of the observable galaxy. We compare our RAR measurements to the predictions of two modified gravity (MG) theories: modified Newtonian dynamics (MOND) and Verlinde's emergent gravity (EG). We find that the measured relation between g obs and g bar agrees well with the MG predictions. In addition, we find a difference of at least 6σ between the RARs of early-and late-type galaxies (split by Sérsic index and u − r colour) with the same stellar mass. Current MG theories involve a gravity modification that is independent of other galaxy properties, which would be unable to explain this behaviour, although the EG theory is still limited to spherically symmetric static mass models. The difference might be explained if only the early-type galaxies have significant (M gas ≈ M ) circumgalactic gaseous haloes. The observed behaviour is also expected in Λ-cold dark matter (ΛCDM) models where the galaxy-to-halo mass relation depends on the galaxy formation history. We find that MICE, a ΛCDM simulation with hybrid halo occupation distribution modelling and abundance matching, reproduces the observed RAR but significantly differs from BAHAMAS, a hydrodynamical cosmological galaxy formation simulation. Our results are sensitive to the amount of circumgalactic gas; current observational constraints indicate that the resulting corrections are likely moderate. Measurements of the lensing RAR with future cosmological surveys (such as Euclid) will be able to further distinguish between MG and ΛCDM models if systematic uncertainties in the baryonic mass distribution around galaxies are reduced. | 10.1051/0004-6361/202040108 | [
"https://arxiv.org/pdf/2106.11677v1.pdf"
]
| 235,593,358 | 2106.11677 | 366cf1dd20e30f0dea70587d80525022476d0aa4 |
The weak lensing radial acceleration relation: Constraining modified gravity and cold dark matter theories with KiDS-1000
June 23, 2021 22 Jun 2021
Margot M Brouwer [email protected]
Kapteyn Astronomical Institute
University of Groningen
PO Box 800NL-9700 AVGroningenthe Netherlands
Institute for Theoretical Physics
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
Kyle A Oman
Kapteyn Astronomical Institute
University of Groningen
PO Box 800NL-9700 AVGroningenthe Netherlands
Institute for Computational Cosmology
Department of Physics
Durham University
South RoadDH1 3LEDurhamUK
Edwin A Valentijn
Kapteyn Astronomical Institute
University of Groningen
PO Box 800NL-9700 AVGroningenthe Netherlands
Maciej Bilicki
Center for Theoretical Physics
Polish Academy of Sciences
Lotników 32/4602-668WarsawPoland
Catherine Heymans
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Faculty of Physics and Astronomy
Ruhr University Bochum
Astronomical Institute (AIRUB)
German Centre for Cosmological Lensing
44780BochumGermany
Henk Hoekstra
Leiden Observatory
Leiden University
P.O.Box 95132300RALeidenThe Netherlands
Nicola R Napolitano
School of Physics and Astronomy
Sun Yat-sen University
519082Guangzhou, Zhuhai CampusP.R. China
Nivya Roy
School of Physics and Astronomy
Sun Yat-sen University
519082Guangzhou, Zhuhai CampusP.R. China
Crescenzo Tortora
INAF -Osservatorio Astronomico di Capodimonte
Salita Moiariello 1680131NapoliItaly
Angus H Wright
Faculty of Physics and Astronomy
Ruhr University Bochum
Astronomical Institute (AIRUB)
German Centre for Cosmological Lensing
44780BochumGermany
Marika Asgari
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Jan Luca Van Den Busch
Faculty of Physics and Astronomy
Ruhr University Bochum
Astronomical Institute (AIRUB)
German Centre for Cosmological Lensing
44780BochumGermany
Andrej Dvornik
Faculty of Physics and Astronomy
Ruhr University Bochum
Astronomical Institute (AIRUB)
German Centre for Cosmological Lensing
44780BochumGermany
Thomas Erben
Argelander-Institut für Astronomie
Auf dem Hügel 7153121BonnGermany
Benjamin Giblin
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Alister W Graham
Centre for Astrophysics and Supercomputing
Swinburne University of Technology
3122HawthornVICAustralia
Hendrik Hildebrandt
Faculty of Physics and Astronomy
Ruhr University Bochum
Astronomical Institute (AIRUB)
German Centre for Cosmological Lensing
44780BochumGermany
Andrew M Hopkins
Australian Astronomical Optics
Macquarie University
105 Delhi Road
North Ryde
2113NSWAustralia
Arun Kannawadi
Department of Astrophysical Sciences
Princeton University
4 Ivy Lane08544PrincetonNJUSA
Konrad Kuijken
Leiden Observatory
Leiden University
P.O.Box 95132300RALeidenThe Netherlands
Jochen Liske
Hamburger Sternwarte
University of Hamburg
Gojenbergsweg 11221029HamburgGermany
Huanyuan Shan
Shanghai Astronomical Observatory (SHAO)
Nandan Road 80200030ShanghaiChina
University of the Chinese Academy of Sciences
Yuquanlu 19A100049BeijingChina
Tilman Tröster
Institute for Astronomy
University of Edinburgh
Royal Observatory
Blackford HillEH9 3HJEdinburghUK
Erik Verlinde
Institute for Theoretical Physics
University of Amsterdam
Science Park 9041098 XHAmsterdamThe Netherlands
Manus Visser
Department of Theoretical Physics
University of Geneva
24 quai Ernest-Ansermet, 1211 Genève 4Switzerland
The weak lensing radial acceleration relation: Constraining modified gravity and cold dark matter theories with KiDS-1000
June 23, 2021 22 Jun 2021Astronomy & Astrophysics manuscript no. RAR_paper_Brouwer_4 Received ...; Accepted ... A&A proofs: manuscript no. RAR_paper_Brouwer_4Gravitational lensing: weak -Methods: statistical -Surveys -Galaxies: haloes -Cosmology: dark mattertheory - Gravitation Article numberpage 1 of 31
We present measurements of the radial gravitational acceleration around isolated galaxies, comparing the expected gravitational acceleration given the baryonic matter (g bar ) with the observed gravitational acceleration (g obs ), using weak lensing measurements from the fourth data release of the Kilo-Degree Survey (KiDS-1000). These measurements extend the radial acceleration relation (RAR), traditionally measured using galaxy rotation curves, by 2 decades in g obs into the low-acceleration regime beyond the outskirts of the observable galaxy. We compare our RAR measurements to the predictions of two modified gravity (MG) theories: modified Newtonian dynamics (MOND) and Verlinde's emergent gravity (EG). We find that the measured relation between g obs and g bar agrees well with the MG predictions. In addition, we find a difference of at least 6σ between the RARs of early-and late-type galaxies (split by Sérsic index and u − r colour) with the same stellar mass. Current MG theories involve a gravity modification that is independent of other galaxy properties, which would be unable to explain this behaviour, although the EG theory is still limited to spherically symmetric static mass models. The difference might be explained if only the early-type galaxies have significant (M gas ≈ M ) circumgalactic gaseous haloes. The observed behaviour is also expected in Λ-cold dark matter (ΛCDM) models where the galaxy-to-halo mass relation depends on the galaxy formation history. We find that MICE, a ΛCDM simulation with hybrid halo occupation distribution modelling and abundance matching, reproduces the observed RAR but significantly differs from BAHAMAS, a hydrodynamical cosmological galaxy formation simulation. Our results are sensitive to the amount of circumgalactic gas; current observational constraints indicate that the resulting corrections are likely moderate. Measurements of the lensing RAR with future cosmological surveys (such as Euclid) will be able to further distinguish between MG and ΛCDM models if systematic uncertainties in the baryonic mass distribution around galaxies are reduced.
Introduction
It has been known for almost a century that the outer regions of galaxies rotate faster than would be expected from Newtonian dynamics based on their luminous, or 'baryonic', mass (Kapteyn 1922;Oort 1932Oort , 1940Babcock 1939). This was also demonstrated by Gottesman et al. (1966) and Bosma (1981) through measurements of hydrogen profiles at radii beyond the optical discs of galaxies, and by Rubin (1983) through measurements of galactic rotation curves within the optical discs. The excess gravity implied by these measurements has generally been attributed to an unknown and invisible substance named dark matter (DM), a term coined more than 40 years prior by Zwicky (1933) when he discovered the so-called missing mass problem through the dynamics of galaxies in clusters. More recently, new methods such as weak gravitational lensing (Hoekstra et al. 2004;Mandelbaum et al. 2006;Clowe et al. 2006;Heymans et al. 2013;von der Linden et al. 2014), baryon acoustic oscillations (Eisenstein et al. 2005;Blake et al. 2011), and the cosmic microwave background (CMB; de Bernardis et al. 2000;Spergel et al. 2003;Planck XVI 2014) have contributed unique evidence to the missing mass problem.
Among many others, these observations have contributed to the fact that cold dark matter 1 (CDM) has become a key ingredient of the current standard model of cosmology: the ΛCDM model. In this paradigm, CDM accounts for a fraction Ω CDM = 0.266 of the critical density ρ crit = 3H 2 0 /8πG in the Universe, while baryonic matter only accounts for Ω bar = 0.049 (Planck VI 2020). The cosmological constant Λ, which is necessary to explain the accelerated expansion of the Universe (Riess et al. 1998;Perlmutter et al. 1999) and is a special case of dark energy (DE), accounts for the remaining Ω Λ = 0.685 in our flat space-time (de Bernardis et al. 2000).
Although the ΛCDM model successfully describes the observations on a wide range of scales, no conclusive direct evidence for the existence of DM particles has been found so far (despite years of enormous effort; for an overview, see Bertone et al. 2005;Bertone & Tait 2018). Combined with other current open questions in physics, such as the elusive unification of general relativity (GR) with quantum mechanics and the mysterious nature of DE, this leaves room for alternative theories of gravity. Two modified gravity (MG) theories that do not require the existence of particle DM are modified Newtonian dynamics (MOND; Milgrom 1983) and the more recent theory of emergent gravity (EG; Verlinde 2017). In these theories all gravity is due to the baryonic matter (or, in the case of EG, the interaction between baryons and the entropy associated with DE). Hence, one of the main properties of these theories is that the mass discrepancy in galaxies correlates strongly with their baryonic mass distribution.
Such a correlation has indeed been observed, such as via the Tully-Fisher relation (Tully & Fisher 1977) between the luminosity of a spiral galaxy and its asymptotic rotation velocity (Pierce & Tully 1988;Bernstein et al. 1994). This relation was later generalised as the baryonic Tully-Fisher relation (McGaugh et al. 2000;McGaugh 2012) to include non-stellar forms of baryonic matter. Even earlier, astronomers had found a strong correlation between the observed rotation velocity as a function of galaxy radius v obs (r) and the enclosed luminous mass M bar (< r) (Sanders 1986(Sanders , 1996McGaugh 2004;Sanders & Noordermeer 2007;Wu & Kroupa 2015). Since M bar (< r) corresponds to the expected gravitational acceleration g bar (r) from baryonic matter, and the observed gravitational acceleration can be calculated through g obs (r) = v 2 obs (r)/r, this relation has also been named the radial acceleration relation (RAR) 2 . McGaugh et al. (2016, hereafter M16) in particular measured the RAR with unprecedented accuracy, using the Spitzer Photometry and Accurate Rotation Curves (SPARC; Lelli et al. 2016) data of 153 late-type galaxies. Their results again showed a tight correlation between g obs and g bar , which they could describe using a simple double power law (eq. 4 in M16) that depends only on g bar and one free parameter: the acceleration scale g † where Newtonian gravity appears to break down. This rekindled the interest of scientists working on alternative theories of gravity (Lelli et al. 2017a,b;Burrage et al. 2017;Li et al. 2018;O'Brien et al. 2019), but also of those seeking an explanation of the RAR within the ΛCDM framework, employing correlations between the masses, sizes, and DM content of galaxies (Di Cintio & Lelli 2016;Keller & Wadsley 2017;Desmond 2017;Ludlow et al. 2017;Navarro et al. 2017;Tenneti et al. 2018). Navarro et al. (2017, hereafter N17) used a range of simplifying assumptions based on galaxy observations and DM simulations in order to create an analytical galaxy model including the baryonic and halo components. With this model they reconstruct the RAR inside galaxy discs, in particular the value of a 0 , the acceleration scale where the relation transitions from the baryondominated to the DM-dominated regime (which is equivalent to g † ), and a min , the minimum acceleration probed by galaxy discs. Based on their results, they claim that the RAR can be explained within the ΛCDM framework at the accelerations probed by galaxy rotation curves (within the galaxy disc, i.e. g obs > a min ). However, since their model relies on the fact that luminous kinematic tracers in galaxies only probe a limited radial range, N17 predicted that extending observations to radii beyond the disc (which correspond to lower gravitational accelerations) would lead to systematic deviations from the simple double power law proposed by M16. Although some progress has been made using globular clusters (Bílek et al. 2019b,a;Müller et al. 2021), using kinematic tracers to measure the RAR beyond the outskirts of visible galaxies remains difficult.
The goal of this work is to extend observations of the RAR to extremely low accelerations that cannot currently be detected through galaxy rotation curves or any other kinematic measurement. To this end, we use gravitational lensing: the perturbation of light inside a gravitational potential as described by relativistic theories such as GR. Both weak and strong gravitational lensing were used by Tian et al. (2020) to measure the RAR from observations of 20 galaxy clusters targeted by the CLASH survey. However, due to the high cluster masses, the accelerations probed by these measurements were of the same order as those measurable with galaxy rotation curves. In this work, we use the method of galaxy-galaxy lensing (GGL): the statistical measurement of the coherent image distortion (shear) of a field of background galaxies (sources) by the gravitational potential of a sample of individual foreground galaxies (lenses; for examples, see e.g. Brainerd et al. 1996;Fischer et al. 2000;Hoekstra et al. 2004;Mandelbaum et al. 2006;van Uitert et al. 2016). Using GGL we can measure the average (apparent) density distribution of isolated galaxies up to a radius of 3 Mpc, roughly 100 times larger than the radius of the luminous disc (∼ 30 kpc). At our stellar mass scale of interest -log(M / h −2 70 M ) ≈ 10.5 -this radius corresponds to g bar ≈ 10 −15 m s −2 , which is three orders of magnitude lower than the baryonic accelerations of the M16 rotation curves 3 .
Our main goal is to use the lensing RAR of isolated galaxies at lower accelerations (beyond the observable galaxy disc) to distinguish which of the aforementioned MG and ΛCDM models best describe this result. To achieve this, we first measure the total and baryonic density profiles of our galaxies through their GGL profiles and luminosities. These measurements will be performed using 1006 deg 2 of weak lensing data from the Kilo-Degree Survey (KiDS-1000;de Jong et al. 2013;Kuijken et al. 2019), and nine-band photometric data from KiDS and the VISTA Kilo-Degree Infrared Galaxy Survey (Edge et al. 2013, VIKING). We then translate these measurements into the observed and baryonic radial accelerations, g obs and g bar . Finally, we compare the resulting RAR to predictions from different MG theories (MOND and EG) and ΛCDM. To test the MG theories, we need to make the assumption that the deflection of light by gravitational potentials (as described in GR) holds in these modified theories, which we motivate in the relevant sections. This work can be seen as an extension of Brouwer et al. (2017), where we tested the predictions of EG using KiDS GGL on foreground galaxies from 180 deg 2 of the Galaxy and Mass Assembly (GAMA) survey. Instead of GAMA, we now use a selection of ∼ 1 million foreground galaxies from KiDS-1000 to achieve a fivefold increase in survey area.
The ΛCDM predictions will not only be provided by the N17 analytical model, but also by mock galaxy catalogues based on two different DM simulations. One is the Marenostrum Institut de Ciències de l'Espai (MICE) Galaxy and Halo Light-cone catalogue Hoffmann et al. 2015), which is based on the MICE Grand Challenge lightcone simulation (Fosalba et al. 2015a,b;Crocce et al. 2015). The other mock galaxy catalogue is based on a suite of large-volume cosmological hydrodynamical simulations, called the BAryons and HAloes of MAssive Systems (BAHAMAS) project (McCarthy et al. 2017).
Having ∼ 1 million foreground galaxies at our disposal allows us to select specific galaxy samples, designed to optimally test the predictions from the aforementioned MG and ΛCDM models. Particularly, we note that the analytical models (MOND, EG and N17) mostly focus on the description of individual, isolated galaxies. In order to test them, we select a sample of galaxies whose GGL profiles are minimally affected by neighbouring galaxies (e.g. satellites) within the radius of our measurement. In contrast, the predictions from simulations can be tested with both isolated and non-isolated galaxy samples.
In addition, our sample of ∼ 350 000 isolated lens galaxies allows us to analyse the RAR as a function of colour, Sérsic index and stellar mass. Because MG and ΛCDM give different predictions regarding the dependence of the RAR on these observables, this allows us to better distinguish between the different models. Specifically: according to the MOND and EG theories the relation between g bar and g obs should remain fixed in the regime beyond the baryon-dominated galaxy disc, and hence be independent of galaxy observables. Within the ΛCDM paradigm, the relation between g bar and g obs is related to the stellar-to-halo-mass relation (SHMR) that is not necessarily constant as a function of galaxy stellar mass or other observables.
Our paper is structured as follows: In Section 2 we describe the methodology behind the GGL measurements and their conversion into the RAR, in addition to the theoretical predictions to which we compare our observations: MOND, EG and the N17 analytical DM model. In Section 3 we introduce the KiDS-1000 and GAMA galaxy surveys used to perform both the GGL and stellar mass measurements. Section 4 describes the MICE and BAHAMAS simulations and mock galaxy catalogues to which we compare our results. In Section 5 we present our lensing RAR measurements and compare them to the different models, first using all isolated galaxies and then separating the galaxies by different observables. Section 6 contains the discussion and conclusion. In Appendix A we validate our isolated galaxy selection, and Appendix B contains a description of the piecewise-powerlaw method of translating the lensing measurement into g obs . Finally, Appendix C shows the comparison of the N17 analytical DM model with our lensing RAR.
Throughout this work we adopt the WMAP 9-year (Hinshaw et al. 2013) cosmological parameters: Ω m = 0.2793, Ω b = 0.0463, Ω Λ = 0.7207, σ 8 = 0.821 and H 0 = 70 km s −1 Mpc −1 , which were used as the basis of the BAHAMAS simulation. When analysing the MICE simulations we use the cosmological parameters used in creating MICE, which are: Ω m = 0.25, σ 8 = 0.8, Ω Λ = 0.75, and H 0 = 70 km s −1 Mpc −1 . Throughout the paper we use the reduced Hubble constant h 70 = H 0 /(70 km s −1 Mpc −1 ). Due to the relatively low redshift of our lens galaxies (z ∼ 0.2) the effect of differences in the cosmological parameters on our results is small.
Theory
Mass measurements with weak gravitational lensing
To estimate the gravitational acceleration around galaxies we used GGL: the measurement of the coherent image distortion of a field of background galaxies (sources) by the gravitational potential of a sample of foreground galaxies (lenses). Because the individual image distortions are very small (only ∼ 1% compared to the galaxy's unknown original shape), this method can only be performed statistically for a large sample of sources. We averaged their projected ellipticity component tangential to the direction of the lens galaxy, t , which is the sum of the intrinsic tangential ellipticity component int t and the tangential shear γ t caused by weak lensing. Assuming no preferential alignment in the intrinsic galaxy shapes ( int t = 0), the average t is an estimator for γ t . By measuring this averaged quantity in circular annuli around the lens centre, we obtained the tangential shear profile γ t (R) as a function of projected radius R. Because our final goal is to compute the observed gravitational acceleration g obs as a function of that expected from baryonic matter g bar , we chose our R-bins such that they corresponded to 15 logarithmic bins between 1×10 −15 < g bar < 5×10 −12 m s −2 . For each individual lens the calculation of these g bar -bins was based on the baryonic mass of the galaxy M gal (see Section 3.3). In real space this binning approximately corresponds to the distance range used in Brouwer et al. (2017): 0.03 < R < 3 h −1 70 Mpc. The lensing shear profile can be related to the physical excess surface density (ESD, denoted ∆Σ) profile through the critical surface density Σ crit :
∆Σ(R) = Σ crit γ t (R) = Σ (< R) − Σ(R) ,(1)
which is the surface density Σ(R) at projected radius R, subtracted from the average surface density Σ (< R) within R. See Section 3.1 for more information on how this is computed.
The error values on the ESD profile were estimated by the square-root of the diagonal of the analytical covariance matrix, which is described in section 3.4 of Viola et al. (2015). The full covariance matrix was calculated based on the contribution of each individual source to the ESD profile, and incorporates the correlation between sources that contribute to the ESD in multiple bins, both in projected distance R and in galaxy observable.
The radial acceleration relation (RAR)
After measuring the lensing profile around a galaxy sample, the next step is to convert it into the corresponding RAR. We started from the ESD as a function of projected radius ∆Σ(R) and the measured stellar masses of the lens galaxies M , aiming to arrive at their observed radial acceleration g obs as a function of their expected baryonic radial acceleration g bar . The latter can be calculated using Newton's law of universal gravitation:
g(r) = G M(< r) r 2 ,(2)
which defines the radial acceleration g in terms of the gravitational constant G and the enclosed mass M(< r) within spherical radius r. Assuming spherical symmetry here is reasonable, given that for lensing measurements thousands of galaxies are stacked under many different angles to create one average halo profile. The calculation of g bar requires the enclosed baryonic mass M bar (< r) of all galaxies. We discuss our construction of M bar (< r) in Section 3.3. The calculation of g obs requires the enclosed observed mass M obs (< r) of the galaxy sample, which we obtained through the conversion of our observed ESD profile ∆Σ(R).
When calculating g obs we started from our ESD profile measurement, which consists of the value ∆Σ(R) measured in a set of radial bins. At our measurement radii (R > 30 h −1 70 kpc) the ESD is dominated by the excess gravity, which means the contribution from baryonic matter can be neglected. We adopted the simple assumption that our observed density profile ρ obs (r) is roughly described by a Singular Isothermal Sphere (SIS) model:
ρ SIS (r) = σ 2 2Gπr 2 .(3)
The SIS is generally considered to be the simplest parametrisation of the spatial distribution of matter in an astronomical system (such as galaxies, clusters, etc.). If interpreted in a ΛCDM context, the SIS implies the assumption that the DM particles have a Gaussian velocity distribution analogous to an ideal gas that is confined by their combined spherically symmetric gravitational potential, where σ is the total velocity dispersion of the particles. In a MG context, however, the SIS profile can be considered to represent a simple r −2 density profile as predicted by MOND and EG in the low-acceleration regime outside a baryonic mass distribution, with σ as a normalisation constant. The ESD derived from the SIS profile is:
∆Σ SIS (R) = σ 2 2GR .(4)
From Brouwer et al. (2017) we know that, despite its simple form, it provides a good approximation of the GGL measurements around isolated galaxies. The SIS profile is therefore wellsuited to analytically model the total enclosed mass distribution of our lenses, which can then be derived as follows:
M SIS (< r) = 4π r 0 ρ SIS (r )r 2 dr = 2σ 2 r G .(5)
Now, for each individual observed ESD value ∆Σ obs,m at certain projected radius R m , we assumed that the density distribution within R m is described by an SIS profile with σ normalised such that ∆Σ SIS (R m ) = ∆Σ obs,m . Under this approximation, we combined equations 4 and 5 to give a relation between the lensing measurement ∆Σ and the deprojected, spherically enclosed mass M obs :
M obs (< r) = 4∆Σ obs (r) r 2 .(6)
Through Eq. 2, this results in a very simple expression for the observed gravitational acceleration:
g obs (r) = G [4∆Σ obs (r) r 2 ] r 2 = 4G∆Σ obs (r) .(7)
Throughout this work, we have used the SIS approximation to convert the ESD into g obs . In Section 4.4 we validate this approach by comparing it to a more elaborate method and testing both on the BAHAMAS simulation.
The RAR with modified Newtonian dynamics
With his theory, MOND, Milgrom (1983) postulated that the missing mass problem in galaxies is not caused by an undiscovered fundamental particle, but that instead our current gravitational theory should be revised. Since MOND is a nonrelativistic theory, performing GGL measurements to test it requires the assumption that light is curved by a MONDian gravitational potential in the same way as in GR. This assumption is justified since Milgrom (2013, while testing the MOND paradigm using GGL data from the Canada-France-Hawaii Telescope Lensing survey), states that non-relativistic MOND is a limit of relativistic versions that predict that gravitational potentials determine lensing in the same way as Newtonian potentials in GR. For this reason GGL surveys can be used as valuable tools to test MOND and similar MG theories, as was done for instance by Tian et al. (2009) using Sloan Digital Sky Survey (SDSS) and Red-sequence Cluster Survey data. MOND's basic premise is that one can adjust Newton's second law of motion (F = ma) by inserting a general function µ(a/a 0 ), which only comes into play when the acceleration a of a test mass m is much smaller than a critical acceleration scale a 0 . This function predicts the observed flat rotation curves in the outskirts of galaxies, while still reproducing the Newtonian behaviour of the inner disc. In short, the force F becomes:
F(a) = m µ a a 0 a , µ(x 1) ≈ 1 , µ(x 1) ≈ x .(8)
This implies that a a 0 represents the Newtonian regime where F N = m a N as expected, while a a 0 represents the 'deep-MOND' regime where F MOND = m a 2 MOND /a 0 . In a circular orbit, this is reflected in the deep-MOND gravitational acceleration g MOND ≡ a MOND as follows:
F MOND = m a 2 MOND a 0 = G Mm r 2 → g MOND = a 0 GM r 2 .(9)
This can be written in terms of the expected baryonic acceleration g bar = GM/r 2 as follows:
g MOND (g bar ) = √ a 0 g bar .(10)
This demonstrates that MOND predicts a very simple relation for the RAR: g obs = g bar in the Newtonian regime (g obs a 0 ) and Eq. 9 in the deep-MOND regime (g obs a 0 ). However, since µ(a/a 0 ), also known as the interpolating function, is not specified by Milgrom (1983), there is no specific constraint on the behaviour of this relation in between the two regimes. In the work of Milgrom & Sanders (2008), several families of interpolation functions are discussed. Selecting the third family (given by their eq. 13) with constant parameter α = 1/2, provides the function that M16 later used to fit to their measurement of the RAR using rotation curves of 153 galaxies. This relation can be written as:
g obs (g bar ) = g bar 1 − e − √ g bar /a 0 ,(11)
where a 0 ≡ g † corresponds to the fitting parameter constrained by M16 to be g † = 1.20 ± 0.26 × 10 −10 m s −2 . Since Eq. 11 (equal to eq. 4 in M16) is also considered a viable version of the MOND interpolation function by Milgrom & Sanders (2008), we will consider it the baseline prediction of MOND in this work. As the baseline value of a 0 , we will likewise use the value of g † measured by M16 since it exactly corresponds to the value of a 0 = 1.2 × 10 −10 m s −2 considered canonical in MOND since its first measurement by Begeman et al. (1991), using the rotation curves of 10 galaxies. One of the main characteristics of the MOND paradigm, is that it gives a direct and fixed prediction for the total acceleration based only on the system's baryonic mass, given by Eq. 11. The main exception to this rule is the possible influence by neighbouring mass distributions through the external field effect (EFE), predicted by Milgrom (1983) and studied analytically, observationally and in simulations by Banik & Zhao (2015); Banik et al. (2020); Chae et al. (2020). Since we explicitly selected isolated galaxies in this work (see Appendix A), this effect is minimised as much as possible. However, since total isolation cannot be guaranteed, a small EFE might remain. In order to describe this effect, we used eq. 6 from Chae et al. (2020):
g MOND (g bar ) = ν e (z) g bar ,(12)
with:
ν e (z) = 1 2 − A e z + 1 2 − A e z 2 + B e z .(13)
Here z ≡ g bar /g † , A e ≡ e(1 + e/2)/(1 + e), and B e ≡ (1 + e). The strength of the EFE is parametrised through: e = g ext /g † , determined by the external gravitational acceleration g ext . Although the interpolation functions differ, the result of Eq. 13 corresponds almost exactly to the M16 fitting function given in Eq. 11 in the limit e = 0 (no EFE). Positive values of e result in reduced values of the predicted g obs at very low accelerations (see Fig. 4 in Section 5.2, and fig. 1 of Chae et al. 2020). It should be noted that this fitting function represents an idealised model and could be subject to deviations in real, complex, 3D galaxies.
The RAR with emergent gravity
The work of Verlinde (2017, V17 hereafter), which is embedded in the framework of string theory and holography, shares the view that the missing mass problem is to be solved through a revision of our current gravitational theory. Building on the ideas from Jacobson (1995Jacobson ( , 2016; Padmanabhan (2010); Verlinde (2011); Faulkner et al. (2014), V17 abandons the notion of gravity as a fundamental force. Instead, it emerges from an underlying microscopic description of space-time, in which the notion of gravity has no a priori meaning. V17 shows that constructing an EG theory in a universe with a negative cosmological constant ('anti-de Sitter') allows for the re-derivation of Einstein's laws of GR. A distinguishing feature of V17 is that it attempts to describe a universe with a positive cosmological constant ('de Sitter'), that is, one that is filled with a DE component. This results in a new volume law for gravitational entropy caused by DE, in addition to the area law normally used to retrieve Einsteinian gravity. According to V17, energy that is concentrated in the form of a baryonic mass distribution causes an elastic response in the entropy of the surrounding DE. This results in an additional gravitational component at scales set by the Hubble acceleration scale a 0 = cH 0 /6. Here c is the speed of light, and H 0 is the current Hubble constant that measures the Universe's expansion velocity.
Because this extra gravitational component aims to explain the effects usually attributed to DM, it is conveniently expressed as an apparent dark matter (ADM) distribution:
M 2 ADM (r) = cH 0 r 2 6G d [M bar (r)r] dr .(14)
Thus the ADM distribution is completely defined by the baryonic mass distribution M bar (r) as a function of the spherical radius r, and a set of known physical constants. Since we measured the ESD profiles of galaxies at projected radial distances R > 30 h −1 70 kpc, we can follow Brouwer et al. (2017) in assuming that their baryonic component is equal to the stars+cold gas mass enclosed within the minimal measurement radius (for further justification of this assumption, see Section 4.3). This is equivalent to describing the galaxy as a point mass M bar , which allows us to simplify Eq. 14 to:
M ADM (r) = cH 0 M bar 6 G r .(15)
Now the total enclosed mass M EG (r) = M bar + M ADM (r) can be used to calculate the gravitational acceleration g EG (r) predicted by EG, as follows:
g EG (r) = GM EG (r) r 2 = GM bar r 2 + cH 0 6 √ GM bar r .(16)
In terms of the expected baryonic acceleration g bar (r) = GM bar /r 2 , this simplifies even further to:
g EG (g bar ) = g bar + cH 0 6 √ g bar .(17)
We emphasise that Eq. 14 is only a macroscopic approximation of the underlying microscopic phenomena described in V17, and is thus only valid for static, spherically symmetric and isolated baryonic mass distributions. For this reason, we selected only the most isolated galaxies from our sample (see Appendix A), such that our GGL measurements are not unduly influenced by neighbouring galaxies. Furthermore, the current EG theory is only valid in the acceleration range g bar < a 0 , often called the deep-MOND regime. Therefore, the prediction of Eq. 17 should be taken with a grain of salt for accelerations g bar > 1.2 × 10 −10 m s −2 . This will not affect our analysis since weak lensing takes place in the weak gravity regime. In addition, cosmological evolution of the H 0 parameter is not yet implemented in the theory, restricting its validity to galaxies with relatively low redshifts. However, we calculated that at our mean lens redshift, z ∼ 0.2, using an evolving H(z) would result in only a ∼ 5% difference in our ESD measurements, based on the background cosmology used in this work.
In order to test EG using the standard GGL methodology, we needed to assume that the deflection of photons by a gravitational potential in this alternative theory corresponds to that in GR. This assumption is justified because, in EG's original (anti-de Sitter) form, Einstein's laws emerge from its underlying description of space-time. The additional gravitational force described by ADM does not affect this underlying theory, which is an effective description of GR. Therefore, we assumed that the gravitational potential of an ADM distribution produces the same lensing shear as an equivalent distribution of actual matter.
The RAR in ΛCDM
To help guide an intuitive interpretation of the lensing RAR within the framework of the ΛCDM theory, we made use of the simple model of N17, which combines a basic model of galactic structure and scaling relations to predict the RAR. We refer to N17 for a full description, but give a summary here. A galaxy of a given stellar (or baryonic -there is no distinction in this model) mass occupies a DM halo of a mass fixed by the abundance matching relation of Behroozi et al. (2013). The dark halo concentration is fixed to the cosmological mean for haloes of that mass (Ludlow et al. 2014). The baryonic disc follows an exponential surface density profile with a half-mass size fixed to 0.2× the scale radius of the dark halo. This model is sufficient to specify the cumulative mass profile of both the baryonic and dark components of the model galaxy; calculating g obs and g bar is then straightforward. However, since the N17 model is merely a simple analytical description, our main ΛCDM test utilised more elaborate numerical simulations (see Section 4).
Data
The Kilo-Degree Survey (KiDS)
We measured the gravitational potential around a sample of foreground galaxies (lenses), by measuring the image distortion (shear) of a field of background galaxies (sources). These sources were observed using OmegaCAM (Kuijken 2011): a 268-million pixel CCD mosaic camera mounted on the Very Large Telescope (VLT) Survey Telescope (Capaccioli & Schipani 2011). Over the past ten years these instruments have performed KiDS, a photometric survey in the ugri bands, which was especially designed to perform weak lensing measurements (de Jong et al. 2013).
GGL studies with KiDS have hitherto been performed in combination with the spectroscopic GAMA survey (see Section 3.2), with the KiDS survey covering 180 deg 2 of the GAMA area. Although the final KiDS survey will span 1350 deg 2 on the sky, the current state-of-the-art is the 4 th Data Release (KiDS-1000; Kuijken et al. 2019) containing observations from 1006 square-degree survey tiles. We therefore used a photometrically selected 'KiDS-bright' sample of lens galaxies from the full KiDS-1000 release, as described in Section 3.3. The measurement and calibration of the source shapes and photometric redshifts are described in Kuijken et al. (2019); Giblin et al. (2021) and Hildebrandt et al. (2021).
The measurements of the galaxy shapes are based on the rband data since this filter was used during the darkest time (moon distance > 90 deg) and with the best atmospheric seeing con-ditions (< 0.8 arcsec). The r-band observations were co-added using the Theli pipeline (Erben et al. 2013). From these images the galaxy positions were detected through the SExtractor algorithm (Bertin & Arnouts 1996). After detection, the shapes of the galaxies were measured using the lensfit pipeline (Miller et al. 2007, which includes a self-calibration algorithm based on Fenech Conti et al. (2017) that was validated in Kannawadi et al. (2019). Each shape is accompanied by a lensfit weight w s , which was used as an estimate of the precision of the ellipticity measurement.
For the purpose of creating the photometric redshift and stellar mass estimates, 9 bands were observed in total. The ugri bands were observed by KiDS, while the VIKING survey (Edge et al. 2013) performed on the VISTA telescope adds the ZY JHK s bands. All KiDS bands were reduced and co-added using the Astro-WISE pipeline (AW; McFarland et al. 2013). The galaxy colours, which form the basis of the photometric redshift measurements, were measured from these images using the Gaussian Aperture and PSF pipeline (GAaP; Kuijken 2008;Kuijken et al. 2015).
The addition of the lower frequency VISTA data allowed us to extend the redshift estimates out to 0.1 < z B < 1.2, where z B is the best-fit photometric redshift of the sources (Benítez 2000;Hildebrandt et al. 2012). However, when performing our lensing measurements (see Section 2.1) we used the total redshift probability distribution function n(z s ) of the full source population. This n(z s ) was calculated using a direct calibration method (see Hildebrandt et al. 2017 for details), and circumvents the inherent bias related to photometric redshift estimates of individual sources.
We note that this is a different redshift calibration method than that used by the KiDS-1000 cosmology analyses Heymans et al. 2021;Tröster et al. 2020), who used a self-organising map to remove (primarily high-redshift) sources whose redshifts could not be accurately calibrated due to incompleteness in the spectroscopic sample (Wright et al. 2020;Hildebrandt et al. 2021). Following Robertson et al. (in prep.) we prioritised precision by analysing the full KiDS-1000 source sample (calibrated using the direct calibration method) since percent-level biases in the mean source redshifts do not significantly impact our analysis.
For the lens redshifts z l , we used the ANNz2 (Artificial Neural Network) machine-learning redshifts of the KiDS foreground galaxy sample (KiDS-bright; see Section 3.3). We implemented the contribution of z l by integrating over the individual redshift probability distributions p(z l ) of each lens. This p(z l ) is defined by a normal distribution centred at the lens' z ANN redshift, with a standard deviation: σ z /(1 + z) = 0.02 (which is equal to the standard deviation of the KiDS-bright redshifts compared to their matched spectroscopic GAMA redshifts). For the source redshifts z s we followed the method used in Dvornik et al. (2018), integrating over the part of the redshift probability distribution n(z s ) where z s > z l . In addition, sources only contribute their shear to the lensing signal when z B + ∆z > z l -when the sum of their best-fit photometric redshift z B and the redshift buffer ∆z = 0.2 is greater than the lens redshift. Hence, when performing the lensing measurement in Section 2.1 the critical surface density 4 (the conversion factor between γ t and ∆Σ, whose inverse is also called the lensing efficiency) was calculated as fol-lows:
Σ −1 crit = 4πG c 2 ∞ 0 D(z l ) ∞ z l D(z l , z s ) D(z s ) n(z s ) dz s p(z l ) dz l .(18)
Here D(z l ) and D(z s ) are the angular diameter distances to the lens and the source respectively, and D(z l , z s ) the distance between them. The constant multiplication factor is defined by Newton's gravitational constant G and the speed of light c. The ESD profile was averaged (or 'stacked') for large samples of lenses to increase the signal-to-noise (S /N) ratio of the lensing signal. We defined a lensing weight W ls that depends on both the lensfit weight w s and the lensing efficiency Σ −1 crit :
W ls = w s Σ −1 crit,ls 2 ,(19)
and used it to optimally sum the measurements from all lenssource pairs into the average ESD:
∆Σ = 1 1 + µ ls W ls t,ls Σ crit,ls ls W ls .(20)
Here the factor (1+µ) calibrates the shear estimates (Fenech Conti et al. 2017;Kannawadi et al. 2019). Extending the method of Dvornik et al. (2017) to the higher KiDS-1000 redshifts, µ denotes the mean multiplicative calibration correction calculated in 11 linear redshift bins between 0.1 < z B < 1.2 from the individual source calibration values m:
µ = s w s m s s w s ,(21)
The value of this correction is µ ≈ 0.014, independent of the projected distance from the lens. We also corrected our lensing signal for sample variance on large scales by subtracting the ESD profile measured around ∼ 5 million uniform random coordinates, 50 times the size of our total KiDS-bright sample. These random coordinates mimic the exact footprint of KiDS, excluding the areas masked by the 'nine-band no AW-r-band' mask that we applied to the KiDSbright lenses (see Section 3.3). In order to create random redshift values that mimic the true distribution, we created a histogram of the KiDS-bright redshifts divided into 80 linear bins between 0.1 < z ANN < 0.5. In each bin, we created random redshift values equal to the number of real lenses in that bin. Because of the large contiguous area of KiDS-1000, we found that the random ESD profile is very small at all projected radii R, with a mean absolute value of only 1.85 ± 0.75% of the lensing signal of the full sample of isolated KiDS-bright galaxies.
The Galaxy and Mass Assembly (GAMA) survey
Although the most contraining RAR measurements below were performed using exclusively KiDS-1000 data, the smaller set of foreground galaxies observed by the spectroscopic GAMA survey functions both as a model and validation sample for the KiDS foreground galaxies. The survey was performed by the Anglo-Australian Telescope with the AAOmega spectrograph, and targeted more than 238 000 galaxies selected from the Sloan Digital Sky Survey (Abazajian et al. 2009, SDSS;). For this study we used GAMA II observations (Liske et al. 2015) from three equatorial regions (G09, G12, and G15) range, the measured ESD profiles are expected to be approximately stationary in proper coordinates. containing more than 180 000 galaxies. These regions span a total area of ∼ 180 deg 2 on the sky, completely overlapping with KiDS.
GAMA has a redshift range of 0 < z < 0.5, with a mean redshift of z = 0.22. The survey has a redshift completeness of 98.5% down to Petrosian r-band magnitude m r,Petro = 19.8 mag. We limited our GAMA foreground sample to galaxies with the recommended redshift quality: n Q ≥ 3. Despite being a smaller survey, GAMA's accurate spectroscopic redshifts were highly advantageous when measuring the lensing profiles of galaxies (see Section 2.1). The GAMA redshifts were used to train the photometric machine-learning (ML) redshifts of our larger sample of KiDS foreground galaxies (see Section 3.3). Also, in combination with its high redshift completeness, GAMA allows for a more accurate selection of isolated galaxies. We therefore checked that the results from the KiDS-only measurements are consistent with those from KiDS-GAMA.
To measure the RAR with KiDS-GAMA, we need individual stellar masses M for each GAMA galaxy. We used the Taylor et al. (2011) stellar masses, which are calculated from ugrizZY spectral energy distributions 5 measured by SDSS and VIKING by fitting them with Bruzual & Charlot (2003) Stellar Population Synthesis (SPS) models, using the Initial Mass Function (IMF) of Chabrier (2003). Following the procedure described by Taylor et al. (2011), we accounted for flux falling outside the automatically selected aperture using the 'flux-scale' correction.
Selecting isolated lens galaxies with accurate redshifts and stellar masses
Because of its accurate spectroscopic redshifts, the GAMA lenses would be an ideal sample for the selection of isolated galaxies and the measurement of accurate stellar masses (as was done in Brouwer et al. 2017). However, since the current KiDS survey area is > 5 times larger than that of GAMA, we selected a KiDS-bright sample of foreground galaxies from KiDS-1000 that resembles the GAMA survey. We then used the GAMA redshifts as a training sample to compute neural-net redshifts for the KiDS-bright lenses (see e.g. Bilicki et al. 2018), from which accurate stellar masses could subsequently be derived. The details of the specific sample used in this work are provided in Bilicki et al. (2021). Here we give an overview relevant for this paper.
To mimic the magnitude limit of GAMA (m r,Petro < 19.8 mag), we applied a similar cut to the (much deeper) KiDS survey. Because the KiDS catalogue does not contain Petrosian magnitudes we used the Kron-like elliptical aperture r-band magnitudes from SExtractor, calibrated for r-band extinction and zero-point offset 6 , which have a very similar magnitude distribution. Through matching the KiDS and GAMA galaxies and seeking the best trade-off between completeness and purity, we decided to limit our KiDS-bright sample to m r,auto < 20.0. In addition we removed KiDS galaxies with a photometric redshift z > 0.5, where GAMA becomes very incomplete.
To remove stars from our galaxy sample, we applied a cut based on galaxy morphology, nine-band photometry and the SExtractor star-galaxy classifier 7 . Through applying the IMAFLAGS_ISO=0 flag, we also removed galaxies that are af-5 The spectral energy distributions were constrained to the rest frame wavelength range 3 000 − 11 000 Å. 6 MAG_AUTO_CALIB = MAG_AUTO + DMAG − EXTINCTION_R 7 Our star-galaxy separation corresponds to applying the following flags: SG2DPHOT=0, SG_FLAG=1, CLASS_STAR<0.5. fected by readout and diffraction spikes, saturation cores, bad pixels, or by primary, secondary or tertiary haloes of bright stars 8 . We applied the recommended mask that was also used to create the KiDS-1000 shear catalogues 9 . In addition, objects that are not detected in all 9 bands were removed from the sample. Our final sample of KiDS-bright lenses consists of ∼ 1 million galaxies, more than fivefold the number of GAMA galaxies. This increased lens sample allowed us to verify the results from Brouwer et al. (2017) with increased statistics, and to study possible dependencies of the RAR on galaxy observables.
To use the KiDS-bright sample as lenses to measure g obs , we needed accurate individual redshifts for all galaxies in our sample. These photometric redshifts z ANN were derived from the full nine-band KiDS+VIKING photometry by training on the spectroscopic GAMA redshifts (see Section 3.2) using the ANNz2 (Artificial Neural Network) machine learning method (Sadeh et al. 2016). When comparing this z ANN to the spectroscopic GAMA redshifts z G measured for the same galaxies, we found that their mean offset (z ANN − z G )/(1 + z G ) = 9.3 × 10 −4 . However, this offset is mainly caused by the low-redshift galaxies: z ANN < 0.1. Removing these reduces the mean offset to δz/(1 + z G ) = −6 × 10 −5 , with a standard deviation σ z = σ(δz) = 0.026. This corresponds to a redshift-dependent deviation of σ z /(1 + z ANN ) = 0.02 based on the mean redshift z ANN = 0.25 of KiDS-bright between 0.1 < z < 0.5, which is the lens redshift range used throughout this work for all lens samples.
In order to measure the expected baryonic acceleration g bar , we computed the KiDS-bright stellar masses M based on these ANNz2 redshifts and the nine-band GAaP photometry. Because the GAaP photometry only measures the galaxy magnitude within a specific aperture size, the stellar mass was corrected using the 'fluxscale' parameter 10 The stellar masses were computed using the LePhare algorithm (Arnouts et al. 1999;Ilbert et al. 2006), which performs SPS model fits on the stellar component of the galaxy spectral energy distribution. We used the Bruzual & Charlot (2003) SPS model, with the IMF from Chabrier (2003, equal to those used for the GAMA stellar masses). LePhare provides both the best-fit logarithmic stellar mass value 'MASS_BEST' of the galaxy template's probability distribution function, and the 68% confidence level upper and lower limits. We used the latter to estimate the statistical uncertainty on M . For both the upper and lower limit, the mean difference with the best-fit mass is approximately:
| log 10 M lim /M best | ≈ 0.06 dex.
Another way of estimating the statistical uncertainty in the stellar mass is to combine the estimated uncertainties from the input: the redshifts and magnitudes. The redshift uncertainty σ z / z G = 0.11 corresponds to an uncertainty in the luminosity distance of: σ(δD L )/ D L = 0.12. We took0 the flux F to remain constant between measurements, such that: 4πD 2 L F ∝ D 2 L ∝ L. Assuming that approximately L ∝ M leads to an estimate: which finally gives our adopted stellar mass uncertainty resulting from the KiDS-bright redshifts: log 10 (1 + δM /M ) = 0.11 dex. The uncertainty resulting from the KiDS-bright magnitudes is best estimated by comparing two different KiDS apparent magnitude measurements: the elliptical aperture magnitudes 'MAG_AUTO_CALIB' from SExtractor and the Sérsic magnitudes 'MAG_2dphot' from 2DPHOT (La Barbera et al. 2008). The standard deviation of their difference, δm = m 2dphot − m calib , is σ(δm) = 0.69, which corresponds to a flux ratio of F 2dphot /F calib = 1.88 (or 0.27 dex). Using the same assumption, now taking D L to remain constant, results in: 4πD 2 L F ∝ F ∝ L ∝ M . This means our flux ratio uncertainty directly corresponds to our estimate of the M uncertainty. Quadratically combining the 0.11 dex uncertainty from the redshifts and the 0.27 dex uncertainty from the magnitudes gives an estimate of the total statistical uncertainty on the stellar mass of ∼ 0.29 dex. This is much larger than that from the LePhare code. Taking a middle ground between these two, we have assumed twice the LePhare estimate: σ M = 0.12 dex. However, we have confirmed that using the maximal estimate σ M = 0.29 dex throughout our analysis does not change the conclusions of this work, in particular those of Section 5.4.
M + δM M = D L (z) + D L (z + δz) 2 D L (z) 2 ,(22)
When comparing M ,ANN with the GAMA stellar masses M ,G of matched galaxies, we found that its distribution is very similar, with a standard deviation of 0.21 dex around the mean. Nevertheless there exists a systematic offset of log(M ,ANN ) − log(M ,G ) = −0.056 dex, which is caused by the differences in the adopted stellar mass estimation methods. In general, it has been found impossible to constrain stellar masses to within better than a systematic uncertainty of ∆M ≈ 0.2 dex when applying different methods, even when the same SPS, IMF and data are used (Taylor et al. 2011;Wright et al. 2017). We therefore normalised the M ,ANN values of our KiDS-bright sample to the mean M ,G of GAMA, while indicating throughout our results the range of possible bias due to a ∆M = 0.2 dex systematic shift in M . We estimated the effect of this bias by computing the RAR with log 10 (M ) ± ∆M as upper and lower limits.
In order to compare our observations to the MG theories, the measured lensing profiles of our galaxies should not be significantly affected by neighbouring galaxies, which we call 'satellites'. We defined our isolated lenses (Appendix A) such that they do not have any satellites with more than a fraction f M ≡ M ,sat /M ,lens of their stellar mass within a spherical radius r sat (where r sat was calculated from the projected and redshift distances between the galaxies). We chose f M = 0.1, which corresponds to 10% of the lens stellar mass, and r sat = 3 h −1 70 Mpc, which is equal to the maximum projected radius of our measurement. In short:
r sat ( f M > 0.1) > 3 h −1 70 Mpc.
We also restricted our lens stellar masses to M < 10 11 h −2 70 M since galaxies with higher masses have significantly more satellites (see Section 2.2.3 of Brouwer et al. 2017). This provided us with an isolated lens sample of 259 383 galaxies. We provide full details of our choice of isolation criterion and an extensive validation of the isolated galaxy sample in Appendix A. Based on tests with KiDS, GAMA and MICE data we found that this is the optimal isolation criterion for our data. The ESD profile of our isolated sample is not significantly affected by satellite galaxies and that our sample is accurate to ∼ 80%, in spite of it being flux-limited. Using the MICE simulation we also estimated that the effect of the photometric redshift error is limited.
Simulations
In order to compare our observations to ΛCDM-based predictions, we used two different sets of simulations: MICE and BA-HAMAS. Here MICE is an N-body simulation, which means that galaxies are added to the DM haloes afterwards, while BA-HAMAS is a hydrodynamical simulation that incorporates both stars and gas through sub-grid physics. MICE, however, has a simulation volume at least two orders of magnitude larger than BAHAMAS. Below we explain the details of each simulation, and how we utilised their unique qualities for our analysis.
MICE mock catalogues
The MICE N-body simulation contains ∼ 7 × 10 10 DM particles in a (3072 h −1 70 Mpc) 3 comoving volume (Fosalba et al. 2015a). From this simulation the MICE collaboration constructed a ∼ 5000 deg 2 lightcone with a maximum redshift of z = 1.4. The DM haloes in this lightcone were identified using a Friend-of-Friend algorithm on the particles. These DM haloes were populated with galaxies using a hybrid halo occupation distribution (HOD) and halo abundance matching (HAM) prescription Crocce et al. 2015). The galaxy luminosity function and colour distribution of these galaxies were constructed to reproduce local observational constraints from SDSS (Blanton et al. 2003b(Blanton et al. ,a, 2005.
In the MICECATv2.0 catalogue 11 , every galaxy had sky coordinates, redshifts, comoving distances, apparent magnitudes and absolute magnitudes assigned to them. Of the total MICE lightcone we used 1024 deg 2 , an area similar to the KiDS-1000 survey. We used the SDSS apparent r-band magnitudes m r as these most closely match those from KiDS (see Brouwer et al. 2018). We could therefore limit the MICE galaxies to the same apparent magnitude as the KiDS-bright sample: m r < 20 mag, in order to create a MICE foreground galaxy (lens) sample. We used the same redshift limit: 0.1 < z < 0.5, resulting in a mean MICE lens redshift z = 0.23, almost equal to that of GAMA and KiDS-bright within this range. The absolute magnitudes of the mock galaxies go down to M r − 5 log 10 (h 100 ) < −14 mag, which corresponds to the faintest GAMA and KiDSbright galaxies. Each galaxy was also assigned a stellar mass M , which is needed to compute the RAR (see Section 2.2). These stellar masses were determined from the galaxy luminosities L using Bell & de Jong (2001) M /L ratios.
In addition, each galaxy had a pair of lensing shear values associated with it (γ 1 and γ 2 , with respect to the Cartesian coordinate system). These shear values were calculated from healpix weak lensing maps that were constructed using the 'onion shell method' (Fosalba et al. 2008(Fosalba et al. , 2015b. The lensing map of MICE-CATv2.0 has a pixel size of 0.43 arcmin. We did not use MICE results within a radius R res corresponding to 3 times this resolution. We calculated R res and the corresponding g bar using the mean angular diameter distance and baryonic mass of the MICE lens sample. For the full sample of isolated MICE galaxies these values are: R res = 0.25 h −1 70 Mpc and g bar = 6.60 × 10 −14 m s −2 . At scales larger than this resolution limit, the MICE shears allowed us to emulate the GGL analysis and conversion to the RAR that we performed on our KiDS-1000 data (as described in Section 2) using the MICE simulation. To create a sample of MICE background galaxies (sources) for the lensing analysis, we applied limits on the MICE mock galaxies' redshifts and apparent magnitudes, which are analogous to those applied to the KiDS source sample: 0.1 < z < 1.2, m r > 20 (see Hildebrandt et al. 2017 and Section 3.1; uncertainties in the KiDS z B are not accounted for in this selection). We also applied an absolute magnitude cut of M r > −18.5 mag, in order to reproduce the KiDS source redshift distribution more closely.
The MICE mock catalogue also features very accurate clustering. At lower redshifts (z < 0.25) the clustering of the mock galaxies as a function of luminosity was constructed to reproduce the Zehavi et al. (2011) clustering observations, while at higher redshifts (0.45 < z < 1.1) the MICE clustering was validated against the Cosmic Evolution Survey (COSMOS; Ilbert et al. 2009). The accurate MICE galaxy clustering allowed us to analyse the RAR at larger scales (> 0.3 h −1 70 Mpc) where clustered neighbouring galaxies start to affect the lensing signal. MICE also allowed us to test our criteria defining galaxy isolation (see Appendix. A).
BAHAMAS mock catalogue
The second set of simulations that we utilised is BAHAMAS (McCarthy et al. 2017). The BAHAMAS suite are smoothedparticle hydrodynamical realisations of (400 h −1 100 Mpc) 3 volumes and include prescriptions for radiative cooling and heating, ionising background radiation, star formation, stellar evolution and chemical enrichment, (kinetic wind) supernova feedback, supermassive black hole accretion, and merging and thermal feedback from active galactic nuclei (AGN). The simulations were calibrated to reproduce the stellar and hot gas content of massive haloes, which makes them particularly well suited for our study of the matter content around haloes out to distances of 1-3 h −1 70 Mpc. The masses of DM and baryonic resolution elements are 3.85×10 9 h −1 100 M and 7.66×10 8 h −1 100 M respectively, and the gravitational softening is fixed at = 4 h −1 100 kpc = 5.71 h −1 70 kpc. Haloes and galaxies were identified in the simulations using the friends-of-friends (Davis et al. 1985) and Subfind (Springel et al. 2001;Dolag et al. 2009) algorithms. We labeled the most massive sub-halo in each Friend-of-Friend group as the 'central' and other sub-haloes as 'satellites'. We constructed an 'isolated' galaxy sample by restricting the selection to central sub-haloes that have no other sub-haloes (satellites or centrals) more massive than 10% of their mass within 3 h −1 70 Mpc. We randomly selected 100 galaxies per 0.25 dex bin in M 200 between 10 12 and 10 13.5 h −2 70 M . In the last two bins there were fewer than 100 candidates, so we selected them all. All galaxies have a redshift z = 0.25. For each selected galaxy we constructed an integrated surface density map, integrated along the line-of-sight for ±15 comoving h −1 100 Mpc around the target halo. We also extracted the cumulative spherically averaged mass profile of each target sub-halo, decomposed into DM, stars, and gas. For both the maps and profiles, we included mass contributions from all surrounding (sub)structures: we did not isolate the haloes from their surrounding environment.
We used the integrated surface density map of each galaxy to calculate its mock ESD profile as a function of the projected distance R from the lens centre, in order to mimic the effect of GGL and the conversion to the RAR on the BAHAMAS results. Each pixel on these maps corresponds to 15 comoving h −1 100 kpc, which in our physical units is: 15/(1 + z) 0.7 −1 h −1 70 kpc = 17.14 h −1 70 kpc. The density maps each have a dimensionality of 400 × 400 pixels. Hence the total area of each map is (6.86 h −1 70 Mpc) 2 . In calculating the lensing profiles and RAR with BAHAMAS we followed, as closely as possible, the GGL procedure and conver-sion to the RAR as described in Section 2. We truncated our lensing profiles at 10 times the gravitational softening length: 10 = 0.057 h −1 70 Mpc, to avoid the numerically poorly converged central region (Power et al. 2003). For a typical galaxy in our sample of isolated BAHAMAS galaxies, this corresponds to g bar ∼ 2.38 × 10 −12 m s −2 .
The BAHAMAS RAR: Quantifying the missing baryon effect
The calculation of the expected baryonic radial acceleration g bar requires the enclosed baryonic mass M bar (< r) within a spherical radius r around the galaxy centre. Since we are dealing with measurements around isolated galaxies at R > 30 h −1 70 kpc, we can approximate M bar (< r) as a point mass M gal mainly composed of the mass of the lens galaxy itself. M gal can be subdivided into stars and gas, and the latter further decomposed into cold and hot gas.
How we obtained the stellar masses of our GAMA, KiDSbright, MICE and BAHAMAS galaxies is described in Sections 3 and 4. From these M values, the fraction of cold gas f cold = M cold /M can be estimated using scaling relations based on H i and CO observations. Following Brouwer et al. (2017) we used the best-fit scaling relation found by Boselli et al. (2014), based on the Herschel Reference Survey (Boselli et al. 2010):
log( f cold ) = −0.69 log(M / h −2 70 M ) + 6.63 .(23)
We applied this equation to all observed and simulated values of M in order to arrive at the total galaxy mass:
M gal = M + M cold = M (1 + f cold ).
The spatial distribution of the stellar and cold gas mass are similar (Pohlen et al. 2010;Crocker et al. 2011;Mentuch Cooper et al. 2012;Davis et al. 2013) and can therefore be considered a single mass distribution, especially for the purposes of GGL, which only measures the ESD profile at scales larger than the galaxy disc (R > 30 h −1 70 kpc). We illustrate this in Fig. 1, which shows the enclosed mass profiles (upper panel) and RAR (lower panel) for different baryonic components in the BAHAMAS simulation. For these mock galaxies, the stellar mass within 30 h −1 70 kpc (red star) gives a good approximation of the M distribution across all radii that we consider. We therefore modeled the baryonic mass of our galaxies as a point mass M gal , containing both the stellar and cold gas mass.
We recognise that the total baryonic mass distribution M bar of galaxies may include a significant amount of additional mass at larger distances, notably in the hot gas phase. This is illustrated in Fig. 1. In the upper panel, we show the average baryonic mass profile for BAHAMAS galaxies with 1 < M 200 /(10 12 h −2 70 M ) < 3. In addition, we show an estimate of the typical baryonic mass profile for galaxies in the same mass range, based on an extrapolation to larger radii of the compilation of observations in Tumlinson et al. (2017); including stars, cold gas (< 10 4 K, traced by absorption lines such as H i, Na i and Ca ii), cool gas (10 4 -10 5 K, traced by many UV absorption lines, e.g. Mg ii, C ii, C iii, Si ii, Si iii, N ii, N iii), warm gas (10 5 -10 6 K, traced by C iv, N v, O vi and Ne vii absorption lines), hot gas (> 10 6 K, traced by its X-ray emission) and dust (estimated from the reddening of background QSOs, and Ca ii absorption). The light blue shaded region therefore illustrates a component of missing baryons predicted by these simulations but not (yet) observed, possibly related to the cosmological missing baryons (e.g. Fukugita et al. 1998;Fukugita & Peebles 2004;Shull et al. 2012). There are several possibilities: (i) there may be additional gas present in a difficult-to-observe phase (e.g. hot, low-density gas, see for instance Nicastro et al. 2018); (ii) the simulations do not accurately Tumlinson et al. (2017). In the inner galaxy the discrepancy (light blue shaded region) between the observed and simulated M bar is relatively small, but in the outer galaxy the majority of the baryons predicted to be present in BA-HAMAS consist of currently unobserved, missing baryons. The orange dashed line shows the expected baryonic mass profile if the baryon density is everywhere equal to a fixed fraction f b = Ω b /Ω m of the local DM density. At large enough radii ( 2 h −1 70 Mpc), the baryon-to-DM ratio converges to the cosmic average. Lower panel: As in upper panel, but in acceleration space. The cosmic baryon fraction provides a strong theoretical upper limit on g bar at low accelerations in the context of the ΛCDM cosmology. reflect reality, for example: galaxies may eject substantially more gas from their surroundings than is predicted by these simulations; (iii) there may be less baryonic matter in the Universe than expected in the standard cosmology based on big bang nucleosynthesis (BBN; Kirkman et al. 2003) calculations and CMB measurements (Spergel et al. 2003;Planck XVI 2014).
The lower panel of Fig. 1 illustrates the magnitude of the resulting systematic uncertainties in g bar . In the ΛCDM cosmology, the expectation at sufficiently large radii is given by (Hinshaw et al. 2013). BAHAMAS, and generically any ΛCDM galaxy formation simulation, converges to this density at low enough accelerations (large enough radii). The most optimistic extrapolation of currently observed baryons falls a factor of ∼ 3 short of this expectation, while the stellar mass alone is a further factor of ∼ 3 lower. The unresolved uncertainty around these missing baryons is the single most severe limitation of our analysis. Given that we are interested in both ΛCDM and alternative cosmologies, we will use the stellar+cold gas mass M gal as our fiducial estimate of the total baryonic mass M bar , which is translated into the baryonic acceleration g bar , throughout this work. This serves as a secure lower limit on g bar . We note that the eventual detection, or robust non-detection, of the missing baryons has direct implications for the interpretation of the results presented in Section 5. In Section 5.2 we address the possible effect of extended hot gas haloes on g bar . We discuss this issue further in Section 6.
g obs = f −1 b g bar where f b is the cosmic baryon fraction f b = Ω b /Ω m = 0.17
Concerning g obs , omitting the contribution of hot gas will not have a large effect on the prediction within the ΛCDM framework (e.g. from simulations) since the total mass distribution at the considered scales is heavily dominated by DM. Within MG frameworks such as EG and MOND, where the excess gravity is sourced by the baryonic matter, it is slightly more complicated. (Brouwer et al. 2017, see section 2.2) carefully modelled the distribution of all baryonic components, based on observations from both GAMA and the literature, including their effect on the excess gravity in the EG framework. They found that, for galaxies with M < 10 11 h −2 70 M , the contribution to the ESD profile (and hence to g obs ) from hot gas and satellites was small compared to that of the stars and cold gas. Although this analysis was done for the EG theory, the effect of these extended mass distributions within MOND are similar or even less. This allows us to use a point mass M gal as a reasonable approximation for the baryonic mass distribution M bar (< r) within our measurement range when computing g obs as predicted by MOND and EG (see Section 2.3 and 2.4).
The BAHAMAS RAR: Testing the ESD to RAR conversion
We used BAHAMAS to test the accuracy of our SIS method (outlined in Section 2.2) in estimating g obs from our GGL measurement of ∆Σ obs , by comparing it against the more sophisticated piece-wise power law (PPL) method outlined in Appendix B. As a test system, we used the 28 galaxies from our BAHAMAS sample with 10 13 < M 200 /( h −2 70 M ) < 10 13.1 . We combined these into a stacked object by averaging the individual ESD profiles as derived from their mock lensing maps. The stacked ESD as measured from the lensing mocks is shown in the left panel of Fig. 2. Since the mock ESD profiles are derived from convergence maps (rather than the shapes of background galaxies), they have no associated measurement uncertainty -for simplicity, we assumed a constant 0.1 dex uncertainty, which is similar to that for the KiDS measurements. We also combined the spherically averaged enclosed mass profiles of the galaxies out to 3 h −1 70 Mpc by averaging them. From this average mass profile we analytically calculated the ESD profile shown in the left panel of Fig. 2. We found that the ∆Σ calculated from the spherically averaged mass profile is ∼ 0.05 dex higher than the direct measurement of the stacked lensing mocks. This primarily results from the fact that the spherically averaged mass profile does not take into account the additional matter outside the 3 h −1 70 Mpc spherical aperture, whereas the mock surface density maps are integrated along the line-of-sight for ±15 comoving h −1 100 Mpc around the lens. The PPL method described in Appendix B attempts to reproduce the ESD profile by converging to an appropriate volume density profile. The resulting recovered ESD profile and its 68% confidence interval is shown with blue points and error bars in the left panel of Fig. 2 -the fit to the mock data is excellent. In the centre panel we show the enclosed mass profile as recovered by both the PPL and SIS methods, in addition to the true enclosed mass profile. Both estimators recover the profile within their stated errors. The PPL method systematically underestimates it by ∼ 0.1 dex across most of the radial range. This is directly caused by the difference between the spherically averaged and mock lensing ESD profiles (left panel). The somewhat wider confidence intervals at small radii are caused by the lack of information in the mock data as to the behaviour of the profile at r < 30 h −1 70 kpc; the PPL model marginalises over all possibilities. Once the enclosed mass is dominated by the contribution at radii covered by the measurement, the uncertainties shrink. To account for the added uncertainty resulting from the conversion to the RAR, we added 0.1 dex to the error bars of our RAR measurements throughout this work.
The SIS method instead slightly underestimates the enclosed mass at small radii, and overestimates it at large radii. The apparent improved performance relative to the PPL method is actually due to a fortuitous partial cancellation of two errors. First, the SIS calculation suffers from the same underestimation of the spherically averaged enclosed mass profile as the PPL method, due to the difference between the mock lensing and spherically averaged ESD profiles. However, in addition to this, the SIS method assumes a density profile ρ(r) ∝ r −2 at all radii. At small radii, the power-law slope is in reality about −2.1. This results in a slight overestimate of the enclosed mass, which partially compensates the underestimate described above, resulting in a net underestimate. At larger radii, the slope of the density profile becomes progressively steeper, such that the assumption of an r −2 profile increasingly overestimates the enclosed mass, eventually resulting in a net overestimate.
The right panel of Fig. 2 illustrates the resulting uncertainty in the measurement of the RAR. To focus on the influence of the method used to recover g obs , we simply used the exact spherically averaged stellar mass profile to calculate g , plotted on the x-axis 12 . We found that, for mock lenses within the BAHAMAS simulation, both the SIS and the PPL method yield acceptable and consistent estimates of g obs . We note that the BAHAMAS g obs (g ) is significantly offset from the RAR as measured by M16; we will return to this point when we compare BAHAMAS to our observations in Section 5.3.
A&A proofs: manuscript no. RAR_paper_Brouwer_4 Fig. 2. Illustration of the recovery of the acceleration profile from simulated weak lensing observations. Left: Average ESD profile of a subset of our sample of BAHAMAS galaxies with 10 13 < M 200 /( h −2 70 M ) < 10 13.1 , derived from the spherically averaged mass profile (red line) and the mock lensing maps (yellow line, with an assumed 0.1 dex Gaussian uncertainty). The PPL method recovery of the ESD profile is shown with the blue points; error bars represent 68% confidence intervals. Centre: The SIS (light blue squares) and PPL (dark blue points) method recover the spherically averaged enclosed mass profile. The uncertainties on the SIS points are derived by sampling the uncertainties on the mock lensing ESD profile. Right: The resulting dynamical acceleration profile g obs and uncertainties, plotted as a function of the acceleration due to stars g = GM (< r)/r 2 .
Results
Tables containing the ESD profile data used to create all results figures (i.e. Figures 3, 4, 5, 8, 9, 10, A.4 and C.1) can be found at: http://kids.strw.leidenuniv.nl/sciencedata.php.
Lensing rotation curves
As a final consistency check between the SIS assumption and the PPL method, we applied both methods to the true KiDS-1000 data. Since these methods are only used to convert ∆Σ(R) into g obs (r), we can leave g bar out of the comparison and plot our results as a function of R. An observable closely related to the RAR that is usually plotted as a function of radius, is the traditional circular velocity curve: v circ (r) = GM obs (< r) r ,
an observable that indeed served as input to the M16 RAR measurement. We applied the SIS method described in Section 2.2 to convert our ESD profiles ∆Σ(R) into v circ (R) since substituting Eq. 6 into Eq. 24 gives:
v circ (r) = G (4∆Σ(r) r 2 ) r = 4G ∆Σ(r) r .(25)
We also applied Eq. 24 to compute v circ (R) from the M(< R) calculated through the PPL method described in Appendix B. We note that both the SIS and PPL method assume spherical symmetry, while in simulations DM haloes are found to deviate from sphericity, which could lead to deviations in the lensing rotation curves (Cuddeford 1993). However, the mean ellipticity of haloes is observed to be small ( | | = 0.174 ± 0.046, Schrabback et al. 2021). The stacking of thousands of lenses with approximately random orientations further reduces the impact on the lensing signal, which means the halo ellipticity will not significantly change our results. Fig. 3 shows the lensing rotation curves for isolated KiDSbright galaxies, divided into four stellar mass bins using the fol- Showing the data in this way allows us to observe for the first time in this intuitive manner how the circular velocity curves of isolated galaxies continue beyond the observable disc (r > 30 h −1 70 kpc). In addition, it provides a consistency check against the SPARC rotation curves ) that form the basis for the M16 RAR measurement. It is remarkable how well the mean of the SPARC rotation curves and our lensing results correspond at their intersection (r ∼ 30 h −1 70 kpc). But most importantly, we find that the 'lensing rotation curves' from the SIS assumption are consistent with the ones from the PPL method. Although the SIS assumption results in slightly more scatter, there is very little systematic bias between the results from the two methods, which have a fractional difference of log(v circ,SIS /v circ,PPL ) = 0.017 dex. Since this measurement is merely a different way of presenting the observed acceleration, which equals g obs (r) = v 2 circ /r, we can easily compute that the expected difference in g obs would be log(g obs,SIS /g obs,PPL ) = 0.038 dex.
The consistency between the two conversion methods allows us to use the SIS assumption throughout this work. The great advantage of this method is that it allows us to convert GGL profiles binned by baryonic acceleration ∆Σ(g bar ), into the RAR: g obs (g bar ). This is not the case for the PPL method, which only works on ∆Σ(R) binned by radius. The former can therefore be applied to any lens sample; the latter only to lenses within a narrow mass range (in order to convert R into g bar using the mean M gal ). As explained in Section 4.4 we added 0.1 dex to the error bars of all RAR measurements in this work, to account for the added uncertainty from the conversion of the ESD to the RAR. After showing that both methods yield acceptable and consistent estimates of g obs , we will show only the SIS measurement when . The black points (with 1σ error bars) show the result calculated using the SIS assumption, while the blue points (with error bars representing the 16th and 84th percentile of the fits) show the result from the more sophisticated PPL method. Our measurements are consistent between the two methods, and also with the rotation curves from SPARC (all data as the blue 2D histogram, the mean as red squares).
The RAR of KiDS compared to MG theories
In Fig. 4 we show the RAR, with the observed radial acceleration computed from our lensing measurements through Eq. 7) on the y-axis. The x-axis shows the expected baryonic (star+cold gas) radial acceleration, where the label serves as a reminder throughout this work that g bar is only computed from the measured stellar masses of the galaxies and an estimate of their cold gas component.
The lensing g obs was measured using the GAMA and KiDSbright isolated galaxy samples, respectively. Due to its smaller survey area (180 vs. 1006 deg 2 ), the error bars using GAMA lenses are larger than those using KiDS-bright lenses. However, as explained in Appendix A, the spectroscopic redshifts of the GAMA survey allow for a more reliable selection of the isolated lenses compared to KiDS (which measures photometric redshifts with a σ z = 0.02 uncertainty). The effect of this uncertainty on the measured lensing profiles is modelled in Fig. A.3, which shows that the ESD profile of the 'offset' MICE sample diverges from the truly isolated MICE galaxies at radius R > 0.3 h −1 70 Mpc. At these large scales, the effect of satellite galaxies on the lensing signal result in a ∼ 30% increase in ∆Σ due to the contribution of satellite galaxies. We translated this radius into a gravitational acceleration value using Eq. 2, based on the average M gal of the lens sample. In this way we estimate that, for the full sample of isolated KiDS-bright galaxies, the isolation criterion is no longer reliable when g bar 10 −13 m s −2 , as indicated by the light blue shaded region in Fig. 4. We note that the GAMA results, which are based on accurate spectroscopic redshift measurements, are still reliable within this region.
The grey band shows the range of possible bias due to a ∆M = ±0.2 dex systematic shift in stellar mass. We estimated this range by performing our analysis assuming stellar masses that are 0.2 dex higher than, and then 0.2 dex lower than, their best-fitting M values (see Section 3.3). We only show this band once, for the KiDS-bright result, but note that this uncertainty equally affects the GAMA stellar masses (and, indeed, any stellar mass measurement; see Wright et al. 2017).
We compare our results to the M16 RAR measurements (both the full dataset: blue 2D histogram, and the mean: red squares), from SPARC galaxy rotation curves, which cover higher accelerations than our lensing measurements (corresponding to smaller scales: R < 30 h −1 70 kpc). At the highestacceleration end (smallest scales), where g obs is dominated by g bar , they follow a one-to-one relation. At lower accelerations (larger scales) their results quickly diverge from unity, signifying the start of the DM dominated regime. We find that these two fully independent RAR observations, respectively from rotation curves and lensing, are in strong agreement 13 . 13 Because the blinding intended to avoid observer bias in the KiDS-1000 cosmological constraints (Asgari et . Measured RAR, which compares the total gravitational acceleration g obs with the expected baryonic acceleration g bar of galaxies. At high accelerations we show the M16 RAR measurements from galaxy rotation curves (all data as the blue 2D histogram, the mean as red squares). Using weak gravitational lensing we were able to extend this measurement to lower accelerations, using both the spectroscopic GAMA and the photometric KiDS-bright isolated lens samples (blue and black points with 1σ error bars). Comparing our lensing observations to two MG models: MOND (the M16 fitting function; grey solid line) and EG (assuming a point mass; red dashed line) we find that GAMA results are in agreement with the two models, while those from KiDS-bright are systematically higher. At very low accelerations (corresponding to R > 0.3 h −1 70 Mpc, light blue shaded region) the uncertainty in the photometric KiDS redshifts affects the isolated lens selection, resulting in systematically higher values of g obs due to the possible contribution of satellites. The results from the spectroscopic GAMA survey, however, are still reliable within this region. The impact of stellar mass uncertainty (∆M = 0.2 dex) on the measurement is shown as the grey band. We show the MOND prediction including the EFE (with e = 0.003, see Eq. 13) as the grey dashed line. In addition, we show the effect on the RAR of KiDS-bright galaxies if g bar contained an additional isothermal hot gas contribution within a 100 h −1 70 kpc radius, with a nominal gas mass equal to the stellar mass (orange crosses with 1σ error bars). We emphasise that this is only a rough order of magnitude estimate of the possible effect of gaseous haloes, which are extremely difficult to observe. Fig. 4 also compares the two MG models, EG and MOND, to our lensing results (for a comparison of these two models with the RAR from SPARC, see Lelli et al. 2017a). As explained in Sections 2.3 and 2.4, we took the MOND prediction to be equal to the extrapolated M16 fitting function (Eq. 11), and that of EG as the prediction from Verlinde (2017) for a point mass (Eq. 17). At high accelerations, the prediction from EG appears to lie above that of MOND and the SPARC data. However, as explained in Section 2.4, the prediction of Eq. 17 should be taken with a grain of salt for accelerations g bar > 1.2 × 10 −10 m s −2 . Within our measurement range, the two predictions are almost indistinguishable. Both models are compatible with the GAMA data. The KiDS-bright data points, however, lie systematically above the MG predictions. Tröster et al. 2020) only has a small effect on GGL observations, this agreement has been present since the start of our analysis (before the data were un-blinded).
To quantify the level of agreement between the acceleration predicted by the different models g mod and the observed g obs , we calculated the χ 2 value:
χ 2 = (g obs − g mod ) · C −1 (g obs − g mod ) ,(26)
where C −1 is the inverse of the analytical covariance matrix (see Section 2.1). We divided this quantity by the number of degrees of freedom N DOF of the model, which gives the reduced χ 2 statistic:
χ 2 red = χ 2 N DOF = χ 2 N data − N param .(27)
Here N data is the number of data points in the measurement and N param is the number of free parameters in the model. Since none of the models have free parameters, N DOF is simply the total number of g bar -bins (in this case N data = 15). Comparing the GAMA data to the two MG models results in χ 2 red -values of 0.8 for both MOND and EG, corresponding to a standard deviation of 0.4σ. This confirms that both models agree well with the GAMA data. When using the KiDS-bright results, neither model provides a good description of the data with: χ 2 red = 4.6 and 5.0 for MOND and EG respectively, corresponding to ∼ 6 standard deviations (∼ 6σ). Taking into account the effect of the photometric redshift uncertainty of KiDS-bright by only using the seven data points within the isolation criterion limit (R < 3 h −1 70 Mpc) we find: χ 2 red = 4.0 for MOND and χ 2 red = 4.4 for EG, ∼ 3.8σ away from a good fit. Considering the ∆M = ±0.2 dex uncertainty shown by the grey band (with the data points beyond the isolation criterion limit still removed) leads to χ 2 red = 1.5 for ∆M = +0.2 dex and χ 2 red = 14 for ∆M = −0.2 dex with respect to MOND, with similar results for EG. Thus, the MOND and EG predictions are able to describe our measurements within the statistical and systematic uncertainties. Whether these models are confirmed or excluded relies heavily on the systematic bias in the stellar mass measurements. This highlights the general point that GGL measurements are now so accurate in determining the total observed mass distribution that improving the RAR measurement primarily depends on obtaining better constraints on the baryonic mass distribution.
This point is highlighted further by the fact that we cannot incorporate measurements of the total baryonic mass distribution into our comparison, in particular those components that have not been detected, such as hot gaseous haloes and missing baryons. This remains a fundamental limitation of all work testing DM or MG theories at large scales (see Section 4.3). Although there have been very recent fruitful attempts at a first detection of this barely visible baryonic component (Macquart et al. 2020;Tanimura et al. 2020), there exist no accurate measurements of its distribution around isolated galaxies. However, we can safely continue as long as all estimates of g bar (in the measurements, models and simulations) are based on the same components (in our case: stars+cold gas). This way our RAR results remain purely observational, based on actual measurements along both axes.
However, a qualitative idea of the possible effect of an additional extended ionised gas component on g bar is depicted in Fig. 4. In addition to our standard stars-and-cold-gas point mass used to calculate g bar , we modeled the hot gas as a simple isothermal density profile (ρ(r) ∝ r −2 ), truncated at the accretion radius R acc . Based on Valentijn (1988), we derived that R acc ≈ 100 h −1 70 kpc for hot gas haloes around galaxies with M ≈ 10 11 h −2 70 M . Finding an accurate estimate of the additional gas mass M gas within this radius is no easy matter. Brouwer et al.
(2017) assumed a total hot gas mass M gas = 3M , based on results from the OWLS hydrodynamical simulations by Fedeli et al. (2014). They found that, in simulations with AGN feedback, OWLS galaxies with a total mass M 200 = 10 12 h −1 70 M (corresponding to M ≈ 10 10 h −2 70 M , a lower limit on the typical stellar masses in our sample) have a gas-to-stellar-mass fraction of M gas /M ≈ 3. One of the few observational scaling relations for hot gas is derived by Babyk et al. (2018), using Chandra X-ray observations of 94 early-type galaxies. In their Fig. 7, which shows the X-ray gas mass versus the total galaxy mass, galaxies with M tot = 10 12 M have gas fractions ranging from 0.1 − 1. However, Babyk et al. (2018) measured both M tot and M gas within 5 effective radii of their galaxies, which means that the hot gas fraction on larger scales could be as high as 3 in extreme cases. These relatively high hot gas masses motivated by the Babyk et al. (2018) observations are possibly biased towards a high X-ray surface brightness and are an order of magnitude higher than the hot gas masses presented in Fig. 7 of Tumlinson et al. (2017). As this gas mass outweighs the possible contribution of various cooler gas and dust components, this case provides a good guide for our evaluation. Based on all these considerations, we assumed a nominal gas-to-stellar-mass fraction of M /M gas = 1, emphasising that this is only an order of magnitude estimate due to the challenging nature of observing circumgalactic gas.
In Fig. 4 we include the RAR of KiDS-bright galaxies with our nominal estimate of the hot gas distribution added to g bar on the x-axis. At the highest accelerations measurable by lensing, we find that these results are almost indistinguishable from the original KiDS-bright measurements. As the acceleration decreases, the g bar values including hot gas shift further to the right (higher values) due to the increased enclosed hot gas mass. This causes a steepening downward slope of the RAR, such that it finally diverges from the g obs ∝ √ g bar relation at very low accelerations (g bar < 10 −14 m s −2 ). The same effect is as also seen in the BAHAMAS results in Fig. 1. As expected, we find that this steepening of the RAR increases for higher assumed gaseous halo masses M gas , and decreases for lower values. This implies that, if gaseous haloes more massive than in our example (M gas M ) were detected directly and incorporated into the measurement, the observed RAR would diverge from the current MOND and EG predictions at low accelerations.
In the case of MOND a steep downward slope at low accelerations is not expected unless, despite our best efforts, our isolated galaxy sample is not truly isolated. In that case undetected satellites might cause an external field effect (EFE). To evaluate this effect we use the results of Chae et al. (2020) for the isolated SPARC galaxies. Based on their results, we have assumed e = g ext /g † = 0.003 as a reasonable estimate of the external gravitational acceleration g ext compared to the critical acceleration scale g † (see Section 2.3) for our isolated lenses. We use the fitting function in Eq. 13, which represents the EFE for an idealised model of galaxies within their environment, to depict the EFE on the predicted MOND RAR in Fig. 4. The extrapolated M16 fitting function represents the MOND prediction without any EFE (e = 0). As expected the MOND prediction including the EFE diverges from the one without, tending towards a steeper downward slope at low accelerations (g bar < 10 −12 m s −2 ). Hence the EFE moves the MOND prediction away from our main observational result: the lensing RAR from the KiDS-bright sample without an estimate for the additional hot gas, which we explore throughout the rest of this work. We will therefore maintain the use of the M16 fitting function as our main MOND prediction since this represents the optimal case considering our observations. Regarding the KiDS-bright result including an estimate for the hot gas, it turns out that the steeper downward slope resulting from the MOND EFE is not steep enough to be consistent with our measured RAR including an estimate of the additional hot gas. This is illustrated by the fact that, for our chosen value e = 0.003, the MOND prediction including EFE and our RAR observation including hot gas reach the same value of g obs at g bar ≈ 10 −15 m s −2 . However, the observation reaches this depth within a much smaller span in g bar (−15 < log 10 (g bar / m s −2 ) < −14). Choosing a different value for the EFE strength e does not solve this problem, and the effect becomes stronger for higher assumed values of M gas . It is therefore unlikely that the MOND EFE can explain the effect of massive (M gas M ) hot gaseous haloes, if such haloes are detected. In the case of EG it is not yet known whether and, if so, how external gravitational fields affect its prediction (Verlinde, priv. comm.).
The RAR of KiDS compared to ΛCDM simulations
In this section we compare the KiDS-1000 RAR with numerical ΛCDM simulations 14 . In order to obtain the predictions from these simulations, we applied the same isolation criterion, GGL procedures and RAR conversion to mock galaxy samples from the MICE and BAHAMAS simulations (see Section 4). In Fig. 5, BAHAMAS (orange band) is shown as the median result of all lens galaxies, with the upper and lower limit of the band representing the 16 th and 84 th percentiles. For MICE (red band) we show the result for isolated lenses selected using the true redshifts (lower limit) and using redshifts with a normally distributed random offset of σ z /(1+z) = 0.02 (upper limit), in order to emulate the effect of the redshift uncertainty in KiDS on the isolated galaxy selection (see Appendix A). This means that the upper limit of the MICE prediction is considered reliable even at high accelerations (blue shaded region), where uncertainties in the galaxy isolation could affect the RAR measurement. The RAR observations are the same KiDS-bright lensing and M16 rotation curve results as shown in Fig. 4, this time compared to the predictions from the two simulations.
We find a good agreement between the MICE simulation and our measurements. The MICE measurements are limited to the low g bar regime, owing to the resolution of the MICE simulations. The MICE scale limit of R > 0.25 h −1 70 Mpc is within the angular scale where satellites missed by the isolation criterion might impact the lensing signal (R > 0.3 h −1 70 Mpc, light blue shaded region). The effect of the KiDS-bright redshift uncertainty σ z on the isolation criterion is however mimicked in the MICE simulation (upper limit of the red band), which means we can safely compare MICE with our low-acceleration measurements. The limited width of red band shows that this effect is relatively small (∼ 30%). The MICE prediction (with the σ z offset) results in a reduced χ 2 value of χ 2 red = 2.3, corresponding to 2.3σ. Figure 5 shows poor agreement between the lensing RAR for isolated BAHAMAS galaxies and the KiDS measurement. The reason for this is straightforward to understand: the BAHAMAS measurement in Fig. 5 runs approximately parallel to both the KiDS and MICE curves, as a result of a constant offset in the stellar-to-halo-mass relation (SHMR) between BAHAMAS and MICE. Both simulations reproduce the observed SHMR in an overall sense, as shown in fig. 6 of McCarthy et al. (2017) and Jakobs et al. (2018) for BAHAMAS, and guaranteed by construction as described in Carretero et al. (2015) for MICE. However, while in MICE our isolated galaxy sample follows essentially the same SHMR as the parent sample, in BAHAMAS isolated galaxies have, on average, triple the stellar mass at fixed halo mass compared to the global BAHAMAS galaxy population. This difference fully accounts for the 0.5 dex horizontal offset between the MICE and BAHAMAS curves in Fig. 5. The failure of BAHAMAS to reproduce the observed lensing RAR could therefore be regarded as a possible shortcoming of the galaxy formation model used in those simulations, rather than a general failure of their cosmological paradigm. However, we note that the offset in the SHMR as a function of local galaxy density is theoretically expected, and (indirectly) observed (e.g. Dutton et al. 2010;Correa & Schaye 2020). It is therefore curious that 14 The first ΛCDM model we test is that of N17, but find that this simple analytical model is not sufficient to describe our data (see Appendix C). MICE, which does not reproduce this observed bias, turns out to be in reasonable agreement with our measurements. The discrepancy between KiDS-bright and BAHAMAS must therefore arise due to some more subtle underlying reason that we have yet to identify; we hope to follow this up in future work. We initially selected BAHAMAS for our analysis due to its large volume -required to produce enough of the rare isolated, relatively massive galaxies of interest -and readily available mock lensing data. It will be interesting to revisit the lensing RAR as cosmological hydrodynamical galaxy formation simulations continue to improve in terms of realism, simulated volume, and resolution.
The RAR for early-and late-type KiDS galaxies
The large size of the KiDS-bright lens sample gives us the opportunity to divide our lenses into different samples based on observed galaxy parameters. We determined the RAR for isolated galaxies split into two types based on either parameter: bulgedominated and disc-dominated based on their Sérsic index, and red and blue based on their u − r colour. Although these selections are far from perfect representations of true morphological types, the red and bulge-dominated samples can roughly be identified with canonically early-type (pressure supported) galaxies and the blue and disc-dominated samples with late-type (rotationally supported) galaxies (Driver et al. 2006) 15 .
The r-band Sérsic indices n of all KiDS galaxies with S /N > 50 (following Roy et al. 2018) were measured using the 2DPHOT multi-purpose environment for 2D wide-field image analysis (La Barbera et al. 2008). For the colour split, we used the u and r magnitudes measured using the GAaP pipeline (see Section 3.1). In Fig. 6 the u − r colour versus stellar mass distribution of isolated galaxies shows the split based on Sérsic index, which defines early-type galaxies as those with n > 2 and late-type disc-dominated galaxies as those with n < 2. Based on the u − r magnitude distribution of these two populations, we defined our split by galaxy colour as follows: galaxies with m u − m r > 2.5 mag are defined as red, and those with m u − m r < 2.5 mag as blue.
In both cases, we aimed to select two samples with the same stellar mass distribution, in order to isolate any possible effect of galaxy type on the RAR from that of M . In Fig. 7 we show the M histogram of the two types (in this case based on galaxy colour). From both samples, we removed galaxies until only the overlapping section of both mass distributions remained. Ideally this should give us two samples (red and blue galaxies) with equal stellar mass distributions, shown by the light shaded blue region. Fig. 8 shows the lensing RAR of equal-mass KiDS-bright galaxies split by Sérsic index (left panel) and u − r colour (right panel). For this result, we focus on establishing whether there exists a significant difference between the RAR of the two types. Contrary to previous plots, the effect of a 0.2 dex global systematic bias in M (normally shown by a grey band) is omitted because this affects both measurements in the same way such that their relative difference does not change (the possibility of a colour-or Sérsic index-dependent M bias is discussed below). KiDS-bright isolated lens galaxies (1000 deg 2 ) Systematic stellar mass uncertainty (M * ± 0.2 dex) MICE isolated mock lens galaxies SPARC rotation curves (mean + 2D histogram) BAHAMAS isolated mock lens galaxies MOND (McGaugh+16,extrapolated) Emergent Gravity (Verlinde+16, point mass) Fig. 5. Measured RAR of the KiDS-bright isolated lens sample (black points with 1σ error bars) compared to two ΛCDM simulations: MICE and BAHAMAS. The accelerations where uncertainty in the photometric KiDS redshifts affects the KiDS-bright isolated lens selection is indicated by the light blue shaded region. The MICE results (red band) emulate the effect of the redshift uncertainty in KiDS, while the BAHAMAS results (orange band) reflect the median and 16 th and 84 th percentiles of the simulated lens galaxies. The MICE simulation, though limited to low accelerations by its resolution, succeeds in reproducing the lensing data. The result from the BAHAMAS simulation runs approximately parallel to the MICE curve, but underestimates our measurement by 0.5 dex due to the biased SHMR of the BAHAMAS isolated galaxies (see Section 5.3).
We indeed observe a significant difference between the RAR measurements of early and late galaxy types. To quantify this difference, we measured the reduced χ 2 between the RAR measurements by replacing g obs and g mod in Eq. 26 with g obs,E and g obs,L from the early-type (red or bulge-dominated) and late-type (blue or disc-dominated) galaxy samples. The χ 2 red equals 67.8/15 = 4.5 for the lenses split by Sérsic index, and 134.2/15 = 8.9 for those split by u − r colour. Taking the full covariance matrix into account we find that even the Sérsic index split, which displays the smallest offset, results in RAR difference with a 5.7σ significance. The mean ratio between the RAR measurements of the two types, log 10 (δg E/L obs ) = log 10 g obs,E /g obs,L , is 0.17 dex and 0.27 dex for the Sérsic and colour splits respectively.
We address the question whether the observed difference of the RAR between early and late types could be caused by any bias in the stellar mass. To this end, we estimated the systematic stellar mass bias between the two types, defined as log 10 (δM E/L ) = log 10 ( M E / M L ), that would be required to resolve the difference between their two RAR measurements. When trying to estimate the effect of this bias on the RAR, we had to take into account that δM E/L affects both the estimated acceleration from baryonic mass g bar (directly) and the observed acceleration g obs (indirectly, through the equal-mass selection). The bias in baryonic acceleration scales linearly with the bias in M , such that: log 10 (δg E/L bar ) = log 10 (δM E/L ). Throughout this work, the observed relation between g bar and g obs at the scales measured by lensing has approximately followed g obs ∝ √ g bar .
This means that we can roughly estimate the effect on g obs as: log 10 (δg E/L obs ) ≈ log 10 (δM E/L ) / 2. Since our measured difference δg obs 0.2 dex, this means log 10 (δM E/L ) should be 2 log 10 (δg E/L obs ) = 0.4 dex. That is, the observed difference could be resolved by a systematic stellar mass bias between the two types 0.4 dex. We will now discuss different sources of a possible systematic bias, and estimate whether they could be the cause of the observed difference.
First, the statistical uncertainty in the M measurements could cause a systematic shift in the two M distributions resulting from Eddington bias (Eddington 1913). We estimated the size of this bias by adding a random offset to the true log 10 (M ) measurements of KiDS-bright before selecting the two 'equal' stellar mass distributions for red and blue galaxies. Based on our estimate of the statistical uncertainty in the KiDS-bright M (see Section 3.3), we drew the random offsets from a lognormal distribution with σ = 0.12 dex. When looking at the underlying A&A proofs: manuscript no. RAR_paper_Brouwer_4 Disc-dominated (n< 2) Fig. 6. 2D histogram of the u − r colour and stellar mass of isolated KiDS-bright galaxies. We divide our galaxies into canonically earlyand late-type galaxies, based on either Sérsic index n or u−r magnitude. When dividing by Sérsic index, we define bulge-dominated (early-type) galaxies as those with n > 2 and disc-dominated (late-type) galaxies as those with n < 2 (red and blue points). When dividing colour we define red (early-type) galaxies as those with m u − m r > 2.5 and blue (latetype) galaxies as those with m u − m r < 2.5 (above and below the dashed horizontal line). Number of galaxies Isolated red galaxies Isolated blue galaxies Selected red & blue galaxies Fig. 7. Stellar mass histogram of the red (early-type) and blue (latetype) isolated KiDS-bright galaxies (red and blue lines), divided by u−r colour (m u − m r ≶ 2.5 mag). To isolate the effect of galaxy type on the RAR from that of M , we select two samples with the same stellar mass distribution by randomly removing galaxies from both samples until only the overlapping region (light blue shaded region) remains.
true stellar mass distributions we found that they are indeed not equal, but that the mean stellar masses M ,E and M ,L of the red and blue samples differ by only 0.025 dex. Of course, this method overlooks the fact that the measured M distribution already contains scatter, and is therefore not the true M distribution. Indeed when we apply the random offset multiple times, we see the Eddington bias decrease by ∼ 5% after every iteration. Therefore, the true Eddington bias is likely to be slightly larger, around 0.027 dex. This is still very small compared to the 0.4 dex bias needed, thus it is very unlikely that the difference we observe is caused exclusively by Eddington bias.
Second, there could be systematic errors in the KiDS-bright M measurements that differ between red and blue galaxies (due to e.g. systematic variation of the IMF, SPS model inaccuracies, or systematic errors in the measured redshifts or magnitudes). In order to estimate the size of any systematic biases in the stellar mass, we compared KiDS-bright's M ,ANN with GAMA's M ,G of exactly the same galaxies. Here M ,ANN is based on the nine-band KiDS+VIKING photometry and photometric redshifts z ANN derived by training the ANNz2 (Artificial Neural Network) machine learning method on the spectroscopic GAMA redshifts (see Section 3.3), while M ,G is based on the ugrizZY SDSS+VIKING photometry combined with the spectroscopic GAMA redshifts (see Section 3.2). After selecting our samples of blue and red galaxies with the same M ,ANN distribution as described above, we indeed found that the M ,G distributions are not exactly equal: M E / M L = 1.4, corresponding to 0.14 dex. This indicates that using different sets of observations and models to measure M can cause a systematic bias between red and blue galaxies, but that this effect is too small to reach the 0.4 dex difference in M needed to explain the 0.2 dex difference in the measured RAR.
In conclusion, even when combined the Eddington plus overall systematic measurement bias is at most 0.17 dex, not even half of what is needed. We note that this bias estimation has been carried out using the types split by u − r colour; when split by Sérsic index, the Eddington and other systematic biases between bulge-and disc-dominated galaxies are even smaller (0.021 and 0.12 dex respectively).
Domínguez Sánchez et al. (2019) reported evidence of a varying IMF in massive early-type galaxies. As seen in fig. 19 of their work, this could cause the global mass-to-light-ratio of these galaxies to increase by as much as 0.09 dex compared to a fixed Chabrier IMF. They find this effect only for their high-mass galaxy sample with a stellar mass of at least M > 2 × 10 11 M , and not for their lower-mass sample. Since we limit all our galaxies to M < 10 11 h −2 70 M (see Section 3.3), the varying IMF is not likely to apply to our early-type galaxy sample. However, even if this had been the case, this 0.09 dex difference in M is small compared to the 0.4 dex needed to explain the difference in the RAR of early-and late-type galaxies.
The higher values of g obs for red and bulge-dominated galaxies that we find in Fig. 8 are in qualitative agreement with earlier GGL studies. A recent KiDS-1000 lensing study by Taylor et al. (2020) found that, within a narrow stellar mass range near the knee of the SHMR (M ∼ 2 − 5 × 10 10 h −2 70 M ), galaxy halo mass varied with galaxy colour, specific star formation rate (SSFR), effective radius R e and Sérsic index n. Although not explicitly mentioned, their figures 1 and 6 reveal that their earlytype (red, low-SSFR) galaxies have larger halo masses than their late-type (blue, low-n, high-SSFR) galaxies of the same stellar mass. Sérsic parameter coupling between n and R e , for a fixed galaxy luminosity, may also contribute towards the trends seen among the early-type galaxies in their M halo -n and M halo -R e diagrams 16 . Much earlier Hoekstra et al. (2005) measured the GGL signal of a sample of 'isolated' Red-sequence Cluster Survey galaxies as a function of their rest-frame B-, V-, and R-band luminosity, and found that early-type galaxies have lower stellar Observed radial acceleration log(g u − r colour (split at 2.5 mag)
GL-KiDS isolated blue/disc-dominated galaxies GL-KiDS isolated red/bulge-dominated galaxies SPARC rotation curves (mean + 2D histogram)
MOND (McGaugh+16, extrapolated)
Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar ) Fig. 8. Measured RAR of the KiDS-bright isolated lenses (points with 1σ error bars) divided into canonically early-and late-type galaxies. In the left panel, the lenses are split by Sérsic index (n ≷ 2) into bulge-dominated (red points) and disc-dominated (blue points) galaxies. In the right panel they are split by u − r colour (m u − m r ≷ 2.5) into red and blue galaxies (with correspondingly coloured points). In both panels we find a significant difference between the RAR measurements of early and late galaxy types. The extrapolated MOND and EG predictions (grey solid and red dashed lines) and the SPARC data (red squares with 2D histogram) are shown as a reference. mass fractions. In contrast, Mandelbaum et al. (2006) found no dependence of the halo mass on morphology for a given stellar mass below M < 10 11 M , although they did find a factor of two difference in halo mass between ellipticals and spirals at fixed luminosity.
Finding a significantly different RAR at equal M would have interesting implications for galaxy formation models in the ΛCDM framework. In the ΛCDM framework it is expected that the galaxy-to-halo-mass relation, and therefore the RAR, can be different for different galaxy types through their galaxy formation history (Dutton et al. 2010;Matthee et al. 2017;Posti et al. 2019;Marasco et al. 2020). Two parameters that correlate heavily with galaxy formation history are Sérsic index and colour. Current MG theories do not predict any effect of galaxy morphological type on the RAR, at least on large scales. The MOND paradigm gives a fixed prediction for the relation between g bar and g obs given by Eq. 11. Since the RAR is the observation of exactly this relation, in principle MOND gives a fixed prediction, independent of any galaxy characteristic. As discussed in Section 2.3, the main exception is the EFE that could be caused by neighbouring mass distributions. However, Fig. 4 shows that an increase in the EFE only predicts an increase in steepness of the downward RAR slope at low accelerations (g bar < 10 −12 m s −2 ), while the observed RAR of both early-and late-type galaxies follow approximately the same slope across all measured accelerations. It is therefore unlikely that their amplitude difference can be explained through the EFE.
We will next discuss whether the observed difference in RAR between early and late types is at odds with EG, but first emphasise three caveats of this discussion.
First, the derivation of the EG formalism assumes a spherical mass distribution. Solutions for non-spherical systems do not exist yet. It is not excluded that solutions for large-scale triaxial ellipticals will differ from rotationally supported spiral galaxies. This requires further theoretical study.
Second, the current EG theory predicts ADM fields based exclusively on the static baryonic mass distribution, although very large-scale dynamics can potentially influence the excess gravitational force predicted by EG. It is unknown whether large-scale pressure supported (virialised) systems create an ADM distribution similar to that of rotationally supported galaxies.
Third, we assume here that, to first order, the uncertainty in the KiDS photometric redshifts affects the isolated galaxy selection of both galaxy types in the same way, allowing us to include the full acceleration range into our comparison. However, the well established morphology-density relation predicts a higher density of satellite and dwarf galaxies around early-type galaxies compared to the late types (Dressler 1980;Goto et al. 2003), although we have minimised this effect by selecting isolated galaxies (see Appendix A). It is not yet known whether and, if so, how these external gravitational fields affect the EG prediction.
To address this last caveat, the light blue shaded region in Fig. 8 shows the acceleration scales beyond the KiDS isolation criterion limit (g bar < 10 −13 m s −2 ), where the presence of satellites might play a role (see Appendix A). But even when we remove all data points inside this region, we obtain a difference log 10 (δg E/L obs ) of 0.14 dex and 0.19 dex for the Sérsic and colour split respectively, where the latter has a significance of 3.2σ. Therefore, even at the scales where isolation is certain (corresponding to R < 0.3 h −1 70 Mpc), the difference remains significant. To evaluate the possible effect of circumgalactic hot gas, we computed the RAR of early and late-type isolated galaxies (of the same stellar mass) while including a rough estimate of the hot gas contribution to g bar . We used the same model of the nominal hot gas distribution around our galaxies as discussed in Sect. 5.2: an isothermal halo within 100 h −1 70 kpc, with a mass M gas = M . When applying the same hot gas model to both early-and late-type galaxies, we find that there remains a > 6σ difference between their RARs, both for the split by Sérsic index and u − r colour. However, for this particular gas model, we find that g bar increases in such a way that the RAR of earlytype galaxies moves to the right, close to the MG predictions where the RAR of late-type galaxies without circumgalactic gas resides. This means that, in the specific case where early-type galaxies have gaseous haloes with M gas = M while late-type galaxies (of the same stellar mass) have negligible hot circumstellar gas, this would reduce the difference in their RARs to ∼ 4σ. Fine-tuning the M gas /M ratio of early-type galaxies to a slightly higher value, while keeping M gas /M ≈ 0 for late types, might remove the difference between their RARs. However, as discussed in Sect. 5.2, unbiased X-ray surveys of circumgalactic gas around isolated galaxies are still lacking, which makes it difficult to obtain representative observational data.
In conclusion, unless early-type galaxies have significant circumgalactic gaseous haloes while late types (of the same stellar mass) do not, the difference we find in the RARs of different galaxy types might prove difficult to explain within MG frameworks. In MOND, g bar and g obs should be directly linked through Eq. 11 without any dependence on galaxy type. In EG the effect might be a consequence of yet unexplored aspects of the theory, such as a non-symmetric mass distribution or the effect of large-scale dynamics. To explore whether this is the case, however, more theoretical work is needed. Through the derivative in Eq. 14, EG does include a dependence on the slope of the baryonic density distribution. A shallower slope of M bar (r) increases M ADM and thus g obs , which might solve the current tension if early-type galaxies have significantly shallower baryonic mass distributions that extend far beyond 30 h −1 70 kpc, such as gaseous haloes (although Brouwer et al. 2017 did not find evidence for a significant effect of the baryonic mass distribution on the EG prediction; see their section 4.3). In addition, EG is currently only formulated for spherically symmetric systems. It would be interesting to investigate whether discs and spheroidal galaxies yield different predictions, and whether these differences would extend beyond 30 h −1 70 kpc. In a ΛCDM context, our findings would point to a difference in the SHMR for different galaxy types. Recently Correa & Schaye (2020) used SDSS data with morphological classifications from Galaxy Zoo to find that, at fixed halo mass (in the range 10 11.7 − 10 12.9 M ), the median stellar mass of SDSS disc galaxies was a factor of 1.4 higher than that of ellipticals. They found this to be in agreement with the EAGLE simulations, where haloes hosting disc galaxies are assembled earlier than those hosting ellipticals, therefore having more time for gas accretion and star formation.
The RAR as a function of stellar mass
In addition to splitting by galaxy type, it is interesting to create the RAR for galaxy samples with different stellar mass M (including very low-mass galaxies, 'dwarfs', in Section 5.6). In the ΛCDM paradigm, where baryonic and dark matter are described as separate substances, there can in theory be a difference in the SHMR depending on galaxy observables such as stellar mass, which could cause a shift in the measured RAR. This is in contrast with most MG models, which predict a fixed RAR (as is the case for MOND, and for EG at scales beyond the galaxy disc). In this section, we separated our isolated KiDSbright lenses into four samples based on M . We selected our M -bins to obtain a similar S /N ratio of the lensing signal in each bin, resulting in the following limits: log 10 (M / h −2 70 M ) = [8.5, 10.3, 10.6, 10.8, 11.0]. Fig. 9 shows the lensing measurements and predictions for isolated galaxies split in four stellar mass bins. For each bin the mean galaxy mass (stars+cold gas) of the lenses, log 10 M gal / h −2 70 M = [10.14, 10.57, 10.78, 10.96], is shown at the top of the panel. Quantifying the difference between MOND (the extended M16 fitting function) and our measurement at all scales results in: χ 2 red = 117.0/60 = 1.9, which (noting that the prediction for EG is very similar) excludes both models at the ∼ 4.5σ level. This result should be taken with caution, however, as at accelerations g bar that correspond to scales larger than R > 0.3 h −1 70 Mpc (light blue shaded region) an increasing signal is to be expected since at these distances satellite galaxies missed by our isolation criterion might affect the measurement. Galaxies with higher stellar masses reside in denser neighbourhoods, and therefore tend to have more satellites (see e.g. Baldry et al. 2006;Bolzonella et al. 2010;Brouwer et al. 2016).
The reduced χ 2 values using only the data within R < 0.3 h −1 70 Mpc are χ 2 red = 49.9/31 = 1.6 for MOND and 51.7/31 = 1.7 for EG respectively (corresponding to a standard deviation of 2.4 and 2.5σ). Considering the stellar mass uncertainty (∆M = ±0.2 dex), which, if it acts to reduce the observed RAR, results in χ 2 red = 0.97 for the extended M16 fitting function (with similar results for EG): a good fit. If the stellar mass uncertainty increases the observed RAR, we find χ 2 red = 4.6: a poor fit. This again highlights the grave importance of accurate baryonic mass measurements in determining the RAR, in addition to deep lensing surveys that can detect satellites down to very faint magnitudes. This could be achieved by future cosmology telescopes such as Euclid (Laureijs et al. 2011) and The Vera C. Rubin Observatory, previously called Large Synoptic Survey Telescope (LSST; Dark Energy Science Collaboration 2012). As for the MICE simulation, it matches our measurements reasonably well in every M bin. For the result that includes the photometric redshift uncertainty σ z in the isolated galaxy selection, we find χ 2 red = 49.7/30 = 1.7 (2.5σ).
5.6. The RAR of low-mass (dwarf) late-type galaxies As a final exploration of different galaxy masses, we attempt to measure the RAR for the lightest lenses in KiDS-bright. Low-mass galaxies are of particular interest to DM and MG researchers as extreme examples that might show eccentric behaviour (e.g. Oman et al. 2016;van Dokkum et al. 2018;Guo et al. 2019), as well as those who attempt to extend the RAR to lower accelerations using galaxy rotation curves (Lelli et al. 2017b;Di Paolo et al. 2019). We therefore select a sample of dwarfs: isolated galaxies with a stellar mass M < 10 10 h −2 70 M (whereas the full sample of isolated galaxies has M < 10 11 h −2 70 M , see Section 3.3). As can be seen in Fig. 6, this sample is dominated by blue, disc-dominated galaxies based on their colours and Sérsic indices (m u − m r > 2.5 mag and n < 2), which means they are likely to be late-type. Since these galaxies are few, and have an even smaller effect on the path of light rays than more massive ones, we needed to reduce the number of bins in g bar from 15 to 5 to obtain sufficient S /N radio in each bin. Fig. 10 shows the resulting RAR measurement of dwarfs compared to the full isolated sample. We do not show the effect of the ∆M = ±0.2 dex systematic uncertainty because this would affect both results in the same way. We find that, within its large error bars, the RAR of the dwarfs is consistent with that of the full isolated sample; they both approximately follow the M. M. Brouwer et al.: The lensing RAR: testing MG and CDM with KiDS-1000 Baryonic (stars+cold gas) radial acceleration log ( KiDS-bright isolated lens galaxies (1000 deg 2 ) Systematic stellar mass uncertainty (M * ± 0.2 dex) MICE isolated mock lens galaxies SPARC rotation curves (mean + 2D histogram) MOND (McGaugh+16,extrapolated) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar ) Fig. 9. Measured RAR of isolated KiDS-bright lenses (black points with 1σ error bars) divided into four stellar mass bins. The mean galaxy mass (stars+cold gas) of the lenses is shown at the top of each panel. At increasing stellar mass, the measurements seem to rise above the predictions from MOND (grey solid line) and EG (red dashed line). However, at scales larger than R > 0.3 h −1 70 Mpc (light blue shaded region) this could be caused by false positives in the isolated galaxy sample due to the KiDS-bright redshift uncertainty. g obs ∝ √ g bar relation expected by the extended MOND and EG predictions, which are shown as a reference. Hence, we do not find a significant difference in the RAR of dwarf galaxies.
Discussion and conclusions
Galaxy-galaxy lensing observations from the fourth data release of the Kilo Degree Survey (KiDS-1000) have extended the RAR of isolated galaxies by nearly 2 orders of magnitude in gravitational acceleration g obs , compared to previous measurements based on rotation curves (most notably McGaugh et al. 2016, M16). To compute the lensing RAR, we converted our ESD profiles ∆Σ(R) into the observed gravitational acceleration g obs , and our galaxy masses (measured using nine-band KiDS+VIKING photometry) into g bar . These measurements allowed us to perform unprecedented tests of two MG models: MOND and EG, as well as tests of DM using the MICE (Nbody + semi-analytic) and BAHAMAS (hydrodynamical) sim-ulations. Our conclusions from these observational tests are as follows:
- Fig. 3: We find that lensing rotation curves of isolated galaxies, as inferred from GGL measurements, remain approximately flat at scales far beyond the visible disc (0.03 < R < 3 h −1 70 Mpc). At the accelerations corresponding to the outskirts of observable galaxies (R ≈ 30 h −1 70 kpc), our lensing results are in excellent agreement with the SPARC rotation curves . These two measurements are obtained by two very different methods, providing independent corroboration of each result. -Fig. 4: At the low accelerations corresponding to GGL scales, the lensing RAR of isolated galaxies approximately follows a g obs ∝ √ g bar relation. This is in agreement with the expectations from EG (Eq. 17) and MOND (which we take to be the M16 fitting function, Eq. 11, extrapolated to larger scales). At low accelerations both these models predict a direct relation between observed and baryonic accel- . Due to the low S /N ratio of the dwarf lensing signal, the number of g bar -bins is reduced from 15 to 5. We find that the RAR of dwarfs is consistent with that of our regular sample, and with the extrapolated MOND and EG predictions (grey solid and red dashed lines), which are shown as a reference. eration of this form, with a very similar proportionality constant 17 of ∼ 1.2 × 10 −10 m s −2 . This reinforces the results of Brouwer et al. (2017), who found that EG provides a good description of ESD profiles measured using 180 deg 2 of KiDS-GAMA data, but with a five times larger survey area. However, this result only remains valid if no massive (M gas M ) extended baryon distributions, such as as-yet undetected gaseous haloes, are common around our isolated lens galaxies.
Article
- Fig. 5: We find that the BAHAMAS simulation underestimates our KiDS-bright lensing RAR. The discrepancy relative to MICE is caused by a bias in the stellar-to-halomass-relation (SHMR) of isolated galaxies in BAHAMAS, which is absent in MICE: BAHAMAS galaxies have stellar masses typically three times higher at fixed halo mass than their non-isolated counterparts. Determining which of the two models more accurately captures the true SHMR is clearly crucial to the interpretation of our measurements in the ΛCDM context. Interestingly, the BAHAMAS RAR still has approximately the correct low-acceleration slope, rather than a steeper slope as would naively be predicted based on the ρ ∝ r −3 outer slopes of the simulated DM haloes. The prediction from MICE (only feasible at low accelerations due to the limited resolution of the simulated lensing measurements) matches our RAR measurements very well. -The additional lensing power at large radii with respect to the prediction from Navarro et al. (2017, see Appendix C) might be caused by large-scale structure along the line-ofsight to the source, in spite of our efforts to select isolated galaxies. This highlights the crucial importance of simulating the entire measurement process (where possible) when making theoretical predictions, both in ΛCDM and MG, before they can be ruled out. In addition, the need for accurate isolated galaxy selection highlights the importance of large spectroscopic surveys, such as the upcoming 4MOST (de Jong et al. 2019) and Dark Energy Spectroscopic Instrument (DESI; Ruiz-Macias et al. 2020) surveys. - Fig. 8: When we split galaxies into two types based on Sérsic index or u − r colour, we find at least a factor of 1.5 ( 0.2 dex) difference between the respective lensing RAR measurements with a significance of at least 5.7σ. This observed difference could be resolved by a 0.4 dex systematic bias between the stellar masses of the two types. However, we calculated that the expected M bias (due to Eddington bias or systematic biases in the M measurement) is at most 0.17 dex. This variation in the RAR based on galaxy type, which is in agreement with Taylor et al. (2020) and Correa & Schaye (2020), could be difficult to explain for MG models that predict a fixed relation between baryonic mass and the total gravitational potential. - Fig. 9: The lensing RAR for galaxy samples split by stellar mass M demonstrated a slight upward trend, away from the fixed predictions of MOND and EG, with increasing M . This could be caused by satellite or companion galaxies missed by the isolated galaxy selection due to the KiDSbright redshift uncertainty, however. With the inclusion of the KiDS isolation criterion limit and accounting for uncertainty in the stellar mass, we find a reasonable agreement between the MG models and observations. This highlights the crucial importance of accurate baryonic mass measurements in determining the RAR, in addition to deep lensing surveys that can detect satellites to down to very faint magnitudes (such as the future Euclid space telescope and Vera C. Rubin Observatory). The MICE prediction, which is corrected for the KiDS-bright redshift uncertainty, again matches well to our data. - Fig. 10: We find no significantly different RAR, relative to the entire isolated lens sample, for a subsample of the lightest KiDS-bright lenses: isolated dwarf (M < 10 10 h −2 70 M ) galaxies.
-Throughout this work, we find that the field of GGL has reached a level of accuracy in the measurement of g obs greater than that of the baryonic acceleration g bar . The fact that we have no accurate measurements of the additional hot gas at large radii, and the ambiguity around the cosmological missing baryons, forces us to limit g bar to the contributions of stars and cold gas. In addition, the current 0.2 dex systematic uncertainty in M prevents us from definitively excluding any of the models we test. This shows that, if we want to have any hope of testing DM and MG models using the next generation of cosmological lensing surveys (such as Euclid and LSST), we also need to focus on the models and observations needed to accurately measure the baryonic mass distribution in and around galaxies.
We find that galaxy lensing rotation curves continue approximately flat out to R = 3 h −1 70 Mpc (where observations are bound to encounter lensing due to surrounding galaxies), which is difficult to explain in a ΛCDM framework that predicts simple NFWlike haloes because of their r −3 outer slope (see the N17 model in Appendix C). However, our analysis of the MICE and BA-HAMAS simulations shows that the combination of the lenses and the additional structure along the line-of-sight can yield an ESD profile consistent with an ∼ r −2 density profile for isolated galaxies, even though the lenses have an intrinsic ∼ r −3 outer profile.
Throughout our analysis we find that the extrapolated M16 fitting function (Eq. 11), which approximately corresponds to the prediction of both MG models (EG and MOND), holds to scales of 3 h −1 70 Mpc for isolated galaxies. A fundamental limitation of this measurement is that the additional diffuse gas surrounding galaxies remains difficult to measure, and has therefore not been included in most of this study. By implementing a rough order of magnitude estimate of the hot gas contribution to g bar , an isothermal distribution with M gas = M within 100 h −1 70 kpc, we found that this causes an overall downward shift of the RAR and a steeper downward slope at very low accelerations (see Fig. 4, and also Fig. 1 for a broader discussion of missing baryons). Although the MOND external field effect (EFE) causes a similar steepening of the RAR, we find that the idealised EFE fitting function of Chae et al. (2020) is not steep enough the explain the effect of gaseous haloes. Therefore, a convincing detection of additional gaseous components with a nominal mass of M gas M would move the observed RAR away from the MG predictions (g bar ∝ √ g obs ) at very low accelerations (g bar < 10 −13 m s −2 ) and towards the DM predictions (where g bar and g obs are independent). A robust non-detection of such massive gaseous haloes in general would likely strengthen the position of MG models. Finding them for early-type galaxies only would reduce the difference between the RAR of early-and late-type galaxies, which otherwise remains unexplained in MG frameworks.
In conclusion, we find that the lensing RAR is a promising method to be used by future cosmological surveys to distinguish between MG and DM models. This can be done by measuring the RAR including large-scale baryonic mass observations; by simply performing the same comparison with even more accurate lensing and stellar mass measurements; or by further exploring the offset that we have found between the RARs of different galaxy types. All these options require that systematic biases in the stellar and other baryonic mass measurements be reduced. GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is www.gama-survey.org.
Acknowledgements
We are indebted to Ian McCarthy, who provided the BA-HAMAS data products used in our analysis. Bob Sanders provided us useful comments about the relation between MOND and the M16 fitting function, and about the deflection of photons in MOND. We would also like to thank Federico Lelli, who provided the idea for the lensing rotation curves shown in Fig. 3. Finally, we would like to thank the anonymous referee for insightful questions and useful comments, which helped to increase the value of this work. This work has made use of CosmoHub (Carretero et al. 2017;Tallada et al. 2020 Fig. A.1. Histogram of the number of isolated galaxies (orange line) compared to the total number of galaxies (red line), as a function of apparent r-band magnitude m r . The dashed vertical line represents the magnitude m bright , below which all satellites with a luminosity fraction larger than f L ≡ L sat /L lens = 0.1 compared to the lens are still detected. Beyond this limit, the fraction of isolated galaxies (blue line) slightly increases because satellites fainter than the flux limit are not detected, which can cause lenses close to the magnitude limit (m lim = 20 mag) to be falsely identified as isolated.
lower for the full isolated sample: log M / h −2 70 M = 10.61. Due to the smaller number of lenses (only 3800), the ESD errors and scatter of the bright isolated sample are much larger than those of the full isolated sample. Nevertheless, it is clear that their ESD profiles show consistent behaviour at both small and large scales. Compared to the total (non-isolated) galaxy sample, both isolated samples show significantly lower lensing signals at large scales (the two-halo term, corresponding to the contribution of satellites). The high level of consistency between the ESD profiles of the full and bright isolated samples indicates that the effect of false positives due to the magnitude limit is limited. In addition, by comparing the expected percentage of true isolated galaxies (32.0%, found in the bright sample) with the higher percentage found in the 'faint' sample (36.8%, for galaxies with m r > 17.5 mag), we estimated that the expected percentage of false positives is less than 20% of the full sample of isolated galaxies.
Nevertheless, we used the MICE simulations to perform one additional test. We selected the isolated sample of MICE lenses using satellite galaxies that extend to m r < 22.5 mag, such that all satellites with f L > 0.1 can be observed. This paints a similar picture as the bright KiDS sample: although the much smaller sample of isolated galaxies selected using the faint satellites greatly increases the scatter, we find no consistent decrease in the lensing signal at > 0.3 h −1 70 Mpc scales compared to the original sample of isolated MICE galaxies. All these tests demonstrate the overall robustness of our isolation criterion. In addition, we note that this issue is only relevant when comparing our observations to the theoretical models (EG, MOND and N17). When comparing to the simulations (BAHAMAS and MICE), applying the same isolation criterion to both data and mocks ensured that any issues with the isolated galaxy selection are mimicked.
The major difference between the isolated galaxy selection of the GAMA and mock galaxies compared to KiDS-bright is that for GAMA and the mocks the true redshift values are Measured ESD profiles of our full sample of isolated KiDSbright galaxies (red points with 1σ error bars), compared to that of a more reliable 'bright' sample (blue, m r < 17.5 mag), which allows us to see all satellites down to luminosity fraction f L ≡ L sat /L lens = 0.1. This is done to assess the effect of the KiDS-bright magnitude limit (m r < 20 mag) on the isolation criterion. Due to the smaller number of lenses, the ESD profile of the bright isolated sample is noisier. Nevertheless, its behaviour on both small and large scales is consistent with the ESD profile of the full isolated sample (orange), indicating that the effect of the magnitude limit is limited. known, whereas the ANNz2 photometric redshifts of KiDSbright are only known within a certain standard deviation σ z (see Section 3.3). Since these photometric redshifts were used to calculate the galaxy distances D(z) (using a flat ΛCDM cosmology, ignoring peculiar velocities) they directly affect the observed spherical distances r between the galaxies, a key ingredient of the isolation criterion. The redshift uncertainty also affects the KiDS-bright stellar mass estimates, which influence both the isolation criterion (through f M ) and the application of the stellar mass limit: log(M / h −2 70 M ) < 11. We assessed the effect of these uncertainties on the isolated galaxy selection by adding a normally distributed random offset with σ z /(1 + z) = 0.02 to the MICE redshifts, and σ M = 0.12 dex to its stellar masses. We find that the effect of the mass uncertainty is negligible, but that of redshift uncertainty is significant. Because the random redshift offset decreases the galaxy clustering, it increases the number of galaxies selected by the isolation criterion, adding galaxies that are not truly isolated to the lens sample (as well as excluding some truly isolated galaxies).
The ESD profile of the offset isolated MICE sample is shown in Fig. A.3, compared to the ESD profiles of all MICE galaxies (without any isolation criterion) and the truly isolated MICE sample. At scales R > 0.3 h −1 70 Mpc, the ESD of the isolated sample selected using the offset MICE data is ∼ 30% higher than that of the truly isolated MICE galaxies. When comparing our KiDS-bright lensing measurements to the MICE simulation, we always take this effect into account by mimicking the redshift offset in the simulation. However, for our comparison with the analytical models (MOND, EG and N17) this process is more difficult. When testing these models, we can only use the ESD profile of isolated KiDS-bright lenses within R < 0. Isolated, offset: σ z /(1 + z) = 0.02, σ M * = 0.12 dex Fig. A.3. Simulated ESD profile of the offset isolated MICE sample (blue line), created by using MICE galaxies with randomly offset redshifts (σ z /[1 + z] = 0.02) and stellar masses (σ M = 0.12 dex) when selecting the isolated lenses. This is done in order to mimic the effect of the KiDS-bright measurement uncertainties on the isolation criterion. Compared to the ESD profile of the truly isolated MICE sample (orange line) the offset sample has a ∼ 30% higher signal at large scales due to the contribution of satellites. We therefore take extra care with KiDSbright results at R > 0.3 h −1 70 Mpc (light blue shaded region). Nevertheless, the ESD of the offset isolated MICE sample is significantly lower than that of all MICE galaxies (red line), created without any isolation criterion. In addition, we show the radius corresponding to three times the resolution of the MICE simulation (dashed vertical line), which in the case of the isolated MICE sample is R < 0.25 h −1 70 Mpc. Throughout this work, we only use the MICE results beyond this radius. the range in g bar where the measurement is reliable, based on the mean M gal of the appropriate lens sample.
Finally, to indicate the effect of selecting isolated galaxies on our lensing RAR measurements, Fig. A.4 shows the RAR of KiDS-bright and MICE galaxies for all lens galaxies without applying the isolation criterion. Because the clustering of galaxies (and hence the effect of the satellite galaxies) correlates with their stellar mass, we divided the lens galaxies into the same four stellar mass bins as used in Section 5.5: log 10 (M / h −2 70 M ) = [8. 5, 10.3, 10.6, 10.8, 11.0]. In that section, Fig. 9 shows the RAR of isolated galaxies in the same stellar mass bins. In both cases, g bar is calculated using only the baryonic masses of the main lens galaxies (i.e. the baryonic masses of the satellites are not included in the x-axis of Fig. A.4). Comparing these two results shows that the effect of our isolated galaxy selection on g obs is very striking: the isolated RAR measurements in Fig. 9 approximately follow the extrapolated M16 and EG predictions, while the non-isolated RAR measurements in Fig. A.4 lie well above these lines at low accelerations (g bar < 10 −13 m s −2 ). As expected, the strength of this two-halo term (which shows the amount of clustering) increases with increasing galaxy stellar mass. Again the MICE simulation was able to predict our measurements: χ 2 red = 51.3/33 = 1.6 (2.3σ). This shows that the clustering simulated within MICE, which drives the lowacceleration upturn due to the two-halo term, is indeed quite accurate. This was to be expected since the clustering in MICE is constructed to reproduce the SDSS observations at z < 0.25 (Zehavi et al. 2011).
The ESD profile was measured in a series of discrete radial bins with edges R m . The representative value at the centre of the bin 18 is ∆Σ m = Σ m − Σ m , where Σ m is the mean surface density within 1 2 (R m + R m+1 ) and Σ m is the surface density averaged over the interval [R m , R m+1 ). We give an expression for this discrete ESD profile in terms of the parametric form for ρ(r) given in Eq. B.4.
The mean enclosed surface density is: (B.8) and the local surface density is given by:
Σ m = 1 πR m R m+1 I 1 (0, R 0 R 1 ,ã 0 ,b 0 ) + m k=0 I 1 ( R m R m+1 , R m+1 R m+2 ,ã m ,b m ) (B.5) a m = ln(Σ m+1 ) − ln(Σ m ) 1 2 (ln(R m+2 ) − ln(R m )) (B.6) b m = ln(Σ m ) − 1 2ã m ln(R m R m+1 ) (B.7) I 1 (R i , R j ,ã,b) = 2πe˜b a + 2 R a+2 j − R a+2 i ,Σ m = N−1 n=0 0 if r n+1 < R m 4e bn R 2 m+1 −R 2 m
(−I 2 (r n+1 , R m , a n )) if r n < R m and R m ≤ r n+1 < R m+1 4e bn R 2 m+1 −R 2 m (I 2 (r n+1 , R m+1 , a n ) − I 2 (r n+1 , R m , a n )) if r n < R m and r n+1 ≥ R m+1 4e bn R 2 m+1 −R 2 m (I 2 (r n+1 , r n , a n ) − I 2 (r n+1 , R m , a n ) +I 2 (r n , R m , a n ) + I 2 (r n+1 , R m+1 , a n ) −I 2 (r n+1 , r n , a n )) if R m ≤ r n < R m+1 and r n ≥ R m+1 4e bn R 2 m+1 −R 2 m (I 2 (r n+1 , R m+1 , a n ) − I 2 (r n+1 , R m , a n ) −I 2 (r n , R m+1 , a n ) + I 2 (r n , R m , a n )) if r n ≥ R m+1 4e bn R 2 m+1 −R 2 m (I 2 (r n+1 , r n , a n ) − I 2 (r n+1 , R m , a n ) +I 2 (r n , R m , a n ) − I 2 (r n+1 , r n , a n )) if r n ≥ R m and r n+1 < R m (B.9)
I 2 (r, R, a) = − 1 3 R a+3 r 2 R 2 − 1 3 2 2 F 1 3 2 , − a 2 ; 5 2 ; 1 − r 2 R 2 if r is finite √ π 2 Γ(− a+1 2 ) Γ(− a 2 ) R a+3 a+3 if r = ∞ , (B.10)
where 2 F 1 (·, ·; ·; ·) is the Gaussian hypergeometric function and Γ(·) is the Gamma function. We assumed that the power law slope in the innermost bin continues to r = 0, and that the slope in the outermost bin continues to r → ∞. When inverting ∆Σ(ρ), we imposed uninformative priors on the power law slopes except those constraints required to guarantee that the total mass is finite and that the calculation is numerically stable. In order to invert ∆Σ m (ρ n ), we took as constant the values R m , ∆Σ obs,m and r n . We then proposed an initial guess ρ n , which we perturbed iteratively, calculating the corresponding ∆Σ m at each iteration and comparing with ∆Σ obs,m via the likelihood function: ln L ∝ − 1 2 (∆Σ obs − ∆Σ) T C −1 (∆Σ obs − ∆Σ) , (B.11)
where C is the covariance matrix for the ∆Σ obs . We used the package emcee (Foreman-Mackey et al. 2013) to estimate the posterior probability distribution of ρ n , and subsequently of the corresponding g obs,n via integration of the volume density profile. 18 Here we define the bin centre as 1 2 (R m + R m+1 ), not the logarithmic centre √ R m R m+1 , which ensures accuracy in the calculation of the mean enclosed surface density.
Fig. 1 .
110 (g bar /m s −2 ), log 10 (g /m s −2 ), log 10 (f b g DM /m s Mass profiles and RAR of BAHAMAS galaxies. Upper panel: Cumulative mass profiles of stars (red dotted line) and total baryons (blue solid line) for BAHAMAS galaxies with 1 < M 200 /(10 12 h −2 70 M ) < 3. The star marker indicates the stellar mass within a 30 h −1 70 kpc aperture, indicative of what is typically regarded as the stellar mass of a galaxy. The blue dash-dotted line shows the typical baryonic mass profile of observed galaxies of similar mass, estimated based on an extrapolation of the compilation infig. 7 of
lowing limits: log 10 (M / h −2 70 M ) = [8.5, 10.3, 10.6, 10.8, 11.0]. For each bin the mean galaxy mass (stars+cold gas) of the lenses, log 10 M gal / h −2 70 M =[10.14, 10.57, 10.78, 10.96], is shown at the top of the panel.
Fig. 3 .
3presenting our results in this section to reduce clutter in the figures. Measured rotation curves -the circular velocity as a function of radius v circ (R) -of the KiDS-bright isolated lens sample, divided into four stellar mass bins. The mean galaxy mass (stars+cold gas) of the lenses is shown at the top of each panel. The light blue shaded region indicates the radii corresponding to R > 0.3 h −1 70 Mpc, where the uncertainty in the photometric KiDS redshifts can affect the isolated lens selection (see Appendix A)
al. 2021; Heymans et al. 2021; Article number, page 13 of 31 A&A proofs: manuscript no. RAR_paper_Brouwer_4 Baryonic (stars+cold gas) radial acceleration log(g bar [h 70 m/s 2 incl. external field effect: e = 0.003) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar ) Fig. 4
stars+cold gas) radial acceleration log(g bar [h 70 m/s 2 ]) Observed radial acceleration log(g
stars+cold gas) radial acceleration log(g bar [h 70 m/s 2 ]) Observed radial acceleration log(g dark matter: g obs = g bar ) Fig. 10. Measured RAR of KiDS-bright lenses (points with 1σ error bars), respectively for isolated dwarfs (log(M / h −2 70 M ) < 10, blue) and the full isolated galaxy sample (log(M / h −2 70 M ) < 11, black)
-bright lens galaxies Isolated: r sat > 3 Mpc/h 70 , M * < 10 11 M /h 2 70 Isolated, bright (m r < 17.5 mag) Fig. A.2.
Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples).
). CosmoHub has been developed by the Port d'Información Científica (PIC), maintained through a collaboration of the Institut de Física d'Altes Energies (IFAE) and the Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT), and was partially funded by the "Plan Estatal de Investigación Científica y Técnica y de Innovación" program of the Spanish government. KAO acknowledges support by the Netherlands Foundation for Scientific Research (NWO) through VICI grant 016.130.338 to M. Verheijen, and support from the European Research Council (ERC) through Advanced Investigator grant to C.S. Frenk, DMIDAS (GA 7786910). MBi is supported by the Polish National Science Center through grants no. 2020/38/E/ST9/00395, 2018/30/E/ST9/00698 and 2018/31/G/ST9/03388, and by the All KiDS-bright lens galaxies Isolated: r sat (f M * > 0.1) > 3 Mpc/h 70 Fraction (isolated/all galaxies)16.0
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3 h −1 70 Mpc. For the mean galaxy mass of the KiDS-bright isolated sample (log[M gal / h −2 70 M ] = 10.69) this corresponds to a baryonic acceleration of g gal > 7.56 × 10 −14 m s −2 . For each RAR measurement resulting from isolated KiDS-bright lenses we will indicate Radius R [Mpc/h 70 ] Excess Surface Density ∆Σ [h 70 M /pc 2 All MICE mock lens galaxies Isolated: r sat > 3 Mpc/h 70 , M * < 10 11 M /h 2]
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DM particles that moved at non-relativistic speeds at the time of recombination, as favoured by measurements of the CMB(Planck XVI 2014) and the Lyman-α forest(Viel et al. 2013).
Another closely related (though slightly different) relation is the mass-discrepancy acceleration relation, which shows the expected baryonic acceleration against the discrepancy between the baryonic and the observed mass: M obs − M bar (seeMcGaugh 2004). Although measuring this relation requires the same data, we prefer the RAR because the two observables (g bar and g obs ) are uncorrelated.Article number, page 2 of 31 M. M. Brouwer et al.: The lensing RAR: testing MG and CDM with KiDS-1000
We note that this value of g bar only takes into account the stellar and cold gas mass of the galaxy. In Section 4.3 we show that the contributions of additional hot gas, dust and 'missing baryons' could increase this value to g bar ≈ 10 −14 m s −2 , which is still two orders of magnitude lower than the accelerations measurable with galaxy rotation curves.
As derived in Appendix C of Dvornik et al. (2018), there are two possible definitions of Σ crit : proper and comoving. In this work we used the proper Σ crit , and we compute ∆Σ(R) as a function of proper transverse separation R. This choice is reasonable because, within a 3 h −1 70 Mpc Article number, page 6 of 31 M. M. Brouwer et al.: The lensing RAR: testing MG and CDM with KiDS-1000
The MICECATv2.0 catalogue is available through CosmoHub (https://cosmohub.pic.es).
We do not include the additional gas, which is predominantly in the hot phase, for consistency with the presentation of the results in Section 5.
In general, the Sérsic index n does not separate early-and late-type galaxies because dwarf early-and late-type galaxies have similar values of n (Graham 2019). However, dwarf early-type galaxies are not abundant in isolation(Janz et al. 2017), which means that our isolated low-n galaxy sample likely consists of late-type galaxies.
The smaller average size for the early-type galaxies, compared to the late-type galaxies, is because of the different 3D-bulge-to-2D-disc ratios: a fixed stellar mass will fit into a smaller volume if distributed in a bulge rather than a disc.
The proportionality constant cH 0 /6 in EG is almost equal to the value of g † found by M16, which is again equal to the a 0 = 1.2 × 10 −10 m s −2 canonical in MOND.
(B.4) Article number, page 28 of 31 M. M. Brouwer et al.: The lensing RAR: testing MG and CDM with KiDS-1000
Article number, page 31 of 31
This work is partly based on tools and data products produced by GAZPAR operated by CeSAM-LAM and IAP.Computations for the N-body simulations were performed in part on the Orcinus supercomputer at the WestGrid HPC consortium (www.westgrid.ca), in part on the GPC supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund -Research Excellence; and the University of Toronto.We are grateful to https://math.stackexchange.com user Paul Enta for providing an expression for one of the integrals needed in Appendix B.This work has made use of python (www.python.org), including the packages numpy (www.numpy.org) and scipy (www. scipy.org). Plots have been produced with matplotlib(Hunter et al. 2007).Author contributions: All authors contributed to the development and writing of this paper. The authorship list is given in three groups: the lead authors (M. Brouwer, K. Oman, E. Valentijn), followed by two alphabetical groups. The first alphabetical group includes those who are key contributors to both the scientific analysis and the data products. The second group covers those who have either made a significant contribution to the data products, or to the scientific analysis.Appendix A: Isolated galaxy selection and validationAfter performing the measurement of the RAR using GGL, our final goal was to compare the results to the different analytical models (Section 2) and N-body simulations (Section 4) that make specific predictions on the galaxy-halo connection. While the simulations were designed to describe galaxies in their cosmological environment, the analytical models mainly focus on the description of individual galaxies. This means that, in order to test these models, we need to select galaxies that are relatively isolated. We defined our isolated lenses such that they do not have any satellites with more than a fraction f M ≡ M ,sat /M ,lens = 0.1 of their stellar mass within a spherical radius r sat = 3 h −1 70 Mpc (see Section 3.3). Here we validate our isolation criterion using the KiDSbright and MICE datasets. We find that increasing the value of r sat does not yield any decrease in the 'two-halo term': the GGL signal at larger scales (> 0.3 h −1 70 Mpc) corresponding to the contribution of satellites. This is true both when all lens masses are considered, and when they are restricted to a specific stellar mass: log(M / h −2 70 M ) = 10.5 ± 0.1. Using both the KiDSbright and MICE galaxies, we reduce the satellite mass fraction to f M = 0 (corresponding to no visible satellites). This also yields no decrease in the two-halo term of the ESD profile since galaxies with f M 0.1 are not likely to be observed in a fluxlimited survey. When we restrict the total stellar mass M ,tot of all satellites within r sat to f M ,tot < 0.1 this does not significantly affect the isolated lens sample (i.e. the samples selected with KiDS-bright are > 99% overlapping). Using the MICE and KiDS data we also experimented with selecting lenses that are isolated within a conical frustum, defined by a projected radius R and line-of-sight distance range ∆D around the lens. However, significantly increasing ∆D beyond 3 h −1 70 Mpc has no effect on the ESD profile, until it reduces our number of selected lenses to the point where the S /N does not allow for a significant measurement. Finally, we apply our isolation criterion to the GAMA survey, to compare our current isolated sample with the 'isolated centrals' that we used inBrouwer et al. (2017). These were selected using a more elaborate isolation criterion, which was driven by the Friends-of-Friends (FoF) group finding algorithm ofRobotham et al. (2011). We find that the two isolated galaxy samples are more than 80% overlapping.However, because both the GAMA survey and the samples designed to mimic it (KiDS-bright and MICE) are fluxlimited, satellites that are fainter than the flux limit are not detected. This can cause lenses that are close to the magnitude limit (m lim = 20 mag) to be falsely identified as isolated. This problem is illustrated inFig. A.1, which shows that the fraction of galaxies assigned to the isolated lens sample increases for higher values of the apparent r-band magnitude m r . The dashed vertical line represents the magnitude m bright , below which all satellites with a luminosity fraction larger than f L ≡ L sat /L lens = 0.1 compared to the lens are still detected. In the case of KiDS-bright: m bright = m lim − 2.5 log 10 ( f F = 0.1) = 17.5 mag .(A.1)Applying m r < m bright provided us with an isolated lens sample that should be free of false positives, allowing us to estimate their effect on the ESD profiles. InFig. A.2we compare the ESD profiles of all galaxies and isolated galaxies with the more reliable 'bright' sample. The mean stellar masses of the lens samples are very similar for the bright and the full sample: log M / h −2 70 M = 10.78 and 10.77 respectively, and slightlyAppendix B: Calculating g obs from an ESD profileTo calculate g obs from the ESD profile throughout this work, we have used a simple analytical method that assumes that DM haloes can be roughly approximated with a singular isothermal sphere density model (see Section 2.2). To make sure this conversion is robust, we compared it to a more elaborate numerical approach that fits a piece-wise power law (PPL) to the stacked ESD profile, without any assumption on the averaged halo shape except for spherical symmetry. We validate both methods using mock surface density maps from the BAHAMAS simulation in Section 4.4. The PPL method assumes a self-consistent form for the volume density profile ρ(r) and parametrises it as a piece-wise power law constrained to be continuous. This comes at the cost of needing to invert the non-linear function ∆Σ(ρ), which we achieve via an iterative method. We chose to parametrize ρ(r) in terms of N pairs of values (r n , ρ n ) such that the slope a n and normalisation b n of the power law profile segments are: ln ρ = a n ln(r) + b n (B.1)A&A proofs: manuscript no. RAR_paper_Brouwer_4Appendix C: The RAR of the N17 modelWe used the lensing RAR to test the analytical prediction from the ΛCDM-based model created by N17. InFig. C.1we show the RAR predicted by this model for a galaxy with a baryonic mass equal to the average stellar + cold gas mass of the lens sample (log 10 M gal = 10.69). At higher accelerations there is a good match between the model and the M16 RAR measurements from galaxy rotation curves, which is expected since the N17 model is designed and confirmed to reproduce these results. However, at the lower accelerations unique to our lensing measurements the N17 model underpredicts the g obs amplitude in comparison to our measurements. Due to their large error bars, the GAMA data can still accommodate the analytical prediction: χ 2 red = 0.90. The KiDS-bright result, however, excludes the N17 prediction with χ 2 red = 4.8, corresponding to 4.3σ. Here we have removed all data points beyond the KiDS isolation limit (R > 3 h −1 70 Mpc); therefore, the strong disagreement between N17 and the data is unlikely to be caused by contamination from satellites.Because of the significant difference in the slope of the model and the data, even taking the ∆M = ±0.2 dex uncertainty into account does not result in a better fit. This strong downward slope results from the r −3 radial dependence of the Navarro-Frenk-White (NFW) density profile at large scales (where an r −2 density profile would instead follow the same slope as the MG predictions inFig. 4). This effect could be slightly mitigated by taking into account the average DM density of the Universe, which would result in an upward turn towards an r −1 slope at g bar < 10 14 m s −2 (as shown by the BAHAMAS prediction of the RAR in the lower panel ofFig. 1). However, components contributing to the large-scale DM profile are not included in the N17 model, which was created to reproduce the RAR at the small scales measured by rotation curves. It is clear from this exercise that, while succeeding to describe the RAR at small scales, this simple model is not sufficient to reproduce the results at the larger scales probed by weak lensing. This requires more elaborate modelling within the ΛCDM paradigm, represented by large cosmological simulations such as BAHAMAS and MICE (see Section 4). In Section 5.3 we made a fairer comparison using these two simulations, which can mimic the measurement more faithfully.
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A&A proofs: manuscript no. A&A proofs: manuscript no. RAR_paper_Brouwer_4
MOND (McGaugh+16, extrapolated) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar ). MOND (McGaugh+16, extrapolated) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar )
This figure primarily shows the effect on the RAR when the isolation criterion is not applied, which is quite significant and depends on the stellar mass of the galaxies (which correlates with galaxy clustering). The extrapolated M16 and EG predictions (grey solid and red dashed lines) function merely as a reference, showing the approximate location of the isolated galaxy RAR. They do not represent predictions in this case because g bar is calculated using only the baryonic masses of the main lens galaxies (without including the baryonic masses of the satellites). The predictions from the MICE simulation (red line) match with our observations. RAR of all GL-KiDS lenses (black points with 1σ error bars) divided into four stellar mass bins, with the mean galaxy mass (stars+cold gas) of the lenses shown at the top of each panel. which shows that the clustering simulated within MICE, driving the low-acceleration upturn due to the two-halo term, is indeed quite accurateFig. A.4. Measured RAR of all GL-KiDS lenses (black points with 1σ error bars) divided into four stellar mass bins, with the mean galaxy mass (stars+cold gas) of the lenses shown at the top of each panel. This figure primarily shows the effect on the RAR when the isolation criterion is not applied, which is quite significant and depends on the stellar mass of the galaxies (which correlates with galaxy clustering). The extrapolated M16 and EG predictions (grey solid and red dashed lines) function merely as a reference, showing the approximate location of the isolated galaxy RAR. They do not represent predictions in this case because g bar is calculated using only the baryonic masses of the main lens galaxies (without including the baryonic masses of the satellites). The predictions from the MICE simulation (red line) match with our observations, which shows that the clustering simulated within MICE, driving the low-acceleration upturn due to the two-halo term, is indeed quite accurate.
M M Brouwer, The lensing RAR: testing MG and CDM with KiDS-1000. M. M. Brouwer et al.: The lensing RAR: testing MG and CDM with KiDS-1000
. GAMA isolated lens galaxies. 1802GAMA isolated lens galaxies (180 deg 2 )
KiDS-bright isolated lens galaxies (1000 deg 2 ) Systematic stellar mass uncertainty (M * ± 0.2 dex) SPARC rotation curves (mean + 2D histogram) Navarro+2017 analytical dark matter model MOND (McGaugh+16, extrapolated) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter. KiDS-bright isolated lens galaxies (1000 deg 2 ) Systematic stellar mass uncertainty (M * ± 0.2 dex) SPARC rotation curves (mean + 2D histogram) Navarro+2017 analytical dark matter model MOND (McGaugh+16, extrapolated) Emergent Gravity (Verlinde+16, point mass) Unity (No dark matter: g obs = g bar )
We compare our results to the analytical ΛCDM-based model created by N17. At higher accelerations there is a good match between the N17 model and the M16 observations, as expected. However, at lower accelerations the model bends down with respect to the lensing measurements, due to the steep outer slope of the NFW density profile (ρ ∝ r −3 ). Large-scale contributions to the total mass distribution, such as the average cosmic DM density, could slightly mitigate this discrepancy. However, these are not implemented into the simple analytical N17 model. Fig. C.1. Measured RAR of the spectroscopic GAMA and photometric KiDS-bright isolated lens samples (blue and black points with 1σ error bars). which was created to reproduce the RAR at the small scales measured by rotation curvesFig. C.1. Measured RAR of the spectroscopic GAMA and photometric KiDS-bright isolated lens samples (blue and black points with 1σ error bars). We compare our results to the analytical ΛCDM-based model created by N17. At higher accelerations there is a good match between the N17 model and the M16 observations, as expected. However, at lower accelerations the model bends down with respect to the lensing measurements, due to the steep outer slope of the NFW density profile (ρ ∝ r −3 ). Large-scale contributions to the total mass distribution, such as the average cosmic DM density, could slightly mitigate this discrepancy. However, these are not implemented into the simple analytical N17 model, which was created to reproduce the RAR at the small scales measured by rotation curves.
| []
|
[
"Analyticity of Entropy Rates of Continuous-State Hidden Markov Models",
"Analyticity of Entropy Rates of Continuous-State Hidden Markov Models"
]
| [
"Vladislav Z B Tadić ",
"Arnaud Doucet "
]
| []
| []
| The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory. | 10.1109/tit.2019.2935777 | [
"https://arxiv.org/pdf/1806.09589v2.pdf"
]
| 49,419,330 | 1806.09589 | b4aeabce85908671ddf98eb262eb972d3b8ff394 |
Analyticity of Entropy Rates of Continuous-State Hidden Markov Models
Aug 2018
Vladislav Z B Tadić
Arnaud Doucet
Analyticity of Entropy Rates of Continuous-State Hidden Markov Models
Aug 2018Hidden Markov ModelsEntropy RateRelative Entropy RateLog-LikelihoodOptimal FilterAnalytic Continuation AMS Subject Classification Primary 94A17; Secondary 62B1094A15
The analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied here. Using the analytic continuation principle and the stability properties of the optimal filter, the analyticity of these rates is shown for analytically parameterized models. The obtained results hold under relatively mild conditions and cover several classes of hidden Markov models met in practice. These results are relevant for several (theoretically and practically) important problems arising in statistical inference, system identification and information theory.
Introduction
Hidden Markov models are a powerful and versatile tool for statistical modeling of complex time-series data and stochastic dynamic systems. They can be described as a discrete-time Markov chain observed through noisy measurements of its states. In this context, the entropy rate can be interpreted as a measure of the information revealed by a model (through noisy measurements of the states), while the relative entropy rate can be viewed as a distance between two models.
The entropy rates of hidden Markov models and their analytical properties have recently gained significant attention by the information theory community. These properties and their links with statistical inference, system identification, stochastic optimization and information theory have extensively been studied in several papers [8] - [11], [12], [17], [18], [20], [21]. However, to the best of our knowledge, the existing results on the analytical properties of the entropy rates of hidden Markov models apply exclusively to the models with finite state-spaces and do not address the continuous-state models models at all. Our results presented here are meant to fill this gap in the literature on hidden Markov models and their entropy rates.
In [21], recursive maximum likelihood estimation in finite-state hidden Markov models has been analyzed and a link between its asymptotic properties (convergence and convergence rate) and the analyticity of the underlying average log-likelihood (i.e., of the underlying relative entropy rate) has been established. In view of the recent results on stochastic gradient search [23], a similar link is expected to hold for continuous-state hidden Markov models (including non-linear state-space models). However, to apply the results of [23] to recursive maximum likelihood estimation in continuous-state hidden Markov models, it is necessary to establish results on the analyticity of the average log-likelihood for these models. Hence, one of the first (and most important) steps in the asymptotic analysis of recursive maximum likelihood estimation in continuous-state hidden Markov models would be establishing the analyticity for entropy rates of such models. The results presented here should provide a theoretical basis for this step.
In this paper, the analyticity of the entropy and relative entropy rates of continuous-state hidden Markov models is studied. Using the analytic continuation principle and the stability properties of the optimal filter, we show the analyticity of these rates for analytically parameterized models. The obtained results hold under (relatively) mild conditions and cover a (relatively) broad class of state-space and continuous-state hidden Markov models met in practice. Moreover, these results generalize the existing results on the analyticity of entropy rates of finite-state hidden Markov models. Further to this, the results presented here are relevant for several (theoretically and practically) important problems related to statistical inference, system identification and information theory.
The paper is organized as follows. In Section 2, the entropy rates of hidden Markov models are specified. In he same section, the main results are presented. Examples illustrating the main results are provided in Sections 3 and 4. In Sections 5 -7, the main results are proved.
Main Results
To define hidden Markov models and their entropy rates, we use the following notation. (Ω, F , P ) is a probability space. d x ≥ 1 and d y ≥ 1 are integers, while X ⊆ R dx and Y ⊆ R dy are Borel sets. P (x, dx ′ ) is a transition kernel on X , while Q(x, dy) is a conditional probability measure on Y given x ∈ X . Then, a hidden Markov model can be defined as the X × Y-valued stochastic process {(X n , Y n )} n≥0 (i.e., X n ∈ X , Y n ∈ Y) which is defined on (Ω, F , P ) and satisfies P ((X n+1 , Y n+1 ) ∈ B|X 0:n , Y 0:n ) = I B (x, y)Q(x, dy)P (X n , dx) almost surely for n ≥ 0 and any Borel set B ⊆ X × Y. {X n } n≥0 are the (unobservable) model states, while {Y n } n≥0 are the state-observations. Y n can be interpreted as a noisy measurement of state X n . States {X n } n≥0 form a Markov chain, while P (x, dx ′ ) is their transition kernel. Conditionally on {X n } n≥0 , state-observations {Y n } n≥0 are mutually independent, while Q(X n , dy) is the conditional distribution of Y n given X 0:n . For more details on hidden Markov models, see [1], [5] and references cited therein.
Besides the model {(X n , Y n )} n≥0 , we also consider a parameterized family of hidden Markov models. To define such a family, we rely on the following notation. d ≥ 1 is an integer, while Θ ⊂ R d is an open set. P(X ) is the set of probability measures on X . µ(dx) and ν(dy) are measures on X and Y (respectively). p θ (x ′ |x) and q θ (y|x) are functions which map θ ∈ Θ, x, x ′ ∈ X , y ∈ Y to [0, ∞) and satisfy X p θ (x ′ |x)µ(dx ′ ) = Y q θ (y|x)ν(dy) = 1 for all θ ∈ Θ, x ∈ X . Then, a parameterized family of hidden Markov models can be defined as a collection of X × Y-valued stochastic processes (X θ,λ n , Y θ,λ n ) n≥0 (i.e., X θ,λ n ∈ X , Y θ,λ n ∈ Y) which are defined on (Ω, F , P ), parameterized by θ ∈ Θ, λ ∈ P(X ) and satisfy P (X θ,λ 0 , Y θ,λ 0 ) ∈ B = I B (x, y)q θ (y|x)λ(dx), P (X θ,λ n+1 , Y θ,λ n+1 ) ∈ B X θ,λ 0:n , Y θ,λ 0:n = I B (x, y)q θ (y|x)p θ (x|X θ,λ n )µ(dx)ν(dy) almost surely for n ≥ 0 and any Borel set B ⊆ X × Y. 1 To define the entropy rates, we need the following notation. r θ (y, x ′ |x) is the function defined by
r θ (y, x ′ |x) = q θ (y|x ′ )p θ (x ′ |x)
for θ ∈ Θ, x, x ′ ∈ X , y ∈ Y. For θ ∈ Θ, λ ∈ P(X ), y 1:n = (y 1 , . . . , y n ) ∈ Y n , n ≥ 1, let q n θ (y 1:n |λ) = · · · n k=1 r θ (y k , x k |x k−1 ) µ(dx n ) · · · µ(dx 1 )λ(dx 0 ). 1 In the context of system identification, {(Xn, Yn)} n≥0 is interpreted as the true system (i.e., the system being identified), while (X θ,λ n , Y θ,λ n ) n≥0 is viewed as a candidate model for {(Xn, Yn)} n≥0 .
Moreover, for θ ∈ Θ, λ ∈ P(X ), n ≥ 1, let
h n (θ, λ) = E 1 n log q n θ Y θ,λ 1:n λ , l n (θ, λ) = E 1 n log q n θ (Y 1:n |λ) .(1)
Then, the entropy rate of model (X θ,λ n , Y θ,λ n ) n≥0 (i.e., the entropy rate of stochastic process Y θ,λ n n≥0 ) is defined as lim n→∞ h n (θ, λ). Similarly, the relative entropy rate between models (X θ,λ n , Y θ,λ n ) n≥0 and {(X n , Y n )} n≥0 (i.e., the relative entropy rate between stochastic processes Y θ,λ n n≥0 and {Y n } n≥0 ) is defined as lim n→∞ l n (θ, λ). For more details on the entropy rate, see [6], [7] and references cited therein. 2 We study here the rates lim n→∞ h n (θ, λ), lim n→∞ l n (θ, λ) and their analytical properties. To formulate the assumptions under which these rates are analyzed, we rely on the following notation. For η ∈ C d , η denotes the Euclidean norm of η. For γ ∈ (0, 1), V γ (Θ) is the open γ-vicinity of Θ in C d , i.e., Assumption 2.1 is related to the stability of the hidden Markov model (X θ,λ n , Y θ,λ n ) n≥0 and its optimal filter, while Assumptions 2.2 -2.4 correspond to the parameterization of the model (X θ,λ n , Y θ,λ n ) n≥0 . Assumption 2.1 (together with Assumptions 2.2, 2.4) ensures that the transition kernel of (X θ,λ n , Y θ,λ n ) n≥0 and its analytical continuation are geometrically ergodic (see Lemma 5.4). Assumption 2.1 (together with Assumptions 2.2 -2.4) also ensures that the optimal filter and its analytical continuation forget initial conditions at an exponential rate (see Lemmas 6.2, 6.6). Assumptions 2.1 -2.4 cover several important classes of hidden Markov models met in practice (for further details, see Sections 3,4). In this or similar form, Assumption 2.1 is an ingredient of a number of asymptotic results on optimal filtering and maximum likelihood estimation in hidden Markov models (see [3], [4], [14], [15]; see also [1], [2], [5] and references cited therein).
Assumption 2.5 corresponds to the stability of the hidden Markov model {(X n , Y n )} n≥0 . According to this assumption, stochastic processes {X n } n≥0 and {(X n , Y n )} n≥0 are geometrically ergodic (for further details on geometric ergodicity, see e.g., [16]).
The main results of the paper are contained in the next two theorems.
Theorem 2.1. Let Assumptions 2.1 -2.3 and 2.5 hold. Then, there exists a function l : Θ → R such that l(θ) is real-analytic for each θ ∈ Θ and l(θ) = lim n→∞ l n (θ, λ) for all θ ∈ Θ, λ ∈ P(X ). is real-analytic for each θ ∈ Θ and h(θ) = lim n→∞ h n (θ, λ) for all θ ∈ Θ, λ ∈ P(X ).
Remark. As Θ can be represented as a union of open balls, it is sufficient to show Theorems 2.1 and 2.2 for the case where Θ is convex. Therefore, throughout the analysis carried out in Sections 5 -8, we assume that Θ is an open convex set.
Theorems 2.1 and 2.2 are proved in Section 7. According to these theorems, for all θ ∈ Θ, λ ∈ P(X ), rates lim n→∞ h n (θ, λ) and lim n→∞ l n (θ, λ) are well-defined. According to the same theorems, for each θ ∈ Θ, rates lim n→∞ h n (θ, λ) and lim n→∞ l n (θ, λ) are independent of λ and real-analytic in θ.
Recently, the analytical properties of the entropy rates of hidden Markov models have extensively been studied in several papers [8] - [12], [17], [18], [20], [21]. However, the results presented therein apply only to the models with finite state-spaces and do not address the continuous-state models at all. To the best of our knowledge, Theorems 2.1 and 2.2 are the first result on the analyticity of the entropy rates of continuous-state hidden Markov models. These theorems also generalize the existing results on the analyticity of the entropy rates of finite-state hidden Markov models. 3 Further to this, Theorems 2.1 and 2.2 are relevant for several (theoretically and practically) important problems arising in statistical inference and system identification. E.g., in [24], we crucially rely on these theorems to analyze recursive maximum likelihood estimation in non-linear state-space models. The same theorems can also be used to study the higher-order statistical asymptotics for maximum likelihood estimation in time-series models (for details on such asymptotics, see [25]).
Example: Mixture of Densities
In this section, the main results are applied to the case when p θ (x ′ |x) and q θ (y|x) are mixtures of densities. More specifically, we consider the case where
p θ (x ′ |x) = Nx i=1 a i θ (x)v i (x ′ ), q θ (y|x) = Ny j=1 b j θ (x)w j (y)(2)
This kind of approximation is involved (implicitly or explicitly) in any numerical implementation of the optimal filter for state-space model (3) (for details see e.g., [1], [2], [5] and references cited therein).
The entropy rates of the hidden Markov model (4) are studied under the following assumptions.
Assumption 4.1. X and Y are compact sets with non-empty interiors.
Assumption 4.2. v(x) > 0 and w(y) > 0 for all x ∈ R dx , y ∈ R dy . Moreover, v(x) and w(y) are real-analytic for each x ∈ R dx , y ∈ R dy . Assumption 4.3. B θ (x) and D θ (x) are invertible for all θ ∈ Θ, x ∈ R dx . Moreover, A θ (x), B θ (x), C θ (x) and D θ (x) are real-analytic in (θ, x) for each θ ∈ Θ, x ∈ R dx .
Assumptions 4.1 -4.3 are relevant for several practically important classes of non-linear state-space models. E.g., these assumptions cover stochastic volatility and dynamic probit models and their truncated versions. For other models satisfying (3) and Assumptions 4.1 -4.3, see [1], [2], [5] and references cited therein.
Using Theorems 2.1 and 2.2, we get the following results.
Results Related to Kernels of
{(X n , Y n )} n≥0 and (X θ,λ n , Y θ,λ n ) n≥0
In this section, an analytical (complex-valued) continuation of the transition kernel of (X θ,λ n , Y θ,λ n ) n≥0 is constructed, and its asymptotic properties (geometric ergodicity) are studied. The same properties of the transition kernel of {(X n , Y n )} n≥0 are studied, too.
Throughout this section and the whole paper, the following notation is used. Z is the set defined by For ζ ∈ M c (Z), ζ denotes the total variation norm of ζ, while |ζ|(dz) is the total variation of ζ(dz). ψ(z) is the function defined by ψ(z) = 1 + | log ϕ(y)| (5) for x ∈ X , y ∈ Y and z = (y, x).r η (y, x ′ |x) is the function defined bỹ
r η (y, x ′ |x) = r η (y, x ′ |x) r η (y ′ , x ′′ |x)ν(dy ′ )µ(dx ′′ ), if r η (y ′ , x ′′ |x)ν(dy ′ )µ(dx ′′ ) = 0 0, otherwise(6)
for η ∈ C d , x, x ′ ∈ X , y ∈ Y. u n η (x 0:n , y 1:n ) is the function defined by u n η (x 0:n , y 1:n ) = n k=1r η (y k , x k |x k−1 )
for x 0 , . . . , x n ∈ X , y 1 , . . . , y n ∈ Y, n ≥ 1 (η has the same meaning as in (6)). δ z (dz ′ ) is the Dirac measure on Z centered at z ∈ Z (i.e., δ z (B) = I B (z) for B ∈ B(Z)). σ(dz) is the measure defined by
σ(B) = I B (y, x)Q(x, dy)π(dx)
for B ∈ B(Z). S(z, dz ′ ), S η (z, dz ′ ) are the kernels defined by
S(z, B) = I B (y ′ , x ′ )Q(x ′ , dy ′ )P (x, dx ′ ), S η (z, B) = I B (y ′ , x ′ )r η (y ′ , x ′ |x)ν(dy ′ )µ(dx ′ ) (8)
for x ∈ X , y ∈ Y, B ∈ B(Z) and z = (y, x) (η has the same meaning as in (6)).
{S n (z, dz ′ )} n≥0 , {S n η (z, dz ′ )} n≥0 are the kernels recursively defined by S 0 (z, B) = S 0 η (z, B) = δ z (B) and S n+1 (z, B) = S n (z ′ , B)S(z, dz ′ ), S n+1 η (z, B) = S n η (z ′ , B)S η (z, dz ′ )
for z ∈ Z, B ∈ B(Z), n ≥ 0 (η has the same meaning as in (6)). {(S η ζ) n (dz)} n≥0 are the measures defined by
(S n η ζ)(B) = S n η (z, B)ζ(dz)
for B ∈ B(Z), ζ ∈ M c (Z), n ≥ 0 (η has the same meaning as in (6)). Under the above notation, we have the following: S(z, dz ′ ), S θ (z, dz ′ ) are (respectively) the transition kernels of {(X n , Y n )} n≥0 , (X θ,λ n , Y θ,λ n ) n≥0 , where θ ∈ Θ, λ ∈ P(X ). We also have (S n η ζ)(B) = · · · I B (y n , x n )u n η (x 0:n , y 1:n )(ν × µ)(dy n , dx n ) · · · (ν × µ)(dy 1 , dx 1 )ζ(dy 0 , dx 0 ) (9)
for η ∈ C d , ζ ∈ M c (Z), B ∈ B(Z), n ≥ 1.
Lemma 5.1. Let Assumption 2.5 hold. Then, there exists a real number
C 1 ∈ [1, ∞) such that ψ(z ′ )S(z, dz ′ ) ≤ C 1 , |S n − σ|(z, B) ≤ C 1 ρ n
for all z ∈ Z, B ∈ B(Z), n ≥ 0 (here, |S n − σ|(z, dz ′ ) denotes the total variation of S n (z, dz ′ ) − σ(dz ′ ), while ρ is specified in Assumption 2.5).
Proof. Let C 1 = 2K (K is specified in Assumption 2.5). Moreover, let x, y be any elements of X , Y (respectively), while z = (y, x) (notice that z is any element of Z). Then, we have
ψ(z ′ )S(z, dz ′ ) = (1 + | log ϕ(y ′ )|) Q(x ′ , dy ′ )P (x, dx ′ ) ≤ 1 + K ≤ C 1 .
We also have
|S n (z, B) − σ(B)| = I B (y ′ , x ′ )Q(x ′ , dy ′ )(P n − π)(x, dx ′ ) ≤ I B (y ′ , x ′ )Q(x ′ , dy ′ )|P n − π|(x, dx ′ ) ≤2Kρ n ≤C 1 ρ n for B ∈ B(Z), n ≥ 0.
Lemma 5.2. Let Assumption 2.2 hold. Then, the following is true:
(i)r θ (y, x ′ |x) = r θ (y, x ′ |x) for all θ ∈ Θ, x, x ′ ∈ X , y ∈ Y.
(ii) There exists a real number δ 1 ∈ (0, δ] such thatr η (y, x ′ |x) is analytic in η and satisfies
|r η (y, x ′ |x)| ≤ 2ϕ(y), r η (y ′ , x ′′ |x)ν(dy ′ )µ(dx ′′ ) = 1 for all η ∈ V δ1 (Θ), x, x ′ ∈ X , y ∈ Y (δ is specified in Assumption 2.2).
Remark. As a direct consequence of Lemma 5.2 (Part (ii)), we have S n η ζ ∈ P c (Z) (i.e., (S n η ζ)(Z) = 1) for η ∈ V δ1 (Θ), ζ ∈ P c (Z), n ≥ 1.
Proof. Due to Assumption 2.2, we have
ϕ(y)ν(dy)µ(dx) = µ ϕ(y)ν(dy) < ∞.
Then, using Assumption 2.2 and Lemma A1.1 (see Appendix 1), we conclude that integral
r η (y, x ′ |x)ν(dy)µ(dx ′ )(10)
is analytic in η for each η ∈ V δ (Θ), x ∈ X . Relying on the same arguments, we deduce
|r η ′ (y, x ′ |x) −r η ′′ (y, x ′ |x)| ≤ d ϕ(y) η ′ − η ′′ δ(11)
for η ′ , η ′′ ∈ V δ (Θ), x, x ′ ∈ X , y ∈ Y (here, d denotes the dimension of vectors in Θ, V δ (Θ)). Throughout the rest of the proof, the following notation is used.C, δ 1 are the real numbers defined bỹ
C = d µ δ ϕ(y)ν(dy), δ 1 = min δ, 1 2C .
θ is any element of Θ, while η, η ′ , η ′′ are any elements in V δ1 (Θ).
x, x ′ are any elements of X , while y is any element in Y. Using (11), we conclude
(r η ′ (y, x ′ |x) −r η ′′ (y, x ′ |x)) ν(dy)µ(dx ′ ) ≤ |r η ′ (y, x ′ |x) −r η ′′ (y, x ′ |x)| ν(dy)µ(dx ′ ) ≤ d µ η ′ − η ′′ δ ϕ(y)ν(dy) =C η ′ − η ′′ .
Consequently, we have
r η (y, x ′ |x)ν(dy)µ(dx ′ ) ≥ r θ (y, x ′ |x)ν(dy)µ(dx ′ ) − (r η (y, x ′ |x) −r θ (y, x ′ |x)) ν(dy)µ(dx ′ ) ≥1 −C η − θ ≥ 1 2 when η − θ ≤ δ 1 . Hence, we get r η (y, x ′ |x)ν(dy)µ(dx ′ ) ≥ 1 2(12)
(since η ∈ V δ1 (Θ), there exists θ ∈ Θ such that η − θ < δ 1 ). Therefore, we havẽ
r η (y, x ′ |x) =r η (y, x ′ |x) r(y ′ , x ′′ |x)ν(dy ′ )µ(dx ′′ ) .(13)
As integral (10) is analytic in η for each η ∈ V δ1 (Θ), we conclude from Assumption 2.2 and (12), (13) that (i), (ii) are true.
Lemma 5.3. Let Assumption 2.2 hold. Then, the following is true: (i) u n η (x 0:n , y 1:n ) is analytic in η for all η ∈ V δ1 (Θ), x 0 , . . . , x n ∈ X , y 1 , . . . , y n ∈ Y, n ≥ 1 (δ 1 is specified in Lemma 5.2).
(ii) There exists a non-decreasing sequence {K n } n≥1 in [1, ∞) such that u n η (x 0:n , y 1:n ) ≤ K n n k=1 ϕ(y k ) , u n η ′ (x 0:n , y 1:n ) − u n η ′′ (x 0:n , y 1:
n ) ≤ K n η ′ − η ′′ n k=1 ϕ(y k )
for all η, η ′ , η ′′ ∈ V δ1 (Θ), x 0 , . . . , x n ∈ X , y 1 , . . . , y n ∈ Y, n ≥ 1.
Proof. Throughout the proof, the following notation is used. {K n } n≥1 are the real numbers defined by K n = 2 n d/δ 1 for n ≥ 1 (here, d denotes the dimension of vectors in Θ, V δ (Θ)). η, η ′ , η ′′ are any elements of V δ1 (Θ). {x n } n≥0 , {y n } n≥1 are any sequences in X , Y (respectively).
Owing to Lemma 5.2, u n η (x 0:n , y 1:n ) is analytic in η for each η ∈ V δ1 (Θ). Due to the same lemma, we have u n η (x 0:n , y 1:n ) ≤ 2 n n k=1 ϕ(y k ) ≤ K n n k=1 ϕ(y k ) for n ≥ 1. Consequently, Lemma A1.1 (see Appendix 1) yields
u n η ′ (x 0:n , y 1:n ) − u n η ′′ (x 0:n , y 1:n ) ≤ 2 n d η ′ − η ′′ δ 1 n k=1 ϕ(y k ) = K n η ′ − η ′′ n k=1 ϕ(y k )
for n ≥ 1.
Lemma 5.4. Let Assumptions 2.1, 2.2 and 2.4 hold. Then, the following is true:
(i) There exist real numbers δ 2 ∈ (0, δ 1 ], C 2 ∈ [1, ∞) such that |S η ′ − S η ′′ | (z, B) ≤ C 2 η ′ − η ′′ , ψ(z ′ ) |S η | (z, dz ′ ) ≤ C 2 for all η, η ′ , η ′′ ∈ V δ2 (Θ), z ∈ Z, B ∈ B(Z) (here, |S η ′ − S η ′′ | (z, dz ′ ) denotes the total variation of S η ′ (z, dz ′ ) − S η ′′ (z, dz ′ ), while δ 1 is specified in Lemma 5.2). (ii) For each η ∈ V δ2 (Θ), there exists a complex measure σ η (dz) on Z such that σ η (B) = lim n→∞ S n η (z, B) for all z ∈ Z, B ∈ B(Z).
(iii) There exists a real number γ 1 ∈ (0, 1), such that
S n η − σ η (z, B) ≤ C 2 γ n 1 for all η ∈ V δ2 (Θ), z ∈ Z, B ∈ B(Z), n ≥ 0 (here, S n η − σ η (z, dz ′ ) stands for the total variation of S n η (z, dz ′ ) − σ η (dz ′ )).
Proof. Throughout the proof, the following notation is used.C 1 ,C 2 are the real numbers defined bỹ
C 1 = 2 µ | log ϕ(y)|ϕ(y)ν(dy),C 2 = 2 µ ϕ(y)ν(dy),
while n 0 is the integer defined as
n 0 = log 4 | log(1 − ε 2 )| (ε, ϕ(y) are specified in Assumptions 2.1, 2.3). {K n } n≥1
are the real numbers defined byK n = (1+C 2 ) n K n for n ≥ 1, while δ 2 , γ 1 ,C 3 , C 2 are the real numbers defined as
δ 2 = δ 1 /(4K n0 ), γ 1 = 2 −1/n0 ,C 3 =K 1 +C 1 +C 2 , C 2 = 16C 3 γ −n0 1 (δ 1 , K n are specified in Lemmas 5.2, 5.3). θ is any element of Θ, while η, η ′ , η ′′ are any elements in V δ2 (Θ).
x, y are any elements of X , Y (respectively), while z = (y, x) (notice that z is any element of Z). ζ, ζ ′ , ζ ′′ are any elements of P c (Z). Relying on Assumption 2.4 and Lemma 5.2, we deduce
ψ(z ′ ) |S η | (z, dz ′ ) ≤ (1 + |log ϕ(y ′ )|) |r η (y ′ , x ′ |x)| ν(dy ′ )µ(dx ′ ) ≤2 µ (1 + |log ϕ(y ′ )|) ϕ(y ′ )ν(dy ′ ) =C 1 +C 2 ≤C 2 (notice that z = (y, x) andC 1 +C 2 ≤C 3 ≤ C 2 )
. On the other side, using Lemma 5.3, we conclude
(S n η ′ ζ)(B) − (S n η ′′ ζ)(B) ≤ · · · I B (y n , x n ) u n η ′ (x 0:n , y 1:n ) − u n η ′′ (x 0:n , y 1:n ) · (ν × µ)(dy n , dx n ) · · · (ν × µ)(dy 1 , dx 1 )|ζ|(dy 0 , dx 0 ) ≤K n µ n ζ η ′ − η ′′ n k=1 ϕ(y k )ν(dy k ) ≤K n ζ η ′ − η ′′ for B ∈ B(Z), n ≥ 1. Therefore, we get S n η ′ ζ − S n η ′′ ζ ≤K n ζ η ′ − η ′′(14)
for n ≥ 1. Hence, we have
|S η ′ − S η ′′ | (z, B) = |S η ′ δ z − S η ′′ δ z | (B) ≤K 1 δ z η ′ − η ′′ ≤ C 2 η ′ − η ′′ for B ∈ B(Z). 6
Let τ θ (dz) be the measure defined by
τ θ (B) = I B (y, x)λ θ (dx|y)ν(dy)
for B ∈ B(Z). Owing to Assumption 2.1, we have
1 = r θ (y, x ′ |x)ν(dy)µ(dx ′ ) ≤ 1 ε λ θ (X |y)ν(dy).
Hence, we get
τ θ (Z) = λ θ (X |y)ν(dy) ≥ ε.
On the other side, due to Assumption 2.1 and Lemma 5.2, we have
S θ (z, B) = I B (y ′ , x ′ )r θ (y ′ , x ′ |x)ν(dy ′ )µ(dx ′ ) ≥ ε I B (y ′ , x ′ )λ θ (dx ′ |y ′ )ν(dy ′ ) = ετ θ (B)
for B ∈ B(Z) (notice that z = (y, x)). Then, standard results in Markov chain theory (see e.g., [16,Theorem 16.0.2]) imply that there exists a probability measure σ θ (dz) on Z such that
|S n θ (z, B) − σ θ (B)| ≤ (1 − ετ θ (Z)) n ≤ (1 − ε 2 ) n for B ∈ B(Z), n ≥ 1. Consequently, we get |(S n θ ζ ′ )(B) − (S n θ ζ ′′ )(B)| = (S n θ − σ θ ) (z, B)(ζ ′ − ζ ′′ )(dz) ≤ |S n θ − σ θ | (z, B)|ζ ′ − ζ ′′ |(dz) ≤(1 − ε 2 ) n ζ ′ − ζ ′′ for B ∈ B(Z), n ≥ 1. 7 Hence, we have S n θ ζ ′ − S n θ ζ ′′ ≤ (1 − ε 2 ) n ζ ′ − ζ ′′(15)
for n ≥ 1.
Owing to (14), we have
S n η ζ ≤ S n θ ζ + S n η − S n θ ζ ≤ 1 +K n η − θ ζ ≤ (1 +K n δ 2 ) ζ ≤ 2 ζ when 1 ≤ n ≤ n 0 , η − θ < δ 2 . 8 Thus, we get S n η ζ ≤ 2 ζ(16)
for 1 ≤ n ≤ n 0 (since η ∈ V δ2 (Θ), there exists θ ∈ Θ such that η − θ < δ 2 ). On the other side, due to (14), (15), we have
S n η ζ ′ − S n η ζ ′′ ≤ S n θ ζ ′ − S n θ ζ ′′ + (S n η − S n θ )(ζ ′ − ζ ′′ ) ≤ (1 − ε 2 ) n +K n η − θ ζ ′ − ζ ′′ ≤ (1 − ε 2 ) n + 1 4 ζ ′ − ζ ′′ when 1 ≤ n ≤ n 0 , η − θ < δ 2 . Hence, we get S n η ζ ′ − S n η ζ ′′ ≤ (1 − ε 2 ) n + 1 4 ζ ′ − ζ ′′ 1 ≤ n ≤ n 0 (since η ∈ V δ2 (Θ), there exists θ ∈ Θ such that η − θ < δ 2 ). Setting n = n 0 , we conclude S n0 η ζ ′ − S n0 η ζ ′′ ≤ ζ ′ − ζ ′′ 2 (notice that (1 − ε 2 ) n0 ≤ 1/4). Consequently, we have S (k+1)n0 η (ζ ′ − ζ ′′ ) = S n0 η S kn0 η ζ ′ − S kn0 η ζ ′′ ≤ 1 2 S kn0 η (ζ ′ − ζ ′′ )(17)
for k ≥ 0. 9 Iterating (17), we get
S kn0 η (ζ ′ − ζ ′′ ) ≤ 1 2 k ζ ′ − ζ ′′(18)
for k ≥ 0. Using (16), (18), we conclude
S (k+1)n0 η ζ − S kn0 η ζ = S kn0 η S n0 η ζ − ζ ≤ 1 2 k S n0 η ζ − ζ ≤ 1 2 k S n0 η ζ + ζ ≤ ζ 2 k−2 (19) for k ≥ 0. Hence, we get ∞ k=0 S (k+1)n0 η ζ − S kn0 η ζ ≤ ∞ k=0 ζ 2 k−2 = 8 ζ < ∞.(20)
Let (S ∞ η ζ)(dz) be the measure defined by
(S ∞ η ζ)(B) = ζ(B) + ∞ k=0 (S (k+1)n0 η ζ)(B) − (S kn0 η ζ)(B) for B ∈ B(Z). Then, due to (20), (S ∞ η ζ)(dz) is well-defined and satisfies S ∞ η ζ ∈ P c (Z).
On the other side, owing to (19), (20), we have
S kn0 η ζ − S ∞ η ζ = ∞ j=k S (j+1)n0 η ζ − S jn0 η ζ ≤ ∞ j=k S (j+1)n0 η ζ − S jn0 η ζ ≤ ∞ j=k ζ 2 j−2 = ζ 2 k−3(21)
for k ≥ 0. Combining this with (18), we get
S ∞ η ζ ′ − S ∞ η ζ ′′ ≤ S kn0 η ζ ′ − S ∞ η ζ ′ + S kn0 η ζ ′′ − S ∞ η ζ ′′ + S kn0 η ζ ′ − S kn0 η ζ ′′ ≤ ζ ′ + ζ ′′ + ζ ′ − ζ ′′ 2 k−3 for k ≥ 0. Therefore, S ∞ η ζ ′ = S ∞ η ζ ′′ for any ζ ′ , ζ ′′ ∈ P c (Z). Consequently, there exists σ η ∈ P c (Z) such that S ∞ η ζ = σ η for any ζ ∈ P c (Z). Then, (16), (21) imply S n η ζ − σ η = S kn0 η (S n−kn0 η ζ) − S ∞ η (S n−kn0 η ζ) ≤ 1 2 k−3 S n−kn0 η ζ ≤ ζ 2 k−4 ≤ C 2 γ n 1 ζ(22)
for (k + 1)n 0 ≥ n > kn 0 , k ≥ 0. 10 Thus, we get (22)).
S n η − σ η (z, B) = S n η δ z − σ η (B) ≤ S n η δ z − σ η ≤ C 2 γ n 1 δ z = C 2 γ n 1 for B ∈ B(Z), n ≥ 0 (set k = ⌊(n − m)/n 0 ⌋ in
Results Related to Optimal Filter
In this section, an analytic (complex-valued) continuation of the optimal filter is constructed, and its asymptotic properties (exponential forgetting) are studied. 9 Notice that S n η ζ ∈ Pc(Z) for each n ≥ 1 (see Lemma 5.2 and the remark immediately after its statement). 10 Notice that ση = S ∞ η (S n−kn 0 η ζ) (as S n−kn 0 η ζ ∈ Pc(Z)). Notice also 2 −(k−4) = 16γ kn 0
1 = (16γ −(n−kn 0 ) 1 )γ n 1 ≤ (16γ −n 0 1 )γ n 1 ≤ C 2 γ n 1 .
In addition to the notation introduced in Section 5, the following notation is used here, too. B(X ) is the collection of Borel sets in X . P(X ) is the collection of probability measures on X , while M p (X ) is the set of positive measures on X . M c (X ) is the collection of complex measures on X . For ξ ∈ M c (X ), ξ denotes the total variation norm of ξ, while |ξ|(dx) is the total variation of ξ(dx). For γ ∈ (0, 1), V γ (P(X )) is the open γ-vicinity of P(X ), i.e.,
V γ (P(X )) = {ξ ∈ M c (X ) : ∃λ ∈ P(X ), ξ − λ < γ}. δ x (dx ′ ) is the Dirac measure on X centered at x ∈ X (i.e., δ x (B) = I B (x) for B ∈ B(X )). R η,y (dx|ξ) is the measure defined by R η,y (B|ξ) = I B (x ′ )r η (y, x ′ |x)µ(dx ′ )ξ(dx)(23)
for
η ∈ C d , ξ ∈ M c (X ), B ∈ B(X ), y ∈ Y (r η (y, x ′ |x) is specified in Lemma 5.2). Φ η,y (ξ) is the function defined by Φ η,y (ξ) = log R η,y (X |ξ), if R η,y (X |ξ) = 0 0, otherwise(24)
(η, ξ, y have the same meaning as in (23)). r m:n η,y (x ′ |x) is the function defined by
r m:n η,y (x ′ |x) = · · · u n−m η (x m:n , y m+1:n )δ x ′ (dx n )µ(dx n−1 ) · · · µ(dx m+1 )δ x (dx m )(25)
for x, x ′ ∈ X , n > m ≥ 0 and a sequence y = {y n } n≥1 in Y (η has the same meaning as in (23), while u n−m η (x m:n , y m+1:n ) is specified in (7)). R m:m η,y (dx|ξ) and R m:n η,y (dx|ξ) are the measures defined by
R m:m η,y (B|ξ) = ξ(B), R m:n η,y (B|ξ) = I B (x ′ )r m:n η,y (x ′ |x)µ(dx ′ )ξ(dx)(26)
for B ∈ B(X ), n > m ≥ 0 (η, ξ, y have the same meaning as in (23), (25)). f m:n η,y (x|ξ), g m:n η,y (x ′ |x, ξ), h m:n η,y (x|x ′ , ξ) are the functions defined by
g m:n η,y (x ′ |x, ξ) = r m:n η,y (x ′ |x) R m:n η,y (X |ξ), if R m:n η,y (X |ξ) = 0 0, otherwise ,(27)
f m:n η,y (x|ξ) = g m:n η,y (x|x ′′ , ξ)ξ(dx ′′ ),
h m:n η,y (x ′ |x, ξ) = g m:n η,y (x ′ |x, ξ) − f m:n η,y (x ′ |ξ) g m:n η,y (x ′′ |x, ξ)µ(dx ′′ )(28)
for n > m ≥ 0 (η, x, x ′ , ξ, y have the same meaning as in (23), (25)). F m:m η,y (dx|ξ) and F m:n η,y (dx|ξ) are the measures defined by
F m:m η,y (B|ξ) = ξ(B), F m:n η,y (B|ξ) = I B (x)f m:n η,y (x|ξ)µ(dx)(30)
for B ∈ B(X ), n > m ≥ 0 (η, ξ, y have the same meaning as in (23), (25)). Throughout this section and the whole paper, measures R m:n η,y (dx|ξ), F m:n η,y (dx|ξ) are also denoted by R m:n η,y (ξ), F m:n η,y (ξ) (short-hand notation), while R m:n η,y (ξ) , F m:n η,y (ξ) are defined by R m:n η,y (ξ) = R m:n η,y (X |ξ), F m:n η,y (ξ) = F m:n η,y (X |ξ).
Under the notation introduced above, we have the following. F m:n θ,y (λ) is the optimal filter for the model (X θ,λ n , Y θ,λ n ) n≥0 , i.e., F 1:n θ,y (B|λ) = P X θ,λ n ∈ B Y θ,λ 1:n = y 1:n for θ ∈ Θ, λ ∈ P(X ), B ∈ B(X ) and a sequence y = {y n } n≥1 in Y.
Lemma 6.1. Let η, ξ be any elements of C d , M s (X ) (respectively), while y = {y n } n≥1 is any sequence in Y. Moreover, let n, m, k be any integers satisfying n ≥ k ≥ m. Then, the following is true:
(i) R m:n η,y (ξ) = R k:n η,y R m:k η,y (ξ) . (ii) R m:n η,y (ξ) = R k:n η,y F m:k η,y (ξ) R m:k η,y (ξ) if R m:k η,y (ξ) = 0. (iii) F m:n η,y (ξ) = F k:n η,y F m:k η,y (ξ) if R m:k η,y (ξ) = 0 and R m:n η,y (ξ) = 0. Proof. (i) When k = m or k = n, (i) is trivially satisfied.
In what follows in this part of the proof, we assume n > k > m.
Owing to (7), we have
u n−m η (x m:n , y m+1:n ) = u n−k η (x k:n , y k+1:n )u k−m η (x m:k , y m+1:k )
for x m , . . . , x n ∈ X . Combining this with (25), it is easy to show
r m:n η,y (x ′ |x) = r k:n η,y (x ′ |x ′′ )r m:k η,y (x ′′ |x)µ(dx ′′ ) for x, x ′ ∈ X .
Then, using (26), we conclude
R m:n η,y (B|ξ) = I B (x ′ )r k:n η,y (x ′ |x ′′ )r m:k η,y (x ′′ |x)µ(dx ′′ )µ(dx ′ )ξ(dx) = I B (x ′ )r k:n η,y (x ′ |x ′′ )R m:k η,y (dx ′′ |ξ)µ(dx ′ ) =R k:n η,y B|R m:k η,y (ξ)
for B ∈ B(X ). Hence, (i) holds when n > k > m.
(ii) We assume R m:k η,y (ξ) = 0 (i.e., R m:k η,y (X |ξ) = 0). Then, using (27), (28), (30), we conclude
F m:k η,y (ξ) = R m:k η,y (ξ) R m:k η,y (ξ) .(31)
Consequently, we have R k:n η,y F m:k η,y (ξ) = R k:n η,y R m:k η,y (ξ) R m:k η,y (ξ) (notice that R k:n η,y (ξ) is linear in ξ). Combining this with (i), we get R m:n η,y (ξ) = R k:n η,y R m:k η,y (ξ) = R k:n η,y F m:k η,y (ξ) R m:k η,y (ξ) .
Thus, (ii) is true.
(iii) We assume R m:k η,y (ξ) = 0, R m:n η,y (ξ) = 0. Therefore, (ii) implies R k:n η,y F m:k η,y (ξ) = 0. Then, using the same arguments as in (ii), we deduce
F m:n η,y (ξ) = R m:n η,y (ξ) R m:n η,y (ξ) , F k:n η,y F m:k η,y (ξ) = R k:n η,y F m:k η,y (ξ) R k:n η,y F m:k η,y (ξ)
.
Combining this with (i) and (31), (32), we get
F m:n η,y (ξ) = R k:n η,y R m:k η,y (ξ) R k:n η,y F m:k η,y (ξ) R m:k η,y (ξ) = R k:n η,y F m:k η,y (ξ) R k:n η,y F m:k η,y (ξ) = F k:n η,y F m:k η,y (ξ) (notice that R k:n η,y (ξ) is linear in ξ). Hence, (iii) holds. Lemma 6.2. Let Assumption 2.1 hold. Then, there exist real numbers δ 3 ∈ (0, δ 1 ], γ 2 ∈ (0, 1), C 3 ∈ [1, ∞) such that F m:n θ,y (λ ′ ) − F m:n θ,y (λ ′′ ) ≤ C 3 γ n−m 2 λ ′ − λ ′′
for all θ ∈ Θ, λ ′ , λ ′′ ∈ V δ3 (P(X )) ∩ M p (X ), n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y (δ 1 is specified in Lemma 5.2).
Proof. Due to [15,
∈ (0, 1), C 3 ∈ [1, ∞) such that F m:n θ,y (λ ′ ) − F m:n θ,y (λ ′′ ) ≤ C 3 γ n−m 2 4 λ ′ λ ′ − λ ′′ λ ′′(33)
for all θ ∈ Θ, λ ′ , λ ′′ ∈ M p (X ), n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y (notice that F m:n θ,y (λ) = F m:n θ,y (λ/ λ ) for each λ ∈ M p (X )). Let δ 3 = min{1/2, δ 1 }, while y = {y n } n≥1 is any sequence in Y. Then, we have λ ≥ 1 − δ 3 ≥ 1/2 for λ ∈ V δ3 (P(X )) ∩ M p (X ). Consequently, (33) implies
F m:n θ,y (λ ′ ) − F m:n θ,y (λ ′′ ) ≤ C 3 γ n−m 2 4 λ ′ − λ ′′ λ ′ − λ ′′ ( λ ′ − λ ′′ ) λ ′ λ ′′ ≤ C 3 γ n−m 2 λ ′ − λ ′′ 2 λ ′ ≤C 3 γ n−m 2 λ ′ − λ ′′ for θ ∈ Θ, λ ′ , λ ′′ ∈ V δ3 (P(X ))∩M p (X ), n ≥ m ≥ 0 (notice that 2 λ ′ ≥ 1, λ ′′ / λ ′′ = 1, | λ ′ − λ ′′ | ≤ λ ′ − λ ′′ ).
Lemma 6.3. Let Assumptions 2.2 and 2.3 hold. Then, the following is true:
(i) R m:n η,y (ξ) is analytic in η for all η ∈ V δ1 (Θ), ξ ∈ V δ1 (P(X )), n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y (δ 1 is specified in Lemma 5.2). (ii) There exists a non-decreasing sequence {L n } n≥1 in [1, ∞) such that R m:n η,y (ξ) ≤ L n−m n k=m+1 ϕ(y k ) , R m:n η ′ ,y (ξ ′ ) − R m:n η ′′ ,y (ξ ′′ ) ≤ L n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) n k=m+1 ϕ(y k )
for all η, η ′ , η ′′ ∈ V δ1 (Θ), ξ, ξ ′ , ξ ′′ ∈ V δ1 (P(X )), n > m ≥ 0 and any sequence y = {y n } n≥1 in Y. for all η ∈ V αn−m (Θ), ξ ∈ V αn−m (P(X )), n > m ≥ 0 and any sequence y = {y n } n≥1 in Y. (iv) There exists a non-decreasing sequence
{M n } n≥1 in [1, ∞) such that max f m:n η,y (x|ξ) , h m:n η,y (x ′ |x, ξ) ≤ M n−m , max f m:n η ′ ,y (x|ξ ′ ) − f m:n η ′′ ,y (x|ξ ′′ ) , h m:n η ′ ,y (x ′ |x, ξ ′ ) − h m:n η ′′ ,y (x ′ |x, ξ ′′ ) ≤ M n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) for all η, η ′ , η ′′ ∈ V αn−m (Θ), ξ, ξ ′ , ξ ′′ ∈ V αn−m (P(X )),
x, x ′ ∈ X , n > m ≥ 0 and any sequence y = {y n } n≥1 in Y.
Proof. (i) and (ii) Throughout these parts of the proof, the following notation is used. {L l } l≥1 , {L l } l≥1 are the real numbers defined byL
l = K l µ + 1 γ l , L l = 2L 2 l(34)
for l ≥ 1 (γ, K l are specified in Assumption 2.3 and Lemma 5.3). m, n are any integers satisfying n > m ≥ 0. In what follows in the proof of (i), (ii), both m, n are kept fixed. θ is any element of Θ, while η, η ′ , η ′′ are any elements in V δ1 (Θ). ξ, ξ ′ , ξ ′′ are any elements of V δ1 (P(X )).
x, x ′ are any elements of X , while y = {y n } n≥0 is any sequence in Y.
Using (25), it is straightforward to verify R m:n η,y (ξ) = · · · u n−m η (x m:n , y m+1:n )µ(dx n ) · · · µ(dx m+1 )ξ(dx m ).
It is also easy to show
· · · n k=m+1 ϕ(y k ) µ(dx n ) · · · µ(dx m+1 )|ξ|(dx m ) = ξ µ n−m n k=m+1 ϕ(y k ) < ∞.
Consequently, Lemmas 5.3, A1.1 (see Appendix 1) imply that R m:n η,y (ξ) is analytic in η for each η ∈ V δ1 (Θ). 11 Hence, (i) holds.
Owing to Lemma 5.3 and (35), we have
R m:n η,y (ξ) ≤ · · · u n−m η (x m:n , y m+1:n ) µ(dx n ) · · · µ(dx m+1 ) |ξ|(dx m ) ≤K n−m µ n−m ξ n k=m+1 ϕ(y k ) ≤2L n−m n k=m+1 ϕ(y k ) ≤L n−m n k=m+1 ϕ(y k )(36)
(since ξ ∈ V δ1 (P(X )), we have ξ ≤ 1 + δ 1 ≤ 2). Due to the same arguments, we have R m:n η ′ ,y (ξ ′ ) − R m:n η ′′ ,y (ξ ′′ ) ≤ · · · u n−m η ′ (x m:n , y m+1:n ) − u n−m η ′′ (x m:n , y m+1:n )
· µ(dx n ) · · · µ(dx m+1 ) |ξ ′ |(dx m ) + · · · u n−m η ′′ (x m:n , y m+1:n ) µ(dx n ) · · · µ(dx m+1 ) |ξ ′ − ξ ′′ |(dx) ≤K n−m µ n−m ( ξ ′ η ′ − η ′′ + ξ ′ − ξ ′′ ) n k=m+1 ϕ(y k ) ≤2L n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) n k=m+1 ϕ(y k ) ≤L n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) n k=m+1 ϕ(y k )(37)
(notice that ξ ′ ≤ 1 + δ 1 ≤ 2). Using (36), (37), we conclude that (ii) is true.
(iii) and (iv) Throughout these parts of the proof, we use the following notation. {L l } l≥1 has the same meaning as in (34), while {α l } l≥1 , {M l } l≥1 , {M l } l≥1 are the numbers defined by
α l = δ 1 8L 2 l ,M l = 10L 4 l , M l = 5M 2 l ( µ + 1)
for l ≥ 1. m, n are any integers satisfying n > m ≥ 0. In what follows in the proof of (iii), (iv), both m, n are kept fixed. θ is any element of Θ, while η, η ′ , η ′′ are any elements of V αn−m (Θ). λ is any element of 11 Notice that u n−m η (xm:n, y m+1:n ) is analytic in η for each η ∈ V δ 1 (Θ), xm, . . . , xn ∈ X , y m+1 , . . . , yn ∈ Y. P(X ), while ξ, ξ ′ , ξ ′′ are any elements of V αn−m (P(X )).
x, x ′ are any elements of X , while y = {y n } n≥1 is any sequence in Y. Using Lemma 5.2 and (25), it is straightforward to verify
R m:k+1 θ,y (λ) = · · · r θ (y k+1 , x k+1 |x k )µ(dx k+1 ) u k−m θ (x m:k , y m+1:k ) · µ(dx k+1 ) · · · µ(dx m+1 )λ(dx m )
for k > m. Consequently, Assumption 2.3 and Lemma 5.2 yield
R m:k+1 θ,y (λ) ≥γ ϕ(y k+1 ) · · · u k−m θ (x m:k , y m+1:k )µ(dx k ) · · · µ(dx m+1 )λ(dx m ) =γ ϕ(y k+1 ) R m:k θ,y (λ)(38)
for k > m. The same arguments also imply
R m:m+1 θ,y (λ) = r θ (y m+1 , x m+1 |x m )µ(dx m+1 ) λ(dx m ) ≥ γ ϕ(y m+1 ) λ = γ ϕ(y m+1 ).
Then, iterating (38), we get
R m:k+1 θ,y (λ) ≥ γ k−m−1 k+1 l=m+2 ϕ(y l ) R m:m+1 θ,y (λ) ≥ γ k−m k+1 l=m+1 ϕ(y l )
for k > m. Hence, we have ) ). Combining this with (37), we get Re R m:n η,y (ξ) ≥R m:n θ,y (λ) − R m:n η,y (ξ) − R m:n θ,y (λ)
R m:n θ,y (λ) ≥ 1 L n−m n k=m+1 ϕ(y k ) (notice thatL n−m ≥ γ −(n−m≥ 1 L n−m − 2L n−m ( η − θ + ξ − λ ) n k=m+1 ϕ(y k ) ≥ 1 L n−m − 4L n−m α n−m n k=m+1 ϕ(y k ) ≥ 1 2L n−m n k=m+1 ϕ(y k ) when η − θ < α n−m , ξ − λ < α n−m . Thus, we have Re R m:n η,y (ξ) ≥ 1 2L n−m n k=m+1 ϕ(y k ) ≥ 1 L n−m n k=m+1 ϕ(y k ) (39) (since η ∈ V αn−m (Θ), ξ ∈ V αn−m (P(X )), there exist θ ∈ Θ, λ ∈ P(X ) such that η − θ < α n−m , ξ − λ < α n−m ).
Owing to Lemma 5.3, we have
r m:n η,y (x ′ |x) ≤ · · · u n−m η (x m:n , y m+1:n ) δ x ′ (dx n )µ(dx n−1 ) · · · µ(dx m+1 )δ x (dx m ) ≤K n−m δ x δ x ′ µ n−m
Then, (27), (39) imply g m:n η,y (x ′ |x, ξ) = r m:n η,y (x ′ |x) R m:n η,y (ξ)
≤ 2L 2 n−m ≤M n−m(41)
(notice that R m:n η,y (ξ) ≥ Re R m:n η,y (ξ) > 0). Consequently, (28) yields
f m:n η,y (x|ξ) ≤ g m:n η,y (x|x ′ , ξ) |ξ|(dx ′ ) ≤M n−m ξ ≤ 2M n−m ≤ M n−m(42)
(since ξ ∈ V αn−m (P(X )), we have ξ ≤ 1 + α n−m ≤ 2). Similarly, we have
g m:n η,y (x ′ |x, ξ) µ(dx ′ ) ≤M n−m µ .(43)
Combining this with (29), (41), (42), we get h m:n η,y (x ′ |x, ξ) ≤ g m:n η,y (x ′ |x, ξ) + f m:n η,y (x ′ |ξ) g m:n η,y (x ′′ |x, ξ) µ(dx ′′ )
≤M n−m + 2M 2 n−m µ ≤M n−m .(44)
Due to Lemma 5.3, we have
r m:n η ′ ,y (x ′ |x) − r m:n η ′′ ,y (x ′ |x) ≤ · · · u n−m η ′ (x m:n , y m+1:n ) − u n−m η ′′ (x m:n , y m+1:n ) · δ x ′ (dx n )µ(dx n−1 ) · · · µ(dx m+1 )δ x (dx m ) ≤K n−m µ n−m−1 η ′ − η ′′ n k=m+1 ϕ(y k ) ≤L n−m η ′ − η ′′ n k=m+1 ϕ(y k ) .
Then, (27), (37), (39), (40) yield g m:n η ′ ,y (x ′ |x, ξ ′ ) − g m:n η ′′ ,y (x ′ |x, ξ ′′ ) ≤ r m:n η ′ ,y (x ′ |x) − r m:n η ′′ ,y (x ′ |x) R m:n η ′ ,y (ξ ′ ) + r m:n η ′′ ,y (x ′ |x) R m:n η ′′ ,y (ξ ′′ )
R m:n η ′ ,y (ξ ′ ) − R m:n η ′′ ,y (ξ ′′ ) R m:n η ′ ,y (ξ ′ ) ≤10L 4 n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤M n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ).(45)
Consequently, (28), (41) imply
f m:n η ′ ,y (x|ξ ′ ) − f m:n η ′′ ,y (x|ξ ′′ ) ≤ g m:n η ′ ,y (x|x ′ , ξ ′ ) − g m:n η ′′ ,y (x|x ′ , ξ ′′ ) |ξ ′ |(dx ′ ) + g m:n η ′′ ,y (x|x ′ , ξ ′′ ) |ξ ′ − ξ ′′ |(dx ′ ) ≤M n−m ξ ′ ( η ′ − η ′′ + ξ ′ − ξ ′′ ) +M n−m ξ ′ − ξ ′′ ≤3M n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤M n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ )(46)
(notice that ξ ′ ≤ 1 + α n−m ≤ 2). Similarly, we get
g m:n η ′ ,y (x ′ |x, ξ ′ ) − g m:n η ′′ ,y (x ′ |x, ξ ′′ ) µ(dx ′ ) ≤M n−m µ ( η ′ − η ′′ + ξ ′ − ξ ′′ ).
Combining this with (29), (42), (43), (45), (46), we get
h m:n η ′ ,y (x ′ |x, ξ ′ ) − h m:n η ′′ ,y (x ′ |x, ξ ′′ ) ≤ g m:n η ′ ,y (x ′ |x, ξ ′ ) − g m:n η ′′ ,y (x ′ |x, ξ ′′ ) + f m:n η ′ ,y (x ′ |ξ ′ ) − f m:n η ′′ ,y (x ′ |ξ ′′ ) g m:n η ′ ,y (x ′′ |x, ξ ′ ) µ(dx ′′ ) + f m:n η ′′ ,y (x ′ |ξ ′′ ) g m:n η ′ ,y (x ′′ |x, ξ ′ ) − g m:n η ′′ ,y (x ′′ |x, ξ ′′ ) µ(dx ′′ ) ≤(M n−m + 5M 2 n−m µ )( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤M n−m ( η ′ − η ′′ + ξ ′ − ξ ′′ ).(47)
Using (39), (42), (44) -(47), we conclude that (iii), (iv) hold.
Lemma 6.4. Let Assumptions 2.2 and 2.3 hold. Then, the following is true: (i) There exists a real number δ 4 ∈ (0, δ 1 ] such that Re {R η,y (X |ξ)} > 0 for all η ∈ V δ4 (Θ), ξ ∈ V δ4 (P(X )), y ∈ Y (δ 1 is specified in Lemma 5.2).
(ii) There exists a real number
C 4 ∈ [1, ∞) such that |Φ η,y (ξ)| ≤ C 4 (1 + | log ϕ(y)|) , |Φ η ′ ,y (ξ ′ ) − Φ η ′′ ,y (ξ ′′ )| ≤ C 4 ( η ′ − η ′′ + ξ ′ − ξ ′′ ) for all η, η ′ , η ′′ ∈ V δ4 (Θ), ξ, ξ ′ , ξ ′′ ∈ V δ4 (P(X )), y ∈ Y.
Proof. Throughout the proof, the following notation is used. δ 4 , C 4 are the real numbers defined by δ 4 = α 1 , C 4 = 5L 2 1 (α 1 , L 1 are specified in Lemma 6.3). η, η ′ , η ′′ are any elements of V δ4 (Θ), while ξ, ξ ′ , ξ ′′ are any elements in V δ4 (P(X )). y is any element of Y.
Using Lemma 6.3, we conclude
Re {R η,y (X |ξ)} ≥ ϕ(y) L 1 ,(48)
|R η,y (X |ξ)| ≤ L 1 ϕ(y),
|R η ′ ,y (X |ξ ′ ) − R η ′′ ,y (X |ξ ′′ )| ≤ L 1 ϕ(y) ( η ′ − η ′′ + ξ ′ − ξ ′′ )(49)
(notice that R η,y (X |y) = R 0:1 η,y (ξ) when y = {y n } n≥1 is a sequence in Y satisfying y = y 1 ). As ϕ(y) > 0 (owing to Assumption 2.2), (48) implies that (i) holds. On the other side, due to (48), (49), we have log |R η,y (X |ξ)| ≤ log L 1 + log ϕ(y) ≤ L 1 (1 + | log ϕ(y)|) , log |R η,y (X |ξ)| ≥ − log L 1 + log ϕ(y) ≥ −L 1 (1 + | log ϕ(y)|) .
Therefore, we get |Φ η,y (ξ)| = |log R η,y (X |ξ)| ≤ |log |R η,y (X |ξ)|| + π ≤ L 1 (1 + | log ϕ(y)|) + π ≤ C 4 (1 + | log ϕ(y)|) . (51) Let φ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) be the function defined by φ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) = log (tR η ′ ,y (X |ξ ′ ) + (1 − t)R η ′′ ,y (X |ξ ′′ )) for t ∈ [0, 1]. Then, due to Assumption 2.2 and (48), we have
|tR η ′ ,y (X |ξ ′ ) + (1 − t)R η ′′ ,y (X |ξ ′′ )| ≥ tRe {R η ′ ,y (X |ξ ′ )} + (1 − t)Re {R η ′′ ,y (X |ξ ′′ )} ≥ ϕ(y) L 1 > 0 (52)
for t ∈ [0, 1]. Hence, φ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) is well-defined and differentiable in t for each t ∈ [0, 1]. We also have
φ ′ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) = ∂ ∂t φ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) = Re {R η ′ ,y (X |ξ ′ )} − Re {R η ′′ ,y (X |ξ ′′ )} tRe {R η ′ ,y (X |ξ ′ )} + (1 − t)Re {R η ′′ ,y (X |ξ ′′ )} for t ∈ [0, 1]. Consequently, (50), (52) yield φ ′ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ ) ≤ L 2 1 ( η ′ − η ′′ + ξ ′ − ξ ′′ )
for t ∈ [0, 1]. Thus, we get
|Φ η ′ ,y (ξ ′ ) − Φ η ′′ ,y (ξ ′′ )| = |φ η ′ ,η ′′ ,y (1|ξ ′ , ξ ′′ ) − φ η ′ ,η ′′ ,y (0|ξ ′ , ξ ′′ )| = 1 0 φ ′ η ′ ,η ′′ ,y (t|ξ ′ , ξ ′′ )dt ≤L 2 1 ( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤C 4 ( η ′ − η ′′ + ξ ′ − ξ ′′ ) .(53)
Using (51), (53), we deduce that (ii) is true.
Lemma 6.5. Let Assumptions 2.1 -2.3 hold. Then, the following is true: (i) There exist real numbers δ 5 , δ 6 ∈ (0, δ 4 ], C 5 ∈ [1, ∞) and an integer n 0 ≥ 1 such that Re R m:n η,y (ξ) > 0, F m:n η,y (ξ) ∈ V δ4 (P(X )), F m:m+n0 η,y (ξ) ∈ V δ6 (P(X )) and
F m:n η,y (ξ ′ ) − F m:n η,y (ξ ′′ ) ≤ C 5 ξ ′ − ξ ′′ ,(54)F m:m+n0 η,y (ξ ′ ) − F m:m+n0 η,y (ξ ′′ ) ≤ ξ ′ − ξ ′′ 2 (55)
for all η ∈ V δ5 (Θ), ξ, ξ ′ , ξ ′′ ∈ V δ6 (P(X )), m + n 0 ≥ n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y (δ 4 is specified in Lemma 6.4).
(ii) There exist real numbers δ 7 ∈ (0, δ 5 ], δ 8 ∈ (0, δ 6 ] such that F m:n η,y (ξ) ∈ V δ6 (P(X )) for all η ∈ V δ7 (Θ), ξ ∈ V δ8 (P(X )), m + n 0 ≥ n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y.
Proof. (i) Throughout this part of the proof, the following notation is used. n 0 is the integer defined by
n 0 = log(4C 3 ) | log γ 2 | ,
while C 5 , δ 5 , δ 6 are the real numbers defined by
C 5 = M n0 (1 + µ ), δ 5 = min{α n0 , δ 4 } 16C 2 5 , δ 6 = 2C 5 δ 5(56)
(γ 2 , C 3 , α n , M n are specified in Lemmas 6.2, 6.3). θ is any element of Θ, while η, η ′ , η ′′ are any elements in V δ5 (Θ). λ is any element of P(X ), while ξ, ξ ′ , ξ ′′ are any elements in V δ6 (P(X )). x is any element of X , while y = {y n } n≥1 is any sequence in Y.
Owing to Lemma 6.3, we have Re R m:n η,y (ξ) > 0 for m + n 0 ≥ n > m ≥ 0 (notice that δ 5 ≤ δ 6 ≤ α n0 ≤ α n−m when m + n 0 ≥ n > m ≥ 0). Hence, we get Re R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) = tRe R m:n η,y (ξ ′ ) + (1 − t)Re R m:n η,y (ξ ′′ ) > 0 (57) for t ∈ [0, 1], m + n 0 ≥ n > m ≥ 0. On the other side, using Lemma 6.3, we conclude
I B (x ′ ) h m:n η ′ ,y (x ′ |x, ξ ′ ) − h m:n η ′′ ,y (x ′ |x, ξ ′′ ) µ(dx ′ ) ≤ I B (x ′ ) h m:n η ′ ,y (x ′ |x, ξ ′ ) − h m:n η ′′ ,y (x ′ |x, ξ ′′ ) µ(dx ′ ) ≤M n−m µ ( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤C 5 ( η ′ − η ′′ + ξ ′ − ξ ′′ )(58)
for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0 (notice that C 5 ≥ M n0 ≥ M n−m when m + n 0 ≥ n > m ≥ 0). Relying on the same lemma, we deduce
F m:n η ′ ,y (B|ξ ′ ) − F m:n η ′′ ,y (B|ξ ′′ ) ≤ I B (x) f m:n η ′ ,y (x|ξ ′ ) − f m:n η ′′ ,y (x|ξ ′′ ) µ(dx) ≤M n−m µ ( η ′ − η ′′ + ξ ′ − ξ ′′ ) ≤C 5 ( η ′ − η ′′ + ξ ′ − ξ ′′ ) for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0. Hence, we have F m:n η ′ ,y (ξ ′ ) − F m:n η ′′ ,y (ξ ′′ ) ≤ C 5 ( η ′ − η ′′ + ξ ′ − ξ ′′ )(59)
for m + n 0 ≥ n > m ≥ 0. Consequently, (54) holds (set η ′ = η, η ′′ = η in (59)). Moreover, when η − θ < δ 5 , ξ − λ < δ 6 , we get
F m:n η,y (ξ) − F m:n θ,y (λ) ≤ C 5 ( η − θ + ξ − λ ) < C 5 (δ 5 + δ 6 ) ≤ δ 4
for m + n 0 ≥ n > m ≥ 0. Therefore, F m:n η,y (ξ) ∈ V δ4 (P(X )) for m + n 0 ≥ n > m ≥ 0. 12 Let φ m:n η,y (t, x|ξ ′ , ξ ′′ ) be the function defined by
φ m:n η,y (t, x|ξ ′ , ξ ′′ ) = f m:n η,y (x|tξ ′ + (1 − t)ξ ′′ )
for t ∈ [0, 1], m + n 0 ≥ n > m ≥ 0. Then, due to (57), we have
φ m:n η,y (t, x|ξ ′ , ξ ′′ ) = r m:n η,y (x|x ′ )(tξ ′ + (1 − t)ξ ′′ )(dx ′ ) R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) = t r m:n η,y (x|x ′ )ξ ′ (dx ′ ) + (1 − t) r m:n η,y (x|x ′ )ξ ′′ (dx ′ ) t R m:n η,y (ξ ′ ) + (1 − t) R m:n η,y (ξ ′′ ) for t ∈ [0, 1], m + n 0 ≥ n > m ≥ 0. Consequently, (27) -(29) imply ∂ ∂t φ m:n η,y (t, x|ξ ′ , ξ ′′ ) = − f m:n η,y (x|tξ ′ + (1 − t)ξ ′′ ) R m:n η,y (ξ ′ ) − R m:n η,y (ξ ′′ ) R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) + r m:n η,y (x|x ′ )(ξ ′ − ξ ′′ )(dx ′ ) R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) = − f m:n η,y (x|tξ ′ + (1 − t)ξ ′′ ) r m:n η,y (x ′′ |x ′ )µ(dx ′′ )(ξ ′ − ξ ′′ )(dx ′ ) R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) + r m:n η,y (x|x ′ )(ξ ′ − ξ ′′ )(dx ′ ) R m:n η,y (tξ ′ + (1 − t)ξ ′′ ) = − f m:n η,y (x|tξ ′ + (1 − t)ξ ′′ ) g m:n η,y (x ′′ |x ′ , tξ ′ + (1 − t)ξ ′′ )µ(dx ′′ )(ξ ′ − ξ ′′ )(dx ′ ) + g m:n η,y (x|x ′ , tξ ′ + (1 − t)ξ ′′ )(ξ ′ − ξ ′′ )(dx ′ ) = h m:n η,y (x|x ′ , tξ ′ + (1 − t)ξ ′′ )(ξ ′ − ξ ′′ )(dx ′ ) for t ∈ [0, 1], m + n 0 ≥ n > m ≥ 0. Hence, we have f m:n η,y (x|ξ ′ ) − f m:n η,y (x|ξ ′′ ) =φ m:n η,y (1, x|ξ ′ , ξ ′′ ) − φ m:n η,y (0, x|ξ ′ , ξ ′′ ) = 1 0 h m:n η,y (x|x ′ , tξ ′ + (1 − t)ξ ′′ )(ξ ′ − ξ ′′ )(dx ′ )dt
for m + n 0 ≥ n > m ≥ 0. Therefore, (30) yields
= 1 0 I B (x)h m:n η,y (x|x ′ , tξ ′ + (1 − t)ξ ′′ )µ(dx)(ξ ′ − ξ ′′ )(dx ′ )dt(60)
for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0. Consequently, we have F m:n θ,y (B|λ + αδ x ) − F m:n θ,y (B|λ) = α
1 0 I B (x ′ )h m:n θ,y (x ′ |x, λ + αtδ x )µ(dx ′ )dt (61)
for B ∈ B(X ), α ∈ (0, δ 6 ), m + n 0 ≥ n > m ≥ 0. 13 On the other side, Lemma 6.2 yields
F m:n θ,y (B|λ + αδ x ) − F m:n θ,y (B|λ) ≤ C 3 γ n−m 2 αδ x = αC 3 γ n−m 2(62)
for B ∈ B(X ), α ∈ (0, δ 6 ), n > m ≥ 0. Combining (61), (62), we get
1 0 I B (x ′ )h m:n θ,y (x ′ |x, λ + αtδ x )µ(dx ′ )dt ≤ C 3 γ n−m 2(63)
for B ∈ B(X ), α ∈ (0, δ 6 ), m + n 0 ≥ n > m ≥ 0. Using (58), (63), we conclude
I B (x ′ )h m:n θ,y (x ′ |x, λ)µ(dx ′ ) ≤ 1 0 I B (x ′ )h m:n θ,y (x ′ |x, λ + αtδ x )µ(dx ′ )dt + 1 0 I B (x ′ ) h m:n θ,y (x ′ |x, λ + αtδ x ) − h m:n θ,y (x ′ |x, λ) µ(dx ′ ) dt ≤C 3 γ n−m 2 + C 5 α for B ∈ B(X ), α ∈ (0, δ 6 ), m + n 0 ≥ n > m ≥ 0 (notice that αtδ x ≤ α). Letting α → 0, we deduce I B (x ′ )h m:n θ,y (x ′ |x, λ)µ(dx ′ ) ≤C 3 γ n−m 2 for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0. Consequently, when η − θ < δ 5 , ξ − λ < δ 6 , (58) yields I B (x ′ )h m:n η,y (x ′ |x, ξ)µ(dx ′ ) ≤ I B (x ′ )h m:n θ,y (x ′ |x, λ)µ(dx ′ ) + I B (x ′ ) h m:n η,y (x ′ |x, ξ) − h m:n θ,y (x ′ |x, λ) µ(dx ′ ) ≤C 3 γ n−m 2 + C 5 ( η − θ + ξ − λ ) ≤C 3 γ n−m 2 + C 5 (δ 5 + δ 6 ) ≤C 3 γ n−m 2 + 1 4
for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0 (notice that C 5 δ 5 ≤ C 5 δ 6 ≤ 1/8). Hence, we have
I B (x ′ )h m:n η,y (x ′ |x, ξ)µ(dx ′ ) ≤C 3 γ n−m 2 + 1 4 for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0 (since η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), there exist θ ∈ Θ, λ ∈ P(X ) such that η − θ < δ 5 , ξ − λ < δ 6 )
. Combining this with (60), we get
F m:n η,y (B|ξ ′ ) − F m:n η,y (B|ξ ′′ ) ≤ 1 0 I B (x ′ )h m:n η,y (x ′ |x, tξ ′ + (1 − t)ξ ′′ )µ(dx ′ ) |ξ ′ − ξ ′′ |(dx)dt ≤ C 3 γ n−m 2 + 1 4 ξ ′ − ξ ′′
for B ∈ B(X ), m + n 0 ≥ n > m ≥ 0. 14 Therefore, we have
F m:n η,y (ξ ′ ) − F m:n η,y (ξ ′′ ) ≤ C 3 γ n−m 2 + 1 4 ξ ′ − ξ ′′ 13
Notice that if α ∈ (0, δ 6 ), t ∈ [0, 1], then λ + αtδx ∈ V δ 6 (P(X )) (as αtδx ≤ αt ≤ α < δ 6 ). Here, δx denotes the Dirac measure centered at x.
14 Notice that tξ ′ + (1 − t)ξ ′′ ∈ V δ 6 (P(X )) since ξ ′ , ξ ′′ ∈ V δ 6 (P(X )) and V δ 6 (P(X )) is convex.
for m + n 0 ≥ n > m ≥ 0. Hence, we get
F m:m+n0 η,y (ξ ′ ) − F m:m+n0 η,y (ξ ′′ ) ≤ C 3 γ n0 2 + 1 4 ξ ′ − ξ ′′ ≤ ξ ′ − ξ ′′ 2 (64)
for m ≥ 0 (notice that C 3 γ n0 2 ≤ 1/4). Consequently, (55) holds. Moreover, when η−θ < δ 5 , ξ−λ < δ 6 , (59) implies
F m:m+n0 η,y (ξ) − F m:m+n0 θ,y (λ) ≤ F m:m+n0 η,y (ξ) − F m:m+n0 η,y (λ) + F m:m+n0 η,y (λ) − F m:m+n0 θ,y (λ) ≤ ξ − λ 2 + C 5 η − θ < δ 6 2 + C 5 δ 5 =δ 6
for m ≥ 0. Thus, F m:m+n0 η,y (ξ) ∈ V δ6 (P(X )) for m ≥ 0. 15 (ii) Let δ 7 , δ 8 be the real numbers defined by δ 7 = δ 5 , δ 8 = δ 5 (δ 5 is specified in (56)). Moreover, let θ, λ, y have the same meaning as in (i), while η, ξ are any elements of V δ6 (Θ), V δ7 (P(X )) (respectively). Consequently, when η − θ < δ 7 , ξ − λ < δ 8 , (59) yields
F m:n η,y (ξ) − F m:n θ,y (λ) ≤ C 5 ( η − θ + ξ − λ ) < C 5 (δ 7 + δ 8 ) ≤ δ 6
for m + n 0 ≥ n > m ≥ 0. Therefore, F m:n η,y (ξ) ∈ V δ6 (P(X )) for m + n 0 ≥ n > m ≥ 0. 16 Lemma 6.6. Let Assumptions 2.1 -2.3 hold. Then, the following is true: (i) Re R m:n η,y (ξ) = 0, F m:n η,y (ξ) ∈ V δ4 (P(X )) for all η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y (δ 4 , δ 5 , δ 6 are specified in Lemmas 6.4, 6.5).
(ii) There exist real numbers γ 3 ∈ (0, 1),
C 6 ∈ [1, ∞) such that F m:n η,y (ξ ′ ) − F m:n η,y (ξ ′′ ) ≤ C 6 γ n−m 3 ξ ′ − ξ ′′(65)
for all η ∈ V δ5 (Θ), ξ ′ , ξ ′′ ∈ V δ6 (P(X )), n ≥ m ≥ 0 and any sequence y = {y n } n≥1 in Y.
Proof. (i) Let n k (m) be the integer defined by n k (m) = m + kn 0 for m, k ≥ 0 (n 0 is specified in Lemma 6.5). Moreover, let y = {y n } n≥1 be any sequence in Y.
First, we show
Re R m:n η,y (ξ) = 0, F m:n η,y (ξ) ∈ V δ4 (P(X )),
F m:n k (m) η,y (ξ) ∈ V δ6 (P(X ))(66)
for each η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k (m) ≥ n ≥ m ≥ 0, k ≥ 0. We prove this by induction in k.
We have R m:n η,y (ξ) = ξ, F m:n η,y (ξ) = ξ for η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n 0 (m) ≥ n ≥ m ≥ 0 (notice that n 0 (m) = n = m when n 0 (m) ≥ n ≥ m ≥ 0). Hence, (66) holds for k = 0 and η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k (m) ≥ n ≥ m ≥ 0. Now, the induction hypothesis is formulated: Suppose that (66) is true for some k ≥ 0 and any η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k (m) ≥ n ≥ m ≥ 0. Then, to show (66) for η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k+1 (m) ≥ n ≥ m ≥ 0, it is sufficient to demonstrate (66) for η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k+1 (m) ≥ n ≥ n k (m), m ≥ 0.
In the rest of the proof of (i), η, ξ are any elements of V δ5 (Θ), V δ6 (P(X )) (respectively). Owing to Lemma 6.5, we have Re R n k (m):n η,y (ξ) > 0, F n k (m):n η,y (ξ) ∈ V δ4 (P(X )), F n k (m):n k+1 (m) η,y (ξ) ∈ V δ6 (P(X )) 15 Since η ∈ V δ 5 (Θ), ξ ∈ V δ 6 (P(X )), there exist θ ∈ Θ, λ ∈ P(X ) such that η − θ < δ 5 , ξ − λ < δ 6 . Notice also F m:m+n 0 θ,y (λ) ∈ P(X ). 16 Since η ∈ V δ 7 (Θ), ξ ∈ V δ 8 (P(X )), there exist θ ∈ Θ, λ ∈ P(X ) such that η − θ < δ 7 , ξ − λ < δ 8 . Notice also F m:n θ,y (λ) ∈ P(X ).
(ξ) ∈ V δ4 (P(X )),(70)F m:n k+1 (m) η,y (ξ) = F n k (m):n k+1 (m) η,y F m:n k (m) η,y (ξ) ∈ V δ6 (P(X ))(71)
for n k+1 (m) ≥ n ≥ n k (m), m ≥ 0. 19 Combining (69) -(71) with the induction hypothesis, we deduce that (66) holds for n k+1 (m) ≥ n ≥ m, m ≥ 0. Then, relying on the principle of mathematical induction, we conclude that (66) is satisfied for each η ∈ V δ5 (Θ), ξ ∈ V δ6 (P(X )), n k (m) ≥ n ≥ m ≥ 0, k ≥ 0. As a direct consequence of this, we have that (i) is true.
(ii) Let γ 3 , C 6 be the real numbers defined by γ 3 = 2 −1/n0 , C 6 = C 5 γ −n0 3 (C 5 , n 0 are specified in Lemma 6.5), while n k (m), y have the same meaning as in (i). Moreover, let η be any element of V δ6 (Θ), while ξ ′ , ξ ′′ are any elements in V δ6 (P(X )).
Owing to Lemmas 6.1, 6.5 and (66), we have
F m:n k+1 (m) η,y (ξ ′ ) − F m:n k+1 (m) η,y (ξ ′′ ) = F n k (m):n k+1 (m) η,y F m:n k (m) η,y (ξ ′ ) − F n k (m):n k+1 (m) η,y F m:n k (m) η,y (ξ ′′ ) ≤ 1 2 F m:n k (m) η,y (ξ ′ ) − F m:n k (m) η,y (ξ ′′ )
for m, k ≥ 0. Consequently, we have
≤C 5 γ n k (m)−m 3 ξ ′ − ξ ′′ ≤C 6 γ n−m 3 ξ ′ − ξ ′′(72)
for n k+1 (m) > n ≥ n k (m), m, k ≥ 0. 20 Thus, (65) holds for each n ≥ m ≥ 0 (set k = ⌊(n − m)/n 0 ⌋ in (72)). 17 Notice that F m:n k (m) η,y (ξ) ∈ V δ 6 (P(X )) (due to the induction hypothesis). 18 Notice that R m:n k (m) η,y (ξ) = 0 (due to the induction hypothesis). 19 Notice that R m:n k (m) η,y (ξ) = 0, R m:n k+1 (m) η,y (ξ) = 0 (due to the induction hypothesis and (69)). Notice also F m:n k (m) η,y (ξ) ∈ V δ 6 (P(X )) (due to the induction hypothesis). 20 Notice that C 5 γ
Proof of Main Results
In this section, Theorems 2.1 and 2.2 are proved. The proofs of these theorems crucially depend on the results related to the kernels S(z, dz ′ ), S η (z, dz ′ ) and the optimal filter F m:n η,y (ξ) (i.e., on Lemmas 5.1, 5.4, 6.6). As the properties of S(z, dz ′ ), S η (z, dz ′ ) are very similar, the proofs of Theorems 2.1 and 2.2 have many elements in common. In order not to consider these elements twice (and to prove Theorems 2.1 and 2.2 as efficiently as possible), we introduce a new kernel T η (z, dz ′ ), where η ∈ V δ (Θ), z ∈ Z. 21 Its purpose is to capture all common features of S(z, dz ′ ), S η (z, dz ′ ) which are relevant for the proof of Theorems 2.1 and 2.2. Using T η (z, dz ′ ), we recursively define kernels T n η (z, dz ′ ) n≥0 by T 0 η (z, B) = δ z (B) and
T n+1 η (z, B) = T n η (z ′ , B)T η (z, dz ′ ) for η ∈ C d , z ∈ Z, B ∈ B(Z), n ≥ 0.
Regarding T η (z, dz ′ ), we assume the following.
for x ∈ X , y ∈ Y and z = (y, x) (η, ξ have the same meaning as in (73), while Φ η,y (ξ) is specified in (24)). 23 Φ n η (ξ, z) is the function defined by Φ n η (ξ, z) = · · · Φ η F n η (ξ, z 1:n ), z n+1 T η (z n , dz n+1 ) · · · T η (z, dz 1 )
for z ∈ Z, n ≥ 0 (η, ξ have the same meaning as in (73)).Ā n η (ξ), A k,n η (ξ, z), B n η (ξ, z) are the functions defined byĀ
n η (ξ) = · · · Φ η F n η (ξ, z 1:n ), z n+1 − Φ η F n−1 η (ξ, z 2:n ), z n+1 · T η (z n , dz n+1 ) · · · T η (z 0 , dz 1 )τ η (dz 0 ), A k,n η (ξ, z) = · · · Φ η F n−k+1 η (ξ, z k:n ), z n+1 − Φ η F n−k η (ξ, z k+1:n ), z n+1 · T η (z n , dz n+1 ) · · · T η (z k , dz k+1 )(T k η − τ η )(z, dz k ), B n η (ξ, z) = Φ η (ξ, z ′ ) (T n+1 η − τ η )(z, dz ′ )
for n ≥ k ≥ 1 (η, ξ, z have the same meaning as in (73), (74)). Under the notation introduced above, we have log q n θ (y 1:n |λ) =
n−1 k=0 Φ θ F k θ (λ, z 1:k ), z k+1(75)
for θ ∈ Θ, λ ∈ P(X ), x 1 , . . . , x n ∈ X , y 1 , . . . , y n ∈ Y, n ≥ 1 and z 1 = (y 1 , x 1 ), . . . , z n = (y n , x n ). We also have
Φ n η (ξ, z ′ ) − Φ n η (ξ, z ′′ ) = n k=1 A k,n η (ξ, z ′ ) − A k,n η (ξ, z ′′ ) + B n η (ξ, z ′ ) − B n η (ξ, z ′′ ),(76)Φ n+1 η (ξ, z) − Φ n η (ξ, z) = n+1 k=1 A k,n+1 η (ξ, z) − n k=1 A k,n η (ξ, z) +Ā n+1 η (ξ) + B n+1 η (ξ, z) − B n η (ξ, z)(77)
for η ∈ V α (Θ), ξ ∈ M c (X ), z, z ′ , z ′′ ∈ Z, n ≥ 1.
Lemma 7.1. Let Assumptions 2.1 -2.3, 7.1 and 7.2 hold. Then, there exist a function φ η mapping η ∈ C d to C and real numbers δ 9 , γ 4 ∈ (0, 1),
C 7 ∈ [1, ∞) such that Φ n η (ξ, z) − φ η ≤ C 7 nγ n 4(78)
for all η ∈ V δ9 (Θ), ξ ∈ V δ9 (P(X )), z ∈ Z, n ≥ 1.
Proof. Throughout the proof, the following notation is used. γ 4 , δ 9 are the real numbers defined by γ 4 = max{β 1/2 , γ 1/2 3 }, δ 9 = min{δ 7 , δ 8 , (1 − γ 4 )/L} (β, δ 7 , δ 8 , γ 3 , L are specified in Assumption 7.2 and Lemmas 6.5, 6.6). θ is any element of Θ, while η is any element in V δ9 (Θ). ξ, ξ ′ , ξ ′′ are any elements of V δ9 (P(X )), while z, z ′ , z ′′ are any elements in Z.
When η − θ < δ 9 , Assumptions 7.1, 7.2 imply
|T η |(z, B) ≤ T θ (z, B) + |T η − T θ |(z, B) ≤ 1 + L η − θ < 1 + Lδ 9 ≤ 1 γ 4
for B ∈ B(Z) (notice that Lδ 9 ≤ 1 − γ 4 ≤ 1/γ 4 − 1). Hence, we get 23 Notice that u n η (z 0:n ), F n η (ξ, z 1:n ), Φη(ξ, z) are just another notation for u n η (x 0:n , y 1:n ), F 0:n η,y (ξ), Φη,y(ξ). However, notation u n η (z 0:n ), F n η (ξ, z 1:n ), Φη(ξ, z) is more suitable (than the original one) for measure-theoretic arguments which the analysis carried out in this section is based on.
|T η |(z, B) ≤ 1 γ 4(79)
for B ∈ B(Z) (since η ∈ V δ9 (Θ), there exists θ ∈ Θ such that η − θ < δ 9 ). Consequently, Assumption 7.1 yields
|τ η |(B) ≤ |T η − τ η |(z, B) + |T η |(z, B) ≤ L + 1 γ 4 (80)
for B ∈ B(Z). LetC 1 = 4C 4 C 6 (C 4 , C 6 are specified in Lemmas 6.4, 6.6). Then, due to Lemmas 6.4, 6.6, we have
Φ η F n η (ξ ′ , z 1:n ), z n+1 − Φ η F n η (ξ ′′ , z 1:n ), z n+1 ≤ C 4 F n η (ξ ′ , z 1:n ) − F n η (ξ ′′ , z 1:n ) ≤ C 4 C 6 γ n 3 ξ ′ − ξ ′′ ≤C 1 γ 2n 4 (81)
for z 1 , . . . , z n+1 ∈ Z, n ≥ 0. 24 On the other side, owing to the Lemmas 6.1, 6.4, 6.6, we have
Φ η F n−k+1 η (ξ, z k:n ), z n+1 − Φ η F n−k η (ξ, z k+1:n ), z n+1 ≤ C 4 F n−k+1 η (ξ, z k:n ) − F n−k η (ξ, z k+1:n ) = C 4 F n−k η F 1 η (ξ, z k ), z k+1:n − F n−k η (ξ, z k+1:n ) ≤ C 4 C 6 γ n−k 3 F 1 η (ξ, z k ) − ξ ≤C 1 γ 2(n−k) 4(82)
for z 1 , . . . , z n+1 ∈ Z, n ≥ k ≥ 1. 25 LetC 2 = 2C 1 L/γ 4 4 . Then, using (79), (81), we conclude
Φ n η (ξ ′ , z) − Φ n η (ξ ′′ , z) ≤ · · · Φ η F n η (ξ ′ , z 1:n ), z n+1 − Φ η F n η (ξ ′′ , z 1:n ), z n+1
· |T η |(z n , dz n+1 ) · · · |T η |(z, dz 1 )
≤C 1 γ 2n 4
· · · |T η |(z n , dz n+1 ) · · · |T η |(z, dz 1 )
≤C 1 γ n−1 4 ≤C 2 γ n 4(83)
for n ≥ 0. Similarly, relying on Assumption 7.2 and (79), (82), we deduce 24 To get the first relation in (81), use Lemma 6.4 and notice that η ∈ V δ 4 (Θ), F n η (ξ ′ , z 1:n ) ∈ V δ 4 (P(X )), F n η (ξ ′′ , z 1:n ) ∈ V δ 4 (P(X )) follow from Lemma 6.6 and η ∈ V δ 9 (Θ) ⊆ V δ 5 (Θ), ξ ′ , ξ ′′ ∈ V δ 9 (P(X )) ⊆ V δ 6 (P(X )). To get the last two relations in (81), use Lemma 6.6 and notice that ξ ′ − ξ ′ ≤ ξ ′ + ξ ′′ ≤ 2(1 + δ 9 ) ≤ 4. 25 To get the first two relations in (82), use Lemmas 6.1, 6.4, and notice that η ∈ V δ 4 (Θ), F n−k+1 η (ξ ′ , z k:n ) ∈ V δ 4 (P(X )), F n−k η (ξ ′′ , z k+1:n ) ∈ V δ 4 (P(X )) follow from Lemma 6.6 and η ∈ V δ 9 (Θ) ⊆ V δ 5 (Θ), ξ ′ , ξ ′′ ∈ V δ 9 (P(X )) ⊆ V δ 6 (P(X )). To get the third relation in (82), use Lemma 6.6 and notice that F 1 η (ξ, z k ) ∈ V δ 6 (P(X )) follows from Lemma 6.4 and η ∈ V δ 9 (Θ) ⊆ V δ 7 (Θ), ξ ∈ V δ 9 (P(X )) ⊆ V δ 8 (P(X )). To get the last relation in (82), notice that F 1
A k,n η (ξ, z) ≤ · · · Φ η F n−k+1 η (ξ, z k:n ), z n+1 − Φ η F n−k η (ξ, z k+1:n ), z n+1 · |T η |(z n , dz n+1 ) · · · |T η |(z k , dz k+1 )|T k η − τ η |(z, dz k ) ≤C 1 γ 2(n−k) 4 · · · |T η |(z n , dz n+1 ) · · · |T η |(z k , dz k+1 )|T k η − τ η |(z, dz k ) ≤C 1 Lβ k γ n−k−1 4 ≤C 2 γ n 4(84)η (ξ, z k ) − ξ ≤ F 1 η (ξ, z k ) + ξ ≤ 2 + δ 6 + δ 9 ≤ 4.
for n ≥ k ≥ 1. On the other side, using (80), (82), we conclude Ā n η (ξ) ≤ · · · Φ η F n η (ξ, z 1:n ), z n+1 − Φ η F n−1 η (ξ, z 2:n ), z n+1 · |T η |(z n , dz n+1 ) · · · |T η |(z 0 , dz 1 )|τ η |(dz 0 )
≤C 1 γ 2(n−1) 4 · · · |T η |(z n , dz n+1 ) · · · |T η |(z 0 , dz 1 )|τ η |(dz 0 ) ≤C 1 L + 1 γ 4 γ n−3 4 ≤C 2 γ n 4(85)
for n ≥ 1.
LetC 3 = C 4 L 2 ,C 4 = 4(C 2 +C 3 )
. Then, owing to Assumption 7.2 and Lemma 6.4, we have
B n η (ξ, z) = Φ η (ξ, z ′′ )T η (z ′ , dz ′′ )(T n η − τ η )(z, dz ′ ) ≤ |Φ η (ξ, z ′′ )| |T η |(z ′ , dz ′′ )|T n η − τ η |(z, dz ′ ) ≤C 4 ψ(z ′′ )|T η |(z ′ , dz ′′ ) |T n η − τ η |(z, dz ′ ) ≤C 4 L 2 β n ≤C 3 γ n 4(86)
for n ≥ 1. Consequently, (77), (84), (85) yield
Φ n+1 η (ξ, z) − Φ n η (ξ, z) ≤ n+1 k=1 A k,n+1 η (ξ, z) + n k=1 A k,n η (ξ, z) + Ā n+1 η (ξ) + B n+1 η (ξ, z) + B n η (ξ, z) ≤2C 2 (n + 1)γ n 4 + 2C 3 γ n 4 ≤C 4 nγ n 4(87)
for n ≥ 1. Hence, we have
∞ n=1 Φ n+1 η (ξ, z) − Φ n η (ξ, z) ≤C 4 ∞ n=1 nγ n 4 ≤C 4 (1 − γ 4 ) 2 < ∞.(88)
On the other side, combining (76), (84), (86), we get
Φ n η (ξ, z ′ ) − Φ n η (ξ, z ′′ ) ≤ n k=1 A k,n η (ξ, z ′ ) + n k=1 A k,n η (ξ, z ′′ ) + B n η (ξ, z ′ ) + B n η (ξ, z ′′ ) ≤2C 2 nγ n 4 + 2C 3 γ n 4
for n ≥ 1. Then, (83) implies
Φ n η (ξ ′ , z ′ ) − Φ n η (ξ ′′ , z ′′ ) ≤ Φ n η (ξ ′ , z ′ ) − Φ n η (ξ ′′ , z ′ ) + Φ n η (ξ ′′ , z ′ ) − Φ n η (ξ ′′ , z ′′ ) ≤C 2 (2n + 1)γ n 4 + 2C 3 γ n 4 ≤C 4 nγ n 4(89)
for n ≥ 1.
Let C 7 =C 4 /(1 − γ 4 ) 2 . Moreover, let φ η (ξ, z) = Φ 0 η (ξ, z) + ∞ n=0 Φ n+1 η (ξ, z) − Φ n η (ξ, z) .· · · ψ(z n+1 ) n+1 k=1 φ(z k ) (ν × µ)(dz n+1 ) · · · (ν × µ)(dz 1 ) = µ n+1 (1 + | log ϕ(y n+1 )|)ϕ(y n+1 )ν(dy n+1 ) n k=1 ϕ(y k )ν(dy k ) < ∞
for n ≥ 1. Consequently, Lemma A1.1 (see Appendix 1) and (93) imply that integral (92) is analytic in η for η ∈ V δ5 (Θ), n ≥ 1.
Proof of Theorem 2.1. Let T η (z, dz ′ ) be the kernel defined by T η (z, B) = S(z, B) for η ∈ C d , z ∈ Z, B ∈ B(Z) (S(z, dz ′ ) is specified in (8)). Moreover, let T n η (z, dz ′ ), Φ n η (λ, z) have the same meaning as in (74), where η ∈ C d , λ ∈ P(X ), z ∈ Z, n ≥ 0. Then, owing to Lemma 7.2, Φ n η (λ, z) is analytic in η for each η ∈ V δ5 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1. On the other side, due to Lemma 5.1, kernel T η (z, dz ′ ) (defined here) satisfies Assumptions 7.1, 7.2. Combining this with Lemma 7.1, we deduce that there exist a function φ η mapping η ∈ C d to C and real numbers δ 9 ∈ (0, δ 5 ], γ 4 ∈ (0, 1), C 7 ∈ [1, ∞) such that (78) holds for η ∈ V δ9 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1. Since the limit of uniformly convergent analytic functions is also analytic (see e.g., [26,Theorem 2.4
.1]), φ η is analytic in η for each η ∈ V δ9 (Θ).
In what follows in the proof, θ, λ, z are any element of Θ, P(X ), Z (respectively). It is straightforward to verify
Φ n θ (λ, z) = E ( Φ θ (F n θ (λ, Z 1:n ), Z n+1 )| Z 0 = z)
for n ≥ 1, where Z n = (Y n , X n ). Therefore, (75) yields
E (log q n θ (Y 1:n |λ)) = n−1 k=0 E Φ k θ (λ, Z 0 )
for n ≥ 1. Then, Lemma 7.1 implies
E 1 n log q θ (Y 1:n |λ) − φ θ ≤ 1 n n−1 k=0 E Φ k θ (λ, Z 0 ) − φ θ ≤ C 7 n n−1 k=0 γ k 4 ≤ C 7 n(1 − γ 4 )
for n ≥ 1. Consequently, there exists a function l : Θ → R with the properties specified in the statement of the theorem.
Proof of Theorem 2.2. Let T η (z, dz ′ ) be the kernel defined by T η (z, B) = S η (z, B) for η ∈ C d , z ∈ Z, B ∈ B(Z) (S η (z, dz ′ ) is specified in (8)). Moreover, let T n η (z, dz ′ ), Φ n η (λ, z) have the same meaning as in (74), where η ∈ C d , λ ∈ P(X ), z ∈ Z, n ≥ 0. Then, due to Lemma 7.2, Φ n η (λ, z) is analytic in η for each η ∈ V δ5 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1. On the other side, Lemma 5.4 implies that Assumptions 7.1, 7.2 hold for kernel T η (z, dz ′ ) (defined here). Combining this with Lemma 7.1, we conclude that there exist a function φ η mapping η ∈ C d to C and real numbers δ 9 ∈ (0, δ 5 ], γ 4 ∈ (0, 1), C 7 ∈ [1, ∞) such that (78) holds for η ∈ V δ9 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1. As the limit of uniformly convergent analytic functions is also analytic (see e.g., [26,Theorem 2.4.1]), φ η is analytic in η for each η ∈ V δ9 (Θ).
In the rest of the proof, θ, λ, z are any element of Θ, P(X ), Z (respectively). It is easy to show Φ n θ (λ, z) = E Φ θ F n θ λ, Z θ,λ 1:n , Z θ,λ n+1 Z θ,λ 0 = z for n ≥ 1, where Z θ,λ n = Y θ,λ n , X θ,λ n . Then, (75) yields E log q n θ Y θ,λ 1:n λ = n−1 k=0 E Φ k θ λ, Z θ,λ 0 for n ≥ 1. Therefore, Lemma 7.1 implies
E 1 n log q θ Y θ,λ 1:n λ − φ θ ≤ 1 n n−1 k=0 E Φ k θ λ, Z θ,λ 0 − φ θ ≤ C 7 n n−1 k=0 γ k 4 ≤ C 7 n(1 − γ 4 )
for n ≥ 1. Consequently, there exists a function h : Θ → R with the properties specified in the statement of the theorem.
(i) {â i η (x)} 1≤i≤Nx , {b j η (x)} 1≤j≤Ny map η ∈ C d , x ∈ C dx to C. (ii)â i θ (x) = a i θ (x),b j θ (x) = b j θ (x) for θ ∈Θ, x ∈ X , 1 ≤ i ≤ N x , 1 ≤ j ≤ N y . (iii) There exists a real number α 1 ∈ (0, 1) such thatâ i η (x),b j η (x) are analytic in (η, x) for η ∈ V α1 (Θ), x ∈ V α1 (X ), 1 ≤ i ≤ N x , 1 ≤ j ≤ N y .
Owing to Assumption 3.2, {â i θ (x)} 1≤i≤Nx , {b j θ (x)} 1≤j≤Ny are positive and uniformly bounded away from zero for θ ∈ clΘ, x ∈ X . Then, due to (iii), there exist real numbers α ∈ (0, α 1 ), β ∈ (0, 1) such that
Re â i η (x) ≥ β, |â i η (x)| ≤ 1 β , Re b j η (x) ≥ β, |b j η (x)| ≤ 1 β (94) for η ∈ V α (Θ), x ∈ V α (X ), 1 ≤ i ≤ N x , 1 ≤ j ≤ N y .
In the rest of the proof, the following notation is used.p η (x ′ |x),q η (y|x), ϕ(y) are the functions defined byp
η (x ′ |x) = Nx i=1â i η (x)v i (x ′ ),q η (y|x) = Ny j=1b j η (x)w j (y), ϕ(y) = Ny j=1 w j (y)
for η ∈ C d , x, x ′ ∈ X , y ∈ Y. r θ (y, x ′ |x),r η (y, x ′ |x) are the functions defined by
r θ (y, x ′ |x) = q θ (y|x ′ )p θ (x ′ |x),r η (y, x ′ |x) =q η (y|x ′ )p η (x ′ |x).
for θ ∈ Θ, η ∈ C d , x, x ′ ∈ X , y ∈ Y.
Owing to (ii), (iii),r η (y, x ′ |x) is analytic in η for each η ∈ V α (Θ), x, x ′ ∈ X , y ∈ Y. Due to the same reasons,r θ (y, x ′ |x) = r θ (y, x ′ |x) for θ ∈Θ, x, x ′ ∈ X , y ∈ Y. On the other side, Assumption 3.3 and (94) imply for η ∈ V α (Θ), x ∈ X , y ∈ Y. Hence, there exists a real number γ ∈ (0, 1) such that γϕ(y) ≤ |r η (y, x ′ |x)| ≤ ϕ(y) γ for η ∈ V α (Θ), x, x ′ ∈ X , y ∈ Y.
|p η (x ′ |x)| ≥ Nx i=1 Re â i η (x) v i (x ′ ) ≥ βεN x , |p η (x ′ |x)| ≤ Nx i=1 |â i η (x)|v i (x ′ ) ≤ N x βε for η ∈ V α (Θ), x, x ′ ∈ X . Similarly,(94)
Owing to Assumption 3.3, we have ϕ(y)ν(dy) = Ny j=1 w j (y)ν(dy) < ∞.
Similarly, due to Assumption 3.4, we get | log ϕ(y)|ϕ(y)ν(dy) ≤ | log w j (y)|w k (y)ν(dy)
<∞.
Then, using Theorems 2.1, 2.2, we conclude that there exist functionsl,h :Θ → R such thatl(θ),h(θ) are real-analytic in θ and satisfy lim n→∞ l n (θ, λ) =l(θ), lim n→∞ h n (θ, λ) =h(θ) for each θ ∈Θ, λ ∈ P(X ) (l n (θ, λ), h n (θ, λ) have the same meaning as in (1)). Consequently, Corollaries 4.1, 4.2 hold (notice that Θ can be represented as the union Θ = ∞ n=1Θ n , where {Θ n } n≥1 are non-empty open balls satisfying clΘ n ⊂ Θ for n ≥ 1). and B ⊂ M i=1 V δ(vi) (v i ). Letĝ i (w) =ĝ(w, v i ), V i = V δ(vi) (v i ) for w ∈ C dw , 1 ≤ i ≤ M . As B is connected, for each 1 ≤ i ≤ M , there exists 1 ≤ j ≤ M , j = i such that V i ∩ V j ∩ R dw = ∅. On the other side, if V i ∩ V j ∩ R dw = ∅, then V i ∩ V j ∩ R dw is a non-empty open set andĝ i (w) =ĝ j (w) = g(w) for w ∈ V i ∩ V j ∩ R dw . Then, by the uniqueness of analytic continuation (see e.g., [13, Corollary 1.2.6]), for each 1 ≤ i ≤ M , there exist 1 ≤ j ≤ M , j = i and a functionĝ ij (w) with the following properties:
(vii)ĝ ij (w) maps w ∈ C dw to C.
(viii)ĝ ij (w) is analytic on V i ∪ V j .
(ix)ĝ ij (w) =ĝ i (w) for w ∈ V i andĝ ij (w) =ĝ j (w) for w ∈ V j . Following these arguments, we conclude that there exists a functionĝ(w) with the following properties: (x)ĝ(w) maps w ∈ C dw to C.
(xi)ĝ(w) is analytic on M i=1 V i . (xii)ĝ(w) =ĝ i (w) for w ∈ V i , 1 ≤ i ≤ M . Now, we drop the assumption that B is connected (i.e., B is any compact set in R dw ). Since B is compact, there exist an integer N ≥ 1 and open sets {W i } 1≤i≤N in R dw such that W i ⊆ C, B ∩W i = ∅, W i ∩W j = ∅ for 1 ≤ i, j ≤ N , i = j and B ⊂ (
xiii) B i ⊂ U i , U i ∩ U j = ∅ for 1 ≤ i, j ≤ N , i = j.
(xiv)ĝ i (w) maps w ∈ C dw to C for 1 ≤ i ≤ N .
(xv)ĝ i (w) = g(w) for w ∈ B i , 1 ≤ i ≤ N .
(xvi)ĝ i (w) is analytic on U i for 1 ≤ i ≤ N . Letĝ(w) be the function defined byĝ(w) =ĝ i (w) for w ∈ U i , 1 ≤ i ≤ N andĝ(w) = 0 for w ∈ N i=1 U i . Due to (xiii),ĝ(w) is well-defined. As B is compact and B ⊂ N i=1 U i (owing to (xiii)), there exists a real number δ ∈ (0, 1) such that B ⊂ V δ (B) ⊂ N i=1 U i . Then, (xv), (xvi) imply thatĝ(w) is analytic on V δ (B) and satisfiesĝ(w) = g(w) for w ∈ B.
Assumption A2.2. G θ (u, y) and h θ (u, y) are real-analytic in (θ, u) for all θ ∈ Θ, u ∈ P N , y ∈ Y. Moreover, G θ (u, y) and h θ (u, y) have complex-valued continuationsĜ η (w, y) andĥ η (w, y) with the following properties:
(i)Ĝ η (w, y) andĥ η (w, y) map η ∈ C d , w ∈ C N , y ∈ Y to C N and C (respectively).
(ii)Ĝ θ (u, y) = G θ (u, y) andĥ θ (u, y) = h θ (u, y) for all θ ∈ Θ, u ∈ P N , y ∈ Y.
(iii) There exists a real number δ ∈ (0, 1) such thatĜ η (w, y) andĥ η (w, y) are analytic in (η, w) for each η ∈ V δ (Θ), w ∈ V δ (P N ), y ∈ Y.
(iv) There exist a real number K ∈ [1, ∞) and a function ψ : Y → [1, ∞) such that exp(ψ(y))ψ(y)ν(dy) < ∞ and Ĝ η (w, y) ≤ K, |ĥ η (w, y)| ≤ ψ(y)
for all η ∈ V δ (Θ), w ∈ V δ (P N ), y ∈ Y.
Assumption A2.1 corresponds to the stability of the hidden Markov model (X θ,λ n , Y θ,λ n ) n≥0 and its optimal filter, while Assumption A2.2 is related to the parameterization of the model (X θ,λ n , Y θ,λ n ) n≥0 . Assumptions A2.1 and A2.2 are the same as the (corresponding) assumptions adopted in [21]. Further to this, Assumptions A2.1 and A2.2 include (as a particular case) all conditions which the results of [10] are based on.
Under the above assumptions, we have the following result on the entropy rate of finite-state hidden Markov models. Proof. For 1 ≤ i ≤ N , let e i be the i-th standard unit vector in R N . For η ∈ C d , x, x ′ ∈ X , y ∈ Y, let ϕ(y) = K exp(ψ(y)), r η y, x ′ |x) = For η ∈ C d , x ∈ X , y ∈ Y, B ∈ B(Z) and z = (y, x), let
T η (z, B) = N i=1
I B (y ′ , x i )r η (y ′ , x i |x)ν(dy ′ ) (here, Z, B(Z) have the same meaning as in Section 5). Then, it is straightforward to show that functionŝ r η (y, x ′ |x), ϕ(y) (defined here) satisfy Assumptions 2.
Z = Y × X , while B(Z) is the collection of Borel sets in Z. P(Z) is the collection of probability measures on Z, while M p (Z) is the set of positive measures on Z. M c (Z) is the collection of complex measures on Z, while P c (Z) is the set defined by P c (Z) = {ζ ∈ M c (Z) : ζ(Z) = 1}.
(iii) There exists a non-increasing sequence {α n } n≥1 in (0, δ 1 ] such that Re R
y k ) .
F
m:n η,y (B|ξ ′ ) − F m:n η,y (B|ξ ′′ ) = I B (x) f m:n η,y (x|ξ ′ ) − f m:n η,y (x|ξ ′′ ) µ(dx)
1 (
1for n k+1 (m) ≥ n ≥ n k (m), m ≥ 0. Combining this with the induction hypothesis, k+1 (m) ≥ n ≥ n k (m), m ≥ 0. 17 Then, the induction hypothesis and Lemma 6.Part k+1 (m) ≥ n ≥ n k (m), m ≥ 0. 18 Therefore, Lemma 6.1 (Part (iii)) and (67),
y (ξ ′ ) − F m:m η,y (ξ ′′ ) = γ n k (m)−m 3 ξ ′ − ξ ′′for m, k ≥ 0. Combining this with Lemma 6.5 and (66), we get F m:n η,y (ξ ′ ) − F m:n η,y (ξ ′′ ) = F n k (m)
LetΘ be any non-empty open set satisfying clΘ ⊂ Θ. As clΘ, X are compact sets, Assumption 3.2 and Lemma A1.2 (see Appendix 1) imply that there exist functions {â i η (x)} 1≤i≤Nx , {b j η (x)} 1≤j≤Ny with the following properties:
i . Let B i = B ∩ W i for 1 ≤ i ≤ N . Hence, {B i } 1≤i≤Nare connected components of B, and thus, {B i } 1≤i≤N are compact and disjoint. Then, according to what has already been shown, there exist open sets {U i } 1≤i≤N in C dw and functions {ĝ i (w)} 1≤i≤N with the following properties:
Theorem A2. 1 .
1Let Assumptions A2.1 and A2.2 hold. Then, all conclusions of Theorem 2.2 are true.
η (e j , y) exp ĥ η (e j , y) I (xi,xj) (x ′ , x), r η (y, x ′ |x)
Corollary 4.1. Let Assumptions 2.5 and 4.1 -4.3 hold. Then, all conclusions of Theorem 2.1 are true.Corollary 4.2. Let Assumptions 4.1 -4.3 hold. Then, all conclusions of Theorem 2.2 are true.
Corollaries 4.1 and 4.2 are proved in Section 8.
for n ≥ 1. On the other side, due to Assumptions 2.2, 2.4, we have
1, 2.2, 2.4. Consequently, Lemma 5.4 implies that kernel T η (z, dz ′ ) (defined here) fulfills Assumptions 7.1, 7.2. On the other side, the conclusions of Lemma 6.6 follow from [21, Lemma 3]. Combining all of these with Lemma 7.1, we deduce that all conclusions of Theorem 2.2 are true (notice that Assumptions 7.1, 7.2 and the conclusions of Lemma 6.6 are sufficient for Lemma 7.1 to hold).
The results presented in[10] can be considered as the strongest result on the analytical properties of the entropy rate of finite-state hidden Markov models. When X has a finite number of elements, Theorem 2.2 includes all results of[10] as a particular case. Theorem 2.2 also simplifies (considerably) the conditions which the results of[10] are based on. Moreover, combining the techniques used in the proof of Theorem 2.2 (in particular, Lemma 5.4, Section 5) with the results of[21], the results presented in[10] can further be generalized (for details, see Appendix 2).
Notice thatK 1 ≤C 3 ≤ C 2 .Here, δz denotes the Dirac measure centered at z.
Notice that σ θ (B)(ζ ′ − ζ ′′ )(dz) = σ θ (B)(ζ ′ (Z) − ζ ′′ (Z)) = 0 (as ζ ′ , ζ ′′ ∈ Pc(Z)).8 Notice thatKnδ 2 ≤Kn 0 δ 2 = δ 1 /4 ≤ 1/4. Notice also that S n θ ζ ≤ ζ (as S n θ (z, dz ′ ) is an element of P(Z)).
Since η ∈ V δ 5 (Θ), ξ ∈ V δ 6 (P(X )), there exist θ ∈ Θ, λ ∈ P(X ) such that η − θ < δ 5 , ξ − λ < δ 6 . Notice also F m:n θ,y (λ) ∈ P(X ).
Notice that detBη(x), detDη(x) are analytic in (η, x) for η ∈ Vα 1 (Θ), x ∈ Vα 1 (X ). Notice also that |detB θ (x)|, |detD θ (x)| are uniformly bounded away from zero for θ ∈ clΘ, x ∈ X .
Assumption 7.1. For each θ ∈ Θ, z ∈ Z, T θ (z, dz ′ ) is a probability measure.Assumption 7.2. (i) There exist real numbers α ∈ (0, δ], L ∈ [1, ∞) such thatfor all η, η ′ , η ′′ ∈ V α (Θ), z ∈ Z, B ∈ B(Z) (here, |T η ′ − T η ′′ |(z, dz ′ ) denotes the total variation of T η ′ (z, dz ′ ) − T η ′′ (z, dz ′ ), while δ, ψ(z) are specified in Assumption 2.2 and(5)).(ii) For each η ∈ V α (Θ), there exists a complex measure τ η (dz) such that lim n→∞ T n η (z, B) = τ η (B) for all z ∈ Z, B ∈ B(Z).(iii) There exists a real number β ∈ (0, 1) such that T n η − τ η (z, B) ≤ Lβ n for all η ∈ V α (Θ), z ∈ Z, B ∈ B(Z), n ≥ 1 (here, T n η − τ η (z, dz ′ ) stands for the total variation of T n η (z, dz ′ ) − τ η (dz ′ )).Remark. According to Lemmas 5.1 and 5.4, both kernels S(z, dz ′ ), S η (z, dz ′ ) satisfy Assumptions 7.1 and 7.2. These assumptions capture all common properties of S(z, dz ′ ), S η (z, dz ′ ) relevant for the proof of Theorems 2.1 and 2.2.Besides the notation introduced in the previous sections, we rely here on the following notation, too. u n η (z 0:n ) and F n η (ξ, z 1:n ) are (respectively) the function and the complex measure defined by u n η (z 0:n ) = u n η (x 0:n , y 1:n ), F n η (ξ, z 1:n ) = F 0:n η,y (ξ)for η ∈ V α (Θ), ξ ∈ M c (X ), x 0 , . . . , x n ∈ X , y 0 , . . . , y n ∈ Y, n ≥ 0 and z 0 = (y 0 , x 0 ), . . . , z n = (y n , x n ),21Tη(z, dz ′ ) can be considered as a mapping with the following properties: (i) Tη(z, B) maps η ∈ C d , z ∈ Z, B ∈ B(Z) to C, (ii) Tη(z, B) is measurable in (η, z) for each B ∈ B(Z), and (iii) Tη(z, B) is a complex measure in B for each η ∈ C d , z ∈ Z.22Here, y 1:0 , z 1:0 denote empty sequences (i.e., sequences without any element). u n η (x 0:n , y 1:n ), F 0:n η,y (ξ) are specified in (7),(30). Notice that F 0:n η,y (ξ) depends only on y ′ 1 , . . . , y ′ n and is independent of other elements of y.Then, due to (88), φ η (ξ, z) is well-defined. On the other side, (87) impliesfor n ≥ 1. Consequently, (89) yieldsfor n ≥ 1. Therefore, φ η (ξ ′ , z ′ ) = φ η (ξ ′′ , z ′′ ) for any ξ ′ , ξ ′′ ∈ V δ9 (P(X )), z ′ , z ′′ ∈ Z. Hence, there exists a function φ η which maps η ∈ C d to C and satisfies φ η = φ η (ξ, z) for all η ∈ V δ9 (Θ), ξ ∈ V δ9 (P(X )), z ∈ Z. Then, using (90), we conclude that (78) holds for η ∈ V δ9 (Θ), ξ ∈ V δ9 (P(X )), z ∈ Z.is analytic in η for all η ∈ V δ5 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1 (δ 5 is specified in Lemmas 6.5, 6.6).(ii) Let Assumptions 2.1 -2.4 hold. Then, integralis analytic in η for all η ∈ V δ5 (Θ), λ ∈ P(X ), z ∈ Z, n ≥ 1.Proof. Throughout the proof, the following notation is used. φ(z) is the function defined by φ(z) = ϕ(y) for x ∈ X , y ∈ Y and z = (y, x). η is any element of V δ5 (Θ), while λ is any element in P(X ). {x n } n≥0 , {y n } n≥0 are any sequences in X , Y (respectively), while {z n } n≥0 is the sequence defined by z n = (y n , x n ) for n ≥ 0 (notice that {z n } n≥0 is any sequence in Z). Using Lemmas 6.1, 6.6, we concludefor n ≥ 1, where y = {y ′ k } k≥1 is any sequence in Y satisfying y ′ k = y k for 1 ≤ k ≤ n + 1. Combining this with Lemmas 5.3, 6.3, 6.6, we deduce that Φ η F n η (λ, z 1:n ), z n+1 , u n η (z 0:n ) are analytic in η for each η ∈ V δ5 (Θ). On the other side, due to Lemmas 5.3, 6.4, 6.6, we havefor n ≥ 1 (ψ(z) is specified in(5)).Owing to Assumption 2.5, we havefor n ≥ 1 (notice that z k = (y k , x k )). Consequently, Lemma A1.1 (see Appendix 1) and (93) imply that integral (91) is analytic in η for each η ∈ V δ5 (Θ), n ≥ 1. Relying on(9), it is easy to show(iii) There exists a real number α 1 ∈ (0, 1) such that η (x),B η (x),Ĉ η (x),D η (x) are analytic in (η, x) for η ∈ V α1 (Θ), x ∈ V α1 (X ).(iv) There exists a real number α 2 ∈ (0, 1) such thatv(x),ŵ(y) are analytic in x, y (respectively) for x ∈ V α2 (X ), y ∈ V α2 (Ỹ).Owing to Assumption 4.3 and (iii), there exists a real number α 3 ∈ (0, α 1 ) such that detB η (x) = 0, detD η (x) = 0 for η ∈ V α3 (Θ), x ∈ V α3 (X ). 26 Therefore,are well-defined and analytic in (η,(Y) (notice that clΘ, X , Y are compact sets and use (95)). Hence,are analytic in (η, x, x ′ , y) for η ∈ V α4 (Θ), x, x ′ ∈ V α4 (X ), y ∈ V α4 (Y). Combining this with Assumption 4.2, we deduce that there exist real numbers α ∈ (0, α 4 ), β ∈ (0, 1) such thatfor η ∈ V α (Θ), x, x ′ ∈ V α (X ), y ∈ V α (Y) (notice that functions (96) are positive and uniformly bounded away from zero for η ∈ clΘ, x, x ′ ∈ X , y ∈ Y). Owing to (97), we haveOn the other side, Lemma A1.1 (see Appendix 1) and (97), (98) imply thatIn the rest of the proof, the following notation is used.p η (x ′ |x),q η (y|x) are the functions defined bŷAs functions (96) and integrals (103) are analytic in (η, x, x ′ , y) for η ∈ V α (Θ), x, x ′ ∈ V α (X ), y ∈ V α (Y), it follows from (99), (101) thatr η (y, x ′ |x) is well-defined and analytic in η for η ∈ V α (Θ), x, x ′ ∈ X , y ∈ Y. Similarly, (99) -(102) imply that there exists a real number γ ∈ (0, 1) such that γ ≤ |r η (y, x ′ |x)| ≤ 1/γ for η ∈ V α (Θ), x, x ′ ∈ X , y ∈ Y. On the other side, (ii) yieldsr θ (y, x ′ |x) = r θ (y, x ′ |x) for θ ∈Θ, x, x ′ ∈ X , y ∈ Y. Then, using Theorems 2.1, 2.2, we conclude that there exist functionl,h :Θ → R such that l(θ),h(θ) are real-analytic in θ and satisfy lim n→∞ l n (θ, λ) =l(θ), lim n→∞ h n (θ, λ) =h(θ) for θ ∈Θ, λ ∈ P(X ) (l n (θ, λ), h n (θ, λ) have the same meaning as in(1)). Consequently, Corollaries 4.1, 4.2 hold (notice that Θ can be represented as the union Θ = ∞ n=1Θ n , where {Θ n } n≥1 are non-empty open balls satisfying clΘ n ⊂ Θ for n ≥ 1).Appendix 1This section contains some auxiliary results which are relevant for the proof of Lemmas 5.2, 5.3, 6.3, 7.2 and Corollaries 3.1 -4.2. Here, we rely on the following notation. d w ≥ 1 and d z ≥ 1 are integers, while A is a bounded convex set in C dw . F (w, z) is a function mapping w ∈ C dw , z ∈ R dz to C, while λ(dz) is a measure on R dz . f (w) is the function defined byLemma A1.1. Assume the following:(i) There exists a real number δ ∈ (0, 1) such that F (w, z) is analytic in w for each w ∈ V δ (A), z ∈ R dz . (ii) There exists a function φ :is well-defined and analytic for all w ∈ V δ (A).Proof. Owing to Cauchy inequality (see e.g.,[26,Proposition 2.1.3]) and (i), (ii), we haveOn the other side, if φ(z)λ(dz) < ∞, then the dominated convergence theorem and (104) imply that f (w) is well-defined and differentiable for w ∈ V δ (A). Consequently, f (w) is analytic for w ∈ V δ (A).In the rest of this appendix, we use the following notation. B is a compact set in R dw , while g(w) is a function mapping w ∈ R dw to R (d w is specified at the beginning in the appendix).Lemma A1.2. Assume that there exists an open set C in R dw such that B ⊂ C and g(w) is real-analytic on C. Then, there exists a functionĝ(w) with the following properties:(i)ĝ(w) maps w ∈ C dw to C.(ii)ĝ(w) = g(w) for all w ∈ B.(iii) There exists a real number δ ∈ (0, 1) such thatĝ(w) is analytic on V δ (B).Proof. First, we assume that B is connected (latter, this assumption is dropped). As g(w) is real-analytic on C, g(w) has an analytic continuation in an open vicinity of any point in C. Hence, there exist functionŝ g(w, v), δ(v) with the following properties:, v ∈ C. Since B is compact, there exist an integer M ≥ 1 and points {v i } 1≤i≤M such that v i ∈ B for 1 ≤ i ≤ MAppendix 2In this section, we explain how Theorem 2.2 can further be extended in the context of finite-state hidden Markov models. Here, we assume that X has a finite number of elements and that µ(x) = 1 for each x ∈ X (in that case, p θ (x ′ |x) is the conditional probability of X θ,λ n+1 = x ′ given X θ,λ n = x). We also rely on the following notation. N is the number of elements in X , while x 1 , . . . , x N are the elements of X (i.e., X = {x 1 , . . . , x N }). P N is the set of N -dimensional probability vectors, while e is the N -dimensional vector whose all elements are one. For θ ∈ Θ, y ∈ Y, let R θ (y) be the N × N matrix whose (i, j)-entry is r θ (y, x i |x j ), where r θ (y, x ′ |x) has the same meaning as in Section 2. For θ ∈ Θ, u ∈ P N , y ∈ Y, let G θ (u, y) and h θ (u, y) be the functions defined by G θ (u, y) = R θ (y)u e T R θ (y)u , h θ (u, y) = log e T R θ (y)u .Regarding functions r θ (y, x ′ |x), G θ (u, y) and h θ (u, y), we assume the following.Assumption A2.1. There exist a real number ε ∈ (0, 1) and a function s θ (y, x) mapping θ ∈ Θ, x ∈ X , y ∈ Y to [0, ∞) such that εs θ (y, x ′ ) ≤ r θ (y, x ′ |x) ≤ s θ (y, x ′ ) ε for all θ ∈ Θ, x, x ′ ∈ X , y ∈ Y.
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| []
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[
"Two-photon correlations in detuned resonance fluorescence",
"Two-photon correlations in detuned resonance fluorescence"
]
| [
"Eduardo Zubizarreta Casalengua ",
"Elena Del Valle ",
"Fabrice P Laussy [email protected] ",
"\nDepartamento de Física Teórica de la Materia Condensada\nFaculty of Science and Engineering\nUniversidad Autónoma de Madrid\n28049MadridSpain\n",
"\nDepartamento de Física Teórica de la Materia Condensada\nUniversity of Wolverhampton\nWulfruna StWV1 1LYWolverhamptonUK\n",
"\nInstitute for Advanced Study\nUniversidad Autónoma de Madrid\n28049MadridSpain\n",
"\nFaculty of Science and Engineering\nTechnische Universität München\n85748GarchingGermany\n",
"\nUniversity of Wolverhampton\nWulfruna St, Russian Quantum Center, Novaya 100WV1 1LY, 143025Wolverhampton, Skolkovo, Moscow RegionUK, Russia\n"
]
| [
"Departamento de Física Teórica de la Materia Condensada\nFaculty of Science and Engineering\nUniversidad Autónoma de Madrid\n28049MadridSpain",
"Departamento de Física Teórica de la Materia Condensada\nUniversity of Wolverhampton\nWulfruna StWV1 1LYWolverhamptonUK",
"Institute for Advanced Study\nUniversidad Autónoma de Madrid\n28049MadridSpain",
"Faculty of Science and Engineering\nTechnische Universität München\n85748GarchingGermany",
"University of Wolverhampton\nWulfruna St, Russian Quantum Center, Novaya 100WV1 1LY, 143025Wolverhampton, Skolkovo, Moscow RegionUK, Russia"
]
| []
| We discuss two-photon correlations from the side peaks that are formed when a two-level system emitter is driven coherently, with a detuning between the driving source and the emitter (quasi-resonance fluorescence). We do so in the context of the theories of frequency-resolved photon correlations and homodyning, showing that their combination leads to a neat picture compatible with perturbative twophoton scattering that was popular in the early days of quantum electrodynamics. This should help to control, enhance and open new regimes of multiphoton emission. We also propose a way to evidence the quantum coherent nature of the process from photoluminescence only, through the observation of a collapse of the symmetry of the lineshape accompanied by a surge of its intensity of emission. We discuss several of our results in the light of recent experimental works. | 10.1088/1402-4896/acc89e | [
"https://export.arxiv.org/pdf/2210.03733v2.pdf"
]
| 252,762,297 | 2210.03733 | 4ebe4d8fecca3033f8c5a6142ea0ed751e6dcd3f |
Two-photon correlations in detuned resonance fluorescence
19 October 2022
Eduardo Zubizarreta Casalengua
Elena Del Valle
Fabrice P Laussy [email protected]
Departamento de Física Teórica de la Materia Condensada
Faculty of Science and Engineering
Universidad Autónoma de Madrid
28049MadridSpain
Departamento de Física Teórica de la Materia Condensada
University of Wolverhampton
Wulfruna StWV1 1LYWolverhamptonUK
Institute for Advanced Study
Universidad Autónoma de Madrid
28049MadridSpain
Faculty of Science and Engineering
Technische Universität München
85748GarchingGermany
University of Wolverhampton
Wulfruna St, Russian Quantum Center, Novaya 100WV1 1LY, 143025Wolverhampton, Skolkovo, Moscow RegionUK, Russia
Two-photon correlations in detuned resonance fluorescence
19 October 2022
We discuss two-photon correlations from the side peaks that are formed when a two-level system emitter is driven coherently, with a detuning between the driving source and the emitter (quasi-resonance fluorescence). We do so in the context of the theories of frequency-resolved photon correlations and homodyning, showing that their combination leads to a neat picture compatible with perturbative twophoton scattering that was popular in the early days of quantum electrodynamics. This should help to control, enhance and open new regimes of multiphoton emission. We also propose a way to evidence the quantum coherent nature of the process from photoluminescence only, through the observation of a collapse of the symmetry of the lineshape accompanied by a surge of its intensity of emission. We discuss several of our results in the light of recent experimental works.
Introduction: historical developments of resonance fluorescence
Resonance fluorescence is the "drosophila" of quantum optics. It is the simplest yet rich enough problem to capture many of the key considerations on light-matter interaction, from quantization of the light field up to the riddle of measurements and observations in quantum mechanics. It consists of driving optically a two-level system (e.g., an arXiv:2210.03733v2 [quant-ph] 18 Oct 2022 atomic transition, a spin, a semiconductor exciton, a superconducting qubit, etc.) with a coherent wave that has the same or a close frequency than that of the spontaneous emission of the emitter. The platform is both of great fundamental interest for the understanding of basic aspects of quantum theory as well as from a technological perspective for its prospects as a quantum emitter, not only as a single-photon source but also in an unsuspected regime of multiphoton emission.
The multiphoton problematic turns out to have been central to theoretical modelling since the early days. As a basic problem of light-matter interaction, the origin of resonance fluorescence goes back to the dawn of quantum electrodynamics (which itself can be dated with Dirac [1]). Pioneering contributions include those of Weisskopf [2,3], for scattering off the ground and excited state of an atom, respectively. A major and recurrent work still of actuality, in the low-driving regime, is that of Heitler [4] who reported his analysis directly in the 3rd edition of his textbook "the Quantum Theory of Radiation", in a chapter (absent in previous editions) titled "resonance fluorescence" ( §20), where he shows that the lineshape of radiation is provided by the driving source itself as opposed to the natural lineshape of the emitter. His analysis follows in essence from the conservation of energy δ(ω − ω L ) of the scattering process so that each photon from the source gets scattered at the energy with which it impinged on the atom, whence the result. The process is actually not as trivial as it looks, with Heitler already observing that the radiation occurs "as if two independent processes, an absorption and a subsequent emission, took place", with the atom "remembering" (his term) "before the emission which quantum it has absorbed". Seen in this way, it is less obvious why the spontaneous emission character of the emitter plays no role. It also brings forward that a two-photon process is involved in this scattering. In fact, as we discuss further below, it turns out to be of central importance for the photon statistics of resonance fluorescence in this low-driving regime [5]: the δ-shaped scattered light itself is uncorrelated and becomes antibunched only if also detecting the weak-but at the two-photon level, essential-incoherent part of the spectrum, that is indeed spread spectrally and originating from multiphoton events. This peak is however very small in intensity as compared to the Rayleigh peak. This led to some confusion in the literature [6,7] that we hope to have clarified [8] (see also [9]): although the incoherent peak vanishes at low intensities in one-photon observables, its contribution rules the photon-statistics (a two-photon observable).
Multiphotons are even more prominent at higher driving. In a nonlinear quantumoptics framework, several degenerate photons from the driving source (which we shall from now on refer to as the "laser") can be redistributed by the emitter at different energies so as to produce a more complex spectral shape, known today to be a triplet with ratio of peak heights 1:3:1 and with a splitting given by the laser intensity. The problem was initially regarded as that of the competition between spontaneous and stimulated emission, with a feeling shared by many theorists of the time that spontaneous emission required quantization of the field for a correct treatment. The exact nature of this spectral shape was the topic of some controvery, in particular it took part in the debates initiated by Jaynes according to whom the light field should not be quantized and his neoclassical theory (relying on a nonlinear feedback from the radiation field back to the emitter) should be used instead. The neoclassicists were also experts in solving the quantized version of a problem to provide what they assumed were the wrong QED predictions, which is how, famously, the Jaynes-Cummings model [10] arose. In this framework, the quantized version of resonance fluorescence by Stroud [11] (part of Jaynes' team) but at the one-photon level, led to incorrect results, such as a 1:2:1 ratio of the peaks, in contrast to a semiclassical treatment by Mollow [12] which was not quantizing the light field but obtained, for the first time, the correct lineshape. For this reason, this characteristic result, that was originally referred to as the AC stark effect, became known as the "Mollow triplet" (it seems that Zoller [13] is the first to have used this denomination). Stroud et al. mention in their conclusions that their analysis is "incomplete in one important aspect": the truncation to one-photon emission. While they recognize that "in the real physical case there will be a cascade emitting many spontaneous-emission photons", they believed that antibunching would make such successive emissions from a quantized model justifying their approximation. Further support for Stroud et al.'s view came from Smithers and Freedhoff [14] who claimed to have included multiphoton effects and yet still arrive at the same (incorrect) result as Stroud et al., but this was disputed by Carmichael and Walls [15] who observed that in their treatment, "Smithers and Freedhoff have not managed to include true photon cascades but have simply followed a series of sequential one-photon emissions". Convincingly, by truncating their quantized version to singlephoton transitions, Carmichael and Walls showed how they downgraded Mollow's spectrum to one with the same attributes as Stroud's. Mollow's result, it must be emphasized, although not quantizing the light-field, is not part of the neoclassical theory, which treats spontaneous emission as a continuous process as opposed to quantum jumps, leading to still further departures between the various models. The reason why Mollow got the correct result is interesting: ironically, it turns out that the semiclassical model is equivalent to a multiphoton quantized model, and that multiphoton effects are responsible for the lineshape, although this is a single-photon observable. This has been recognized and commented by various people at the time but the most insightful discussion seems to be that of Mollow himself, in his 1975 follow-up paper [16]. He carried on such a fully-quantum treatment, including multiphoton contributions of all orders, and showed that the c-number description of the laser does not spoil the fullyquantum nature of the problem, as long as multiphoton effects are included. These correspond to the back-reaction in Jaynes' neoclassical theory and to what a modern treatment would qualify as virtual photons, i.e., the atom re-absorbing photons that it has just emitted. In this context, the problem can be understood as a scattering one. In the words of Mollow [16] " the individual multiphoton scattering processes [. . . ] are concealed from view, with only their accumulated effect exhibited". We will come back to this important observation later on. Another key contribution to that approach of resonance fluorescence comes from Cohen-Tannoudji [17] and his co-workers [18,19,20,21], who provided both the so-called dressed-atom and a perturbative scattering pictures. From this viewpoint, spontaneous emission is not deemed central but is relegated to a secondary plane. Instead, dressing the atom is considered first, yielding new eigenstates for the system with an exact (all-order) treatment of the lightmatter coupling. Then spontaneous emission is brought back to make the dressed atom cascade down its energy diagram and in this process replacing photons from the laser by fluorescence photons from the atom. This remains the most picturesque way to understand the spectral shape of the Mollow triplet and can also account for a lot, although not all, the phenomenology of correlations between the peaks.
We now turn to the detuned resonance fluorescence, i.e., when the driving laser is close to but not right at the energy of the two-level system. In the earlier treatments, such as from Heitler, exact resonance implied divergence and one of Heitler's inputs was precisely to damp the system [22] so as to arrive to a physical response for exact resonance. In a modern quantum-optical, master equation approach, resonance is actually simpler while off-resonance comes with additional subtleties but also with several advantages. Not least is the fact that detuning helps the splitting of what always remains a triplet. In fact, the first neatly resolved Mollow triplet was out of resonance [23] and if one would stick to resonance, it would then be apparently the improved setup of Walthers that has reported the first resonant Mollow triplet, albeit in a conference proceedings [24] (the first report of an even better triplet in a leading journal came however only a few months later [25]). Detuning also weakens the efficiency of the coupling and what determines whether one is in the Heitler (low-driving) or Mollow (high-driving) regime in this context is an interesting question that we address elsewhere [26]. As was already described by Mollow in his magnum opus [12], when the 2LS is detuned from the laser, one gets at low driving a doublet with a peak centered on the atom and the other peak shifted by twice the detuning, with the Rayleigh-scattered laser sitting in between, that is to say, one always has a triplet in the non-detuned case. This is shown in Fig. 1(e). Here it must be appreciated that the two side peaks are vanishing with Ω σ → 0 as compared to the coherent peak in the center, with a ratio 8Ω 2 σ /(γ 2 σ +4∆ 2 σ )) for their respective intensities, i.e., most of the emission comes from the central peak, just as the case of resonance where the Lorentzian foothill is dwarfed by the Rayleigh peak. Here too, however, two-photon observables, such as photon statistics, are ruled by the interplay of the coherent and incoherent emission [27], regardless of their relative intensities. The only, but striking, difference is that this central incoherent peak has now split in two. With increasing driving, a central incoherent peak grows at the laser position, becoming of identical height with the side peaks when Ω σ ≈ ∆ σ / √ 2 (exactly so in the limit γ σ ∆ σ , Ω σ ) and for higher-still driving converging towards a resonant Mollow triplet, since detuning now becomes negligible as compared to driving. There is therefore a smooth transition between the various cases [26].
A triplet structure comes with an obvious opportunity to correlate photons from the various peaks. This was highlighted by Cohen-Tannoudji and Reynaud [19] and implemented by Aspect et al. [28]. At low-enough driving but with detuning to [20]. Regardless of detuning, the spectral shape is the same: the central peak is the Rayleigh δ peak that elastically scatters the laser photon (depicted as a thin blue line topped by the ∞ symbol). The detuning places the emitter on one side (upper row) or the other (lower row), while two-photon energy conservation places a copycat peak on the other side of the laser as result of the virtual states created to fill-in the loop (second column). The spectral shape is shown, in (e), for the Heitler regime of vanishing driving Ω σ → 0 and, in (f), at nonzero driving, producing an emerging fluorescence peak at the center of the triplet and slightly increasing the triplet splitting. maintain a multi-peak structure, the perturbative treatment is appealing and provides a remarkable and compelling picture for the triplet, that is shown in Figs. 1(a-d). Here again, scatterings between the states of the system gives the best phenomenological explanation for the observed doublet (on the one hand) surrounding the central peak (on the other hand). This was discussed in details by Dalibard and Reynaud [20]. In addition to the Rayleigh peak, that elastically scatters the photons and thus pins the peak at the energy of the laser (1st column), the scheme also involves two virtual states and a two-photon transition. One of the photons originates from the excited-to-ground state transition with energy ω σ , but the other, not expected a priori, originates from the virtual states created by the detuned laser which needs them to close the two-photon emission process. This is at energy ω ν ≡ 2ω L − ω σ . Note that, although the states are virtual, both photons are real. We will clarify this statement in the following. The prediction from such a picture, indeed confirmed experimentally [28], is that such a twophoton emission comes as a cascade: first the virtual-state's photon then the emitter's. While such considerations have been made and confirmed a long time ago, recent reports by Masters et al. [29] of the observation of bunching from the incoherent part of the spectrum of the low-driving detuned resonance fluorescence, and by Long Ng et al. [30] of cross-correlations from the high-driving but also detuned Mollow triplet, bring back such important questions in the limelight of modern setups and the improved accuracy affordable today. In particular, modern authors envision quantum-optical technological prospects, which were not at the core of the preoccupations of the founding fathers who were worrying, instead, on experimental validations of one or the other model of the theory of light-matter interactions, from Dirac's quantum electrodynamics to dissipative quantum optics. Such recent works can also participate to the experimental validation of, or discrimination between, the more refined theories available today, e.g., compare Figure 5 of Ref [30] with Figure 4a of Ref. [31] that itself includes, in addition to the assumed exact cross-correlations between the peaks, results from earlier works [32]. More importantly, they also allow to test more recent and new proposals regarding multiphoton emission from resonance fluorescence. In the following, we discuss our own input to this problem, which starts with the theory of frequency-resolved photon correlations [33]. We articulate our discussion along the experimental findings of Long Ng, Masters et alii.
(a) (b) (c) (d) (e) (f)
Frequency-resolved photon-correlation and homodyning
Correlating the peaks in Fig. 1 can be done by filtering them first and directing the respective outputs to a correlator (e.g., an Hanbury Brown-Twiss setup). This was already discussed in precisely these terms by Cohen Tannoudji. Masters et al. chose a different strategy of filtering out the central Rayleigh peak and auto-correlating the output. We will come back to their approach but discussing first the more traditional one that has been considered several times [28,34], we must highlight the input by Schrama et al. [32]. They implement the photodetection theory [35], though at, or close to, resonance and in the high (Mollow) driving regime, but more importantly, with some approximations in the model to undertake the complex calculations involved in the way the theory was then formulated (as nested, time-ordered, high-dimensional integrals). We provided an alternative, numerically exact as well as efficient, formulation of the problem that allows us to compute faithfully frequency-resolved n-photon correlations [33], in some cases even analytically. This consists in enlarging the original problem (in this case resonance fluorescence) with n two-level systems ς i (in this case, n = 2) that are coupled both through an Hamiltonian H n and Liouvillian L n to the system (in units ofh = 1):
∂ t ρ = −i[ω σ σ † σ + Ω(σ + σ † ), ρ] + γ σ 2 L σ ρ − i 2 j=1 [ω i ς † j ς j , ρ] + [σς † j + σ † ς j , ρ] + 2 j=1 Γ j 2 L ς j ρ(1)
where the first line in Eq. (1) is the resonance fluorescence problem in its simplest possible and modern quantum-optical formulation, while the second line implements the sensor formalism where ω i set the frequencies which are to be correlated while Γ i set the filters' bandwidths. One can then (easily) solve this master equation without worrying for the complicated frequency variables that otherwise enter at the level of parameters in complex integrals, and which have been upraded here to an operator. This makes the original evaluations in terms of folded integrals turn to standard intensity-intensity correlations:
g Γ 1 ,Γ 2 (ω 1 , t 1 , ω 2 , t 2 ) = lim →0 :ς † 1 ς 1 (t 1 )ς † 2 ς 2 (t 2 ): ς † 1 ς 1 (t 1 ) ς † 2 ς 2 (t 2 ) ,(2)
where the limit limit → 0 is to be taken, providing finite values for Glauber's correlators since both numerators and denominator are of the same order. Given that we will deal with steady-states only, we will compute g Γ 1 ,Γ 2 (ω 1 , ω 2 ; τ ) with τ ≡ t 2 − t 1 the time-delay between the photons, which can be done with the quantum regression theorem. We need not elaborate further here on the method itself, only emphasize again that it provides the exact (according to the established theory of photo-detection) correlations between the filtered photons, by detectors with the respective bandwidths at the given time delay. Note that if ω 1 = ω 2 , this describes filtered auto-correlations. In the following, we shall consider that Γ 1 = Γ 2 . We discuss the results in next Section. We need however to first introduce another idea that relates to the contribution of the Rayleigh peak. The nature of antibunching in resonance fluorescence is very different in the lowdriving (Heitler) and high-driving (Mollow) regimes [8]. In the latter, it follows from the more straightforward and popular picture of the two-level system being with some probability in its excited state and, in its transition back to the ground state, releasing a single photon. The density matrix in the limit of infinite driving is ρ = 1 2 (|0 0| + |1 1|). In stark contrast, in the low-driving regime, antibunching arises from an interference between the displaced squeezed thermal state in which the two-level system is driven to leading-order, and the coherent state that is imprinted by the driving laser [27]. The presence of squeezing as well as a strong Poisson content from the laser means that the antibunched emission is of a more subtle multiphoton character in this case. Photodetection tampers with the squeezing in a way that is fundamentally equivalent to frequency-filtering. This disturbs the interference that otherwise yields a perfect antibuching. We have shown with López Carreño how one can, however, restore perfect antibunching by correcting for the excess coherence from the laser as a result of filtering the tails of the incoherent spectrum [5]. We later addressed the case of detuned resonance fluorescence in the context of filtered-homodyned correlations [36], but we did so for antibunching. In the next Section, we will instead phase-shift our focus to consider the case of multiphoton emission, which is the one revived by Masters et al. [29] The formalism is exactly the same, the regime and interpretations however deserve considerations of their own. We also later generalized the scheme to a large range of platforms where coherence is involved in some form [37], in which case we have shown that such multiphoton interferences are key to explain the structure and type (conventional or unconventional) of a wide range of correlations, from antibuching to superbunching. As a general statement, one can in the framework that we have just laid down, seize additional control of the system by homodyning, i.e., externally changing the nature (constructive or destructive) of the interference and further selecting auto or cross-correlations to characterize the system, resulting in a much enhanced versatility and performance of its emission. We do this in the following in the case of two-photon emission in detuned resonance fluorescence.
Removing the central peak
We now turn to the exact computation of correlations in the case where the laser is (with no loss of generality, since solutions are symmetric) red-detuned at ω L as compared to the atom at ω σ = ω L + ∆ σ (cf. Fig. 1(c,e)). The main question addressed by Masters et al. is whether the side peaks are bunched, what they interpret as simultaneous two-photon emission. We come back to qualify further this interpretation but first consider the question itself. In their case, they used a narrowband notch filter to attenuate the central peak, and let otherwise pass the other photons, i.e., from the side peaks, which they autocorrelate. The ideal version of this experiment is precisely our homodyning scheme, unfiltered, where we remove the central peak by destructive interference. Indeed, we have checked that the two-photon coincidence for the notch filter of width Γ and centered at ω σ when filtering out the coherent Rayleigh peak, agrees in the limit Ω σ → 0 with the full-homodyned zero-delay coincidence, that is given by g (2) σ,F =1 (0) = (γ 2 σ + 4∆ 2 σ )(γ 2 σ + 4[8Ω 2 σ + ∆ 2 σ ])/(64Ω 4 σ ). The general and time-resolved case is shown in Fig. 2 from, bottom, no homodyning (corresponding to F = 0), up to, top, full-destructive homodyning (F = 1) that completely removes the laser contribution. In the first case, we are looking there at the full autocorrelation of the light itself. This is a well-known result: the antibunching is perfect but oscillates strongly, which is usually (and correctly) interpreted as the effect of detuning. As the coherent peak is removed, one can see how antibunching is gradually lost, along with the oscillations (although the two peaks remain detuned) to give rise to bunching. This is the same result as obtained by Masters et al. who, both in their experiment and theoretical treatments, report a more complicated structure, due to the filtering. For more stringent filtering their result reverts in character but this needs not concern us here as this involves even more complicated but technical effects that are well described by the authors. We believe that our description of their idea provides its cleanest and ideal formulation. Now turning to its interpretation, one must note that the interference is perfect between the two splitted incoherent peaks and the central coherent peak: regardless of the detuning, detecting all frequencies yield perfect antibunching. This must be kept in mind when considering the interpretation of the perturbative picture of Fig. 1. The coherent peak, that plays no role in the diagram itself, is fundamental to decide of the outcome of the actual detection. A possible interpretation is that the coherent peak is responsible for the synchronization of the two scattered photons from the diagram, so that they do not appear together in the emission but are modulated in time, so as to ensure perfect antibunching at τ = 0 but resulting in strong-bunching shortly afterwards. That is another, more sophisticated but equally valid, way to understand the oscillations in g (2) (τ ). Still another understanding [36] is that such a detection at all frequencies removes the indistinguishability between the photons and one can thus no longer assert that their detuning leads to a spectral distinction.
Cross-correlations of the side peaks
So far, we have not actually filtered the emission. Instead, we used homodyning to remove a particular part of the spectrum, namely, the central scattering peak. We now bring filtering to similarly remove this central peak, by placing two detectors (or filters) on the side peaks, although what we actually gain in this way is the possibility to turn to cross-correlations. This can tell us about the cascaded emission, i.e., whether one photon comes before the other, which was the actual prediction from the pioneering papers of the two-photon character in this case, as opposed to simultaneous emission. We first provide the ideal two-photon cascade cross-correlation g (2) (τ ), defined as an uncorrelated (Poisson) stream (1) of photons with emission rate γ 1 , each photon of which triggers the emission of another photon in another stream (2), in the good time order, i.e., emitted at a later time, with probability p and with decay rate γ 2 , or in the wrong time-order, i.e., emitted before the triggering photon, wih probability 1 − p and decay rateγ 2 . In this case, we find that:
g (2) (τ ) = 1 + (1 − p)γ 2 γ 1 exp(γ 2 τ ) if τ < 0, p γ 2 γ 1 exp(−γ 2 τ ) if τ > 0.(3)
Note that there is a discontinuity at τ = 0 which prevents to define g (2) (0), that is however of no concern since this occurs at a single point (in a practical context, one can take either limit ±τ → 0 or an average). A physical mechanism approaching this ideal scenario would besides connect smoothly the two domains. In our case, the exact numerical results for the filtered two-photon correlations of resonance fluorescence are shown in Fig. 3 for various critical parameters being varied one at a time. Taken in turns, we find that:
(a) With no (or very broad, Γ γ σ ) filtering, there is no asymmetry and there is a strong antibuching, recovering in this case the full correlation of the complete spectrum. As the filtering tightens around the peaks (i.e., Γ decreases) an asymmetry develops in the form that is typical of a two-photon cascade, cf. Eq. (3), with a transition to bunching that is maximum at positive τ . The correlations also exhibit strong oscillations. For smaller still values of Γ (narrow filters), correlations increase but the asymmetry reduces. In the limit of Γ → 0, correlations diverge and become τ independent (flat).
(b) As a function of detuning ∆ σ , one goes from antibunching at resonance (∆ σ = 0) to a growth of the asymmetry and increase of the correlations. In this case, the trend is consistent and both the asymmetry and correlations increase to realize the case of an increasingly better two-photon cascade.
(c) As a function of driving Ω σ , one goes from the Heitler (Ω σ → 0) to the detuned Mollow (growing an incoherent central peak) and ultimately to the resonant Mollow triplet Ω σ γ σ as detuning becomes negligible. In this case, the cascade asymmetry remains constant while the strength of the correlations decrease and the oscillations dampen.
(d) Finally, as a function of pure dephasing γ φ , the asymmetry also remains constant as the correlations weaken, although in this case better retaining the oscillations and also recovering some antibunching for very large dephasing, due to spectral overlap induced by the line broadening.
These observations are informative regarding the general character of the twophoton emission, which we can summarize as follows: there is a clear cascade, i.e., one photon comes before the other, as is consistent with the two-photon scattering picture; this is better resolved when photons are adequately detected, i.e., there is an optimum filter width to optimize the effect; the cascade gets better with increasing detuning, forcing the system into the perturbative two-photon limit; correlations are stronger for weaker driving, i.e., the Heitler regime is more correlated than the Mollow one but oscillations are also more pronounced; dephasing damages the process, and even reverts to the opposite regime of single-photon emission. Also, it should be emphasized once more that these results are supposedly exact, since they have been obtained with no approximations from the filtered and photo-detection side once given the model (which is that from Mollow). It becomes particularly interesting, however, to now combine homodyning and cross-correlation filtering, which we do in the next Section.
Cross-correlations of the homodyned side peaks
We now repeat the same procedure as previously, cross-filtering the two peaks, but removing the central peak. We do that by homodyning, since this is the ideal scenario, but this could experimentally be approached by cross-correlating the output of Masters et al.'s narrowband notch filter. The results are shown in Fig. 4 and are to be contrasted one-to-one with those of Fig. 3, which was including the central coherent peak. While one could assume that cross-correlating the side peaks is tantamount in the first place to remove the central peak, interference filters have tails and their suppression is not perfect, therefore some residual physics of interference leaks through and impact the results. This is why Fig. 2 differs from Masters et al.'s filter approach: their suppression of the coherent peak in this way is partial only. The impact turns out to be, maybe surprisingly, momentous. One need only compare the strong, both qualitative and quantitative, departures between the non-homodyned and homodyned versions to appreciate the importance and value of this modus operandi. First, notice that the scale up to ≈ 10 6 in Fig. 3 need be inflated to over 10 10 in Fig. 4: correlations are considerably stronger. Also, the various cases split neatly the ones from the others, at the exception of filtering, which tends to now feature more similar magnitudes of correlations, while it retains an optimum filter width to maximize the asymmetry. Finally, oscillations are very much, sometimes completely suppressed. Understanding these oscillations as a way for the system to provide one-photon emission from a two-photon mechanism by synchronizing them, this means that, by removing the central peak, we manage to enter more deeply into a more genuine two-photon physics. The shapes observed in this case are very close to the ideal two-photon cascade, Eq. (3), with identical slopes and thus decay rates γ 2 =γ 2 for the ordered and out-of-order photons, as expected for a biphoton. The rate is γ 2 = γ σ − 8γ σ Ω 2 σ /(γ 2 σ + 4∆ 2 σ ) ≈ γ σ in the Heitler regime while the fast-rising slope bridging the gap is given by Γ, the filters' width. For γ 2 =γ 2 , this gap allows from Eq. (3) to estimate both the ratio γ 2 /γ 1 (absolute gap obtained by the differences between the ±τ → 0 limits) and the probability p (relative gap obtained by their ratio for high enough correlations), from which one can see that the time-ordering is very good, with the ω υ photon arriving with high probabiliy before the ω σ one. It should be noted, however, that two-photon emission is not fully captured by the two-photon correlations alone. For instance, the probability that the emission of one photon successfully heralds the other is not captured by a g (2) measurement, which is not affected by photon-losses, although the symmetry of the peaks in luminescence suggests that this also is very high. Still, a thorough description of the two-photon emission in this regime is a separate problem. Nevertheless, all these facts together confirm that an ideal two-photon cascade can be approached in detuned resonance fluorescence, at large detunings, for filter widths commensurable with the width of the peaks and when suppressing the coherent Rayleigh peak.
We conclude with an unexpected result. While the process is more efficient without dephasing, its correlations are not seriously affected by it, but the impact of dephasing on the signal itself is dramatic. Two-photon emission from the side peaks of detuned resonance fluorescence is very weak in intensity, being second-order in the driving. Most of the emission originates from the central peak, and removing it, naturally leaves only little emission. It is interesting that a tiny amount of pure dephasing changes considerably both the spectral shape and the amount of incoherent emission: the perfectly symmetric spectral shape collapses with very small dephasing, at the same time as the intensity grows considerably. The symmetric peak at ω υ remains the same (i.e., remains very small), but a channel is now opened for bare, incoherent emission from the real transition at ω σ . This means that the two-photon diagrammatic picture breaks down and get substituted by a more straightforward picture of a classical oscillator driven out of resonance and emitting at its natural frequency, which is a first-order process, and therefore of considerably higher intensity than the two-photon emission. What is maybe more remarkable is that such first-order processes get cancelled in the coherent picture, i.e., the interferences leading to one-photon emission contrive to suppress this channel, that would be expected in a classical picture. Dephasing plays the role of scrambling this interference by marring the multiphoton wave interference, thus re-opening a strong classical channel. We do not think that this feature has been previously noted in this particular context, although it seems to be of general validity and is certainly occuring in other systems and configurations. In our case, it achieves the demonstration that two-photon emission is a fragile quantum coherent process, of weak intensity. The loss of the satellite peak on the other side from the emitter is also more dramatic as Ω σ goes to zero, showing that this is a Heitler, two-photon effect. Experimentally, it is likely that some amount of dephasing is present but is counteracted by some amount of driving. Observing a collapse of the satellite peak-the more abruptly and with a greater gain of intensity from the other (emitter) peak, the lower the driving-would be an experimental demonstration of two-photon physics directly at the level of photoluminescence.
Summary and Conclusions
We discussed the two-photon correlations in detuned resonance fluorescence, whose spectral shape-two-side peaks sitting around a central Rayleigh scattered peak (at low enough driving)-lends itself to a natural interpretation in terms of two-photon scattering, as presented by Cohen-Tannoudji, Reynaud, Dalibard and others. In particular, we contrasted our theoretical results to those aimed by Masters et al. of supressing the coherent peak. They achieved this by filtering it out and otherwise auto-correlating the emission that passes through, finding a clear transition from antibunching with strong-oscillations to bunching with reduced-but always presentoscillations. In our case, we can completely remove the coherent peak by perfect destructive interferences with an external (homodyning) laser. We find essentially the same result as Masters et al. of a transition from perfect antibunching with strong oscillations, when the coherent peak is present, to bunching when removing it, although in our case there is also a disappearance of the oscillations. We then turn to crosscorrelations of the peaks, showing how they bear strong features of a two-photon cascade indeed, that get considerably improved by removing the coherent peak also in this case. This shows that homodyning is a valuable additional concept, rather than supplement, to frequency filtering, and can lead, in the case of resonance fluorescence, to an almost ideal cascaded two-photon emission. These considerations are only a small part of the general picture of multiphoton correlations and emission from resonance fluorescence, which generalizes to higher photon numbers, involving other types of scattering processes (prominently, leapfrog processes [38,39]) and calling for further investigation of their fundamental character as well as relevance for technology. The interpretations of the underlying physics are interesting but intrinsically limited by one's mental picture(s). The two-photon diagrams in Fig. 1 provide a compelling mechanism of a coherent quantum character, but they do not include the subtle multiphoton interferences with the Rayleigh peak and the detection process, which are crucial for a comprehensive description. As befits quantum mechanics, the best one can do is to compute observables. All interpretations will be constrained in one way or another, although useful applications could derive from them. In that regard, we have provided a formalism to achieve exact results. It is remarkable that the simplest treatment of the simplest system, already at the time of Mollow, yields the best results while more sophisticated techniques, more suitable for more complex systems, come with some approximations. We ourselves suspect that the driven two-level system could be the richest and most fundamental source of multiphoton physics and that more complex systems could, instead of bringing additional physics, result in imperfect limits of this ultimate realization. There are many platforms to implement this physics, from atoms [40] to ions [41] and molecules [42] passing by semiconductors (quantum dots [43], spin [44], NV centers [45], etc.), superconducting qubits [46] and circuits [47], and still others [48,49,50]. While the interest for basic and fundamental multiphoton emission originates from the early days of quantum electrodynamics, recent progress from both material, technological and theoretical sides makes it a particularly burgeoning field of study.
Figure 1 .
1Scattering-picture of detuned resonance fluorescence, in the perturbative treatment of Dalibard and Reynaud
Figure 2 .
2Two-photon correlation (no spectral filtering) as a function of the homodyning field that gradually suppresses the coherent (Rayleigh-scattered) central peak, from no supression (F = 0) to complete suppression (F = 1). The latter case is the ideal realization of Masters et al.'s suppression of the coherent part. We find a good agreement with their result but with the more straightforward form of supressing the oscillations.
Figure 3 .
3Cross-correlations of the two peaks of detuned resonance fluorescence in the cases where, (a) the filter width is varied, (b) the detuning is varied, (c) the driving intensity is varied or (d) the dephasing rate is varied, all other parameters being otherwise kept fixed to Γ/γ σ = 10, Ω = 0.1, ∆ σ /γ σ = 20, γ φ = 0 and γ σ setting the unit. The black trace is the same in all panels. This figure is discussed in more details in the text but is particularly important in connection toFig. 4.
Figure 4 .
4Same asFig. 3but without the coherent peak, i.e., with homodyning F = 1 to suppress it. This results in a considerable improvement in the two-photon cascade character of the emission.
AcknowledgmentsEdV acknowledges support from the CAM Pricit Plan (Ayudas de Excelencia del Profesorado Universitario), TUM-IAS Hans Fischer Fellowship and projects AEI/10.13039/501100011033 (2DEnLight) and Sinérgico CAM 2020 Y2020/TCS-6545 (NanoQuCo-CM).
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[
"Quantum simulator for many-body electron-electron Coulomb interaction with ion traps",
"Quantum simulator for many-body electron-electron Coulomb interaction with ion traps"
]
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"Da-Wei Luo \nBeijing Computational Science Research Center\n100084BeijingChina\n\nDepartment of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain\n\nBasque Foundation for Science\n48011Ikerbasque, BilbaoSpain\n",
"P V Pyshkin \nBeijing Computational Science Research Center\n100084BeijingChina\n\nDepartment of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain\n\nBasque Foundation for Science\n48011Ikerbasque, BilbaoSpain\n",
"Michele Modugno \nDepartment of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain\n\nBasque Foundation for Science\n48011Ikerbasque, BilbaoSpain\n",
"Mike Guidry \nDepartment of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA\n",
"J Q You \nBeijing Computational Science Research Center\n100084BeijingChina\n",
"Lian-Ao Wu \nDepartment of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain\n\nBasque Foundation for Science\n48011Ikerbasque, BilbaoSpain\n"
]
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"Beijing Computational Science Research Center\n100084BeijingChina",
"Department of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain",
"Basque Foundation for Science\n48011Ikerbasque, BilbaoSpain",
"Beijing Computational Science Research Center\n100084BeijingChina",
"Department of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain",
"Basque Foundation for Science\n48011Ikerbasque, BilbaoSpain",
"Department of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain",
"Basque Foundation for Science\n48011Ikerbasque, BilbaoSpain",
"Department of Physics and Astronomy\nUniversity of Tennessee\n37996KnoxvilleTennesseeUSA",
"Beijing Computational Science Research Center\n100084BeijingChina",
"Department of Theoretical Physics and History of Science\nThe Basque Country University UPV/EHU\n48080BilbaoSpain",
"Basque Foundation for Science\n48011Ikerbasque, BilbaoSpain"
]
| []
| We propose an analog quantum simulator that uses ion traps to realize the many-body electron-electron Coulomb interaction of an electron gas. This proposal maps a system that is difficult to solve and control to an experimentally-feasible setup that can be realized with current technologies. Using a dilatation transform, we show that ion traps can efficiently simulate electronic Coulomb interactions. No complexity overhead is added if only the energy spectrum is desired, and only a simple unitary transform is needed on the initial state otherwise. The runtime of the simulation is found to be much shorter than the timescale of the corresponding electronic system, minimizing susceptibility of the proposed quantum simulator to external noise and decoherence. This proposal works in any number of dimensions, and could be used to simulate different topological phases of electrons in graphene-like structures, by using ions trapped in honeycomb lattices. | null | [
"https://arxiv.org/pdf/1512.05375v1.pdf"
]
| 118,543,158 | 1512.05375 | 0d264e38d691edec14e67c1204c8b044fd196d4b |
Quantum simulator for many-body electron-electron Coulomb interaction with ion traps
16 Dec 2015
Da-Wei Luo
Beijing Computational Science Research Center
100084BeijingChina
Department of Theoretical Physics and History of Science
The Basque Country University UPV/EHU
48080BilbaoSpain
Basque Foundation for Science
48011Ikerbasque, BilbaoSpain
P V Pyshkin
Beijing Computational Science Research Center
100084BeijingChina
Department of Theoretical Physics and History of Science
The Basque Country University UPV/EHU
48080BilbaoSpain
Basque Foundation for Science
48011Ikerbasque, BilbaoSpain
Michele Modugno
Department of Theoretical Physics and History of Science
The Basque Country University UPV/EHU
48080BilbaoSpain
Basque Foundation for Science
48011Ikerbasque, BilbaoSpain
Mike Guidry
Department of Physics and Astronomy
University of Tennessee
37996KnoxvilleTennesseeUSA
J Q You
Beijing Computational Science Research Center
100084BeijingChina
Lian-Ao Wu
Department of Theoretical Physics and History of Science
The Basque Country University UPV/EHU
48080BilbaoSpain
Basque Foundation for Science
48011Ikerbasque, BilbaoSpain
Quantum simulator for many-body electron-electron Coulomb interaction with ion traps
16 Dec 2015(Dated: December 18, 2015)PACS numbers: 0365Ge, 3280Qk
We propose an analog quantum simulator that uses ion traps to realize the many-body electron-electron Coulomb interaction of an electron gas. This proposal maps a system that is difficult to solve and control to an experimentally-feasible setup that can be realized with current technologies. Using a dilatation transform, we show that ion traps can efficiently simulate electronic Coulomb interactions. No complexity overhead is added if only the energy spectrum is desired, and only a simple unitary transform is needed on the initial state otherwise. The runtime of the simulation is found to be much shorter than the timescale of the corresponding electronic system, minimizing susceptibility of the proposed quantum simulator to external noise and decoherence. This proposal works in any number of dimensions, and could be used to simulate different topological phases of electrons in graphene-like structures, by using ions trapped in honeycomb lattices.
Introduction.-Many-body interactions are fundamental to understanding a variety of interesting phenomena. Quantum many-body problems are notoriously difficult to solve for the full energy spectrum, and are even more problematic for dynamical properties. Few realistic exactly-solvable models exist, necessitating approximations that typically are valid only in some regions of the parameter space. The root of this difficulty is that the Hilbert space grows exponentially as the number of quantum particles increases [1], so on a classical computer the resources and time required to solve a problem exhibit a corresponding exponential growth. Some powerful numerical tools such as quantum Monte Carlo methods and the density matrix renormalization group have had some success for some systems, but the complexity involved prevents an efficient numerical study of many other interesting problems, especially for quantum dynamics and higher-dimension systems.
A powerful alternative to calculation is quantum simulation [1][2][3]. A universal quantum computer has not yet been realized but quantum simulations of specific systems have enjoyed considerable success, aided by experimental advances such as the realization of high-fidelity quantum gates and increasingly precise measurement [4,5]. They have been used to study quantum phase transitions [6], open quantum systems [7], and pairing Hamiltonians [8], to name just a few applications. Quantum simulators follow the laws of quantum mechanics, with an exponentially-growing computing capability [1,2] that can match the exponential growth of problem size with particle number. The idea to simulate one quantum system with another has been proven [9] to be efficient, at least for any many-body system having only few-body particle correlations.
A quantum simulator works by designing a custom Hamiltonian H s so that the evolution operator U = T exp[−i t 0 H s dτ] behaves like the physical system one wishes to simulate. Then a precise measurement at the end of the time evolution gives the physical quantity of interest. A useful quantum simulator requires high-fidelity Hamiltonian engineering and initialization, and precise measurement. A number of systems have been realized experimentally to implement the quantum simulation task, including ion traps, ultracold atoms in optical lattices, NMR nuclear spins, and superconducting qubits.
Many-body electron-electron (e-e) Coulomb interactions play a critical role in many important phenomena such as the fractional quantum Hall effect [10,11] and high T c superconductors [12]. Dealing with the mutual Coulomb interactions between all the electrons is daunting but essential to a deep understanding of such systems. In this Letter, we propose to use quantum simulations to obtain the properties of systems exhibiting many-body e-e Coulomb interactions. The quantum simulator may be realized using an ion trap [2,13,14]. Control methods for ion traps may be implemented with current technologies [4,5] and they have been used widely in quantum simulation and quantum information tasks [15], with applications in areas such as quantum chemistry [16] and mass spectrometry [17].
Electrons in the physical system to be emulated and ions in the trap carry different masses and charges, so a direct simulation is not possible. However, we have employed a dilatation transform to establish an explicit mapping between the simulator on one timescale and the interacting electrons on a different timescale. Therefore, the ion trap at rescaled times can be used to simulate the e-e Coulomb interactions at physical times. To read out the results, we propose imaging of the ions to record their positions, which then have a direct mapping to the positions of the interacting electrons. Remarkably, if only the energy spectrum is desired the unitarity of the dilatation transform implies that it will not alter the spectrum so it is not necessary to generate the transform physically.
This proposal is dimensionality-agnostic. For example, it could be employed for simulating the behavior of 2D electron gases. For charged particles confined in 2D, some interesting phenomenon can arise, such as various quantum Hall effects [10,11]. Moreover, by loading the ions in honeycomb lattices, one could also realize different topological phases characterizing the behavior of electron in graphenelike structures [32]. The ion trap simulator allows for a study of the finite-size boundary effects for the interacting electrons, where bound states are possible. As a concrete example of this proposal, we shall illustrate the experimental setup of the quantum simulation by using calcium ions carrying one positive charge.
The interacting electron gas.-An interacting electron gas can be described by the Hamiltonian (h ≡ 1) [18]
H eg = N ∑ i=1 p p p 2 i 2m e + N ∑ i< j e 2 |q q q i − q q q j | ,(1)
where m e is the electron mass and p p p i and q q q i are the momentum and position operators, respectively, for the i-th electron. The first term represents the kinetic energy and the second term represents the potential energy resulting from mutual Coulomb interactions among the electrons. Because of the many-body nature of the problem, the Hilbert space grows exponentially with electron number, making an exact solution of the Schrödinger equation difficult. Although in some cases it may be possible to consider only the average effect of the electrons (mean field approximations), a complete description requires accounting for the interaction between all electrons. Traditionally, perturbation theory approximations or numerical tools have been used to tackle this problem, with mixed success. In this Letter, we take a fundamentally different approach: we propose an experimentally-feasible quantum simulator that can faithfully represent the Coulomb interaction between the electrons, thus bypassing cumbersome calculations.
Ion-trap simulator.-Typically quantum simulation consists of three stages: (1) preparation of an initial state, (2) time evolution under a specifically-engineered Hamiltonian, and (3) readout of the result, which can be achieved through quantum phase estimation procedures or measurement. Assuming the quantum simulator to be well-controlled experimentally, the most critical task is to design a Hamiltonian of the quantum simulator leading to the required propagator, so that a map exists between the initial and final states of the simulator and those of the system under consideration [2]. Ion traps are widely-used and experimentally well-controlled systems, with the ions either localized or undergoing axial and cyclotron motions as in Penning traps [19]. Since the fermionic ions exhibit mutual Coulomb interaction, an ion trap loaded with identical ions may be a good candidate for the simulation of e-e Coulomb interactions.
To simulate the electron gas, we propose an analogue quantum simulator using an ion trap with N identical ions interacting by Coulomb interactions, as illustrated schematically in
H s = N ∑ i=1 p p p 2 i 2m ion + N ∑ i< j Q 2 e 2 |q q q i − q q q j | ,(2)
where m ion is the mass of the ion and Q is the degree of ionization. This Hamiltonian is formally similar to Eq. (1) but there are two significant differences: (1) The ion to electron mass ratio is of order 10 4 − 10 5 for ions such as 9 Be + or 111 Cd + commonly used in traps, and Q may be greater than 1. Since the kinetic term and the Coulomb interaction terms do not commute, these disparities rule out direct use of the ion trap as an analog simulator for this problem. For the ion trap to faithfully simulate the e-e Coulomb interaction, an explicit mapping between the two systems is required. We shall now show that this can be achieved by the introduction of a scaled evolutionary time for the simulator. Dilatation operator. Let us consider the (unitary) dilatation operator [20] S(r r r) = exp N ∑ j=1 ir j 2 q q q j · p p p j + p p p j ·q q q j ,
where r r r = (r 1 , . . . , r N ), with r k being a real dilatation parameter for the kth particle. Using the Baker-Campbell-Hausdorff formula
exp[αA]B exp[−αA] = B + ∞ ∑ m=1 α m m! [ m A, B](4)
where is the commutator, one finds that for the kth particle the position and momentum operators transform as S † (r r r)q q q k S(r r r) = exp(−r k )q q q k , S † (r r r)p p p k S(r r r) = exp(r k )p p p k .
Thus the dilatation transform scales the momentum and position terms differently, allowing the ratio between the two terms in the Hamiltonian (2) to be tuned.
We take r k ≡ r for k = 1, . . . , N and denote the corresponding dilatation transform as S(r). Because S(r) is unitary, S † (r) f k (p p p k ,q q q k )S(r) = f k (exp(r)p p p k , exp(−r)q q q k )
for any operator functions f k and
H s ≡ S † (r)H s S(r) = exp(r)Q 2 N ∑ i=1 p p p 2 i 2m eff + N ∑ i< j e 2 |q q q i − q q q j | ,(6)
where m eff = exp(−r)Q 2 m ion represents the the effective mass after the dilatation. Then, by requiring m eff = m e , one recovers exactly the e-e Hamiltonian in Eq. (1). This relation fixes the dilatation parameter r and the scaled runtimẽ t = t/(exp(r)Q 2 ), so that the evolution operator is
U(t) = exp(−iH eg t) = S † (r)U (t) S(r), U (t) = exp(−iH st ).(7)
This establishes a one-to-one mapping between the initial state |ψ(0) of the interacting electrons and the initial state |φ (0) of the ion-trap quantum simulator, as well as between the corresponding final states |ψ(t f ) and |φ (t f ) , so that quantum simulation of the e-e Coulomb interactions using ion traps is possible.
Physically the map is a rotation described by the dilatation operator and a corresponding rescaling of time for the propagator. The dilatation parameter and the new time scale are determined solely by the mass ratio between the ion and electron, and the degree of ionization. As a bonus, since the mass of the electron is much less than that of the ions and Q ≥ 1, we have Q 2 exp(r) ≫ 1. Therefore, to simulate the propagator U(t) the simulator runtimet is much less than the physical runtime t. This efficiency is particularly beneficial since shorter runtimes decrease the susceptibility of the quantum simulator to external noise and decoherence. When we are interested in boundary effects or need to take into consideration the finite size nature of the simulator, the dilatation parameter also dictates the mapping between the boundaries of the electron gas and the ion trap size. For example, with a hard wall boundary of width w for the electron gas, the dilation transform will map that to a hard wall boundary of width exp(−r)w for the ion trap. A similar scaling would apply to any external potential (as in the case of an honeycomb lattice), namely V s (q q q) = Q 2 exp(r)V ext (exp(r)q q q), where V s and V ext are the potentials for the ion-trap simulator and electron gas, respectively.
Readout. Results may be read out using an imaging technique to measure the position of the ions [21] in the trap and build up a history of the position over time. Assume that one prepares an arbitrary initial state of the simulator as |ϕ(0) = c v |v , where |v labels the eigenvector with an eigenvalue of E v . Then, a position measurement would return
n(q q q) = dvdv ′ c * v c v ′ exp[i(E v − E v ′ )t] v|n(q q q)|v ′ ≡ dvdv ′ F(v, v ′ ) exp[i(E v − E v ′ )t],
where n is the density of state at position q q q. The dilatation transformation is unitary, so that U(t) and U (t) have the same spectrum, which can be extracted by means of a simple Fourier transformation [8],
S(ω) = dvdv ′ F(v, v ′ )δ [ω − (E v − E v ′ )]
with the sharpness of the delta function being related to the sampling frequency of the measurements.
In principle, if one would like to simulate the dynamical evolution of a precise electronic initial state |ψ(0) , one should prepare the corresponding initial state of the simulator as |ϕ(0) = S(r)|ψ(0) . This could be achieved by propagating the initial state |ψ(0) using the Hamiltonian
H ′ = − 1 2 N ∑ j=1
q q q j · p p p j + p p p j ·q q q j (8) for a time duration of r, according to Eq. (3). We then let the state propagate for a duration of t, followed by an inverse propagation of (8). Then, a measurement of the ions would give expectation values for the electron gas according to
ϕ(0)|U † ( t)OU ( t)|ϕ(0) = ψ(t)|O|ψ(t) ,
where O is the observable under consideration and U ( t) =
S(r)U(t)S † (r).
We also envisage a more general way to obtain the spectrum, through the quantum phase estimation algorithm [22,23]. This would require the coupling of the simulator to a quantum circuit capable of generating a controlled-U operation that takes a qubit state as control and applies the unitary operation on the wave function only if the control qubit is in the |1 state. For an estimated phase of n-bit precision, one needs n Hadamard gates to transform the n ancillary qubits from |0 to |+ = (|0 + |1 )/ √ 2. Denoting the controlled-U operation as
K j = |1 1| j U 2 j−1 (t) + |0 0| j 1,
for the jth qubit |q j = c 0 |0 j + c 1 |1 j , we have
K j |q j |ψ = [c 0 |0 j + c 1 exp(i2πϕ2 j−1 )|1 j ] ⊗ |ψ
where U (t)|ψ = exp(i2πϕ)]|ψ . The phase ϕ is what one should read out for the simulation. Treating the product basis for the qubits |v 1 , v 2 , . . . , v n , where v i = {0, 1} as an n bit binary number b so that the basis can be denoted as |b , b = 0, . . . , 2 n−1 , the inverse quantum Fourier transform (IQFT) performs the mapping
2 n −1 ∑ j=0 exp(i2π jϕ) 2 n/2 | j |ψ → |M |ψ ,
where M denotes an n-bit estimate for the phase ϕ after measurement. Experimental setup.-A realistic experimental setup for simulation of e-e Coulomb interactions is afforded by trapped 40 Ca + ions [13]. The number of 40 Ca + ions that can be loaded with present technology ranges from a few to tens of thousands [24], and individual addressing of the ions has been achieved [13]. From the mass ratio between a calcium ion and an electron, the timescalet for this simulator is related to the physical electronic timescale t byt ≈ 1.37 × 10 −5 t.
To simulate an N-electron system the trap is loaded with N ions. The duration for propagation depends on the precision required in the simulation but is limited by the trap decay time, which is around 1µs for a radio-frequency ion trap that incorporates an optical cavity [24]. Since the physical electronic time in this example is about 10 5 times the trap evolution time, a relatively long electronic time (∼ 0.1 s) can be attained for a trap evolution time that is less than the trap decay time.
To read out the simulation results the ions can be imaged to build a record of position measurements. Then the records can either be mapped to the real electron positions for appropriate initial states, or a Fourier transform can be used to obtain the energy spectrum. Alternatively, a phase-estimation algorithm with a phase precision of n bits can be implemented by sending n qubits initially prepared in the |0 state through Hadamard gates, which is also realizable using ion traps [4,25]. This readout procedure is capable of high precision since the realization IQFT can be scalable in a semiclassical way [26][27][28].
Conclusion.-We have proposed a dimensionality-agnostic quantum simulation of the Coulomb interactions in an electron gas by using ion traps loaded with positive ions. The disparity between the masses of electrons and ions, as well as the different charges that the ions may carry, preclude a direct simulation. However, we have shown through a dilatation transform that the propagator of the electron gas at time t gives a spectrum that is mapped one-to-one to the spectrum of the ion trap at a rescaled timet, with the scaling factor betweent and t specified completely by the mass of the ion and its degree of ionization. An imaging on the ions can be used to build a measurement record of their positions, which is mapped to the measurement record for electron positions, and a Fourier transform yields the energy spectrum. As a concrete example we have illustrated the experimental setup for this approach using 40 Ca + ions, for which we find that the constraint set by trap decay time should permit electron propagation for as long as ∼ 0.1 seconds to be studied.
When only the energy spectrum is required, no additional complexity overhead is added in that we do not need to simulate the dilatation operation on the trapped ions explicitly (it is unitary and does not affect the spectrum). If the wave function is also required, only a simple unitary rotation on the initial state is necessary. Moreover, because of the rescaling of time the runtime of the simulator is much shorter than the timescale for evolution of the electron gas, minimizing the susceptibility of the quantum simulator to external noise and decoherence.
Straightforward extension of this proposal can incorporate different geometries of the trapped ions; for example ion chains [14], ion rings [29], and even periodic lattices [30,31] are feasible. Particularly interesting applications involve trapped ions loaded in two-dimensional honeycomb lattices. This would permit emulation of electrons in graphene-like structures, allowing massless Dirac quasiparticles and associated topological phases to be studied [32].
This technique can be extended to many other interesting systems to solve the problem of scale difference between the kinetic and position-dependent terms or, more generally, the problem of terms having different powers of momentum and position dependence.
FIG. 1 :
1(Color online) Schematic representation of the ion-trap simulator for Coulomb interactions: N identical ions, mutually interacting via the Coulomb force.
Fig. 1 .
1The Hamiltonian is given by
[ m A, B] = [A, [ m−1 A, B]] and [ 1 A, B] = [A, B]
* Electronic address: lianaowu@gmail. * Electronic address: [email protected]
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| []
|
[
"A neural network-based scale-adaptive cloud-fraction scheme for GCMs",
"A neural network-based scale-adaptive cloud-fraction scheme for GCMs"
]
| [
"Guoxing Chen [email protected] \nDepartment of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina\n\nShanghai Qi Zhi Institute\nShanghaiChina\n\nShanghai Frontier Science Center of Atmosphere-Ocean Interaction\nFudan University\nShanghaiChina\n",
"Wei-Chyung Wang \nAtmospheric Sciences Research Center\nUniversity at Albany\nState University of New York\nNew YorkUSA\n",
"Shixi Yang \nDepartment of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina\n",
"Yixin Wang \nDepartment of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina\n",
"Feng Zhang \nDepartment of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina\n\nShanghai Qi Zhi Institute\nShanghaiChina\n\nShanghai Frontier Science Center of Atmosphere-Ocean Interaction\nFudan University\nShanghaiChina\n",
"Kun Wu \nMinistry of Education/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters\nKey Laboratory of Meteorological Disaster\nNanjing University of Information Science and Technology\nNanjingChina\n"
]
| [
"Department of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina",
"Shanghai Qi Zhi Institute\nShanghaiChina",
"Shanghai Frontier Science Center of Atmosphere-Ocean Interaction\nFudan University\nShanghaiChina",
"Atmospheric Sciences Research Center\nUniversity at Albany\nState University of New York\nNew YorkUSA",
"Department of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina",
"Department of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina",
"Department of Atmospheric and Oceanic Sciences\nInstitute of Atmospheric Sciences\nFudan University\nShanghaiChina",
"Shanghai Qi Zhi Institute\nShanghaiChina",
"Shanghai Frontier Science Center of Atmosphere-Ocean Interaction\nFudan University\nShanghaiChina",
"Ministry of Education/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters\nKey Laboratory of Meteorological Disaster\nNanjing University of Information Science and Technology\nNanjingChina"
]
| []
| Key Points (<140 characters):• A neural network-based scheme for parameterizing sub-grid cloud fraction in climate models is developed using the CloudSat data.• The scheme considers the effects of both horizontal and vertical grid sizes on cloudfraction parameterization.• The scheme better predicts total cloud-fraction spatial distribution and cloud vertical structure than the Xu-Randall scheme.AbstractCloud fraction significantly affects the short-and long-wave radiation. Its realistic representation in general circulation models (GCMs) still poses great challenges in modeling the atmosphere.Here, we present a neural network-based diagnostic scheme that uses the grid-mean temperature, pressure, liquid and ice water mixing ratios, and relative humidity to simulate the sub-grid cloud fraction. The scheme, trained using CloudSat data with explicit consideration of grid sizes, realistically simulates the observed cloud fraction with a correlation coefficient (r) > 0.9 for liquid-, mixed-, and ice-phase clouds. The scheme also captures the observed non-monotonic relationship between cloud fraction and relative humidity and is computationally efficient, and robust for GCMs with a variety of horizontal and vertical resolutions.For illustrative purposes, we conducted comparative analyses of the 2006-2019 climatologicalmean cloud fractions among CloudSat, and simulations from the new scheme and the Xu-Randall scheme (optimized the same way as the new scheme). The network-based scheme improves not only the spatial distribution of the total cloud fraction but also the cloud vertical structure (r > 0.99). For example, the biases of too-many high-level clouds over the tropics and too-many low-level clouds over regions around 60°S and 60°N in the Xu-Randall scheme are significantly reduced. These improvements are also found to be insensitive to the spatio-temporal variability of large-scale meteorology conditions, implying that the scheme can be used in different climate regimes.Plain Language SummaryThe clouds reflecting shortwave radiation and absorbing/emitting longwave radiation are all sensitive to the cloud fraction. However, the simulation of cloud fraction in GCMs has been difficult, because most clouds are smaller than the typical scales of GCM grids and cannot be resolved by the grid-scale physics, while the physical understanding of sub-grid processes is still inadequate. Thus, this study uses a data-driven approach, i.e., the neural network, to parameterize the sub-grid cloud fraction in climate models. The database for training and evaluating this new scheme is obtained by upscaling the CloudSat (quasi-) observational data and emulating the GCM 'grid-mean' properties required for cloud-fraction parameterization, minimizing the dataoriented biases. Moreover, the effects of the GCM horizontal and vertical grid sizes are both considered in the network, increasing the scheme adaptivity for use in GCMs with different resolutions. Results show that the new scheme correctly predicts observed features of cloudfraction variation with cloud condensate content and relative humidity for clouds of different phases and better predicts total cloud-fraction spatial distribution and cloud vertical structure than the conventional Xu-Randall scheme. This suggests that the new scheme has great potential to reduce the biases of cloud radiative effects existing in current GCMs. | null | [
"https://export.arxiv.org/pdf/2304.01879v1.pdf"
]
| 257,921,895 | 2304.01879 | cf293a2b8ba226d90d3f2d7129e682499db4f37b |
A neural network-based scale-adaptive cloud-fraction scheme for GCMs
Guoxing Chen [email protected]
Department of Atmospheric and Oceanic Sciences
Institute of Atmospheric Sciences
Fudan University
ShanghaiChina
Shanghai Qi Zhi Institute
ShanghaiChina
Shanghai Frontier Science Center of Atmosphere-Ocean Interaction
Fudan University
ShanghaiChina
Wei-Chyung Wang
Atmospheric Sciences Research Center
University at Albany
State University of New York
New YorkUSA
Shixi Yang
Department of Atmospheric and Oceanic Sciences
Institute of Atmospheric Sciences
Fudan University
ShanghaiChina
Yixin Wang
Department of Atmospheric and Oceanic Sciences
Institute of Atmospheric Sciences
Fudan University
ShanghaiChina
Feng Zhang
Department of Atmospheric and Oceanic Sciences
Institute of Atmospheric Sciences
Fudan University
ShanghaiChina
Shanghai Qi Zhi Institute
ShanghaiChina
Shanghai Frontier Science Center of Atmosphere-Ocean Interaction
Fudan University
ShanghaiChina
Kun Wu
Ministry of Education/Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters
Key Laboratory of Meteorological Disaster
Nanjing University of Information Science and Technology
NanjingChina
A neural network-based scale-adaptive cloud-fraction scheme for GCMs
manuscript submitted to Journal of Advances in Modeling Earth Systems manuscript submitted to Journal of Advances in Modeling Earth Systems 21 * Corresponding author: Guoxing Chen
Key Points (<140 characters):• A neural network-based scheme for parameterizing sub-grid cloud fraction in climate models is developed using the CloudSat data.• The scheme considers the effects of both horizontal and vertical grid sizes on cloudfraction parameterization.• The scheme better predicts total cloud-fraction spatial distribution and cloud vertical structure than the Xu-Randall scheme.AbstractCloud fraction significantly affects the short-and long-wave radiation. Its realistic representation in general circulation models (GCMs) still poses great challenges in modeling the atmosphere.Here, we present a neural network-based diagnostic scheme that uses the grid-mean temperature, pressure, liquid and ice water mixing ratios, and relative humidity to simulate the sub-grid cloud fraction. The scheme, trained using CloudSat data with explicit consideration of grid sizes, realistically simulates the observed cloud fraction with a correlation coefficient (r) > 0.9 for liquid-, mixed-, and ice-phase clouds. The scheme also captures the observed non-monotonic relationship between cloud fraction and relative humidity and is computationally efficient, and robust for GCMs with a variety of horizontal and vertical resolutions.For illustrative purposes, we conducted comparative analyses of the 2006-2019 climatologicalmean cloud fractions among CloudSat, and simulations from the new scheme and the Xu-Randall scheme (optimized the same way as the new scheme). The network-based scheme improves not only the spatial distribution of the total cloud fraction but also the cloud vertical structure (r > 0.99). For example, the biases of too-many high-level clouds over the tropics and too-many low-level clouds over regions around 60°S and 60°N in the Xu-Randall scheme are significantly reduced. These improvements are also found to be insensitive to the spatio-temporal variability of large-scale meteorology conditions, implying that the scheme can be used in different climate regimes.Plain Language SummaryThe clouds reflecting shortwave radiation and absorbing/emitting longwave radiation are all sensitive to the cloud fraction. However, the simulation of cloud fraction in GCMs has been difficult, because most clouds are smaller than the typical scales of GCM grids and cannot be resolved by the grid-scale physics, while the physical understanding of sub-grid processes is still inadequate. Thus, this study uses a data-driven approach, i.e., the neural network, to parameterize the sub-grid cloud fraction in climate models. The database for training and evaluating this new scheme is obtained by upscaling the CloudSat (quasi-) observational data and emulating the GCM 'grid-mean' properties required for cloud-fraction parameterization, minimizing the dataoriented biases. Moreover, the effects of the GCM horizontal and vertical grid sizes are both considered in the network, increasing the scheme adaptivity for use in GCMs with different resolutions. Results show that the new scheme correctly predicts observed features of cloudfraction variation with cloud condensate content and relative humidity for clouds of different phases and better predicts total cloud-fraction spatial distribution and cloud vertical structure than the conventional Xu-Randall scheme. This suggests that the new scheme has great potential to reduce the biases of cloud radiative effects existing in current GCMs.
Introduction
Clouds play important roles in the Earth climate system. They dominate the energy budget by reflecting shortwave radiation and trapping longwave radiation (Wild et al., 2019), participate in the hydrological cycle via precipitation, and alter mass and energy vertical profiles by cloud venting (Yin et al., 2005;Chen et al., 2012a) and latent-heat release. On the other hand, clouds are strongly coupled with aerosols and meteorology (Stevens and Feingold, 2009), involving complex feedback that spans several temporal and spatial scales (e.g., Lau et al., 2006;Xue et al., 2008;Chen et al., 2012b;Chen et al., 2018aChen et al., , 2018bSong et al., 2019). However, clouds are sub-grid scale in nature for current climate models, and cloud macro-and microphysical properties in models have to be parameterized using the grid-mean atmospheric properties, making clouds the main source of most uncertainties in studies on climate change and climate modeling (e.g., Caldwell et al., 2016;Stevens et al., 2016).
Specifically, cloud fraction parameterization poses a major challenge in climate modeling. Findings in the recent Coupled Model Intercomparison Project phase 6 (CMIP6) revealed that current general circulation models (GCMs) have biases in both the daily-mean cloud fraction and the cloud diurnal variation. For example, for the former, the simulated cloud fraction is too small over the tropics, extratropics, and midlatitude regions (Vignesh et al., 2020;Li et al., 2021); and for the latter, the cloud fraction over land is too small during the daytime and too large during the nighttime (Chen & Wang, 2016b;. As the shortwave cloud radiative effect (SWCRE) occurs only during the daytime while the longwave cloud radiative effect (LWCRE) persists throughout the daytime and nighttime, the biases in cloud fraction can affect not only the energy balance but also the energy diurnal variation, which may have implications on atmospheric variations of longer time scales (e.g., Slingo et al., 2003;Ruppert, 2016).
Current parameterization schemes of cloud fraction can be divided into diagnostic and prognostic approaches. In the diagnostic approach, the sub-grid cloud fraction is a function of the instantaneous grid-mean relative humidity and/or cloud water content (e.g., Sundqvist et al., 1989;Xu & Randall, 1996;Shiu et al., 2021). These schemes are easy to implement and computationally efficient, but too empirical. On the other hand, the prognostic approach, which explicitly simulates the temporal variate of cloud fraction using source and sink terms associated with advection, cumulus convection, stratiform condensation, evaporation, and precipitation (Tiedtke, 1993;Wilson et al., 2008;Park et al., 2016), seems more physical. However, all source/sink terms are empirically related to the grid-mean properties, which are difficult to verify with observations. In addition, climate models usually employ different horizontal and vertical resolutions, which may affect the sub-grid statistical characteristics and thus the cloud-fraction parameterization.
In recent years, deep-learning methods have been demonstrated to be an effective alternative approach in atmospheric modeling (Chantry et al., 2021;Schultz et al., 2021). They can help reveal associations among atmospheric parameters by learning directly from data, regardless of inadequate or even none prior domain knowledge. For example, the feedforward neural network (also called multiple-layer perceptron, hereafter denoted as neural network for short) is good at emulating complex functions and can be used to replace certain modules for either accounting for poorly-understood processes or saving computational cost. It has been used to parameterize processes such as cloud cover (Grundner et al., 2022), radiation (Krasnopolsky & Fox-Rabinovitz, 2006), convection Han et al., 2020;T. Zhang et al., 2021), and boundary-layer turbulence . All yield promising outcomes. For the same reasons, the networks with complex architectures exhibit superiorities in building tools for specific predictions such as El Niño (Nooteboom et al., 2018;Ham et al., 2019), precipitation (Shi et al., 2015;Ravuri et al., 2021; and clouds , and for data processing (Rasp & Lerch, 2018;Leinonen et al., 2021;Kim et al., 2021;Pan et al., 2019Pan et al., , 2021Pan et al., , 2022.
In this study, we introduce a neural network-based diagnostic scheme for simulating cloud fraction in climate models. This scheme has advantages in at least three aspects. First, using the neural network avoids the non-physical assumption of formula forms in the parameterization; second, the data for developing the scheme were obtained from CloudSat observations of cloud profiles and the associated large-scale meteorology conditions, minimizing the data-oriented scheme biases; and third, the scale adaptivity (both horizontally and vertically) is considered while developing the scheme so that the scheme has higher robustness across models with different resolutions and in models with varying resolutions (e.g., the Model for Prediction Across Scales-Atmosphere, Skamarock et al., 2012; the Global-to-Regional Integrated forecast SysTem atmosphere model, Zhou et al., 2020).
The rest of this manuscript is arranged as follows: Section 2 describes the data preparation; Sections 3 gives the description of the new scheme and the conventional Xu-Randall scheme (Xu & Randall, 1996), the latter of which is used as a baseline for evaluating the new scheme; Section 4 tests the new scheme in aspects of accuracy and scale adaptivity within the context of a machine-learning study; Section 5 evaluates the new scheme with multiple-year CloudSat data using an offline application; and last the summary and discussion are given in Section 6.
Data preparation
CloudSat
We use the CloudSat data for training, validation, and testing of the network-based cloudfraction scheme. CloudSat is a polar-orbiting satellite launched by NASA in April 2006. It observes cloud vertical structure along its track using a 95 GHz radar onboard (Stephens et al., 2002). Its data have been widely used previously in studies on cloud climatology (e.g., Weisz et al., 2007;Kato et al., 2010;Chen et al., 2016;Tang et al., 2020), model development (e.g., Di Giuseppe & Tompkins, 2015Li et al., 2018), and model evaluations (e.g., Bodas-Salcedo et al., 2008;Greenwald et al., 2012;Kodama et al., 2012;Wang & Zhang, 2018;Vignesh et al., 2020).
This study uses 6 variables from 3 CloudSat products: cloud liquid and ice water contents from 2B-CWC-RO (Austin & Wood, 2018); CPR (cloud profiling radar) cloud mask from 2B-GEOPROF (Marchand et al., 2008); atmospheric pressure, temperature, and specific humidity from ECMWF-AUX (Partain, 2022). The data all have a horizontal resolution of 1.1 km and a vertical resolution of 240 m. Figure 1 presents the availability of CloudSat data at the time of our analysis. As CloudSat completes around 14.6 revolutions around Earth per day, there are around 5000 granules of observations annually in normal years like 2007-2016. In 2011and 2017, the CloudSat observation was interrupted occasionally due to reasons such as battery anomalies or orbit maneuvers (https://cloudsat.atmos.colostate.edu/news/CloudSat_status), yielding fewer data in these years. Meanwhile, the granule amount is relatively larger in the boreal summer and autumn than in the winter and spring. This may imply that the data is less representative of cloud climatology in the winter and spring seasons, but it does not affect the robustness of our scheme. It is shown below that the available data are more than enough for the scheme development and evaluation.
Figure 1.
Year-by-year variation of CloudSat data amount used in this study. The colored segments indicate the portions of winter (December-January-February), spring (March-April-May), summer (June-July-August), and autumn (September-October-November), respectively.
Data preprocessing
Typical GCM grids have horizontal sizes (∆x) of 10-100s km (much larger than the CloudSat horizontal resolution) and vertical sizes (∆z) of 0.1-1s km (mostly larger than the CloudSat vertical resolution). Thus, the CloudSat clouds can be considered as sub-grid clouds at GCM grids, and the CloudSat data can be upscaled/aggregated to emulate atmospheric properties and cloud fraction simulated at GCM grids for the input and output of a cloud-fraction parameterization scheme.
In upscaling, the 'grid-mean' temperature, specific humidity, atmospheric pressure, and liquid/ice water contents are calculated by averaging the raw CloudSat data adjacent within the range of Δx × Δz, while the sub-grid cloud fraction is calculated as the horizontal area cloud fraction (i.e., each profile in the cloudy subarea must have one or more of the 240-m layers with CPR cloud mask ≥ 30 within Δx × Δz). The definition of sub-grid cloud fraction is consistent with that in most radiation parameterizations, and the resulting values are usually larger than the volume cloud fraction required by certain microphysical parameterizations (Grundner et al., 2022). More details of the upscaling process can be found in Wang et al. (2023). In addition, to account for the different horizontal and vertical resolutions of GCMs, the upscaling is carried out for 42 different ∆x-∆z combinations. Therein, ∆x spans 40-100 km with an increment of 10 km while ∆z spans 10-60 hPa with an increment of 10 hPa, covering most grid sizes that may be seen in current GCMs.
It is noticed that the amount of available CloudSat data is far more than enough for the scheme development. For example, upscaling the data in 2015 to the resolution of 100 km × 20 hPa (hereafter x100z20 for short) can yield more than 10 million cloudy samples, which is more than sufficient for training a neural network with only 100s trainable parameters (see Section 3.1). Therefore, to save training time, we randomly draw 100 granules from the data in 2015 when upscaling to each set of resolutions. The choice of 2015 is arbitrary and using the data in any other year or multiple years does not affect the training results. Upscaling to 42 sets of resolutions yields more than 10 million cloudy samples in total. These samples are randomly split into three parts: 60% for scheme training, 20% for scheme validation, and 20% for scheme testing.
Data uncertainty
The CloudSat data uncertainty exists in two aspects. First, the uncertainty of satellite data retrieval has been discussed in previous studies (e.g., Weisz et al., 2007;Kotarba & Solecki, 2021). Notably, the radar-only product of 2B-GEOPROF (2B-CWC-RO) is less reliable for cloud mask (cloud water contents) than the radar-lidar (radar-visible) combined product of 2B-GEOPROF-Lidar (2B-CWC-RO), respectively. Here, we intentionally avoid using the latter two products to ensure the data inherent consistency to the most degree. The radar-lidar combined cloud mask is not consistent with cloud water contents in 2B-CWC-RO while the radar-visible combined water contents are only available for the daytime and thus less representative. An isolation-forest model is trained for clouds of each phase (i.e., ice, mixed, and liquid).
The inputs of the models are the relative humidity of the respective phases and the sub-grid cloud fraction, and the contamination ratio is set to 0.01.
Second, the uncertainty of atmospheric states from the ECMWF-AUX is remarkable but seldom discussed. Particularly, it was revealed that the upscaled CloudSat data have some anomalous samples with large cloud cover at very-low relative humidity (Wang et al., 2023), which is counterintuitive and unrealistic. These are attributed to biases in the ECMWF-AUX data. For example, the ECMWF model tends to underestimate the inversion layer height in the eastern subtropical ocean (e.g., the southeast Pacific, Andrejczuk et al., 2012;Chen et al., 2015;Chen and Wang, 2016a), causing underestimated relative humidity in the upper cloud layer. As there is not a well-defined quantitative relationship between the sub-grid cloud fraction and the gridmean relative humidity, it is difficult to find a physical-based method to detect these anomalous data.
Therefore, to facilitate the process of detecting anomalous samples in the upscaled data, we used an off-the-shelf machine-learning method-isolation forest (Liu et al., 2008(Liu et al., , 2012. The method assumes that anomalous samples are isolatedly distributed from normal samples in the sample space and thus easier to be isolated than normal samples through random partitioning of the data. The algorithm builds a collection of binary trees from random subsets of data, where the anomalous samples should lead to shorter paths to leaves than normal samples, and aggregates the anomaly score from each tree to come up with a final anomaly score for determining the anomalous samples.
In this study, we train an isolation-forest model for clouds of each phase (i.e., ice, mixedphase, and liquid) using 100,000 samples randomly drawn from the CloudSat data upscaled at the resolution of x100z20 for the whole year of 2015. The inputs for the models are sub-grid cloud fraction and relative humidity for the respective phases, where the saturated water vapor mixing ratio for the mixed phase is set to the weighted mean of those over ice and liquid surfaces. The contamination ratio, a parameter setting the proportion of anomalous samples in the dataset, is set to 0.01 for clouds of all three phases. To further demonstrate the effect of screening, we examine an example of the upscaled CloudSat observation by the granule No. 03307. Shown in Fig. 3, many samples in the upper portion of the cloud layer have the sub-grid cloud fraction (numbers in the figure) larger than 0.5 and the relative humidity (colored shadings in the figure) lower than 0.2, and the screening successfully distinguishes these anomalous samples (marked with red strikethroughs).
Note that the data screening is only to remove samples that we consider to be unrealistic. The data screening certainly does not affect the resulting scheme simply because the anomalous samples take a very small fraction of the CloudSat data (Fig. 2). Likewise, increasing the contamination ratio also has little effect on the scheme results over the normal samples that dominate the data population. For example, setting the contamination ratio to 0.05 further reduces the number of samples having large cloud fraction at small relative humidity but does not cause any evident changes to the variation of cloud fraction with either relative humidity or cloud condensate mixing ratio (shown in Figs. S1-S2 in the Supplementary Materials).
Scheme description
This section describes the neural network architecture and the algorithm of the Xu-Randall scheme. The database is the upscaled CloudSat data for 42 sets of resolutions (Section 2.2). The training, validation, and testing datasets include around 5.4, 1.8, and 1.8 million samples, respectively. The former two datasets are used for the NSA scheme development and the Xu-Randall scheme tuning, while the testing dataset is used for the scheme testing in Section 4.
3.1 Architecture of the network-based scale-adaptive cloud-fraction scheme Figure 4 presents the architecture of the network-based scale-adaptive (NSA for short) cloud-fraction scheme (Fig. 4a). The network is composed of one input layer, a tunable number n of hidden layers, one output layer, and the ReLU activation layers (not shown in the figure) following the input layer and hidden layers. The loss function is the mean square error.
The input layer consists of 8 variables: air pressure (P), air temperature (T), liquid/ice water mixing ratios (QL/QI), relative humidity over liquid/ice surfaces (RL/RI, calculated with the upscaled atmospheric pressure, temperature, and specific humidity), and the horizontal/vertical grid sizes (∆x/∆z) of the host GCM. Therein, the first 6 variables are chosen because of their close association with cloud formation based on domain knowledge. They are also used in the Xu-Randall scheme, making it fair to compare results from the NSA and the Xu-Randall schemes in the following sections. ∆x and ∆z are to implicitly include the effects of grid sizes on the sub-grid cloud 3D structure (e.g., cloud sizes and vertical overlap) and thus the sub-grid cloud cover. As shown below, it provides the scheme with the flexibility of use at a variety of GCM resolutions.
We are aware that the input size could be reduced by replacing RL and RI using the specific humidity (QV). However, using QV causes a larger burden for the neural network, and the resulting network should have more hidden layers or more neurons to get training accuracy close to that of the network using RL and RI. Thus, using RL and RI is a choice out of computation efficiency. The output layer has only one neuron, i.e., the sub-grid cloud fraction (CF).
Figure 4.
Architecture of the network-based scale-adaptive (NSA) cloud-fraction parameterization (a) and the variation of scheme performance with the neuron amounts in hidden layers for the four-layer version of the scheme (b). In (b), the black symbols and curve (with respect to the left y axis) give the root mean square error (RMSE) estimated on the validation dataset, while the red symbols and curve (with respect to the right y axis) give the time consumption of the network-based scheme, where the scheme is implemented in Fortran, compiled with the GNU Fortran compiler (i.e., gfortran), and executed for 3,000,000 times; and the cross symbols indicate results of the NSA scheme with 10 neurons in both hidden layers. There are 211 trainable parameters in the cross-signed scheme. The output of the network is bounded to 0-1 before the analysis.
The number of hidden layers (n) and the neuron amounts in each hidden layer (Nhi, i = 1, 2, …, n) are tunable hyperparameters in the network. Unlike most deep-learning studies that determine the hyperparameters based on only the validation accuracy, this study takes a compromise between the validation accuracy and the computational efficiency, as the scheme is planned to be executed every time step at each grid of the host GCM. As shown in Fig 4b, when n equals 2, the validation root-mean-square error (RMSE) decreases by about 6% as Nh1,h2 increases from 8 to 64, while the time consumption increases by more than 20 times. Clearly the computational efficiency is not an ignorable factor. In addition, using one hidden layer increases the RMSE too much, while using three hidden layers leads to lower computation efficiency. Therefore in this study, we set n to 2, Nh1 and Nh2 both to 10. This setting yields a validation RMSE of 0.135 (black cross) and a time consumption of 1.48 s for 3,000,000 executions (red cross), which is about 5 times that of the Xu-Randall scheme and still acceptable. Below, the scheme testing and the offline application of the NSA scheme are both based on this setting.
We utilize one single network to parameterize the CF of different phases for simplicity. As the CF variation with QC and relative humidity is sensitive to the cloud phase (Wang et al., 2023; also shown below in Fig. 6), using separate networks for clouds of different phases can yield better results. However, using multiple networks increases the memory cost for storing the network parameters and reduces the computational efficiency. Thus, the multiple-network approach is not taken in this study.
Tuned Xu-Randall scheme
The Xu-Randall scheme is a conventional diagnostic cloud-fraction parameterization scheme for use in climate models. It is the default cloud-fraction scheme in the Weather Research and Forecast (WRF) Model and has been used in many weather and climate studies (e.g., Chen et al., 2018bChen et al., , 2021Yang et al., 2018;Song et al., 2019). The scheme is also employed in the FGOALS-f3 GCM developed by the Institute of Atmospheric Physics, Chinese Academy of Sciences (Zhou et al., 2015). Therefore, this scheme is used as a baseline for evaluating the NSA scheme. Below we briefly introduce the Xu-Randall scheme, and more details can be found in the paper by Xu & Randall (1996).
The Xu-Randall scheme was built based on the simulations of a cloud ensemble model using a curve-fitting approach. It assumes the sub-grid cloud fraction is a function of the gridmean relative humidity (R) and cloud condensate mixing ratio (QC, i.e., QI for ice clouds, QL for liquid clouds, and QI + QL for mixed-phase clouds). The function formula is
( ) 1 exp , if < 1 1* C CF R R RQ Q = − − − (1), 1, if 1 CF R = (2),
where Q* is the grid-mean water vapor saturation mixing ratio, α = 100, β = 0.25, and γ = 0.49 are empirical parameters determined through curve-fitting based on the simulation dataset.
As the Xu-Randall scheme was built on a completely different dataset than the NSA scheme, it would be unfair to expect the scheme produces results close to the CloudSat observations. Therefore, we have tuned the scheme using the same procedure and the same datasets (i.e., the aforementioned training and validation datasets) that are used for training the NSA scheme, yielding α = 70.3378, β = 0.0507, and γ = 0.6315. Below we mainly present results from the tuned Xu-Randall scheme, and results of the original Xu-Randall scheme are also included in the Supplementary Materials for interested readers.
Scheme testing
This section evaluates the NSA scheme using the testing dataset within the context of a machine-learning study. Results of the cloud spatial and temporal characteristics predicted by the scheme are given in Section 5. Figure 5 presents the joint probability density distribution of CFs from the scheme predictions and the CloudSat observations for clouds of different phases, estimated based on the testing dataset. The NSA scheme is shown to well predict CFs for clouds of all three phases and has relatively larger biases (underestimation) when the observed CF is close to 1 (Figs. 5a, 5c, 5e). This could suggest the effect of the data imbalance. However, we found that balancing the CF frequency distribution by undersampling the training data could not much improve the results. This implies that the large-CF samples may have larger uncertainties or larger inherent inconsistencies, making the scheme difficult to learn. It is noticed that the correlation coefficient (r, larger is better) is the largest for the mixedphase clouds, while the RMSE (smaller is better) is the smallest for the liquid clouds. The two metrics are not consistent for two reasons. First, the fraction of large-CF samples is much larger in ice and mixed-phase clouds than in liquid clouds. Large-CF samples contribute more to the RMSE than small-CF samples, causing the relatively larger RMSE in ice and mixed-phase clouds. Second, the CloudSat data for liquid clouds might have large uncertainties. Liquid clouds are mostly sited in the lower troposphere, where the atmospheric properties are dominated by turbulence and difficult to simulate by GCMs. Therefore, air temperature and humidity from the ECMWF-AUX may have large uncertainties in the lower troposphere, causing the smaller r for the CF prediction of liquid clouds.
Accuracy
In contrast, the tuned Xu-Randall scheme yields larger RMSE and smaller correlation coefficient for clouds of all three phases. The biases over ice clouds are the largest, where the RMSE is 0.366, more than twice that of the NSA scheme, and the correlation coefficient is less than 0.6. It is noticed that the original Xu-Randall scheme reaches better results over the liquid clouds but worse results over the mixed and ice clouds (shown in Fig. S3) than the tuned one. This indicates the tuning process has opposite effects on the scheme performance over liquid vs. mixed and ice clouds and suggests that the sub-grid statistics could be sensitive to the cloud phases. Moreover, even when we tune the scheme individually for each upscaled resolution, the scheme still exhibits obvious inferiorities to the NSA scheme at all resolutions (shown in Fig. S4). Figure 6 shows the variation of cloud fraction with relative humidity and cloud condensate mixing ratio from the CloudSat observations and the scheme predictions. In the observations, the ice CF is dominated by R (Fig. 6a), the liquid CF is dominated by QC (Fig. 6i), and the mixed CF is sensitive to both factors (Fig. 6e). It is worth highlighting that holding QC unchanged, the CF variation with R is non-monotonic for clouds of all three phases. The liquid CF decreases with R until R equals around 0.8, and increases afterward. In contrast, the ice CF increases with R until R equals around 1, and decreases afterward. The mixed CF exhibits a complex pattern perhaps due to the phase partition of QC. It is also noticed that the supersaturation (i.e., R >1), which seldom occurs in liquid clouds, is not infrequent in ice clouds.
The NSA scheme prediction presents patterns quite close to the observed ones. Particularly, the scheme well captures the non-monotonic feature of the CF variation with R. For samples that dominate the data population (i.e., samples below the black contours), the scheme results are almost identical to the observations. The scheme may underestimate CF when QC is large (especially for ice and mixed-phase clouds, Fig. 6a vs. Fig. 6b, Fig. 6e vs. Fig. 6f). Nevertheless, the occurrence frequency of the associated samples (i.e., samples above the black contours in Figs. 6d, 6h, and 6l) is very low.
The tuned Xu-Randall scheme prediction presents similar patterns for clouds of three phases, in line with the same formula (i.e., Eqs. 1-2) across different cloud phases. The patterns for liquid and mixed clouds (Figs. 6g and 6k) are close to the CloudSat observations, consistent with the relatively larger correlation coefficients and smaller RMSEs in Figs. 5d and 5f. However, the pattern for ice clouds (Fig. 6c) is quite different from the observed one, showing the scheme underestimates CF for QC < 50 mg kg -1 and overestimates CF for R > 1. Results for the original Xu-Randall scheme are similar to the tuned one (figure not shown). Neither the original nor the tuned Xu-Randall scheme can capture the non-monotonic variation of CF with R, which is not included in the parameterization formula. Figure 6. Comparisons of cloud fraction as a function of cloud condensate mixing ratio and relative humidity between the CloudSat observations (a, e, i), the NSA scheme (b, f, j), and the tuned Xu-Randall scheme (c, g, k) for clouds of different phases (ice-only, mixed, and liquid only), estimated based on the testing dataset. The right column (d, h, l) indicates the joint probability density distribution of respective cloud types in the testing dataset. Black lines indicate contours of the probability density of 0.001 and 0.0001, below which lie more than 88% and 97% of the corresponding sample populations, respectively.
Scale adaptivity
To examine whether considering ∆x and ∆z improves the NSA scheme performance, we take two additional network-based schemes as baselines: N and Nx100z20. Both schemes have the same network architecture as the NSA scheme except for excluding ∆x and ∆z from the input layer. The N scheme is trained based on the same dataset as the NSA scheme, while the Nx100z20 is trained based on the upscaled CloudSat at the resolution of x100z20 for the whole year of 2015 (no random drawing), where the sample amount is similar to that of the database for training the NSA scheme. Below we compare the three network-based schemes based on the same testing dataset that is used above in Figs. 5-6.
Figure 7.
Comparisons of RMSE at different horizontal and vertical resolutions between the predictions of three neural network-based schemes estimated based on the testing dataset. Both N and Nx100z20 schemes have identical architecture to the NSA scheme except for excluding ∆x and ∆z from the input layer. The N scheme is trained with the same database as the NSA scheme, while the Nx100z20 scheme is trained based on the upscaled CloudSat data at the resolution of x100z20 for the whole year of 2015. Figure 7 compares the RMSE of the three network-based schemes for samples with different horizontal and vertical resolutions. In the NSA scheme, the RMSE is not sensitive to ∆x or ∆z for ice clouds (Fig. 7a), slightly sensitive to ∆z for mixed clouds (Fig. 7d), and sensitive to both resolutions for liquid clouds (Fig. 7g), where the RMSE tends to increase with ∆z and decreases with ∆x. When ∆x and ∆z are excluded from the input layer, the N scheme has larger RMSEs for clouds of all three phases, especially the ice and mixed clouds (Figs. 7b, 7e). As the deficiency of the N scheme may be attributed to the inherent inconsistency of the training database associated with different resolutions, we further examine the results of the Nx100z20, where the database inherent consistency is warranted. However, it shows that the RMSEs of the Nx100z20 scheme are even larger (Figs. 7c, 7f, 7i), and close to those of the NSA scheme only when ∆x is the largest and ∆z is the smallest. Therefore, we can conclude that both horizontal and vertical resolutions affect the sub-grid statistics of cloud fraction, and that including ∆x and ∆z in the scheme greatly increases the scale adaptivity of the NSA scheme, leading to higher robustness for use in GCMs with different horizontal and vertical resolutions. Figure 8 presents the mean cloud fraction at different horizontal and vertical resolutions from the CloudSat observations and the scheme predictions to show more details of how ∆x and ∆z may affect the cloud fraction parameterization. In the CloudSat observations (Figs. 8a, 8e, and 8i), the cloud fraction tends to increase with ∆z and decrease with ∆x. The NSA scheme well captures this feature, and the biases are less than 0.01 at most resolutions (Figs. 8b, 8f, and 8j). In contrast, the biases of the N scheme are larger than 0.01 for most resolutions and can have values up to 0.09). Meanwhile, the biases are generally symmetric about the diagonal, i.e., negative in the upper left and positive in the lower right. This is not a surprise. The scheme does not consider the resolution variability in the training database, and consequently the scheme biases tend to be the smallest in the middle of the resolution ranges and increase when the resolutions go toward either end. When we remove the inherent inconsistency associated with resolutions from the training database, the Nx100z20 scheme only shows better performance for samples with resolutions close to x100z20 (i.e., ∆x in 80-100 km and ∆z in 10-20 hPa). The bias increases rapidly with the decrease of ∆x and the increase of ∆z, with the largest value of 0.14. Hence, it is further confirmed that the NSA scheme has good scale adaptivity.
Offline application
This section compares the climatology of cloud spatial and temporal variability simulated by the NSA scheme with those from the CloudSat observations and the tuned Xu-Randall scheme using an offline method. The NSA and the tuned Xu-Randall schemes both take the upscaled CloudSat data at the resolution of x100z20 for the period of 2006-2019 as inputs to make the respective predictions. We first get an intuitive understanding of the scheme biases ( Fig. 9), then examine the generalizability of the NSA scheme (Fig. 10), and last assess the scheme performance based on 2006-2019 multiple-year mean results (Figs. 11-13).
We take the offline mode rather than incorporating the two schemes into a host model to exclude the distractions that could be caused by other model components (e.g., the simulated cloud condensate mixing ratios in the host model may deviate too much from the observations, Jiang et al., 2012) and possible feedbacks. Hence, the scheme-observation discrepancies can be mostly attributed to the scheme deficiencies, and the inter-scheme differences are not blurred by the possible compensating biases within the host model. October 2018 (a-b) compared with those from the NSA (c) and the tuned Xu-Randall (d) schemes. The dark gray regions at lower levels indicate the surface terrain. In (a), the light blue indicates overcast cloud coverage (CPR cloud mask equals 30 or 40), in which the light-gray regions are grids where the CloudSat does not have a valid observation for one or more of gridmean properties that are required for parameterizing the sub-grid cloud fraction. Figure 9 presents the observed and modeled cloud fractions in a CloudSat granule to get a better understanding of the data upscaling and an intuitive view of the scheme biases. In the raw CloudSat data (Fig. 9a), a binary CF is determined by the CPR cloud mask: 1 for CPR cloud mask ≥ 30 and 0 otherwise. When upscaled (Fig. 9b), the sub-grid CF has decimal values within 0-1. The upscaled CF may not have valid values at certain grids (indicated by light grey in Fig. 8a) due to two reasons. First, the raw CloudSat does not have valid estimates for liquid or ice cloud water content; and second, none of the raw CloudSat data falls in the range of ∆z because the pressure change in 240 m exceeds ∆z (this situation occurs mainly in the lower atmosphere where the vertical gradient of atmospheric pressure is large).
The NSA and the tuned Xu-Randall schemes both well predict the general distribution of the sub-grid CF (Figs. 9c-d) but exhibit different characteristics in the scheme-observation discrepancies. The NSA scheme tends to underestimate CF when CF is close to 1 (consistent with results in Figs. 5-6), while the Xu-Randall scheme overestimates CF at many grids (particularly the high-level CF that are mostly ice clouds) because of its treatment of CF at supersaturation. The Xu-Randall scheme assumes that CF equals 1 when the grid-mean RH equals or exceeds 1 (Eq. 2). This assumption works well for liquid clouds where the supersaturation is usually very low (Fig. 6l), but not for ice clouds where the occurrence of large supersaturation is not scarce (Fig. 6d). (2006, 2012, and 2018), where the CloudSat data are dominated by granules in winter and spring (summer and autumn), present no unique features distinguished from results in other years. Therefore, it is inferred that the generalizability of the NSA scheme is very good and at least similar to that of the Xu-Randall scheme, warranting its robustness in different climates. Figure 11. 2006-2019 mean global distribution of total cloud fraction (assuming maximumrandom overlap) from the CloudSat observation, the NSA scheme, and the tuned Xu-Randall scheme. Numbers at the upper right corners of (b) and (c) indicate the correlation coefficient (r; before the slash) and RMSE (after the slash) of the respective scheme predictions with respect to the CloudSat observations. The global-mean total cloud fraction is 0.46, 0.42, and 0.52 for the CloudSat observations, the NSA, and the tuned Xu-Randall scheme predictions, respectively. Figure 11 compares the 2006-2019 averaged global distributions of total CF from the observation and the scheme predictions, assuming the maximum-random overlap. The observed CF is larger in the tropics and the regions around 60° of both hemispheres and smaller in the subtropical regions. The two schemes both capture the general patterns. The tuned Xu-Randall prediction has a global mean of 0.52, larger than that from the original Xu-Randall scheme (0.46; Fig. S5a), which is closer to the observation (0.46) than the NSA prediction (0.42). However, the original/tuned Xu-Randall prediction has too large CF over the tropics (especially the India-Pacific warm pool) and too small/large CF over the Southern Ocean, where the NSA prediction is closer to the observation. Consequently, the NSA prediction has a larger spatial correlation coefficient (0.995) and a smaller RMSE (0.037) than the Xu-Randall predictions. Using the maximum overlap assumption reaches similar results while using the random overlap suggests that the tuned Xu-Randall scheme prediction is closer to the observation (shown in Table S1 in the Supplementary Materials). These conflicting results imply that the cloud vertical structure could be quite different between the scheme predictions, as examined below. Figure 12 presents the zonal-mean vertical structure of CF from the observations and the scheme predictions. In the tuned Xu-Randall scheme, the CF overestimation in the tropics is associated with the too-large high-level CF (clouds above 440 hPa), while the CF overestimation in regions around 60°S and 60°N is due to the too-large low-level CF (clouds below 680 hPa). The original Xu-Randall scheme overestimates the high-level CF in the tropics as well but underestimates the low-level CF in regions around 60°S and 60°N (Fig. S5b). The tuning causes larger CF at all levels and yields better mid-level CF but worse low-level CF, which is consistent with the results on mixed and liquid clouds shown in Fig. 5 and Fig. S3. The NSA scheme predicts smaller high-level CF in the tropics and smaller mid-and low-level CFs in regions around 60°S and 60°N and reaches a larger spatial correlation (0.998) and a smaller RMSE (0.007) than both the original and tuned Xu-Randall schemes. Figure 13. Annual variation of cloud vertical distribution averaged at different latitudes from the CloudSat observations (left), the NSA (middle), and the tuned Xu-Randall (right) scheme predictions. Numbers in middle and right columns indicate the correlation coefficient (r) and RMSE of the respective scheme predictions with respect to the CloudSat observations. Dashed lines indicate the heights of 440 hPa and 680 hPa, respectively. Figure 13 further examines the seasonal variations of the cloud vertical structure at different latitudes. In the observations, the cloud vertical structure varies with the seasonal migration at all latitudes. While both schemes can predict the general variation, the NSA scheme shows better results (i.e., larger correlation coefficient and smaller RMSE) than the original (Fig. S6) and tuned Xu-Randall schemes in all seasons and all latitudes. This indicates that the NSA scheme generally exhibits superiorities in the spatial distribution of total CF and the cloud vertical structure, and further confirms that the superiorities are not sensitive to the spatiotemporal variability of the large-scale meteorology conditions, suggesting the scheme has enough robustness to simulate CF over different climate regimes.
Summary and discussion
This study presents a neural network-based cloud-fraction parameterization scheme for use in climate models. The scheme is developed using the CloudSat (quasi-)observational data, and for the first time considers explicitly the effects of both horizontal and vertical grid sizes on the cloud-fraction parameterization. It not only simulates realistically the cloud characteristics but also captures the observed non-monotonic functional relationship of cloud fraction with relative humidity. In addition, our preliminary study shows that using the network-based scheme in the WRF Model hardly increases the computation time as compared with using the Xu-Randall scheme. The utility of the scheme in GCMs shows promising features in accuracy, scale adaptivity, and computational efficiency.
In the case of comparing with the Xu-Randall parameterization, the network-based scheme simulates more-realistic total cloud fraction spatial distribution and cloud vertical structure. Particularly, the biases of too-large high-level cloud fraction over the tropics and toosmall low-level cloud fraction over regions around 60°S and 60°N in the original Xu-Randall scheme, which agrees with the too-strong LWCRE and too-weak SWCRE over the respective regions as shown in current GCMs (Flato et al., 2013;Schuddeboom and McDonald, 2021;, are much eased.
We are aware that the grid-mean atmospheric properties in GCMs could differ markedly from the CloudSat data (e.g., Jiang et al., 2012), hindering the above findings from being fully justified in GCMs. Hence, one of our undergoing studies is to further evaluate the NSA scheme using the ERA-Interim reanalysis (Dee et al., 2011), where the grid-mean relative humidity is quasi observed due to the assimilation while the cloud condensates are fully model simulated. The preliminary results show that although cloud condensate mixing ratios in the reanalysis are much smaller than the CloudSat observations, using the NSA scheme still yields better mid-and high-level cloud fractions than the original Xu-Randall scheme (figure not shown). Therefore, it is believed that incorporating the network-based scheme has great potential to improve the cloud radiative effects and the energy budget in GCMs.
One important aspect of cloud fraction is the vertical overlap, which has significant implications to cloud radiative effects (e.g., Liang & Wang, 1997;Zhang & Jing, 2016;. As shown by Zhang et al. (2013), the inter-GCM spreads in cloud radiative effects can be largely attributed to the different treatments of cloud vertical overlap. In our scheme, this aspect, at least in the sub-grid scale, is implicitly considered, which reduces the simulation biases in cloud fraction (shown in Figs. 7-8). Therefore, further investigation of the grid-scale cloud vertical overlap using similar approaches is plausible and warranted. Therein, the pred and obs stand for the prediction and observation datasets, respectively, i for the sample index, and M for the sample amounts.
Open Research
The CloudSat data is available at https://www.cloudsat.cira.colostate.edu, and the code for the Xu-Randall scheme was obtained from the Weather Research Forecast Model Version 3.7.1 (https://www2.mmm.ucar.edu/wrf/users/download/get_source.html). The network was implemented using the Pytorch application programming interface (https://pytorch.org/). The codes for upscaling CloudSat data, scheme training, and result analysis together with the databases for training NSA and Nx100z20 schemes are preserved at https://doi.org/10.5281/zenodo.7539634.
Figure 2 .
2Effect of the isolation-forest screening on the data population. The shadings indicate sample numbers in the upscaled CloudSat data at the resolution of x100z20 for the whole year of 2015.
Figure 2
2compares the sample number distributions of the non-screened and screened datasets. The anomalous samples exist in clouds of all three phases (Figs. 2a, 2c, and 2e). The screening only removes samples with large cloud fraction but small relative humidity and does not affect the rest samples(Figs. 2b, 2d, and 2f).
Figure 3 .
3Effect of the data screening on data samples, demonstrated using the CloudSat observation at 12°-28°N by the granule No. 03307 on 11 December 2018, upscaled at x100z20. Numbers in the figure indicate observed sub-grid cloud fraction, colored shadings indicate relative humidity over the liquid surface, and grey shadings indicate no valid observations. The detected anomalous samples are marked with red strikethroughs.
Figure 5 .
5Joint probability density distribution between cloud fractions from the scheme predictions and the CloudSat observations for clouds of different phases (ice-only, mixed, and liquid-only), estimated based on the testing dataset: (left) results from the NSA scheme; and (right) results from the tuned Xu-Randall scheme. The lower-right numbers indicate the correlation coefficient (r) and RMSE of the predictions with respect to the observations for each type of clouds.
Figure 8 .
8Comparisons of mean cloud fraction at different horizontal (∆x) and vertical (∆z) resolutions between the CloudSat observations and the predictions of three neural network-based schemes, estimated based on the testing dataset. The numbers indicate the prediction-observation differences (prediction minus observation) with absolute values larger than 0.01.
Figure 9 .
9CloudSat observed cloud fraction at 60°-79°N by the granule No. 66364 on 13
Figure 10 .
10Cloud-fraction RMSE year-by-year variation of the NSA and the tuned Xu-Randall scheme predictions with respect to the CloudSat observations. Only cloudy grids are included in the calculation.
Figure 10
10gives the year-by-year variation of the cloud-fraction RMSE, calculated with cloudy samples of all phases. Despite that the schemes are trained/tuned based on only data from the year of 2015, the year-by-year variations of both RMSEs are very small (both within the range of ± 4% of the multiple-year mean) and do not show clear sensitivity to the interannual climate variability. Meanwhile, the RMSEs in 2011
Figure 12 .
122006-2019 averaged zonal-mean vertical distribution of cloud fraction from the CloudSat observations, the NSA and the tuned Xu-Randall scheme predictions. Numbers in (b) and (c) indicate the correlation coefficient (r) and RMSE of the respective scheme predictions with respect to the CloudSat observations. Dashed lines indicate the heights of 440 hPa and 680 hPa, respectively.
AcknowledgmentsThis study is supported by the National Key R&D Program of China (2021YFC3000801) and the National Natural Science Foundation of China (42275074). WCW acknowledges the support from the SUNY Research Foundation Fixed-Price Balance Account (31972). The authors thank the editor (Dr. Tapio Schneider) and the two anomalous reviewers for their kind and patient comments, which greatly help clarify and improve this manuscript.AppendixThe root-mean square error (RMSE) and correlation coefficient (r) of a prediction with respect to the observation are calculated as follows:
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| []
|
[
"Rapid Microwave-Assisted Synthesis of Dextran-Coated Iron Oxide Nanoparticles for Magnetic Resonance Imaging",
"Rapid Microwave-Assisted Synthesis of Dextran-Coated Iron Oxide Nanoparticles for Magnetic Resonance Imaging"
]
| [
"Elizabeth A Osborne \nDepartment of Chemistry\n\n",
"Tonya M Atkins \nDepartment of Chemistry\n\n",
"Dustin A Gilbert \nDepartment of Physics\n\n",
"Susan M Kauzlarich \nDepartment of Chemistry\n\n",
"Kai Liu \nDepartment of Physics\n\n",
"Angelique Y Louie *[email protected] \nDepartment of Biomedical Engineering\nUniversity of California\n95616DavisCaliforniaUSA\n"
]
| [
"Department of Chemistry\n",
"Department of Chemistry\n",
"Department of Physics\n",
"Department of Chemistry\n",
"Department of Physics\n",
"Department of Biomedical Engineering\nUniversity of California\n95616DavisCaliforniaUSA"
]
| []
| Currently, magnetic iron oxide nanoparticles are the only nano-sized magnetic resonance imaging (MRI) contrast agents approved for clinical use, yet commercial manufacturing of these agents has been limited or discontinued. Though there is still widespread demand for these particles both for clinical use and research, they are difficult to obtain commercially, and complicated syntheses make in-house preparation infeasible for most biological research labs or clinics. To make commercial production viable and increase accessibility of these products, it is crucial to develop simple, rapid, and reproducible preparations of biocompatible iron oxide nanoparticles. Here, we report a rapid, straightforward microwave-assisted synthesis of superparamagnetic dextran-coated iron oxide nanoparticles. The nanoparticles were produced in two hydrodynamic sizes with differing core morphologies by varying the synthetic method as either a two-step or single step process. A striking benefit of these methods is the ability to obtain swift and consistent results without the necessity for air, pH, or temperature sensitive techniques; therefore, reaction times and complex manufacturing processes are greatly reduced as compared to conventional synthetic methods. This is a great benefit for cost-effective translation to commercial production. The nanoparticles are found to be superparamagnetic and exhibit properties consistent for use in MRI. In addition, the dextran coating imparts the water-solubility and biocompatibility necessary for in vivo utilization.Iron oxide particles have been widely studied for commercial applications in magnetic storage media, catalysis, magnetic inks, pigments, ferrofluid seals, biotechnology, and medical diagnostics. 1 In particular, advances in the fabrication of nanosized magnetic particles has led to novel and rapidly evolving uses in biomedical applications such as cell tracking and separation, hyperthermia treatment, immunoassays, drug delivery vehicles, and magnetic resonance imaging (MRI) contrast agents. 1-3 For such applications in vivo, the ideal iron oxide nanoparticles should exhibit a small size (< 100 nm) with minimal polydispersity, a nontoxic, biocompatible surface coating, and high magnetization in the presence of a magnetic field. Development of superparamagnetic iron oxide nanoparticles (SPIO) with a variety of differing cores and surface coatings for molecular imaging has resulted in numerous contrast agents that are currently in commercial use or under clinical investigation. 2,4 Yet, consistency of successful preparations that are time-, energy-, and cost-efficient is always of concern.The scientific literature presents a plethora of techniques to prepare magnetic iron oxide nanoparticles including coprecipitation, 5-15 thermal decomposition, 16-19 sonolysis, 20-24 electrochemical deposition, 25 sol-gel processes, 26-28 spray and laser pyrolysis, 29-31 flow injection synthesis, 32 hydrothermal and high temperature syntheses, 33-38 and nanoreactors such as protein cages, 39-42 vesicles, 43,44 and microemulsions [45][46][47][48] . While each of these methods may enjoy certain advantages depending on the desired properties of the iron oxide nanoparticles, many of them possess severe disadvantages for clinical translation such as the necessity for costly, specialized equipment, low yielding, excessively complicated synthetic schemes, and uncontrollable or inconsistent results. An example of a circumstance where manufacturing process vs. cost was critical, is in the case of Feridex®, the only U.S. FDA approved SPIO | 10.1088/0957-4484/23/21/215602 | [
"https://arxiv.org/pdf/1210.1827v1.pdf"
]
| 5,434,748 | 1210.1827 | 40ea53b80d721fd373a1f98bb9de4d10d230a539 |
Rapid Microwave-Assisted Synthesis of Dextran-Coated Iron Oxide Nanoparticles for Magnetic Resonance Imaging
Elizabeth A Osborne
Department of Chemistry
Tonya M Atkins
Department of Chemistry
Dustin A Gilbert
Department of Physics
Susan M Kauzlarich
Department of Chemistry
Kai Liu
Department of Physics
Angelique Y Louie *[email protected]
Department of Biomedical Engineering
University of California
95616DavisCaliforniaUSA
Rapid Microwave-Assisted Synthesis of Dextran-Coated Iron Oxide Nanoparticles for Magnetic Resonance Imaging
Currently, magnetic iron oxide nanoparticles are the only nano-sized magnetic resonance imaging (MRI) contrast agents approved for clinical use, yet commercial manufacturing of these agents has been limited or discontinued. Though there is still widespread demand for these particles both for clinical use and research, they are difficult to obtain commercially, and complicated syntheses make in-house preparation infeasible for most biological research labs or clinics. To make commercial production viable and increase accessibility of these products, it is crucial to develop simple, rapid, and reproducible preparations of biocompatible iron oxide nanoparticles. Here, we report a rapid, straightforward microwave-assisted synthesis of superparamagnetic dextran-coated iron oxide nanoparticles. The nanoparticles were produced in two hydrodynamic sizes with differing core morphologies by varying the synthetic method as either a two-step or single step process. A striking benefit of these methods is the ability to obtain swift and consistent results without the necessity for air, pH, or temperature sensitive techniques; therefore, reaction times and complex manufacturing processes are greatly reduced as compared to conventional synthetic methods. This is a great benefit for cost-effective translation to commercial production. The nanoparticles are found to be superparamagnetic and exhibit properties consistent for use in MRI. In addition, the dextran coating imparts the water-solubility and biocompatibility necessary for in vivo utilization.Iron oxide particles have been widely studied for commercial applications in magnetic storage media, catalysis, magnetic inks, pigments, ferrofluid seals, biotechnology, and medical diagnostics. 1 In particular, advances in the fabrication of nanosized magnetic particles has led to novel and rapidly evolving uses in biomedical applications such as cell tracking and separation, hyperthermia treatment, immunoassays, drug delivery vehicles, and magnetic resonance imaging (MRI) contrast agents. 1-3 For such applications in vivo, the ideal iron oxide nanoparticles should exhibit a small size (< 100 nm) with minimal polydispersity, a nontoxic, biocompatible surface coating, and high magnetization in the presence of a magnetic field. Development of superparamagnetic iron oxide nanoparticles (SPIO) with a variety of differing cores and surface coatings for molecular imaging has resulted in numerous contrast agents that are currently in commercial use or under clinical investigation. 2,4 Yet, consistency of successful preparations that are time-, energy-, and cost-efficient is always of concern.The scientific literature presents a plethora of techniques to prepare magnetic iron oxide nanoparticles including coprecipitation, 5-15 thermal decomposition, 16-19 sonolysis, 20-24 electrochemical deposition, 25 sol-gel processes, 26-28 spray and laser pyrolysis, 29-31 flow injection synthesis, 32 hydrothermal and high temperature syntheses, 33-38 and nanoreactors such as protein cages, 39-42 vesicles, 43,44 and microemulsions [45][46][47][48] . While each of these methods may enjoy certain advantages depending on the desired properties of the iron oxide nanoparticles, many of them possess severe disadvantages for clinical translation such as the necessity for costly, specialized equipment, low yielding, excessively complicated synthetic schemes, and uncontrollable or inconsistent results. An example of a circumstance where manufacturing process vs. cost was critical, is in the case of Feridex®, the only U.S. FDA approved SPIO
MRI contrast agent. Feridex® was approved by the FDA in 1996 and likewise, obtained approval in overseas markets as well. Despite worldwide distribution and use of this SPIO contrast agent, Feridex® is no longer being marketed by its proprietary owner and lone manufacturer, AMAG Pharmaceutical, Inc.
The company announced on their website that production of the contrast agent was halted in November 2008. Though a reason was not specified, one can infer that the company's net profits from Feridex® were not sufficient to continue commercial manufacture of the agent. Perhaps more time-and costefficient production would have changed this outcome.
By far, the most common and classically used method for synthesis of SPIO is coprecipitation of Fe 3+ and Fe 2+ salts in a basic medium. 4 The coprecipitation method offers some advantages in that a large quantity of particles may be prepared and the size, shape, and composition of the nanoparticles can be modified depending upon the ratio and type of iron salts used, the reaction pH, and the ionic strength of the medium. [49][50][51][52][53] In addition, researchers have demonstrated the ability to apply surface coatings to the particles both during and following the coprecipitation process. Nonetheless, a major drawback of this method is limited control of size distribution. 4,54,55 Despite the ease and efficiency of coprecipitation, extensive purification or size sorting may be necessary. 2,56 For surfactant stabilized particles, Perales-Perez and colleagues used physicochemical methods to effectively select 4-7nm magnetite particles from polydisperse powder with sizes ranging from 4-40nm. 57 For polymer coated particles, Rheinlander et al. demonstrated using flow field-flow fractionation to sort a 10-50nm mixture of dextran-coated iron oxide particles into approximately 8 different batches according to size. 58 Magnetic properties vary depending on nanoparticle size; therefore, the ability to control the dispersity of a sample is crucial. 59 Several methods have been proposed for particle size sorting in recent years, including common physical processes such as ultracentrifugation, microfiltration, or size exclusion chromatography. 57,60 Also, size sorting may be performed by adding an electrolyte solution to the particle solution to disrupt its stability and cause precipitation of large particles. 61 Less common methods such as magnetic chromatography 56 have also been explored. Each of these methods lengthens and complicates the synthetic process and reduces the yield of the desired product. As a result, researchers are seeking new means that offer even greater advantages without the inherent shortcomings of the current methodology. A potential answer lies in the emergent microwave synthesis technology.
Microwave-assisted synthesis has been utilized since the late 1980s in the preparation of organics, organometallics, and peptides. 62 Just in the last decade, exploration of this methodology for inorganic material synthesis has been gaining momentum. Microwave synthesis has been shown to significantly reduce reaction time, increase yields, reduce side reactions, enhance reproducibility, and provide a more energy efficient, greener process. 62,63 Microwave heating presents significant benefits over traditional heating methods (i.e. oil bath), which rely on conduction and convection for heat distribution. Microwave radiation heats materials through much more efficient dielectric heating; this occurs as molecular dipoles attempt to align with the alternating electric field. Thus, the heating phenomenon depends on a substance's ability to absorb microwaves and convert the energy to heat; generally, more polar solvents, Dextran is an abundant, inexpensive polymer that consists solely of alpha-D-glucopyranosyl monomers. The polysaccharide has a large number of hydroxyl groups that foster chelation and hydrogenbonding interactions with iron oxide surfaces as well as provide locations for modification. 2 Dextran coating provides stability and biological compatibility to the iron oxide nanoparticles, thereby making them suitable MRI contrast agents. 4 Due to their superior biocompatibility properties, this platform is a common motif for MR T2 agents. Previous pharmacokinetic studies have established that dextran-coated iron oxide nanoparticle are sequestered by the reticuloendothelial system and metabolized to molecular iron for incorporation into hemoglobin within 5-40 days. 75,76 Herein, we report the synthesis of superparamagnetic dextran-coated iron oxide nanoparticles by two separate microwave heating methods.
In varying the synthetic strategy by either a two-step or single step process, we were able to consistently obtain iron oxide nanoparticles in two size regimes. The resulting nanoparticles were characterized with transmission electron microscopy (TEM), dynamic light scattering (DLS), powder X-ray diffraction (XRD), Fourier transform infrared spectroscopy (FT-IR), inductively coupled plasma mass spectrometry (ICP-MS), vibrating sample magnetometry (VSM), superconducting quantum interference device (SQUID) magnetometry and magnetic resonance relaxometry.
RESULTS AND DISCUSSION
Microwave-assisted synthesis of dextran-coated iron oxide nanoparticles (DIO) was successfully accomplished by two methods. In the first approach, the nanoparticles were synthesized via a two-step reaction. First, uncoated iron oxide nanoparticles, basically iron cores with no coating, were prepared by hydrazine reduction of ferric chloride with microwave heating at 100C for 10 minutes. Subsequent coating of the nanoparticles with dextran was achieved in a second stage of microwave heating at 100C for 2 minutes in the presence of additional ferric chloride, sodium hydroxide, and reduced dextran. The coating step was more challenging than first anticipated, as numerous attempts were unsuccessful. The initial trials, comprising reactions of either reduced dextran with bare nanoparticles or reduced dextran with bare nanoparticles in acidic or basic conditions, resulted in large aggregates of particles that quickly fell out of solution. Successful dextran coating was finally achieved by inclusion of an additional 400 L of 1.0M ferric chloride in the second coating step. Presumably, the Fe 3+ ions associate with the nanoparticle surface and function to provide a charged particle shell, thus sustaining the particle dispersion in aqueous solution and encouraging dextran passivation. 77 In the second synthetic approach, the nanoparticles were synthesized in a one-pot single step microwave reaction. Ferric chloride and reduced dextran were reacted with hydrazine in the microwave at 100C for 10 minutes. The two methods result in different sized nanoparticles.
TEM was used to characterize the size and shape of the iron oxide cores. TEM images in Figure 1 demonstrate the differing morphology obtained with the different synthetic schemes. The single step synthesis resulted in relatively monodisperse particles with an iron core size of 6.5 ± 1.2 nm. In contrast, two-step synthesis results in larger size particles with the iron oxide cores measuring 18.0 ± 4.1 nm in a mixture of shapes ranging from spheroid to cubes. Predictably, the dextran coating step did not change the observed core size as uncoated particles after step 1 similarly measured 17.7 ± 6.6 nm (Fig. 1b). However, the shape has changed and the large square looking particles are noticeably missing from the TEM image.
The histograms beneath each image in Figure 1 provide a look at the entire population distribution of the nanoparticle core diameters used to calculate the mean size ± one standard deviation. Dynamic light scattering (DLS) was used to determine the hydrodynamic size of the particles in aqueous solution. While the dextran coating is translucent to TEM, the DLS measurement provides the size of the particles with coating included. The two-step synthesis produced nanoparticles with an overall hydrated diameter of 67 ± 17 nm, while the single step method yielded 39 ± 8 nm particles. These DLS results are consistent with values previously obtained in our lab for dextran coating thickness and iron core size with conventional heating methods. 78-80 Therefore, compared with synthetic methods that yield wide size ranges, there is promising efficiency for production by the microwave method. In this case, on average, 85% of the onestep product is between 30-50 nm and 62% of the two-step product is 50-80 nm in size. In addition, the reduced reaction time and reduced complexity of the technique make it an attractive alternative. Furthermore, the smaller iron core size for nanoparticles formed while dextran is in situ versus subsequent addition has been reported in literature. [81][82][83] Dextran is thought to limit the growth of the iron oxide core by confining the space available for crystal growth. At high dextran concentration, the randomly folded polymer provides small spaces for nanoparticle nucleation and growth that is halted upon dextran adsorption to the iron oxide surface. 82,83 The crystal structures of representative samples of the dextran-coated and uncoated nanoparticles were analyzed by powder X-ray diffraction (XRD). Though line broadening slightly obscures the precision of the diffraction patterns, the line position and intensity features (Fig. 2) were consistent with the spinel phases of magnetite (Fe 3 O 4 ) and/or maghemite (-Fe 2 O 3 ). These two phases are not distinguishable from the obtained XRD patterns; therefore, we suggest that the product may contain either or both of these iron oxide phases. The reference peaks for magnetite and maghemite are provided (in Fig. 2d/e) for comparison to the prepared nanoparticles. The line broadening observed for XRD patterns of nanosized crystalline particles results from diffraction of X-rays over volumes with size analogous to the Fig. 3. For the one-step synthesis, the sample is superparamagnetic with no hysteresis (Fig. 3a). On the contrary, a small ferromagnetic component was observed for the two-step product (Fig. 3b), consistent with the larger iron oxide core observed by TEM. Notably, small but appreciable remanent magnetization only appeared after the dextran coating step for the two-step synthesis; the magnetization curve for the uncoated nanoparticles after step 1 of the two-step synthesis (not shown) was similar to that of the onestep product with no hysteresis. The dextran coating step likely caused further grain growth, albeit small, that pushed the superparamagnetic blocking temperature above room temperature.
The temperature dependence of the magnetization was analyzed by SQUID magnetometry. For the one-step DIO sample, the zero-field-cooled (ZFC) curve displays a maximum in the temperature dependence of the magnetization, corresponding to a blocking temperature of ~25 K (Fig. 4a). In comparison, the two-step DIO sample is ferromagnetic at room temperature. Saturation magnetization at 10 K for the one-step and two-step sample is 71 emu/g Fe (Fig. 4b) and 42 emu/g Fe , respectively.
Following the method laid out in Cho et al., 85,86 from the blocking temperature, we estimate the size of the one-step particles to be 5.7 nm, consistent with the TEM imaging. Similarly, the size of the two-step a) b)
Figure 3. Room temperature magnetic hysteresis loops of (a) one-step iron oxide nanoparticles and (b) two-step iron oxide nanoparticles with an inset showing the small hysteresis loop, indicating a ferromagnetic component.
particles is >13 nm, again consistent with TEM measurements. Despite the larger iron oxide core, the two-step microwave prepared particles exhibited lower magnetization suggesting the potential presence of phases that are less magnetic. In literature, reduced magnetization due to surface characteristics and coatings has also been observed and attributed to several mechanisms, including spin-canting at the particle surface or a surface layer that is magnetically ineffective. 3 Further examination beyond the scope of this paper is required to illuminate the effect of the nanoparticle surface coating on magnetization.
Infrared spectroscopy (IR) confirmed the presence of dextran coating as revealed by comparison of both uncoated and dextran coated iron oxide nanoparticles ( Figure 5). The uncoated nanoparticles (red), obtained after step 1 of the two-step synthesis, display the Fe-O vibration band at 586cm -1 .
Following the addition of dextran, characteristic absorption bands for hydroxyl at 3387cm -1 (broad), C-O at 1018cm -1 , and C-H at 2932cm -1 were observed for both the one-(blue) and two-step (green) dextran coated nanoparticles, while the Fe-O band was slightly obscured. In concurrence with the measured hydrodynamic size, these results further verified the coating of dextran onto the iron oxide nanoparticles. Table 1 includes the calculated relaxivities for the two iron oxide microwave preparations. The one-step product yielded an r 2 relaxivity of 58.1 ± 3.0 mM -1 s -1 while the two-step DIO was 39.3 ± 7.4 mM -1 s -1 . While we initially expected the larger core nanoparticles to produce the greater relaxivity, these results are consistent with the magnetization data in which the two-step particles generate smaller magnetization. As a reference, the formerly commercially manufactured SPIO, Feridex®, was reported to have an r 1 of 10.1 mM -1 s -1 and an r 2 of 120 mM -1 s -1 at 1.5 T field strength. 2 Though the relaxivity of Feridex® is roughly twice that of our one-pot DIO, the wide size distribution of Feridex® (60-150nm) is not ideal for uniform biodistribution.
Figure 5. IR spectra of one-step iron oxide nanoparticles (B, blue), two-step iron oxide nanoparticles (A, green), and uncoated IO particles (C, red).
A B C One-step DIO 7.8 ± 0.3 58.1 ± 3.0
CONCLUSIONS
In conclusion, this work describes two methods for the microwave-assisted synthesis of superparamagnetic dextran coated iron oxide nanoparticles. These simple and rapid preparations allow for consistent synthesis of dextran-coated iron oxide nanoparticles without the necessity for techniques such as prior chilling or degassing of reagents, strict pH control, or use of Schlenk line. Therefore, reaction times and complex manufacturing processes are greatly reduced, and the majority of the product is in a desirable size range, as compared to conventional synthetic methods. This is of utmost importance for cost-effective translation to commercial production. Based on the results described herein, the single-step,
one-pot microwave synthesis shows greater promise over the two-step method as a rapid and consistent method to produce iron oxide nanoparticles for MRI. The one-step method yields dextran passivated iron oxide nanoparticles that are fairly monodisperse and of favorable size (hydrodynamic diameter of ~39 nm and 85% of particles are 30-50 nm in size) to allow longer blood retention; also, its magnetization and relaxivity profiles are superior to that of the two-step product. In future work, we will determine what effect a change in synthetic parameters, such as stoichiometry or microwave reaction time and temperature, will have on the agent's morphology and MR characteristics.
Microwave synthesis of iron oxide nanoparticles offers exciting possibilities for rapid production of clinically relevant contrast agents and the ability to rapidly explore new platforms (i.e. Core-shell, doped, etc.) for future MR contrast agents with nearly instant results. In addition, in our previously published work, we have demonstrated the ability to modify dextran coatings to allow for further functionalization or targeting ability. 78-80, 87 Thus, the potential to develop these nanoparticles, beyond blood-pool agents, into activatable MRI contrast agents is within reach.
MATERIALS AND METHODS
General
All reagents were purchased from commercial sources and used without further purification. Nanopure water (18.0 Mcm) from a Barnstead nanopure filtration unit was used throughout experiments.
Microwave synthesis was carried out in a Discover SP/Explorer-12 Automated Microwave Synthesis System (CEM Corporation, USA) using glass microwave reaction vessels with Teflon caps purchased from CEM Corp. The temperature of each microwave reaction was precisely controlled by the Discover system; however, the reaction pressure was not controlled beyond setting the maximum at 250psi and allowing the vessels to self-vent as needed. FT-IR spectra were obtained using a Shimadzu IR Prestige 21 equipped with a diffuse reflectance accessory. Lyophilized nanoparticle powder was mixed with KBr, and the measurements were performed under ambient conditions. Two-step synthesis of dextran-coated iron oxide particles (DIO) a.
Step 1: Preparation of uncoated iron oxide nanoparticles. In a typical microwave synthesis, ferric chloride (FeCl 3 6 H 2 O, 78 mg) was dissolved in 8 mL of water in a 35 mL microwave reaction vessel equipped with a stir bar. Immediately prior to microwave heating, hydrazine hydrate (N 2 H 4 H 2 O, 1 mL) was added to the vessel at room temperature with stirring. 67 The mixture was heated by microwave at 100 ± 5 C for 10 min (300W max power, 250psi max pressure) with rapid stirring. The black product was collected by centrifugation, washed with water five times, and carried forward to step 2 or freeze-dried to yield a black crystalline powder.
b.
Step 2: Dextran coating of particles. In preparation for dextran coating, the dextran (10,000 MW) was reduced with sodium borohydride according to a literature method. 88 Briefly, an aqueous solution of dextran was stirred with sodium borohydride (26 equivalents) at room temperature for 12 h. The reaction was stopped by the dropwise addition of conc. HCl until pH 7 and then the reduced dextran was dialysed (Spectra/Por 6, MWCO 10,000) against nanopure water. The dextran was applied to freshly prepared uncoated iron oxide particles following the water washings. Reduced dextran (100 mg) was dissolved in an aqueous nanoparticle suspension (5 mL) by sonication. Then, 1.0 M ferric chloride (400 L) and 1.0 M NaOH (1.0 mL) were added to the mixture preceding microwave heating at 100 C for 2 min. The resulting solution was purified by membrane dialysis (Spectra/Por 6, MWCO 15,000) against nanopure water and freeze-dried to yield a brown crystalline solid.
Single step, one pot preparation of dextran-coated iron oxide particles (DIO)
Dextran coated iron oxide nanoparticles were synthesized in a single step by a microwave method similar to the uncoated nanoparticles described above. Ferric chloride (76 mg) and reduced dextran (100 mg)
were dissolved in 8 mL of water with stirring in a 35 mL microwave reaction vessel. The ratio of dextran:Fe used (1:27) was chosen based on our previous experience in dextran coating preparation. 79,80 Immediately prior to microwave heating, hydrazine hydrate (1 mL) was added to the vessel at room temperature with stirring. The mixture was heated by microwave at 100 ± 5 C for 10 min with rapid stirring. The product could not be collected by centrifugation due to increased stability in aqueous media.
The black solution was purified by membrane dialysis (Spectra/Por 6, MWCO 15,000) against nanopure water and freeze-dried to yield a brown crystalline solid.
Characterization of size, iron content, and crystalline structure Hydrated particle size was determined by dynamic light scattering (DLS) on a Nanotrac 150 particle size analyzer (Microtrac, Inc., Montgomeryville, PA). A geometric eight-root regression, with no residuals, was used to fit the data. The Nanotrac 150 has a built-in thermometer to measure the cell temperature, from which the viscosity is calculated; the nanorange option was enabled and scan time of three times 30 s was used. Particle size is expressed as the mean diameter ± one standard deviation. Transmission electron microscopy (TEM) images were utilized to determined iron oxide core size and shape (Phillips CM120, operating at 80kV and equipped with a Gatan Megascan digital camera). Sample solution was dropped onto carbon-coated 300 mesh copper grids. Core particle diameter was determined by taking the mean diameter of at least 300 particles for 3 separate syntheses as measured from TEM images in Image J software (National Institutes of Health). Iron content was determined by inductively coupled plasma mass spectrometry (ICP-MS) (Agilent Technologies 7500ce). The samples were prepared by digestion in 3% nitric acid. Powder X-ray diffraction (XRD) patterns were obtained on a Bruker D8 Advance X-ray diffractometer with Cu K α radiation (1.5418 Å). The diffraction patterns were collected between 20° < 2θ < 68 with a Time/Step of 1.61 s and a step size of 0.006 (2θ). Data were analyzed with MDI Jade Plus 6.1.1 software. Average crystallite size was determined by the Whole Profile Fitting within the MDI Jade suite of programs for the freeze-dried product from the one-step synthesis. The other products were analyzed for crystallite size by the Scherrer equation for the (hkl) peak.
Magnetic Measurements
A Princeton Measurement Corp. MicroMag vibrating sample magnetometer (VSM) and a Quantum
Design superconducting quantum interference device (SQUID) magnetometer were used to evaluate the magnetic properties of the iron oxide nanoparticle samples. The lyophilized nanoparticle powders were prepared by first measuring their mass, then placing each in a gel-cap with cotton stopper. The gel-cap was mounted on a quartz rod for insertion to the VSM. Each measurement was conducted in a field range of ±1.8T, with variable step size down to 2 Oe near remnance and a measurement time of 0.1 sec.
Background measurements of the gel-cap plus stopper were conducted and found to be < 3% of the signal from the powders. A SQUID magnetometer was used to measure the zero-field-cooled (ZFC) and fieldcooled (FC) temperature dependence of the magnetization in an external field of 10 Oe, as well as magnetic hysteresis loops at low temperatures (measurement time ~30s).
Relaxometry
Longitudinal (T 1 ) and transverse (T 2 ) relaxation times were measured on a Bruker Minispec mq60 relaxometer (Billerica, MA) at 60 MHz and 37 C. T 1 values were measured using an inversion recovery sequence with 10-15 data points, and T 2 was measured using a Carr-Purcell-Meiboom-Gill (CPMG) sequence with τ = 1 ms, and 200 data points. Longitudinal and transverse relaxivity was determined as the slope of the line for plots of 1/T 1 or 1/T 2 , respectively, against increasing iron concentration with a correlation coefficient greater than 0.99.
Figure 1 .
1Representative TEM images of (a) one-pot synthesized dextran-coated iron oxide nanoparticles (DIO) with an iron oxide core measuring 6.5 ± 1.2 nm, (b) uncoated iron oxide nanoparticles (obtained after step 1 of the two-step synthesis) with a core diameter of 17.7 ± 6.6 nm, and (c) two-step DIO with a core diameter of 18.0 ± 4.1 nm. Scale bar = 50 nm. The histograms included below each image indicate the population size distributions for three batches of each, respectively.X-rays wavelength. 84 This effect allows for calculation of the crystallite size using the Scherrer equation assuming one product and a Gaussian size distribution. Calculation of the crystallite sizes yielded 4.3 nm ± 0.2 (whole pattern fitting assuming all Fe 3 O 4 ) for the one-step iron oxide, and this is slightly smaller than that determined by TEM. The powder diffraction patterns of the uncoated iron oxide nanoparticles and the two-step DIO product show reflections that are composed of a broad base indicative of small particles superimposed with a narrow peak indicative of much larger particles, consistent with the large size distribution observed in the TEM. This is especially apparent for the powder diffraction of the twostep DIO. Whole pattern fitting is only appropriate for obtaining the average crystallite size when the samples are fairly uniform. While the average crystallite size is consistent with the TEM for the one-step method, in the case of the uncoated particles and two-step DIO, crystallite size from powder X-ray diffraction would be misleading because of the very large size distribution observed in the TEM.Percent iron content for the DIO nanoparticles was determined by ICP-MS in order to allow calculation of magnetic properties per mass of iron. Given the percent of iron per mass of sample, the
Figure 2 .
2XRD patterns for (a) one-step DIO, (b) uncoated iron oxide particles (obtained after step 1 of the two-step synthesis), (c) two-step DIO, and reference patterns for (d) Fe 3 O 4 PDF #65-3107 and (e) -Fe 2 O 3 PDF # 25-1402. the particles was evaluated by VSM and plotted with respect to the amount of iron (emu/gram of Fe). Room temperature magnetic hysteresis loops of dextran-coated iron oxides are shown in
Figure 4 .
4Temperature dependence of magnetization of the one-step dextran-coated iron oxide nanoparticles measured in a 10 Oe external field, (a) ZFC (black squares) and FC (red circles) curves and (b) magnetic hysteresis loop at 10 K, normalized to the mass % of Fe. a) b) Relaxivity, a concentration-independent measure of the effectiveness of a paramagnetic material, was determined as the slope of inverse relaxation time versus iron concentration. Superparamagnetic iron oxide nanoparticles are commonly considered T 2 contrast agents due to their high transverse relaxivities. 2 Longitudinal and transverse relaxation time of aqueous iron oxide nanoparticle samples were measured at 60 MHz and 37 C.
reagents, and catalysts are more efficiently heated. Beyond the previous noted benefits, in the case of nanoparticle synthesis, microwave preparation has yielded greater control of size and dispersity as well asenhanced crystallinity. 64 Microwave syntheses for superparamagnetic iron oxide nanoparticles have been reported that yield Fe 2 O 3 particles 65-68 , mixed Fe 2 O 3 and Fe 3 O 4 particles 69 , and Fe 3 O 4 particles 67,70-74 . Only a small fraction of those reported indicate passivation of the nanoparticle surface, which is necessary for water solubility and biocompatibility. In 2007, Zhu and coworkers reported microwave synthesis of irregular shaped Fe 3 O 4 nanoparticles and ellipsoidal Fe 2 O 3 nanoparticles that were coated with polyethylene glycol (PEG-20000). 67 In 2009, Yang et al published the microwave synthesis of spherical, nanoporous Fe 3 O 4 nanoparticles also coated with PEG. 71 In 2010, Edrissi and colleague reported the microwave-assisted synthesis of spheroid Fe 2 O 3 nanoparticles coated with linoleic or palmitoleic acid. 74 To date, no microwave methods to prepare dextran-coated iron oxide nanoparticles have been reported.
Table 1 .
1Longitudinal and Transverse Relaxivity of Dextran-coated Iron Oxide Nanoparticlesr 1 (mM -1 s -1 )
r 2 (mM -1 s -1 )
Two-step DIO
2.3 ± 0.3
39.3 ± 7.4
AcknowledgmentThe authors wish to acknowledge funding from the Department of Energy (DESC0002289) and NSF (DMR-1008791) for support of this work.
. H Pardoe, W Chua-Anusorn, St, T G Pierre, Dobson, J. Journal of Magnetism and Magnetic Materials. 22541Pardoe, H.; Chua-anusorn, W.; St. Pierre, T. G.; Dobson, J. Journal of Magnetism and Magnetic Materials 2001, 225, 41.
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. S J Cho, A M Shahin, G J Long, J E Davies, K Liu, F Grandjean, S M Kauzlarich, Chem. Mater. 18960Cho, S. J.; Shahin, A. M.; Long, G. J.; Davies, J. E.; Liu, K.; Grandjean, F.; Kauzlarich, S. M. Chem. Mater. 2006, 18, 960.
. E A Osborne, B R Jarrett, C Tu, A Y Louie, Journal of the American Chemical Society. 5934Osborne, E. A.; Jarrett, B. R.; Tu, C.; Louie, A. Y. Journal of the American Chemical Society 2010, 132, 5934.
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[
"THE X-RAY SPECTRUM OF COMPTON-THICK SEYFERT 2 GALAXIES",
"THE X-RAY SPECTRUM OF COMPTON-THICK SEYFERT 2 GALAXIES"
]
| [
"Giorgio Matt \nDipartimento di Fisica\nUniversità degli Studi \"Roma Tre\"\nVia della Vasca Navale 84I-00146RomaItaly\n"
]
| [
"Dipartimento di Fisica\nUniversità degli Studi \"Roma Tre\"\nVia della Vasca Navale 84I-00146RomaItaly"
]
| []
| Current ideas on the X-ray spectrum of Compton-thick Seyfert 2 galaxies are reviewed, and a brief description of the four presently known sources of this class are given. | null | [
"https://arxiv.org/pdf/astro-ph/9612002v1.pdf"
]
| 14,418,729 | astro-ph/9612002 | 09fd3ea02ae40f33beea7c06d564f5bf188cb2be |
THE X-RAY SPECTRUM OF COMPTON-THICK SEYFERT 2 GALAXIES
arXiv:astro-ph/9612002v1 29 Nov 1996
Giorgio Matt
Dipartimento di Fisica
Università degli Studi "Roma Tre"
Via della Vasca Navale 84I-00146RomaItaly
THE X-RAY SPECTRUM OF COMPTON-THICK SEYFERT 2 GALAXIES
arXiv:astro-ph/9612002v1 29 Nov 1996
Current ideas on the X-ray spectrum of Compton-thick Seyfert 2 galaxies are reviewed, and a brief description of the four presently known sources of this class are given.
Introduction
In the unified model for Seyfert galaxies (Antonucci 1993 and references therein), type 2 sources are believed to be intrinsically identical to type 1's, but observed at inclination angles, with respect to the symmetry axis of the molecular torus (see e.g. Ward 1994), greater than the torus half-opening angle. Therefore, at least in optical/UV, the nucleus of Seyfert 2 galaxies is not directly visible.
In X-rays the situation is more complex. In this band, the two most important interactions between photons and (cold) matter are photoabsorption and Compton scattering. The photoabsorption cross section strongly depends on energy (decreasing approximately as E −3.5 ), while the Compton cross section is constant, at least up to the Klein-Nishina decline. The two cross sections, for solar chemical composition, are equal at about 10 keV. If the column density of the torus is smaller than σ −1 T ∼ 1.5 × 10 24 cm −2 , the nucleus turns out to be visible above a few keV, and the sources are named Compton-thin, because the matter is optically thin to Compton scattering. If, on the contrary, the column density exceeds that value, the matter is optically thick to Compton scattering; for Compton optical depths of ∼a few, the nucleus becomes practically invisible also in hard X-rays because, after a few scatterings, photons are redshifted down to the photoabsorption dominated regime. These sources are called Comptonthick Seyfert 2 galaxies, and can be observed in X-rays only in scattered light, as will be discussed in the next section. Of course, they are very faint in X-rays and only a handful (four, excluding the borderline source NGC 4945) of them are presently known.
The X-ray spectrum of Compton-thick sources
Even if the type-1 nucleus is, in Compton-thick sources, completely obscured in the whole X-ray band, its existence is revealed by scattered light. Two scattering media are thought to be present in the vicinity of the nucleus: (a) the optically thin matter responsible for the scattering and polarization of the optical broad lines in several Seyfert 2's (the so-called "warm mirror"), and (b) the inner surface of the torus itself which, in these sources, is optically thick by definition. (a) Scattering from the warm mirror of the nuclear continuum radiation produces a spectrum which is practically identical to the nuclear spectrum up to energies at which Compton recoil is important, where a cut-off occurs (Matt 1996;Poutanen et al. 1996). Therefore, its shape is expected to be, at least below say a few tens of keV, a power law with photon index ∼2 (Nandra & Pounds 1994). Line emission from ionized atoms is also expected, iron Kα lines being among the most prominent (other lines, e.g. from carbon, oxygen and neon, can also be very intense; however, they are often diluted by other continua arising in soft X-rays, e.g. from starburst regions. Fe xxv (6.7 keV) and Fe xxvi (6.97 keV) Kα line emission from such a medium is discussed in detail by Matt, Brandt & Fabian (1996). Their equivalent width with respect to the continuum reflected from the warm mirror itself can be as high as a few keV. (b) As shown by Ghisellini, Haardt & Matt (1994) and Krolik, Madau &Życki (1994), the spectrum of the nuclear radiation scattered from the inner surface of the torus is similar to that observed in many Seyfert 1 galaxies (Mushotzky, Done & Pounds 1993 and references therein), where the reflecting matter is believed to be the inner accretion disc. It is usually referred to as the Compton reflection component. It is very flat in the classical 2-10 keV band (photon index less than 1) due to the increasing ratio of Compton scattering to photoabsorption cross sections (e.g. Lightman & White 1988). A fluorescent Kα line at 6.4 keV from neutral iron, with an equivalent width with respect to the continuum reflected by the same matter of about 1-2 keV (e.g. Matt et al. 1991;Ghisellini, Haardt & Matt 1994;Reynolds et al. 1994) is also expected, as well as fainter lines from lighter elements (Reynolds et al. 1994;Matt, Fabian & Reynolds 1996).
The intensity of the radiation from the warm mirror depends mainly on the optical depth of the mirror itself, while that of the torus component depends basically on the inclination angle of the system (see Matt, Brandt & Fabian 1996). Their relative ratio can therefore be very different from source to source. Three cases are then possible: i) The warm mirror component dominates over the torus component. A 2-10 keV spectrum composed by a power law with photon index about 2 plus strong ionized lines is expected. No sources of this kind have been discovered yet to our knowledge.
ii) The two components are of the same order. A spectrum with an intermediate or flat photon index (1-1.5 or less) and both neutral and ionized lines is expected, and it has indeed been observed in NGC 1068 (Ueno et al. 1994; see below for a revised analysis) and NGC 6240 (Iwasawa, private communication). It is worth stressing that it is possible to observe in one and the same source a continuum dominated, above a few keV, by the torus component and also a complex iron line, as the lines from ionized matter can be very bright and then visible even if the relative continuum is much smaller than the torus one (this actually seems to be the case for the two sources of this group).
iii) The torus component dominates over the warm mirror component. A very flat (less than one) spectrum with a strong 6.4 keV iron line is expected, and actually observed in NGC 6552 ) and in the Circinus Galaxy .
In the next section we will briefly discuss each of the four currently known Comptonthick sources.
Group II sources
NGC 1068 is surely the best studied among Compton-thick Seyfert 2's (and maybe among all Seyfert 2's). Its X-ray spectrum is composed by thermal-like emission, probably due to a starburst region (Wilson et al. 1994) and dominating below 3-4 keV, and a hard power law with superposed a complex iron line, composed by a neutral component at 6.4 keV and contributions from He-and H-like iron (Ueno et al. 1994 and references therein). Iwasawa, Fabian & Matt (1996) have recently re-analysed the ASCA PV observation and found that: a) the photon power law index is smaller than previously estimated, being less than 0.4. This value is consistent with a pure reflection continuum. b) A feature just redwards of the neutral iron line is present, with a flux of about 10 percent of that of the line, and interpreted as the line Compton shoulder, i.e. line photons scattered once before escaping from the matter. This feature is predicted by the reflection models (see e.g. Matt, Perola & Piro 1991) but never detected so far. c) The ionized iron lines are both redshifted; the derived velocity of the emitting matter is about 4000-5000 km/s. The interpretation of the spectrum above a few keV is that we are looking at reflection from both the warm mirror and the obscuring torus, with the former component smaller but not completely overwhelmed by the latter one, so that the continuum is dominated by the torus reflection but the lines from the ionized matter (possibly forming a wind, as suggested by its relatively high velocity) are still visible. NGC 6240 is one of the most famous merging system and a prototype ultraluminous infrared galaxy. In X-rays, the source has been observed by both ROSAT-PSPC (Fricke & Papaderos 1996) and ASCA (K. Iwasawa, private communication). The ROSAT-PSPC revealed extended soft X-ray emission, which can be explained by a starburst model. The ASCA spectrum is complex, consisting of: a thermal-like component, consistent with the ROSAT observation and dominating in soft X-rays; a quite hard power law; both neutral and ionized iron lines. The hard spectrum is very similar to that of NGC 1068, and can be explained in the same way.
Group III sources
NGC 6552 was serendipitously observed by ASCA (mainly thanks to the very prominent 6.4 keV iron line; Fukuzawa et al. 1994), and then identified with a Seyfert 2 galaxy. Its X-ray spectrum is well described by a pure Compton reflection continuum (Reynolds et al. 1994), and then interpreted as reflection from cold matter (possibly the torus) of an otherwise invisible X-ray nucleus. A heavily absorbed power law maybe also be present, but the source is too faint to permit any strong conclusion on this and other spectral details. The Circinus Galaxy is a nearby (4 Mpc) Seyfert 2 with a very prominent ionization cone, strong coronal lines and water maser emission (see Oliva et al. 1994 and references therein). Before ASCA, it was never observed in X-rays, apart from a detection in the ROSAT all sky survey. The ASCA spectrum is quite remarkable, showing a very prominent iron Kα line at 6.4 keV, many other lines from lighter elements as well as the Kβ iron line, and a very hard continuum . Below about 2 keV, a further component emerges, possibly due to a starburst region. Above that energy, the spectrum is well described by a pure reflection component. Lines from elements lighter than iron are also expected in this model (Reynolds et al. 1994;Matt, Fabian & Reynolds 1996). However, their observed intensities and centroid energy are not entirely consistent with reflection by cold matter; improved ASCA-SIS calibration matrices have to be waited for before further addressing this issue.
Conclusions
The model described in Sec.2 for the X-ray spectrum of Compton-thick Seyfert 2 galaxies is fully consistent with the observations of the four known sources of this class. A prediction of this model is a spectral cut-off at a few tens of keV due to Compton recoil. BeppoSAX and RXTE observations will then be valuable in testing the model.
AcknowledgementsI thank all the colleagues who collaborate with me on this topic. I'm particularly indebted to K. Iwasawa for allowing me to quote his results on NGC 6240 before publication.
. R R J Antonucci, Ann. Rev. Astron. Astrophys. 31473Antonucci, R.R.J., 1993: Ann. Rev. Astron. Astrophys. 31, 473.
K J Fricke, P Papaderos, n.263Proc. of the conference "Roentgenstrahlung from the Universe. of the conference "Roentgenstrahlung from the Universe377MPE ReportFricke K.J., Papaderos P.: 1996, Proc. of the conference "Roentgenstrahlung from the Universe". MPE Report n.263, p.377.
. Y Fukazawa, Publ. Astron. Soc. Japan. 46141Fukazawa Y., et al.: 1994, Publ. Astron. Soc. Japan 46, L141.
. G Ghisellini, F Haardt, J H Matt G ; Krolik, P Madau, P T Życki, Mon. Not. R. Astr. Soc. 26757Astrophys. J.Ghisellini G., Haardt F., Matt G.: 1994, Mon. Not. R. Astr. Soc. 267, 743. Krolik J.H., Madau P.,Życki P.T.: 1994 Astrophys. J. 420, L57.
. K Iwasawa, A C Fabian, A P Matt G ; Lightman, T R White, Mon. Not. R. Astr. Soc. 33557Astrophys. J.Iwasawa K., Fabian A.C., Matt G: 1996, Mon. Not. R. Astr. Soc., submitted. Lightman, A.P., White, T.R.: 1988, Astrophys. J. 335, 57.
G Matt, n.263Proc. of the conference "Roentgenstrahlung from the Universe. of the conference "Roentgenstrahlung from the Universe479MPE ReportMatt, G.: 1996, Proc. of the conference "Roentgenstrahlung from the Universe". MPE Report n.263, p.479.
. G Matt, W N Brandt, A C Fabian, Mon. Not. R. Astr. Soc. 280823Matt G., Brandt W.N., Fabian A.C.: 1996, Mon. Not. R. Astr. Soc. 280, 823.
. G Matt, A C Fabian, C Reynolds, G S ; Matt, G C Perola, L ; Piro, G Matt, Mon. Not. R. Astr. Soc. 24569Mon. Not. R. Astr. Soc.Matt G., Fabian A.C., Reynolds C.S: 1996, Mon. Not. R. Astr. Soc., submitted. Matt G., Perola G.C., Piro L.: 1991, Astron. Astrophys. 245, 25. Matt, G., et al.: 1996, Mon. Not. R. Astr. Soc. 281, L69.
. R F Mushotzky, C Done, K A Pounds, K Nandra, K A Pounds, Ann. Rev. Astron. Astrophys. 31405Mon. Not. R. Astr. Soc.Mushotzky R.F., Done C., Pounds K.A.: 1993, Ann. Rev. Astron. Astrophys. 31, 717. Nandra, K., Pounds, K.A.: 1994, Mon. Not. R. Astr. Soc. 268, 405.
. E Oliva, M Salvati, A F M Moorwood, A Marconi, Astron. Astrophys. 288457Oliva E., Salvati M., Moorwood A.F.M., Marconi A., 1994: Astron. Astrophys. 288, 457.
. J Poutanen, M Sikora, M C Begelamn, P ; Magdziarz, C S Reynolds, A C Fabian, K Makishima, Y Fukazawa, T Tamura, Mon. Not. R. Astr. Soc. 46555Astrophys. J.Poutanen J., Sikora M., Begelamn M.C., Magdziarz P: 1996, Astrophys. J. 465, L107. Reynolds C.S., Fabian A.C., Makishima K., Fukazawa Y., Tamura T.: 1994, Mon. Not. R. Astr. Soc. 268, L55.
. S Ueno, R F Mushotzky, K Koyama, K Iwasawa, H Awaki, I Hayashi, Publ. Astron. Soc. Japan. 4671Ueno S., Mushotzky R.F., Koyama K., Iwasawa K., Awaki H., Hayashi I.: 1994, Publ. Astron. Soc. Japan 46, L71.
M J Ward, Oxford, A S Wilson, M Elvis, A Lawrence, J Bland-Hawthorn, Proceedings of the Workshop on "Evidence for the Torus. the Workshop on "Evidence for the Torus39175Ward M.J, ed.: 1994, Proceedings of the Workshop on "Evidence for the Torus", Oxford. Wilson A.S., Elvis M., Lawrence A., Bland-Hawthorn J.: 1992, Astrophys. J. 391, L75.
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"On the frequency of height values",
"On the frequency of height values"
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"Gabriel A Dill [email protected] \nCorrespondence\nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building, Radcliffe Observatory Quarter, Woodstock RoadOX2 6GGOxfordUK\n"
]
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"Correspondence\nMathematical Institute\nUniversity of Oxford\nAndrew Wiles Building, Radcliffe Observatory Quarter, Woodstock RoadOX2 6GGOxfordUK"
]
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"Dill Res. Number Theory"
]
| We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k ∈ {0, d} or gcd(k, d) = 1. We therefore study the behaviour in the case where 0 < k < d and gcd(k, d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function. | 10.1007/s40993-021-00261-1 | null | 229,211,199 | 2012.09085 | e7e16ac00a43fbc48d97144881949afa7b138c93 |
On the frequency of height values
2021
Gabriel A Dill [email protected]
Correspondence
Mathematical Institute
University of Oxford
Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock RoadOX2 6GGOxfordUK
On the frequency of height values
Dill Res. Number Theory
733202110.1007/s40993-021-00261-1R E S E A R C HHeightMahler measureCounting Mathematics Subject Classification: 11G50
We count algebraic numbers of fixed degree d and fixed (absolute multiplicative Weil) height H with precisely k conjugates that lie inside the open unit disk. We also count the number of values up to H that the height assumes on algebraic numbers of degree d with precisely k conjugates that lie inside the open unit disk. For both counts, we do not obtain an asymptotic, but only a rough order of growth, which arises from an asymptotic for the logarithm of the counting function; for the first count, even this rough order of growth exists only if k ∈ {0, d} or gcd(k, d) = 1. We therefore study the behaviour in the case where 0 < k < d and gcd(k, d) > 1 in more detail. We also count integer polynomials of fixed degree and fixed Mahler measure with a fixed number of complex zeroes inside the open unit disk (counted with multiplicities) and study the dynamical behaviour of the height function.
Introduction
LetQ denote the algebraic closure of Q in C. For an algebraic number α ∈Q, let H(α) denote the (absolute multiplicative Weil) height of α, as defined in Section 1.5 of [3]. We have H(α) ∈Q ∩ [1, ∞) for all α ∈Q. By a well-known theorem of Northcott [18] (see also Theorem 1.6.8 in [3]), there are at most finitely many algebraic numbers of bounded degree and bounded height. This article seeks to answer the question: "How many α ∈Q are there of fixed degree d and fixed height H?" In particular, we are interested in whether the height assumes many values, but each single value is assumed only rarely, or whether there are only few values that are however assumed very often. For fundamental properties of the height, we refer to Section 1.5 of [3].
Much is known about counting algebraic numbers or more generally points in P n (Q) of fixed degree (over Q or over any fixed number field) and bounded height: Schanuel first proved, in [21], an asymptotic for the number of algebraic points of bounded height that are defined over a fixed number field. Further results, including the asymptotic for the number of quadratic points (over Q) of bounded height, were obtained by Schmidt in [22] and [23]. If n is larger than the degree of the point (over Q), then Gao found and proved the correct asymptotic in [12]. He also determined the correct order of magnitude for any n and any degree (over Q). Masser and Vaaler then counted algebraic numbers of fixed degree and bounded height in [17] (over Q) and [16] (over any fixed number field). If the degree of the point (over any fixed number field) is at most slightly less than 2n 5 , then Widmer obtained the correct asymptotic in [28]. More recently, Guignard [14] counted quadratic points (over any fixed number field) if n ≥ 3; he also counted points whose degree (over any fixed number field) is an odd prime less than or equal to n − 2. However, Guignard uses a slightly different height, corresponding to another choice of norm at the infinite places.
The same problem has also been studied for integral points, i.e. elements ofQ n whose coordinates are algebraic integers: In Theorem 5.2 in Chapter 3 of [15], Lang gives an asymptotic for the number of algebraic integers of bounded height that lie in a fixed number field (with an unspecified constant in the main term). The work [7] of Chern and Vaaler, which was also used crucially in [17], yields an asymptotic for the number of algebraic integers of fixed degree over Q and bounded height. In [1], Barroero extended the results of Lang and Chern and Vaaler to count algebraic integers of fixed degree over any fixed number field and bounded height. Widmer counted, in [29], integral points of fixed degree (over any fixed number field) and bounded height under the assumption that the degree of the point is either 1 or at most slightly less than n. In [13], Grizzard and Gunther counted (among other things) algebraic integers of fixed degree (over Q), fixed norm, and bounded height. This last result is somewhat related to our work in that the d-th power of the height of an algebraic integer of degree d (over Q) with no conjugate inside the open unit disk is equal to the absolute value of its norm.
We emphasize that all these results give much more precise asymptotics than the ones obtained in this article. However, already when counting rational numbers of fixed height, Euler's phi function appears, so it is clear that such precise asymptotics cannot be obtained in general when counting algebraic numbers of fixed degree and fixed height. Instead, we strive to obtain an asymptotic for the logarithm of the associated counting function.
In order to state our results, we have to introduce some notation: The conjugates (over Q) of an algebraic number are the complex zeroes of its minimal polynomial over Q. While there is no nice asymptotic for the logarithm of the counting function associated to our question from the beginning, we have managed to obtain such an asymptotic in many cases if the number of conjugates that lie inside the open unit disk is also prescribed. if these limits exist. It will follow from Lemma 3.1 that A(k, d) is an infinite set. Thus, B(k, d) contains arbitrarily large elements and at least the limit superior and inferior corresponding to a(k, d) certainly exist. We remark that it is not clear if the conjugates inside the open unit disk are the right thing to take into account here. The Galois group of the normal closure of Q(α), the degree [Q H(α) d : Q], and the normal closure of Q H(α) d also seem to play an important role as will become apparent. Of course, these objects are not independent of one another (e.g. k ∈ {0, d} is equivalent to [Q H(α) d : Q] = 1).
The main results of this article can be summarized as follows: Demanding that the action of the Galois group of the normal closure of Q(α) on the conjugates of α ∈ A(k, d, H) is sufficiently generic implies that there are few such α if 0 < k < d. The following is our strongest result in this direction: We also show that the height function together with the degree is in some sense "almost injective" if the degree is at least 2:
Theorem 1.3 (= Theorem 4.3). Let d ≥ 2. For every > 0, there is H 0 = H 0 (d, ) ∈ R such that |{α ∈ C; [Q(α) : Q] = d, H(α) ≤ H}| |{H(α); α ∈ C, [Q(α) : Q] = d, H(α) ≤ H}| ≤ H for all H ≥ H 0 .
If d = 4 and k = 2, then we obtain finer results than those given by In the construction in the proof of Theorem 6.2, the field Q(H 4 ) is made to vary in an infinite set unless κ = 4. This suggests that in general fixing the field Q(H d ) might lead to more uniform growth behaviour. The following is a simplified version of Theorem 7.1: Theorem 1.5 Let δ ∈ (0, 1) and > 0 and let K ⊂Q be a fixed Galois extension of Q. Let k, d ∈ N such that 0 < k < d and let H ∈Q ∩ [1, ∞) such that the normal closure of Q(H d ) is equal to K .
Suppose that α ∈ A(k, d, H). Set L = Q(α) ∩ K and l = d[L : Q] −1 and let β ∈ L be the Q(α)/L-norm of α. There exists a constant C = C(k, d, K, δ, ) > 0 such that if for every field embedding σ :
Q(β) → C, we have either |σ (β)| ≥ (1 − δ) −1 or |σ (β)| ≤ 1 − δ, then |A(k, d, H)| ≥ CH d(l−1)− .
Theorem 7.1 is then used together with upper bounds for |A(k, d, H)| from Theorem 5.2 for the determination of the limit superior corresponding to a(k, d) in Theorem 7.5. We also give examples that show the necessity of the dependence of C on K and δ.
In Sect. 8, we count polynomials with integer coefficients of fixed degree d and fixed Mahler measure M as defined in Section 1.6.4 of [3]. Among these polynomials, those that are irreducible in Z[t] are in a 1-to-d correspondence with the algebraic numbers of degree d and height M 1 d . However, we also count the polynomials that are reducible in Z[t] and this leads to somewhat simpler results although even fewer of the considered limits exist. We obtain Theorem 8.1, an analogue of Theorem 1.1 in this context.
Following a suggestion of Norbert A'Campo, we study the dynamical behaviour of the height function in Sect. 9. The dynamical behaviour of the Mahler measure has been studied initially by Dubickas in [9] and [10] and subsequently by Zhang in [31] as well as by Fili, Pottmeyer, and Zhang in [11]. We obtain the following result: Theorem 1.6 (= Theorem 9.3). Let α ∈Q and define inductively α 0 = α, α n = H(α n−1 ) (n ∈ N). Then either there exist N, a ∈ N and b ∈ Q, b > 0, such that α n = a b for all n ≥ N or lim n→∞ α n = 1.
In particular, the periodic points of H are precisely the a b for a ∈ N and b ∈ Q, b > 0. Our proofs are mostly elementary. Our constructions of many algebraic numbers of a given height rely on point counting results for lattices by (in the proofs of Theorem 2.1 and Lemma 3.1) and Widmer in [30] (in the proof of Theorem 7.1). The first of these results generalizes a theorem of Davenport in [8] while the second one generalizes a theorem of Skriganov in [25].
The main result of [2] is formulated in an arbitrary o-minimal structure; we will however apply it only in the structure of semialgebraic sets, where a subset of R n (n ∈ N) is called semialgebraic or definable (in the structure of semialgebraic sets) if it is a finite union of sets defined by a finite number of polynomial equations and inequalities with real coefficients. By the Seidenberg-Tarski theorem, the structure of semialgebraic sets is o-minimal, which implies that besides polynomial equations and inequalities with real coefficients, we can also use existential and universal quantifiers to define semialgebraic sets. For a general introduction to o-minimal structures, see [26].
For a real number ξ , we denote by [ξ ] the largest integer which does not exceed ξ . We use φ to denote Euler's phi function and μ to denote the Möbius function. For a finite field extension L/K , we denote the corresponding field norm by N L/K . If K is a number field, then we denote its ring of integers by O K . The norm of an ideal I of O K is denoted by N (I). The imaginary unit in C is denoted by √ −1 and the real and imaginary part of a complex number are denoted by Re and Im respectively. For a real-valued function f on S ⊂ R n , we write O(f ) for any function g : S → R such that there exists a constant C = C(f, g) ≥ 0 with |g(s)| ≤ Cf (s) for all s ∈ S. If n = 1, S is unbounded, and f (s) > 0 for |s| large enough, we say that a function g :
S → R is of growth order o(f ) if lim s∈S,|s|→∞ |g(s)| f (s) = 0.
If α is an algebraic number of degree d, a minimal polynomial of α in Z[t] is an irreducible element of Z[t] that has α as a zero. There are two choices for a minimal polynomial of α in Z[t] as (Z[t]) * = {±1}. The following simple observation will be used at different places throughout this article: If a is the leading coefficient of a minimal polynomial of α in Z[t] and α 1 , . . . , α d−k are the conjugates of α that lie outside the open unit disk, then H(α) d = |a||α 1 | · · · |α d−k | = ±aα 1 · · · α d−k (see Propositions 1.6.5 and 1.6.6 in [3]). We can write ±α i instead of |α i | (i = 1, . . . , d − k) since the non-real conjugates appear in complex conjugate pairs and the real conjugates are equal to their absolute value up to sign.
The case k ∈ {0, d}
In this section, we treat the case where k ∈ {0, d}, which is the easiest one to resolve. Theorem 2.1 Let d ∈ N. The following hold: (ii) Let us define
(i) b(0, d) = b(d, d) = d,Z = (w 0 , . . . , w d−1 , T ) ∈ R d × R; w 0 > 0, ∃x 1 , . . . , x d , y 1 , . . . , y d ∈ R : x 2 j + y 2 j ≥ 1 ∀j = 1, . . . , d, g j (x 1 , y 1 , . . . , x d , y d ) = 0 ∀j = 0, . . . , d − 1, and f j (x 1 , y 1 , . . . , x d , y d , T ) = w j ∀j = 0, . . . , d − 1 , (2.1) where f j (x 1 , y 1 , . . . , x d , y d , T ) = Re (−1) d−j T (x 1 + √ −1y 1 ) · · · (x d + √ −1y d ) σ j (x 1 + √ −1y 1 , . . . , x d + √ −1y d ) ,
and g j (x 1 , y 1 , . . . , x d , y d ) =
Im 1 (x 1 + √ −1y 1 ) · · · (x d + √ −1y d ) σ j (x 1 + √ −1y 1 , . . . , x d + √ −1y d ) for j = 0, . . . , d − 1, where σ j is the j-th elementary symmetric polynomial in d variables.
The set Z is definable in the o-minimal structure of all semialgebraic subsets of R n (n ∈ N). Let π : R d × R → R d be the canonical projection. For T ∈ R, T = 0, the set Z T = π(Z ∩(R d ×{T })) parametrizes polynomials w 0 t d +· · ·+w d−1 t +T of degree d with real coefficients and positive leading coefficient that have no complex zeroes inside the open unit disk and whose constant coefficient is equal to T . Note that Z T = |T | · Z T /|T | (T = 0) and that the coordinates of a point in Z T can all be bounded by some constant multiple of |T |, depending on d. It follows that the volume of Z T is |T | d times the volume of Z T /|T | (T = 0) and that the volume of any orthogonal projection of Z T on some j-dimensional coordinate subspace of R d has j-dimensional volume at most a constant multiple of |T | j , depending on d.
We then deduce from Theorem 1.3 in [2] that
Z T ∩ Z d − V T /|T | |T | d = O(|T | d−1 ) for |T | ≥ 1, where V u is the volume of Z u for u ∈ {±1}.
Here and in the rest of this proof, the implicit constants in the O notation depend only on d. The volume V u is positive for
u ∈ {±1} since |w 0 z d + · · · + w d−1 z + u| > 1 − d 2d= Z T ∩ Z d + Z (−T ) ∩ Z d = (V 1 + V −1 )T d + O(T d−1 ) for T ∈ N. If we definẽ N d (T ) = |{P(t) = at d + · · · ± T ∈ Z[t]
; a > 0, gcd(a, . . . , ±T ) = 1, all complex zeroes of P are at least 1 in absolute value}|, then we have N d (T ) = S|TÑ d (S). Using Möbius inversion together with an elementary bound for the divisor function, we deduce that
N d (T ) = S|T μ(S)N d T S = (V 1 + V −1 )T d ⎛ ⎝ S|T μ(S) S d ⎞ ⎠ + O T d− 1 2 .
Here S|T μ(S)
S d = p|T 1 − 1 p d is at most 1 and at least φ(T ) T .
In fact, for d ≥ 2, the product is at least ∞ k=2 1 − 1 k 2 = 1 2 , so bounded from below uniformly. What we really want iŝ
N d (T ) = |{P(t) = at d + · · · ± T ∈ Z[t]; a > 0, gcd(a, . . . , ±T ) = 1, P is irreducible in Q[t],
and all complex zeroes of P are at least 1 in absolute value}|
since |A(0, d, H)| = dN d (H d ) if H d ∈ N, but the contribution of the reducible polynomials toÑ d (T ) is at most d 2 e=1 R|TÑ e (R)Ñ d−e T R = d 2 e=1 R|T O(R 2e−d T d−e ) = O T d− 1 2 .
Hence, we obtain that If [Q(α) : Q] = d ≥ 2 and H(α) = 1 for some α ∈Q, then α is a root of unity by Kronecker's theorem (Theorem 1.5.9 in [3]), so α ∈ A(0, d, 1) and d = φ(n) for some n ∈ N. On the other hand, if d = φ(n) for some n ∈ N, then any primitive n-th root of unity belongs to A(0, d, 1). It follows that 1 never belongs to B(d, d) if d > 1 and that 1 belongs to B(0, d) if and only if d ∈ φ(N).
d(V 1 + V −1 ) φ(H d ) H d H d 2 − O H d d− 1 2 ≤ |A(0, d, H)| ≤ d(V 1 + V −1 )H d 2 + O HB(0, d) = N 1 d \{1} if d / ∈ φ(N), N 1 d if d ∈ φ(N), and B(d, d) = N 1 d \{1} if d > 1, N 1 d if d = 1.
Let now N be a natural number that is greater than or equal to 2. We want to show that the positive real d-th root N 1 d of N belongs to B(0, d) ∩ B (d, d). For this, we define a natural number p as follows: If N = 2, we set p = 1. If N ≥ 3, then we let p ∈ N be a prime number such that p < N and p does not divide N . Such a prime number always exists: If N = 3, we set p = 2. If N ≥ 4 and no such prime number existed, then N would be divisible by the product of all prime numbers that are smaller than N . Now − 1 ≥ 5 must have a prime factor and this prime factor must be greater than or equal to N . It follows that ≤ N ≤ − 1, a contradiction.
The
Some useful lemmas
In this section, we collect some simple but useful lemmas. The first one shows that specifying the number of conjugates inside the open unit disk does not change the growth rate obtained by Masser and Vaaler in [17]. We can again apply Theorem 1.3 from [2] to the following definable family of semialgebraic sets:
Z = (w 0 , . . . , w d , T ) ∈ R d+1 × R; T ≥ 1, w 0 > 0, ∃x 1 , . . . , x d ,y 1 , . . . , y d ∈ R : x 2 j + y 2 j < 1 ∀j = 1, . . . , k, x 2 j + y 2 j ≥ 1 ∀j = k + 1, . . . , d, g j (x 1 , y 1 , . . . , x d , y d ) = 0 ∀j = 1, . . . , d, w 0 f j (x 1 , y 1 , . . . , x d , y d ) = w j ∀j = 1, . . . , d, w 2 0 d j=k+1 (x 2 j + y 2 j ) ≤ T 2 ,(3.1)
where
f j (x 1 , y 1 , . . . , x d , y d ) = (−1) j Re σ j (x 1 + √ −1y 1 , . . . , x d + √ −1y d )
and
g j (x 1 , y 1 , . . . , x d , y d ) = Im σ j (x 1 + √ −1y 1 , . . . , x d + √ −1y d )
for j = 1, . . . , d and the σ j are again the elementary symmetric polynomials in d variables.
If againZ T = π(Z ∩ (R d+1 × {T })) for the projection π : R d+1 × R → R d+1 and T ≥ 1, then it is easy to see that all coordinates of a point inZ T are bounded by some constant multiple of T , depending on d, thatZ T = T ·Z 1 , and thatZ 1 has non-empty interior. Similarly as above, N d,k (T ) := |Z T ∩Z d+1 | counts the number of polynomials P(t) ∈ Z[t] of degree d with positive leading coefficient and precisely k complex zeroes inside the open unit disk (counted with multiplicities) such that the product of the leading coefficient and the absolute values of the complex zeroes outside the open unit disk, each absolute value raised to the power of the respective zero's multiplicity, is at most T . IfÑ d,k (T ) denotes the number of such polynomials with coprime coefficients, then we have that
N d,k (T ) = ∞ n=1Ñ d,k T n .
Using another Möbius inversion and Theorem 1.3 from [2], we deduce that
N d,k (T ) = C ∞ n=1 μ(n) n d+1 T d+1 + O(T d log max{2, T })
for some constant C > 0, where C as well as the implicit constant in the O notation depend only on d and k. The proof of Lemma 2 in [17] shows that the number of reducible polynomials that we count in this way is of lower growth order. We can therefore deduce the lemma by setting T = H d .
The next lemma follows straightforwardly from Lemma 3.1.
Lemma 3.2 Let d ∈ N and k ∈ {0, . . . , d}. If the limits a(k, d) and b(k, d) both exist, then a(k, d)
+ b(k, d) = d(d + 1).
Proof If they added up to some smaller number, we would immediately obtain a contradiction with Lemma 3.1 for H big enough, so suppose they add up to some bigger number. If b(k, d) = 0, then a(k, d) > d(d + 1) and we immediately get a contradiction with Lemma 3.1 for H big enough. So we can assume that b(k, d) > 0.
We can find some
∈ (0, 1) such that (1 − )b(k, d) + (1 − )a(k, d) > d(d + 1) and then we can find δ ∈ (0, ) such that (1 − δ)b(k, d) > (1 + δ)(1 − )b(k, d) and (1 − )b(k, d) + (1 − δ)(1 − )a(k, d) > d(d + 1). For H ≥ 1 large enough, it follows from the definitions of a(k, d) and b(k, d) that H ∈B(k,d,H) |A(k, d, H )| ≥ H ∈B(k,d,H) H ≥H 1− |A(k, d, H )| ≥ H (1−δ)b(k,d) − H (1+δ)(1− )b(k,d) H (1−δ)(1− )a(k,d) . As (1 − δ)b(k, d) > (1 + δ)(1 − )b(k, d)
and δ < , the right-hand side grows asymptotically faster than
H (1− )b(k,d)+(1−δ)(1− )a(k,d) . Since (1 − )b(k, d) + (1 − δ)(1 − )a(k, d) > d(d + 1), this again contradicts Lemma 3.1.
The next lemma is the first and weakest in a series of results saying that for k ∈ {1, . . . , d− 1}, there cannot exist too many α ∈ A(k, d, H) whose Galois group is "large". Furthermore, we have
Lemma 3.3 Let d ∈ N, > 0, and k ∈ {1, . . . , d−1}. There exists a constant C = C(k, d, ) such that |{α ∈ A(d − k, d,|A(k, d, H)| ≤ CH for all H ≥ 1 with [Q(H d ) : Q] = d k . Proof Let α ∈ A(d − k, d, H)
and assume either that the Galois group of the normal closure of Q(α) acts transitively on the k-element subsets of the set of conjugates of α
or that [Q(H d ) : Q] = d d−k . Now, for such an α we have H d = H(α) d = ±aα 1 · · · α k , where a > 0 is the leading coefficient of a minimal polynomial of α in Z[t]
and α 1 , . . . , α k are the conjugates of α that do not lie inside the open unit disk. By assumption, we have 0 < k < d. We can assume without loss of generality that α = α 1 since α 1 determines α up to d possibilities. Now note that
aα k = aα k 1 = a k j=1 α j k i=2 aα k+1 k j=1 j =i α j a k+1 j=2 α j k−1 ,
where α k+1 is a conjugate of α, distinct from the α j (j = 1, . . . , k) (here we use that k < d).
The numerator and denominator of the right-hand side are products of conjugates of ±H d by our assumption on either the Galois group of the normal closure of Q(α) or the degree of H d . So aα k is determined by H up to finitely many possibilities (bounded in terms of only d and k), so it can be assumed fixed. The same holds for aα k j for all j = 1, . . . , d by conjugating. And aα k together with a determines α up to k possibilities (here we need that k > 0), so it remains to bound the number of possibilities for a.
But a d−k |b| k = d j=1 a|α j | k is already determined up to finitely many possibilities
(bounded independently of H), where b is the constant coefficient of a minimal polynomial of α in Z[t]
, and a has to divide this natural number as k < d. Since | d j=1 aα k j | ≤ H d 2 , it follows from well-known bounds for the divisor function that there are at most C (d, )H possibilities for a.
The next two lemmas contain general facts from algebraic number theory that will be useful at several places in this article. Proof Let v be a finite place of Q(α 1 , . . . , α d ) and | · | v an associated absolute value. We have
a s∈S α s v ≤ |a| v d i=1 max{1, |α i | v }.
From the Gauss lemma (Lemma 1.6.3 in [3]) and the definition of a, we deduce that
|a| v d i=1 max{1, |α i | v } = 1.
As v was arbitrary, the lemma follows.
Lemma 3.5 Let K ⊂Q be a number field, N ∈ Z, H ≥ 1, and > 0. Set D = [K : Q].
There exists a constant C = C(D, ) such that
|{α ∈ O K ; N K /Q (α) = N, H(α) ≤ H}| ≤ CH .
Lemma 3.5 essentially follows from the proof of Proposition 2.5 in [5]. For the reader's convenience, we reproduce the proof here.
Proof We can assume without loss of generality that 0 < |N | ≤ H D since otherwise the set whose cardinality we wish to bound has at most one element.
Let U K denote the group of algebraic units in K . We call two elements of K \{0} associate if their quotient belongs to U K . It follows from [4], pp. 219-220, our bound for |N | in terms of H, and elementary bounds for the divisor function that the number of pairwise non-associate elements of
O K \{0} with K /Q-norm N is bounded by C (D, )H 2 .
Hence we can assume that α = α 0 ξ for some ξ ∈ U K and fixed α 0 ∈ O K with N K /Q (α) = N K /Q (α 0 ) = N and max{H(α), H(α 0 )} ≤ H. It follows that H(ξ ) ≤ H 2 . We want to bound the number of possibilities for ξ .
Let σ i : K → C denote the distinct embeddings of K in C (i = 1, . . . , D) and set
ν(η) = (log |σ 1 (η)|, . . . , log |σ D (η)|)
for η ∈ U K . Then ν is a group homomorphism from the multiplicative group U K to the additive group R D and its image ν(
U K ) ⊂ R D is a discrete free Z-module. Since H(ξ ) ≤ H 2 , we have that ν(ξ ) belongs to the cube [−2D log H, 2D log H] D .
This cube can be covered by at most C (D, )H 2 translates of the unit cube [0, 1] D . Since ν is a group homomorphism, it therefore suffices to show that the number of
η ∈ U K with ν(η) ∈ [−1, 1] D is bounded by a constant depending only on D. For η ∈ U K define P η (t) = D i=1 (t − σ i (η)) ∈ Z[t] so that P η (η) = 0. If ν(η) ∈ [−1, 1] D ,
then the absolute value of each coefficient of P η is bounded by exp(D(1 + log 2)). This completes the proof of the lemma.
The case k ∈ {1, d − 1} or d prime
In this section, we completely resolve the cases where k ∈ {1, d − 1} or d is prime. We also determine b(k, d) for all k and d. Although many of the results in this section will be superseded by Theorem 5.2, we have included them because they can be proved in a different, somewhat easier way.
(i) a(1, d) = a(d − 1, d) = 0 if d ≥ 2, (ii) b(k, d) = d(d + 1) if d ≥ 2 and 0 < k < d, and (iii) a(k, d) = 0 if d is prime and 0 < k < d.
(In particular, all these limits exist.)
Theorem 4.1(ii) implies together with Lemma 3.2 that for d ≥ 2 and 0 < k < d, a(k, d) must be equal to 0 if it exists.
Proof (i) This follows from Lemma 3.3 as the Galois group of the normal closure of Q(α) always acts transitively on the 1-element and the (d − 1)-element subsets of the set of conjugates of α.
(ii) It follows from Lemma 3.
1 that |B(k, d, H)| = O H d(d+1) for H ≥ 1. If the equality in (ii) is false or the limit b(k, d) does not exist, it follows that there is some > 0 such that there exist arbitrarily large H ≥ 1 such that |B(k, d, H)| ≤ H d(d+1)− . Lemma 3.3 implies that for H ∈ [1, ∞), the number of α ∈ A(k, d, H )
with Galois group isomorphic to the full symmetric group S d is bounded from above by CH 2 for some constant C = C(k, d, ). Furthermore, the number of α ∈ H ≤H A(k, d, H ) with Galois group not isomorphic to the full symmetric group is of growth order o H d(d+1) (see [27]). But by Lemma 3.1, the number of α of degree d with precisely k conjugates inside the open unit disk and height at most H grows asymptotically like some constant positive multiple of H d(d+1) , which yields a contradiction for H large enough.
(iii) We follow a similar strategy as in the proof of Lemma 3.3. Let d be a prime number, 0 < k < d, H ∈ [1, ∞), and α ∈ A(k, d, H). The Galois group of the normal closure of Q(α) must contain an element of order d since d is prime and the Galois group acts transitively on the d-element set of conjugates of α. Since d is prime, such an element of order d must act as a d-cycle on the conjugates of α. If these conjugates are α 1 , . . . , α d , we can assume without loss of generality that this d-cycle acts on them by acting on the indices as (
12 · · · (d − 1)d). We have H d = H(α) d = ±a i∈I α i for some I ⊂ {1, . . . , d} with |I| = d − k and a ∈ N the leading coefficient of a minimal polynomial of α in Z[t].
We aim to write some l-th power of aα d−k 1 as a quotient of products of conjugates of ±H d , where l ∈ N and the number of conjugates that appear are bounded in terms of k and d only. Once this is achieved, we can conclude as in the proof of Lemma 3.3.
To
a (formal) product d i=1 α e i i with e i ∈ Z we associate a vector (e 1 , . . . , e d ) ∈ Z d . Let v ∈ Z d be the vector associated to i∈I α i . Consider the Z-module generated by A i v (i = 0, . . . , d − 1)
, where A is a permutation matrix corresponding to the cycle (12 · · · d).
If finite (which we will later prove it to be), the index [Z d : ] can be bounded by (d − k) d 2 through an application of Hadamard's determinant inequality.
Assuming for the moment that [Z d : ] < ∞, we deduce that (n, 0, . . . , 0) ∈ for some natural number n ≤ (d − k)
l ≤ (d − k) d 2 −1 ,
where the number of conjugates that appears is bounded in terms of k and d only as we wanted.
It remains to prove that [Z d : ] < ∞. Equivalently, we can show that the vector
subspace V of C d generated by the A i v (i = 0, . . . , d − 1) has dimension d. Over C, the matrix A is diagonalizable and we have C d = d−1 i=0 W ζ i , where ζ is a primitive d-th root of unity and W λ = {w ∈ C d ; Aw = λw} (λ ∈ C). The vector subspace V is A-invariant and so V = d−1 i=0 (V ∩ W ζ i ). It cannot be contained in W 1 since Av = v (here we use that 0 < k < d), so there exists some j ∈ {1, . . . , d − 1} with V ∩ W ζ j = {0}. As dim W ζ i = 1 for all i, it follows that W ζ j ⊂ V . Since V is defined over Q, it follows by conjugating that d−1 i=1 W ζ i ⊂ V . But 0 = d−1 i=0 A i v ∈ V ∩ W 1 , so W 1 ⊂ V as well. It follows that V = d−1 i=0 W ζ i = C d .
By adapting the proof of Theorem 4.1(iii), we can now strengthen Lemma 3.3. . The Galois group of the normal closure of Q(α) can be identified with a subgroup G of the symmetric group S d . To a (formal) product d i=1 α e i i with e i ∈ Z we again associate a vector (e 1 , . . . , e d ) ∈ Z d . The group G then acts on Q d by permuting the coordinates. We will denote the vector associated to i∈I α i by v. As we have seen in the proof of Theorem 4.1(iii), it suffices to show that the vector space V generated over Q by the gv for g ∈ G must be Q d in order to prove the lemma.
Lemma 4.2 Let d ∈ N, > 0, and k ∈ {1, . . . , d−1}. There exists a constant C = C(k, d, ) such that |{α ∈ A(k, d,
Certainly, this vector space is G-invariant. Since G acts 2-transitively, we know that there are only 4 G-invariant vector subspaces of Q d , i.e. {0}, Q(1, 1, 1, . . . , 1),
Q(1, −1, 0, . . . , 0) ⊕ Q(0, 1, −1, 0, . . . , 0) ⊕ · · · ⊕ Q(0, . . . , 0, 1, −1),
and Q d (see [24], Exercise 2.6). We can immediately exclude the first two since neither of them contains the vector v. Furthermore, the vector g∈G gv is non-zero and lies in Q (1, 1, 1, . . . , 1), so we can also exclude the third one. It follows that V = Q d and we are done.
The next theorem shows that the height function together with the degree is in some sense "almost injective" if the degree is at least 2. By Theorem 4.1(i) each summand here is bounded by CH 2 for some C = C(d, ) and we are done.
The case gcd(k, d) = 1
In this section, we prove Theorem 5.2, which will give a useful unconditional upper bound for |A(k, d, H)|. Theorem 5.2 also provides a further strengthening of Lemmas 3.3 and 4.2. We first prove an auxiliary lemma that will also be useful later.
α s < |α 1 · · · α d−k | for every subset S of {1, . . . , d} of cardinality d − k that is not equal to {1, . . . , d − k}.
We deduce that the coefficients of the polynomial d−k i=1 (t − α i ) belong to Q(H d ). In order to prove the first part of the lemma, we will make use of the following simple facts: If K 2 /K 1 is a finite Galois extension of fields of characteristic 0 within a fixed algebraic closure K 1 and ξ ∈ K 1 , then [K 2 (ξ ) :
K 2 ] divides [K 1 (ξ ) : K 1 ]. Furthermore, if η is a conjugate of ξ over K 1 , then [K 2 (η) : K 2 ] = [K 2 (ξ ) : K 2 ].
We In particular, a(k, d) = 0 if gcd(k, d) = 1.
We will see later that the exponent in the bound for |A(k, d, H)| is indeed sharp for every choice of (k, d). Let us also note at this stage that one might hope a priori to prove that a(k, d) = 0 for all d ∈ N and k ∈ {1, . . . , d − 1} with gcd(k, d) = 1 by showing the following: For any transitive subgroup G of the symmetric group S d and any vector v ∈ Q d with exactly k entries equal to 1 and d − k entries equal to 0, the set Gv generates Q d . Unfortunately, this statement is wrong. One can construct a counterexample with G equal to the subgroup generated by the d-cycle (12 · · · d) from any counterexample to the following statement: Any sum of k distinct d-th roots of unity is non-zero. If we denote e 2π √ −1 n by ζ n for n ∈ N, then a construction by Rédei (see [19], Satz 9) yields counterexamples like
0 = (−1) + (−1)(−1) = ζ 2 + 2 i=1 ζ i 3 ⎛ ⎝ 6 j=1 ζ j 7 ⎞ ⎠ ,
where the right-hand side is a sum of 13 distinct 42-nd roots of unity. If G is a 2-transitive subgroup of S d , then it follows from the proof of Lemma 4.2 that the statement is correct.
We see that Theorem 5.2(i) yields an upper bound with exponent as soon as the normal closure of Q(H d ) contains α. If we restrict ourselves to α such that Q(α) is Galois over Q, we can for example obtain such a bound as soon as [Q(H d ) : Q] = d. In Theorem 6.1, we will see another case where Theorem 5.2(i) can be applied with l = 1. A(k, d, H). Let K be the normal closure of Q(H d ) and set l = [K (α) : K ]. By Lemma 5.1, l divides gcd(k, d). Thus, part (ii) of the theorem directly follows from part (i), after adjusting the constant C. Since l ≤ gcd(k, d) < d, we must have l = 1 if Gal(Q/Q) acts primitively on the set of conjugates of α, so part (iii) also follows from part (i).
Proof of Theorem 5.2 Let H ∈Q ∩ [1, ∞) and let α ∈
We now fix l and prove part (i): Let α 1 , . . . , α d be the conjugates of α, numbered so that
|α i | ≥ 1 if and only if 1 ≤ i ≤ d − k.
Let a be the (non-zero) leading coefficient of a minimal polynomial of α in Z[t], chosen such that H(α) d = aα 1 · · · α d−k . It follows from Lemma 5.1 that
N Q(H d )/Q (H d ) = a [Q(H d ):Q]N Q(H d )/Q (H d ) = a [Q(H d ):Q] (α 1 · · · α d ) 1− k d [Q(H d ):Q] . (5.1) In particular, d divides (d − k)[Q(H d ) : Q]. Since k > 0 and aα 1 · · · α d ∈ Z, we have that a divides N Q(H d )/Q (H d ) in Z.
So the number of possibilities for a is bounded by C 1 H 3 for some constant C 1 , depending only on d and . Hence we can assume that a ∈ Z\{0} is fixed.
Let I ⊂ {α 1 , . . . , α d } be the subset of conjugates of α over K (of cardinality l). For j ∈ {1, . . . , l}, we set γ j = a J ⊂I,|J |=j β∈J β.
All the γ j lie in the fixed number field K that is determined uniquely by H and d. We The algebraic integers γ j (j = 1, . . . , l) lie in the given number field K of degree at most d! and their height is bounded by C 4 H d , where C 4 depends only on d and k. It therefore follows from Lemma 3.5 that the number of possibilities for each of them, if their K /Qnorm is fixed, is bounded by C 5 H 3l , where C 5 depends only on d, k, and . Part (i) of the theorem now follows since aα l + l j=1 (−1) j γ j α l−j = 0 and so α is determined up to Galois conjugation by l, a, and the γ j (j = 1, . . . , l).
The case (k, d) = (2, 4)
One might be tempted to conjecture that a(k, d) = 0 for all d ≥ 2 and 0 < k < d, but this is not true. We begin our investigations by studying the simplest non-trivial case, namely (k, d) = (2, 4). In this case, there are three possibilities for [Q(H 4 ) : Q], namely 2, 4, or 6. In the last case, we can apply Lemma 3.3 to obtain that |A(2, 4, H)| grows more slowly than H for every > 0. We now show in the next theorem that the same holds in the middle case, where [Q(H 4 ) : Q] = 4. In the first case, i.e. if the normal closure of Q(α) is Q(α), K coincides with the normal closure of Q(α) as both are equal to Q(α).
In the second case, i.e. if the normal closure of Q(α) is a number field of degree 8, the Galois group of the normal closure of Q(α) is isomorphic to the dihedral group D 4 and Q(H 4 ) is a quartic subfield of that normal closure. If K is not equal to the normal closure of Q(α), then the extension Q(H 4 )/Q is Galois. Suppose now that the conjugates of α are the α i (i = 1, . . . , 4) and that the Galois group is generated by field automorphisms acting on the conjugates α i by acting on their indices as the cycle (1234) and the transposition (13). Since [Q(H 4 ) : Q] = 4, we can assume after a cyclic renumbering that H 4 = ±aα 1 α 2 , where a ∈ N is the leading coefficient of a minimal polynomial of α in Z[t]. The only subfield of the normal closure of Q(α) of degree 4 that is Galois over Q corresponds under the Galois correspondence to the cyclic normal subgroup of D 4 generated by (13) (24). But this element does not fix H 4 since |α 1 α 2 | ≥ 1 > |α 3 α 4 |. So Q(H 4 ) cannot be Galois over Q and it follows also in this case that K is equal to the normal closure of Q(α).
The theorem now follows from Theorem 5.2(i) with l = 1.
In the case where [Q(H 4 ) : Q] = 2, it follows from Theorem 5.2 that we have |A(2, 4, H)| ≤ C( )H 4+ for all such H. However, the next theorem shows that one cannot always expect this growth and in fact one cannot obtain a uniform growth rate in H even after partitioning A(2, 4) into an arbitrary finite number of subsets. In Sect. 7, we will prove that |A(2, Proof Let κ ∈ [0, 4]. We fix m ∈ N prime with m ≡ 1 mod 4 and denote its positive square root by √ m. We define u 1 + u 2 √ m = u 1 − u 2 √ m (u 1 , u 2 ∈ Q). If κ < 4, we apply a theorem of Chebyshev [6] (Bertrand's postulate) to find a prime number b 2 ∈ N such that
m κ 8−2κ ≤ |b 2 | ≤ 2m κ 8−2κ . (6.1) We then set β = b 1 + b 2 √ m, where b 1 ∈ {[b 2 √ m], [b 2 √ m] + 1} is not divisible by b 2 . After maybe replacing β, b 1 , b 2 by −β, −b 1 , −b 2 ,
which preserves (6.1), we can assume that 0 < β < 1.
If κ = 4, we take m = 2 and β = (3 + 2 √ 2) r for some r ∈ N. The integers b 1 , b 2 are then defined by β = b 1 + b 2 √ 2. We automatically have that 0 < β < 1. If κ > 0, we assume that
|β| ≥ 4 √ m + 8 (6.2)
by choosing m or r sufficiently large. If κ = 0, we assume that m ≥ 5. We set H = |β| 1 4 . We record that
H 4 = |β| ≤ 3 √ m|b 2 | ≤ 6m 1 2 + κ 8−2κ = 6m 4 2(4−κ) (κ < 4) (6.3)
as well as A(2, 4, H). Let α 1 , . . . , α 4 be the conjugates of α, ordered such that |α 1 |, |α 2 | ≥ 1, and let a > 0 be the leading coefficient of a minimal polynomial of α in Z[t]. It follows that β = ±H 4 ∈ {±aα 1 α 2 }.
Let F be the fixed field of the stabilizer H of {α 1 , α 2 } in the Galois group G of the normal closure of Q(α). By Lemma 5.1, every σ ∈ G which fixes β must lie in H. Since the converse implication holds trivially, it follows that F = Q(β) and β ∈ {±aα 3 α 4 }. We deduce from Lemma 3.4 that a divides a 2 4 j=1 α j = ββ in Z. Since |β| < 1, it follows from well-known bounds for the divisor function that the number of possibilities for a is bounded by C 1 H 4 for a certain constant C 1 that depends only on .
From now on, we assume that a is fixed and count the number of possibilities for α. We have aα 2 1 − γ α 1 ± β = 0, where γ = a(α 1 + α 2 ). For a given α 1 , there are exactly four possible α. From now on, we assume that α = α 1 . It then suffices to bound the number of possibilities for γ . Now γ lies in F , so γ ∈ Q(β). Furthermore, we have that γ = a(α 1 + α 2 ) ∈ Q(β) is an algebraic integer by Lemma 3.4. Since m ≡ 1 mod 4, we have γ ∈ Z + Z √ m, so c 2 , a),c 1 =ã −1 c 1 , andc 2 =ã −1 c 2 . By the usual bound for the divisor function, the number of possibilities forã is bounded from above by C 2 a 16 ≤ C 2 H 4 with a constant C 2 that depends only on . In the following, we assume thatã is fixed.
γ = c 1 + c 2 √ m for some c 1 , c 2 ∈ Z. Letã = gcd(c 1 ,
As γ = a(α 3 + α 4 ), the integer c 2 1 − mc 2 2 = γ γ is divisible by a thanks to Lemma 3.4. It follows thatã 2 gcd(a,ã 2 ) −1 (c 2 1 −mc 2 2 ) is divisible by a = a gcd(a,ã 2 ) −1 . Asã 2 gcd(a,ã 2 ) −1 and a are coprime, we deduce thatc 2 1 − mc 2 2 is divisible by a . By construction, we have that gcd(c 1 ,c 2 , a ) = 1. It follows thatc 2 = 0 unless a = 1.
Furthermore, we know that
|c 2 | =ã −1 |c 2 | ≤ |γ | + |γ | 2ã √ m ≤ 2|β| + 2a 2ã √ m ≤ 2|β| a √ m
since |α 3 |, |α 4 | < 1, |α 1 |, |α 2 | ≥ 1, and a|α 1 α 2 | = H 4 = |β|. Thanks to (6.3) and (6.4), it follows that (6.5) at least if κ < 4. If κ = 4, the same follows from |β| = H 4 and √ 2 ≤ 12. For a givenc 2 , we have to bound the number ofc 1 ∈ Z such that |c 1 −c 2 √ m| =ã −1 |γ | < 2aã −1 andc 2 1 − mc 2 2 ≡ 0 mod a . Setm = gcd(m, a ), thenm is squarefree and must dividec 1 . Furthermore,m is uniquely determined by m, a, andã, so we can assume it fixed. We set c 1 =c 1m −1 , m = mm −1 , and a = a m −1 . It follows thatmc 2 1 ≡ m c 2 2 mod a . By construction, we have gcd(m , a ) = 1. We also have gcd(c 2 2 , a ) = 1 since a common prime divisor of a andc 2 would have to divide a and thereforec 1 , but gcd(c 1 ,c 2 , a ) = 1. It follows that gcd(mc 2 1 , a ) = 1 as well. The number of square roots modulo a of a number coprime to a is bounded by 2 s+1 , where s is the number of distinct prime factors of a . The number of c 1 satisfying |c 1 − c 2 √ mm −1 | =m −1 |c 1 −c 2 √ m| < 2a(ãm) −1 that lie in a given congruence class modulo a is at most 4 gcd(a,ã 2 )ã −1 since gcd(a,ã 2 )ã −1 is a natural number and a gcd(a,ã 2 )ã −1 = a(ãm) −1 . It follows that the number of c 1 for a givenc 2 is at most 2 s+3 gcd(a,ã 2 )ã −1 . If a ≥ 3, we have s < 7 5 log a log log a by Théorème 11 in [20]. As the function x → log x log log x is strictly monotonically increasing for x ≥ 16, all natural numbers less than 16 have at most 2 distinct prime factors, and a ≤ a ≤ H 4 , we have s ≤ max 2, 28 5 log H log log max{3, H} .
|c 2 | ≤ 12 m 4 2(4−κ) a √ m = 12 m κ 2(4−κ) a ≤ 12 H κ a ,
Recall thatc 2 can only be 0 if a = 1, in which case gcd(a,ã 2 )ã −1 = aã −1 ≤ a 1 2 . Thanks to (6.5) and the above, the number of possibilities for the pair (c 1 ,c 2 ) is then bounded from above by 24 H κ a gcd(a,ã 2 )ã −1 + a If κ < 2, we have to study more closely the case thatc 2 = 0. We use that a is the leading coefficient of a minimal polynomial of α in Z[t]. Ifc 2 = 0 and γ = c 1 , we can therefore conclude that a divides all coefficients of the polynomial
(at 2 − a(α 1 + α 2 )t + aα 1 α 2 )(at 2 − a(α 3 + α 4 )t + aα 3 α 4 ) = (at 2 − c 1 t ± β)(at 2 − c 1 t ± β) ∈ Z[t].
Here, the sign of β is the same as that of β. In particular, a divides c 1 (β + β) = 2b 1 c 1 as well as
ββ = b 2 1 − mb 2 2 . Set δ = gcd(b 1 , b 2 1 − mb 2 2 ) = gcd(b 1 , mb 2 2 )
. It follows from (6.1) that 0 < |b 1 | ≤ |b 2 | √ m + 1 ≤ 2m 4 2(4−κ) + 1. As κ < 2, this implies together with (6.3) that 0 < |b 1 | < m for H ≥ H 1 = H 1 (κ). We assume from now on that H ≥ H 1 . Since m is prime, it then follows that δ = gcd(b 1 , b 2 2 ). But b 2 is prime and does not divide b 1 , so δ = 1. Since any common divisor of a and b 1 must also divide δ, it follows that gcd(a, b 1 ) = 1.
We deduce that c 1 must be divisible by a gcd(a, 2) −1 . Since |c 1 | = |γ | < 2a, there are at most 8 possibilities for c 1 .
Putting everything together, we obtain that the number of possibilities for α is bounded by (2,4). Since H( √ β) = H, the lower bound holds with c = 1. We now assume that κ > 0. We choose γ = c 1 + c 2 √ m with 2 4 − β > 0 and an arbitrary complex square root otherwise. It follows that α is an algebraic integer of degree dividing 4. Note that α = 0 and γ = β+α 2 α is uniquely determined by α.
2 · 4 · C 1 H 4 · C 2 H 4 · 25H κ · 8 · max{4,c 2 ∈ 1, . . . , |β| 2 √ m − 2 √ m and c 1 = [c 2 √ m] + 1. It follows that 0 < γ ≤ 2 √ mc 2 + 1 ≤ |β| − 3. (6.6) We set α = γ 2 + γ 2 4 − β, so α 2 − γ α + β = 0, where γ 2 4 − β denotes the positive square root if γ
We begin by controlling the cases where [Q(α) : Q] < 4. If α were a rational integer, then α would be a common divisor of b 1 and b 2 . As b 1 and b 2 are coprime by construction, it would follow that α = ±1 and therefore
|γ | = β + α 2 α = |β + 1| ≥ |β| − 1.
This contradicts (6.6). So α cannot be a rational integer. √ m] + 1 − c 2 √ m < 1 and 0 < β < 1 together with (6.6) and fundamental properties of the height, we can bound the height of γ ± δ from above by
2H(γ )H(δ) = 2H(γ )H(γ 2 − 4β) 1 2 ≤ 4 √ 2H(γ ) 2 H(β) = 4 √ 2|γ ||β| 1 2 ≤ 4 √ 2|β| 3 2 .
If we write η = ηζ u l , where η and η are two possible values for γ + δ, then it follows that h Hence there are at most C 5 log |β| possibilities for the unit and hence for γ +δ, where C 5 is an absolute constant. Now γ +δ determines γ −δ since (γ +δ)(γ −δ) = 4β and β is fixed. And γ + δ together with γ − δ determines γ , so there are at most C 5 log |β| possibilities for γ as well. It follows that α is quadratic for at most C 6 |β| κ 8 = C 6 H κ 2 choices of γ , where C 6 = C 6 (κ) depends only on κ.
Summarizing, we find that α has degree < 4 for at most C 6 H κ 2 choices of γ .
If α has degree 4 over Q, which we from now on assume, its conjugates are γ 2 ± γ 2 4 − β and γ 2 ± γ 2 4 − β, where γ 2 4 − β also denotes the positive square root if γ 2 4 − β > 0 and an arbitrary complex square root otherwise.
If β > 0 and γ 2 4 < |β|, then
γ 2 ± γ 2 4 − β = |β| 1 2 > 1.
If β > 0 and γ 2 4 ≥ |β|, we have
γ 2 ± γ 2 4 − β ≥ γ 2 − γ 2 4 − β > 1,
since γ < |β| + 1 and γ ≥ 2|β|
1 2 ≥ 2. If β < 0, we have γ 2 ± γ 2 4 − β ≥ γ 2 4 + |β| − γ 2 > 1, since γ < |β| − 1.
Recall that β > 0. If |γ | < 2β 1 2 , then γ 2 4 − β is purely imaginary and
γ 2 ± γ 2 4 − β = β 1 2 < 1.
Otherwise, we have
γ 2 ± γ 2 4 − β ≤ γ 2 + γ 2 = |γ | = [c 2 √ m] + 1 − c 2 √ m < 1.
So in any case, α has two conjugates inside and two conjugates outside the open unit disk. Finally, we can compute that
H(α) 4 = ⎛ ⎝ γ 2 + γ 2 4 − β ⎞ ⎠ ⎛ ⎝ γ 2 − γ 2 4 − β ⎞ ⎠ = |β|,
so H(α) = |β| 1 4 = H. Thanks to (6.2), the number of choices for γ can be estimated as
|β| 2 √ m − 2 √ m ≥ |β| − 2 √ m − 4 2 √ m ≥ |β| 4 √ m = H 4 4 √ m .
We can then use (6.4) to deduce that the number of choices for γ is equal to at least H κ 4 if κ < 4. If κ = 4, we get that the number of choices for γ is equal to at least H κ
The case gcd(k, d) > 1
In fact, the situation is even worse than Theorem 6. Before the proof, we make some remarks on this theorem: The number of possibilities for L given K is bounded in terms of d and k. As β is a product of l conjugates of α, we can bound its height by H l . Since K /Q is a Galois extension, we have that [K (α) : K ] = [Q(α) : L] = l for any α ∈ A L,β (k, d, H). An upper bound for |A L,β (k, d, H)| of the same growth order as (7.2) (up to H 2 ) is therefore provided by Theorem 5.2(i). However, the following examples show that it is not possible in general to prove the lower bound (7.2) with C depending on k, d, δ, and , but not on K , or with C depending on k, d, K , and , but not on δ. For reasons of space, we grudgingly leave it to the reader to work out the details in the examples.
The necessity of the dependence on K is shown by the following example: The necessity of the dependence on δ is shown by the following example:
b √ 2 − 1 < a ≤ b √ 2 + 1.
We deduce that a = a. This implies that α satisfies an equation
aα 2 + γ α ± (1 + b √ 2) = 0 with γ ∈ O K . Since N (I) = a for I = aO K + (1 + b √ 2)O K ,
we must have γ ∈ I. Let γ denote the image of γ under the non-trivial field automorphism of K . Then one can show that max{|γ |, |γ |} ≤
2(1 + b √ 2) while min{|γ |, |γ |} ≤ max{1 + b √ 2 − a, a − b √ 2 + 1} ≤ 2.
Applying Theorem 2.1 in [30] with S = ((1, 1), (1, 1)) and C = {(0, 0)} to the image of I under a Minkowski embedding (cf. the proof of Lemma 7.4 below and note that the K /Q-norm of every element of I is divisible by a) shows that the number of such γ is bounded independently of (a, b), but H → ∞ as b → ∞.
We now prove Theorem 7.1.
Proof of Theorem 7. 1 We first prove ( We can also deduce that |σ (β)| < 1 for precisely k l embeddings σ : L → C. Recall that l = d[L : Q] −1 . We can and will assume without loss of generality that l ≥ 2. For γ = (γ 1 , . . . , γ l−1 ) ∈ I l−1 , define the polynomials
P γ (t) = at l + γ l−1 t l−1 + · · · + γ 1 t + (−1) l aβ ∈ O L [t]
and Q γ = σ :L →C σ (P γ ) ∈ Z[t].
Suppose that γ satisfies the following:
(1) I is generated by a, γ 1 , and aβ, It follows from (1), (7.3), and the Gauss lemma (Lemma 1.6.3 in [3]
(2) L = Q(γ 1 ),(3)) that Q γ = a [L:Q]−1 Q γ , where Q γ ∈ Z[t] is primitive with leading coefficient a.
If |σ (β)| < 1 for σ : L → C and α γ ,σ is some complex zero of σ (P γ ), then it follows from (3) that
a|α γ ,σ | l ≤ ⎛ ⎝ a|σ (β)| + l−1 i=1 |σ (γ i )| ⎞ ⎠ max{1, |α γ ,σ |} l−1 < a(|σ (β)| + δ) max{1, |α γ ,σ |} l−1 . (7.5)
As |σ (β)| ≤ 1 − δ by our hypothesis, this implies that |α γ ,σ | < 1.
If |σ (β)| ≥ 1 for σ : L → C and α γ ,σ is some complex zero of σ (P γ ), then it follows from (4) that
a|σ (β)| ≤ ⎛ ⎝ a + l−1 i=1 |σ (γ i )| ⎞ ⎠ max{1, |α γ ,σ |} l < a(1 + δ|σ (β)|) max{1, |α γ ,σ |} l . (7.6)
As |σ (β)| ≥ (1 − δ) −1 by our hypothesis, this implies that |α γ ,σ | ≥ 1.
It follows from (2) that the σ (P γ ) for σ : L → C are pairwise distinct. Together with (5) and the fact that K /Q is Galois, this implies that Q γ is irreducible in Q[t] and therefore in Z[t].
Let α γ be a complex zero of P γ . It follows that (5), we must have L = Q(α γ ) ∩ K . We also have that N Q(α γ )/L (α γ ) = β so that α γ ∈ A L,β (k, d, H). Since γ is uniquely determined by α γ , a, and L, we have reduced the proof of (7.2) to proving the following Lemma 7.4: Proof We will use c 1 , c 2 , . . . for positive constants that depend only on k, d, K , δ, and .
Recall that we have assumed that l ≥ 2. We can assume without loss of generality that < 1 2 . We want to use Theorem 2.1 in [30]. Let r and s denote the number of real embeddings and pairs of complex conjugate embeddings of L respectively. For each pair of complex conjugate embeddings of L, we choose one element of the pair. Furthermore, we order both the real embeddings of L and the pairs of complex conjugate embeddings of L in fixed ways each. Let denote the image of I inside R r × C s under the thus obtained Minkowski embedding. We identify C with R 2 by identifying (v, w) ∈ R 2 with v + w √ −1 ∈ C and we identify each γ i with its image in (i = 1, . . . , l − 1). In the following, we use the notation of [30]: We set n = r + s, N = [L : Q], C = {0} ⊂ R N , m j = β j = 1 for 1 ≤ j ≤ r, and m j = β j = 2 for r + 1 ≤ j ≤ r + s. For each j ∈ {1, . . . , n}, let σ j : L → C denote the associated embedding used to define the Minkowski embedding.
We have Nm β ( ) ≥ a for μ( , B), where B > 0 is arbitrary. We set Q j = aδ l if |σ j (β)| < 1 and Q j = aδ|σ j (β)| l otherwise (j = 1, . . . , n). Conditions (3) and (4) for a fixed i define a product Z Q of intervals and disks satisfying conditions (1) and (2) on p. 480 of [30] for our choice of Q j , y j = 0 ∈ R m j , and (κ, M) = (8N 5/2 , 1) (cf. [30], p. 479).
Thanks to ( [30] then yields a main term which is greater than or equal to c 2 H d for the number of γ i satisfying (3) and (4), for fixed i. Choosing B = Q max and using our lower bound for μ( , B), we find that the corresponding error term is bounded from above by
c 3 H d−l .
We turn to (1). The ideal I = aO L + aβO L is contained in I and [I : I ] divides [I : (1) is not satisfied, then γ 1 is contained in IP for some prime ideal P such that IP divides I . Note that N (P) then divides a. Applying Theorem 2.1 in [30] as above to each ideal IQ instead of I with Q a product of pairwise distinct such P and then using the inclusion-exclusion principle, we see that imposing (1) means that the main term gets multiplied by a factor while the error term gets multiplied by 2 u , where u is the number of possibilities for P.
aO L ] = a [L:Q] N (I) −1 = a. If
As a ≤ H d , the factor in the main term can be bounded from below by c 4 H − while u is bounded from above by log H + c 5 thanks to Théorème 11 in [20]. We next consider (2). If (2) is not satisfied, then σ (γ 1 ) = σ (γ 1 ) for two distinct embeddings σ , σ of L in C and so γ 1 lies in a lower-dimensional linear subspace of R N , obtained by equating two coordinates or setting a coordinate equal to 0. The intersection of such a subspace with Z Q is a bounded convex set of volume 0 that is contained in Z Q and so Theorem 2.1 in [30] shows that the number of such γ 1 can be absorbed into the error term.
It remains to be shown that the number of γ which satisfy conditions (1) to (4), but not (5) is of lower growth order than H d(l−1)− . LetP be some monic irreducible factor of P γ in K [t]. Setl = degP. We definep 1 , . . . ,p˜l ∈ K by aP(t) = at˜l +p 1 t˜l −1 + · · · +p˜l and set K = Q(p 1 , . . . ,p˜l). SinceP is irreducible in K [t],P divides Q γ ∈ Z[t], and K /Q is Galois,
we deduce that σ :K →C σ (P) is irreducible in Q[t] and divides Q γ ∈ Z[t].
LetQ be a minimal polynomial in Z[t] of some complex zero ofP and letã denote the leading coefficient ofQ, thenQ =ã σ :K →C σ (P). SinceQ is primitive and divides Q γ in Sinceã divides a, it follows from Lemma 3.4 thatq i ∈ Z. SinceQ divides Q γ , we have |q i | ≤ c 6 a Q γ (ζ )=0 max{1, |ζ |}. Thanks to (7.4), (7.5), and (7.6), this implies that
Q[t],|q i | ≤ c 6 a σ :L →C |σ (β)|≥1 |σ (β)| = c 6 H d . (7.8)
SinceQ divides Q γ , we can also use (7.4), (7.5), and (7.6) to estimate
H(p i ) ≤ a ⎛ ⎝ σ ∈Gal(K /Q) max 1, |σ (p i )| a ⎞ ⎠ 1 [K :Q] ≤ c 7 a Q γ (ζ )=0 max{1, |ζ |} ≤ c 7 a σ :L →C |σ (β)|≥1 |σ (β)| = c 7 H d (7.9) for i = 1, . . . ,l.
It then follows from (7.7), (7.8), (7.9), and Lemma 3.5 that the number of possibilities for p i ∈ O K is bounded from above by c 8 H d+ l (i = 1, . . . ,l). The leading coefficient of aP is of course always equal to a. Furthermore, suppose that P γ = aP 1 · · ·P m withP 1 =P and allP i monic and irreducible in K [t]. The above argument forP shows that aP i ∈ O K [t] for all i. Since a m−1 P γ = (aP 1 ) · · · (aP m ), we deduce thatp˜l divides a m β in O K . As m ≤ l, it follows that N K /Q (p˜l) divides a [K :Q](l−1) N K /Q (aβ) in Z. Lemma 3.5 then shows together with (7.4), (7.9), and elementary bounds for the divisor function that there are at most c 9 H l possibilities forp˜l.
This implies that the number of possibilities forP is bounded from above by c 10 H d(l−1)+ l l . If γ satisfies conditions (1) to (4), but not (5), then a −1 P γ is equal to a product of at least two such factorsP. Furthermore, γ is uniquely determined by P γ , so it follows that the number of such γ is less than or equal to c 11 H d(l−2)+ . This completes the proof of Lemma 7.4 and thereby completes the proof of Theorem 7.1.
It is now easy to show that a(k, d) does not exist if 0 < k < d and gcd(k, d) > 1. We can even determine the corresponding limit superior and limit inferior. Proof Set l = gcd(k, d). The limit superior is less than or equal to d(l − 1) by Theorem 5.2(ii). If l = 1, this already proves the theorem, so let us assume that l ≥ 2. We want to show that the limit superior is also greater than or equal to d(l − 1). We fix a totally real number field L of degree d l that is a Galois extension of Q. Such an L can be constructed as a subfield of Q cos 2π p , where p is prime and p ≡ 1 mod 2d l . We will prove the following analogue of Theorem 1.1 for the Mahler measure instead of the height: We see that e ≤ max i {k i − 1} ≤ k − 1 < max{k, d − k} and f ≤ max{k, d − k}. It follows that the number of possibilities for A is bounded from above by CM max{k,d−k}+(s+2) . This proves that the limit superior in (8.4) is less than or equal to max{k, d − k}.
For the inequality in the other direction, we consider M ∈ N such that M We can deduce from Theorem 4.1(ii) that the limit in (8.5) has to be greater than or equal to d + 1 (if it exists). For the inequality in the other direction (which will also imply the existence of the limit), we can use that for M ∈
Dynamics of the height function
In this section, we study the dynamics of the restriction of the height function toQ ∩ R.
We start by classifying the periodic points. We define inductively H 0 = id and H n = H • H n−1 (n ∈ N). Theorem 9.1 If n ∈ N and α ∈Q are such that H n (α) = α, then α = a b for some a ∈ N and b ∈ Q, b > 0, and H(α) = α. Conversely, H(a b ) = a b for all a ∈ N and b ∈ Q, b > 0.
The proof of this theorem will be essentially achieved by the following lemma: We can now prove Theorem 9.1.
For d ∈ N = {1, 2, . . .}, k ∈ {0, . . . , d}, and H ∈ [1, ∞), we set A(k, d, H) = {α ∈ C; [Q(α) : Q] = d, H(α) = H, and precisely k conjugates of α lie inside the open unit disk}, A(k, d) = H≥1 A(k, d, H), B(k, d, H) = {H(α); α ∈ C, [Q(α) : Q] = d, H(α) ≤ H, and precisely k conjugates of α lie inside the open unit disk}, and B(k, d) = {H(α); α ∈ A(k, d)}.By Northcott's theorem, the setsA(k, d, H) and B(k, d, H)are finite for all H ≥ 1. The main goal of this article is to measure the growth of |A(k, d, H)| and |B(k, d, H)| as functions of H.As A(k, d, H)is empty if H / ∈ B(k, d), we consider |A(k, d, H)| only for H ∈ B(k, d). We set a(k, d) = lim H∈B(k,d) H→∞ log |A(k, d, H)| log H and b(k, d) = lim H→∞ log |B(k, d, H)| log H
Theorem 1. 1
1Let d ∈ N and k ∈ {0, . . . , d}. Then the following hold:(i) b(0, d) = b(d, d) = d (Theorem 2.1(i)), (ii) a(0, d) = a(d, d) = d 2 (Theorem 2.1(ii)), (iii) b(k, d) = d(d + 1) if 0 < k < d (Theorem 4.1(ii)), (iv) a(k, d) = 0 if 0 < k < d and gcd(k, d) = 1 (Theorem 5.2), and (v) a(k, d)does not exist if 0 < k < d and gcd(k, d) > 1, but the corresponding limit superior and limit inferior are equal to d(gcd(k, d)−1) and 0 respectively (Theorem 7.5 and Lemma 7.6).
Theorem 1.2 (= Theorem 5.2(iii)). Let d ∈ N, > 0, and k ∈ {1, . . . , d − 1}. There exists a constant C, depending only on d, k, and , such that for all H ∈ [1, ∞) we have |{α ∈ A(k, d, H); the Galois group of the normal closure of Q(α) acts primitively on the set of conjugates of α}| ≤ CH .
(w 0 , . . . , w d−1 ) ∈ [1/(4d), 1/(2d)] × [−1/(2d), 1/(2d)] d−1 and all z ∈ C with |z| < 1 and therefore [1/(4d), 1/(2d)] × [−1/(2d), 1/(2d)] d−1 ⊂ Z u . We have N d (T ) := |{P(t) = at d + · · · ± T ∈ Z[t]; a > 0, all complex zeroes of P are at least 1 in absolute value}|
Proof (As suggested by G. Rémond.) It follows from Propositions 1.6.5 and 1.6.6 in[3] that B(0, d) and B(d, d) are both contained in N 1 d . In the case d = 1, the lemma follows from H(n) = H(n −1 ) = n for all n ∈ N together with H(0) = 1, so we assume that d ≥ 2.
Lemma 3. 1
1Let d ∈ N and k ∈ {0, . . . , d}. The limit lim H→∞ H ≤H |A(k, d, H )| H d(d+1)exists and is positive.
Lemma 3.1 implies that A(k, d) is infinite and so B(k, d) contains arbitrarily large elements. Proof We first remark that H ≤H |A(k, d, H )| = |{α ∈ C; [Q(α) : Q] = d, H(α) ≤ H, and precisely k conjugates of α lie inside the open unit disk}|.
H); the Galois group of the normal closure of Q(α) acts transitively on the k-element subsets of the set of conjugates of α}| ≤ CH for all H ≥ 1.
Lemma 3. 4
4Suppose that α ∈Q with [Q(α) : Q] = d. Let α 1 , . . . , α d be the conjugates of α and let a ∈ Z be the leading coefficient of a minimal polynomial of α in Z[t]. Let S be a subset of {1, . . . , d}. Then a s∈S α s is an algebraic integer.
Theorem 4. 1
1Let d ∈ N and k ∈ {0, . . . , d}. Then the following hold:
d 2 .
2We see that d − k must divide n since d − k divides the sum of the coordinates of every element of . Hence we have n = (d − k)l with l ∈ N bounded by (d − k) d 2 −1 . The expression of (n, 0, . . . , 0) as a linear combination of the A i v is necessarily unique and the coefficients of the A i v in this linear combination can also be bounded in absolute value in terms of k and d only (i = 0, . . . , d − 1). Translating all of this into terms of products of conjugates of ±H d yields that (aα d−k 1 ) l can be written as a quotient of products of conjugates of ±H d for some natural number
H); the Galois group of the normal closure of Q(α) acts 2-transitively on the conjugates of α}| ≤ CH for all H ≥ 1. Proof Let H ∈ [1, ∞) and α ∈ A(k, d, H) such that the Galois group of the normal closure of Q(α) acts 2-transitively on the conjugates of α. Let α 1 , . . . , α d be the conjugates of α. We want to mimick the proof of Theorem 4.1(iii). We have H d = ±a i∈I α i for some I ⊂ {1, . . . , d} with |I| = d − k and a ∈ N the leading coefficient of a minimal polynomial of α in Z[t]
Theorem 4. 3
3Let d ≥ 2. For every > 0, there is H 0 = H 0 (d, ) ∈ R such that |{α ∈ C; [Q(α) : Q] = d, H(α) ≤ H}| |{H(α); α ∈ C, [Q(α) : Q] = d, H(α) ≤ H}| ≤ H for all H ≥ H 0 . Theorem 4.3 is patently wrong for d = 1, where the left-hand side of the inequality in the theorem grows linearly in H. Proof First, we can replace the numerator in the inequality by the cardinality of the set {α ∈ C; [Q(α) : Q] = d, H(α) ≤ H, precisely one conjugate of α lies outside the open unit disk}. Why? By Lemma 3.1, the number of α of degree d with precisely one conjugate outside the open unit disk and height at most H grows asymptotically like some constant positive multiple of H d(d+1) . Because of Lemma 3.1, applied for all k ∈ {0, . . . , d} (or thanks to the main result of [17]), demanding that α is in this set then changes the left-hand side of the inequality in the theorem by a factor bounded from below by some c = c(d) > 0 for H large enough in terms of d. Let B(d; H) = {H(α); α ∈ C, [Q(α) : Q] = d, H(α) ≤ H}, then we can rewrite our new numerator as H ∈B(d;H) |{α ∈ C; [Q(α) : Q] = d, H(α) =H, precisely one conjugate of α lies outside the open unit disk}|.
deduce that [K (α) : K ] = [K (α i ) : K ] divides [Q(H d , α i ) : Q(H d )] (i = 1, . . . , d − k) and divides [Q(α) : Q] = d. But by the above, d − k is the sum of some of the [Q(H d , α i ) : Q(H d )], namely one for each irreducible factor ofd−k i=1 (t − α i ) in Q(H d )[t]. So [K (α) : K ] divides d − k and d, hence divides gcd(k, d).This completes the proof of the lemma.We can now prove Theorem 5.2.Theorem 5.2 Let d ∈ N, > 0, and k ∈ {1, . . . , d − 1}. There exists a constant C, depending only on d, k, and , such that for all H ∈Q ∩ [1, ∞) the following hold: (i) Let K denote the normal closure of Q(H d ) and let l ∈ N divide gcd(k, d), then |{α ∈ A(k, d, H); [K (α) : K ] = l}| ≤ CH d(l−1)+ , (ii) |A(k, d, H)| ≤ CH d(gcd(k,d)−1)+ , and (iii) |{α ∈ A(k, d, H); the Galois group of the normal closure of Q(α) acts primitively on the set of conjugates of α}| ≤ CH .
where I is the orbit of {α 1 , . . . , α d−k } under the Galois group of the normal closure of Q(α) and the cardinality of I is [Q(H d ) : Q]. Since the Galois group acts transitively on {α 1 , . . . , α d }, we have that
J
deduce from Lemma 3.4 that the γ j are algebraic integers (j = 1, . . . , l). The orbit of I under Gal(Q/Q) consists of d l pairwise disjoint sets I = I 1 , …, I d l. We calculate that N K /Q (γ j ) is equal to ⊂I s ,|J |=j β∈J β is a rational integer and together with a, it completely determines N K /Q (γ j ).If j = l, then N j = N l divides N Q(H d )/Q (H d ) by (5.1) since k < d. Therefore, N l is already determined up to C 2 H 3 possibilities, where C 2 depends only on d, k, and . If j ∈ {1, . . . , l − 1}, then N j is at least bounded in absolute value by C 3 H d , where C 3 depends only on d and k.
Theorem 6. 1
1Let > 0. There exists a constant C = C( ) such that |A(2, 4, H)| ≤ CH for all H ≥ 1 with [Q(H 4 ) : Q] = 4. Proof If [Q(H 4 ) : Q] = 4 and α ∈ A(2, 4, H), then the normal closure of Q(α) is either Q(α) or a number field of degree 8; otherwise, its Galois group would be naturally isomorphic to the symmetric or the alternating group on 4 elements and we would get [Q(H 4 ) : Q] = 6 (recall that by Lemma 5.1, a Galois automorphism ofQ can only fix H 4 if it fixes the set of conjugates of α that lie outside the open unit disk). We denote the normal closure of Q(H 4 ) by K .
4, H)| ≥ C H 4− if [Q(H 4 ) : Q] = 2 and there exists α ∈ A(2, 4, H) satisfying a certain additional condition depending on Q(H 4 ) and a parameter δ (cf. Theorem 1.5), but the constant C will also depend on Q(H 4 ) and δ. Theorem 6.2 The limit a(2, 4) does not exist. For every κ ∈ [0, 4], there exists a sequence (H n ) n∈N in B(2, 4) such that [Q(H 4 n ) : Q] = 2 for all n ∈ N, lim n→∞ H n = ∞, and lim n→∞ log |A(2, 4, H n )| log H n = κ.
Let > 0 .
0We will prove that there exist positive constants H 0 , c, H 1 , C such that the constants H 0 and c depend only on κ, the constants H 1 and C depend only on κ and , |A(2, 4, H)| ≤ CH κ+ if H ≥ H 1 , and |A(2, 4, H)| ≥ cH κ if H ≥ H 0 . The theorem then follows since [Q(H 4 ) : Q] = 2 and H tends to infinity as m or r respectively tend to infinity. Upper bound. We first prove the upper bound. Let α ∈
√ m)/Q (4β)| ≤ 16|β| that the norms of the ideals generated by γ + δ and γ − δ lie in a set of cardinality at most C 3 |β| κ 32 for some constant C 3 = C 3 (κ). Of course, these norms are also at most equal to |N Q(√ m)/Q (4β)| ≤ 16|β|. The number of ideals of norm N in a quadratic number field is bounded by the number of natural numbers dividing N . It follows that the ideals themselves lie in a set of cardinality at most C 4 |β| κ 16 for a constant C 4 that depends only on κ, so we can assume them to be fixed. This determines γ + δ and γ − δ up to multiplication by a unit of Z[ √ m]. This unit is of the form ζ u l , where ζ = ±1, l ∈ Z, and u is fixed (depending on m) and satisfies H(u) > 1, so H(u) ≥ h 2 = min{H(ξ ); ξ ∈ C, [Q(ξ ) : Q] ≤ 2, H(ξ ) > 1} > 1. Using the fact that |γ | = [c 2
|l| 2 ≤
2H(u) |l| = H(u l ) ≤ H(η)H(η ) ≤ 32|β|3 and so |l| is bounded from above by log(32)+3 log |β| log h 2 .
.
Since γ is uniquely determined by α, the lower bound is proven with c
2 suggests: The limit a(k, d) never exists if 0 < k < d and gcd(k, d) > 1. In this section, we consider the general case where gcd(k, d) > 1 and first prove the following more refined result about the frequency of the corresponding height values. It is valid for all k ∈ {1, . . . , d − 1}, but of interest mostly in the case where gcd(k, d) > 1. Theorem 7.1 Let k, d ∈ N such that 0 < k < d. Let δ ∈ (0, 1) and > 0 and let K ⊂Q be a fixed Galois extension of Q. Let H ∈Q ∩ [1, ∞) such that the normal closure of Q(H d ) is equal to K . For a subfield L ⊂ K such that [L : Q] | d | [L : Q] gcd(k, d) and β ∈ L, set A L,β (k, d, H) = {α ∈ A(k, d, H); Q(α) ∩ K = L, N Q(α)/L (α) = β}. Then we have that A(k, d, H) = L⊂K [L:Q]|d|[L:Q] gcd(k,d) β∈L A L,β (k, d, H). (7.1) There exists a constant C = C(k, d, K, δ, ) > 0 such that if A L,β (k, d, H) = ∅ for a subfield L ⊂ K as above and β ∈ L and if furthermore for every field embedding σ : Q(β) → C, we have either |σ (β)| ≥ (1 − δ) −1 or |σ (β)| ≤ 1 − δ, then |A L,β (k, d, H)| ≥ CH d(l−1)− , (7.2) where l = d[L : Q] −1 .
Example 7. 2
2Let m ∈ N be even such that m − 1 and m + 1 are both squarefree and m > 2. The asymptotic count of squarefree integers shows that there exist arbitrarily large such m. Set α = m + √ m 2 − 1, where √ · denotes the positive square root. Onecan show that α 2 is not a square in Q( √ m 2 − 1) and so [Q(α) : Q] = 4. We find that α ∈ A(2, 4, H) with H 4 = m + √ m 2 − 1. We have K = L = Q( √ m 2 − 1), l = 2, and β = N Q(α)/L (α) = −(m + √ m 2 − 1) in Theorem 7.1.We can take δ = 1 2 for m large enough. One can show that any α ∈ A(2, 4, H) with |α | ≥ 1 is an algebraic integer and satisfies an equation α 2 + γ α ± β = 0 with γ ∈ O K . Let γ denote the image of γ under the non-trivial field automorphism of K . Then one can show that |γ | ≤ 2|β| while |γ | ≤ 2. This implies that the number of such γ is bounded independently of m, but H → ∞ as m → ∞.
Example 7. 3
3Let (a, b) ∈ N 2 be a solution to a 2 − 2b 2 = −1 and set α = 1+b √ 2 a , where √ · again denotes the positive square root. The Q( √ 2)/Q-norm of α 2 is −1, which implies that α / ∈ Q( √ 2) and so [Q(α) : Q] = 4. We find that a minimal polynomial of α in Z[t] is at 4 − 2t 2 − a and α ∈ A(2, 4, H) with H 4 = 1 + b √ 2. We have K = L = Q( √ 2), l = 2, and β = N Q(α)/L (α) = − 1+b √ 2 a in Theorem 7.1. Let α ∈ A(2, 4, H) such that |α | ≥ 1 and let a > 0 be the leading coefficient of a minimal polynomial of α in Z[t]. One can show that a divides N K /Q (H 4 ) = −a 2 and that
7.1): Let α ∈ A(k, d, H). The normal closure of Q(H d ) is equal to K . Set l = [K (α) : K ]. By Lemma 5.1, l divides gcd(k, d). Set L = K ∩ Q(α). Since K /Q is Galois, we have that [Q(α) : L] = [K (α) : K ] = l. So [L : Q] = d l divides d and is divisible by d gcd(k,d). We set β = N Q(α)/L (α) ∈ L and it follows that α ∈ A L,β (k, d, H). This proves (7.1).Next, we prove (7.2): We fix L and β and suppose that A L,β (k, d, H) = ∅ and that for every field embedding σ : Q(β) → C, we have either |σ (β)| ≥ (1 − δ) −1 or |σ (β)| ≤ 1 − δ. It follows that there exists some α ∈ A L,β (k, d, H).Let a denote the leading coefficient of a minimal polynomial of α in Z[t], chosen such that a > 0. If P denotes the (monic) minimal polynomial of α in L[t], then Lemma 3.4 shows that aP ∈ O L [t]. Let I denote the ideal of O L generated by the coefficients of aP.Since K /Q is Galois and L ⊂ K , every field embedding σ : L → C factors through K . Thus, we can setJ = σ :L →C σ (I)O K , it is an ideal of O K . Since Q(α) ⊃ L,we have that σ :L →C σ (P) is the minimal polynomial of α in Q[t] and a σ :L →C σ (P) is a minimal polynomial of α in Z[t]. In particular, the sets of complex zeroes of the σ (P) form a partition of the conjugates of α. This implies together with Lemma 3.4 that the ideal J is divisible by a [L:Q]−1 O K . At the same time, J contains a [L:Q]−1 q for every coefficient q of a minimal polynomial of α in Z[t]. It follows that J = a [L:Q]−1 O K , which implies that N (I) = a [L:Q]−1 . (7.3) Let α 1 , . . . , α d be the conjugates of α, numbered so that |α i | ≥ 1 if and only if 1 ≤ i ≤ d − k. We deduce that H(α) d = ±aα 1 · · · α d−k . It follows from Lemma 5.1 that the coefficients of the polynomial d−k i=1 (t − α i ) belong to Q(H d ). By the proof of (7.1), we have l = [Q(α) : L] = [K (α) : K ]. In particular, P is also the minimal polynomial of α in K [t]. Let I ⊂ {α 1 , . . . , α d } be the subset of conjugates of α over K (of cardinality l). The orbit of I under Gal(Q/Q) consists of d l pairwise disjoint sets I = I 1 , …, I d l . As the coefficients of the polynomial d−k i=1 (t − α i ) belong to K and K /Q is Galois, we must have {α 1 , . . . , α d−k } = j∈S I j for some S ⊂ 1, . . . , d l . This implies that for every σ : L → C, σ (P) has either all complex zeroes inside or all complex zeroes outside the open unit disk. As β is the constant coefficient of P up to sign and σ :L →C σ (P) is the minimal polynomial of α in Q[t], we have that H d = a
|σ (γ i )| ≤ aδ l for all σ : L → C such that |σ (β)| < 1 (i = 1, . . . , l − 1),(4) |σ (γ i )| ≤ aδ|σ (β)| l for all σ : L → C such that |σ (β)| ≥ 1 (i = 1, . . . , l − 1), and (5) P γ is irreducible in K [t].
[Q(α γ ) : Q] = [L : Q]l = d and H(α γ ) d = a σ :L →C |σ (β)|≥1 |σ (β)|, which equals H d by (7.4). Since |σ (β)| < 1 for precisely k l embeddings σ : L → C, we have that precisely k conjugates of α γ lie inside the open unit disk. Furthermore, we deduce from (5) that [L(α γ ) : Q] = [L(α γ ) : L][L : Q] = l[L : Q] = [Q(α γ ) : Q], which implies that Q(α γ ) ⊃ L. It follows that L ⊂ Q(α γ ) ∩ K . Since [Q(α γ ) : Q(α γ ) ∩ K ] = [K (α γ ) : K ] and [K (α γ ) : K ] = l = [Q(α γ ) : L] by
Lemma 7. 4
4In the above setting, there exists a constant C = C(k, d, K, δ, ) > 0 such that the number of γ ∈ I l−1 satisfying (1) to (5) is greater than or equal to CH d(l−1)− .
:Q] = a 1− l d since the L/Q-norm of any non-zero element of I is non-zero and divisible by N (I) = a [L:Q]−1 . The same lower bound holds
7.4), we have Q = a 1− l d δH l l −1 . It follows from (7.4) that the volume of Z Q is greater than or equal to c 1 a [L:Q]−1 H d . Let L denote the discriminant of L, then the determinant of is equal to 2 −s |
we denote its Mahler measure by M(A). If α is an algebraic number of degree d, its (multiplicative) height is equal to the d-th (positive real) root of the Mahler measure of any one of its two minimal polynomials in Z[t]. Together with the properties that M(a) = |a| (a ∈ Z) and M(AB) = M(A)M(B) (A, B ∈ Z[t]), this characterizes the Mahler measure uniquely. For given d ∈ N, k ∈ {0, . . . , d}, and M ∈ [1, ∞), we define A(k, d, M) = {A ∈ Z[t]; deg A = d, M(A) = M, and precisely k complex zeroes of A (counted with multiplicities) lie inside the open unit disk}, B(k, d) = M≥1 {M(A); A ∈ A(k, d, M)}, and B(k, d, M) = B(k, d) ∩ [1, M].
α i ) d i for a 0 ∈ N and algebraic numbers α i of degree [Q(α i ) : Q] = d i with precisely k i conjugates inside the open unit disk. Of course, we then have s i=1 d i = d and s i=1 k i = k. The number of possibilities for s and the d i and k i is bounded in terms of only d and k, so we can assume that s and the d i and k i are fixed. Set F = Q(M). We claim that H(α i ) d i ∈ F for all i = 1, . . . , s. If not, there would exist some σ∈ Gal(Q/Q) such that σ (M) = M, but σ H(α i ) d i = H(α i ) d i for some i. But then it follows that σ H(α i ) d i < |H(α i ) d i |, while σ H(α j ) d j ≤ |H(α j ) d j | for all j = i, and so |M| = |σ (M)| < |M|, a contradiction. Now H(α i ) d i ∈ F isan algebraic integer by Lemma 3.4, its height is bounded by H(α i ) d i ≤ M, and N F /Q H(α i ) d i divides N F /Q (M). Since [F : Q] is bounded in terms of only d and k, we can use Lemma 3.5 together with elementary bounds for the divisor function to deduce that H(α i ) d i is determined up to CM possibilities, so we can assume thatH i = H(α i ) d i is fixed. But then α i is determined up to C i H d i + i possibilities if k i ∈ {0, d i } and up to C i H gcd(k i ,d i )−1+ i possibilities if 0 < k i < d iby Theorems 2.1(ii) and 5.2(ii). Note that s i=1 H i ≤ M. Since a 0 divides N F /Q (M), it is determined up to C M possibilities. All in all, the number of possibilities for A (given a fixed s and fixed d i and k i ) is bounded from above by CM (s+2) +max{e,f } with e = max i {gcd(k i , d i ) − 1; 0 < k i < d i } and f = max i {d i ; k i ∈ {0, d i }}.
1 k
1∈ B(k, k) (if k = 0) and M 1 d−k ∈ B(0, d − k) (if k = d). By Lemma 2.2, all M ∈ N\{1} satisfy these conditions. We can then apply Theorem 2.1(ii) to find many productsA(t)(t − 1) d−k ∈ A(k, d, M) with A equal to a minimal polynomial (in Z[t]) of some α ∈ A k, k, M 1 k (if k = 0) and A(t)t k ∈ A(k, d, M) with A equal to a minimal polynomial (in Z[t]) of some α ∈ A 0, d − k, M 1 d−k (if k = d).Note that α is determined by A up to k or d − k possibilities respectively. This establishes that the limit superior in (8.4) is greater than or equal to max{k, d − k}. Hence, equality holds in(8.4).If k ∈ {0, d} and M ∈ B(k, d), then M ∈ N automatically. It then follows from the above thatlim inf M∈ B(k,d) M→∞ log | A(k, d, M)| log M ≥ d as well as lim sup M∈ B(k,d) M→∞ log | A(k, d, M)| log M ≤ d.We deduce(8.1). In that case, we also have B(k, d,M) = {n ∈ N; n ≤ M} (for M ∈ [1, ∞)) since M nt d = M n(t − 1) d = n for n ∈ N,so (8.2) follows as well. Suppose now that k ∈ {1, . . . , d −1}. We first prove (8.3). It follows from Lemma 3.1 and [27] that we can find α ∈ A(k, d) of arbitrarily large height such that the Galois group of the normal closure of Q(α) is isomorphic to the full symmetric group S d . Any product of k conjugates of such an algebraic number α has degree d k . We can therefore find arbitrarily large M = H(α) d ∈ B(k, d) such that [Q(M) : Q] = d k . Let now M ∈ B(k, d) be arbitrary with [Q(M) : Q] = d k and let A ∈A(k, d, M). Suppose that A decomposes in Z[t] as the product of a non-zero integer a 0 and s irreducible factors A i of degree d i and with k i complex zeroes inside the open unit diskrespectively (i = 1, . . . , s). We can then bound the degree of M from above byif d , d ∈ N, k ∈ {0, . . . , d }, k ∈ {0, . . . , d }, and (k , k ) / ∈ {(0, 0), (d , d )}. If s > 1, this implies that s i=1 d ik i is smaller than d k , and we obtain a contradiction. We deduce that s = 1. Therefore, A must be equal to the product of a non-zero integer a 0 and a minimal polynomial (in Z[t]) of some algebraic number α of degree d. We want to bound the number of possibilities for a 0 and α.SinceH(α) d is an algebraic integer, a 0 divides N Q(M)/Q (M) and so the number of possibilities for a 0 is bounded by C M . Since [Q H(α) d : Q] = [Q(M) : Q] = d k , the number of possibilities for α, given a 0 , is bounded by C H(α) ≤ C M thanks to Lemma 3.3. We deduce (8.3).
[ 1 ,
1∞), any M ∈ B(k, d, M) is equal to M(A) for some A ∈ Z[t] of degree d, and that the d + 1 coefficients of this A are all bounded by 2 d M(A) ≤ 2 d M in absolute value thanks to Lemma 1.6.7 in [3].
Lemma 9. 2
2If α ∈Q, then H(α) ≥ H(H(α)) with equality if and only if H(α) = a b for some a ∈ N and b ∈ Q, b > 0. Proof Let β = H(α). We have β [Q(α):Q] = a |γ |≥1 γ , where a is the leading coefficient of a minimal polynomial of α in Z[t] and the product runs over all complex zeroes γ of that minimal polynomial that are at least 1 in absolute value. It is now clear that any conjugate of β [Q(α):Q] that is not equal to β [Q(α):Q] is less in absolute value than β [Q(α):Q] . Furthermore, β [Q(α):Q] is an algebraic integer by Lemma 3.4. It follows that H(β) [Q(α):Q] = H β [Q(α):Q] < β [Q(α):Q] and hence H(H(α)) < H(α) unless [Q β [Q(α):Q] : Q] = 1, in which case β = a b for some a ∈ N and b ∈ Q, b > 0. Furthermore, it is clear that H(a b ) = a b for all a ∈ N and b ∈ Q, b > 0.
Theorem 1.4 Let > 0. There exists a constant C, depending only on , such that the following hold:(i) |A(2, 4, H)| ≤ CH for all H ∈ B(2, 4) such that [Q(H 4 ) : Q] = 6 (Lemma 3.3), (ii) |A(2, 4, H)| ≤ CH for all H ∈ B(2, 4) such that [Q(H 4 ) : Q] = 4 (Theorem 6.1), and (iii) for every κ ∈ [0, 4], there exists a sequence (H n ) n∈N in B(2, 4) such that [Q(H 4 n ) : Q] = 2 for all n ∈ N, lim n→∞ H n = ∞, and lim n→∞Theorem 1.1
according to whether [Q(H 4 ) : Q] equals 2, 4, or 6:
log |A(2,4,H n )|
log H n
= κ (Theorem 6.2).
Proof (i) Eisenstein's criterion shows that all real positive d-th roots of integers between 2 and H d that are congruent to 2 modulo 4 belong to B(0, d, H). Using that the height of a non-zero algebraic number is equal to the height of its inverse, we deduce that they also belong toB(d, d, H). Also,every element of B(0, d, H) or B(d, d, H)is a real positive d-th root of some integer between 1 and H d . So H d ≥ |B(0, d, H)| ≥ 1 5 H d for H large enough and the same holds for |B(d, d, H)|. It follows that (i) holds.and
(ii) a(0, d) = a(d, d) = d 2 .
(In particular, all these limits exist.)
Let d ∈ N. We haved d− 1
2
for H d ∈ N. Since H d ∈ N for all H ∈ B(0, d), we deduce (ii) from elementary lower
bounds for φ(H d ), at least for a(0, d). For a(d, d) we can repeat the same argument, but
counting 1
α instead of α and replacing x 2
j + y 2
j ≥ 1 by x 2
j + y 2
j > 1 in (2.1).
One can say even more about the sets B(0, d) and B(d, d). We denote by N
1
d the set of
the positive real d-th roots of all natural numbers.
Lemma 2.2
polynomial Nt d − p is irreducible in Z[t] by the coprimality of p and N together with Eisenstein's criterion (applied to pt d − N if N = 2). The complex zeroes of this polynomial belong to A d, d, N This completes the proof of the lemma.1
d
and their inverses belong to A 0, d, N
1
d . It follows
that N
1
d ∈ B(0, d) ∩ B(d, d).
Lemma 5.1 Let k, d ∈ N such that 0 < k < d and let H ∈Q ∩ [1, ∞). Let K denote the normal closure of Q(H d ) and suppose that α ∈ A(k, d, H). Then [K (α) : K ] divides gcd(k, d). Furthermore, let α 1 , . . . , α d be the conjugates of α, numbered so that |α i | ≥ 1 if and only if 1 ≤ i ≤ d − k. Then any element σ ∈ Gal(Q/Q) that fixes H d also fixes the set {α 1 , . . . , α d−k }.Proof We first prove the second part of the lemma: If it were false, there would exist an element σ ∈ Gal(Q/Q) that fixes H d , but does not fix the set {α 1 , . . . , α d−k }. But this immediately yields a contradiction since H d = aα 1 · · · α d−k for some non-zero integer aand
s∈S
If κ ≥ 2, we can estimate H 2 ≤ H κ .1
2
· 8 · max 4, H
28
5 log log max{3,H}
≤ (24H κ + H 2 ) · 8 · max 4, H
28
5 log log max{3,H}
.
H 28 5
28log log max{3,H} } ≤ CH κ+ for H ≥ H 1 with a constant C that depends only on . Lower bound. For the lower bound, we first treat the case κ = 0, so β = ±(b 1 + 2√
m)
withb 1 ∈ {[2
√
m], [2
√
m] + 1} odd. Let
√
β denote an arbitrary complex square root of
β. The degree of
√
β is 4: Otherwise,
√
β would have to be an element a 1 + a 2
√
m of
Z[
√
m], which implies that a 2
1 + a 2
2 m + 2a 1 a 2
√
m = β, so a 1 , a 2 ∈ {±1} and m + 1 ∈
{±[2
√
m], ±([2
√
m] + 1)}. This yields a contradiction with m ≥ 5. Therefore, we have
√
β ∈ A
but leaves α unchanged to the defining equation of α and obtain a contradiction). Therefore, γ 2 −4β is a square in Z[If α is quadratic, we have α ∈ Z[
√
m] (if not, we could apply an automorphism ofQ that
sends
√
m to −
√
m, √
m]. It follows that (γ +δ)(γ −δ) = 4β
for a certain δ ∈ Z[
√
m]. By using an elementary bound for the divisor function, we deduce
from |N Q(
it follows thatQ divides Q γ in Z[t] and soã divides a. Lemma 3.4 now implies thataP ∈ O K [t], sop i ∈ O K for all i = 1, . . . ,l.We have|N K /Q (p i )| = |NK /Q (p i )| [K :K ] = a [K :Q]−1 |q i | σ :K →C Ĩ ⊂{ξ ∈C;σ (P)(ξ )=0} |Ĩ|=i ζ ∈Ĩ ζ (i =1, . . . ,l).[K :K ]
,
(7.7)
wherẽ
q i = a
Theorem 7.5 Let k, d ∈ N such that 0 < k < d. Then we havelim sup
H∈B(k,d)
H→∞
log |A(k, d, H)|
log H
= d(gcd(k, d) − 1).
Theorem 8.1 Let d ∈ N. If k ∈ {0, d}, then If k ∈ {1, . . . , d − 1}, we have Proof Let d ∈ N, k ∈ {0, . . . , d}, M ∈ B(k, d), and > 0. All unspecified constants will depend only on d, k, and . We first bound | A(k, d, M)| from above: Let A ∈ A(k, d, M). By factoring A into irreducible factors in Z[t], we see that M = a 0lim
M∈ B(k,d)
M→∞
log | A(k, d, M)|
log M
= d
(8.1)
and
lim
M→∞
log | B(k, d, M)|
log M
= 1.
(8.2)
lim inf
M∈ B(k,d)
M→∞
log | A(k, d, M)|
log M
= 0,
(8.3)
lim sup
M∈ B(k,d)
M→∞
log | A(k, d, M)|
log M
= max{k, d − k},
(8.4)
and
lim
M→∞
log | B(k, d, M)|
log M
= d + 1.
(8.5)
AcknowledgementsThis article has grown out of an appendix to my PhD thesis. I thank my PhD advisor Philipp Habegger for his constant support and for many helpful and interesting discussions. I thank Philipp Habegger and Gaël Rémond for helpful comments on the thesis and I thank Gaël Rémond for suggesting the proof of Lemma 2.2. I thank Fabrizio Barroero for useful comments on an earlier version of this article and I thank Martin Widmer for correspondence on his work. I thank the referee for their helpful suggestions for improving the exposition. When I had the initial idea for this article, I was supported by the Swiss National Science Foundation as part of the project "Diophantine Problems, o-Minimality, andi=1 v i < 0. As every algebraic unit has norm 1, it follows that there exists an algebraic unit u ∈ L such that |σ i (u)| < 1 (i ≤ k l ) and. We fix a prime q that does not ramify in L. For n ∈ N sufficiently large, we can suppose that the algebraic integer β = qu n satisfies We now deduce the theorem by applying Theorem 7.1 with K = L, δ = 1 2 , and l = gcd(k, d) and letting n and thereby H go to infinity.Determining the corresponding limit inferior is even easier.Proof Thanks to Lemma 3.1 and[27], we can find α ∈ A(k, d) of arbitrarily large height such that the Galois group of the normal closure of Q(α) is isomorphic to the full symmetric group S d . The degree of any product of k conjugates of such an algebraic number α is equal to d k . We can therefore find arbitrarily largeThe lemma now follows from Lemma 3.3.Counting polynomials of given Mahler measureIn this section, we consider polynomials of degree d with integer coefficients of a given Mahler measure instead of algebraic numbers of degree d of a given height. The difference is of course that we also consider reducible polynomials. For a polynomial A ∈ Z[t], Proof of Theorem 9.1 Suppose that H n (α) = α for some n ∈ N and α ∈Q. It follows from Lemma 9.2 that H n (α) = H n (H n (α)) ≤ H(H(H n−1 (α))) ≤ H(H n−1 (α)) = H n (α).We deduce that equality must hold everywhere. Hence, Lemma 9.2 implies that α = H n (α) is of the desired form and we have H(α) = α. The converse implication is again obvious.Next, we study the possibilities for the forward orbit of a given element. : Q] −1 times in this product, then α d! n n+1 must be a rational integer and so α n+1 is of the form a b for some a ∈ N and b ∈ Q, b > 0. Otherwise, α d! n n+1 is equal to a product of at most d! − 1 conjugates of α d! n−1 n up to sign and therefore equal to a product of at most (d! − 1) n conjugates of α 1 up to sign.If no α n is of the form a b for some a ∈ N and b ∈ Q, b > 0, then it follows directly thatfor all n ∈ N since every conjugate of α 1 is less than or equal to α 1 in absolute value. We deduce that lim n→∞ α n = 1.
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| []
|
[
"Learning Parameter Distributions to Detect Concept Drift in Data Streams",
"Learning Parameter Distributions to Detect Concept Drift in Data Streams"
]
| [
"Johannes Haug \nUniversity of Tuebingen\nTuebingenGermany\n",
"Gjergji Kasneci [email protected] \nUniversity of Tuebingen\nTuebingenGermany\n"
]
| [
"University of Tuebingen\nTuebingenGermany",
"University of Tuebingen\nTuebingenGermany"
]
| []
| Data distributions in streaming environments are usually not stationary. In order to maintain a high predictive quality at all times, online learning models need to adapt to distributional changes, which are known as concept drift. The timely and robust identification of concept drift can be difficult, as we never have access to the true distribution of streaming data. In this work, we propose a novel framework for the detection of real concept drift, called ERICS. By treating the parameters of a predictive model as random variables, we show that concept drift corresponds to a change in the distribution of optimal parameters. To this end, we adopt common measures from information theory. The proposed framework is completely model-agnostic. By choosing an appropriate base model, ERICS is also capable to detect concept drift at the input level, which is a significant advantage over existing approaches. An evaluation on several synthetic and real-world data sets suggests that the proposed framework identifies concept drift more effectively and precisely than various existing works. | 10.1109/icpr48806.2021.9412499 | [
"https://arxiv.org/pdf/2010.09388v1.pdf"
]
| 224,703,420 | 2010.09388 | cff49a0a06629d8ece126126fdee6669842e9168 |
Learning Parameter Distributions to Detect Concept Drift in Data Streams
Johannes Haug
University of Tuebingen
TuebingenGermany
Gjergji Kasneci [email protected]
University of Tuebingen
TuebingenGermany
Learning Parameter Distributions to Detect Concept Drift in Data Streams
Data distributions in streaming environments are usually not stationary. In order to maintain a high predictive quality at all times, online learning models need to adapt to distributional changes, which are known as concept drift. The timely and robust identification of concept drift can be difficult, as we never have access to the true distribution of streaming data. In this work, we propose a novel framework for the detection of real concept drift, called ERICS. By treating the parameters of a predictive model as random variables, we show that concept drift corresponds to a change in the distribution of optimal parameters. To this end, we adopt common measures from information theory. The proposed framework is completely model-agnostic. By choosing an appropriate base model, ERICS is also capable to detect concept drift at the input level, which is a significant advantage over existing approaches. An evaluation on several synthetic and real-world data sets suggests that the proposed framework identifies concept drift more effectively and precisely than various existing works.
I. INTRODUCTION
Data streams are a potentially unbounded sequence of observations. As such, data streams are subject to a number of external factors, e.g. seasonal or catastrophic events. Hence, the distributions of a data stream are usually not stationary, but change over time, which is known as concept drift.
Concept drift can seriously affect the quality of predictions, if it goes unnoticed. Concept drift detection models help identify and handle distributional changes, allowing us to maintain a high predictive performance over time. Ideally, concept drift detection models are sensitive enough to detect drift with only a short delay. However, concept drift detection should also be robust against small perturbations of the input in order to avoid false positives and thus be reliable.
Let X and Y be random variables that correspond to the streaming observations and the associated labels. According to [1], concept drift resembles a difference in the joint probability P (Y, X) at different time steps t, u ∈ {1, .., T }, i.e. P t (Y, X) = P u (Y, X) ⇔ P t (Y |X)P t (X) = P u (Y |X)P u (X).
We call P t (Y, X) the active concept at time step t. Moreover, we distinguish between real and virtual concept drift. Virtual concept drift describes a change in P (X), i.e. P t (X) = P u (X). Hence, virtual concept drift is independent from the target distribution and does not change the decision boundary [2]. On the other hand, real concept drift, sometimes called concept shift, corresponds to a change in the conditional target distribution, i.e. P t (Y |X) = P u (Y |X). Real concept drift shifts the decision boundary, which may influence subsequent predictions [2]. It is therefore crucial to detect changes of P (Y |X) in time to avoid dramatic drops in predictive performance. In this paper, we investigate the effective and robust identification of real concept drift.
Unfortunately, concept drift does not follow a clear pattern in practice. Instead, we might observe large differences in the duration and magnitude of concept drift. To this end, we distinguish between different types of concept drift [1]- [3]: Sudden drift describes an abrupt change from one concept to another. Incremental drift is a steady transition of concepts over some time period. In a gradual drift, the concepts alternate temporarily, until a new concept ultimately replaces the old one. Sometimes we also observe mixtures of different concept drift types and recurring or cyclic concepts. For further information, we refer the fellow reader to [1]. In general, concept drift detection models should allow timely and accurate detection of all types of concept drift.
In a data stream, we can only access a fraction of the data at every time step t. To detect real concept drift, we thus need to approximate P t (Y |X), by using a predictive model f θt . Accordingly, we get P t (Y |X) ≈ P (Y |X, θ t ), with parameters θ t = (θ tk ) K k=1 . We optimize the model parameters, given the new observations in every time step. Consequently, θ t represents our most current information about the active concept at time step t. A concept drift detection model should therefore adhere to changes of the model parameters through the following two properties: Property 1. Model-Aware Concept Drift Detection. Let θ t , θ u be the parameters of a predictive model f θ at two time steps t and u. Let further D be a statistical divergence measure (e.g., Kullback-Leibler, Jensen-Shannon, etc.). Concept drift detection is model-aware, if for a detected drift between any two time steps t and u, we observe D(θ t , θ u ) > 0.
Accordingly, we associate concept drift with updates of the predictive model f θ . Given that f θ is robust, model-awareness reduces the sensitivity of a concept drift detection scheme to random input perturbations, which in turn reduces the risk of false alarms. Property 2. Explainable Concept Drift Detection. Concept drift detection at time step t is explainable with respect to the predictive model f θt , if the concept drift can be associated with individual model parameters, i.e. each dimension of θ t .
If we associate concept drift with individual parameters, we can make more targeted model updates. Hence, we may avoid unnecessary and costly adaptations of the predictive model. Moreover, some parameter distributions even allow us to relate concept drift to specific input features. In this way, concept drift becomes much more transparent.
In this paper, we propose a novel framework for Effective and Robust Identification of Concept Shift (ERICS). ERICS complies with the Properties 1 and 2. We use the probabilistic framework introduced in [4] to model the distribution of the parameters θ at every time step. Specifically, we express real concept drift in terms of the marginal likelihood and the parameter distribution P (θ; ψ), which is itself parameterized by ψ. Unlike many existing models, ERICS does not need to access the streaming data directly [5]. Instead, we detect concept drift by investigating the differential entropy and Kullback-Leibler (KL) divergence of P (θ; ψ) at different time steps. In this context, we show that concept drift corresponds to changes in the distributional uncertainty of model parameters. In other words, real concept drift can be measured as a change in the average number of bits required to encode the parameters of the predictive model. By specifying an adequate parameter distribution, we can identify concept drift at the input level, which offers a significant advantage over existing approaches in terms of explainability. In fact, the proposed framework can be applied to almost any parameter distribution and online predictive model. For illustration, we apply ERICS to a Probit model. In experiments on both synthetic and realworld data sets, we show that the proposed framework can detect different types of concept drift, while having a lower average delay than state-of-the-art methods. Indeed, ERICS outperforms existing approaches with respect to the recall and precision of concept drift alerts.
In summary, we propose a generic and flexible framework that leverages the uncertainty patterns of model parameters for more effective concept drift detection in data streams. An open source version of ERICS is available at https://github.com/haugjo/erics.
II. ERICS: A CONCEPT DRIFT DETECTION FRAMEWORK
Real concept drift corresponds to a change of the conditional target distribution P (Y |X) [1]. However, data streams are potentially infinite and so the true distribution P (Y |X) remains unknown. Hence, we may use a predictive model f θ to approximate P (Y |X). Since we update the model parameters θ for every new observation, θ t represents our most current information about the active concept at time step t. Consequently, we may identify concept drift by investigating changes in θ over time.
To this end, we adopt the general framework of [4] and treat the parameters θ as a random variable, i.e. θ ∼ P (θ; ψ). Analogously, we optimize the distribution parameters ψ at every time step with respect to the log-likelihood. This optimization problem can be expressed in terms of the marginal likelihood P (Y |X, ψ) [4]. Hence, the marginal likelihood relates to the optimal parameter distribution under the active concept. Accordingly, we may associate concept drift between two time steps t and u with a difference of the marginal likelihood for the distribution parameters ψ t and ψ u :
P (Y |X; ψ t ) = P (Y |X; ψ u ) ⇔ |P (Y |X; ψ t ) − P (Y |X; ψ u )| > 0 ⇔ P (Y |X, θ) P (θ; ψ t ) − P (θ; ψ u ) dθ > 0. (1)
From (1), we may obtain a general scheme for concept drift detection. To this end, we rephrase (1) in terms of the differential entropy and KL-divergence, which are common measures from information theory. The entropy of a random variable corresponds to the average degree of uncertainty of the possible outcomes. Besides, entropy is often described as the average number of bits required to encode a sample of the distribution. On the other hand, the KL-divergence measures the difference between two probability distributions. It is frequently applied in Bayesian inference models, where it describes the information gained by updating from a prior to a posterior distribution. We can derive the following proportionality:
P (Y |X, θ) P (θ; ψ t ) − P (θ; ψ u ) dθ ∝ P (θ; ψ t ) dθ − P (θ; ψ u ) dθ ∝ P (θ; ψ t ) log P (θ; ψ t ) dθ − P (θ; ψ u ) log P (θ; ψ t ) dθ = H[P (θ; ψ u ), P (θ; ψ t )] − h[P (θ; ψ t )] = h[P (θ; ψ u )] − h[P (θ; ψ t )] + D KL [P (θ; ψ u ) P (θ; ψ t )],(2)
where h[P (θ; ψ t )] is the differential entropy of the parameter distribution at time step t. Note that we have rephrased the cross entropy H[P (θ; ψ u ), P (θ; ψ t )] by using the KLdivergence D KL . We may now substitute (2) into (1) to derive a general scheme for concept drift detection:
h[P (θ; ψ u )] − h[P (θ; ψ t )] ∆Uncertainty + D KL [P (θ; ψ u ) P (θ; ψ t )] ∆Distribution > 0
(3) Intuitively, real concept drift thus corresponds to a change in the uncertainty of the optimal parameters and a divergence of the parameter distribution. On the other hand, stable concepts are characterized by a static parameter distribution and uncertainty.
Note that (3) has another interpretation in the context of Bayesian inference. As mentioned before, the KL-divergence D KL [P (θ; ψ u ) P (θ; ψ t )] can be interpreted as the information gained from inferring the posterior P (θ; ψ u ) from a prior P (θ; ψ t ). According to (3), we thus find that every difference in parameter uncertainty (entropy) between time step t and u, which can not be attributed to the inference of posterior parameters, may be traced back to a concept drift.
Finally, we show that the proposed concept drift detection scheme adheres to the Properties 1 and 2.
Proof that ERICS is model-aware (Property 1): By construction, we model the parameters θ through a distribution P (θ; ψ). According to (3), we write
P (θ; ψ t ) log P (θ; ψ t ) dθ − P (θ; ψ u ) log P (θ; ψ t ) dθ ,
which is 0 iff P (θ; ψ t ) = P (θ; ψ u ). Consequently, we find that Equation (3) evaluates to true, iff P (θ; ψ t ) = P (θ; ψ u ). By definition, for any sensible statistical divergence measure D, we know that D(P (θ; ψ t ), P (θ; ψ u )) = 0 ⇔ P (θ; ψ t ) = P (θ; ψ u ). Equation (3) holds true, and thus P (θ;
ψ t ) = P (θ; ψ u ) ⇔ D(P (θ; ψ t ), P (θ; ψ u )) > 0
Proof that ERICS is explainable (Property 2): By construction, any parametric distribution P (θ; ψ) used in Equation (3) can be evaluated for each parameter individually, i.e. we have P (θ k ; ψ k ) ∀k.
A. Continuous Concept Drift Detection
Based on the general scheme (3), we are able to identify concept drift between any two time steps t and u. In practice, we are mainly interested in concept drifts between successive time steps t − 1 and t. However, if we were to study (3) for two time steps only, our concept drift detection model might become too sensitive to random variations of the predictive model. To be more robust, we examine the moving average of (3) instead. Specifically, we compute the moving average at time step t over M time steps as
MA t = 1 M t i=t−M +1 h[P (θ; ψ i )] − h[P (θ; ψ i−1 )]+ D KL [P (θ; ψ i ) P (θ; ψ i−1 )] . (4)
As before, the moving average contains our latest information on the model parameters and the active concept. We can adjust the sensitivity of our framework by selecting M appropriately. In general, the larger we select M , the more robust the framework becomes. However, a large M might also hide concept drifts of small magnitude or short duration. So far we have treated all changes of the parameter distribution as an indication of concept drift. Indeed, this is in line with the general definition of concept drift [1]. Still, we argue that only certain changes in the parameter distribution have practical relevance. For example, suppose that we use stochastic gradient descent (SGD) to optimize the model parameters at every time step. If we start from an arbitrary initialization, the distribution of optimal parameters usually changes significantly in early training iterations. However, given that the concept P (Y |X) is stationary, SGD will almost surely converge to a local optimum. Consequently, we will ultimately minimize the entropy and KL-divergence of P (θ; ψ) in successive time steps. In other words, (4) will tend to decrease as long as we optimize the parameters ψ with respect (a) β = 0.01
(b) β = 0.001 (c) β = 0.0001 Fig. 1.
Updating the α-Threshold. The proposed framework uses a dynamic threshold α (red line) to detect concept drift. According to (4), we track a moving average of the divergence of the parameter distribution (dark blue line). If the total divergence in a sliding window (green line) exceeds the threshold, ERICS detects a concept drift (black vertical lines). By adjusting the hyperparameter β, we can control the iterative updates of α and thus regulate the sensitivity of ERICS after a drift is detected. Generally, the larger we choose β, the more sensitive ERICS becomes to changes of the parameter distribution. Here, we depict different β for the KDD data set [6]. We artificially generated four sudden concept drifts (blue vertical lines). In this example, small update steps (i.e. small β) are preferable to give the predictive model enough time to adapt to the new concept. Note that the early alerts correspond to the initial training phase of the predictive model. Hence, we would ignore them in practice.
to the active concept. However, if the decision boundary changes due to a real concept drift, SGD-updates will aim for a different optimum. This change of the objective will temporarily lead to more uncertainty in the model and thus increase the entropy of the parameter distribution.
We exploit this temporal pattern for concept drift detection. To this end, we measure the total change of (4) in a sliding window of size W :
t j=t−W +1 MA j − MA j−1 > α t ⇔ Drift at t,(5)
where α t ≥ 0 is an adaptive threshold. As before, we may control the robustness of the concept drift detection with the sliding window size W . Whenever we detect concept drift, i.e. (5) evaluates to true, we redefine α t as
α t = t j=t−W +1 MA j − MA j−1 .(6)
In this way, we temporarily tolerate all changes to the predictive model up to a magnitude of (6). We consider these changes to be the after-effects of the concept drift. We then update α t in an iterative fashion. Let β be a user-defined hyperparameter in the interval [0, 1]. Each update depends on the current αvalue, the β-hyperparameter and the time elapsed since the last concept drift alert, which we denote by ∆ Drif t :
α t = α t−1 − (α t−1 * β * ∆ Drif t )(7)
Note that α t will asymptotically approach 0 over time, if there is no concept drift. In this way, we gradually reduce the tolerance of our framework after a drift is detected. The choice of a suitable β usually depends on the application at hand. By way of illustration, we applied ERICS with different β to the KDD data set [6]. We used [7]'s method to induce sudden concept drift after every 20% of observations. For more information, see Section V. Figure 1 illustrates the components of ERICS for three different β-values. Notably, the larger we chose β, the more drifts we detected. Since we were dealing with a sudden concept drift in this particular example, we could be less sensitive and apply smaller update steps. For β = 0.0001, we achieved good first results in all our experiments. Therefore, this value can generally be used as a starting point for further optimization.
To conclude our general framework, we provide a pseudo code implementation in Figure 2.
B. Limitations and Advantages
The proposed framework does not access streaming observations directly, but uses the parameters of a predictive model instead. Accordingly, our approach is much more memory efficient than many related works. Yet, if the parameter distribution does not change in a drift period, concept drift may go unnoticed. In general, however, ERICS can detect all concept drifts that affect the predictive outcome.
One should also be aware that some predictive models are prone to adversarial attacks. Accordingly, ERICS can only be as robust as its underlying predictive model. This sensitivity to the predictive model is shared by most existing works. With ERICS, the possibility of misuse is drastically reduced, as we closely monitor the distribution of the model parameters at all times.
III. ILLUSTRATING ERICS
ERICS is model-agnostic. This means that the framework can be applied to different predictive models f θ and parameter distributions P (θ; ψ). In this way, we enable maximum flexibility with regard to possible streaming applications. By way of illustration, we adopt a Probit model with independent normally distributed parameters. This setup has achieved stateof-the-art results in online feature selection [4]. Besides, it offers dramatic computational advantages due to its low complexity. In line with [4], we optimize ψ at every time step with respect to the log-likelihood for the Probit model.
Require: [ψ t , .., ψ t−M ]; [MA t−1 , .., MA t−W ]; α t−1 ; ∆ Drif t α t ← Eq. (7) ∆ Drif t ← ∆ Drif t + 1 MA t ← Eq. (4) sumW indow ← t j=t−W +1 MA j − MA j−1 if sumW indow > α t then α t ← sumW indow ∆ Drif t ← 1 end if return α t ; MA t ; ∆ Drif t
The assumption of independent model parameters may appear restrictive, but in practice it often leads to good results, e.g. in the case of local feature attributions [8], [9] or feature selection [4], [10]. In fact, the independence assumption allows us to identify the parameters affected by concept drift and thus to comply with Property 2. Since the Probit model comprises one parameter per input feature, we can readily associate concept drift with individual input variables.
Accordingly, let P (θ; ψ t ) = N (ψ t = (µ t , Σ t )), where µ t = (µ tk ) K k=1 is a vector of mean values and Σ t is the diagonal covariance matrix, where the diagonal entries correspond to the vector σ 2 t = (σ 2 tk ) K k=1 . The differential entropy of P (θ; ψ t ) is
h P (θ; ψ t ) = 1 2 K + K ln(2π) + ln K k=1 σ 2 tk .
The KL-divergence between P (θ; ψ t ) and P (θ; ψ t−1 ) is
D KL [P (θ; ψ t ) P (θ; ψ t−1 )] = 1 2 K k=1 σ 2 tk + (µ t−1,k − µ tk ) 2 σ 2 t−1,k − K + ln K k=1 σ 2 t−1,k K k=1 σ 2 tk .
According to (4), we then write the moving average as
MA t = 1 2M t i=t−M +1 K k=1 σ 2 ik + (µ i−1,k − µ ik ) 2 σ 2 i−1,k − K . (8)
Note that (8) scales linearly with the number of parameters K, i.e. it has O(K) time complexity. In order to identify concept drift at individual parameters (which is equivalent to examining individual features, since we use a Probit model), we can investigate the moving average of a specific parameter θ k :
MA tk = 1 2M t i=t−M +1 σ 2 ik + (µ i−1,k − µ ik ) 2 σ 2 i−1,k − 1(9)
In this case, we maintain a different threshold α k per parameter. Note that (9) has a constant time complexity.
IV. RELATED WORK
In this section, we briefly introduce some of the most prominent and recent contributions to concept drift detection.
DDM monitors changes in the classification error of a predictive model [11]. Whenever the observed error changes significantly, DDM issues a warning or an alert. We find various modifications of this general scheme, including [12] and [13]. Another well-known method for concept drift adaptation is ADWIN [14]. Here, the authors maintain a sliding window, whose size changes dynamically according to the current rate of distributional change. [15] also employ a sliding window approach and provide a feasible implementation of Fisher's Exact test, which they use for concept drift detection. Similar to our framework, [16] use a sliding window and the entropy to detect concept drift. However, they examine entropy with regard to the predictive result and disregard the model parameters. FHDDM applies a sliding window to classification results and tracks significant differences between the current probability of correct predictions and the previously observed maximal probability [17]. To this end, FHDDM employs a threshold that is based on the Hoeffding bound. In a later approach, the same authors instead use McDiarmid's inequality to detect concept drift [18]. EWMA is a method that monitors an increase in the probability that observations are misclassified [19]. The authors use an exponentially weighted moving average, which places greater weight on the most recent instances in order to detect changes. [20] also focus on the predictive outcome. Specifically, they investigate the distribution of the loss function via resampling. Likewise, the LFR method uses certain test statistics to detect concept drift by identifying changes through statistical hypothesis testing [21]. Finally, [22] compare the labels of close data points in successive batches to detect concept drift.
In addition, we find various approaches that examine ensembles of online learners to deal with concept drift. For example, [23] compare two models; one that is trained with all streaming observations and another that is trained only with the latest observations. Likewise, [24] analyze the density of the posterior distributions of an incremental and a static estimator.
More information about the progress in concept drift detection can be found in [1]- [3], [25].
Conceptually, our work differs substantially from the remaining literature. Instead of directly examining the streaming observations or the predictive outcome, ERICS monitors changes in the parameters of a predictive model.
V. EXPERIMENTS
We evaluated ERICS in multiple experiments. All experiments were conducted on an i5-8250U CPU with 8 Gb of RAM, running 64-bit Windows 10 and Python 3.7.3. We compared our framework to the popular concept drift detection methods ADWIN [14], DDM [11], EWMA [19], FHDDM [17], MDDM [18] and RDDM [13]. We used the predefined implementations of these models as provided by the Tornado framework [26]. Besides, we applied the default [27] in an interleaved test-then-train evaluation. The VFDT is a state-of-the-art online learner, which uses the Hoeffding bound to incrementally construct a decision tree for streaming data. We used the VFDT implementation of scikit-multiflow [28] in our experiments. Note that we consider a simple binary classification scenario in all our experiments, since it should be handled well by all models. We optimized the hyperparameters of ERICS in a grid search. The search space was either chosen empirically or according to [4]. Table II lists all hyperparameters per data set. The hyperparameters "Epochs", "LR (learning rate) µ" and "LR σ" control the training of the Probit model, which we adopted from [4].
A. Data Sets
In order to evaluate the timeliness and precision of a concept drift detection model, we require ground truth. Consequently, we generated multiple synthetic data sets using the scikitmultiflow package [28]. Detailed information about each generator can be obtained from the corresponding documentation. We exhibit the properties of all data sets in Table I. Note that we simulated multiple types of concept drift. Specifically, we produced sudden concept drifts with the SEA generator. To this end, we specified a drift duration (width parameter) of 1. We alternated between the classification functions 0-3 to produce the different concepts. With the Agrawal generator, we simulated gradual drift of different duration. Again, we alternated between the classification functions 0-3 to shift the data distribution. With the rotating Hyperplane generator, we simulated an incremental drift over the full length of the data set. We generated 20 features with the Hyperplane generator, out of which 10 features were subject to concept drift by a magnitude of 0.5. Finally, we produced a Mixed drift using the Agrawal generator. The Mixed data contains both sudden and gradual drift, which we obtained by alternating the classification functions 0-4. All synthetic data sets contain 10% noisy data. We obtained 100,000 observations from each data stream generator.
In addition, we evaluated the proposed framework on real world data. However, since real world data usually does not provide any ground truth information, we had to artificially induce concept drift. For this reason, we applied the methodology of [7] to induce sudden concept drift in five wellknown data sets from the online learning literature. First, we randomly shuffled the data to remove any natural (unknown) concept drifts. Next, we ranked all features according to their information gain. We then selected the top 50% of the ranked features and randomly permuted their values. In this way, we generated sudden drifts after every 20% of the observations. Specifically, we introduced concept drift to the real-world data sets Spambase, Adult, Human Activity Recognition (HAR), KDD 1999 and Dota2, which we took from the UCI Machine Learning repository [6]. Note that we drew a random sample of 100,000 observations from the KDD 1999 data to allow for feasible computations.
Besides, we used the MNIST data set to evaluate partial concept drift detection at the input level. We selected all observations that are either labelled 3 or 8, since these numbers are difficult to distinguish. In the first half of the observations, we treated 3 as the true class. In the second half of the observations, we switched the true class to 8. In this way, we simulated a sudden concept drift of all input features.
For all real world data sets, we normalized the continuous features to the range [0, 1] and one-hot-encoded the categorical features. In the Adult data set, we imputed all NaN-values by a new category unknown. Moreover, we altered the labels of the HAR data set to simulate binary classification between the class moving (original labels walking, walking downstairs and walking upstairs) and non-moving (original labels sitting, laying and standing). We trained the online predictive models (Probit and VFDT) in batches of the following size: For Spambase and HAR we chose a batch size of 10. Adult was processed in batches of size 50. For all remaining data sets, we trained on batches of 100 observations.
B. Delay, Recall and Precision
In our first experiment, we applied the concept drift detection models to all synthetic and real-world data sets. Figure 3 exhibits the drift alerts of every model. The blue vertical lines and shaded areas indicate periods of concept drift. Each black vertical line corresponds to one drift alert. Most models identify concept drift in early iterations. This is due to the initial training phase of the predictive model and therefore has no practical relevance. For the upcoming evaluations, we have therefore ignored all drift alerts in the first 80 batches. By Figure 3, the proposed framework ERICS performs well in all data sets. Given the low complexity of the underlying Probit model, some concept drifts do not infer a change of the parameter distribution immediately. This can be seen in small delays, such as for the Agrawal data, for example. Still, ERICS achieves the smallest average delay of all concept drift detection models, which is shown in Table III.
Strikingly, ERICS generally seems to produce fewer false alarms than related models. We find support for this intuition by examining the average recall ( Figure 4) and precision ( Figure 5) over all data sets. Similar to [29], we evaluated the detected drifts for different detection ranges. The detection range corresponds to the number of batches after a known drift, during which we consider an alert as a true positive. Whenever there is no drift alert in the detection range, we count this as a false negative. Besides, all drift alerts outside of the detection range are false positives. We used these scores to compute the recall and precision values. Again, we find that ERICS tends to struggle in the early stages, right after a drift happens. As mentioned before, we attribute this to the slowly updating Probit model that we used for illustration. The VFDT, which is used by all related models, is much more complex and can thus adapt to changes faster. Additionally, we must treat some recall scores with care. For example, in four data sets, the DDM model detects drift in almost every time step. Hence, it achieves perfect recall, although the drift alerts are not reliable at all. Still, ERICS ultimately outperforms all related models in terms of both recall and precision. The superiority of our framework is even more apparent, if we look at the harmonic mean of precision and recall, which is the F1 score that we show in Figure 6.
C. Detecting Drift at the Input Level
As mentioned before, by using a Probit model and treating parameters as independently Gaussian distributed, we are able to associate concept drift with specific input features. By means of illustration, we apply ERICS to a sample of the MNIST data set, which we induced with concept drift. In Figure 7, we exhibit the mean of all observations corresponding to the true class before and after the concept drift (left For all data sets, we illustrate the drift alerts obtained from each concept drift detection model. The blue vertical lines and shaded areas correspond to known concept drifts. Each black marker stands for one drift alert. Early drift alerts can be attributed to the initial training of the predictive model and were therefore ignored. Notably, ERICS (ours) seems to detect most concept drifts, while triggering considerably fewer false alarms than most related models. We find support for this intuition in the remaining figures. Fig. 4. Recall. We show the average recall over all data sets for different detection ranges. ERICS (ours) ultimately detects more than 90% of the known concept drifts. The apparent disadvantage of ERICS in early batches can be attributed to the slower update speed of the Probit model as compared to the VFDT [27], which was used by the remaining concept drift detection methods.
Besides, the recall scores should be considered with care, since some methods tend to detect drift at almost every time step and are thus not reliable.
subplots). We also show the absolute difference between those mean values. In the outer most subplot on the right, we illustrate the drift alerts per input feature in the first 15 batches after the concept drift. The color intensity corresponds to the number of drift alerts (where many alerts correspond to darker patterns). Strikingly, the frequency of drift alerts closely maps the absolute difference between the two concepts. This shows that ERICS is generally able to identify the input features that Fig. 5. Precision. We show the average precision over all data sets for different detection ranges. ERICS (ours) tends to identify concept drift later than some related models. Therefore, we count fewer true positives in small detection ranges, which leads to lower precision. However, for larger detection ranges, our framework is more precise than any other model in the evaluation.
are most affected by concept drift. We expect this pattern to become even clearer, when using more complex base models.
VI. CONCLUSION
In this work, we proposed a novel and generic framework for the detection of concept drift in streaming applications. Our framework monitors changes in the parameters of a predictive model to effectively identify distributional changes of the input. We exploit common measures from information theory, by showing that real concept drift corresponds to Fig. 6. F1: We illustrate the F1 measure, which is the harmonic mean of the precision and recall shown in earlier plots. Here, the advantage of ERICS (ours) is most apparent, since it significantly outperforms all related methods for a detection range greater than 30 batches. In the left subplots, we exhibit the mean of the true class before and after the concept drift. The third subplot depicts the absolute difference of these mean values. In the right subplot, we show the alerts of ERICS in the first 15 batches after the concept drift. The color intensity corresponds to the frequency of drift alerts per input feature. Strikingly, the drift alerts seem to map the absolute difference between both concepts. This suggests, that ERICS does indeed identify concept drift for those input features that are most affected by a distributional change.
changes of the uncertainty regarding the optimal parameters. Given an appropriate parameter distribution, the proposed framework can also attribute drift to specific input features. In experiments, we highlighted the advantages of our approach over multiple existing methods, using both synthetic and realworld data. Strikingly, ERICS detects concept drift with less delay on average, while outperforming existing models in terms of both recall and precision.
Fig. 2 .
2Pseudo Code. Concept drift detection with ERICS at time step t.
Fig. 3 .
3Drifts Alerts.
Fig. 7 .
7Partial Drift Detection. By choosing an appropriate base model and parameter distribution, ERICS can attribute concept drift to individual input features. We selected all observations of MNIST with the label 3 or 8 and induced concept drift by changing the true class after half of the observations.
TABLE I SYNTHETIC
IAND REAL WORLD DATA SETSName
#Samples #Features Data Types
SEA (synth.)
100,000
3
cont.
Agrawal (synth.)
100,000
9
cont.
Hyperplane (synth.) 100,000
20
cont.
Mixed (synth.)
100,000
9
cont.
Spambase
4,599
57
cont.
Adult
48,840
54
cont., cat.
HAR (binary)
7,450
562
cont.
KDD (sample)
100,000
41
cont., cat.
Dota
102,944
116
cat.
MNIST (binary)
10,398
784
cont.
TABLE II HYPERPARAMETERS
IIOF ERICS PER DATA SET set of parameters throughout all experiments. Note that all related models require classifications of a predictive model. To this end, we trained a Very Fast Decision Tree (VFDT)Data Set
M
W
β
Epochs LR µ LR σ
SEA
75
50
0.0001 10
0.01
0.01
Agrawal
100 50
0.001
10
0.01
0.01
Hyperplane 100 50
0.0001 10
0.01
0.01
Mixed
100 50
0.0001 10
0.1
0.01
Spambase
35
25
0.001
10
0.1
0.01
Adult
50
50
0.001
10
0.1
0.01
HAR
25
50
0.001
10
0.1
0.01
KDD
50
50
0.0001 10
0.01
0.01
Dota
75
50
0.0001 10
0.01
0.01
MNIST
25
20
0.001
50
0.1
0.01
TABLE III AVERAGE
IIIDELAY IN NUMBER OF BATCHESDrift Detection Models
Datasets
ERICS
ADWIN DDM
EWMA FHDDM
MDDM RDDM
SEA
52.75
34.55
16.45
0.26
178.61
178.58
7.42
Agrawal
71.33
16.80
0.00
0.31
0.023
0.20
137.22
Hyperplane 2.00
42.27
2.23
2.36
11.24
11.22
2.42
Mixed
34.00
27.95
0.34
11.55
75.26
75.25
227.98
Spambase
7.35
79.0
60.23
29.96
71.63
71.58
117.45
Adult
2.61
63.15
488.39
172.57
488.39
488.39
488.39
HAR
13.05
372.45
372.45 1.55
372.45
372.45
77.10
KDD
22.01
50.35
0.00
0.55
100.25
100.21
13.21
Dota
44.04
25.32
514.72 43.40
3.56
3.54
116.46
Mean
27.68
79.09
161.65 29.17
144.60
144.60
131.96
Rank
1
3
7
2
5
5
4
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| [
"https://github.com/haugjo/erics."
]
|
[
"Mul-timodal Emergent Fake News Detection via Meta Neural Process Networks",
"Mul-timodal Emergent Fake News Detection via Meta Neural Process Networks"
]
| [
"Yaqing Wang ",
"Fenglong Ma ",
"Haoyu Wang ",
"Kishlay Jha ",
"Jing Gao ",
"\nPurdue University\nWest LafayetteIndianaUSA\n"
]
| [
"Purdue University\nWest LafayetteIndianaUSA"
]
| [
"Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery"
]
| Fake news travels at unprecedented speeds, reaches global audiences and puts users and communities at great risk via social media platforms. Deep learning based models show good performance when trained on large amounts of labeled data on events of interest, whereas the performance of models tends to degrade on other events due to domain shift. Therefore, significant challenges are posed for existing detection approaches to detect fake news on emergent events, where large-scale labeled datasets are difficult to obtain. Moreover, adding the knowledge from newly emergent events requires to build a new model from scratch or continue to fine-tune the model, which can be challenging, expensive, and unrealistic for real-world settings. In order to address those challenges, we propose an end-to-end fake news detection framework named MetaFEND, which is able to learn quickly to detect fake news on emergent events with a few verified posts. Specifically, the proposed model integrates meta-learning and neural process methods together to enjoy the benefits of these approaches. In particular, a label embedding module and a hard attention mechanism are proposed to enhance the effectiveness by handling categorical information and trimming irrelevant posts. Extensive experiments are conducted on multimedia datasets collected from Twitter and Weibo. The experimental results show our proposed MetaFEND model can detect fake news on never-seen events effectively and outperform the state-of-the-art methods.CCS CONCEPTS• Computing methodologies → Artificial intelligence; • Information systems → Web applications. KEYWORDS meta-learning; fake news detection; natural language processing ACM Reference Format: | 10.1145/3447548.3467153 | [
"https://arxiv.org/pdf/2106.13711v1.pdf"
]
| 224,835,228 | 2106.13711 | 7a24ba5f8fb525145fc951301035b6051ce7f25f |
Mul-timodal Emergent Fake News Detection via Meta Neural Process Networks
2021
Yaqing Wang
Fenglong Ma
Haoyu Wang
Kishlay Jha
Jing Gao
Purdue University
West LafayetteIndianaUSA
Mul-timodal Emergent Fake News Detection via Meta Neural Process Networks
Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery
the 27th ACM SIGKDD Conference on Knowledge Discovery202110.1145/3447548.3467153Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). KDD '21, August 14-18, 2021, Virtual Event, Singapore and Data Mining (KDD'21), August 14-18, 2021, Virtual Event, Singapore. ACM, New York, NY, USA, 9 pages. https://
Fake news travels at unprecedented speeds, reaches global audiences and puts users and communities at great risk via social media platforms. Deep learning based models show good performance when trained on large amounts of labeled data on events of interest, whereas the performance of models tends to degrade on other events due to domain shift. Therefore, significant challenges are posed for existing detection approaches to detect fake news on emergent events, where large-scale labeled datasets are difficult to obtain. Moreover, adding the knowledge from newly emergent events requires to build a new model from scratch or continue to fine-tune the model, which can be challenging, expensive, and unrealistic for real-world settings. In order to address those challenges, we propose an end-to-end fake news detection framework named MetaFEND, which is able to learn quickly to detect fake news on emergent events with a few verified posts. Specifically, the proposed model integrates meta-learning and neural process methods together to enjoy the benefits of these approaches. In particular, a label embedding module and a hard attention mechanism are proposed to enhance the effectiveness by handling categorical information and trimming irrelevant posts. Extensive experiments are conducted on multimedia datasets collected from Twitter and Weibo. The experimental results show our proposed MetaFEND model can detect fake news on never-seen events effectively and outperform the state-of-the-art methods.CCS CONCEPTS• Computing methodologies → Artificial intelligence; • Information systems → Web applications. KEYWORDS meta-learning; fake news detection; natural language processing ACM Reference Format:
INTRODUCTION
The recent proliferation of social media has significantly changed the way in which people acquire information. According to the 2018 Pew Research Center survey, about two-thirds of American adults (68%) get news on social media at least occasionally. The fake news on social media usually take advantage of multimedia content which contain misrepresented or even forged images, to mislead the readers and get rapid dissemination. The dissemination of fake news may cause large-scale negative effects, and sometimes can affect or even manipulate important public events. Recent years have witnessed a number of high-impact fake news spread regarding terrorist plots and attacks, presidential election and various natural disasters. Therefore, there is an urgent need for the development of automatic detection algorithms, which can detect fake news as early as possible to stop the spread of fake news and mitigate its serious negative effects. Task Challenges. Thus far, various fake news detection methods, including both traditional learning [5,32] and deep learning based models [21-23, 26, 28, 35] have been exploited to identify fake news. Despite the success of deep learning models with large amounts of labeled datasets, the algorithms still suffer in the cases where fake news detection is needed on emergent events. Due to the domain shift in the news events [38], the model trained on past events may not achieve satisfactory performance and thus the new knowledge from emergent events are needed to add into fake news detection models. However, adding the knowledge from newly emergent events requires to build a new model from scratch or continue to fine-tune the model on newly collected labeled data, which can be challenging, expensive, and unrealistic for real-world settings.
Moreover, fake news usually emerged on newly arrived events where we hardly obtain sufficient posts in a timely manner. In the early stage of emergent events, we usually only have a handful of related verified posts (An example is shown in the Fig. 1). How to leverage a small set of verified posts to make the model learn quickly to detect fake news on the newly-arrived events is a crucial challenge.
Limitations of Current Techniques. To overcome the challenge above, the few-shot learning, which aims to leverage a small set of data instances for quick learning, is a possible solution. One promising research line of few-shot learning is meta-learning [6,19], whose basic idea is to leverage the global knowledge from previous tasks to facilitate the learning on new task. However, the success of existing meta-learning methods is highly associated with an important assumption: the tasks are from a similar distribution and the shared global knowledge applies to different tasks. This assumption usually does not hold in the fake news detection problem as the writing style, content, vocabularies and even class distributions of news on different events usually tends to differ. As it can be observed from Figure 2, the ratios of fake news on events are significantly different. The significant difference across events posts serious challenges on event heterogeneity, which cannot be simply handled by globally sharing knowledge [40]. Another research line of few-shot learning is neural processes [7,8,16], which conduct inference using a small set of data instances as conditioning. Even though neural processes show better generalizablity , they are based on a fixed set of parameters and usually suffer from the limitations like underfitting [16], thereby leading to unsatisfactory performance. These two research lines of models are complementary to each other: the parameter adaptation mechanism in meta-learning can provide more parameter flexibility to alleviate unfitting issues of the neural process. Correspondingly, the neural processes can help handle the heterogeneity challenge for MAML by using a small set of data instances as conditioning instead of encoding all the information into parameter set. Although it is promising to integrate two popular few-shot approaches together, the incompatible operations on the given small set of data instances is the main obstacle for developing the model based on these two. Our Approach. To address the aforementioned challenges, in this paper, we propose a novel meta neural process network (namely MetaFEND) for emergent fake news detection. MetaFEND unifies the incompatible operations from meta-learning and neural process via a simple yet novel simulated learning task, whose goal is to adapt the parameters to better take advantage of given support data points as conditioning. Toward this end, we propose to conduct leave-one-out prediction as shown in the Fig. 3, i.e., we repeatedly use one of given data as target data and the rest are used as context set for conditioning on all the data in support set. Therefore, the proposed model can handle heterogeneous events via event adaption parameters and conditioning on event-specific data instances simultaneously. Furthermore, we incorporate two novel components -label embedding and hard attention -to handle categorical characteristics of label information and extract the most informative instance as conditioning despite imbalanced class distributions of news events. Experimental results on two large real-world datasets show that the proposed model effectively detect fake news on new events with a handful of posts and outperforms the state-of-the-art approaches. Our Contributions. The main contributions of this paper can be summarized as follows:
• We recognize the challenges of fake news detection on emergent events and formulate the problem into a few-shot learning setting. Towards this end, we propose an effective meta neural process framework to detect fake news on emergent events with a handful of data instances. • The proposed MetaFEND method fuses the meta-learning method and neural process models together via a simulated learning task design. We also propose two components label embedding and hard attention to handle categorical information and select the formative instance respectively. The effects of two components are investigated in the experiments. • We empirically show that the proposed method MetaFEND can effectively identify fake news on various events and largely outperform the state-of-the-art models on two realworld datasets.
BACKGROUND
We define our problem and introduce preliminary works in this section.
Problem Formulation
There are many tasks related to fake news detection, such as rumor detection [14] and spam detection [29]. Following the previous work [28,30], we specify the definition of fake news as news which is intentionally fabricated and can be verified as false. In this paper, we tackle fake news detection on emergent events and make a practical assumption that a few labeled examples are available per event. Our goal is to leverage the knowledge learned from past events to conduct effective fake news detection on newly arrived events with a few examples. More formally, we define the fake news detection following the few-shot problem.
Few-shot Fake News Detection Let E denote a set of news events. In each news event ∼ E, we have a few labeled posts on the event . The core idea of few-shot learning is to use episodic classification paradigm to simulate few-shot settings during model training. In each episode during the training stage, the labeled posts are partitioned into two independent sets, support set and query set. Let
Preliminary Work
MAML. We first give an overview of MAML method [6], a representative algorithm of gradient-based meta-learning approaches, and take few-shot fake news detection as an example. The meta-learning procedure is split into two stages: meta-training and meta-testing.
During the meta-training stage, the baseline learner is adapted to specific event as with the help of the support set {X , Y }. Such an event specific learner is evaluated on the corresponding query set {X , Y }. The loss L ( , {X , Y }) on {X , Y } is used to update the parameters of baseline learner . During the meta-testing stage, the baseline learner is adapted to the testing event ′ using the procedure in meta-training stage to obtain event specific parameters ′ , which is employed to make predictions on the query set {X ′ , Y ′ } of event ′ .
MAML update parameter vector using one or more gradient descent updates on event . For example, when using one gradient update:
= ( , {X , Y }) = − ▽ L ( , {X , Y }).
The model parameters are trained by optimizing for the performance of with respect to across events sampled from (E). More concretely, the meta-objective is as follows:
min ∑︁ ∼E L ( ) = ∑︁ ∼E L ( − ▽ L ( ,{X ,Y }) , {X , Y }).
Limitations of MAML. The MAML can capture task uncertainty via one or several gradient updates. However, in fake news detection problem, when events are heterogeneous, the event uncertainty is difficult to encode into parameters via one or several gradient steps. Moreover, even if given support data and query data of interest are from the same event, there is no guarantee that they are all highly related to each other. In such a case, the parameter adaption on fake news detection loss on support set may be misleading for some posts. Conditional Neural Process (CNP). The CNP includes four components: encoder, feature extractor, aggregator and decoder. The basic idea of conditional neural process is to make predictions with the help of support set {X , Y } = { , , , } =1 as context. The dependence of a CNP on the support set is parametrized by a neural network encoder, denoted as (·). The encoder (·) embeds each observation in the support set into feature vector, and the aggregator agg(·) maps these feature vectors into an embedding of fixed dimension. In CNP, the aggregation procedure is a permutation-invariant operator like averaging or summation. The query data of interest , is fed into feature extractor ℎ(·) to get the feature vector. Then the decoder (·) takes the concatenation of aggregated embedding and given target data , as input and output the corresponding prediction as follows:
( , |{X , Y }, , ) = agg( ({X , Y })) ⊕ ℎ( , ) .
where ⊕ is concatenation operator. Limitations of CNP. One widely recognized limitation of CNP is underfitting [16]. For different context data points, their importance is usually different in the prediction. However, the aggregator of CNP treats all the support data equally and cannot achieve query-dependent context information. Moreover, the CNP simply concatenates the input features and numerical label values of posts together as input, ignoring the categorical characteristics of labels.
METHODOLOGY
In this paper, we study how to develop an effective model which can identify fake news on emergent events with a small set of labeled data. To this end, we propose a meta neural process framework which can fuse meta-learning and neural process methods together via a simulated task. To tackle the challenges brought by heterogeneous news events, we further propose a label embedding component to handle categorical labels and a hard attention component, which can select the most informative information from the support set with imbalanced class distributions. In the next subsection, we introduce our overall design and architecture.
Meta-learning Neural Process Design
As shown in Figure 3, our proposed framework includes two stages: event adaptation and detection. The event adaptation stage is to adapt the model parameters to specific event with the help of the support set. The detection stage is to detect fake news on the given event with the help of the support and the adapted parameter set. Event adaption. We take the -th support data { , , , } as an example, in the event adaption stage, the { , , , } is used as target data and the rest of support set {X , Y } \ { , , , } are used as context set accordingly. The context set {X , Y } \ { , , , } and target data , are fed into the proposed model to output the prediction. The loss can be calculated between the predictionˆ, and the corresponding label , . For simplicity, we use to represent all the parameters included in the proposed model. Then, our event adaption objective function on the support set can be represented as follows:
L = ∑︁ log ( , |{X , Y } \ { , , , }, , ).(1)
We then update parameters one or more gradient descent updates on L for event . For example, when using one gradient update:
= − ▽ L .(2)
Detection stage. The proposed model with event-specific parameter set takes query set X and entire support set {X , Y } as input and outputs predictionsŶ for query set X . The corresponding loss function in the detection stage can be represented as follows:
L = log ( | , , ).(3)
Through this meta neural process, we can learn an initialization parameter set which can rapidly learn to use given context inputoutputs as conditioning to detect fake news on newly arrived events. Neural Network Architecture. From Figure 3, we can observe that the network structures used in these two stages are the same, Figure 3: The proposed framework MetaFEND. The proposed framework has two stages: event adaption and detection. During the event adaption stage, the model parameter set is updated to event-specific parameter set . During the detection stage, the event-specific parameter set is used to detect fake news on event . ⊕ denotes concatenation operation and ⊗ means element-wise product.
including feature extractor, label embedding, aggregator and detector. The feature extractor is a basic module which can take posts as input and output corresponding feature vectors. Label embedding component is to capture semantic meanings of labels. Then we use an aggregator to aggregate these information into a fixed dimensional vector, namely context embedding, which is used as reference for fake news detection. Thereafter both the context embedding and target feature vector are fed into detector to output a vector. The final prediction is based on the similarities between this output vector and label embeddings. In the following subsections, we use event adaption to introduce the details of each component in our proposed model. For simplicity, we omitted superscript and in the illustrations about components.
Feature Extractor
From Figure 3, we can observe that feature extractor is a basic module to process raw input. Following the prior works [35,38], our feature extractor consists of two parts: textual feature extractor and visual feature extractor. For a minor note, the feature extractor is a plug-in component which can be easily replaced by other stateof-the-art models. Textual feature extractor. We adopt convolutional neural network [17], which is proven effective in the fake news detection [35,38], as textual feature extractor. The input of the textual feature extractor is unstructured news content, which can be represented as a sequential list of words. For the -th word in the sentence, we represent it by the word embedding vector which is the input to the convolutional neural network. After the convolutions neural network, we feed the output into a fully connected layer to adjust the dimension to dimensional textual feature vector. Visual feature extractor. The attached images of the posts are inputs to the visual feature extractor. In order to efficiently extract visual features, we employ the pretrained VGG19 [31] which is used in the multi-modal fake news works [13,35]. On top of the last layer of VGG19 network, we add a fully connected layer to adjust the dimension of final visual feature representation to the same dimension of textual feature vector . During the joint training process with the textual feature extractor, we freeze the parameters of pre-trained VGG19 neural network to avoid overfitting.
For a multimedia post, we feed the text and image of the example into textual and visual feature extractor respectively. The output of two feature extractors are concatenated together to form a feature vector. For the target data , , we denote its feature vector as h , . For the context data , where ≠ , we denote its feature vector as c , ∈ C .
Aggregator
To construct context embedding for target data, we need to design an aggregator which satisfies two properties: permutation-invariant and target-dependent. To satisfy the two properties, we choose to adopt the attention mechanism which can compute weights of each observations in context set with respect to the target and aggregates the values according to their weights to form the new value accordingly. Attention mechanism. In this paper, we use scaled dot-product attention mechanism [33]. This attention function can be described as mapping a query and a set of key-value pairs to an output, where the query Q, keys K, values V, and output are all vectors. In our problem, for the target data , and the context set X \{ , } = { , } =1, ≠ on event . We use the the target feature vector h , ∈ R 1× after linear transformation as query vector Q , the context feature vector C = [ ,1 , ..., , ] ∈ R × after linear transformation as the Key vector K. For the context set, we represent its label in-
formation Y \ { , } = { , } =1, ≠ by semantic embeddings as vec = {vec , } =1, ≠ .
The details of label embedding are introduced in the next subsection. Then we concatenate context feature vector and label embedding as C e ⊕ vec = [c ,1 ⊕ vec ,1 , ..., c , ⊕ vec , ] ∈ R ( −1)×2 . The concatenated embedding after linear transformation is used as value vector V. We represent Q , V, K as follows:
Q = W h , , K = W C , V = W (C ⊕ vec ), where W ∈ R × , W ∈ R × and W ∈ R 2 × .
The output is computed as a weighted sum of the values, where the weight assigned to each value is computed by dot-product function of the query with the corresponding key. More specifically, attention function can be represented as follows:
a = softmax( Q K √ )(4)
Attention(Q , K, V) := V.
(5) Limitation of Soft-Attention. The attention mechanism with soft weight values is categorized into soft-attention. However, softattention cannot effectively trim irrelevant data especially when we have a context set with an imbalanced class distribution shown in Fig. 2. Moreover, we show a case study in the experimental section for a better illustration. Hard-Attention. To overcome the limitation of soft-attention, we propose to select the most related context data point instead of using weighted average. To enable argmax operation to be differentiable, we use Straight-Through (ST) Gumbel SoftMax [12] for discretely sampling the context information given target data. We introduce the sampling and arg max approximations of ST Gumbel SoftMax procedure next.
The Gumbel-Max trick [9] provides a simple and efficient way to draw samples z from a categorical distribution with class probabilities. In our problem, for the -th target data point , with context set X \ { , } = { , } =1, ≠ , the class probabilities can be obtained from the weight vector a = [ ,1 , ..., , ] from dotproduct attention mechanism according to Eq. 4. Because arg max operation is not differentiable, we use the softmax function as a continuous, differentiable approximation to arg max, and generate K-dimensional sample vectors P = [ ,1 , ,2 .., , ] as follows:
where is a temperature parameter, = − log(− log( )) is the Gumbel noise and is generated by a certain noise distribution (e.g., ∼ N (0, 1)). As the softmax temperature approaches 0, the Gumbel-Softmax distribution becomes identical to the categorical distribution. Moreover, Straight-Through (ST) gumbel-Softmax takes different paths in the forward and backward propagation, so as to maintain sparsity yet support stochastic gradient descent. Through gumbel-softmax, the hard-attention mechanism is able to draw the most informative sample based on weight vectors from P for given target sample , .
The hard-attention can trim the irrelevant data points and select the most related data point, denoted as c , ⊕ v , ∈ R 2 . Besides the hard-attention mechanism, the aggregator includes an additional fully connected layer on top of hard-attention to adjust the dimension. The c , ⊕ v , is fed into this fully connected layer to output context embedding r , ∈ R .
Detector based on Label Embedding
Categorical characteristic of label information. The context information includes posts and their corresponding labels. The existing works like CNP [7] and ANP [16] usually simply concatenate the input features and numerical label values together as input to learn a context embedding via a neural network. Such operation discards the fact that label variables are categorical. Moreover, this operation tends to underestimate the importance of labels as the dimension of input features is usually significantly larger than that of single dimensional numerical value. To handle categorical characteristic, we propose to embed labels into fixed dimension vectors inspired by word embedding [24]. We define two embeddings vec(fake) and vec(real) for the labels of fake news and real news respectively. For example, given the -th post , on event , the corresponding label is fake and its label embedding vector is vec(fake), and we denote the label embedding of , as vec , . To ensure that the label embedding can capture the semantic meanings of corresponding labels, we propose to use embeddings vec(fake) and vec(real) in the detector as metrics and output predictions are determined based on metric matching.
The detector is a fully-connected layer which takes target feature vector and context embedding as inputs and outputs a vector that has the same dimensionality as that of the label embedding. More specifically, for -th target data, the context embedding r , and target feature vector h , are concatenated. Then the detector takes r , ⊕ h , ∈ R 2 as input and produces a output vector o , ∈ R . The similarities between output o , from our model and label embeddings vec(fake) and vec(real) are calculated as follows:
similarity(o , , vec(fake)) = o , • vec(fake) ,(7)
similarity(o , , vec(real)) = o , • vec(real) .
The two similarity scores are then mapped into [0, 1] as probabilities via softmax. The trainable label embedding capture semantic meaning of labels and can generalize easily to new events with the help of adaptation step according to Eq. 2.
Algorithm Flow
After introducing the meta-learning neural process design, feature extractor, label embedding, aggregator and detector components, we present our algorithm flow. As it can be observed from Figure 3, when tackling an event , our proposed framework MetaFEND has two stages: event adaption and detection. In more details, our proposed model adapts to the specific event according to Eq. 2 and then the event-specific parameter is used in the fake news detection on given event. The algorithm flow is same in the two stages and we use event adaption stage as an example to illustrate this procedure.
Our input includes handful instances as context set {X , Y } \ { , , , } and , as target data. We first feed X \ { , } into feature extractor and get context feature representations C . The context feature representations C is then concatenated with label embedding vec of Y . In the target side, the target data , is also fed into feature extractor to get representation as h , . The aggragator component aggregates h , , C and vec as introduced in section 3.3 to output context embedding r , ∈ R . Then we concatenate r , with target feature vector h , ∈ R . The concatenated feature goes through the detector which is consisted of a fully connected layer to output a vector o , . The similarity scores between o , and vec(fake), vec(real) are calculated according to Eq. 7 and Eq. 8 respectively. In the end, the similarity scores are mapped to probability values for fake news detection via softamax operation.
EXPERIMENTS
In this section, we introduce the datasets used in the experiments, present the compared fake news detection models, validate the effectiveness and explore some insights of the proposed framework.
Datasets
To fairly evaluate the performance of the proposed model, we conduct experiments on datasets collected from two real-world social media datasets, namely Twitter and Weibo. The detailed description of the datasets are given below: The Twitter dataset is from MediaEval Verifying Multimedia Use benchmark [2], which is used in [13,35] for detecting fake content on Twitter. The Weibo dataset 1 is used in [13,27,35] for detecting multi-modal fake news. The news events are included in the Twitter dataset and we follow the previous works [13,27,35] to obtain events on Weibo via a single-pass clustering method [14]. In the two datasets above, we only keep the events which are associated with more than 20 posts and randomly split the posts on same event into support and query data. To validate performance of the models on newly emergent events, we ensure that the training and testing sets do not contain any common event. We adopt Accuracy and F1 Score as evaluation metrics. These two datasets cover diverse news events and thus can be used as good test-grounds for evaluation of fake news detection on heterogeneous events.
Baselines
To validate the effectiveness of the proposed model, we choose baselines from multi-modal models and the few-shot learning models. For the multi-modal models, we fine-tune them on support set from events in the testing data for a fair comparison. In the experiments, we have the 5-shot and 10-shot settings. In our problem, 5-shot setting refers to that 5 labeled posts are provided as support set. Fine-tune models. All the multi-modal approaches take the information from multiple modalities into account, including VQA [1], att-RNN [13] and EANN [35]. In the fine-tune setting, the training data including labeled support data and labeled query data is used to train the baselines. In the testing stage, the trained models are first fine-tuned on the labeled support data of given event, and then make predictions for testing query data. (1) VQA [1]. Visual Question Answering (VQA) model aims to answer the questions based on the given images and is used as a baseline for multimodal fake news in [13]. (2) att-RNN [13]. att-RNN is the state-of-theart model for multi-modal fake news detection. It uses attention mechanism to fuse the textual, visual and social context features.
In our experiments, we remove the part dealing with social context information, but the remaining parts are the same. (3) EANN [35]. EANN is one of the state-of-the-art models for fake news detection. It consists of three components: feature extractor, event discriminator and fake news detector. It captures shared features across different events of news to improve generlziation ability. Few-shot learning models. We use CNP [7], ANP [16], MAML [6] and Meta-SGD [19] as few-shot learning baselines. (1) CNP [7]. Conditional neural process is the state-of-the-art model for few-shot learning. It combines neural network and gaussian process by using a small set of input-output pairs as context to output predication for given input of data. (2) ANP [16]. Attentive neural process belongs to the family of neural process which outputs prediction based on concatenation of learned distribution of context, context features and given input. (3) MAML [6]. Model-aganostic Meta-learning is a representative optimization-based meta-learning model. The mechanism of MAML is to learn a set of shared model parameters across different tasks which can rapidly learn novel task with a small set of labeled data. (4) Meta-SGD [19]. Meta-SGD is one of the state-of-the-art meta learning method for few-shot learning setting. Besides a shared global initialized parameters as with MAML, it also learns step sizes and update direction during the training procedure.
The proposed model share the same feature extractor backbone with EANN, CNP, ANP, MAML, Meta-SGD to study the effects of other designs in addition to benefits of the feature extractor backbone. Implementations In the proposed model, the 300 dimensional FastText pre-trained word-embedding weights [3] are used to initialize the parameters of the embedding layer. The window size of filters varies from 1 to 5 for textual CNN extractor. The hidden size of the fully connected layer in textual and visual extractor and dimension are set as 16 which is searched from options {8, 16, 32, 64}. decays from 1 to 0.5 as the suggested way in [12]. The gradient update step is set to 1 an inner learning rate is set to 0.1 for fine-tune models: MAML, Meta-SGD and our proposed framework MetaFEND. We implement all the deep learning baselines and the proposed framework with PyTorch 1.2 using NVIDIA Titan Xp GPU. For training models, we use Adam [18] in the default setting. The learning rate is 0.001. We use mini-batch size of 10 and training epochs of 400. Table 2 shows the performance of different approaches on the Twitter and Weibo datasets. We can observe that the proposed framework MetaFEND achieves the best results in terms of most of the evaluation metrics in both 5-shot and 10-shot settings. Twitter. On the Twitter dataset in 5-shot setting, compared with CNP, ANP incorporates the attention mechanism and hence can achieve more informative context information. Due to the heterogeneity of events, it is not easy for Meta-SGD to learn a shareable learning directions and step size across all events. Thus, Meta-SGD's performance is lower than MAML's in terms of accuracy. Compared with all the baselines, MetaFEND achieves the best performance in terms of most the metrics. Our proposed model inherits the advantages of MAML to learn a set of parameters which can rapidly learn to detect fake news with a small support set. Moreover, MetaFEND Table 2: The performance comparison of models for fake news detection on the Twitter and Weibo datasets under 5-shot and 10-shot settings. Accuracy and F1 score of models are followed by standard deviation. The percentage improvement (↑) of MetaFEND over the best baseline per setting is in the last row. EANN, CNP, ANP, MAML, Meta-SGD and MetaFEND share the same feature extractor as the backbone. can use the support data as conditioning set explicitly to better capture the uncertainty of events and thus it is able to achieve more than 5% improvement compared with MAML in terms of accuracy. In the 10-shot setting, as the size of give support data increases, the soft attention mechanism of ANP unavoidably incorporates the irrelevant data points. In contrast, the proposed model MetaFEND employs the hard-attention mechanism to trim irrelevant data points from context set and significantly outperforms all the baselines in terms of all the metrics. Weibo. Compared with the Twitter data, the Weibo dataset has different characteristics. On the Weibo dataset, most of the posts are associated with different images. Thus, we can evaluate the performance of models under the circumstance where support datasets do not include direct clues with query set. As EANN tends to ignore event-specific features, it achieves the lowest accuracy among finetune models in 10-shot setting. For the few-shot models, ANP and CNP achieves better performance compared with gradient-based meta-learning methods MAML and Meta-SGD. This is because the parameter adaptation may not be effective when support data set and query set do not share the same patterns. Compared with ANP in 5-shot setting, our proposed method MetaFEND achieves 4.39% improvement in terms of accuracy and 5.51% improvement in terms of F1 score. The reason is that our MetaFEND can learn a base parameter which can rapidly learn to use a few examples as reference information for fake news detection. Thus, our proposed model enjoys the benefits of neural process and meta-learning model families.
Performance Comparison
Ablation Study
We show ablation study to analyze the role of Hard-Attention and label embedding components. soft-attention mechanism which unavoidably incorporates unrelated data points and significantly outperforms the soft-attention in terms of accuracy score. Thus, we can conclude that hard-attention mechanism can take effectively advantage of support set, and the superiority is more significant as we enlarge size of support set. w/o Label Embedding v.s. w/ Label Embedding. To analyze the role of label embedding in the proposed model, we design MetaFEND's corresponding reduced model by replacing label embedding with label value 0 or 1. Accordingly, we change the multiplication between output with label embedding to a binary-class fully connected layer to directly output the probabilities. Figure 4b shows the results in terms of accuracy score. In Figure 4b, "w/o label embedding" denotes that we remove the label embedding, and "w label embedding" denotes the original approach. We can observe that the accuracy score of "w label embedding" is greater than "w/o label embedding" in 5-shot and 10-shot settings, demonstrating the effectiveness of label embedding
Case Study
In order to illustrate the challenges of emergent fake news detection and how our model handles challenges, we show one example in 5-shot learning setting as case study in Fig. 5. As it can be observed, the four of five news examples in the support set are real news. Due to imbalanced class condition in the support set, it is difficult for Soft-Attention to provide correct prediction for news of interest in the query set. More specifically, Fig. 5 shows the attention score values (red color) between examples in support set and query set based on multi-modal features. Although the first example with largest attention score value is most similar to news example in the query set, the majority of context information is from the other four examples due to imbalanced class distribution. Such an imbalanced class distribution leads to incorrect prediction for Soft-Attention. The Hard-Attention mechanism can achieve correct result by focusing on the most similar sample in the support set. Through this example, we can also observe the necessity of event adaption stage. The posts and images for the same event are very similar and difficult to distinguish. Without event adaption stage, the model cannot capture informative clues to make correct predictions.
RELATED WORK
In this section, we briefly review the work related to the proposed model from fake news detection and few-shot learning.
Fake News Detection
Many fake news detection algorithms try to distinguish news according to their features, which can be extracted from social context and news content. (1) Social context features represent the user engagements of news on social media [30] such as the number of followers, hash-tag (#), propagation patterns [39] and retweets. However, social context features are very noisy, unstructured and labor intensive to collect. Especially, it cannot provide sufficient information for newly emerged events. (2) Textual features are statistical or semantic features extracted from text content of posts, which have been explored in many literatures of fake news detection [4,10,30]. Unfortunately, linguistic patterns are not yet well understood, since they are highly dependent on specific events and corresponding domain knowledge [28]. To overcome this limitation, approaches like [20][21][22][23]26] propose to use deep learning models to identify fake news and have shown the significant improvements.
(3) Visual features have been shown to be an important indicator for fake news detection [15,30]. The basic features of attached images in the posts are explored in the work [11,15,25].
In this paper, we consider multi-modal features when identifying fake news on social media. To tackle multi-modal fake news detection, in [13], the authors propose a deep learning based fake news detection model, which extracts the multi-modal and social context features and fuses them by attention mechanism. To detect fake news on never-seen events, Wang et al. [35] propose an event-adversarial neural network (EANN) which can capture eventinvariant features for fake news detection. However, EANN cannot take advantage of a small set of labeled data to further capture event specification and thus is not suited for our task.
Few-Shot Learning
Meta-learning has long been proposed as a form of learning that would allow systems to systematically build up and re-use knowledge across different but related tasks [34,36,37]. MAML [6] is to learn model initialization parameters that are used to rapidly learn novel tasks with a small set of labeled data. Following this direction, besides initialization parameters, Meta-SGD [19] learns step sizes and updates directions automatically in the training procedure. As tasks usually are different in the real setting, to handle task heterogeneity, HSML [40] customizes the global shared initialization to each cluster using a hierarchical clustering structure. The event heterogeneity is widely observed for fake news detection, where nonexistence of hierarchical relationship in news events makes this task more challenging. Neural process approaches [7,8,16] combine stochastic process and neural network to handling task heterogeneity by conditioning on a context set. Conditional Neural Process (CNP) [7] and Neural Process (NP) [8] use neural networks to take input-output pairs of support set as conditioning for inference, incorporating task specific information. However, these two works aggregate the context set by average or sum, ignoring different importance among context data samples and thereby leading to unsatisfactory performance. Attentive Neural Process (ANP) [16] incorporates attention mechanism into Neural Process to alleviate such a issue. However, ANP still suffers from underfitting issue due to fixing parameters for different tasks. Additionally, ANP directly concatenates the label numeric values with feature representation, discarding the categorical characteristics of label information.
Different from existing works, our proposed framework maintains the parameter flexibility following the principle of metalearning and inherits generalization ability to handle event heterogeneity from neural processes. Moreover, we incorporate label embedding component to handle categorical characteristics of label information and utilize hard attention to extract most informative context information. Thus, our proposed model enjoys the benefits of two model families without suffering their limitations.
CONCLUSIONS
In this work, we study the problem of fake news detection on emergent events. The major challenge of fake news detection stems from newly emerged events on which existing approaches only showed unsatisfactory performance. In order to address this issue, we propose a novel fake news detection framework, namely MetaFEND, which can rapidly learn to detect fake news for emergent events with a few labeled examples. The proposed framework can enjoy the benefits of meta-learning and neural process model families without suffering their own limitations. Extensive experiments on two large scale datasets collected from popular social media platforms show that our proposed model MetaFEND outperforms the state-of-the-art models.
Fake news can manipulate important public events and becomes a global concern. If the fake news detection algorithm can function as intended, it is beneficial to prevent the spread of fake news in the early stage and correspondingly many negative public events caused by fake news may be avoided. However, we are also aware that automatic detection may suppress the public discussion. The failure modes may lie in the negation cases: if someone tries to spot the fake news by citing false information contents, the automatic algorithm may not understand the logic behind the post and incorrectly identify it as fake news. The bias may be unavoidable included in the dataset especially when the events are controversial or lacking a clear standard for annotation. Our proposed model explicitly uses the labeled sample as reference information and thus it is possible to replace the incorrect annotated support set by correct ones to correct the bias. To reduce harm brought by the automatic algorithm, both technology and human review are needed and an effective user appeal system should be employed in case the incorrect detection happened.
Figure 1 :
1Fake news examples on an emergent event Boston Bombing from Twitter.
Figure 2 :
2The number of events with respect to different percentages of fake news.
{X , Y } = { , , , } =1 represent the support set, and {X , Y } = { , , , } = +1 be the query set. The model is trained to learn to conduct fake news detection on the query set {X , Y } given the support set {X , Y }. During the inference stage, labeled posts are provided per event. For each event , the model leverages its corresponding labeled posts as support set {X , Y } = { , , , } =1 to conduct fake news detection on given event .
(log( , ) + )/ ) , ≠ exp((log( , ) + )/ )
Figure 4 :
4Soft-Attention v.s. Hard-Attention. To intuitively illustrate the role of hard-attention mechanism in the proposed model, we show ablation study by replacing hard-attention with soft-attention. Then we repeatedly run the new designed model on the Twitter dataset five times in 5-shot and 10-shot settings respectively and report the average of accuracy values. The results are show in the Figure 4. From Figure 4a, we can observe that accuracy scores of "Hard-Attention" in 5-shot and 10-shot settings are greater than those of "Soft-Attention" respectively. As the number of support set increases, hard-attention mechanism does not have the limitation of The ablation study about (a) Soft-Attention and Hard-Attention and (b) Label Embedding.
Figure 5 :
5Fake news examples missed by Soft-Attention but spotted by Hard-Attention
Table 1 :
1The Statistics of the Datasets.Twitter Weibo
# of fake News
6,934
4,050
# of real News
5,683
3,558
# of images
514
7,606
https://github.com/yaqingwang/EANN-KDD18
ACKNOWLEDGMENTThe authors thank the anonymous referees for their valuable comments and helpful suggestions. This work is supported in part by the US National Science Foundation under grants NSF IIS-1553411 and IIS-1956017. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
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| While non-volatile memories (NVMs) provide several desirable characteristics like better density and comparable energy efficiency than DRAM, DRAM-like performance, and disk-like durability, the limited endurance NVMs manifest remains a challenge with these memories. Indeed, the endurance constraints of NVMs can prevent solutions that are commonly employed for other mainstream memories like DRAM from being carried over as-is to NVMs. Specifically, in this work we observe that, Oblivious RAM (ORAM) primitive, the state-ofart solution to tackle memory bus side channel vulnerability, while widely studied for DRAMs, is particularly challenging to implement as-is for NVMs as it severely affects endurance of NVMs. This is so, as the inherent nature of ORAM primitive causes an order of magnitude increase in write traffic and furthermore, causes some regions of memory to be written far more often than others. This non-uniform write traffic as manifested by ORAM primitive stands to severely affect the lifetime of non-volatile memories (1% of baseline without ORAM) to even make it impractical to address this security vulnerability.To address this challenge, in this work, we propose Egalitarian ORAM (E-ORAM), which discusses an ORAM design for NVMs while achieving close to ideal lifetime of non-volatile memory. To do so, we observe that the inherent nature of ORAM primitive can be exploited to design a wear-levelling algorithm which spreads the additional write traffic generated by ORAM more uniformly across the entire non-volatile memory space. We demonstrate that in comparison to existing state-of-art wear-levelling algorithm which in presence of ORAM can only attain 2.8% lifetime, proposed E-ORAM design attains approximately 91% lifetime allowing it to perform orders of magnitude more reads/writes before system failure while introducing less than 0.02% performance overhead. | 10.48550/arxiv.2304.09411 | [
"https://export.arxiv.org/pdf/2304.09411v1.pdf"
]
| 258,213,007 | 2304.09411 | 70359b374e369d35158a6a5da86f8de6e0e03e89 |
Egalitarian ORAM: Wear-Leveling for ORAM
Yi Zheng
Pennsylvania State University
Aasheesh Kolli [email protected]@amd.com
Pennsylvania State University
Google, Inc
Shaizeen Aga
Advanced Micro Devices
Inc
Egalitarian ORAM: Wear-Leveling for ORAM
While non-volatile memories (NVMs) provide several desirable characteristics like better density and comparable energy efficiency than DRAM, DRAM-like performance, and disk-like durability, the limited endurance NVMs manifest remains a challenge with these memories. Indeed, the endurance constraints of NVMs can prevent solutions that are commonly employed for other mainstream memories like DRAM from being carried over as-is to NVMs. Specifically, in this work we observe that, Oblivious RAM (ORAM) primitive, the state-ofart solution to tackle memory bus side channel vulnerability, while widely studied for DRAMs, is particularly challenging to implement as-is for NVMs as it severely affects endurance of NVMs. This is so, as the inherent nature of ORAM primitive causes an order of magnitude increase in write traffic and furthermore, causes some regions of memory to be written far more often than others. This non-uniform write traffic as manifested by ORAM primitive stands to severely affect the lifetime of non-volatile memories (1% of baseline without ORAM) to even make it impractical to address this security vulnerability.To address this challenge, in this work, we propose Egalitarian ORAM (E-ORAM), which discusses an ORAM design for NVMs while achieving close to ideal lifetime of non-volatile memory. To do so, we observe that the inherent nature of ORAM primitive can be exploited to design a wear-levelling algorithm which spreads the additional write traffic generated by ORAM more uniformly across the entire non-volatile memory space. We demonstrate that in comparison to existing state-of-art wear-levelling algorithm which in presence of ORAM can only attain 2.8% lifetime, proposed E-ORAM design attains approximately 91% lifetime allowing it to perform orders of magnitude more reads/writes before system failure while introducing less than 0.02% performance overhead.
Introduction
Memory system innovations continue to be fueled by the everpresent and ever-growing demands for increased memory capacity and bandwidth. One such innovation is Non-volatile memories (NVMs) (including PCM [22], STTRAM [20], and ReRAM [39]). These memories offer several desirable characteristics like better density and comparable energy efficiency than DRAM, DRAM-like performance, and disk-like durability. Consequently, NVMs potentially stand to play an important role in future memory systems.
However, deploying NVMs widely is not devoid of challenges. One of the challenges associated with NVMs is the limited endurance NVMs manifest, that is, the physical properties of memory cells in NVMs dictate a limit on number of writes to the memory cell (typically between 10 7 to 10 8 [13]). Beyond this limit, the memory cell may lose the ability to change state causing data errors. This in turn can lead to system failure when enough memory lines reach endurance limit. This happens when the number of spare memory lines is lower than memory lines that have incurred more writes than dictated by the endurance limit.
We observe in this work that the endurance constraints of NVMs make it impractical to carry-over solutions which are widely studied for mainstream memories like DRAM to NVMs. Specifically, we observe this to be true in the context of ORAM primitive, which is typically studied to address memory bus side channel vulnerability for DRAM systems. To exploit this vulnerability, an attacker taps the memory bus to learn the memory address trace of the program. Prior work [40] has shown that even in presence of data encryption, an attacker can learn sensitive information about the program simply by observing the memory address trace of the program. To tackle this vulnerability, the ORAM primitive accesses an order of magnitude more memory locations on each memory access and further shuffles memory on each access. As such, under ORAM, any memory address trace is computationally indistinguishable from any other address trace of the same length.
However, this very nature of ORAM primitive while effective in addressing memory bus side channel severely affects endurance of NVMs. This is so, as memory lines are less likely to reach endurance limit in presence of uniform memory write traffic (write traffic uniformly distributed across the entire memory space) as compared to non-uniform memory traffic where few memory lines are written more often than others. The most state-of-art implementation of ORAM, Path-ORAM [37], however, manifests exponential write distribution severely affecting NVM endurance. This happens as Path-ORAM organizes memory as a binary tree and on each memory access reads and writes a entire path in this tree. Fig. 1 shows the memory layout of Path ORAM. As shown in Fig. 1, Path ORAM arranges memory as a balanced binary tree structure, where each node is written equally at the same level. From the root to leaf, the expected number of writes to each node decreases exponentially. This leads to an exponential write distribution, where nodes closer to the root node in the memory tree are written far often than leaf nodes in the tree. This non-uniform write traffic as manifested by ORAM primitive stands to severely affect the lifetime of non-volatile memories (1% of baseline without ORAM) so as to even make it impractical to address memory bus side channel for NVMs.
Note that, prior works [28] have observed that even typical application behavior can exacerbate the endurance constraints of NVMs as they manifest non-uniformity in write traffic. As such, to address this, several wear-levelling algorithms have been proposed by prior works [7,11,28,14,5]. The tenet of these algorithms is to spread application write traffic as uniformly as possible across the non-volatile memory space by employing memory line remapping as necessary. However, as we show in this work, the ORAM primitive severely stresses these prior solutions and as such, existing state-of-art wearlevelling algorithm [28] in presence of ORAM can only attain 2.8% lifetime.
To tackle NVM endurance constraints while addressing memory bus side channel vulnerability with ORAM primitive, in this work we make the observation that the inherent nature of ORAM primitive can be exploited to design a wear-levelling algorithm which directly tackles the endurance pressure of ORAM primitive. Specifically, while the Path-ORAM [37] implementation of ORAM primitive by its very nature manifests an exponential write distribution, it does so in a deterministic manner. That is, the expected number of writes to a given memory line is statically known depending on where in the Path-ORAM tree the memory line belongs (e.g., root note is written twice as much as its children). We use this information provided by the inherent nature of the ORAM primitive to augment a state-of-art wear-levelling algorithm, Start-Gap [28] and make it ORAM primitive aware. We term this ORAM design for NVMs which directly tackles NVM endurance constraints, E-ORAM. We show how our proposed design can attain 91% lifetime and perform orders of magnitude more reads/writes before system failure while introducing less than 0.02% performance overhead.
The contributions of this work are: • We observe in this work that the endurance constraints of non-volatile memories (NVMs) make it impractical to implement Oblivious-RAM (ORAM) primitive for NVMs, a state-of-art defense against memory-bus side channel vulnerability, as it leads to a system lifetime of 1% of baseline system without ORAM. • To address this challenge, we make the observation that the inherent nature of ORAM primitive can be exploited to design a wear-levelling algorithm which directly tackles the endurance pressure of ORAM primitive. • We harness the above observation in our proposed ORAM design for NVMs, termed, E-ORAM, in which we augment a state-of-art wear-levelling algorithm, Start-Gap [28] with ORAM primitive awareness. • We demonstrate that while state-of-art wear-levelling algorithm [28] in presence of ORAM can only attain 2.8% lifetime, proposed E-ORAM design attains 91% lifetime allowing it to perform orders of magnitude more reads/writes before system failure while introducing less than 0.02% performance overhead.
Background and Motivation
In this section, we will introduce necessary background to understand this work. To that end, first, we discuss background on non-volatile memories (NVMs) and the endurance limit they manifest followed by a discussion of Start-Gap [28], a state-of-the-art wear-leveling mechanism. Then, we discuss the threat model we assume in this work. After that, we provide background on the most state-of-art implementation of ORAM primitive, Path-ORAM [37]. Finally, we discuss how the Path-ORAM algorithm stresses the endurance of NVMs.
Non-Volatile Memories
Emerging non-volatile memories (NVMs), including PCM, STTRAM and ReRAM [22,20,39] can provide several desirable properties like high density, non-volatility and an access latency close to DRAM [31]. Further, as NVMs are byteaddressable (they provide load/store interface [43]) and nonvolatile, they can make it possible to host recoverable data structures in NVMs instead of accessing them from disk at steeper performance and energy costs. While NVMs have several desirable characteristics making their inclusion in future memory systems possible, several challenges remain. One of the challenges is about their limited write endurance. A typical write endurance ranges from 10 7 to 10 8 [13] due to physical properties of NVM memory cells. Once the limit is passed, data errors may raise because memory cells may no longer be able to change their states, which can in turn lead to system failure if sufficient memory lines reach endurance limit. This happens when the number of spare memory lines is lower than memory lines that have incurred more writes than dictated by the endurance limit.
Several prior works have been proposed to solve the limited write endurance problem of NVMs. These approaches can typically be categorized into write reduction techniques and wear-leveling techniques. Write reduction techniques focuses on reducing the number of NVM writes. For example, Qureshi et al. [29] proposed using PCM as an extended storage of DRAM. Gogte et al. [14] proposed using software management system to migrate frequently accessed pages from NVM to DRAM. On the other hand, wear-leveling techniques focus (a) Initial Status of the ORAM tree, stash, and PosMap. A read request to block C is issued by the application.
(b) Status of the ORAM tree, stash, and PosMap after read accesses to all nodes on the path from root to leaf 0. Block C and D are loaded into the stash. Then, ORAM remaps block C to leaf 1.
(c) Status of the ORAM tree, stash, and PosMap after write back operation. The write back path is the same as the read path. Block D is mapped to leaf 0, so it is pushed to leaf 0. Block C is mapped to leaf 1, as such, unlike before it cannot be placed in leaf block. Block A can be drained from stash to the tree. on spreading memory writes uniformly across all NVM memory space. We discuss several prior wear-leveling techniques and compare them to our proposed work in Section 8.
Baseline Start-Gap
Start-Gap divides NVMs into equal-sized groups. Each group reserves an empty cache line. The CPU maintains a start register and a gap register for each group. Each memory address will be translated based on the value of start and gap registers. The CPU maintains a counter for each group to store the number of writes to that group so far. Upon the counter reaching a threshold, the gap register will be updated, so one of the cache lines in that group will be mapped to a new address. Once the gap register finishes looping through the entire group, the start register will be updated. The algorithm also employs an address randomizer which maps logical addresses to intermediate addresses before further translating them to physical addresses with start and gap registers.
Threat Model
We assume a secure processor where applications can harness isolated execution [23]. We assume that the execution of an application along with its data in the processor structures (registers, caches, on-chip networks) are secure and isolated from other applications. We assume a system where NVM is employed as the main memory. We assume the NVM is untrusted and to defend against cold (re)boot attacks [17], data is encrypted in memory. Further, we also assume an adversary with physical access to the computers and as such, capable of probing the off-chip memory bus [10] to learn the address trace and data being communicated between the processor and NVM. Prior works [40] have shown that even with encrypted data, an attacker can ascertain sensitive secrets about an application simply by observing the address trace of the application. We assume prior solutions for mitigating page-fault side-channel [2,8]. Finally, prior works which address other important vulnerabilities such as power [19], thermal [27], program execution time [41] side-channels, and leaks via communication patterns over the network [35,26] do exist and we assume addressing these vulnerabilities to be outside the scope of this paper.
Path Oblivious RAM
Oblivious RAM [15], is a cryptographic primitive which makes a memory access trace computationally indistinguishable from a random access trace of the same length. While several realizations of this primitive have been proposed to date [9,16,16,21], a backbone for them is Path ORAM [37], the most practical implementation of ORAM primitive. In this section, we provide relevant background on Path-ORAM algorithm (henceforth simply referred to as ORAM).
ORAM arranges memory into a balanced binary tree structure, and each node in the ORAM tree contains a fixed number of slots where each slot can store a single data block (typically a single cache block). ORAM tree has associated with it a utilization factor, referring to the fraction of actual data blocks over total blocks in the tree. Non-data blocks are filled with dummy blocks. Each real data block has a corresponding leaf ID, which indicates that this data block is stored on the path from the root node to the leaf corresponding to the leaf ID. This mapping information is maintained by a position-map (PosMap) structure. All blocks, regardless of real data block or dummy block, are encrypted before being stored into memory. Encryption metadata, such as block ID (the slot within each node), leaf ID, as well as encryption counter, are also maintained by the ORAM tree. ORAM also uses an on-chip structure, stash, to aid in its working.
Next, we use a 3-level ORAM tree in Fig. 2 as an example to show the workings of ORAM. Consider the scenario when the application issues a read request to data block C. Fig 2a shows the initial status of the ORAM tree, stash and PosMap. On receiving the read request, the algorithm first looks up PosMap to find the leaf ID that the block C corresponds to, which is 0 (leaf nodes in the tree are labeled from 0 to 3). Per the ORAM primitive, each program memory access is translated to a read and write of a path from root node to the leaf associated with the block. As such, the algorithm proceeds to read and write path to leaf 0 as depicted in Fig. 2a. To do so, ORAM issues read operations to all nodes along the path from the root node to leaf 0 and loads real data blocks (C and D), into the stash. Post the path read the needed data block is in the stash and is decrypted and supplied to the processor. At this moment, ORAM will remap block C to a newly generated random leaf ID, which is 1 in this case, as shown in Fig. 2b. This remapping confirms that the next access to block C accesses a completely random path in the ORAM tree. Finally, ORAM writes back the same path it read from, draining re-encrypted blocks from the stash as deep as possible. In this example, block D is mapped to leaf 0, which is fine to be pushed into leaf 0. Block A and C are mapped to leaf 1, so the deepest node they can be stored in is the parent node of leaf 0 and 1.
By re-mapping the data block each time it is accessed to a random leaf, ORAM allows for a completely new set of blocks are accessed each time the same data block is accessed. This, along with the fact that each access causes re-encryption of all accessed blocks helps hide address trace of the application.
Challenges in realizing ORAM for NVMs
We discuss in this section how, by its very nature, the ORAM primitive severely stresses endurance of NVMs. This is so, as ORAM manifests a skewed write access pattern as shown in Figure 3. As discussed in section 2.4, memory is organized as a binary tree with ORAM and memory accesses are transformed into read/writes of paths in the tree. This translates to the scenario as depicted in Figure 3, where the nodes closer to the root (lower Node-ID) are written far more often than nodes closer to leaf nodes (higher Node-ID). From the root node, its two child nodes are accessed with equal probability due to the nature of ORAM, so each child node is accessed with half frequency of the root node. This access property applies to all nodes, which causes the number of writes per node in level to decrease exponentially as the level increases linearly. Therefore, we see that leaf nodes are written far less frequently than the root node, while occupying half of the tree.
We depict the stress ORAM's write distribution causes on NVM endurance in Fig. 4 (please see Section 7 for details on our methodology). In Fig. 4 we depict system lifetime for a NVM that implements ORAM. We define system lifetime to be:
Li f etime = #memory writes × ORAM tree path length #cache lines × Wmax
As every program memory access in ORAM translates to read and write of a path in ORAM tree (as explained in Section 2.4), in the above equation, the numerator calculates the total reads/writes to NVM which implements ORAM before system failure. The denominator calculates the ideal number of reads/writes possible to memory with perfect wear-levelling, where Wmax refers to the endurance of a cache line, e.g., the number of writes to a cache line before it fails. We chose Wmax to be 10 7 . We consider a NVM failure and consequent system failure when more than 1% of cache lines have reached endurance limit, similar to a mechanism used in [14]. Further, we show both the lifetime without wear-leveling and in presence of state-of-art wear-levelling solution, Start-Gap [28], As Fig. 4 depicts, both configurations achieve less than 4% lifetime and furthermore, lifetime drops with larger ORAM tree size (= larger NVM memory size). Emerging NVMs are deployed in terabytes [1], which would translate to a very large tree, e.g., 32-level tree. As a result, it is expected that the lifetime will be poor. This makes it impractical to implement ORAM for NVMs and consequently, impractical to address memory bus side channel for NVMs.
Egalitarian ORAM -Insight
In this section, we propose the inspiration for E-ORAM, our proposed wear-levelling algorithm to tackle endurance pressure of ORAM primitive for NVMs. Fig. 5 shows a simple 5-level ORAM tree with 31 nodes. In this example, we label each node with a unique node ID. Nodes closer to the root node are assigned smaller node IDs. N ORAM accesses to this tree result in a total of 5 * N node reads and writes, N for each of the 5 levels. However, since levels 0 to 4 contain 1, 2, 4, 8, 16 nodes respectively, the number of reads and writes to each of the nodes in these levels are N,
N 2 , N 4 , N 8 , N 16 respectively.
The goal of an ideal wear-leveling algorithm would be to confirm that each of the 31 physical node locations see the same number of writes, i.e., 5 * N 31 . This goal can be achieved by periodically remapping logical nodes to different physical nodes so that the writes are as close to uniformly distributed as possible. To minimize performance overheads from the re-mappings, we have to ensure that there are (i) only as many re-mappings as necessary and (ii) the indirection costs associated with routing accesses to logical nodes to the appropriate physical nodes must be kept low.
E-ORAM is based on the observation that nodes in levels 0, 1, and 2 are accessed more frequently than the ideal and the remaining nodes in levels 3 and 4 are accessed less frequently. If we were to form a group with the root node from level 0 and eight nodes from level 4 and assume ideal wear-leveling within the group, we can achieve average number of writes per node of N 6 , coming within 3% of the ideal ( 5 * N 31 ). Similarly, a group formed with one node from level 1 and four nodes from level 4 comes within 7% of the ideal. As we discuss in detail later, continuing this process of grouping one frequently written node (referred to as the Most Frequently Accessed Node, MFAN, henceforth) from a level close to the root with some partner nodes (referred to as the Partner Nodes, PNs, henceforth) from a relatively infrequently written level and wear-leveling within each of the groups can get us very close to ideal wear-leveling. We define such in-group wear-leveling as MFAN movement.
Note that prior state-of-art wear-levelling solution, Start-Gap [28], also proposed breaking down memory space into groups and performing wear-levelling in each group. However, we augment this basic idea in several important ways. When handling ORAM's exponential write distribution, Start-Gap falls short due to its design choice of breaking down the memory addresses into randomized, fixed-sized, wearleveling groups. With fixed sized groups, if a group happens to contain a very heavily written ORAM node (say the root node), then that group is likely to wear out much sooner than other groups that contain only relatively infrequently written nodes (say only the leaf nodes). This imbalance limits the lifetime achieved with Start-Gap on ORAM-style access patterns. However, one of the advantages of ORAM-style access pattern is that given its randomization and the binary-tree arrangement, the fraction of total writes incurred by each node is known apriori just by knowing the size of the ORAM tree. With this knowledge, it is possible to statically partition the ORAM tree into groups such that average number of writes per node incurred across all the groups is similar. Our central idea is to form variable sized groups instead of fixed-size groups in an ORAM primitive aware manner (unlike randomized in Start-Gap). This design choice allows very frequently written nodes like the root node to be grouped with a large number of infrequently nodes and reduce the average number of writes per node. However, having variable sized groups creates challenges in keeping track of which nodes belong to which groups and the wear-leveling movements within each group. We discuss our solutions to these challenges next.
Egalitarian ORAM
In this section, we discuss two parts E-ORAM is composed of: 1) the partition algorithm to group ORAM tree nodes and 2) the node remapping algorithm, which efficiently remaps nodes within the same group.
Static group partitioning
The first part of E-ORAM involves a static partition of the ORAM tree. We partition the ORAM tree into many groups with the following goals: (i) the average number of writes per node in each group approaches the average number of writes per node in the entire tree and (ii) within each group, there exists exactly one node being written more frequently than all other nodes (MFAN as defined in section 3), and all other nodes are expected to be accessed equally (partner nodes, as defined in section 3). This goal is to simplify the hardware implementation of E-ORAM as we will explain in §4.2.1.
Given the two principles, we can design an algorithm to partition our ORAM tree. For an ORAM tree with L levels (with level 0 being the root node), we will choose a threshold level, t, such that all nodes belonging to levels 0 to (t − 1) are grouped with level (L − 1), which is the leaf level. Exactly how we choose the value of t will be described later in this section. Then, nodes in the leaf level are further divided into t equal-size chunks, and assign them with chunk IDs, defined as C i , where 0 ≤ i ≤ t. Nodes in C i are paired with nodes in level i. Use Fig. 5 as an example, we will choose t = f loor(5/2) = 2 as the first step, and then divide nodes in level 4 into 2 chunks, C 0 and C 1 . C 0 nodes (yellow nodes in level 4) are grouped with the root node (also yellow), while the two nodes in level 1 (nodes 1 and 2, red and green respectively) are grouped with half of the nodes in C 1 each (red and green respectively). We denote each subset nodes in C 1 as Parts and assign them with Part IDs. So far, we have formed groups for nodes in level 0 and 1 and level 4, that leaves the nodes in levels 2 and 3 to be grouped. As can be seen from Fig. 5, the nodes in levels 2 and 3 can be viewed as four separate, smaller ORAM trees (in blue, grey, purple, and orange colors). Our partitioning strategy can be recursively applied to these intermediate, smaller ORAM trees until the entire larger tree has been partitioned. One important characteristic in the groups formed by our algorithm is that each group has one MFAN (most frequently accessed node) and the rest of the partner nodes all belong to the same, less frequently accessed level. With this characteristic, we define K as the highest level that contains MFAN nodes, called MFAN level. In this example, K = 2. This value is crucial when discussing how to trigger MFAN movement in section 4.2.2 and performance overhead in section 5.1.
The last aspect that we have not yet discussed is how to choose an appropriate threshold level t. Assuming perfect intra-group wear-leveling, we experimented with different values of t. We found that threshold level value of L 2 gives us the lowest number of writes per node in the most vulnerable group, making it the optimal threshold level value in our algorithm.
Node Remapping
While the groups are decided statically, at runtime, the wearleveling algorithm has to perform the following actions: (i) decide when to perform the MFAN movements for each of the groups and (ii) accurately maintain the mapping of logical node IDs and corresponding physical locations accounting for all the MFAN movements performed. Apart from being functionally correct, we also aim to perform the above two actions with the high performance and low storage overheads. Next, we describe how we perform both actions.
MFAN Movement Algorithm
In this section, we will discuss how MFAN movements are handled. Notice for a group g, when an MFAN movement occurs, we need to perform a data swap between MFAN and one of its partner nodes. We will use Fig. 5 as an example to illustrate the MFAN movement algorithm.
Consider the red group, it has fives nodes in total: the MFAN (node 1 in level 1) and four partner nodes (P1, P2, P3, P4, nodes 23, 24, 25, 26 respectively in level 4). For illustration purposes, we can visualize these five nodes arranged in an array in the order P1, P2, P3, P4, MFAN, as shown in Fig. 6a. After a threshold number of writes are performed to this group, we swap the MFAN with the node on its left in the array (wrapping around when necessary). Fig. 6b shows the state of the array after swapping P4 and MFAN. Before the swap, all the writes to MFAN were handled by the physical node 1 and all the writes to P4 were handled by the physical node 26. After the swap, all the writes to MFAN will be handled by the physical node 26 and vice-versa. So, this swap effectively wear-levels between physical node 1 and 26. Similarly, once we hit the threshold number of writes in the new arrangement, the MFAN, now at physical node 26, will be swapped with the partner node to its left (P3), now at physical node 25, as shown in Fig. 6c. When we are ready to make the next swap, physical nodes 25, 26, and 1 all would have seen about the same number of writes, in other words they have been wearlevelled. We continue this MFAN movement within the group to provide effective intra-group wear-levelling. A few important things to notice about this intra-group wearlevelling approach: Simple and ideal intra-group wear-levelling: Given the way in which we construct our groups, each group has one MFAN and many partner nodes where each partner node is expected to incur the same number of writes. The partner nodes all being equivalent (from a write frequency point-of-view) allows us to achieve ideal intra-group wear-levelling by simply moving the MFAN, no other nodes need to be moved unless they are being swapped with the MFAN. Deterministic locations: One requirement for swap-based wear-leveling approaches is to be able to accurately locate the nodes after a series of swaps. One approach to tracking the physical locations of logical nodes is to maintain a mapping table for the indirection. However, given that we move the MFAN in a deterministic pattern and at fixed intervals (every threshold number of writes), just tracking the number of writes to the group is enough to exactly determine the physical location to which the MFAN is currently mapped to. Furthermore, since the MFAN is the only node that gets swapped, the location of the partner nodes can also be formulaic-ly determined simply based on the number of number of writes incurred by the group. Specifically, the following two formulae will allow us to accurately determine the exact physical locations of the MFAN and each of the partner nodes based on the number of writes to the group.
For the new array index of MFAN, we can compute it with the following equation:
MFAN = (N g − 1) − s g %N g(1)
where N g is the size of this group, and s g refers to the number of swaps happened in this group. For any partner node whose initial array index is y, its new array index y can be computed with
y = y + f loor( s g + y N g − 1 ) %N g (y = N g − 1 )(2)
Notice that for both equation 1 and 2, the input and output value are both array indices. For example, if we want to compute the location of MFAN shown in Fig. 6a, we can achieve this by feeding N g = 5 and s g = 0 into equation 1, which gives us 4. This value 4 needs to be mapped to physical node 1, because it is the last element of the array, which is mapped to node 1. Similarly, after 1 swap operation, equation 1 will tell us that MFAN locates in array index 3. Since element 3 is not the last element in the array, we know that this value must be mapped to one of the partner nodes in the group. We can find its physical location by adding 3 with a OFFSET value, where OFFSET denotes the the node ID that array index 0 being mapped to. In this example, OFFSET = 23, so MFAN now locates in physical node 26.
The OFFSET value varies among groups. It is unacceptable if we store all OFFSET values on chip, considering the huge number of groups we have. Fortunately, it is not hard to show that the value of OFFSET for an arbitrary group can be computed by the following information: 1) the level that all partner nodes reside in, denoted as l p , which is 4 in this example; 2) the MFAN ID, specified by (i, j), which refers to the node in i th level, j th node in that level; 3) the number of levels sharing the same partner level, denoted as l s . We propose storing such information in an on-chip lookup table, as shown in Fig. 7. In Fig. 7, column 'IsMFAN' contains a single bit indicating whether nodes in a level are MFANs in their groups. Column 'PartnerLevel' denotes the level containing partners nodes, which is l p . Column 'LevelFrom' refers to the starting level being paired with l p , and column 'LevelTo' is the ending level being paired with l p . Essentially, variable l s can be computed by subtracting 'LevelTo' by 'LevelFrom' for a given level ID. Then, OFFSET can be easily computed by
(3) OFFSET = 2 l p − 1 + ChunkID × ChunkSize + PartID × PartSize
where term 2 l p − 1 is the first node in level l p ; ChunkID can be computed by value i, which is 1 in our example; ChunkSize can be found by the value of l s by dividing the number of nodes in level l p by l s , which is 8 in our example; PartID can be found with the value of j, which is 1 in our example; and finally, PartSize can be found by dividing ChunkSize by the number of nodes in level i, which is 4 in this example. Plug all values into equation 3, we get OFFSET = 23. This deterministic movement pattern implies that we do not need an expensive mapping table for each group, instead we simply can implement the circuitry necessary to calculate the exact locations of each of the nodes in the ORAM controller.
Triggering MFAN Movement
To trigger gap movement, Start-Gap [28] reserves an on-chip counter for each group in the NVM. Each counter stores the number of writes to its corresponding group. When a counter reaches a threshold, X, Start-Gap will trigger gap movement on that group. Start-Gap defines X as wear-leveling frequency [28]. We will use the same definition here. In our case, we employ a similar mechanism in our design. Ideally, we could store on-chip counters for all groups, but this is unrealistic because of the extremely large number of groups generated by our partition algorithm. Fortunately, we observe strong correlations among the number of writes to groups. Recall that ORAM assumes a random leaf node access. This random number is assumed to be uniformly distributed. Therefore, we can compute the expected number of writes to each node, and therefore, to each group given the static ORAM tree partitions. Then, if we reserve one global counter which stores the total number of ORAM accesses so far, then we could compute the expected number of memory writes to any group. In addition, we found that for a group whose MFAN is located in level i, denoted as g(i) (g(i) may refer to 2 i groups), the expected number of writes to group g(i + 1) is around half of g(i). This is true because the MFAN node in g(i + 1) receives half number of writes compared to the MFAN node in g(i), and the number of partner nodes in g(i + 1) is also half of the number of partner nodes in g(i). If partner nodes in g(i) and those in g(i + 1) are located in the same level, then the expected number of writes to group g(i + 1) is exactly half of g(i).
Since the expected number of writes to a group is determined by its MFAN, then we could simply reserve 1 on-chip counter, Ctr, to store the total number of writes to the NVM. For g(0), we perform MFAN movement with a period of X, and for g(1), we perform MFAN movement with half frequency compared to g(0), etc. The equivalent expression is that for every X ORAM accesses, all g(0) will perform MFAN movements; for every 2X ORAM accesses, all g(0) and g(1) will perform MFAN movements, etc. Therefore, the MFAN movement triggering algorithm can be formulated as shown in algorithm 1:
Algorithm 1: MFAN Movement Triggering Algo- rithm.
Data: Counter Ctr, MFAN Level K, Wear-Leveling Frequency X Result: A set, S, which contains level IDs. All groups whose MFANs locates in these levels needs to perform MFAN movements On an ORAM request:
if Ctr % X = 0 then k ← 0 ; while k < K & Ctr % 2 k+1 X = 0 do k ← k + 1 ; end end S ← {0, 1, ..., k} ;
Ctr ← Ctr + 1 ; return (S, Ctr) ;
In algorithm 1, Ctr refers to the on-chip counter; K refers to the MFAN level. In Fig. 5, the value of K will be 2. The return value, S, essentially states that MFAN movements need to be performed on g(0), g(1) ... g(k).
One of the drawbacks of algorithm 1 is that when the counter hits 2 K X, then all groups need to perform MFAN movements. The large number of MFAN movements performed at the same time could starve out actual ORAM accesses that need to be performed. For example, there are 2 28 − 1 groups for a 32-level ORAM tree. To avoid this concern, we improve our MFAN movement triggering algorithm to stagger the MFAN movements. We shall work through an example to illustrate the problem and our solution. For simplicity, we define a checkpoint as each time Ctr%X = 0 satisfies. The first time Ctr reaches X, we define it as checkpoint 0. Fig. 8 shows such an example. Suppose we have an ORAM tree with MFAN level K = 2. Then, at checkpoint 0, only the group whose MFAN node is the root node needs to perform MFAN movement. At checkpoint 1, 3 groups colored in yellow needs to perform MFAN movement. At checkpoint 2, only the root group needs to perform MFAN movement. This is because Ctr = 3X, so k remains 0. At checkpoint 3, since Ctr = 4X, so k is 2 now, we need to perform MFAN movement for all 7 yellow groups. This imbalance of the number of MFAN movements at each checkpoint could cause an ORAM request to be blocked for a long time. Alternatively, we observe that we can solve this problem simply by rearranging the order of doing these MFAN movements, as shown in Fig. 9. We notice that we can perform exactly three MFAN movements at each checkpoint. At checkpoint 0, we perform MFAN movement for the groups whose MFAN node is the first node in each level. At checkpoint 1, we perform MFAN movement for those with MFAN node being the second node in each level. Essentially, the MFAN node ID that we need to perform MFAN movement can be computed by taking checkpoint modulo the number of nodes in each MFAN level. This will give us MFAN (0, 0, 2) for checkpoint 2, which corresponds to nodes (0, 1, 5), respectively; and (0, 1, 3) for checkpoint 3, which corresponds to nodes (0, 2, 6), respectively.
While this looks good enough, we still observe the similar drawback. That is, at each checkpoint, the corresponding ORAM request will suffer K + 1 more MFAN movements compared with an ordinary ORAM request. We can leverage this by spreading the K + 1 MFAN movements across the X ORAM requests. That is, when each time Ctr hits X K+1 , we will perform exactly one MFAN movement for one group. In this way, one ORAM request will see at most one additional MFAN movement operation.
Finally, we need to choose the appropriate MFAN movement frequency X. A small X causes too many MFAN movements and increases the overhead of the wear-leveling algorithm due to additional memory accesses incurred while a very large X causes too few MFAN movements, hindering efficient wear-leveling. After experimenting with different values of X, we observed that for a Wmax of 10 7 [14], that a frequency of 10000 incurs about 0.1% increase memory accesses while also helping us achieve significant lifetime improvements. The detailed performance impact of X will be discussed in 5.1.
Algorithm Overhead
In this section, we discuss the performance and storage overheads incurred by E-ORAM.
Performance Overhead
The performance overhead incurred by E-ORAM can be broken down into two components: (i) overhead due to additional memory reads and writes incurred for MFAN movements and (ii) overhead due to additional logic in the ORAM controller for logical node ID to physical node ID translation, etc.
MFAN movement overhead: Systems with ORAM are usually bottlenecked by memory due to the sheer number of memory accesses that need to be performed for each ORAM access. So, calculating the number of additional memory accesses performed due to E-ORAM gives us a reasonable approximation of the performance overhead incurred.
For each MFAN movement, a total of two node reads and two node writes need to be performed. From our node remapping algorithm in section 4.2, we know that an MFAN movement occurs every X K+1 ORAM accesses, where X is the wearleveling frequency and K is the MFAN level. The total number of node reads (or writes) performed for every X K+1 ORAM requests (not counting the additional number of reads or writes caused by wear-leveling) is XL K+1 and our algorithm introduces an additional two reads (or writes), bringing the overhead to:
MFAN overhead = 2 XL K+1 = 2(K + 1) XL(4)
For a 32-level ORAM tree, its MFAN level is 27, and E-ORAM would cause a 0.0175% increase in the number of reads and writes, a negligible overhead.
ORAM controller overhead: The other performance overhead is from additional operations performed at the ORAM controller, like lookup table reconstruction, logical node ID to physical node ID translation, etc. We did RTL level synthesis of such hardware, which contains combinational logic, and the result shows that the overhead is small and can be parallelized with other ORAM controller actions on the critical path of an ORAM access. Hence this part of the performance overhead is also negligible.
Storage Overhead
The storage overhead incurred by our algorithm is entirely onchip for: (i) a counter to track the number of ORAM accesses performed and (ii) a lookup table for storing the static partition information, which is used to compute node remapping in 4.2. For the counter, a 64-bit counter is enough to track 2 64 ORAM accesses, well over the lifetime of a 32-level ORAM tree even with a Wmax of 10 8 .
The other storage overhead is the lookup table where each entry contains four elements: one Boolean variable 'MFAN level', and 3 integers used to store level information. The Boolean variable occupies 1 bit per entry. For the three integers, 6 bits each is enough for trees with up to 64 levels. Hence, the lookup table overhead is
LookupTable overhead = (6 × 3 + 1) × L = 19L(bits) (5)
For a 32-level ORAM tree, the lookup table overhead is merely 76 bytes, leading to 84 bytes overhead in total. This concludes that E-ORAM incurs minimal storage overhead.
Discussion
Security implications: While our algorithm significantly improves NVM lifetime, it breaks away from ORAM algorithm in that under ORAM, every program memory access needs to be translated into an ORAM access. In E-ORAM, when an MFAN movement is triggered, we do not further translate this into ORAM accesses. We argue that this change does not break ORAM's obliviousness nor does it add any new security vulnerabilities. This is so, as with this, the attacker now only additionally knows that an MFAN movement is triggered, and X K+1 ORAM accesses have been executed since last MFAN movement. This does not leak information regarding the application unique behavior apart from the number of ORAM accesses that have been performed, which the attacker is already privy to.
We will now discuss how E-ORAM could be potentially integrated with existing variations of ORAM that aim to improve ORAM performance.
Ring ORAM: Ring ORAM [32] saves ORAM bandwidth by fetching a single block per path read, instead of the entire path, and writing back a deterministic path every A path read operations. Every path read operation will incur an update to that block metadata. This significantly saves ORAM bandwidth and extends NVM lifetime (measured in days) by A times. However, Ring ORAM still incurs skewed access pattern, and E-ORAM could work well with it. Further optimizations could be made if we incorporate E-ORAM with Ring ORAM, but this is out of the scope of this paper.
DRAM+NVM [29]: Qureshi et al. [29] proposed a DRAM and NVM hybrid memory system. If we apply ORAM on this hybrid memory system, we could 1) cache some blocks on smaller tree on the DRAM (called ρ), and store all blocks on the NVM, (This architecture is proposed in [25].) or 2) split the address space across both DRAM and NVM, such that the top more frequently written nodes are mapped to DRAM. For the first scenario, the lifetime of the NVM could be extended by receiving less frequently writes, but since the tree size of the NVM does not change, and the access pattern to the NVM does not change, so the lifetime (percentage lifetime) of the NVM does not change. For the second scenario, the DRAM acts as a bigger tree-top cache [24]. If the DRAM size is 3% of NVM size [29], and the ORAM tree in the NVM has 32 levels (NVM tree), then the ORAM tree in the DRAM (DRAM tree) has 27 levels. This extremely large tree-top cache leads to the access pattern to the NVM to be 2 27 smaller ORAM tree with five levels each. Admittedly, if we apply baseline startgap on these 2 27 5-level ORAM trees, the lifetime could be significantly boosted without E-ORAM. However, we claim that the performance of setup 1) outperforms setup 2). If we define the DRAM access latency as T D , and define NVM access latency as T N , then we could compute the average memory access time (AMAT) for both setups:
AMAT 1 = 27T D + r m × 32T N AMAT 2 = 27T D + 5T N
where r m refers to the ρ miss rate. It is not hard to show that as long as r m is less than 5 32 = 15.6%, AMAT 1 will be less than AMAT 2 .
Since Setup 1) could give user better performance, and its NVM has a bigger tree, then E-ORAM will certainly outperforms Start-Gap as being measured by lifetime.
Processor
ISA
UltraSPARC III ISA CMP Size and Core Freq. ORAM, which is our baseline; 2) ORAM algorithm accommodated with Start-Gap [28]; 3) ORAM algorithm accommodated with Start-Gap and a 1MB treetop cache [24]; and 4) Fork Path algorithm [42] augmented with Start-Gap and a 1MB merge-aware-cache (MAC). For Start-Gap, we use 256 equal-size groups to maximize lifetime.
Results
NVM lifetime for varying ORAM tree sizes: Fig. 10 shows the simulated lifetime for different ORAM tree sizes. As shown in Fig. 10, E-ORAM achieves a 87.45% lifetime for a 16-level ORAM tree, and the lifetime gradually increases as we scale the ORAM tree. The lifetime is able to reach 91.04% for a 28-level tree. We are not able to simulate a bigger ORAM tree because of the very long simulation time. We can induce that as the NVM size grows, lifetime continues to grow.
Comparison with the state-of-the-art: Fig. 11 shows the simulation results when comparing E-ORAM with other four comparison designs mentioned in §7.1. We simulate the five configurations with different ORAM tree sizes and collect the number of ORAM accesses being done before device fails. The y-axis is computed by normalizing the total number of ORAM accesses collected from a configuration with the number collected from the baseline configuration. We plot the result in log scale because we observe a huge lifetime gain from E-ORAM. As shown in Fig. 11, the baseline configuration has a number of 1. Start gap performs roughly 4x better compared to the baseline. With the help of 1MB treetop cache, start gap could further improve the number of ORAM accesses to around 8x. If we apply Fork Path on top of start gap, and employ a MAC, due to Fork Path's nature of merging redundant node accesses, we could further improve the number of ORAM accesses to 9x maximum. Compared to all these configurations, E-ORAM could significantly boost the number of ORAM accesses to 90x. This is even 10x compared to configuration 4), the Fork Path configuration. The reason that E-ORAM works so well is that E-ORAM dynamically adjust the grouping based on the expected number of writes per node. This information is embedded in the nature of ORAM and is not employed by previous wear-leveling algorithms.
Performance impact: Fig. 12 shows the slowdown caused by E-ORAM for SPEC CPU ® 2017 benchmarks. As shown in Fig. 12, E-ORAM incurs negligible performance overhead, ranging from 0.119% (538.imagick_r) to 0.187% (554.roms_r). We observed similar results with GAP benchmarks [3]. We also measured that additional memory accesses generated by E-ORAMare about 0.22%.
Related Work
ORAM: Several works [33,12,24,42,25] build and improve upon Path ORAM [37], considered one of the most state-of-art implementation of ORAM primitive. Our work is orthogonal and complementary to these proposals.
Wear-leveling and write reduction: Prior solutions to tackle NVM endurance either reduce write traffic to NVM (write reduction) or employ wear-levelling to even out write traffic to memory. Our proposed solution E-ORAM is orthogonal to write reduction techniques such as LEO [30] which reduce NVM writes by decreasing block encryptions.
Park et al. [7] proposed an age-based adaptive swapping and shifting wear-leveling algorithm that keeps track of the number of writes. Based on the tracked access pattern, they swap pages and shift lines in a page. However, this design introduces relatively large on-chip storage overhead compared to E-ORAM. Ferreira et al. [11] proposed a wear-leveling algorithm that randomly picks two pages to be swapped. While this design is simple, it fails to achieve uniform write pattern in the case of ORAM, because the most frequently written node (the root node) is unlikely to be picked due to the large number of nodes in the ORAM tree. Hu et al. [18] proposed a software-based wear-levelling algorithm to spread data across PCM uniformly. However, this technique cannot be applied in the presence of ORAM, because the ORAM controller will still create an exponentially distributed memory access pattern. Chang et al. [5] proposed a wear-levelling technique using sliding window with dynamic window size. Memory writes to addresses within the window are tracked. The memory lines are swapped based on the number of observed writes.
Conclusions
ORAM incurs an exponentially distributed memory write pattern causing significant lifetime reductions in NVM devices due to their limited write endurance. We present E-ORAM, that leverages this skewed, but deterministic memory access pattern to mitigate NVM lifetime reductions. E-ORAM uses a pre-computed static partition grouping technique to determine wear-leveling granularity and an efficient intra-group wearleveling algorithm. Finally, we show that E-ORAM introduces less than 0.2% performance overhead, negligible storage overhead (84 bytes for a 32-level ORAM tree), with 90x lifetime boost compared to no wear-leveling.
Figure 1 :
1Traditional memory layout (left) vs Path ORAM layout (right). Path ORAM leads to an exponentially distributed memory access pattern. The numbers on the right shows the expected number of writes per node at each level.
Figure 2 :
2Illustration of Path ORAM accesses and important hardware structures involved.
Figure 3 :
3ORAM's exponential write distribution: The x-axis shows the node ID. The y-axis shows the corresponding expected number of writes to each node in log scale normalized to the expected number of writes to a leaf node. We assume that the tree has 10 levels and that nodes closer to root are assigned with smaller IDs.
Figure 4 :
4ORAM significantly reduces NVM lifetime: This figure shows the lifetime of an ORAM enabled NVM device with and without Start-Gap [28]. We assume a region-based Start-Gap with randomized grouping and 256 groups as that performs the best wear-leveling in our sensitivity analysis.
Figure 5 :
5Example static group partitioning with E-ORAM: A ORAM tree with 5 levels and 31 nodes partitioned into 7 groups, each identified with a different color. MFAN level, K (defined in section 4.1), is 2.
Status of the red group. (b) Status of the red group after swapping MFAN with P4.(c) Status of the red group after swapping MFAN with P3.
Figure 6 :
6Illustration of MFAN movement in a group with one MFAN and four partner nodes.
Figure 7 :
7A lookup table illustration: A sample lookup table storedon chip, which is used to compute OFFSET .
Figure 8 :
8Illustration of the naive MFAN movement triggering algorithm: For different checkpoints, there is a significant imbalance in the number of groups that need to perform MFAN movements.
Figure 9 :
9Illustration of a balanced MFAN movement triggering algorithm: In the new algorithm, exactly 3 groups need to perform MFAN movements at each checkpoint.
Figure 11 :
11E-ORAM vs state-of-the-art: This figure shows the expected number of ORAM accesses before device failure normalized to baseline no-wear-leveling case, for various stateof-the-art ORAM and wear-leveling configurations and ORAM trees. The results are plotted in log scale, with a value of 100 being ideal. E-ORAM achieves near ideal lifetime while the state-of-the-art is at least 10x worse.
Figure 12 :
12E-ORAM performance overhead: E-ORAM introduces negligible slowdowns (≤ 0.2%) versus no wear-leveling, across memory intensive SPEC CPU ® 2017 workloads.
Table 1: Simulated System configurationFigure 10: Simulated NVM lifetime with E-ORAM for different tree sizes. E-ORAM achieves 90% or higher of the ideal lifetime for different trees sizes versus 4% with Start-Gap [28].1-core, 2.6 GHz
Re-Order-Buffer
128 entry
NVM Parameters
NVM Device Parameters
2 ranks per channel,
8 banks per chip,
32768 rows per bank
NVM Bus Frequency
2666 MHz
NVM Write Queue Size
64 entries
NVM Cache Line Endurance
10 7 writes
ORAM Configuration
Block Size
64B
Bucket Size
4 cache lines
ORAM Tree Level
24
PLB Size
64KB
Encryption/Decryption Latency
21 cycles
Number of Recursive PosMaps
4
Stash Size
200 entries
More information about SPEC CPU ® 2017 can be obtained from https://www.spec.org/cpu2017.2 SPEC CPU ® is a registered trademark of the Standard Performance Evaluation Corporation.
EvaluationMethodologyWe evaluate the performance overhead of E-ORAM by applying USIMM, a trace-based cycle accurate memory simulator[6]. We implemented Freecursive ORAM[12]as well as our wear-leveling engine on USIMM. On the backend, we employed an FR-FCFS memory scheduling policy[34]. For the ORAM tree, we simulated a relatively small 24-level tree, due to very long simulation time for larger trees.Table 1shows the configuration of our simulator and the ORAM tree. For traces, we employed a similar mechanism with[38]. We select 15 workloads from state-of-art benchmarks, 15 from SPEC CPU ® 2017[4] 1 2 . These workloads are selected for the fact that these workloads are the most memory-intensive workloads[36,38]. The traces are then collected for 5 million memory reads and writes, after workloads being fast-forwarded to warm up the LLC.State-of-the-art comparisons: We compare E-ORAM with five ORAM settings/configurations: 1) ORAM algorithm without any wear-leveling algorithm nor any variants of
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ACM/SPEC International Conference on Performance Engineering, ICPE '19. New York, NY, USAAssociation for Computing MachineryACM/SPEC International Conference on Performance Engineering, ICPE '19, page 285-292, New York, NY, USA, 2019. Association for Computing Machinery.
Path oram: An extremely simple oblivious ram protocol. Emil Stefanov, Marten Van Dijk, Elaine Shi, Christopher Fletcher, Ling Ren, Xiangyao Yu, Srinivas Devadas, Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13. the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13New York, NY, USAAssociation for Computing MachineryEmil Stefanov, Marten van Dijk, Elaine Shi, Christopher Fletcher, Ling Ren, Xiangyao Yu, and Srinivas Devadas. Path oram: An extremely simple oblivious ram protocol. In Proceedings of the 2013 ACM SIGSAC Conference on Computer & Communications Security, CCS '13, page 299-310, New York, NY, USA, 2013. Association for Computing Machinery.
Compact leakage-free support for integrity and reliability. Meysam Taassori, Rajeev Balasubramonian, Siddhartha Chhabra, Alaa R Alameldeen, Manjula Peddireddy, Rajat Agarwal, Ryan Stutsman, Proceedings of the ACM/IEEE 47th Annual International Symposium on Computer Architecture, ISCA '20. the ACM/IEEE 47th Annual International Symposium on Computer Architecture, ISCA '20IEEE PressMeysam Taassori, Rajeev Balasubramonian, Siddhartha Chhabra, Alaa R. Alameldeen, Manjula Peddireddy, Rajat Agarwal, and Ryan Stutsman. Compact leakage-free support for integrity and reliability. In Proceedings of the ACM/IEEE 47th Annual International Sympo- sium on Computer Architecture, ISCA '20, page 735-748. IEEE Press, 2020.
Overcoming the challenges of crossbar resistive memory architectures. C Xu, D Niu, N Muralimanohar, R Balasubramonian, T Zhang, S Yu, Y Xie, IEEE 21st International Symposium on High Performance Computer Architecture (HPCA). C. Xu, D. Niu, N. Muralimanohar, R. Balasubramonian, T. Zhang, S. Yu, and Y. Xie. Overcoming the challenges of crossbar resistive memory architectures. In 2015 IEEE 21st International Symposium on High Performance Computer Architecture (HPCA), pages 476-488, 2015.
Controlled-channel attacks: Deterministic side channels for untrusted operating systems. Y Xu, W Cui, M Peinado, 2015 IEEE Symposium on Security and Privacy. Y. Xu, W. Cui, and M. Peinado. Controlled-channel attacks: Deter- ministic side channels for untrusted operating systems. In 2015 IEEE Symposium on Security and Privacy, pages 640-656, 2015.
Predictive mitigation of timing channels in interactive systems. Danfeng Zhang, Aslan Askarov, Andrew C Myers, Proceedings of the 18th ACM Conference on Computer and Communications Security, CCS '11. the 18th ACM Conference on Computer and Communications Security, CCS '11New York, NY, USAAssociation for Computing MachineryDanfeng Zhang, Aslan Askarov, and Andrew C. Myers. Predictive mitigation of timing channels in interactive systems. In Proceedings of the 18th ACM Conference on Computer and Communications Security, CCS '11, page 563-574, New York, NY, USA, 2011. Association for Computing Machinery.
Fork path: Improving efficiency of oram by removing redundant memory accesses. Xian Zhang, Guangyu Sun, Chao Zhang, Weiqi Zhang, Yun Liang, Tao Wang, Yiran Chen, Jia Di, Proceedings of the 48th International Symposium on Microarchitecture, MICRO-48. the 48th International Symposium on Microarchitecture, MICRO-48New York, NY, USAAssociation for Computing MachineryXian Zhang, Guangyu Sun, Chao Zhang, Weiqi Zhang, Yun Liang, Tao Wang, Yiran Chen, and Jia Di. Fork path: Improving efficiency of oram by removing redundant memory accesses. In Proceedings of the 48th International Symposium on Microarchitecture, MICRO-48, page 102-114, New York, NY, USA, 2015. Association for Computing Machinery.
A study of application performance with non-volatile main memory. Y Zhang, S Swanson, 31st Symposium on Mass Storage Systems and Technologies (MSST). Y. Zhang and S. Swanson. A study of application performance with non-volatile main memory. In 2015 31st Symposium on Mass Storage Systems and Technologies (MSST), pages 1-10, May 2015.
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[
"arXiv:hep-ph/0009044v1 5 Sep 2000 ASTROPHYSICAL NEUTRINOS: 20th CENTURY AND BEYOND",
"arXiv:hep-ph/0009044v1 5 Sep 2000 ASTROPHYSICAL NEUTRINOS: 20th CENTURY AND BEYOND"
]
| [
"IUPAP Centennial LecturerJ N Bahcall \nInstitute for Advanced Study\n08540PrincetonNJUSA\n"
]
| [
"Institute for Advanced Study\n08540PrincetonNJUSA"
]
| []
| ∼jnb I summarize the first four decades of solar neutrino research and suggest what may be possible to learn with extragalactic neutrinos and with solar neutrinos in the next decade. | 10.1142/s0217751x01005948 | [
"https://export.arxiv.org/pdf/hep-ph/0009044v1.pdf"
]
| 29,814 | hep-ph/0009044 | b425b23eda815761a9b834253df686e61788054a |
arXiv:hep-ph/0009044v1 5 Sep 2000 ASTROPHYSICAL NEUTRINOS: 20th CENTURY AND BEYOND
IUPAP Centennial LecturerJ N Bahcall
Institute for Advanced Study
08540PrincetonNJUSA
arXiv:hep-ph/0009044v1 5 Sep 2000 ASTROPHYSICAL NEUTRINOS: 20th CENTURY AND BEYOND
1
∼jnb I summarize the first four decades of solar neutrino research and suggest what may be possible to learn with extragalactic neutrinos and with solar neutrinos in the next decade.
Introduction
I was asked by Art McDonald to give one of the opening talks on the subject " Neutrino Astrophysics in the 20th Century and Beyond." Feeling very honored, I readily accepted. But, as I started thinking about what I should say to so many knowledgeable friends, I got really worried. I realized that it would be impossible in twentyfive minutes to discuss intelligently all of neutrino astrophysics of the 20th century. There is just too much important material to cover even if I spoke very fast, unintelligibly fast, and even if I did not say anything about our goals for the future. So, I decided to limit my remarks to two specific topics: solar neutrinos and extragalactic neutrinos. I will not say anything about the enormous achievements in the prediction and detection of supernova neutrinos and in the calculations of neutrino cooling processes for stars that are not exploding. I will also not discuss the role of neutrinos in Big Bang nucleosynthesis nor in cosmology. There are lots of grand things to say about these subjects, and many other topics in neutrino astrophysics, but I will not discuss them today.
I will take a somewhat historical approach and emphasize those aspects of the development of our subject which may help guide our thinking about what we should do in the future. I will begin with solar neutrinos and then switch abruptly to extragalactic neutrinos. 2. Solar neutrinos 2.1. Bruno Pontecorvo and Ray Davis I want to begin by paying tribute to two of the great scientists and pioneers of neutrino astrophysics, Ray Davis and Bruno Pontecorvo. Bruno first suggested using chlorine as a detector of neutrinos in a Chalk River report written in 1946. Ray followed Bruno's suggestion and the careful unpublished feasibility study of Louie Alvarez. Using with care and skill a chlorine detector and reactor neutrinos, Ray showed in 1955Ray showed in -1958 that ν e andν e were different. About a decade later, Ray first detected solar neutrinos, laying the foundation for the studies we shall hear about today.
In 1967, one year before the first results of Ray's chlorine solar neutrino experiment were announced, Bruno published a prophetic paper entitled: 'Neutrino Experiments and the Problem of Conservation of Leptonic Charge ' [Zh. Exp. Teor. Fiz. 53, 1717(1967]. In this paper, Bruno suggested many different experiments that could test whether leptonic charge was conserved. The grandchildren of most of these experiments are being discussed in this conference, Neutrino 2000.
Bruno included a short section in his paper that he called 'Oscillations and Astronomy.' In this section, Bruno wrote: "From the point of view of detection possibilities, an ideal object is the sun," What a wonderfully contemporary statement! Bruno, like most particle physicists of the 1960's and perhaps 1970's and 1980's, did not believe astrophysical calculations could be reliable. He wrote in this same section on oscillations and astronomy: "Unfortunately, the weight of the various thermonuclear reactions in the sun, and the central temperature of the sun are insufficiently well known in order to allow a useful comparison of expected and observed solar neutrinos, from the point of view of this article." [This was 30 years before the precise confirmation of the standard solar model by helioseismology.] To support his claim, Bruno referenced only his 1946 Chalk River report, which mentioned the sun in just two sentences. Bruno did cite our calculations of the solar neutrino fluxes elsewhere in his 1967 paper, but they seem not to have affected his thinking. Figure 2. The energy Spectrum of neutrinos from the pp chain of interactions in the Sun, as predicted by the standard solar model. Neutrino fluxes from continuum sources (such as p − p and 8 B) are given in the units of counts per cm 2 per second. The percentage errors are the calculated 1σ uncertainties in the predicted fluxes. The p−p chain is responsible for more than 98% of the energy generation in the standard solar model. Neutrinos produced in the carbon-nitrogen-oxygen CNO chain are not important energetically and are difficult to detect experimentally. The arrows at the top of the figure indicate the energy thresholds for the ongoing neutrino experiments. This spectrum is from BP98: J. N. Bahcall, S. Basu, and M. H. Pinsonneault, Phys. Lett. B, 433, 1 (1998).
What can we learn from this bit of history? When Ray and I wrote our PRL papers arguing that a chlorine detector of 600 tons could observe solar neutrinos, we never discussed the possibility of using neutrinos to learn about particle physics. The only motivation we gave was "...to see into the interior of a star and thus verify directly the hypothesis of nuclear energy generation in stars." [PRL 12, 300 (1964)].
Why did we not discuss using neutrinos for particle physics? Frankly, because we never thought about it. And even if we had, we would have known better than to mention it to our particle physics friends. Bruno had the insight and the vision and indeed the courage to argue that astronomical neutrinos could potentially give us unique information about neutrino characteristics. His paper is all the more remarkable because it was published a year before the first results of the chlorine experiment showed that the rate Ray observed was less than our calculated rate.
We learn from these events that pioneering experiments can lead to important results in areas that are unanticipated. We will come back to this conclusion at the end of this talk. Figure 2 shows the calculated solar neutrino spectrum predicted by the Standard solar model. The percentage errors are the calculated 1σ uncertainties in the predicted fluxes, based upon the published errors of the measured quantities and on many calculations of standard solar models. As you will hear from the talks in the later parts of this morning session, the total intensities and the energy spectra shown in Fig. 2 are now widely used to interpret, and indeed to plan, solar neutrino experiments such as those discussed in today's sessions: chlorine, Super-Kamiokande, SNO, SAGE, GALLEX, GNO, and BOREXINO. Figure 3 compares the calculated versus the measured rates for the six solar neutrino experiments for which results have been reported. Assuming nothing happens to the neutrinos after they are created, the measured rates range from 33% ±5% of the calculated rate (for chlorine) to 58% ±7%. As is now well known, the observed rates cannot be fit (at a C.L. of about 99%) with any linear combination of undistorted solar neutrino energy spectra.
Standard Model Predictions
Today we know that there are three reasons that the calculations of solar neutrino fluxes are robust: 1) the availability of precision measurements and precision calculations of input data that have been gradually refined over four decades; 2) the intimate connection between neutrino fluxes and the measured solar luminosity; and 3) the measurement of the helioseismological frequencies of the solar pressure-mode (p-mode) eigenfrequencies. Figure 3. Comparison of measured rates and standard-model (BP98) predictions for six solar neutrino experiments. The unit for the radiochemical experiments (chlorine and gallium) is SNU (10 −36 interactions per target atom per sec); the unit for the water-Cerenkov experiments (Kamiokande and Super-Kamiokande) is the rate predicted by the standard solar model plus standard electroweak theory. The experimental results are described by Lande, Suzuki, Gavrin, and Belotti in these proceedings. Could the solar model calculations be wrong by enough to explain the discrepancies between predictions and measurements shown in Fig. 3? Helioseismology, which confirms predictions of the standard solar model to high precision, suggests that the answer is "No." Figure 4 shows the fractional differences between the most accurate available sound speeds measured by helioseismology and sound speeds calculated with our best solar model (with no free parameters). The horizontal line corresponds to the hypothetical case in which the model predictions exactly match the observed values. The rms fractional difference between the calculated and the measured sound speeds is 1.1 × 10 −3 for the entire region over which the sound speeds are measured, 0.05R ⊙ < R < 0.95R ⊙ . In the solar core, 0.05R ⊙ < R < 0.25R ⊙ (in which about 95% of the solar energy and neutrino flux is produced in a standard model), the rms fractional difference between measured and calculated sound speeds is 0.7 × 10 −3 .
Figure 4. Predicted versus Measured Sound
Speeds. This figure shows the excellent agreement between the calculated (solar model BP98, Model) and the measured (Sun) sound speeds, a fractional difference of 0.001 rms for all speeds measured between 0.05R ⊙ and 0.95R ⊙ . The vertical scale is chosen so as to emphasize that the fractional error is much smaller than generic changes in the model, 0.09, that might significantly affect the solar neutrino predictions. The measured sound speeds are from S. Basu et al., Mon. Not. R. Astron. Soc. 292, 234 (1997); The figure is taken from BP98.
The arrow in Fig. 4 shows how different the solar model sound speeds would have to be from the observed sound speeds if one wanted to use solar physics to reduce the 7 Be neutrino flux. The position of the arrow is fixed by artificially reducing the predicted 7 Be neutrino flux that is not observed in the gallium experiments, SAGE and GALLEX plus GNO (see Fig. 3). if the p − p neu-trinos are present. Remember, we believe we can calculate the p − p flux to ±1%.
The discrepancy with the hypothetical new solar physics was estimated by using the temperature dependence of the 7 Be neutrino flux (∝ T 10 ) and the sound speeds (∝ T 1/2 ). The agreement with the hypothetical solar physics is more than 100 times worse than the agreement with the Standard Model physics. Figure 4 has contributed to the consensus view that the experimental results shown in Fig. 3 require new particle physics for their explanation.
Summing up and looking ahead
I want now to look back and then look ahead. I will begin by giving my view of the principal accomplishments of solar neutrino research to date (Sect. 2.3.1). Then I will discuss two of the expected highlights of the next decade of solar neutrino research, the measurement of the neutral current to charge current ratio for 8 B neutrinos ((Sect. 2.3.2) and the detection of solar neutrinos with energies less than 1 MeV ((Sect. 2.3.3).
Principal achievements
What are the principal achievements of the first four decades of solar neutrino research? I give below my personal list of the 'top three achievements.'
• Solar neutrinos have been detected. The chlorine, Kamiokande, Super-Kamiokande, GALLEX, SAGE, GNO, and SNO experiments have all measured solar neutrino events. This is the most important achievement. The detection of solar neutrinos shows empirically that the sun shines by the fusion of light elements.
• Evidence for new physics has been found. For more than thirty years, beginning with the fact that Ray's first measurements in 1968 indicated a flux lower than the standard model predictions, we have had evidence for new physics in the solar neutrino arena. This evidence has steadily deepened as new solar neutrino experiments have confirmed and extended the neutrino discrepancies and helioseismology has confirmed the standard solar model. The fact that neutrino oscillations have now been observed in atmospheric neutrino phenomena fur-ther strengthens the case that oscillations occur for solar neutrinos. We are still looking for a 'smoking gun' single effect that shows up in just one solar neutrino experiment, rather than combining the results of two or more different experiments. I will discuss some possibilities below.
• Neutrino fluxes and energy spectra are approximately as predicted by the standard solar model. If you had told me in 1964 that six solar neutrino experiments would give results within a factor of three of the predicted standard model results, I would have been astonished and delighted. This is especially so considering that the crucial 8 B neutrino flux depends upon the 25th power of the central temperature of the sun. This agreement exists without making any corrections for neutrino oscillations.
If we correct the observed solar neutrino event rates for the effects of neutrino oscillations using the six currently allowed two-neutrino oscillation scenarios, the inferred 8 B neutrino flux at the source is rather close to the best-estimate predicted flux. At the 99% CL, one infers (see hepph/9911248): 0.55 ≤ φ( 8 B)/(Standard prediction) ≤ 1.32, (1) which is a slightly tighter range than the 3σ prediction of the standard solar model. Figure 5 shows the predictions of the currently allowed neutrino oscillation solutions for the double ratio, [NC]/[CC], of neutral current to charged current event rates in the deuterium detector SNO. Art McDonald will describe later this morning the experimental characteristics of this great observatory and outline for us the extensive program of SNO measurements. The important message of Fig. 5 is that all of the currently allowed oscillation solutions for active neutrinos predict a value for the double ratio that is different from the no oscillation value of 1.0 by at least nine times the estimated non-statistical measurement uncertainty.
SNO and the [NC]/[CC] ratio
We all eagerly look forward to this crucial and decisive measurement. The dashed error bar labeled "Measure 3σ" represents the net estimated uncertainty in interpreting the measurements, including the energy resolution, energy scale, 8 B neutrino energy spectrum, neutrino cross section, counting statistics, and the hep flux. This is Fig. 7a of Bahcall, Krastev, and Smirnov, hep-ph/0002293.
Solar neutrinos below 1 MeV
More than 98% of the calculated standard model solar neutrino flux lies below 1 MeV. The rare 8 B neutrino flux is the only solar neutrino source for which measurements of the energy have been made, but 8 B neutrinos constitute a fraction of less than 10 −4 of the total solar neutrino flux.
The great challenge of solar neutrino astronomy is to measure neutrino fluxes below 1 MeV. We must develop experiments that will measure the 7 Be neutrinos (energy of 0.86 MeV) and the fundamental p-p neutrinos (< 0.43 MeV). A number of promising possibilities were discussed at the LowNu workshop that preceded this confer-ence. The BOREXINO observatory, which can detect ν − e scattering, is the only approved solar neutrino experiment which can measure energies less than 1 MeV.
The p-p neutrinos are overwhelmingly the most abundant source of solar neutrinos, carrying about 91% of the total flux according to the standard solar model. The 7 Be neutrinos constitute about 7% of the total standard model flux.
We want to test and to understand neutrino oscillations with high precision using solar neutrino sources. To do so, we have to measure the neutrino-electron scattering rate with 7 Be neutrinos, as will be done with the BOREXINO experiment, and also the CC (neutrino-absorption) rate with 7 Be neutrinos (no approved experiment). With a neutrino line as provided by 7 Be electroncapture in the sun, unique and unambiguous tests of neutrino oscillation models can be carried out if one measures both the charged-current and the neutral current reaction rates.
I believe that we have calculated the flux of p-p neutrinos produced in the sun to an accuracy of ±1%. This belief should be tested experimentally. Unfortunately, we do not yet have a direct measurement of this flux. The gallium experiments only tell us the rate of capture of all neutrinos with energies above 0.23 MeV.
The most urgent need for solar neutrino research is to develop a practical experiment to measure directly the p-p neutrino flux and the energy spectrum of electrons produced by weak interactions with p-p neutrinos. Such an experiment can be used to test the precise and fundamental standard solar model prediction of the p-p neutrino flux. Moreover, the currently favored neutrino oscillation solutions all predict a strong influence of oscillations on the low-energy flux of ν e . Figure 6 shows the calculated neutrino survival probability as a function of energy for three global best-fit MSW oscillation solutions. You can see directly from this figure why we need accurate measurements for the p-p and 7 Be neutrinos. The currently favored solutions exhibit their most characteristic and strongly energy dependent features below 1 MeV. Naturally, all of the solutions give similar predictions in the en-½ Figure 6. Survival probabilities for MSW solutions. The figure presents the yearly-averaged survival probabilities for an electron neutrino that is created in the sun to remain an electron neutrino upon arrival at the Super-Kamiokande detector. ergy region, ∼ 7 MeV, where the Kamiokande and Super-Kamiokande data are best. The survival probability shows a strong change with energy below 1 MeV for all the solutions, whereas in the region above 5 MeV (accessible to Super-Kamiokande and to SNO) the energy dependence of the survival probability is at best modest.
The p-p neutrinos are the gold ring of solar neutrino astronomy. Their measurement will constitute a simultaneous and critical test of stellar evolution theory and of neutrino oscillation solutions.
Extragalactic neutrinos
Experimentalists often like to describe the power of their experiments in terms of the expected or observed number of events per year and L/E, where L is the distance between the accelerator and the detector and E is the beam energy. The quantity L/E determines, together with the square of the mass difference, the survival probability for vacuum neutrino oscillations. More generally, L/E represents the time of flight in the rest Figure 7. Very longbaseline neutrino oscillation experiments. The figure shows that experiments such as ANTARES, BAIKAL, ICECUBE, and NESTOR, which may detect high-energy neutrinos from distant gamma-ray bursts, have extraordinary sensitivity to vacuum neutrino oscillations. Neutrinos of 10 5 GeV from gamma-ray bursts located at cosmological distances were used to locate the positions of ANTARES, BAIKAL, ICECUBE, and NESTOR in the figure.
frame of the particle, the time for rare events to occur. Figure 7 shows the extraordinary sensitivity to neutrino oscillation of experiments like ANTARES, BAIKAL, ICECUBE, and NESTOR that can detect neutrinos from distant extragalactic sources. The accelerator experiments that will be discussed at Neutrino 2000 lie in the left-hand side of Fig. 7, L/E < 10 4 km/GeV . Solar neutrino experiments like Super-Kamiokande, SNO, and BOREXINO can reach to 10 10 km/GeV and, for the lower energy experiments, even 10 11 km/GeV. Extragalactic sources such as gammaray-bursts (GRBs) have such a long baseline (∼ 10 10 lyrs) that the new generation of extragalactic experiments, ANTARES, BAIKAL, ICECUBE, and NESTOR will extend to the right-hand side of Fig. 7, to L/E > 10 18 km/GeV. I want to say a few words about the possi-bilities for detecting GRB neutrinos, which will be discussed in more detail in these proceedings by Eli Waxman. I believe that GRBs offer the best chance for detecting extragalactic neutrinos among all the known sources of astronomical photons.
The phenomenology of the photons observed from gamma-ray bursts is now relatively well understood. Many different types of observations have been carried out and the results are well summarized by the expanding fireball model. Using this model, one can work out the flux of neutrinos from shocks. Figure 8 shows the neutrino energy spectra that Waxman and I have estimated to be produced by GRBs, both from the direct burst (energies ∼ 10 6 GeV) and from the afterglow (energies ∼ 10 8 GeV to ∼ 10 19 GeV). The observed population of GRBs should give rise to ∼ 10 events per km 2 per year from neutrinos with characteristic energies of order 10 14 eV. We shall hear on the last day of this conference that the calculated GRB flux may be detectable in ANTARES, ICECUBE, or NESTOR. The fundamental assumption used in calculating the GRB neutrino flux is that GRBs produce the observed flux of high-energy cosmic rays, an assumption for which Eli Waxman has provided a strong plausibility argument.
GRBs occur at modest to large redshifts. We know the time of the explosion to an accuracy ∼ 10 sec (from the gamma rays). Therefore, GRBs can be used to test special relativity to an accuracy of 1 part in 10 16 and to test the weak equivalence principle to an accuracy of 1 part in 10 6 . If special relativity is right, the photons and the neutrinos should arrive at the same time (to an accuracy of about 10 sec, the duration of the burst). If the weak equivalence principle is valid, the arrival times of neutrinos (which traverse significant gravitational potentials) from distant sources should be independent of neutrino flavor.
GRBs can also be used to probe the weak interactions to an extraordinary level of precision. Gamma-ray bursts are expected to produce only ν e and ν µ . The large area detectors of extragalactic neutrinos are in principle sensitive to vacuum neutrino oscillations with mass differences as Figure 8. The Waxman-Bahcall upper bound on muon neutrino intensities (ν µ +ν µ ). This figure is from Bahcall and Waxman, hep-ph/9902383. The numerical value of the bound assumes that 100% of the energy of protons is lost to π + and π 0 and that the π + all decay to muons that also produce neutrinos. The dot-dash line gives the upper bound corrected for neutrino energy loss due to redshift and for the maximum known evolution (QSO or star-formation evolution). The lower line is obtained assuming no evolution. The solid curves show the predictions of representative AGN jet models taken from the earlier papers of Mannheim (marked M95B in the figure), Protheroe (P97), and Halzen and Zas (HZ97). The AGN models were normalized so that the calculated gamma-ray flux from π 0 decay fits the observed gamma-ray background. small as ∆m 2 ≥ 10 −17 eV 2 (from ν µ → ν τ ).
Not everything is encouraging in Fig. 8. The figure also shows the upper limit that is allowed for astrophysical neutrino production from (γ, π) interactions on high energy protons. The upper limit is established by using the observed cosmic ray flux of high energy protons. Prior to the recognition of this limit a number of authors had suggested much more optimistic models (also shown in the figure), that were normalized by fit-ting π 0 decay to the observed gamma-ray background.
Goals for Astrophysical neutrinos: 2000-2010
It seems to me that we have three principal goals for this next decade.
• Determine the mixing angles and mass differences that are important for solar neutrino phenomena.
• Test precisely stellar evolution by observing p − p and 7 Be neutrinos, and by determining the total flux of 8 B neutrinos.
• Discover extragalactic neutrinos, perhaps from gamma-ray bursts.
From time to time, friends ask me to compare the search for solar neutrinos with the search for neutrinos from GRBs. They are very different. From photon studies, we know more observationally about the sun than about any other astronomical source, certainly much more than about the mysterious GRBs. Moreover the sun is in the simplest stage of stellar evolution, in quasi-static equilibrium with a characteristic time scale for evolution of 10 9 yr (10 16 s ). We do not even know the energy source of GRBs. We do know that GRBs are far from equilibrium, evolving explosively on a time scale of order 10 −3 s.
We want to do extragalactic neutrino astronomy because it is truly an exploration of the universe. We do solar neutrino astronomy to test fundamental theories of physics and astronomy. But, perhaps solar neutrino research and extragalactic neutrino research may in the end share a fundamental characteristic: surprise. Remember, that we undertook solar neutrino research to test stellar evolution and unexpectedly (at least for everybody except Bruno Pontecorvo) we found evidence for new neutrino physics.
In a sense, we are returning to our original goal in neutrino astronomy, but by a round-about path. We must first understand neutrino oscillation phenomena in order to be able to use solar neutrino observations to test precisely the theory of stellar evolution, our original goal. Perhaps with extragalactic astronomy we will participate in a similar cycle of astronomical exploration and physical clarification. T. S. Elliot in 'The Four Quartets' described the cycle succinctly and beautifully:
We shall not cease from exploration And the end of all of our exploring Will be to arrive where we started And know the place for the first time.
Figure 1 .
1Bruno Pontecorvo wrote in 1967: 'From the point of view of detection possibilities, an ideal object is the sun.' Figure courtesy of S. Bilenky .
Figure 5 .
5The neutral current to charged current double ratio, [NC]/[CC] . The standard model value for [NC]/[CC] is 1.0. The figure shows, for a 5 MeV threshold for the CC measurement, the predicted double ratio of Neutral Current to Charged Current for different neutrino scenarios. The solid error bars represent the 99% C.L. for the allowed regions of the six currently favored neutrino oscillation solutions.
AcknowledgmentsI acknowledge support from NSF grant #PHY95-13835. | []
|
[
"Pose Estimation for Non-Cooperative Spacecraft Rendezvous Using Convolutional Neural Networks",
"Pose Estimation for Non-Cooperative Spacecraft Rendezvous Using Convolutional Neural Networks"
]
| [
"Ph.DSumant Sharma [email protected] \nDepartment of Aeronautics & Astronautics\nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA\n",
"Candidate \nDepartment of Aeronautics & Astronautics\nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA\n",
"Ph.DConnor Beierle [email protected] \nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA\n",
"Candidate \nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA\n",
"Assistant ProfessorSimone D'amico [email protected] \nStanford University\n496 Lomita Mall94305StanfordCA\n"
]
| [
"Department of Aeronautics & Astronautics\nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA",
"Department of Aeronautics & Astronautics\nDepartment of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA",
"Department of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA",
"Department of Aeronautics & Astronautics\nStanford University\n496 Lomita Mall94305StanfordCA",
"Stanford University\n496 Lomita Mall94305StanfordCA"
]
| []
| On-board estimation of the pose of an uncooperative target spacecraft is an essential task for future on-orbit servicing and close-proximity formation flying missions. However, two issues hinder reliable on-board monocular vision based pose estimation: robustness to illumination conditions due to a lack of reliable visual features and scarcity of image datasets required for training and benchmarking. To address these two issues, this work details the design and validation of a monocular vision based pose determination architecture for spaceborne applications. The primary contribution to the state-of-the-art of this work is the introduction of a novel pose determination method based on Convolutional Neural Networks (CNN) to provide an initial guess of the pose in real-time on-board. The method involves discretizing the pose space and training the CNN with images corresponding to the resulting pose labels. Since reliable training of the CNN requires massive image datasets and computational resources, the parameters of the CNN must be determined prior to the mission with synthetic imagery. Moreover, reliable training of the CNN requires datasets that appropriately account for noise, color, and illumination characteristics expected in orbit. Therefore, the secondary contribution of this work is the introduction of an image synthesis pipeline, which is tailored to generate high fidelity images of any spacecraft 3D model. In contrast to prior techniques demonstrated for close-range pose determination of spacecraft, the proposed architecture relies on neither hand-engineered image features nor a-priori relative state information. Hence, the proposed technique is scalable to spacecraft of different structural and physical properties as well as robust to the dynamic illumination conditions of space. Through metrics measuring classification and pose accuracy, it is shown that the presented architecture has desirable robustness and scalable properties. Therefore, the proposed technique can be used to augment the current state-ofthe-art monocular vision-based pose estimation techniques used in spaceborne applications. | 10.1109/aero.2018.8396425 | [
"https://arxiv.org/pdf/1809.07238v1.pdf"
]
| 49,535,593 | 1809.07238 | a1887657d8c7f0f5f9e6d88c07e9c1ece029dabc |
Pose Estimation for Non-Cooperative Spacecraft Rendezvous Using Convolutional Neural Networks
Ph.DSumant Sharma [email protected]
Department of Aeronautics & Astronautics
Department of Aeronautics & Astronautics
Stanford University
496 Lomita Mall94305StanfordCA
Candidate
Department of Aeronautics & Astronautics
Department of Aeronautics & Astronautics
Stanford University
496 Lomita Mall94305StanfordCA
Ph.DConnor Beierle [email protected]
Department of Aeronautics & Astronautics
Stanford University
496 Lomita Mall94305StanfordCA
Candidate
Department of Aeronautics & Astronautics
Stanford University
496 Lomita Mall94305StanfordCA
Assistant ProfessorSimone D'amico [email protected]
Stanford University
496 Lomita Mall94305StanfordCA
Pose Estimation for Non-Cooperative Spacecraft Rendezvous Using Convolutional Neural Networks
On-board estimation of the pose of an uncooperative target spacecraft is an essential task for future on-orbit servicing and close-proximity formation flying missions. However, two issues hinder reliable on-board monocular vision based pose estimation: robustness to illumination conditions due to a lack of reliable visual features and scarcity of image datasets required for training and benchmarking. To address these two issues, this work details the design and validation of a monocular vision based pose determination architecture for spaceborne applications. The primary contribution to the state-of-the-art of this work is the introduction of a novel pose determination method based on Convolutional Neural Networks (CNN) to provide an initial guess of the pose in real-time on-board. The method involves discretizing the pose space and training the CNN with images corresponding to the resulting pose labels. Since reliable training of the CNN requires massive image datasets and computational resources, the parameters of the CNN must be determined prior to the mission with synthetic imagery. Moreover, reliable training of the CNN requires datasets that appropriately account for noise, color, and illumination characteristics expected in orbit. Therefore, the secondary contribution of this work is the introduction of an image synthesis pipeline, which is tailored to generate high fidelity images of any spacecraft 3D model. In contrast to prior techniques demonstrated for close-range pose determination of spacecraft, the proposed architecture relies on neither hand-engineered image features nor a-priori relative state information. Hence, the proposed technique is scalable to spacecraft of different structural and physical properties as well as robust to the dynamic illumination conditions of space. Through metrics measuring classification and pose accuracy, it is shown that the presented architecture has desirable robustness and scalable properties. Therefore, the proposed technique can be used to augment the current state-ofthe-art monocular vision-based pose estimation techniques used in spaceborne applications.
Abstract-On-board estimation of the pose of an uncooperative target spacecraft is an essential task for future on-orbit servicing and close-proximity formation flying missions. However, two issues hinder reliable on-board monocular vision based pose estimation: robustness to illumination conditions due to a lack of reliable visual features and scarcity of image datasets required for training and benchmarking. To address these two issues, this work details the design and validation of a monocular vision based pose determination architecture for spaceborne applications. The primary contribution to the state-of-the-art of this work is the introduction of a novel pose determination method based on Convolutional Neural Networks (CNN) to provide an initial guess of the pose in real-time on-board. The method involves discretizing the pose space and training the CNN with images corresponding to the resulting pose labels. Since reliable training of the CNN requires massive image datasets and computational resources, the parameters of the CNN must be determined prior to the mission with synthetic imagery. Moreover, reliable training of the CNN requires datasets that appropriately account for noise, color, and illumination characteristics expected in orbit. Therefore, the secondary contribution of this work is the introduction of an image synthesis pipeline, which is tailored to generate high fidelity images of any spacecraft 3D model. In contrast to prior techniques demonstrated for close-range pose determination of spacecraft, the proposed architecture relies on neither hand-engineered image features nor a-priori relative state information. Hence, the proposed technique is scalable to spacecraft of different structural and physical properties as well as robust to the dynamic illumination conditions of space. Through metrics measuring classification and pose accuracy, it is shown that the presented architecture has desirable robustness and scalable properties. Therefore, the proposed technique can be used to augment the current state-ofthe-art monocular vision-based pose estimation techniques used in spaceborne applications.
INTRODUCTION
The on-board determination of the pose, i.e., the relative position and attitude, of a noncooperative target spacecraft using a monocular camera is a key enabling technology for future on-orbiting servicing and debris removal missions such as e.Deorbit and PROBA-3 by ESA [1], ANGELS by US Air Force [2], PRISMA by OHB Sweden [3], OAAN [4] and Restore-L by NASA [5], and CPOD by Tyvak [6]. The knowledge of the current pose of the target spacecraft during proximity operations enables real-time approach trajectory generation and control updates [7]. This aspect is crucial in noncooperative maneuvers, since little knowledge about the kinematic characteristics of the target is available before the mission and, therefore, the rendezvous and docking trajectory must be generated on-board using the current state estimates. In contrast to systems based on LiDAR and stereo camera sensors, monocular navigation ensures pose determination under low power and mass requirements [8], making it a natural sensor candidate for navigation systems in future formation flying missions.
The current state-of-the-art monocular pose determination methods for spaceborne applications depend on classical image processing algorithms that identify visible target features [9], [10], [11], [12], [13], [14], [15]. These handengineered features (e.g., edges, corners, and lines) are then matched against a reference texture model of the spacecraft to determine the pose. This routine is executed in closedloop for pose tracking using filtering techniques. Generally, the pose solver is an iterative algorithm that minimizes a certain fit error between the features detected in the image and the corresponding features of a reference model. The main advantage of such methods is a high level of interpretation at each step of the image processing and pose determination pipeline. However, these methods are disadvantaged due to the lack of robustness in the presence of adverse illumination conditions and the computational complexity resulting from the evaluation of a large number of possible pose hypotheses. To overcome these two disadvantages, Oumer et al. [16] proposed a method based on appearance learning by creating an offline database of feature points and clusters using a vocabulary tree. However, the main drawback of their work is the reliance on a mock-up of the target satellite for training purposes.
In contrast, enabled by availability of large image datasets and cheap computation, the current state-of-the-art pose de-termination methods for terrestrial applications are shifting towards deep learning techniques [17], [18], [19], [20], [21], [22], [23], [24], [25]. In particular, recent work [20] proposes to solve a classification problem to determine the pose as opposed to a regression problem. The method is exhibited for a variety of 3D models present in terrestrial environments. This approach can be combined with a sliding-window over an image to solve the detection problem as shown by Romano [19] and Sermanet et al. [22]. In comparison to methods used in spaceborne applications, these methods are scalable to tackle multiple types of target in various visual scenes since they do not require the selection of specific hand-engineered features. However, availability of image datasets containing space imagery hinders their use in spaceborne applications. Moreover, unlike imagery captured for terrestrial applications, space imagery is characterized by high contrast, low signal-to-noise-ratio, and low sensor resolution.
The main contribution of this paper is a deep Convolutional Neural Network (CNN) based pose determination method for spaceborne applications. This work leverages transfer learning and learning based on synthetic space imagery datasets. Additionally, the paper investigates the relationship between the performance of this method and factors such as the size of the training datasets, sensor noise, and the level of pose-space discretization. Finally, the paper also compares the performance of this method against state-of-the-art pose determination methods currently employed for spaceborne applications. The method consists of an off-line training phase and an on-line prediction phase. During the training phase, the method automatically generates several thousand synthetic images of a target spacecraft and uses them to train a CNN. During the prediction phase, the input to the method is a grayscale image of a target satellite taken at close proximity (∼10 [m] inter-satellite separation). The trained CNN is then used to predict a pose label corresponding to a region in the four-dimensional space. Of these four dimensions, three correspond to the attitude of the camera reference frame w.r.t. the target's body reference frame and one corresponds to the distance from the origin of the camera reference frame to the origin of the target's body reference frame. Note that this reduces the problem of estimating the full three-dimensional relative position to a unidimensional relative range. Practically, this implies that the architecture requires a sliding-window based approach [22] to detect the region of the image where the target is present and then use the resulting bearing angle information to re-construct the three-dimensional relative position. Since low-level features (e.g., edges, blobs, etc.) for both terrestrial and spaceborne applications can be hypothesized to be similar, the five convolutional layers of the CNN are trained with images from the ImageNet dataset [26] while the fully connected layers of the network are trained with synthetically generated images of the Tango satellite of the PRISMA mission. The architecture of the AlexNet network is adopted as the baseline architecture [27] for this work.
The paper is organized as follows: Section 2 describes the framework for the synthetic dataset generation and the CNN architecture; Section 3 describes the various combinations of datasets used for training, validation experiments, and accompanying results; and Section 4 presents conclusions from this study and presents directions for further work and development.
METHODS
Formally, the problem statement for this work is the determination of the attitude and position of the camera frame, C, with respect to the body frame of the target spacecraft, B. In particular, t BC is the relative position of the origin of the target's body reference frame w.r.t. the origin of the camera's reference frame. Similarly, q(R BC ) is the quaternion associated with the rotation matrix that aligns the target's body reference frame with the camera's reference frame. Training a CNN usually requires extremely large labeled image datasets such as ImageNet [26] and Places [28], which contain millions of images. Collecting and labeling such amount of actual space imagery is extremely difficult. Therefore, this work employs two techniques to overcome this limitation:
• a pipeline for automated generation and labeling of synthetic space imagery.
• transfer learning which pre-trains the CNN on the large ImageNet dataset.
These two techniques are discussed in detail in the following subsections.
Synthetic Dataset Creation
The automated pipeline for generation and labeling of space imagery is based on discretizing the four-dimensional viewspace around a target spacecraft. Three degrees of freedom result from the attitude of the target spacecraft relative to the camera and one degree of freedom results from the distance of the camera from the target spacecraft.
Uniformly locating a set of n camera locations around the target spacecraft is akin to solving for a minimum-energy configuration for charged particles on a sphere of radius r. The determination of a stable configuration of particles constrained on a sphere and being acted by an inverse square repelling force is known as the Thomson problem [29]. The solution is a set of n(n − 1)/2 separations s i,j that minimizes
E = n−1 i=1 n j=1+1 1 s i,j .(1)
Effectively, a locally optimal solution to the problem can be found by iteratively updating the particle positions along the negative gradient of E. A small mesh of camera locations generated in such a manner can be successively subdivided until n camera locations are present on the sphere. Camera locations thus obtained account for two of the four degrees of freedom in the view-space. The third degree of freedom is the rotation of the camera about the boresight direction, which can be uniformly discretized in m − 1 intervals from zero to 360 • . Finally, the degree of freedom corresponding to the distance of the camera relative to the target can be simulated by generating spheres of varying radii. Hence, the inputs to the pipeline are:
• Sphere radii, |t BC | • Number of camera locations per sphere, n • Number of rotations about the camera boresight per camera location, m • 3D texture model of the target spacecraft along with the reflective properties of each of its surfaces and a coarse knowledge of the location of the illumination sources Figure 2. Illustration of the pose space discretization using multiple spheres with uniformly distributed camera locations. This scenario shows two spheres with ten camera locations each. Figure 2 shows a mock scenario with |t BC | = 2, n = 10, m = 1. To create a total of 125,000 images for the purpose of this paper, the following values were chosen as inputs for the pipeline: |t BC | = [8,9,10,11,12,13] meters, n = 500, m = 50. For each of these images, three additional copies were produced with varying levels of Zero Mean White Gaussian Noise (ZMWGN). In particular, the variance of the three levels of noise was selected as 0.01, 0.05, and 0.1 (note that image pixel intensity varies from 0 to 1). Typical images taken in spaceborne applications suffer from high levels of noise due to small sensor sizes and high dynamic range imaging. Therefore, it is imperative to create synthetic images that also possess similar noise characteristics. For each of the 125,000 noise-free images, three additional copies were created in which the target satellite was not aligned with the center of the image plane. This simulates cases where the target spacecraft is in one corner of the image plane, possibly with a few of its features outside the viewing cone of the camera. Finally, a dataset of 25 images (referred to as "Imitation-25" in Table 2) was rendered to imitate 25 actual images of the Tango spacecraft from the PRISMA mission [30]. Specifically, flight dynamics products from the PRISMA mission consisting of on-ground precise relative orbit determination based on GPS (accurate to about 2 [cm] 3D rms) [31] is used as the relative position. On-board attitude estimates from the Tango spacecraft (accurate to about 3 • 3D [14] are used to obtain the relative attitude. Note that the Tango spacecraft employed sun sensors and magnetometers while the Mango spacecraft employed a star tracker. Figure 3 shows a montage of the synthetically generated images part of this dataset compared against their real counterparts.
In total, a superset of 500,000 images were created. The images were rendered using C++ language bindings of OpenGL.
Although the pipeline was used to generate synthetic images of the Tango spacecraft used in the PRISMA mission [32], it can easily accommodate any other spacecraft. The camera field of view was selected to be 31.5 degrees, modeling after the close range camera flown aboard the Mango spacecraft of the PRISMA mission. The generated images were resized to be 227 pixels by 227 pixels to match the input size of the AlexNet architecture [27] as well as to conserve disk space and RAM usage during the training process.
After the generation of images, each image must be assigned a pose label that best approximates the true pose of the camera relative to the target spacecraft while capturing the image. This approximate pose label will be used to train the CNN with the expectation that the CNN will learn the visual features associated with the cluster of images belonging to each pose label. More importantly, it is expected that the CNN will learn the correlation between these learned features and the approximate pose label associated with those images. Since the CNN solves a classification problem to determine a pose that exists in a continuous domain, it is important to clarify the distinction between classification and pose estimation accuracies. It is expected that the level of discretization of the pose space used during training drives the accuracy of the on-line pose estimation. Thus, the choice of the number of pose labels used during training depends on the required pose estimation accuracy. For the purpose of this paper, four levels of discretization were used resulting in 6, 18, 648, and 3000 pose labels. The pose labels were generated using the same procedure as described above for the image generation. The input values |t BC |, n, m used for each of these pose labels is presented in Table 1. Each generated image is then assigned to a pose label for each of the four levels of discretization using a simple search algorithm. First, all pose labels associated with the same camera distance relative to the target as the image are selected. Then, for each possible pose label an axis-angle parametrization of the attitude change required to match the camera attitude associated with the image can be calculated. Finally, the pose label that minimizes this angular change is selected as that image's pose label. The pseudo-code for this search algorithm is presented below as Algorithm 1.
Note that Algorithm 1 is repeated for each level of discretization. For the purpose of this paper, this allowed us to compose the superset of 500,000 images into 10 datasets. Table 2 presents the details for each of these datasets. Each for label in allPoseLabels do 5: if image.dist == label.dist then 6: quatDiff = quatmult(image.quat, quatinv(label.quat)) 7: angDiff = quat2axang(quatDiff) 8: if angDiff < minAngDiff then 9: minAngDiff = angDiff 10: image.label = label dataset was further divided into a training, validation, and test set, which represented 60%, 20%, and 20% images of the dataset, respectively. Figure 5 shows a montage of images associated with four different pose labels of the Clean-648 training dataset.
Convolutional Neural Network
The CNN used in this work adopts the structure of the AlexNet architecture [27]. AlexNet was chosen over networks such as VGG-16 [33] and Inception [34] due to the relatively lower number of operations required for inference [35]. Moreover, since the number of images in the synthetically generated datasets is not as high as typical datasets used to train these networks, the proposed method relies on transfer learning. The hypothesis is that low level features detected by the first few layers of a CNN are the same across the terrestrial and spaceborne domains. Therefore, only the parameters in the last few layers need to be determined to adapt the network for space imagery. The AlexNet architecture was used to train eight networks with varying sizes and compositions of the training set. The description of the eight networks is presented in Table 3. The AlexNet architecture is shown in Figure 6, it contains eight layers with weights, the first five are convolutional layers and the remaining three are fully-connected layers. The output of the last fully-connected layer is used in an xway softmax loss function which produces a distribution over the class labels (where x is the number of pose labels in the dataset used to train the network). The formula to compute the softmax loss function is presented below in equations 2 and 3. The network maximizes the multinomial logistic regression objective, which is equivalent to maximizing the average across training cases of the log-probability of the correct label under the prediction distribution.
Loss softmax = x i=1 L i(2)
where
L i = − log e fy i j e fj(3)
In equation 3 the notation f j refers to the j-th element of the vector of values output by the last fully-connected layer, f .
The "dropout" technique [36] was used while training the fully connected layers. This technique consists of setting to zero the output of each hidden neuron with probability of 0.5. The "dropped" neurons do not contribute to the forward pass and do not participate in back-propagation. The output of the remaining neurons is scaled by a factor of 2 such that the expected sum remains unchanged. This technique reduces the possibility of co-adaptations of neurons, i.e., neurons cannot rely on the presence of particular other neurons but instead learn more robust features. Secondly, horizontal reflection of images was utilized for "net1","net2","net3","net4", and "net5", which effectively increased the size of the training set by a factor of two. This data augmentation technique was hypothesized to reduce over-fitting on the image data by artificially enlarging the dataset.
EXPERIMENTS
This section presents results of two types of experiments. The first type of experiments were carried to understand the training process of the CNN's on small sets of images and evaluating the network's test accuracy as a factor of:
• Number of images used in training • Amount of ZMGWN in test images • Amount of displacement of the target from the center of the image plane
The second set of experiments were carried to understand the feasibility of training CNN's for a realistic on-orbit servicing mission and evaluating its performance on imagery imitating actual space imagery from the PRISMA mission. Both of these experiments and their results are discussed in the following subsections.
Type 1
The Type 1 experiments involved comparing the performance of net1, net2, net3, net4, net5, and net6 on the Clean-6, Clean-18, Gaussian-1, Gaussian-5, Gaussian-10, Centered-1, Centered-2, and Centered-3 datasets. The comparison is based on the accuracy of the predictions and the F-Measure (F M ) of the classifications. The results presented here are based on testing the networks on the test set of the datasets described in Table 2. In particular, the accuracy is defined as the percentage of the test images that were correctly classified by the network. F M of the classifications is based on the precision and recall. These metrics are based on the number of false positives (F P ), false negatives (F N ), and true positives (T P ) over several samples. In order to compute these values, we treat each class as a binary classification problem, defining a positive sample when it belongs to that class, and negative otherwise.
precision = T P (T P + F P )(4)recall = T P (T P + F N )(5)
These are then used to calculate the F-Measure:
F M = 2 · precision · recall precision + recall(6)
There are several trends as seen in Figure 7, which shows the classification accuracy of five separately trained networks on six different datasets. Firstly, all networks are more or less equally capable in classifying images in the Clean-6 dataset where images are free from ZMGWN and the target is centered in the image plane (all networks trained in this work were trained with centered targets). This most likely shows that the networks have a vast number of parameters and can easily over-fit the features seen in the images. Secondly, as the ZMGWN is added, all networks show a decline in the classification accuracy. Notably, "net5" fares quite well as compared to the other networks as it used some noisy images during training. This implies that as long as sensor noise is known and can be modeled beforehand, the CNN can be made to be more robust to noise through the augmentation of the training data with noise. Thirdly, the classification accuracy of the networks correlates with the number of training images used during the training since "net1", which was trained with more images than "net2", "net3", and "net4" has higher accuracy for Clean-6, Uncentered-1, Uncentered-2, and Uncentered-3 datasets.
Lastly, "net6" was trained and tested using the Clean-18 dataset without data augmentation. The network produced a classification accuracy of 99.4%, which was significantly higher than any other networks trained on smaller datasets with data augmentation. To visualize how the network had learned to separate the 18 different classes, the test set images of Clean-18 were embedded according to their features from the penultimate fully connected layer. The t-Distributed Stochastic Neighbor Embedding (t-SNE) technique was used for dimensionality reduction [37]. This technique represents images as nodes in a graph and arranges the nodes in two dimensions in a manner that respects the high-dimensional L2 distances between their features. In other words, t-SNE arranges images that have similar features nearby in a 2D embedding. This can be visualized in Figure 8 for the Clean-18 test set images, where images from three different intersatellite ranges are represented by different marker types. It can be easily seen that the network has learned to differentiate images from separate ranges. Moreover, for each range, certain classes are learned to be closer as compared to the others. This is to be expected since for example, two pose labels visually do look similar (in fact, they are close to being horizontal mirrors of each other).
Type 2
The Type 2 experiments involved evaluating the performance of net6 and net7 on the Clean-648, Clean-3k, and Imitation-25 datasets. The goal of these experiments was to stresstest all key aspects of this method, from conceiving pose labels, to training and testing. Unlike the Type 1 experiments, these experiments were run with a high number of pose labels and large training datasets in order to achieve a higher "pose estimation accuracy". In particular, the pose estimation accuracy is defined by two metrics: E R and E T , which are differences between true and estimated values of relative attitude and position, respectively. In particular,
E T = t BC,est − t BC,true(7)E R = 2 cos −1 (z s ), where (8) z = [z s z v ] = q true * conj(q est ).
Here q true and q est are true and estimated values of the quaternion associated with the rotation matrix that aligns the target's body reference frame and the camera's reference frame. Table 4 shows the test set accuracy for net7 and net8 on the Clean-648 and Clean-3k datasets, respectively. Note that net7 has a much higher classification accuracy as compared to net8 since it only needs to pick the correct pose label out of a set of 648 pose labels compared to 3000 for net7. In addition, net7 was trained on the Clean-648 dataset which contained approximately 45 images per pose label as compared to 25 images per pose label for the Clean-3k dataset used for net8. However, due to the larger number of classes in Clean-3k dataset, net8 produced higher pose estimation accuracy compared to net7.
Since the output of the fully connected layer of the net8 is used in a 3000-way softmax (648-way softmax for net7), the values can be interpreted as the probability of the image being associated to each pose label. Further, this allows the setting up of a confidence metric to classify the pose solutions. For example, this paper classifies the pose solution to be of "high confidence" if the ratio of the highest and the nexthighest probability values is greater than 2. Figures 9 and 10 present a few of these high and low confidence pose solutions provided by net8 on the Imitation-25 dataset. Note that the Imitation-25 dataset was generated using the PRISMA flight dynamics products, independent of the datasets used in training and validating these networks. Table 5 shows the pose estimation accuracy of the high confidence solutions provided by net8 alongside all solutions provided by net7 and net8 on the Imitation-25 dataset. Table 5 also shows the pose estimation accuracy of two other architectures, namely, the Sharma-Ventura-D'Amico (SVD) architecture [15] and an architecture based on the EPnP [38] and RANSAC algorithms [39]. These two architectures rely on the conventional method of hypothesizing and verifying poses based on the extraction of edge features from the image. Table 5 shows that net8 has a much higher accuracy compared to net7 due to the finer discretization of the pose space in the Clean-3k dataset. Further, net8 is also more accurate than the architecture based on EPnP and RANSAC algorithms but less accurate than the SVD architecture. However, note that the high confidence solutions of SVD are only available on 20% of the images of the Imitation-25 dataset whereas net8 provides a high confidence solution on 68% of the images. Hence, this suggests the potential use of net8 (or similar CNN based approaches) to provide a coarse initial guess for the SVD architecture (or similar feature based approaches).
CONCLUSIONS
In this work, we successfully set up a framework for pose determination using convolutional neural networks, and exhaustively tested it against different datasets. Some interesting conclusions can be drawn from these experiments, which can be used as the building blocks for the development of navigation systems in future formation flying missions. First, the size of training set is shown to correlate with accuracy. This warrants the generation of even larger synthetic datasets by introducing slight variations in the target location and orientation. Second, as compared to networks trained on noise-free images, the classification accuracy of the network trained with images containing small amounts of Gaussian white noise had a much better performance on test images containing high amounts of Gaussian white noise. This proves that as long as the sensor noise could be modeled beforehand or removed using pre-processing techniques, the CNN have a good potential for pose determination using actual space imagery. Third, all networks were trained using the transfer learning approach, which only required training of the last few layers. This proves that several low level features of spaceborne imagery are also present in terrestrial objects and there is no need to train a network completely from scratch for spaceborne applications. Lastly, the network trained using 75000 images associated with 3000 pose labels showed the highest pose estimation accuracy on the Imitation-25 dataset. In fact, its accuracy was better than an architecture based on classical feature detection algorithms. As compared to the best performing feature detection based algorithm, the network provided high confidence solutions for three times as many images. Therefore, the network clearly has potential to be used as an initializer for the current state-of-the-art pose determination algorithms.
However, there are several caveats in the presented work and significant potential for future development and enhancements. First, the networks need to be tested not just using synthetic imagery but also actual space imagery. Further, a much larger dataset is required for a comprehensive comparative assessment of the CNN-based pose determination architectures with the conventional pose determination architectures. Second, since the architecture is modular and scalable to any target spacecraft, its accuracy and robustness potential with other spacecraft and orbit regimes need to be evaluated. Third, the relationship of the fidelity of the synthetic training dataset with the navigation performance at test time needs to be further assessed. In particular, it needs to be determined how accurate the assumptions of the illumination environment, target texture, and reflectance properties need to be during the CNN training to guarantee reliable and accurate pose solutions at test time. Lastly, there is potential of an increase in the pose estimation accuracy if a larger number of pose labels and larger datasets are used during training. Therefore, more layers of the current network architecture would be trained with space imagery instead of terrestrial imagery. However, the benefit of a larger network would drive up memory requirements on-board the servicer spacecraft.
OF CONTENTS 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. METHODS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3. EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 4. CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 BIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 978-1-5386-2014-4/18/$31.00 c 2018 IEEE
Figure 1 .
1Illustration of the pose determination problem.
Figure 3 .
3Comparison of the synthetically generated images from the Imitation-25 dataset (top row) with actual space imagery (bottom row) from the PRISMA mission. Relative position and orientation of the camera used for image generation were obtained from actual flight data for this dataset.Algorithm 1 Assigns a pose label to an image for each of the four levels of pose discretization 1: procedure ASSIGNLABEL(image,
Figure 4 .
4Visualization of a uniform distribution of 10242 camera locations (small colored markers) and 162 pose labels (large black markers) around a unit sphere. Camera locations associated to the same pose label are denoted by the same colored marker.
Figure 5 .
5Montage of a few images from four different pose labels of the Clean-648 dataset.
Figure 6 .
6An illustration of the architecture of AlexNet, used as the baseline for all eight networks in this paper. The network's input is 154587-dimensional, and the number of neurons in the network's remaining layers is given by 145200-93312-32448-32448-21632-2048-2048-x. The last layer contains as many neurons as the number of pose labels in the dataset used to train the particular network.
Figure 7 .
7Classification accuracy [%] for seven datasets using five separate networks.
Figure 8 .
8The t-Distributed Stochastic Neighbor Embedding (t-SNE)representation of the Clean-18 test set images.
Figure 9 .
9Montage of a few images from the high confidence pose solutions produced by net8 on the Imitation-25 dataset.
Figure 10 .
10Montage of a few images from the low confidence pose solutions produced by net8 on the Imitation-25 dataset.
TABLE
Table 1 .
1Summary of discretization levels used to generate the pose labels.# Pose Labels
|t BC | [m]
n
m
6
{3}
6
1
18
{3, 5, 9}
6
1
648
{8,9,10,11}
162 1
3000
{8,9,10,11,12} 300 10
rms) and the Mango spacecraft (accurate to about 0.1 • 3D
rms)
Table 2 .
2Description of the ten datasets created from the synthesized images.Dataset
Description
# Pose Labels # Images
Clean
No noise, centered target
6
1601
Clean-18
No noise, centered target
18
4803
Clean-648
No noise, centered target
648
40968
Clean-3k
No noise, centered target
3000
125000
Gaussian-1 ZMGWN with variance of 0.01, centered target
6
1601
Gaussian-5 ZMGWN with variance of 0.05, centered target
6
1601
Gaussian-10 ZMGWN with variance of 0.1, centered target
6
1601
Centered-1
No noise, t = [0.2, 0.0, 3.0] meters
6
1601
Centered-2
No noise, t = [0.0, 0.2, 3.0] meters
6
1601
Centered-3
No noise, t = [0.2, 0.2, 3.0] meters
6
1601
Imitation-25
No noise, PRISMA flight data
3000
25
Table 3 .
3Description of the eight networks trained for this work. Note that the columns represent the number of training images used from the particular dataset.Network # Pose Labels Clean Gaussian-1 Clean-18 Clean-648 Clean-3k
Table 4 .
4Performance of the net7 and net8 networks on
test sets of the Clean-648 and Clean-3k datasets,
respectively.
Metric
net7
net8
Mean E R (deg)
22.91 11.94
Mean E T (m)
0.53
0.12
Mean Classification Accuracy (%) 83.3
35
Table 5 .
5Performance of net7 and net8 networks compared against conventional pose determination methods using the Imitation-25 dataset. Min. E R , Max. E R (deg) 19.16, 174.98 5.05, 175.05 Min. E T , Max. E T (m)Metric
net7
net8
net8 (high conf.) SVD (high conf.) EPnP+RANSAC
Mean E R (deg)
82.18
30.75
14.35
2.76
140.71
5.05, 38.32
0.67, 4.94
22.9, 215.9
Std. Dev. E R (deg)
55.04
48.62
9.99
1.98
54.9
Mean E T (m)
1.79
1.12
0.83
0.53
1.45
0.28, 5.04
0.34, 3.06
0.34, 2.20
0.14, 0.71
0.19, 5.01
Std. Dev. E T (m)
1.39
0.77
0.52
0.10
1.25
Solution Availability (%)
100
100
68
20
100
ACKNOWLEDGMENTSThe authors would like to thank The King Abdulaziz City for Science and Technology (KACST) Center of Excellence for research in Aeronautics & Astronautics (CEAA) at Stanford University for sponsoring this work. The authors would also like to thank OHB Sweden, the German Aerospace Center (DLR), and the Technical University of Denmark (DTU) for the PRISMA images used in this work. The authors would like to thank Keanu Spies of the Space Rendezvous Laboratory for his technical contributions to the generation of the virtual spacecraft imagery.BIOGRAPHY[Sumant Sharma is a Ph.D. student in the Space Rendezvous Laboratory of Stanford University and a Systems Engineer at NASA Ames Research Center. He graduated from the Georgia Institute of Technology and Stanford University with a Bachelor of Science and a Master of Science degree in aerospace engineering, respectively. His current research focus is to devise algorithms based on monocular computer vision to enable navigation systems for on-orbit servicing and rendezvous missions requiring close proximity.Connor Beierle is a Ph.D. student in the Space Rendezvous Laboratory. He graduated from Stony Brook University with a Bachelor of Engineering degree in mechanical engineering. He has experience working for NASA and Google. He has also worked on the Hemispherical Anti-Twist Tracking System in the Space Systems Development Laboratory. His current research is focused on the high-fidelity validation of advanced optical navigation techniques for spacecraft formation-flying and rendezvous. and PRISMA (Sweden/Germany/France), for which he received several awards. Dr. D'Amico's research lies at the intersection of advanced astrodynamics, GN&C, and space system engineering to enable future distributed space systems. He has over 100 scientific publications including conference proceedings, peer-reviewed journal articles, and book chapters. He has been a Program Committee Member
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| []
|
[
"Reduced order prediction of rare events in unidirectional nonlinear water waves",
"Reduced order prediction of rare events in unidirectional nonlinear water waves"
]
| [
"Will Cousins [email protected] \nDepartment of Mechanical Engineering\nMassachusetts Institute of Technology\n77 Massachusetts AveCambridgeMAUSA\n"
]
| [
"Department of Mechanical Engineering\nMassachusetts Institute of Technology\n77 Massachusetts AveCambridgeMAUSA"
]
| []
| We consider the problem of short-term prediction of rare, extreme water waves in unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually-intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the dispersive mixing of different harmonics, can grow significantly due to localized nonlinear focusing. Here we show how the interplay between i) statistical properties captured through linear information such as the waves power spectrum and ii) nonlinear dynamical properties of focusing localized wave groups defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the 'trigger' of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the Modified Nonlinear Schrodinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms. | 10.1017/jfm.2016.13 | [
"https://arxiv.org/pdf/1501.05001v1.pdf"
]
| 14,763,838 | 1501.05001 | cbd9ea901d25628e752db1f5576465cf148eacfa |
Reduced order prediction of rare events in unidirectional nonlinear water waves
20 Jan 2015
Will Cousins [email protected]
Department of Mechanical Engineering
Massachusetts Institute of Technology
77 Massachusetts AveCambridgeMAUSA
Reduced order prediction of rare events in unidirectional nonlinear water waves
20 Jan 2015
We consider the problem of short-term prediction of rare, extreme water waves in unidirectional fields, a critical topic for ocean structures and naval operations. One possible mechanism for the occurrence of such rare, unusually-intense waves is nonlinear wave focusing. Recent results have demonstrated that random localizations of energy, induced by the dispersive mixing of different harmonics, can grow significantly due to localized nonlinear focusing. Here we show how the interplay between i) statistical properties captured through linear information such as the waves power spectrum and ii) nonlinear dynamical properties of focusing localized wave groups defines a critical length scale associated with the formation of extreme events. The energy that is locally concentrated over this length scale acts as the 'trigger' of nonlinear focusing for wave groups and the formation of subsequent rare events. We use this property to develop inexpensive, short-term predictors of large water waves. Specifically, we show that by merely tracking the energy of the wave field over the critical length scale allows for the robust, inexpensive prediction of the location of intense waves with a prediction window of 25 wave periods. We demonstrate our results in numerical experiments of unidirectional water wave fields described by the Modified Nonlinear Schrodinger equation. The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by uncertainty and potentially strong nonlinear mechanisms.
Introduction
Understanding and predicting nonlinear water waves, especially those characterized by large magnitude, is one of the most challenging topics for ocean engineering both because of the catastrophic impact they can have on ocean engineering structures (e.g. for ships and offshore platforms) and naval operations, but also because of the serious lack of specialized mathematical tools for the analysis of the underlying physics [17,3,31,40]. This is because nonlinear water wave dynamics are characterized both by the existence of inherent uncertainty (expressed in the form of phase uncertainty between different Fourier modes) and also in the form of strong nonlinearities and associated energy transfers between modes. The latter can be activated locally and intermittently, leading to unusually high magnitude waves that emerge out of the complex background wave field.
An extreme form of such dynamical evolution is the case of freak or rogue waves, which can be as large as four times the standard deviation of the surrounding wave field [32,17]. However, waves of even lower magnitude (but larger compared with the surrounding wave field) can still occur with large probability, causing the heavy-tailed statistics for water wave elevations [32]. Waves of this magnitude have caused considerable damage to ships, oil rigs, and human life [23,29]. In addition, many naval operations e.g. transfer of cargo between ships moored together in a sea base, landing on aircraft carriers, or path planning of high-speed surface vehicles require short-term prediction of the surrounding wave field. To make such predictions, unusually high wave elevations must be forecasted reliably and inexpensively.
One mechanism for the occurrence of such rare, unusually-intense wave elevations is nonlinear wave focusing [25,34,11]. For deep water waves a manifestation of this focusing is the well-known Benjamin-Feir instability of a plane wave to small sideband perturbations. This instability, which has also been demon-strated experimentally [8], generates huge coherent structures by soaking up energy from the nearby field [5,42,35]. In [11], it was recently demonstrated that even imperfect background conditions, i.e. completely different from the idealized plane wave setup of the Benjamin-Feir instability, can still lead to important wave focusing and rare events. In particular, it was analytically shown and numerically demonstrated for unidirectional wavetrains that there is a critical combination of wave group length scales and amplitudes which will lead to wave focusing and thus unusually high elevations. In contrast to the standard BF mechanism these instabilities have an essentially localized character.
Random relative phases between different harmonics can potentially lead to the 'triggering' of this localized focusing mechanism. Such phase randomness is mainly introduced by the mixing of different harmonics due to their dispersive propagation. Here we show how the interplay between i) statistical properties captured through linear information such as the waves power spectrum and ii) nonlinear focusing properties of localized wave groups and in particular nonlinear focusing properties, defines a critical length scale that is associated with the occurrence of strongly nonlinear interactions and the formation of extreme events. The energy that is locally concentrated over this length scale acts essentially as the 'trigger' of nonlinear focusing of wave groups.
We use this property to develop short-term predictive schemes for the occurrence of large water waves. Specifically, we first demonstrate that using a scale-selection algorithm [28] allows for the partition of the current wave field into a set of wave groups having different length scale and amplitude. A direct computation of the analytical nonlinear stability criterion for focusing of localized wavegroups [11] allows for a very inexpensive forecast of nonlinear focusing and subsequent growth for each wave group of the field. In a second stage, we demonstrate that merely tracking the energy of the wave field over the critical length scale defined by the interplay between statistics and nonlinearity allows for an even cheaper and robust forecast of upcoming intense nonlinear wave elevations, with a prediction window on the order of 25 wave periods. We demonstrate our results in numerical experiments involving unidirectional water wave fields described by the Modified Nonlinear Schrodinger equation.
The proposed predictive method reveals and directly utilizes the low-dimensional character for the domain of attraction to these rare water waves. In particular, despite the distribution of the background energy over a wide range of scales, the 'trigger' of nonlinear focusing is essentially low-dimensional and to this end it can be used as an inexpensive way to estimate the probability for a rare event in the near future. In addition, the association of the predictor with the energy over a specific length scale gives strong robustness properties against measurement errors and energy in other spatial scales of the wave field. The presented approach introduces a new paradigm for handling spatiotemporal rare events in dynamical systems with inherent uncertainty by providing an efficient description of the 'trigger' that leads to those rare events through the careful study of the synergistic action between uncertainty and nonlinearity.
Extreme Events in Envelope Equations
In this work, we consider waves traveling on the surface of a fluid of infinite depth. A typical approach for modeling this phenomena is to assume incompressible, irrotational, inviscid flow, which gives Laplace's equation for the velocity potential. This equation is paired with two boundary conditions: a pressure condition and a kinematic one (a particle initially on the free surface remains so). This model agrees well with laboratory experiments [39], and faithfully reproduces the classical k −5/2 spectral tail observed in deep water [33]. Although some care is required to numerically deal with the free surface, this fully nonlinear model may be solved numerically with reasonable computational effort, particularly in one space dimension [15,13,16,9]. However, the presence of the free surface makes analysis of the underlying dynamics challenging.
Thus, in this work we consider approximate equations governing the evolution of the wave envelope, the Nonlinear Schrodinger Equation (NLS) [42] and the Modified Nonlinear Schrodinger Equation (MNLS) of [19]. Both NLS and MNLS can be derived via a perturbation approach from the fully nonlinear model under assumptions of small steepness and slow variation of the wave envelope. Although forms of these equations exist in a full two-dimensional setting, here we consider wave fields varying only in the direction of propagation. The NLS equation, in nondimensionalized coordinates, reads
∂u ∂t + 1 2 ∂u ∂x + i 8 ∂ 2 u ∂x 2 + i 2 |u| 2 u = 0 (1)
where u(x, t) is the wave envelope. To leading order the surface elevation is given by η(
x, t) = ℜ[u(x, t)e i(x−t) ].
Equation (1) has been nondimensionalized with x = k 0x , t = ω 0t , u = k 0ũ , wherex,t, andũ are physical space, time and envelope. k 0 is the dominant spatial frequency of the surface elevation, and ω 0 = √ gk 0 . Our primary interest in this work is the Modified NLS equation, which is a higher order approximation of the fully nonlinear model,
∂u ∂t + 1 2 ∂u ∂x + i 8 ∂ 2 u ∂x 2 − 1 16 ∂ 3 u ∂x 3 + i 2 |u| 2 u + 3 2 |u| 2 ∂u ∂x + 1 4 u 2 ∂u * ∂x + iu ∂φ ∂x z=0 = 0 (2)
where φ is the velocity potential and ∂φ/∂x
z=0 = −F −1 |k|F [|u| 2 ] /2. F denotes the Fourier transform.
The MNLS equation has been shown to reproduce laboratory experiments reasonably well [30,21]. There are even higher order envelope equations, such as the Broadband Modified NLS Equation (BMNLS) [37]. However, we do not discuss these equations here. Although there are considerable differences between NLS and MNLS, we found minimal differences between simulations of MNLS and BMNLS. Dysthe et. al. also found similar agreement between MNLS and BMNLS [18]. These envelope equations, as well as the fully nonlinear water wave model, admit periodic plane wave solutions. Interestingly, these plane wave solutions are unstable to sideband perturbations. This instability, termed the Benjamin-Feir instability after its discoverers ( [5], see also Zakharov [42]), has a striking manifestation. Energy is "sucked up" from the nearby field to produce a large amplitude coherent structure, containing a wave 2.4-3 times larger than the surrounding background field [35]. This behavior is been shown numerically in envelope equations [41,20] as well as the fully nonlinear formulation [24]. Furthermore, a number of experiments confirm these numerical predictions [8,7].
However, in realistic physical settings, the water surface is not merely a plane wave-energy is distributed over a range of frequencies. Thus, in this work, we consider extreme waves emerging out of a background with Gaussian spectra and random phases, that is
u(x, 0) = N/2 −N/2+1 2∆ k F (k∆ k )e i(ω k x+ξ k ) , F (k) = ǫ 2 σ √ 2π e −k 2 2σ 2
where ξ k are independent, uniformly distributed random phases. Here we adopt the definition of an extreme wave as any instance where |u| > H E = 4ǫ (as defined above ǫ is the standard deviation of the surface elevation). In these irregular wave fields, it is well known that the critical quantity for extreme event formation is the Benjamin-Feir Index, which is the ratio of the energy level of the field to its bandwidth [25]. If the Benjamin-Feir Index is large enough, then nonlinear interactions dominate, leading to the appearance of large amplitude coherent structures and heavy-tailed statistics for the elevation [4,14,18,25,32]. We solve the MNLS equation numerically using a Fourier method in space. The use of periodic boundary conditions is of course artificial, but is a standard convention. We take our spatial domain to be 128 wavelengths (256π), large enough to avoid any box-size effects. We use a 4th order Runge-Kutta exponential time differencing scheme [12,22]. This scheme requires evaluation of the function φ(z) = (e z − 1)/z. Naive computation of φ can suffer from numerical cancellation error for small z [26]. We use a Pade approximation code from the EXPINT software package, which does not suffer from such errors [6]. We use 2 10 Fourier modes with a time step of 0.025; results in this work were insensitive to further refinement in grid size. Initially the density is Gaussian but eventually develops heavy tails due to nonlinear interactions, which also cause the spectrum to change shape (bottom left). These heavy tails are due to large amplitude coherent structures (bottom right), which emerge via focusing of localized wave groups (top right).
Localized Wave Group Evolution
A large Benjamin-Feir Index indicates that extreme events are more likely than Gaussian statistics would suggest. However, a large Benjamin-Feir Index does not provide any specific information on precisely where an extreme event will occur. Thus, in order to develop a scheme providing precise spatiotemporal predictions, we must develop a more precise indicator than the BFI. We observed in simulations of high BFI fields, extreme events appear to be triggered by the focusing of localized wave groups (Ruban has also made a similar observation [36]). Figure 1 displays such an example of extreme event formation by focusing of localized groups. In this example we see that a localized group focuses, narrowing in width and doubling in amplitude, yielding an extreme event.
To better understand this mechanism, in [11] we studied the evolution of isolated wave groups. We briefly review these results describing the evolution of hyperbolic secant initial data:
u(x, 0) = Asech(x/L)
Due to the invariance of the envelope equations we take A real with no loss of generality. We investigated the evolution of such groups as a function of amplitude A and length scale L. In particular, we are interested in whether or not a group will focus and, if it does, the degree by which the group amplitude is magnified. Similar questions asked by Adcock et al. in the context of the one-dimensional NLS [2] as well as NLS and the fully nonlinear model in two dimensions [1]. To answer these questions, we numerically evolved hyperbolic secant initial data for many values of A and L for NLS and MNLS. To measure the degree of focusing a group undergoes, we computed the value of the first spatiotemporal local maximum of |u|, and termed this value u max (A, L). For a defocusing group (middle row, Figure 2), this local maximum will occur at x = 0, t = 0 and we trivially have u max (A, L) = A. For a focusing group (bottom row, Figure 2), the group will contract and increase in amplitude. The group amplitude will eventually reach a maximum and then demodulate, decreasing in amplitude. To make the focusing behavior clear, in the top two panes of Figure 2 we plot the amplitude growth factor 1 A u max (A, L)
for NLS (top left pane) and MNLS (top right pane). This amplitude growth factor describes the degree of focusing which has occurred, with a value of 1 indicating that the group does not grow in amplitude. We observe that for both NLS and MNLS, there are a range of groups that focus considerably. However, there are stark differences between group evolution in the two equations owing to the lack of scale invariance in the MNLS equation. In MNLS, the set of focusing groups is smaller compared with NLS, and many groups that do focus do so to a smaller degree (see example in bottom row, Figure 2). Particularly, in MNLS there is a smallest focusing length scale where groups thinner than this scale do not focus, regardless of how large their initial amplitude may be. The scale invariance of NLS, however, precludes such behavior. A similar lack of focusing behavior at small group length scales was observed by Henderson et. al. in numerical simulations of the fully nonlinear model in one space dimension [24].
The nonlinear behavior of focusing groups can also be described analytically through an adaptive projection of the governing equations on a family of basis elements that respect, by design, certain conservation properties of NLS and MNLS equations. In particular, following [11] we formulate a reduced-order model that captures the behavior of a localized wave group that has a hyperbolic secant form for as long as this assumption is valid. In addition, the adaptive character of the projection allows for the consideration of the constraints between the amplitude A and length scale L that follow from the conservation properties of NLS and MNLS. This approach gives the following reduced-order model for MNLS [11]
d 2 |A| 2 dt 2 = K |A| 2 d|A| 2 dt 2 − 3|A| 2 2048L 6 (196|A| 4 L 4 − 64|A| 2 L 4 + 168|A| 2 L 2 + 32L 2 + 27).
We stress that the main takeaway from these experiments and reduced order models is the construction of the map u max (A, L). That is, given a wave group of amplitude A and length scale L, we know the maximally focused amplitude group after its nonlinear evolution via interpolation of our numerical experiments, or approximately using the reduced-order model. This map will be an essential component of our extreme event predictive scheme, developed in Section 5.
Probability of Critical Wave Groups in Irregular Wave Fields
The analysis presented above illustrates the family of unstable wave groups that will focus and eventually lead to an extreme event. In a particular sea state, dispersion effects create random mixing of different harmonics. To this end, all groups do not occur with equal likelihood-the probability of a particular group occurring is determined by the spectral properties of the field. For example, in a Gaussian spectrum the spatial field will contain groups unlikely to focus if the energy level ǫ is small (low amplitude groups) or if the spectral bandwidth σ is large (small length scale groups). Thus, the frequency and nature of the unstable wave groups in a particular field results from the interplay of the nonlinear dynamical properties of the system (determined by the underlying physics) and the statistical properties of the background field (captured by the spectrum), where dispersion is the dominant mechanism.
We now describe a procedure to directly quantify how the statistics of the background field interact with the underlying nonlinear dynamics to create extreme events. To do so, for a variety of Gaussian spectra we compute the joint density of group amplitude and length scale. To compute this density, we use a scale selection algorithm to identify groups as well as their associated length scales and amplitudes (see Appendix A for details on the scale selection algorithm). In the top pane of Figure 3, we display a particular field |u(x)| as well as the groups identified by the group detection algorithm, showing that this algorithm appropriately picks out the dominant groups. We then compute the joint probability density of group amplitude and length scale by applying this group identification algorithm to 50,000 realizations of Gaussian spectrum random fields. For each length scale L, we determine the smallest group amplitude required to trigger an extreme event. That is, if the extreme event threshold is H E = 4ǫ, we find, for each L, the smallest A such that u max (A, L) ≥ H E . Denoting this amplitude as A * , we trivially have A * ≤ H E . This procedure describes a curve A * (L), where isolated groups located above this curve in the (L, A) plane would yield an extreme event, and those below this curve would not. We overlay this curve over the joint densities of group amplitude and length scale displayed in Figure 3.
For a given spectrum, we may determine the frequency and nature of extreme event-triggering groups (i.e. those lying above the respective curves in Figure 3. As expected, increasing the energy level or decreasing the spectral bandwidth increases the number of these extreme-triggering groups. This analysis provides a concrete, wave-group based explanation of the development of heavy tails via nonlinear interactions in high BFI regimes. However, unlike traditional BFI-based analysis, this wave group analysis directly incorporates the lack of scale invariance in MNLS. This fact can be observed in Figure 3 (middle and bottom rows) where we display (A, L) densities for two different spectra with the same BFI. We note the scale invariance property of NLS (note that the two black curves are identical, albeit the axes rescale) and how this contrasts to the corresponding curves (red dashed lines) for the MNLS, which change between the two spectra even though the BFI index remains the same.
This statistical instability analysis allows us to characterize the properties of extreme event-triggering groups in a maximum likelihood sense. Specifically, for a given spectrum, we can compute the density of the length scale of groups that would generate extreme events (i.e. those lying above the curves in the left column of Figure 3). In Figure 3 (right column) we also display examples of these length scale densities for the two different spectra. The concentration of these densities around their peak value suggests that for a given spectrum there is a most likely extreme event triggering length scale, L E . As the distribution is fairly narrow, we expect that the majority of extreme events that occur will be triggered by localization at length scales close to L E . In Section 5 we will use this fact to develop a predictive scheme based on projecting the field onto an appropriately tuned set of Gabor wavelets.
Prediction of Extreme Events
In this section, we describe the central result of this paper, schemes for prediction of extreme events. Our goal is to develop schemes that provide reasonably precise spatiotemporal predictions of upcoming extreme events. These schemes also must require less computational cost then solving the full envelope equations.
We describe two predictive schemes that meet these criteria. First, we develop an algorithm to predict extreme events by identifying the dominant wave groups in a given field, and use our results from Section 3 regarding evolution of localized groups to predict whether this group will trigger an extreme event. Second, we develop a scheme based on projecting the field onto a carefully tuned set of Gabor modes. We find that large values of a certain Gabor coefficient indicate that an upcoming extreme event is likely. This Gabor-based scheme is nearly as reliable as the group detection scheme, yet requires a remarkably small computational cost.
Prediction by Wave Group Identification
Here we describe a straightforward scheme for advance prediction of extreme events via wave group identification. For a given field, we apply the group identification algorithm described in Appendix A to the envelope. This gives the spatial location, amplitude A, and length scale L of each wave group. We then predict the future focused amplitude of the group by evaluating u max (A, L), where u max is the numerically constructed function from our prior study of localized groups in Section 3. If u max (A, L) is at least 95% of the extreme event threshold H E , then we predict that an extreme event will occur. We choose this conservative prediction threshold in order to minimize the number of false negatives (extreme events that we fail to predict). In Figure 4, we display an example output of our predictive scheme for a simulation of MNLS with initial conditions generated via a Gaussian spectrum with random phases (ǫ = 0.05, σ = 0.1, BFI = 1.4, H E =0.2). We display the spatial dependence of the surface elevation for three different values of time. In this simulation, the surface elevation first exceeds H E around t = 200 near x = 300. After exceeding this threshold, the extreme event continues to focus, eventually reaching a maximum of approximately 0.3 at t ≈ 385, x ≈ 390 (Figure 4, bottom pane).
We highlight each wave group with a rectangle whose height is equal to the predicted focused amplitude of the group, with a red colored rectangle indicating that we predict that the group will focus to form an extreme event. We see that our scheme identifies the group that will trigger the extreme event far in advance. The initial prediction occurs at the beginning of the simulation at t = 0, 200 time units (≈32 temporal wave periods) before the elevation crosses the extreme event threshold H E = 0.2, and nearly 400 time units before the extreme event reaches its maximal amplitude. Perhaps most importantly, the prediction occurs while the elevation is at the relatively modest value of 0.147.
We emphasize that our scheme accurately gives the spatiotemporal location where the extreme event will occur. The group that we predict will focus to an extreme event (red rectangle in Figure 4, top pane), has a length scale of 10.3 and amplitude 0.147. In MNLS, a localized hyperbolic secant initial profile with these characteristics will focus to a maximum amplitude of 0.264 after 351 time units, meaning that our scheme predicts a wave of this amplitude at t = 351, x = 361. To predict this spatial location we use the linear group velocity and the fact that the group is located at x = 185 initially. This agrees well with the observed dynamics of the simulation of the full field using MNLS-the identified group reaches an actual maximum of 0.289.
To test the reliability of this scheme, we implemented it on 100 simulations each of NLS and MNLS with BFI = 1.4 (see Appendix A for details on these simulations). Here we only discuss the MNLS results, as the NLS results are similar and MNLS is the more physically relevant equation. In these 100 simulations, there were 336 extreme events. We predicted all of these extreme events in advance-there were no false negatives. There were 91 instances where we predicted an extreme event but one did not occur, giving a false positive rate of 21.3%. For our correct predictions, the average warning time (the amount of time before the prediction began and the onset of the extreme event) was 153 time units (≈24 temporal wave periods). Figure 5: Left: scatterplot of predicted/actual amplitudes as well as the line predicted=actual (red). Note that the vertical dashed line is located at 0.95H E to reduce false negatives (as discussed in the text). Right: spatiotemporal dependence of |u| (red) and predicted future amplitude (blue).
Our scheme has value beyond a binary predictor of extreme events. In the left pane of Figure 5, we display a scatterplot showing the relationship between the predicted fugure amplitude and the actual future amplitude of the wave field. We observe that our predictor reliably estimates the future amplitude in a continuous sense. In particular, in addition to predicting when an upcoming extreme event is likely, our scheme predicts when a particularly large extreme event is upcoming.
To further illustrate the skill of our scheme, in the right pane of Figure 5, we plot the surface |u(x, t)| in red, as well as the predicted future group shape in blue. The field displayed here is the same field displayed in Figure 4 in a coordinate frame moving with the linear group velocity. The surface plots in Figure 5 provide a visualization of the skill of our scheme in predicting the future amplitude of the future extreme wave, as well as the spatial location at which it will occur. The reason the blue surface near the extreme event decays and vanishes around t = 200 is because we locally turn off the predictor while an extreme event is occurring (|u| > H E ).
Prediction by Gabor Projection
We now describe an alternative reliable prediction scheme that requires negligible computational cost. In this scheme, we predict upcoming extreme events by projecting the field onto a set of carefully tuned Gabor modes. This approach is similar in spirit to our extreme event predictive scheme for the model of Majda, McLaughlin, and Tabak (MMT) [10]. This projection requires only a single convolution integral, so its computation is extremely cheap. Even at this low cost, this scheme reliably predicts upcoming extreme events with spatiotemporal skill.
As we showed in Section 3, for a given spectrum we can compute the joint density of wave group amplitude and length scale. Using our study of isolated localized groups, we can then compute the conditional density of wave group properties for groups that will trigger extreme events. This gives, among other things, the density of group length scales for groups that would focus to form an extreme event (refer to Figure 3, right column). From this, we can compute the spatial length scale L G with the maximum likelihood of triggering an extreme event. Due to the narrowness of the distribution of extreme event-triggering group length scales, we expect that extreme events will be preceded by energy localization at a length scale close to L G .
To predict extreme events, we estimate the energy concentrated in scale L G . To do so, we "project" the field onto the set of Gabor basis functions v n (x; x c ), which are complex exponentials multiplied by a Gaussian window function. Projecting the envelope onto these Gabor functions gives the the Gabor coefficients Y n (x c , t):
v n (x; x c ) = e iπn(x−xC)/LG exp − (x − x C ) 2 2L 2 G , Y n (x c , t) = u(x, t), v n (x; x c ) / v n (x; x c ), v n (x; x c ) .
We expect that a large value of Y 0 at a spatial point x c indicates that an extreme event is likely in the future near x c in space (in a frame moving with the group velocity). To confirm this, we compute the following family of conditional distributions.
F Y0 (U) P max |x * −xc|<LG t * ∈[t+tA,t+tB ] |u(x * , t * )| > U |Y 0 (x c , t)| = Y 0 .(3)
That is, given a particular current value of Y 0 , we examine what are the statistics of the envelope u in the future. Here we choose t A = 50 and t B = 350 from the time required for a group of length scale of L G to focus to form an extreme event. We compute the statistics (3) In the left pane of Figure 6, we display the family of conditional densities of future |u| for a range of values of |Y 0 |. These densities show that when |Y 0 | is large, |u| is likely to be large in the future. From these conditional statistics, we compute the probability of an upcoming extreme event P EE as a function of current Y 0 by integrating over (H E , ∞). This function is displayed in the right panel of Figure 6. P EE has a sigmoidal dependence on Y 0 : if Y 0 is large enough than an upcoming extreme event is nearly guaranteed, while if Y 0 is small enough than an upcoming extreme event is highly unlikely. We emphasize that Y 0 becomes large distinctly before the extreme event occurs. The conditional statistics shown in Figure 6 pair a value of Y 0 with a maximum value of |u| which occurs at least t A = 50 time units in the future. To further illustrate this point, we statistically investigate the energy exchanges between the various Gabor modes. To do so, we compute the statistics of the Gabor coefficients during, before, and far from an extreme event. We display these Gabor statistics in Figure 7. We see that away from extreme events, we have nearly Gaussian statistics for the coefficients. In this regime the coefficients also appear to be uncorrelated. Before an extreme event, the coefficients are larger-Y 0 in particular is quite large. During an extreme event, Y 0 is on average smaller than before extremes (see Figure 7, bottom left). In the formation of the extreme events, Y 0 decreases by transferring energy to Y 1 , which is largest during the extreme events. During the extreme events, the Gabor coefficients are also strongly correlated (Figure 7, top left).
To predict extreme events, we first compute Y 0 by convolving the field u with a Gaussian with the length scale L G tuned to the particular spectrum. After computing Y 0 , we then compute the probability of an upcoming extreme event via our pre-computed conditional statistics (Figure 6, right). We predict that an extreme event will occur if P EE > P * , where P * is a threshold probability that we choose. Choosing a large P * will result in few false positives and many false negatives, while choosing a small P * will result in few false negatives and many false positives. We choose P * = 0.5 as it gives a low rate of false negatives (meaning we predict almost all extreme events) with a reasonably low false positive rate.
In Figure 8, we display the output of the Gabor predictive scheme applied to an example extreme event in a simulation of MNLS (this is the same example we investigated for the group detection-based predictive scheme in Figure 4. The extreme event that occurs at x ≈ 390, t ≈ 385 is predicted from the initial conditions by our Gabor-based predictive scheme, with a high level of confidence (P EE = 0.965). This prediction occurs 200 time units (32 wave periods) before the amplitude exceeds H E , and 385 time units (61 wave periods) before the maximally focused amplitude. The spatial location of the upcoming extreme event (in a moving coordinate frame) is predicted with very good accuracy as well. To assess the reliability of this scheme we tested it on the 100 NLS and MNLS simulations used to test the group detection-based scheme in the previous section. None of these simulations were used to generate the conditional statistics in Figure 6. We give data describing the performance of both predictive schemes in Table 1. The Gabor scheme predicted 316 of the 336 extreme events giving a low false negative rate of 5.9%. There were 108 instances where the Gabor scheme predicted an extreme event but one did not occur, giving a false positive rate of 25.5%. Thus, the Gabor scheme has a slightly higher false positive/negative rate compared with the group detection scheme but overall it performs nearly as well while requiring extremely little computational effort. Additionally, the average warning time for the Gabor scheme was larger than the group detection scheme (245 vs 153 time units, 39 vs 24 temporal wave periods).
Discussion
We have demonstrated how phase uncertainty (induced by the dispersive mixing of waves) and strong local nonlinearity interact with each other in nonlinear water waves resulting in the triggering of extreme waves. More specifically, we formulated a predictive scheme for rare events based on i) a nonlinear stability result that quantifies the focusing of localized wave groups and ii) the detection of wave groups that satisfy this localized stability criterion using a scale-selection algorithm applied much earlier, when the wave group still shows no obvious features that indicate a rare event.
In a second stage we utilized the scale-selection algorithm to quantify, for a given wave spectrum, the probability for the formation of critical wave groups that can evolve into rare events. The combined analysis revealed a spatial length scale that has the highest probability to 'trigger' a rare event. Based on this result we formulated an even simpler predictor that tracks the energy which can randomly accumulate over this length scale. We applied the two predictive schemes in the MNLS equation to forecast rare events in directional water waves. The results indicate very high accuracy of prediction, as well as robustness to the background wavenumbers, allowing for reliable predictions of, on average, 25-39 wave periods before the occurrence of an extreme wave.
The presented approach introduces a new paradigm for understanding and predicting intermittent and localized events in dynamical systems characterized by important uncertainty and potentially strong nonlinear mechanisms. Future efforts include extension to two-dimensional water waves, study of the effects of bathymetry, as well as combination of the presented approach with standard filtering schemes formulated for linear and weakly non-linear systems in order to extend even further the prediction window.
field |u|, we similarly find wave groups by computing s (2) and subsequently find the local minima. If we find a local minima at (x C , L * ), we conclude that there is a wave group at x = x C having a length scale L * /2. We determine the amplitude of the wave group, A, by computing the local maxima of |u| near x C .
In some instances, scale space extrema do not correspond to actual wave groups. Consider u(x) = e −x 2 /2L 2 0 − e −x 2 /2L 2 1 , where L 1 = 0.8L 0 , displayed in the left pane of Figure 9. There are two distinct peaks, which the scale selector detects. However, there is an additional local minimum of s (2) near x = 0 that does not correspond to a wave group. To eliminate such false positives, for each local minimum of s 2 , we compute the quantity C, which measures how close u is to a Gaussian-like blob:
C = 1 − ||f (x) − Ae −(x−xC) 2 /2(L * ) 2 || 2 ||Ae −(x−xC) 2 /2(L * ) 2 || 2
if |u| is an exact Gaussian, then C is 1. Thus, we take small values of C as evidence that the local minimum of s (2) do not correspond to a wave group. In the two-humped case displayed in Figure 9, the two local minima around x = −1, 1 have C ≈ 0.9, while C ≈ 0.5 at x = 0. In practice, we ignore local minima of s (2) where C < 0.75. An example output of the scale selection algorithm with this criteria applied to an irregular wave field is displayed in the top pane of Figure 3. We see that the algorithm successfully identifies the dominant wave group in the field. Right: Negative part of s (2) . The extremum at x = 0, L/2 = 2 is eliminated via the procedure described in the text.
To compute the local minima of s (2) , we first generate initial guesses for the minima by computing s (2) on a grid with N x,SS spatial points and N L points in the length scale dimension. We then refine the local minima of the grid-evaluated s (2) by Newton's method. For a particular length scale value L, computation of of s (2) (·, L) requires two fast Fourier transforms. Thus, the cost is O(N L N x,SS log N x,SS ). We have found that a relatively small N L is adequate to generate reliable initial guesses for the minima (we use N L = 20).
For the Newton iteration, we compute the gradient and Hessian of s (2) analytically (these analytic expressions do contain integral terms, which we evaluate numerically). For each wave group, we refine (x C , L) to five digits of precision, which requires no more than 3 Newton iterations in almost all cases. Note that compared to the grid computation of s (2) , the cost of the Newton iterations is low. The reason for this is that at each iteration the gradient and Hessian of s (2) need only be calculated at a single point. This means that the associated integration is only over a small subset of the full spatial domain.
Prediction via this scale selection algorithm is considerably cheaper than solving the envelope PDE. In the example considered in Figure 55.1, we predict an extreme event 200 time units in advance using the scale selection-based algorithm. Evolving the field this many time units with the PDE would require thousands of time steps, each costing O(N x,P DE log N x,P DE ) to compute the nonlinear terms, where N x,P DE is the number of spatial grid points in the numerical PDE solver. By contrast, the scale selection algorithm only requires N L = 20 evaluations of s (2) (·, L), with each evaluation of s (2) (·, L) requiring O(N x,SS log N x,SS ) operations. To accurately resolve the small scale dynamics and the nonlinear terms, N x,P DE must be considerably greater than N x,SS , which demonstrates clearly the computational gain of the proposed approach (for the considered setting we found that N x,SS can be 16 times smaller than N x,P DE with no loss of reliability.
Figure 1 :
1Top Left: Probability density function of the surface elevation in numerical simulations of MNLS for various times.
Figure 2 :
2Top: Amplitude growth factors for localized groups in NLS (left) and MNLS (right). Middle: examples of defocusing, amplitude decreasing groups for NLS (left) and MNLS (right). Bottom: examples of focusing, amplitude increasing groups for NLS and MNLS. The initial conditions for the simulations displayed in the bottom left and bottom right are identical. Although both groups focus, the amplitude grows considerably less in MNLS than in NLS.
Figure 3 :
3Top: Dominant wave groups (red, dashed) determined by applying the group detection algorithm to the field |u(x)| (blue, solid). Middle Row: Joint density of group amplitude and length scale (left) and length scale density for extreme triggering groups (right), BFI = 1.4, ǫ = 0.05, σ = 0.1. Bottom row: idem for BFI = 1.4, ǫ = 0.1, σ = 0.2. In each joint density plot we overlay the curve that separates groups that would focus (if isolated) to form extreme events, NLS curve is black, solid, MNLS curve is red, dashed.
Figure 4 :
4Top: Initial conditions for a simulation of MNLS. Our scheme identifies a group around x = 190 which we predict will grow to form a large extreme event. Middle: Group in initial stages of focusing, breaks the extreme event threshold H E = 0.2 near x = 290. Bottom: group is fully focused and attains its maximum amplitude near x = 390.
from 200 simulations of NLS/MNLS with Gaussian spectra and random phases with ǫ = 0.05, σ = 0.1, BFI = 1.4.
Figure 6 :
6Left: Family of conditional densities of current |Y 0 | and future |u|, corresponding to (3). Right: probability of upcoming extreme event as a function of |Y 0 |
Figure 7 :
7Top: Level surfaces of the joint density of the real parts of Y 0 , Y 1 , and Y 2 during (P 1 , left), before (P 2 , middle) and far from (P 3 , right) an extreme event. Bottom: marginal statistics of Y 0 (left), Y 1 (middle), and Y 2 (right) during (P 1 ), before (P 2 ), and far from (P 3 ) an extreme event.
Figure 8 :
8Surface elevation (blue) and probability of upcoming extreme event (red) for t = 0 (top), t = 202.5 (middle), and time at which maximum elevation attained, t = 384.5 (bottom).
Figure 9 :
9Left: Plot of double-humped field |u| and the groups identified via the scale selection algorithm.
Predictive Scheme Extreme Events False Neg.False Pos.
Avg. Warning Time
Gabor
336
20 (5.9%) 108 (25.5%)
245 (39 periods)
Group Detection
336
0
91 (21.3%)
153 (24 periods)
Table 1 :
1Performance of Gabor and group detection prediction schemes on 100 simulations of MNLS.
AcknowledgmentsThis research has been partially supported by the Naval Engineering Education Center (NEEC) grant 3002883706 and by the Office of Naval Research (ONR) grant ONR N00014-14-1-0520. The authors thank Dr. Craig Merrill (NEEC Technical Point of Contact) for numerous stimulating discussions.A Scale Selection AlgorithmHere we describe the wave group detection algorithm used in the prediction scheme discussed in Section 55.1. To find the dominant wave groups in a given irregular wave field, we look for Gaussian-like "blobs" in |u(x)|. To find these blobs, we use an existing algorithm based on the scale normalized derivatives of |u|[27,38,28]. These scale-normalized derivatives, s (m) , are normalized spatial derivatives of the convolution of |u| with the heat kernel:where g(x, L) is the heat kernel:in the above, x is the spatial variable and L is the length scale variable. Following[28]we choose m = 2 for optimal blob detection. We illustrate this approach by computing the scale-normalized derivatives of a single Gaussian: u(x) = Ae −x 2 /2L 2 0 . It is straightforward to show that s (2) has a local minimum at x = 0, L = 2L 2 0 . In an arbitrary
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| []
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[
"Bypassing Pauli's Theorem",
"Bypassing Pauli's Theorem"
]
| [
"José M Isidro [email protected] \nInstituto de Física Corpuscular (CSIC-UVEG) Apartado de Correos\n22085, 46071ValenciaSpain\n"
]
| [
"Instituto de Física Corpuscular (CSIC-UVEG) Apartado de Correos\n22085, 46071ValenciaSpain"
]
| []
| We define a quantum-mechanical time operator that is selfadjoint and compatible with the energy operator having a spectrum bounded from below. On their common domain, the operators of time and energy satisfy the expected canonical commutation relation. Pauli's theorem is bypassed because the correspondence between time and energy is not given by the standard Fourier transformation, but by a variant thereof known as the holomorphic Fourier transformation. | 10.1016/j.physleta.2004.12.012 | [
"https://export.arxiv.org/pdf/quant-ph/0410070v2.pdf"
]
| 5,601,949 | quant-ph/0410070 | 597525b425effefdec68ebe5cffceb7675b7b944 |
Bypassing Pauli's Theorem
14 Dec 2004 April 1, 2022
José M Isidro [email protected]
Instituto de Física Corpuscular (CSIC-UVEG) Apartado de Correos
22085, 46071ValenciaSpain
Bypassing Pauli's Theorem
14 Dec 2004 April 1, 2022
We define a quantum-mechanical time operator that is selfadjoint and compatible with the energy operator having a spectrum bounded from below. On their common domain, the operators of time and energy satisfy the expected canonical commutation relation. Pauli's theorem is bypassed because the correspondence between time and energy is not given by the standard Fourier transformation, but by a variant thereof known as the holomorphic Fourier transformation.
Introduction
The definition of a time operator in quantum mechanics is an outstanding problem ever since Pauli's theorem [1]; see ref. [2] for a brief account and ref. [3] for a detailed treatment. This has prompted attempts to introduce arrival time and time-of-flight operators [4], and to provide a physical interpretation for quantum theories based on nonhermitian operators [5] or on positive, operator-valued measures [6]. Critical assessments of the technical aspects of Pauli's theorem have also appeared [7].
In this letter we present an alternative definition of a quantum-mechanical time operator that bypasses the technical objections raised by Pauli to the existence of a quantum-mechanical time operator. Briefly, a selfadjoint Hamiltonian operator H that is bounded from below is placed in canonical correspondence with a nonhermitian time operator T via the holomorphic Fourier transformation (HFT) [8]. The latter differs substantially from the standard Fourier transformation used in quantum mechanics. Perhaps its most striking feature is the appearance of a nonhermitian time operator T that is canonically conjugate, via the HFT, to the Hamiltonian H. However, the square T 2 admits a selfadjoint Friedrichs extension T 2 F . Finally, T 2 F admits a selfadjoint square root that serves as a bona fide time operator. After the technical presentation of sections 2 and 3, we present some examples and discuss our conclusions in section 4.
The holomorphic Fourier transformation
Background material on the HFT, summarised in the following, can be found in ref. [8]. Let H denote the upper half plane: the set of all z ∈ C such that Im(z) > 0. Let f ∈ L 2 (0, ∞). For z = x + i y ∈ H, the function ϕ defined as
ϕ(z) := 1 √ 2π ∞ 0 ds f (s) e isz ,(1)
the integral understood in the sense of Lebesgue, is holomorphic on H. Its restrictions to horizontal straight lines y = const > 0 in H are a bounded set in L 2 (R). Conversely, let ϕ be holomorphic on H, and assume that
sup 0<y<∞ ∞ −∞ dx |ϕ(x + iy)| 2 = C < ∞.(2)
Then the function f defined by
f (s) := 1 √ 2π ∞ −∞ dz ϕ(z) e −isz ,(3)
the integration being along any horizontal straight line y = const > 0 in H, satisfies the following properties. First, f (s) is independent of the particular horizontal line y = const > 0 chosen. Second, f ∈ L 2 (0, ∞). Third, for any z ∈ H, eqn. (1) holds, with
∞ 0 ds |f (s)| 2 = C.(4)
We call f the holomorphic Fourier transform of ϕ. Some features of the HFT on H are worth mentioning. Let Ω(H) denote the space of all holomorphic functions on H, and let Ω 0 (H) denote the proper subspace of all ϕ ∈ Ω(H) such that the supremum C introduced in eqn. (2) is finite. Then C defines a squared norm ||ϕ|| 2 on Ω 0 (H). The subspace Ω 0 (H) is complete with respect to this norm. This norm is Hilbert, i.e., it verifies the parallelogram identity. Hence the scalar product ϕ|ψ defined on Ω 0 (H) through
4 ϕ|ψ := ||ψ + ϕ|| 2 − ||ψ − ϕ|| 2 + i ||ψ + i ϕ|| 2 − i ||ψ − i ϕ|| 2(5)
turns the complete normed space Ω 0 (H) into a Hilbert space with respect to the scalar product (5). In fact, via the HFT, the subspace Ω 0 (H) is isometrically isomorphic to the Hilbert space L 2 (0, ∞).
Quantum operators from the HFT
Introducing Planck's constant , the HFT reads
ϕ(z) = 1 √ 2π ∞ 0 ds f (s) e i sz f (s) = 1 √ 2π ∞ −∞ dz ϕ(z) e − i sz .(6)
In this section we promote the variables z ∈ H and s ∈ (0, ∞) to quantum operators Z and S, respectively, and study their properties. We define operators S and Z as
(Sf )(s) := s f (s), (Zf )(s) := i df ds .(7)
Equation (6) implies that a conjugate representation for them is given by their HFT transform,
(Sϕ)(z) = −i dϕ dz , (Zϕ)(z) = z ϕ(z).(8)
Irrespective of the representation chosen we have that the Heisenberg algebra
[Z, S] = i 1(9)
holds on the intersection D(S) ∩ D(Z) of their respective domains. Next we make precise what these domains are.
On the domain
D(S) = {f ∈ L 2 (0, ∞) : ∞ 0 ds s 2 |f (s)| 2 < ∞},(10)
which is dense in L 2 (0, ∞), the operator S is symmetric,
f |S|g * = g|S|f .(11)
A closed, symmetric, densely defined operator admits a selfadjoint extension if and only if its defect indices d ± are equal. Moreover, such an operator is essentially selfadjoint if and only if its defect indices are both zero [9]. The operator S turns out to be essentially selfadjoint, with point, residual and continuous spectra given by
σ p (S) = φ, σ r (S) = φ, σ c (S) = [0, ∞).(12)
The properties of the conjugate operator Z are subtler. One finds
f |Z|g * = i f (0)g * (0) + g|Z|f ,(13)
so Z is symmetric on the domain
D(Z) = {f ∈ L 2 (0, ∞) : f abs. cont., ∞ 0 ds | df ds | 2 < ∞, f (0) = 0}.(14)
(f is absolutely continuous). The adjoint Z † also acts as i d/ds, with a domain D(Z † )
D(Z † ) = {f ∈ L 2 (0, ∞) : f abs. cont., ∞ 0 ds | df ds | 2 < ∞},(15)
where the boundary condition f (0) = 0 has been lifted. On the space
L 2 (0, ∞) we have d + (Z) = 0, d − (Z) = 1.
We conclude that Z admits no selfadjoint extension. Its point, residual and continuous spectra are
σ p (Z) = φ, σ r (Z) = H ∪ R, σ c (Z) = φ.(16)
The domain D(Z) is strictly contained in D(Z † ). This implies that the operators X := (Z + Z † )/2 and Y := (Z − Z † )/2i which one would naively construct out of Z are ill defined. There is no way to define selfadjoint operators X and Y corresponding to the classical coordinates x = Re z and y = Im z. This is compatible with the fact that, the defect indices of Z being unequal, Z does not commute with any complex conjugation on L 2 (0, ∞) [9]. However, we will see presently that one can make perfectly good sense of a quantum-mechanical operator Z admitting no selfadjoint extension.
With our choice of domain D(Z), which makes Z symmetric, Z 2 is also symmetric. One proves that d − (Z 2 ) = 1 = d + (Z 2 ). Hence Z 2 , although not essentially selfadjoint, admits a selfadjoint extension. A popular choice is the Friedrichs extension [9]. Given an operator A, this extension is characterised by a boundedness condition
ψ|A|ψ ≥ −α ||ψ|| 2 ∀ψ ∈ D(A)(17)
for a certain α ≥ 0. Now the operator Z 2 admits a Friedrichs extension Z 2 F with a lower bound α = 0:
f |Z 2 F |f ≥ 0, ∀f ∈ D(Z 2 F ).(18)
The point, residual and continuous spectra of this extension are
σ p (Z 2 F ) = φ, σ r (Z 2 F ) = φ, σ c (Z 2 F ) = [0, ∞).(19)
Now the crucial point is that the square root of the Friedrichs extension allows us to define a selfadjoint momentum operator. Let us define the new operator Z √
Z √ := + Z 2 F .(20)
Z √ is selfadjoint, with a domain D(Z √ ) univocally determined by the spectral decomposition of Z [9]. The point, residual and continuous spectra of Z √ are
σ p (Z √ ) = φ, σ r (Z √ ) = φ, σ c (Z √ ) = [0, ∞).(21)
We observe that the operation of taking the Friedrichs extension does not commute with the operation of taking the square root. Finally let us consider transforming the operators S and Z under SL(2, R). We can reparametrise the coordinate z ∈ H by means of a Möbius transformation z →z = (az + b)(cz + d) −1 , with ad − bc = 1. Thenz ∈ H. We now write the HFT as
ϕ(z) = 1 √ 2π ∞ 0 dsf (s) e i sz f (s) = 1 √ 2π ∞ −∞ dzφ(z) e − i sz ,(22)
wheres ∈ (0, ∞) is the variable conjugate toz under (22). One can define quantum operatorsS andZ satisfying the Heisenberg algebra (9). Hence this is a canonical transformation from S, Z toS,Z. In terms of the transformed variabless,z, the transformed operatorsS andZ have the same spectra as before.
Examples and discussion
The standard Fourier transformation maps (a subspace of) L 2 (R) into (a subspace of) L 2 (R). It is also an isospectral transformation between selfadjoint operators such as the position operator X and its conjugate momentum operator P for a particle on the whole real line R. In the context of the standard Fourier transformation on L 2 (R), coordinate and momentum are sometimes referred to as a Schrödinger pair. On the contrary, the HFT is not an isospectral transformation: the operators S and Z do not have identical spectra. Furthermore, the very choice of the dynamical variable to be represented by complex variable z of the HFT is a nontrivial choice in itself.
Since the Hamiltonian H is bounded from below it makes sense to take, in section 3, the selfadjoint operator S as the Hamiltonian H and the nonhermitian operator Z as the time operator T . In this way we arrive at a selfadjoint time operator T √ := T 2 F with the semiaxis (0, ∞) as its continuous spectrum. It is this latter operator T √ that we take to define (positive) time. We further observe that we have an additional SL(2, R) symmetry at our disposal, generated by translations, dilations and inversions acting on H and hence also on its boundary R. Under dilations x → λx, where λ > 0, the semiaxis (0, ∞) transforms into itself, while we can shift it into any desired interval (k, ∞), k ∈ R, by means of a translation x → x − k. Under an inversion x → −1/x, the semiaxis (0, ∞) transforms into its opposite (−∞, 0). Convening that the inversion maps the point at infinity into zero, and viceversa, it suffices to consider the inversion transformation and its corresponding operatorT in order to obtain the whole real line R as the (joint) continuous spectrum of the two time operators T √ andT √ . Overall there is a whole SL(2, R)'s worth of time operators to choose from. This fits in well with the multiplicity of existing time-of-arrival operators in the literature [3,4,7], although a general criterion to map a given SL(2, R)-time, as proposed here, with those of refs. [3,4,7], is lacking.
The existenc of a whole SL(2, R)'s worth of time operators brings us to a related question, namely, whether or not our formalism also works in the presence of degeneracy. The answer is affirmative. SL(2, R) acts as per eqn. (22). This group has finite-dimensional representations in all real dimensions, as well as a continuous series of infinite-dimensional representations. Its action commutes with the operator S in the sense that the transformed operatorS has the same spectrum as S, even if the corresponding eigenfunctions (eqns. (24), (25) below) get exchanged under an SL(2, R)-transformation. Thus picking a representation of SL(2, R) with the desired dimension (i.e., with the desired degeneracy), eventually infinite, the commutativity of the SL(2, R)-action with the Hamiltonian H = S ensures that our formalism remains valid also in the presence of degeneracy.
As an example let us work out the case of a free particle moving on the whole real line. Standard quantum mechanics tells us that the Hamiltonian H f = P 2 /2m is twofold degenerate. In coordinate representation, the eigenfunctions corresponding to the eigenvalue E p = p 2 /2m are u ±p (x) = exp (±ipx). This twofold degeneracy can be understood in terms of SL(2, R)-transformations as follows. The spectrum of H f is the semiaxis [0, ∞), so one naturally identifies H f with the operator S of eqn. (7). Now let us use the HFT and set S = H f in eqn. (7). As usual we denote by E, t the variables corresponding to the operators H f , T √ , respectively. In energy representation the eigenfunction of Z = T √ corresponding to the eigenvalue t ′ is
f t ′ (E) = exp − i Et ′ ,(23)
while that of H f corresponding to the eigenvalue E ′ is
f E ′ (E) = δ(E − E ′ ).(24)
We can HFT-transform the eigenfunction (24) to obtain its expression in time representation,
ϕ E ′ (t) = exp i E ′ t .(25)
The twofold degeneracy of the eigenfunctions (25) is recovered once one remembers that the operator T √ is defined as the positive square root (20); negative times are obtained from the SL(2, R)-transformed operatorT √ discussed previously. We further notice that the eigenvalue equation satisfied by the eigenfunctions (25),
H f ϕ E ′ (t) = −i dϕ E ′ dt = E ′ ϕ E ′ (t),(26)
is actually equivalent to the time-dependent Schrödinger equation, since the minus sign multiplying i can be flipped by changing to the SL(2, R)-transformed timeT √ . This is in agreement with the fact that the time-dependent Schrödinger equation can be understood as the operator identity
i ∂ ∂t = − 2 2m ∇ 2 + V = H.(27)
Beyond the particular case when H = H f , however, the identification H = S must be made with some care. Usually the basic operators, i.e., those satisfying the Heisenberg algebra, are position X and momentum P , and the Hamiltonian H is a function of the latter, typically H = P 2 /2m + V (X). As X and P do not commute, the time-independent Schrödinger equation Hu(x) = Eu(x) is nontrivial, but there are two cases when diagonalising the Hamiltonian reduces to diagonalising one of the two basic operators X, P , as in eqns. (23)-(24). The first case is when the kinetic term P 2 /2m can be neglected in comparison with the potential term V (X); then it suffices to diagonalise X. The second case is when V = 0 identically, so we need only diagonalise P : this is the free particle. As a rule, however, our particle will not be free, and the spectrum of H will only share one property in common with the spectrum of S, namely, that they are both bounded from below. While a constant shift in energy (an SL(2, R)-transformation) can bring the ground state of H to zero energy, in general the spectrum of S will be larger than that of H. All this notwithstanding, the spectral objections just raised to the identification S = H can also arise (and actually do arise) for the usual Schrödinger pair X, P in the presence of a potential V , or in the presence of boundary conditions. Thus, e.g., while P = −i d/dx on its own has all of R as its spectrum, this is generally no longer true when V = 0 (e.g., a particle inside an infinite potential well). In other words, the previous caveat concerning the spectra must also be borne in mind when approaching quantum mechanics from the point of view based on the usual Fourier transformation and the Heisenberg algebra [X, P ] = i .
To summarise, the HFT allows one to construct a selfadjoint Hamiltonian bounded from below that is conjugate to a time operator whose spectrum is the whole real line. As opposed to the usual Fourier transformation, which is an isospectral transformation between X and P , the HFT is not an isospectrality between H and T . In compensation, there is an SL(2, R)-symmetry acting on T which the usual Fourier transformation lacks. Our formalism is based on the Heisenberg algebra [T, H] = i rather than the usual [X, P ] = i . We have proved the equivalence between the approach based on the HFT with the Heisenberg algebra [T, H] = i and the approach based on the usual Fourier transformation with the Heisenberg algebra [X, P ] = i .
AcknowledgementsIt is a great pleasure to thank J. de Azcárraga for encouragement and support, I. Egusquiza for technical discussions, and the referees for useful suggestions to improve the manuscript. The author thanks Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut (Potsdam, Germany), where this work was was begun, for hospitality. This work has been partially supported by European Union network MRTN-CT-2004-005104, by research grant BFM2002-03681 from Ministerio de Ciencia y Tecnología, by research grant GV2004-B-226 from Generalitat Valenciana, by EU FEDER funds, by Fundación Marina Bueno and by Deutsche Forschungsgemeinschaft.
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[
"Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes",
"Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes"
]
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"Kevin Yang \nStanford University\nFebruary 142020\n"
]
| [
"Stanford University\nFebruary 142020"
]
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| Within this article, the Kardar-Parisi-Zhang stochastic PDE is both considered and established as the continuum model for the height function associated to the long-range asymmetric exclusion process. Precisely, we demonstrate that the lattice Cole-Hopf transform of the height function, or equivalently the Gartner transform, converges in the appropriate topology to the solution of a linear multiplicative stochastic heat equation for arbitrarily long-range exclusion processes living on the infinite-volume lattice , thereby furthering the result of Dembo-Tsai in [8]. The primary technical novelty that we develop towards establishing this KPZ continuum limit is a robust dynamical variation and improvement upon the classical one-block estimate of [14], crucially exploiting the temporal ergodicity of the lattice dynamics. 7 | 10.1007/s00220-022-04628-y | [
"https://arxiv.org/pdf/2002.05176v1.pdf"
]
| 211,097,080 | 2002.05176 | 3587eb670b54f3985c074b240c93685892dd090e |
Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes
12 Feb 2020
Kevin Yang
Stanford University
February 142020
Kardar-Parisi-Zhang Equation from Long-Range Exclusion Processes
12 Feb 2020
Within this article, the Kardar-Parisi-Zhang stochastic PDE is both considered and established as the continuum model for the height function associated to the long-range asymmetric exclusion process. Precisely, we demonstrate that the lattice Cole-Hopf transform of the height function, or equivalently the Gartner transform, converges in the appropriate topology to the solution of a linear multiplicative stochastic heat equation for arbitrarily long-range exclusion processes living on the infinite-volume lattice , thereby furthering the result of Dembo-Tsai in [8]. The primary technical novelty that we develop towards establishing this KPZ continuum limit is a robust dynamical variation and improvement upon the classical one-block estimate of [14], crucially exploiting the temporal ergodicity of the lattice dynamics. 7
INTRODUCTION
The current article concerns the universality behind the Kardar-Parisi-Zhang stochastic PDE among interacting particle systems, namely the asymmetric exclusion processes. Precisely, the primary objective for this article is to establish the KPZ equation as the universal family of continuum models parameterized by its coefficients for the height function associated to a large family of long-range asymmetric exclusion processes under the weak-type scaling for the asymmetry.
The Kardar-Parisi-Zhang SPDE is the following family of stochastic PDEs parameterized by both its diffusivity constant α ∈ >0 and effective drift α ′ ∈ :
∂ T H T,X = α 2 ∂ 2 X H T,X − α ′ 2 ∂ X H T,X 2 + α 1 2Ẇ
T,X ; (T, X ) ∈ >0 × X .
(1.1)
The space X can be either the torus 1 or the line , among other examples, though in this article we address the situation of X = ; the compact framework X = 1 yields the same behaviors as X = for this article, though we do not address this explicitly. Moreover, the random fieldẆ T,X is a space-time white noise on >0 × X defined to be a centered Gaussian field on >0 × X with covariance kernel E[Ẇ T,XẆS,Y ] = δ 0 (T − S)δ 0 (X − Y ) for all (T, X ), (S, Y ) ∈ 0 × X , where δ 0 denotes the Dirac point mass supported at the origin.
The KPZ equation (1.1) was originally derived via renormalization group (RG) considerations in the seminar PRL article of Kardar, Parisi, and Zhang [18]. The equation has then been conjectured to be the universality class for slope-dependent growth processes, such as paper-wetting, crack formation, burning fronts, epidemics; these processes include the ballistic deposition and Eden models, respectively. We refer the reader to [5] for any details. However, giving part of the motivation of this article, understanding the KPZ equation (1.1) itself is a nontrivial task already, as explained below.
The analytic problem with the KPZ stochastic PDE is the quadratic nonlinearity which is, a priori, ill-defined given the spatial roughness of the space-time white noise. The classical solution theory to (1.1) is through the Cole-Hopf transform Z T,X , defined to be the linearization
∂ T Z T,X = α 2 ∂ 2 X Z T,X + λα 1 2 Z T,XẆT,X , (T, X ) ∈ >0 × X ,(1.2)
with λ = α ′ α ∈ ; indeed, the field Z T,X is formally obtained through exponentiating the solution H T,X to the corresponding KPZ equation, and (1.2) itself admits a classical solution theory via Ito's calculus. Moreover, by a theorem of Mueller [21], positive solutions remain positive under the SHE flow (1.2) with probability 1, which therefore provides a solution theory for the KPZ equation via the SHE. 1 To illustrate the problem, consider the simple or nearest-neighbor models with dynamics on some lattice X , interpreted as either the full line or a torus of growing volume, given by the following data:
• Any initial particle configuration must have at most one particle per site in X . • Provided any initial particle configuration, distinct particles perform symmetric simple random walks on X with an asymmetric drift of order N − 1 2 . • If any particle attempts a jump that would yield a site with multiple particles, suppress that jump; this preserves the total particle number and ensures each site has at most one particle during the entire lifetime of the process.
We formulate the above dynamics more precisely later in this section for the long-range processes. Delaying also a precise formulation of the height function until later in this section, within the seminal article [3] of Bertini-Giacomin, the authors establish the family of KPZ stochastic PDEs, again parameterized by its coefficients, as the continuum model for the height function associated to the nearest-neighbor model described earlier. Roughly speaking, the key insight is inspired by fluid dynamics and Gartner's analysis towards establishing the hydrodynamic limit of the height function associated to an ASEP with weaker-asymmetry; the employed tool referred to as the Gartner transform solves a microscopic version of a stochastic heat equation, a linear equation for which the analysis is almost entirely PDE-based.
In [7], the observation leading to a microscopic stochastic heat equation is reinterpreted as a self-duality of the Gartner transform with respect to an exponential-type function of the associated height function; in particular, much probabilistic detail in the work of Bertini-Giacomin in [3] is hidden behind the aforementioned duality, and more generally the quantum integrable structure, of the ASEP. In a successful first attempt towards establishing the KPZ equation as a continuum model for the height function associated to long-range exclusion processes, Dembo-Tsai prove the analogous result in [8] under two constraints:
• The maximal range of interaction between any two sites within the lattice X , or equivalently the maximal jumprange for any given particle, is bounded above by 3.
• The vector of asymmetry coefficients belongs to a sufficiently small neighborhood of a specialized one-dimensional space of possible asymmetry coefficients, to be made precise shortly. Here, sufficiently small refers to at the scale for which the microscopic features behind the particle system are identical to those for the particle system evolving with the specialized asymmetry.
A possibly valid, common perspective towards the above pair of assumptions is that neither assumption above carries with it a blatant physical or probabilistic lesson, so neither assumption should be necessary in establishing the KPZ equation as a continuum model for the associated height function. However, a consequence of the exclusion component of the particle system, in [8] the authors encounter multiple nonlinearities of the Gartner transform in deriving an approximation for the duality that arises in the nearest-neighbor models. All but two nonlinearities are successfully analyzed via a hydrodynamic theory for the particle system; the remaining two nonlinearities are addressed by employing the two assumptions above, which, via algebraic formulas, transforms the problem of their analysis into a similar problem of hydrodynamic analysis.
Within the current article, we establish the family of KPZ equations parameterized by their coefficients as the continuum model for long-range exclusion processes without any constraint on the maximal range of interaction, and while improving significantly the second assumption concerning the asymmetry. Moreover, this result remains valid for all types of the initial data considered in [1], [3], and [8], including the narrow-wedge initial condition. Equivalently, the primary result within this article furthers the primary result within [8] to arbitrarily long-range interactions and a significantly larger family of allowable asymmetries.
Meanwhile, the primary technical innovation and contribution within this article is a dynamical analog of the classical one-block strategy, whose ingredients consist of a log-Sobolev inequality for the dynamic and an energy estimate of Kipnis-Varadhan. We address this point with additional detail after we have introduced enough preliminary discussion.
1.1. The Model. We review the particle system from [8] of primary interest in this article. We employ a scaling parameter N ∈ >0 , and the desired continuum limits will emerge in the limit N → ∞ which itself will be referred to as the large-N limit. 2 A precise description of the stochastic particle dynamics requires an underlying space-time geometry, some static system specifying the state-space, and a suitable operator to serve as the generator. We list these three components in this order.
• Our particles will move on the infinite-volume lattice . Our temporal coordinates will be [0, ∞) = >0 . Considering approximations of this half-line by intervals with finite time-horizons, our dynamics will occur on [0, T f ]× N for arbitrary T f ∈ >0 .
• The state space is exactly the state space of ASEP on . More precisely, using the spin-notation adopted previously in [8], for any sub-lattice Λ ⊆ we define Ω Λ = {±1} Λ . Observe this association prescribes a mapping Λ → Ω Λ for which any containment Λ ′ ⊆ Λ induces the canonical projection operator Ω Λ → Ω Λ ′ given by
Π Λ→Λ ′ : Ω Λ → Ω Λ ′ , (η x ) x∈Λ → (η x ) x∈Λ ′ . (1.3)
We implicitly adopt the physical interpretation that η x = 1 indicates the presence of a particle located at x ∈ N and that η x = −1 indicates the absence of any particle. The aforementioned relationship between the sub-lattices and corresponding state-spaces thus enables us to write the Greek letter η to denote a particle state in Ω Λ without explicitly referring to exact sub-lattice Λ ⊆ .
We conclude by mentioning that the global state space Ω is equipped with the σ-algebra generated by cylinder sets, namely the σ-algebra induced by taking pullbacks of measurable sets under the projections Ω → Ω Λ with Λ ⊆ ranging over all finite sub-lattices of . • To specify the stochastic dynamic, we must first introduce the following two sets of coefficients, the first of which encodes the underlying symmetric mechanism and the second of which encodes the asymmetric component:
A N := α N k ∈ >0 : ∞ k=1 α N k = 1 ∞ k=1 , Γ N := γ N k ∈ ∞ k=1 . (1.4)
For convenience, let us assume γ N k 0 for all k ∈ >0 , though this assumption is certainly removable by straightforwardly adapting the analysis within this article.
For any pair of sites x, y ∈ , we denote by L x, y the generator for the symmetric exclusion process on the bond {x, y}. We specify the generator L N ,! T of our dynamic via its action of a generic functional f : Ω → given by
L N ,! T f : η · → N 2 ∞ k=1 α N k x∈ 1 + γ N k N 1 + η x 2 1 − η x+k 2 L x,x+k f. (1.5)
To completely specify the data of our stochastic process, we let F • denote the canonical filtration.
We proceed by presenting structural assumptions concerning our particle dynamic. The first assumption consists of two sub-components; one of these sub-components is to ensure that the predicted KPZ-limit as discussed within the introduction in [8] is well-defined, and the other component concerns moments of these coefficients. Assumption 1.1. For any k ∈ >0 , there exist deterministic parameters α k , γ k so that
lim N →∞ ∞ k=1 k 2 α N k − α k = 0, lim N →∞ ∞ k=1 k γ N k − γ k = 0. (1.6)
Moreover, for all p ∈ >1 , we have
sup N ∈ >0 ∞ k=1 k p α N k p 1, sup N ∈ >0 ∞ k=1 k p |γ N k | p 1. (1.7)
The next assumption is significantly more interesting concerning the physics behind our model. To present the assumption, we require some notation.
Notation 1.2.
For any N , k ∈ >0 we define the specialized asymmetry coefficients by the formula
α N k γ N , * k = 2λ N ∞ ℓ=k+1 ℓ − k k α N ℓ + λ N α N k , λ N · = ∞ k=1 k 2 α N k −1 ∞ k=1 kα N k γ N k .
(1.8) 3 Consequence of definition of λ N ∈ >0 , the coefficients (γ N , * k ) ∞ k=1 and (γ N k ) ∞ k=1 are intimately connected in an important fashion; see Lemma 2.1 below for details.
Concerning the coefficients of the asymmetry component of our dynamic, the following constraint is of primary interest; it provides a priori comparison between γ N k and γ N , * k for all k ∈ >0 that is both stable in the large-N limit and a significant improvement on the parallel assumptions in [8]. Assumption 1.3. There exists some β c ∈ >0 sufficiently small but universal so that
∞ k=1 k γ N k − γ N , * k N − 1 2 +β u (1.9)
for any β u < β c .
The assumption within [8] concerning the asymmetry coefficients for the system requires the a priori estimate of N −1 rather than N − 1 2 ; thus the improvement within this article relaxes the necessary bound from inverse of the hydrodynamic scale to inverse of the finer fluctuation scale.
1.2. Lattice Cole-Hopf Transform. Following both of [3] and [8], we construct a space-time random field as a functional of the entire process for the particle system which converges to the continuum SHE in the large-N limit. This approach is due originally to Gartner in the derivation of the hydrodynamical limit for the ASEP with weaker asymmetry. We now define the associated height function and Gartner transform to be
h N T,X • = h N T,0 + N − 1 2 0< y<X η N T, y , (1.10a) Z N T,X • = exp −λ N h N T,X + ν N T (1.10b) where ν N • = 1 4 N ∞ k=1 k γ N k u(N −1 ) − α N k v(N −1 ) + λ 2 N N ∞ k=1 α N k k + λ 2 6k 2 − 5k 12N (1.11)
with u(x) = x − 1 2 sinh(2λ N x) and v(x) = x −1 [cosh(2λ N x) − 1]. Lastly, we linearly interpolate both the associated height function and Gartner transform to obtain continuous functions on 0 × .
Main Theorem.
The primary result within this article is a continuum limit for the Gartner transform Z N T,X . To present the result precisely, we introduce the following two types of initial particle configurations, which we often refer to as initial data from the PDE perspective.
For any p ∈ >0 , we define X L p ω = [E |X | p ] 1 p to be the L p -norm with respect to the underlying randomness. Definition 1.5. We say the particle system is at near stationarity, or equivalently that a probability measure µ 0,N on Ω is near stationary, if the following moment bounds when taken with respect to µ 0,N : above, p ∈ >1 is any arbitrary finite moment, and we have u ∈ (0, 1 2 ). Moreover, we require the spatial fields Z N 0,N X → N →∞ Z 0,X with respect to the locally uniform topology on C( ) for some continuous function Z 0,X ∈ C( ). 4 Further, we define the narrow-wedge initial condition, or narrow-wedge initial data, to be the probability measure on Ω supported on the configuration
sup X ∈ e N −κ,X Z N 0,X Lη NW y • = 1 >0 ( y) − 1 0 ( y), y ∈ .
(1.14)
Before we state the primary result, we recall the exponent β c ∈ >0 from Assumption 1.3. Second, the upcoming result relies on the topology of the Skorokhod spaces; for a reference, see [4].
Theorem 1.6. Suppose β c ∈ >0 is sufficiently small, and moreover define the constants α = ∞ k=1 k 2 α k and λ = lim N →∞ λ N . Then the following hold:
• Under near-stationary initial data, the space-time field Z N T,N X is tight in the large-N limit with respect to the Skorokhod topology on D( 0 , C( )). Moreover, all limit points concentrate on the unique weak solution to the SHE (1.2) with parameters α, λ ∈ defined above and with initial data Z 0,X .
• Under narrow-wedge initial data, consider instead
Z N T,X • = 1 2λ N N 1 2 Z N T,X . (1.15)
Then the space-time field Z N T,N X is tight in the large-N limit with respect to the Skorokhod topology on D( >0 , C( )). Moreover, all limit points concentrate on the unique weak solution to the SHE (1.2) with parameters α, λ ∈ defined above and with initial data the Dirac point mass δ 0 supported at the origin. Equivalently, all limit points concentrate on the fundamental solution to the SHE (1.2), in distribution.
In proving Theorem 1.6, for simplicity we assume β c = 0; the interested reader is invited to compute the optimal value of β c ∈ >0 provided by our analysis.
Consequence of Theorem 1.6 within the setting of the narrow-wedge initial data, we obtain limiting exact statistics for the height function h N T,X which indicate its membership in the KPZ universality class. Because such consequence concerning the limiting exact statistics is identical to Theorem 1.3 in [8], we refer therein for the precise result.
Lastly, in proving Theorem 1.6, we will organize our analysis by explicitly writing the bulk of the analysis when proving results necessary for establishing Theorem 1.6 with near-stationary initial data. Meanwhile, the necessary adjustments for proving Theorem 1.6 for narrow-wedge initial data will be ultimately organized and combined in the final section of this article, as the strategies are almost identical.
1.4.
Background. This subsection contains information that is considered "well-known" to experts on the KPZ equation, though we include this discussion anyway for a sense of completeness.
In [15], Hairer developed the famous, robust theory of regularity structures in a successful effort to construct an intrinsic solution theory for the KPZ equation; Hairer's theory of regularity structures was afterwards furthered to singular stochastic PDEs beyond the KPZ equation in [16]. However, Hairer's theory of regularity structures is a stochastic analytic theory; in particular it provides a stochastic analytic toolbox, and thus the problem of implementing regularity structures into lattice dynamics and interacting particle systems remains open. For progress in this direction, we refer to [17].
An alternative approach to understanding the KPZ equation is through the stochastic Burgers equation, whose solution is formally defined via u = ∂ X H . In [10], Goncalves and Jara successfully introduced the nonlinear martingale problem formulation for the stochastic Burgers equation known as energy solutions. The primary advantage of the energy solution theory is its robust nature when analyzing fluctuation fields of interacting particle systems; coupled with the uniqueness result for energy solutions obtained by Gubinelli and Perkowski within [13], Goncalves and Jara, along with Sethuraman in additional papers, successfully established the energy solution theory for the stochastic Burgers equation as the continuum model for the analog of the derivative of the height function for a general class of interacting particle systems. Moreover, the energy solution theory was established to agree with the Cole-Hopf theory of solutions to the KPZ equation as a byproduct in [13]. The primary disadvantage to the energy solution theory, however, is its limited scope; the analysis depends heavily on analyzing particle systems beginning at the stationary measure, and thus cannot study the initial particle configurations 5 of primary interest outside stationarity such as the narrow-wedge, flat, or the one-sided Brownian initial data. See [5] for a further discussion on initial conditions, and [11] and [12] for the application of energy solution theory.
Lastly, although not directly related to the main problem of the KPZ equation, we remark briefly on the seminal article [6] in which Chang and Yau establish the additive SHE as the continuum model for fluctuations of the empirical measure associated to a reversible Ginzburg-Landau model. The methods employed within [6] rely on substantially little algebraic structure of the Ginzburg-Landau model outside of its Gibbs-type structure which yields both a log-Sobolev inequality and large-deviations principles. In particular, Chang and Yau reduce the continuum model problem into an N -body eigenvalue problem via the classical Feynman-Kac estimate (see Lemma 7.2 in Appendix 1 in [19]). In principle, an important although certainly not all-encompassing component of our proof of Theorem 1.6 may be replaced with a similar approach, namely reducing our analysis to an eigenvalue estimate via the Feynman-Kac formula, but at this point only if we study the particle system in the finite-volume framework on the one-dimensional torus.
1.5. Outline. Although the strategy behind our proof of Theorem 1.6 will not be discussed until after we have developed enough framework, we present a structural outline for the remainder of the article now for presentational clarity.
In Section 2, we re-develop the framework established in Section 2 of [8] though from a perspective that will be useful for this article; the strategy for the proof of Theorem 1.6 is also presented towards the end of this section. Within Section 3, we introduce a novel entropy production estimate in the infinite-volume and asymmetric regime and further preliminary estimates. In Section 4, we compactify dynamics of the Gartner transform via detailed heat kernel analysis. Within Section 5, we develop the primary technical contribution for this article, which we refer to as the dynamical variation of the oneblock strategy. In Section 6, we establish preliminary temporal regularity estimates for the Gartner transform. In Section 7, we employ the aforementioned dynamical one-block strategies and regularity estimates to implement a multiscale analysis and establish the fundamental estimate. For Section 8, we organize results in the previous section to establish Theorem 1.6 for near-stationary initial data via a continuity method. Lastly, within Section 9, we comment on the necessary adjustments to analyze narrow-wedge initial data with the same strategy.
1.6. Acknowledgements. The author is grateful for constant encouragement, discussion, and advice from Amir Dembo.
The author is further grateful for A.Dembo's careful listening and feedback concerning some technical details of this article. 1.7. Notation. For any probability measure µ, we let E µ denote the expectation with respect to this measure. Further, for any σ-algebra F , we let E µ F denote the expectation conditioning on F . Moreover, for any function F : → and for any k ∈ , we define the appropriately-scaled lattice-differential operators
N −1 D k F (X ) • = F (X + k) − F (X ), (1.16) ∆ k F • = D k D −k F. (1.17)
Finally, for any space-time function F (T, X ) : 0 × → , we define the following two-parameter family of norms
F (T, X ) L ∞ T,X (κ;δ) • = sup (T,X )∈[0,T f ]× T 1 2 −δ e −κ,X |F (T, X )|. (1.18)
We emphasize that after Section 4, the definition of these norms change, precisely by replacing the infinite volume lattice with a compact sub-domain of size N 5 4 +ǫ , where ǫ ∈ >0 is an arbitrarily small but universal constant. Next, for any s, T ∈ >0 , we define ̺ s,T = |T − s| simply for compact notation.
Lastly, for any (T, X ) ∈ × , we define the space-time shift operator τ T,X f (s, y, η N r,z ) = f (T + s, X + y, η N T +r,X +z ).
STOCHASTIC INTEGRAL EQUATION
This section is primarily devoted towards providing a recapitulation of the framework developed in [8] as we will adopt an identical framework with technical adjustments for this article. Afterwards, we provide an outline towards overcoming the obstacles discussed in [8] that limit the allowed maximal range of interaction in [8], thus proving Theorem 1.6. 6 Before we begin, we remark that this section requires the notions of canonical ensembles as invariant probability measures for the exclusion process. The definition of these canonical ensembles will not be used outside defining hydrodynamictype quantities; we present these probability measures in Section 3.
Specialized Asymmetry.
Before we record the dynamics of the Gartner transform, we require this next key observation concerning the relationship between the original asymmetry coefficients and these specialized ones.
Lemma 2.1. We have ∞ k=1 kα N k γ N k − γ N , * k = 0. (2.1) Proof. By definition of λ N ∈ >0 , it suffices to prove ∞ k=1 kα N k γ N , * k = λ N ∞ k=1 k 2 α N k . (2.2)
By definition of γ N , * k , the LHS of the desired identity is given by
∞ k=1 kα N k γ N , * k = 2λ N ∞ k=1 ∞ ℓ=k (ℓ − k)α N ℓ + λ N ∞ k=1 kα N k (2.3) = 2λ N ∞ k=1 ∞ ℓ=k ℓα N ℓ − 2λ N ∞ k=1 k ∞ ℓ=k α N ℓ + λ N ∞ k=1 kα N k . (2.4)
We now rewrite both the double summations by accumulating the resulting coefficients for each α N k with k ∈ >0 . In the first double summation, we obtain kα N k a total of k-times for any k ∈ >0 . For the second double summation, we obtain a copy of jα N k for each j ∈ 1, k . Combining these two observations gives
2λ N ∞ k=1 ∞ ℓ=k ℓα N ℓ − 2λ N ∞ k=1 k ∞ ℓ=k α N ℓ = 2λ N ∞ k=1 k 2 − k j=1 j α N k (2.5) = λ N ∞ k=1 k 2 − k α N k , (2.6) from which we see ∞ k=1 kα N k γ N , * k = 2λ N ∞ k=1 ∞ ℓ=k (ℓ − k)α N ℓ + λ N ∞ k=1 kα N k (2.7) = 2λ N ∞ k=1 ∞ ℓ=k ℓα N ℓ − 2λ N ∞ k=1 k ∞ ℓ=k α N ℓ + λ N ∞ k=1 kα N k (2.8) = λ N ∞ k=1 k 2 − k α N k − λ N ∞ k=1 kα N k (2.9) = λ N ∞ k=1 k 2 α N k . (2.10)
This completes the proof.
2.2.
Hydrodynamic-Type Quantities. The current subsection is organizational; we present the two classes of quantities appearing in the dynamics for the Gartner transform.
First, we define a class of weakly vanishing quantities whose analysis was a primary contribution of [8].
Definition 2.2.
A space-time random field w T,X (η) : 0 × × Ω → is a weakly vanishing random field if the following conditions are satisfied:
• For all (T, X , η) ∈ 0 × × Ω , we have w T,X (η) = τ T,X w 0,0 (η). • We have E µ gc w 0,0 (η) = 0,
where µ gc is the grand-canonical ensemble on of parameter 1 2 .
• For some universal constant κ ′ ∈ >0 , we have w 0,0 (•) L ∞ ω 1 uniformly in N ∈ >0 .
Although the definition of weakly vanishing quantity provided in [8] differs from the above definition, Lemma 2.5 and its straightforward extension to higher-degree polynomials of the occupation variables show that the class of random fields introduced above are weakly vanishing quantities in the sense that is presented in [8].
The second class of quantities introduced as follows are those of primary interest concerning the technical innovations within this article.
Definition 2.3.
A space-time random field g T,X (η) : 0 × × Ω → is said to be a pseudo-gradient field if the following conditions are satisfied:
• For all (T, X , η) ∈ 0 × N × Ω N , we have g T,X (η) = τ T,X g 0,0 (η). • For any canonical-ensemble parameter ̺ ∈ [−1, 1], we have E µ can ̺, N g 0,0 (η) = 0. • We have the universal bound sup N ∈ >0 g 0,0 (•) L ∞ ω 1.
• The support of g T,X ,η has its size bounded above by N ǫ PG ∈ >0 for some arbitrarily small but universal constant
ǫ PG ∈ >0 .
Remark 2.4. The constant ǫ PG ∈ >0 will be made explicit and precise in Proposition 2.6 below; therein, we may indeed take ǫ PG ∈ >0 an arbitrarily small yet universal exponent.
For example, any honest spatial gradient of a functional of the particle system is a pseudo-gradient field, provided the necessary a priori estimates are satisfied. In providing a nontrivial example, the cubic nonlinearity appearing in Proposition 2.3 in [8] is a pseudo-gradient field that is not an honest spatial gradient assuming the maximal jump-range in the particle system is greater than 3.
The following refinements of pseudo-gradient fields will also serve an important role towards the proof of Theorem 1.6. Definition 2.5. A given space-time random fieldḡ T,X (η) : 0 × × Ω → is said to admit a pseudo-gradient factor if it is uniformly bounded andḡ T,X (η) = g T,X (η) · f T,X (η), (2.11) where the following constraints are satisfied:
• We have f T,X (η) = τ T,X f 0,0 (η) for all (T, X , η) ∈ 0 × N × Ω N , and f 0,0 (•) L ∞ ω 1. • The factor g T,X (η) is a pseudo-gradient field.
• The η-wise supports of g T,X (η) and f T,X (η) are disjoint.
Approximate Lattice-SHE.
We now present the result of [8] giving the stochastic dynamics of the Gartner transform though presented in the framework we have developed thus far. Proposition 2.6. Define β X = 1 3 + ǫ 1 for ǫ 1 ∈ >0 an arbitrarily small but universal constant. Moreover, denote by ǫ c ∈ >0 any arbitrarily small but universal constant.
Under Assumption 1.1 and Assumption 1.3, for any (T,
X ) ∈ [0, T f ] × we have Z N T,X = U N 0,T • Z N 0, y + T 0 U N s,T • Z N s, y dQ N s, y + T 0 U N s,T • N 1 2 · − N β X w=1 τ −w g N s, y · Z N s, y ds + T 0 U N s,T • N β X g N s, y Z N s, y ds + T 0 U N s,T • w N s, y Z N s, y ds + T 0 U N s,T • N ǫ c k=1 c k D k w N ,k s, y Z N s, y , (2.12)
where • The family of operators U N s,T defines the semigroup associated to the elliptic-type differential operator
L N ,! T · = 1 2 N ǫ c k=1 α N k ∆ k , α N k = α N k + N −1 λ 2 N k 2 − 1 α N k − k −1 ∞ ℓ=k+1 (2ℓ − k)α N k +ᾱ N k (2.13) 8 for some vector (ᾱ N k ) ∞ k=1 satisfying |ᾱ N k | N − 3 2 α N k with universal implied constant, • The integrator dQ N s,
y is a martingale increment given by the linear combination of compensated Poisson processes
dQ N s, y · = e 2λN − 1 2 − 1 ∞ k=1 y−k<z< y 1 + η N s,z 2 1 − η N s,z+k 2 dQ N ,k,+ s, y − α N k + γ N k N dt (2.14) + e −2λN − 1 2 − 1 ∞ k=1 y<z< y+k 1 + η N s,z 2 1 − η N s,z−k 2 dQ N ,k,− s, y − α N k dt .
Above, the collection of Poisson processes (Q N ,k,± •, y ) y∈ N ,k∈ >0 are jointly independent with rates so that each quantity inside either summation on the RHS above is a martingale difference.
• The field g N T,X is a pseudo-gradient field. • The field g N T,X admits a pseudo-gradient factor whose support is contained in a sub-lattice whose size is bounded above by N ǫ c ∈ >0 ; ifḡ N T,X denotes this pseudo-gradient factor, then [ḡ N T,X ] −1 g N T,X is some average of monomial functionals in the occupation variables. Proof. The desired equation (2.12) is the integral equation associated to a stochastic differential equation that is immediate from Proposition 2.3 in [8] combined with the derivation of the stochastic dynamics prior and the following observations.
• In principle, the SDE-dynamics of the Gartner transform obtained in Section 2 of [8] are matched with the infiniterange Laplacian-type operator within the proof of Proposition 2.3 in [8]. However, given the moment bounds in Assumption 1.1, we may match the SDE-dynamics with the Laplacian-type operator or range N ǫ c ∈ >0 with an error whose N -dependent prefactor is bounded above by κ p N −p for any p ∈ >1 . We therefore compute the action of this Laplacian-type operator on the Gartner transform for ranges at most N ǫ c ∈ >0 . • The cubic nonlinearity in Proposition 2.3 is a pseudo-gradient field, because all canonical ensembles are invariant under permutations of the lattice-sites. Moreover, if g N T,X denotes the total contribution from these cubic nonlinearities, an application of summation-by-parts yields the following expansion for any ℓ ∈ >0 :
T 0 U N s,T • N 1 2 g N s, y Z N s, y ds = T 0 U N s,T • N 1 2 τ −ℓ g N s, y Z N s, y ds (2.15) + T 0 U N s,T • N 1 2 g N s, y Z N s, y − Z N s, y−ℓ ds − T 0 N − 1 2 D −ℓ U N s,T • g N s, y Z N s, y−ℓ ds.
Upon Taylor expansion similar to that within the proof of Proposition 2.3 in [8], within the final quantity we may replace the shifted Gartner transform Z N s, y−ℓ by the unshifted Gartner transform Z N s, y at the cost of some additional weakly vanishing quantity in (2.12). Furthermore, concerning the second quantity within the RHS of (2.15), we recall that the cubic nonlinearity within Proposition 2.3 in [8] is a linear combination of cubic nonlinearities such that cubic nonlinearities whose support is of length ℓ ∈ >0 come with coefficients bounded by κ p ℓ −p ∈ >0 with any p ∈ >1 , and the number of these nonlinearities is bounded above by κℓ 2 ∈ >0 with κ ∈ >0 some universal constant. Thus, upon organizing the cubic nonlinearities according to length of support, another Taylor expansion analogous to that within the proof of Proposition 2.3 in [8] shows that the second quantity on the RHS of (2.15) corresponds to a weakly vanishing quantity.
It remains to analyze the resulting first quantities on the RHS of (2.15). To this end, for total clarity we remark that this remaining quantity is of the form where {e k } ∞ k=1 have all moments uniformly bounded in N ∈ >0 , and g N ,k s, y is some pseudo-gradient field whose support is contained in y + −2k, 0 ⊆ , courtesy of the spatial shift implemented in (2.15). Another application of summation-by-parts yields
N ǫ c k=1 e k T 0 U N s,T • N 1 2 g N ,k s, y · Z N s, y ds = N ǫ c k=1 e k T 0 U N s,T • N 1 2 · N β X w=1 τ −w g N ,k s, y · Z N s, y ds (2.17) + N ǫ c k=1 e k T 0 U N s,T • N 1 2 g N ,k s, y · N β X w=1 Z N s, y − Z N s, y+w ds − N ǫ c k=1 e k N β X w=1 T 0 N − 1 2 D w U N s,T • g N ,k s, y · Z N s, y+w ds.
Observe the second quantity on the RHS of (2.17) yields another Taylor expansion as in the proof of Proposition 2.3 in [8]. Because the consequential polynomials in the occupation variables have supports disjoint from that of g N ,k s, y due to the spatial shift employed within (2.15), we are left with analyzing the third quantity within the RHS of (2.17). To this end, for each w ∈ 1, N β X , we may first rewrite D w U N s,T as a sum of D 1 -operators acting on the heat kernel U N s,T evaluated at w-many different points. For each resulting operator, we employ a change-of-variables by shifting the lattice , afterwards re-centering each Gartner transform via the Taylor expansion calculation from Proposition 2.3 in [8]. Ultimately, we obtain a negligible gradient-type quantity, or equivalently another quantity of type the last term in (2.12).
• Because γ N k = γ N , *
k in general, consequence of following the proof of Proposition 2.3 in [8], in addition to the RHS (2.12) we are missing a quantity of the form
N 1 2 T 0 U N s,T • Q N s, y Z N s, y ds, Q N s, y · = κ λ N ∞ k=1 α N k γ N k − γ N , * k z 1 < y<z 2 z 2 −z 1 =k η N s,z 1 η N s,z 2 (2.18)
for some constant κ λ N ∈ >0 that is uniformly bounded in N ∈ >0 depending only on λ N ∈ >0 . Upon rewriting this quadratic nonlinearity as
Q N s, y = κ λ N ∞ k=1 α N k γ N k − γ N , * k z 1 < y<z 2 z 2 −z 1 =k η N s,z 1 η N s,z 2 − η N s, y η N s, y+1 + κ λ N ∞ k=1 kα N k γ N k − γ N , * k η N s, y η N s, y+1 ,(2.19)
we observe the first quantity on the RHS is a pseudo-gradient term given the constraint in Assumption 1.3. Moreover, by Lemma 2.1, the last quantity on the RHS vanishes deterministically. We may therefore repeat the procedure of the previous bullet point for the quantities corresponding to an index k N ǫ c per the first bullet point. showing such quantities contribute an negligible amount in the large-N limit, we propose the following "preliminary" yet unsuccessful strategy.
• First, we mention that these long-range asymmetric exclusion processes do not admit any known duality except in the situation of the ASEP. Moreover, approaches via duality are unlikely to give a simple argument showing the vanishing of the pseudo-gradient quantities on the RHS of (2.12); see [7], for example.
Similarly, the strategy introduced in [3] of interpreting particle functionals as products of gradients of the Gartner transform are seemingly unhelpful given that the pseudo-gradient quantities are higher-degree polynomials in the occupation variables and we have only one copy of the Gartner transform.
• The remaining approach is the perspective of hydrodynamic theory and analysis. Following the classical approach, we rely entirely on the spatial-averaging procedure introduced within the proof of Proposition 2.6. However, given the regularity of the Gartner transform, via either its continuum SHE model or by its definition, we may only hope to replace a pseudo-gradient by its spatial average on the scale N 1 2 ∈ >0 . Unfortunately, via CLT considerations, if the average particle density on this scale is anything close to the physically predicted 1 2 given the near-stationary initial data, pointwise-in-time this spatial average is expected to be roughly N − 1 4 ∈ >0 , which is certainly far from sufficient to counter the N -dependent prefactor.
The failure of the classical one-block strategy outlined in the second bullet point is its heavy reliance on spatial averaging. In particular, as explained in the seminal work [6], the proof of Theorem 1.6 depends heavily on a successful implementation of time-averaging. The perspective taken within the current article is that if the hydrodynamical theory and toolbox were suitable to prove Theorem 1.6 from (2.12), then the classical one-block estimate is inherently open for improvement; this motivates our development of a dynamical variation of the one-block strategy, which we spend almost the entire remainder of this section explaining.
Our implementation of time-averaging contrasts from that within [6]. Briefly, the strategy therein relies on a Feynman-Kac estimate which then reduces their problem into an N -body eigenvalue problem, solved using large-deviations estimates for the Ginzburg-Landau model. Unfortunately, we cannot adopt such an approach because the integrals of primary interest include the Gartner transform, which is not only a non-local function of the particle system but it further depends on the entire path of the exclusion process because of its dependence on the flux h N T,0 . Furthermore, within [6] the problem is set on the torus and its lattice approximations, whereas in this current article we analyze the more probabilistically interesting infinite-volume lattice and thus require suitably strong entropy production estimates like in Theorem 2.1 in [20], though this latter point is likely irrelevant.
Contrary to [6], the approach taken within the current article towards implementing time-averaging consists of replacing the spatial average of the pseudo-gradient quantity on the RHS of (2.12) by its temporal average on some mesoscopic time-scale. Considerations with respect to any of the invariant probability measures, essentially generated by the canonical ensembles, show that the desired time-scale is roughly τ ∼ N −1 ; see Lemma 3.15. Indeed, this roughly matches the scale that we ultimately arrive at.
Before addressing the problem of analysis outside the invariant probability measure, we note that even this replacement with a mesoscopic time-average is nontrivial because of the "insufficient" temporal regularity behind the Gartner transform; this can be deduced by looking at the temporal regularity of the continuum SHE model. In order to overcome this hurdle is a multiscale analysis, we employ the following algorithm:
• Replace the pseudo-gradient field, or a suitable spatial-average of it, by its time-average on some time-scale above the microscopic time-scale. • Replace this newly obtained time-average by another time-average on a time-scale that is larger than the previous time-scale by a factor of Nβ ∈ >0 , whereβ ∈ >0 is a universal constant.
• Repeatedly iterate until you reach the desired time-scale.
We refrain from introducing any precise details, but we emphasize that the success behind this algorithm relies heavily on the powerful estimate obtained from time-averaging and the temporal ergodicity of the lattice dynamic under the diffusive regime; in particular, the analogous multiscale strategy for spatial averages fails upon considerations with respect to the invariant probability measures.
Although the aforementioned strategy succeeds with respect to the invariant probability measures of the particle dynamic, we are interested in the problem of non-equilibrium probability measures, including deterministic initial data. To this end, we employ the classical one-block strategy though with two important changes.
• First, like in [6], we require a quantitative version of the one-block strategy which originally relied on qualitative compactness arguments, and to this end we employ the optimal log-Sobolev inequality, or LSI, within [22].
• Second, we require a dynamical variation of the one-block strategy to address moments of a path-space observable, namely a time-average. The key observation employed is that any expectation taken with respect to the path-space probability measure may be decomposed into an expectation over the dynamic conditioning on the initial data, and afterwards an expectation over the initial data. Because our time-averages are taken with respect to the same dynamic in distribution at each space-time point but with an initial condition that is sampled from its macroscopic space-time average, we may apply the classical one-block strategy and obtain the desired replacement.
Somewhat more precisely, this previous paragraph yields this next observation; for any path-wise functional f, for any sub-lattice Λ ⊆ , and for any T ∈ >0 , we have
E − T 0 − X ∈Λ f(τ s,X η N •,0 ) ds = E E path η N 0 f(η N •,0 ) ,(2.20)
where the E-operator is an expectation taken over the initial data η N 0 , and the expectation E path η N 0 is an expectation with respect to the path-space probability measure conditioning on the initial data η N 0 , and is therefore a function of this initial data; crucially, we note the path-space probability measure is a space-time homogeneous probability measure, so that the above identity actually holds. In particular, the E-operator above is taken with respect to the space-time average of the probability measures at the different space-time points in [0, T ] × Λ as in the classical one-block scheme. At this point, we proceed as in the one-block strategy with a log-Sobolev inequality.
We briefly remark that the above identity between expectations is actually useless at face-value; this is a simple consequence of the observation that the E-operator on the RHS is the expectation with respect to a particle configuration on the infinitevolume lattice . However, for the class of functionals f depending only on mesoscopic space-time-scales, we may replace the function f by its values with respect to the exclusion process on some mesoscopic neighborhood equipped with periodic boundary conditions; this is roughly due to the diffusive nature with weak asymmetry of particles in the exclusion process, or more precisely of the discrepancies between two exclusion processes, and a coupling argument.
Moreover, we remark that the aforementioned strategy succeeds on the torus, and to the author's knowledge, it actually fails in the infinite-volume setting. To remedy this issue, we require a compactification scheme consisting of the following three ingredients:
• Via precise heat kernel estimates, for the purposes of understanding the Gartner transform on arbitrary compact domains, we replace the infinite-volume Gartner transform evolution (2.12) by the same evolution equation on some torus of size roughly N 5 4 ∈ >0 . This consists of replacing both the (stochastic) heat flow propagating (2.12) by its periodification and replacing the initial data by its periodification.
• Observe the previous "analytic" compactification does not replace the entire particle system by its periodification.
In particular, to implement a quantitative and precise version the one-block estimate, we are left with a problem of estimating entropy production in the infinite-volume setting. This has been performed in several settings, such as that of [20]. However, the asymmetry within our particle system presents an honest obstruction to employing the results therein or of related articles. Regardless, we obtain a sufficient entropy production estimate by combining the procedure in [20] and another coupling argument.
• The third and final ingredient is another technical multiscale analysis towards estimating the time-average outside of the invariant probability measures; we do not discuss any details for now.
ENTROPY PRODUCTION, THE LSI, AND SPECTRAL ANALYSIS
The primary objective for the current section is a suitable preliminary discussion consisting of tools from hydrodynamical analysis that will be crucial in our proof of Theorem 1.6. Beyond the entropy production estimate in Proposition 3.5, the remainder of the preliminary bounds are well-established if not classical.
Canonical Ensembles.
We briefly review the fundamental invariant probability measures for the long-range exclusion processes of primary interest. Definition 3.1. For any sub-lattice Λ ⊆ and ̺ ∈ [−1, 1], we define the (̺, Λ)-canonical ensemble and grand-canonical ensemble to be the following probability measures on Ω Λ , respectively:
µ can ̺,Λ • = Unif H ̺,Λ , H ̺,Λ • = η ∈ Ω Λ : − x∈Λ η x = ̺ , (3.1) µ gc ̺,Λ • = X ∈Λ [̺δ +1 + (1 − ̺)δ −1 ] . (3.2)
The following consistency-type property of canonical ensembles on different sub-lattices will be crucial to our analysis in this section; the proof is a direct consequence of conditioning.
Lemma 3.2. Suppose Λ i ⊆ Λ o ⊆ is a nested pair of sub-lattices, and fix any ̺ o ∈ [−1, 1]. For any functional ϕ : Ω Λ i → whose support is contained in Λ i ⊆ , we have E µ can ̺, o Λ o ϕ = ̺∈[−1,1] p Λ i ,Λ o ̺ E µ can ̺,Λ i ϕ, p Λ i ,Λ o ̺ • = P µ can ̺ o ,Λ o − x∈Λ i η x = ̺ . (3.3)
Equivalently, the pushforward of µ can
̺ o ,Λ o under the projection map Ω Λ o → Ω Λ i
is a convex combination of canonical ensembles
on Ω Λ i with the above coefficients.
Our last discussion concerning the canonical and grand-canonical ensembles concerns the relative entropy and Dirichlet form; these quantities are standard in both the theory and toolbox for hydrodynamical limits, but we define them as follows for the reader's convenience. 1], and suppose that f ∈ L 1 (Ω Λ ) is a probability density with respect to the canonical ensemble µ can ̺,Λ . • The relative entropy with respect to µ can ̺,Λ is defined as
Definition 3.3. Consider any sub-lattice Λ ⊆ and parameter
̺ ∈ [−1,H can ̺,Λ ( f ) • = E µ can ̺,Λ f log f . (3.4)
Similarly, we define the relative entropy with respect to µ gc ̺,Λ to be
H can ̺,Λ ( f ) • = E µ gc ̺,Λ f log f . (3.5)
• The Dirichlet form with respect to µ can ̺,Λ is defined as
D can ̺,Λ ( f ) • = E µ can ̺,Λ f 1 2 · x, y∈Λ α N |x− y| L x, y f 1 2 , (3.6)
where we recall L x, y is the generator for the symmetric exclusion process on the graph consisting of only the bond {x, y}. Similarly, we define the Dirichlet form with respect to µ gc ̺,Λ as
D gc ̺,Λ ( f ) • = E µ gc ̺,Λ f 1 2 · x, y∈Λ α N |x− y| L x, y f 1 2 . (3.7)
The analytic statement that relates the relative entropy and Dirichlet form is the log-Sobolev inequality within Theorem A of [22]. Another important inequality that concerns the relative entropy is the upcoming entropy inequality which holds true for any random variable X ∈ L 1 µ can ̺,Λ , for any probability density f with respect to µ can ̺,Λ , and for any κ ∈ >0 :
E µ can ̺,Λ ( f X ) κ −1 H can ̺,Λ ( f ) + κ −1 log E µ can ̺,Λ exp [κ f ] . (3.8)
The same inequality certainly holds for µ gc ̺,Λ in place of µ can ̺,Λ , as well, though the LSI of Theorem A in [22] applies only to the canonical ensembles.
We finally recall that the Dirichlet form is convex; see Appendix 1 in [19] for the proof. 13 3.2. Entropy Production. The current subsection is dedicated towards obtaining an entropy production estimate. This requires establishing the following preliminary notation.
Notation 3.4.
For any probability density f ∈ L 1 (Ω ), and any sub-lattice Λ ⊆ , we let f Λ = E Λ f denote the projection of the probability density onto the compact sub-lattice via conditional expectation.
Given any probability measure µ 0,N on Ω , we define µ T,N = e T L N,! T µ 0,N to be the probability measure on Ω obtained via evolving µ 0,N under the long-range exclusion dynamic by time T ∈ >0 . Moreover, for any T ∈ 0 , we define
f T,N • = dµ T,N dµ gc 1/2 , (3.9a) f T,N • = − T 0 f t,N dt. (3.9b)
The main result for the current subsection is the following Dirichlet energy estimate.
Proposition 3.5.
Consider any time-scale T ∈ >0 independent of N ∈ >0 , and consider any arbitrarily small but universal
constant ǫ ∈ >0 . Define N = −N 5 4 +ǫ , N 5 4 +ǫ . Then we have a universal constant ǫ ′ ǫ such that for any initial data µ 0,N on Ω , we have D gc 1 2 , N (f N T,N ) T N − 3 4 +ǫ ′ .
(3.10)
Supposing the particle dynamic lived on the torus N with a periodic boundary condition, the result of Proposition 3.5 would be classical; see Appendix 1 in [19], for example. In particular, the problem arises from the non-compact geometry.
Although our treatment roughly follows the strategy of [20], first, we present a preliminary reduction which provides some precise treatment of the non-compact infinite-volume issue in presence of the asymmetry. To present this preliminary estimate, we first introduce the following generator for a particle dynamic with large but N -dependent finite-range jumplengths:
L N ,! T f : η → N 2 Nǫ k=1 α N k x∈ 1 + γ N k N 1 + η x 2 1 − η x + k 2 L x,x+k f; (3.11)
above,ǫ ∈ >0 is arbitrarily small but universal.
µ 0,N • = Π N µ 0,N ⊗ Π C N µ gc 1 2 , , (3.12) µ T,N • = e T L N,! T µ 0,N , (3.13) g T,N • = d µ T,N dµ gc 1/2 , (3.14) g T,N • = − T 0 g t,N dt. (3.15)
Then for any D ∈ >0 , we have
D gc 1 2 , N (f N T,N ) D D gc 1 2 , N (ḡ N T,N ) + N −D . (3.16)
Moreover, we have
− T 0 D gc 1 2 , (g t,N ) dt N − 1 2 +ǫ ′′ , (3.17)
where the implied constant is universal.
14 Remark 3.7. Observe that convexity of the Dirichlet form provides a sub-optimal and weaker version of Proposition 3.5.
This basic estimate, however, is insufficient for the purposes of this article.
It will serve convenient for the remainder of this section to establish the following notation.
Notation 3.8.
For any x, y ∈ , we define the following involution σ x, y : Ω → Ω given by
σ x, y η(z) • = η( y) z = x, η(x) z = y, η(z) z = x, y. (3.18)
The action of σ x, y extends to functionals of the particle system, defined by σ x, y f(η) = f(σ x, y η).
Proof. We first remark that (3.17) is consequence of replacing the infinite-volume particle dynamic with the particle dynamic on arbitrarily large intervals with periodic boundary condition, applying the entropy production inequality of Theorem 9.2 in Appendix 1 in [19], and passing to the infinite-volume limit.
It remains to establish (3.16). To this end, we first assume that for ǫ c ∈ >0 an arbitrarily small but universal constant, the maximal jump-length is bounded by N ǫ c ∈ >0 . We now construct a coupling given as follows.
• Upon realizing the symmetric long-range exclusion process as attaching a Poisson clock to each bond connecting two sites, and the step in the process corresponding to a ringing of this Poisson clock as swapping the occupation variable values at these two sites, for any bonds shared between the global dynamic and the local periodic dynamic we couple the bond-clocks together. We emphasize that this is not the basic coupling. • The remaining Poisson clocks associated to the lower-order totally asymmetric long-range exclusion process are then equipped with the basic coupling, that any of these remaining clocks, if are shared between the two dynamics, are coupled together. Equivalently, any particles that occupy any site shared in the two domains N and jump together whenever possible.
We make three further observations.
(1) There are no present discrepancies between these two initial particle configurations on N ⊆ N by assumption.
Moreover, the coupling constructed above cannot create discrepancies. In particular, any discrepancies that appear in N ⊆ N must be consequence of some interaction that is present in one dynamic and absent in the other. These interactions are those coming from either the periodic boundary condition defining the periodic process or from the interaction of sites within the larger torus N ⊆ to sites outside this sub-lattice in the original infinite-volume process. Ultimately, discrepancies must travel the distance of N 3 2 +ǫ ′′ in order to create discrepancies in N .
(2) Under the above semi-basic coupling, the dynamics of any tagged trajectory are a free and unsuppressed symmetric long-range random walk plus a randomly suppressed and randomly killed totally asymmetric long-range random walk, where the randomness is a function of the environment of particles. The latter totally asymmetric component
occurs with a dampened speed of scale N 3 2 ∈ >0 .
(3) Thus, by the Azuma martingale inequality and a large-deviations-estimate for the Poisson clocks, the probability we find a discrepancy between the respective particle configurations on N ⊆ N ⊆ given by the two different dynamics we are coupling is exponentially small. More precisely, we have an exponentially small upper bound on the total variation distance betweenf N T,N andḡ N T,N , which then gives the next L 1 ω -estimate with ǫ ′′′ ∈ >0 another universal constant:
f N T,N −ḡ N T,N L 1 ω e −N ǫ ′′′ . (3.19)
It remains to estimate the Dirichlet form in terms of the L 1 ω -above. In what follows, all expectations are taken with respect to the invariant measure µ . 15 For any x, y ∈ N , we have
E σ x, yf N T,N − f N T,N 2 − E σ x, yḡ N T,N − ḡ N T,N 2 = 2 E σ x, yf N T,N ·f N T,N − 2 E σ x, yḡ N T,N ·ḡ N T,N (3.20) E σ x, yf N T,N f N T,N − ḡ N T,N (3.21) + E σ x, yḡ N T,N f N T,N − ḡ N T,N E f N T,N − ḡ N T,N 2 , (3.22)
where the last upper bound follows from the Cauchy-Schwarz inequality and the observation thatf N T,N andḡ N T,N are probability densities with respect to the invariant measure µ . We now observe the following general inequality for a, b ∈ >0 :
| a − b| 2 |a − b| | a + b| | a − b| |a − b|; (3.23) if a = 0 or b = 0,E f N T,N − ḡ N T,N 2 f N T,N −ḡ N T,N L 1 ω (3.24) e −N ǫ ′′′ . (3.25)
Lastly, because the Dirichlet form over N is bounded by N D -many of these last quantities with D ∈ >0 arbitrarily large but universal, the result follows.
Courtesy of Lemma 3.6, it remains to prove Proposition 3.5 upon replacingf N T,N byḡ N T,N , the latter of which admits a description as a "pre-compactified" cousin off N T,N with cut-off in the maximal jump-length. We now briefly explain the utility of Lemma 3.6. The classical strategy within [20] behind obtaining entropy production bounds in the infinite-volume setting is performed for symmetric reversible systems. In particular, the strategy therein does not obviously extend to asymmetric systems, let alone particle systems with an asymmetry of order as large as N − 1 2 on the macroscopic time-scale of primary interest in this article. Therefore, our proof of Proposition 3.5 is the combination of the strategy for reversible systems within [20] combined with an additional estimate for asymmetric dynamics which requires Lemma 3.6 as an a priori estimate.
Proof of Proposition 3.5. Following the proof of Theorem 2.1 in [20], consider any smooth function ζ : → such that ζ(x) ∼ |x| as |x| → +∞; its utility will appear towards the end of this proof. It will also serve convenient to establish the following notation.
g ℓ • = g ℓ N T,N , g ℓ,x, y • = g ℓ N ∪{x, y} T,N .
(3.26)
Moreover, we define the boundary of this sub-lattice by
∂ ℓ N • = (x, y) ∈ : x < y, x ∈ N , y ∈ N , |x − y| Nǫ . (3.27)
Applying the Kolmogorov forward equation for g T,N , we see
d dT H gc 1 2 , ℓ N (g ℓ ) −N 2 D gc 1 2 , ℓ N (g ℓ ) + N 2 E (x, y)∈∂ ℓ N α N | y−x| 1 + γ N | y−x| N 1 + η x 2 1 − η y 2 · g ℓ,x, y · L x, y log g ℓ .
(3.28) 16 For the symmetric component of the second quantity on the RHS of (3.28), we will compute as in Theorem 2.1 in [20]:
N 2 E (x, y)∈∂ ℓ N α N | y−x| · g ℓ,x, y · L x, y log g ℓ = N 2 (x, y)∈∂ ℓ N α N | y−x| E g ℓ,x, y · L x, y log g ℓ = N 2 (x, y)∈∂ ℓ N α N | y−x| E L x, y g ℓ,x, y L x, y log g ℓ = N 2 (x, y)∈∂ ℓ N α N | y−x| E L x, y g 1 2 ℓ,x, y σ x, y g 1 2 ℓ,x, y + g 1 2 ℓ,x, y L x, y log g ℓ N 11 4 (x, y)∈∂ ℓ N α N | y−x| E L x, y g 1 2 ℓ,x, y 2 (3.29) + N 5 4 (x, y)∈∂ ℓ N α N | y−x| E σ x, y g 1 2 ℓ,x, y + g 1 2 ℓ,x, y L x, y log g ℓ 2 .
The second quantity on the RHS of (3.29) admits the following upper bound courtesy of the general convexity inequality
log b − log a 2a − 1 2 b − a , valid for all a, b ∈ >0 : N 5 4 (x, y)∈∂ ℓ N α N | y−x| E σ x, y g 1 2 ℓ,x, y + g 1 2 ℓ,x, y L x, y log g ℓ 2 N 5 4 (x, y)∈∂ ℓ N α N | y−x| E σ x, y g ℓ,x, y σ x, y g ℓ σ x, y g ℓ L x, y log g ℓ 2 (3.30) + N 5 4 (x, y)∈∂ ℓ N α N | y−x| E g ℓ,x, y g ℓ g ℓ L x, y log g ℓ 2 N 5 4 (x, y)∈∂ ℓ N α N | y−x| g ℓ,x, y g ℓ L ∞ ω E g ℓ L x, y log g ℓ 2 (3.31) N 5 4 (x, y)∈∂ ℓ N α N | y−x| E σ x, y g ℓ + g ℓ (3.32) N 5 4 . (3.33)
Indeed, g ℓ is the equally-weighted average over two values of g ℓ,x, y , which gives a straightforward bound for the L ∞ ω -norm of their ratio; moreover, we have employed invariance of µ gc 1/2 under the action of σ x, y repeatedly, and the final inequality is consequence of moment assumptions on the coefficients {α N k } ∞ k=1 . Lastly, all implied constants are universal constants. We now address the first quantity on the RHS of (3.29). Because the Dirichlet form is convex, we know
N 11 4 (x, y)∈∂ ℓ N α N | y−x| E L x, y g 1 2 ℓ,x, y 2 N 3 2 (x, y)∈∂ ℓ N α N | y−x| N 5 4 j=1 E L x, y g ℓ+ j 2 (3.34)
We return back to the asymmetric component of the operator within the second quantity on the RHS of (3.28). Proceeding along a similar calculation, defining β N k = α N k γ N k 0 just for this proof, observe
N 2 E (x, y)∈∂ ℓ N β N | y−x| N 1 + η x 2 1 − η y 2 · g ℓ,x, y · L x, y log g ℓ N 2 (x, y)∈∂ ℓ N β N | y−x| E g ℓ,x, y · L x, y log g ℓ + = N 3 2 (x, y)∈∂ ℓ N β N | y−x| E g ℓ,x, y g ℓ g ℓ · L x, y log g ℓ + N 3 2 (x, y)∈∂ ℓ N β N | y−x| E L x, y g 1 2 ℓ + N 3 2 (x, y)∈∂ ℓ N β N | y−x| E σ x, y g 1 2 ℓ − g 1 2 ℓ 2 1 2 .
(3.35) 17 Applying once more the convexity of the Dirichlet form, given the assumed moment bounds on {β k } ∞ k=1 we obtain
N 2 E (x, y)∈∂ ℓ N β N | y−x| N 1 + η x 2 1 − η y 2 · g ℓ,x, y · L x, y log g ℓ N 3 2 D gc 1 2 ,
(g) 1 2 .
(3.36)
Ultimately, for any ℓ ∈ >0 , we obtain the following upper bound with κ ∈ >0 a universal constant:
d dT H gc 1 2 , ℓ N (g ℓ ) −N 2 D gc 1 2 , ℓ N (g ℓ ) + κN 3 2 (x, y)∈∂ ℓ N α N | y−x| N 5 4 j=1 E L x, y g ℓ+ j 2 + N 5 4 + N 3 2 D gc 1 2 ,(g) 1 2
.
(3.37)
Following the proof of Theorem 2.1 in [20], we obtain
N − 5 4 ∞ ℓ=0 exp −ζ(N − 5 4 ℓ) − T 0 D gc 1 2 , ℓ N (g ℓ N T,N ) N −2− 5 4 ∞ ℓ=0 exp −ζ(N − 5 4 ℓ) · H gc 1 2 , ℓ N (g ℓ N 0,N ) − H gc 1 2 , ℓ N (g ℓ N T,N ) (3.38) + N − 3 4 + N − 1 2 − T 0 D gc 1 2 , (g t,N ) 1 2 dt (3.39) T N − 3 4 +ǫ + N − 1 2 − T 0 D gc 1 2 , (g t,N ) dt 1 2 (3.40) T N − 3 4 +ǫ + N − 3 4 +ǫ ′′ , (3.41)
where the penultimate inequality is consequence of Jensen's inequality and the straightforward bound on relative entropy H gc 1 2 ,Λ ( f ) |Λ|, and the ultimate inequality is a consequence of Lemma 3.6. Considering only the summands |ℓ| N 5 4 +ǫ ′′′ for ǫ ′′′ ∈ >0 arbitrarily small but universal, the result follows upon convexity of the Dirichlet form, again.
Entropy Inequalities.
We recall the general entropy inequality in (3.8). The second inequality of primary importance is the log-Sobolev inequality of [22]; we refer the reader to Theorem A therein.
Combining these two entropy inequalities with the Dirichlet form estimate in Proposition 3.5, we obtain an inequality which we use repeatedly throughout this article. However, it will be convenient to first establish some notation.
p Λ,̺ T,N · = E µ can ̺ N , N f Λ T,N 1 − x ∈Λ η x =̺ . (3.43)
We further define the following post-projection relative entropy and Dirichlet form:
H can ̺,Λ f Λ T,N · = H can ̺,Λ d µ Λ T,N dµ can ̺,Λ p Λ,̺ T,N −1 1 − x ∈Λ η x =̺ , (3.44a) D can ̺,Λ f Λ T,N · = D can ̺,Λ d µ Λ T,N dµ can ̺,Λ p Λ,̺ T,N −1 1 − x ∈Λ η x =̺ . (3.44b)
Lemma 3.12. Fix β ∈ >0 , and suppose ϕ has a support Λ ϕ ⊆ . Then for any κ ∈ >0 , we have
E µ gc 1/2, ϕ f Λ ϕ T,N κ ′ κ −1 N − 7 4 |Λ ϕ | 3 + κ −1 sup ̺∈[−1,1] log E µ can ̺,Λ ϕ e κϕ , (3.45) where κ ′ ∈ >0 is a universal constant.
Proof. Appealing to Lemma 3.2 and (3.8), we have the following for any κ ∈ >0 :
E µ gc 1/2, ϕ f Λ ϕ T,N = ̺∈[−1,1] p Λ ϕ ,̺ T,N E µ can ̺,Λ ϕ f Λ ϕ T,N − x∈Λ η x = ̺ (3.46) ̺∈[−1,1] p Λ ϕ ,̺ T,N κ −1 H can ̺,Λ ϕ f Λ ϕ T,N + κ −1 log E µ can ̺,Λ ϕ e κϕ . (3.47)
The second quantity admits the straightforward bound
̺∈[−1,1] p Λ ϕ ,̺ T,N · κ −1 log E µ can ̺,Λ ϕ e κϕ κ −1 sup ̺∈[−1,1] log E µ can ̺,Λ ϕ e κϕ . (3.48)
Concerning the relative entropy, we apply the LSI within Theorem A in [22] to obtain
κ −1 ̺∈[−1,1] p Λ ϕ ,̺ T,N H can ̺,Λ ϕ f Λ ϕ T,N κ −1 |Λ ϕ | 2 ̺∈[−1,1] p Λ ϕ ,̺ T,N D can ̺,Λ ϕ ( f Λ ϕ T,N ) (3.49) = κ −1 |Λ ϕ | 2 D gc 1 2 ,Λ ϕ ( f Λ ϕ T,N ), (3.50)
where the final identity follows from the definition of the D-functional, or equivalently by definition of conditional probability and conditional expectation. Again by convexity, as in the classical one-block estimate within Theorem 2.4 in [14],
we obtain κ −1 |Λ ϕ | 2 D gc 1 2 ,Λ ϕ ( f Λ ϕ T,N ) κ −1 |Λ ϕ | 3 | N | −1 D gc 1 2 , N (f Λ ϕ T,N ) (3.51) κ −1 N −2 |Λ ϕ | 3 , (3.52)
where the final inequality is consequence of Proposition 3.5. This completes the proof.
3.4. Spectral Analysis. The dynamical one-block analysis for pseudo-gradient fields depends on the following inequality of Kipnis-Varadhan; roughly speaking, the inequality provides an estimate for the temporal-average of any pseudo-gradient field in terms of an energy-type quantity, when the process starts with respect to a stationary measure which, in the situation of the exclusion processes within this article, are generated by the canonical ensembles.
To state the estimate, we introduce the following Sobolev space.
Definition 3.13. Consider any sub-lattice Λ ⊆ . We first define the following operator given by the global generator L ! T restricted to Λ ⊆ and without the N -dependent prefactor:
L Λ ϕ • = x, y∈Λ α N |x− y| + γ N |x− y| N η x∧ y (1 − η x∨ y ) L x, y ϕ, ϕ : Ω Λ → ,(3.53)
For any fixed parameter ̺ ∈ [−1, 1], we define the spaceḢ −1 ̺,Λ via the norm
ϕ 2 H −1 ̺,Λ • = sup ψ∈L ∞ (Ω Λ ) D N ϕ,ψ , D N ϕ,ψ • = 2 E µ can ̺,Λ (ϕψ) − N 2 D can ̺,Λ (ψ 2 ). (3.54a)
Demonstrating the utility of the above Sobolev-type spaces consists of two ingredients. First, we illustrate their relevance by presenting the Kipnis-Varadhan inequality, for whose proof we cite Appendix 1 in [19]. To state this result, we establish notation for a periodic dynamic that will be used throughout the article.
Notation 3.14. Consider any sub-lattice Λ ⊆ , and consider the Markov process s → η N ,per s given by the generator
L ! T Λ,per ϕ • = ∞ k=1 x∈Λ 1 x,x +k∈Λ α N k + γ N k N η x (1 − η x +k ) L x,x+k ϕ, ϕ : Ω Λ → . (3.55)
The Markov process s → η N ,per s is the long-range exclusion process on Λ ⊆ upon imposing periodic boundary conditions. Lemma 3.15. Consider any sub-lattice Λ ⊆ and any parameter ̺ ∈ [−1, 1]. For any bounded function g : Ω Λ → N and for any T ∈ >0 , we have
E µ can ̺,Λ T 0 g(η N ,per s ) ds 2 • = E µ can ̺,Λ E η N,per 0 T 0 g(η N ,per s ) ds 2 20T g 2 H −1 ̺,Λ , (3.56)
with the inner expectation on the RHS taken with respect to the path-space measure induced by the Markov process s → η N ,per s .
The second ingredient demonstrating the utility of the Sobolev-type spacesḢ −1 ̺,Λ consists of estimating the associated norm; this depends largely on a spectral gap for the symmetrization of L ! T Λ,per and an orthogonality-type property for the norm itself. We record both of these estimates in the following lemma. • For each j ∈ 1, J , the support of ϕ j is contained in some further sub-lattice Λ j ⊆ Λ; alternatively, the function ϕ j is realizable as a function ϕ j :
Ω Λ j → ; • The supports Λ 1 , . . . , Λ J ⊆ Λ are mutually disjoint; • For each j ∈ 1, J and for any ̺ ∈ [−1, 1], we have E µ can ̺,Λ j ϕ j = 0.
Then we have the following two estimates:
sup j=1,...,J |Λ j | −2 ϕ j 2 H −1 ̺,Λ N −2 sup ̺ j ∈[−1,1] E µ can ̺ j ,Λ j |ϕ j | 2 , (3.57) J j=1 ϕ j 2 H −1 ̺,Λ J j=1 ϕ j 2 H −1 ̺,Λ . (3.58)
Proof. See the proofs of Proposition 3.3 and Proposition 3.4, respectively, in [10].
Observe the remarkably small N -dependent factor in the estimate within Lemma 3.16; indeed, this N -dependent factor reflects the diffusive scaling underlying the particle system dynamic and its speed of convergence to statistical equilibrium.
Our last spectral estimate addresses pseudo-gradient fields whose supports may be relatively large, but whose pseudogradient factors have small supports.
Lemma 3.17. Let ϕ : Ω → denote a pseudo-gradient field with support Λ ϕ ⊆ of the form
ϕ = g · ℓ i=1 η x i , (3.59)
where g is a pseudo-gradient field with support Λ ⊆ disjoint from {x 1 , . . . , x ℓ }, and where ℓ ∈ 0 denotes any non-negative integer. Then
ϕ 2 H −1 ̺,Λ ϕ g 2 H −1 ̺,Λ .
(3.60)
Proof. Provided any functional ψ :
Ω Λ ϕ → , let us define ψ = ψ · ℓ i=1 η x i . By definition of theḢ −1 ̺,Λ ϕ
-norm and positivity of the Dirichlet form, we have
ϕ 2 H −1 ̺,Λ ϕ sup ψ∈L ∞ (Ω Λ ϕ ) 2 E µ ̺ ̺,Λ ϕ (ϕψ) + N 2 E µ ̺ ̺,Λ ϕ ψL Λ ϕ ψ (3.61) sup ψ∈L ∞ (Ω Λ ϕ ) 2 E µ ̺ ̺,Λ ϕ (ϕψ) + N 2 E µ ̺ ̺,Λ ϕ ψL Λ ψ (3.62) = sup ψ∈L ∞ (Ω Λ ϕ ) 2 E µ ̺ ̺,Λ ϕ (g ψ) + N 2 E µ ̺ ̺,Λ ϕ ψL Λ ψ . (3.63)
Indeed, because the operator L Λ does not act on those sites outside Λ ⊆ , and since η 2
x i = 1 for all x i ∈ , the last bound follows. But this last bound is bounded above by g 2
H −1 ̺,Λ
upon applying the convexity of the Dirichlet form and replacing ψ in the last bound above by its conditional expectation onto Λ ⊆ , so the result follows. 20 3.5. Martingale Estimates. Applying martingale analysis, we now exploit the spatial ergodicity of pseudo-gradient fields through the following result; the setting is quite similar to the framework of Lemma 3.16. • For each j ∈ 1, J , the support of ϕ j is contained in some further sub-lattice Λ j ⊆ Λ; alternatively, the function ϕ j is realizable as a function ϕ j :
Ω Λ j → ; • The supports Λ 1 , . . . , Λ J ⊆ Λ are mutually disjoint; • For each j ∈ 1, J and for any ̺ ∈ [−1, 1], we have E µ can ̺,Λ j ϕ j = 0. • For each j ∈ 1, J , we have ϕ L ∞ Ω Λ j 1.
Then for some universal constant κ ′ ∈ >0 , for any C ∈ >0 , and for any ̺ ∈ [−1, 1], we have
P µ can ̺,Λ − J j=1 ϕ j C e −κ ′ J C 2 . (3.64)
Proof. Consider the filtration of σ-algebras defined by F j • = σ(Λ 1 , . . . , Λ j ) for j ∈ 1, J , and observe the following process is a discrete-time martingale with respect to the filtration under the canonical ensemble µ can ̺,Λ by assumption:
J 0 → J 0 j=1 ϕ j , J 0 ∈ 1, J . (3.65)
The result now follows from the Azuma martingale inequality and a union bound over J 0 ∈ 1, J .
Φ J • = − J j=1 ϕ j , Φ N J ,ǫ • = Φ J 1 |Φ J | N ǫ J − 1 2 (3.66)
There exists some universal constant κ ∈ >0 such that for any ǫ ∈ >0 , we have
E µ can ̺,Λ exp J 1 2 Φ N J ,ǫ 1 + exp −κN 2ǫ . (3.67)
Proof. Applying Lemma 3.18, the desired estimate follows from a straightforward computation involving Gaussian probability densities.
ANALYTIC COMPATIFICATION
The present section provides an analytic strategy to replace the integral equation (2.12) on the infinite-volume lattice by the solution to a periodized integral equation on a compact torus of divergent size. To make this precise, we introduce
notation. In what follows, ǫ 0 ∈ >0 is an arbitrarily small but universal constant. +ǫ 0 ⊆ . Further, for any X ∈ , denote byX ∈ N the reduction of X ∈ modulo this torus. Lastly, for any functional of the particle system, denoted by
f X = τ X f for example, we define f p X = fX . We define Z N ,p
T,X as the unique solution to the following stochastic differential equation:
dZ N ,p T,X = L N Z N ,p T,X + Z N ,p T,X dQ N ,p T,X + N 1 2 Av β X ,p g N T,X Z N ,p T,X dt + N β X g N ,p T,X Z N ,p T,X dt + w N ,p T,X Z N ,p T,X dt + ∞ k=1 c N k D k w N ,k,p T,X Z N ,p T,X dt, (4.1) with initial data Z N ,p 0,X • = Z N 0,X ; we remark dQ N ,p T,X = dQ N ,p
T,X for total transparency. Moreover, recall β X = 1 3 + ǫ 1 with ǫ 1 ∈ >0 arbitrarily small but universal, and {c N k } ∞ k=1 are the coefficients for the gradient quantity appearing in (2.12).
Similarly, we have the integral representation
Z N ,p T,X = U N 0,T • Z N ,p 0, y + T 0 U N s,T • Z N s, y dQ N ,p s, y + T 0 U N s,T • N 1 2 Av β X ,p g N s, y Z N ,p s, y ds + T 0 U N s,T • N β X g N ,p s, y Z N ,p s, y ds (4.2) + T 0 U N s,T • w N ,p s, y Z N ,p s, y ds + ∞ k=1 c N k T 0 D −k U N s,T • w N ,k,p s, y Z N ,p s, y ds.
The primary result for the current section is a comparison between the honest Gartner transform Z N T,X and its compactification Z N ,p T,X on compact domains.
e −κ,X E Z N T,X − Z N ,p T,X 2 T e −κ 0 N 1 2 . (4.3)
Moreover, consider any constantǭ 0 ∈ (0, ǫ 0 ). The following estimate holds uniformly over X ∈ −N 5 4 +ǭ 0 , N 5 4 +ǭ 0 with universal constants κ 1 , κ 2 ∈ >0 :
P sup T ∈[0,T f ] e −κ,X Z N T,X − Z N ,p T,X T f e −κ 1 N 1 2 T f e −κ 2 N 1 2 . (4.4)
In particular, we obtain the following straightforward consequence and reduction for proving Theorem 1.6, which we state rather loosely. Moreover, for the remainder of the article after this section, we will simply denote Z N ,p T,X by Z N T,X . In particular, after this section, the topologies L ∞ T,X (κ; δ) are taken with respect to N as the spatial domain rather than the infinite-volume lattice . We reemphasize this at the beginning of each section to follow for the remainder of this article for total clarity. Proof. Upon iteration, it suffices to prove that for any T 1 ∈ >0 and T 0 ∼ N − 1 2 , we have instead the estimate
sup (T,X )∈[T 1 ,T 1 +T 0 ]× e −κ,X Z N T,X L 2p ω κ,T f sup X ∈ e −κ,X Z N T 1 ,X L 2p ω . (4.6)
The constant κ 0 ∈ >0 would therefore depend only on κ, T f ∈ >0 . The above estimate, however, follows from exactly the considerations from Proposition 3.2 in [8].
We now briefly explain the content of Proposition 4.2. The heat kernel underlying the semigroup U N s,T admits Gaussian and Poisson estimates, recorded shortly in the following subsection. In particular, the validity of Proposition 4.2 remains if we replace N by any compact sub-lattice containing the origin and whose boundary is some distance from the origin sufficiently large so that the heat kernel estimate effectively "counters" the moment bound obtained in Lemma 4.4 above.
Establishing Proposition 4.2 now consists entirely of precisely executing the idea presented in this paragraph 22 4.2. Heat Kernel Estimates. Our first preliminary heat kernel estimate is the following two-sided estimates in the Gaussian and Poisson regimes, respectively. Proposition 4.5. Suppose T |X |. Then for some universal constants κ ± ∈ >0 , we have
T − 1 2 exp − |X | 2 κ − T U N 0,T (0, X ) T − 1 2 exp − |X | 2 κ + T , (4.7)
where the implied constants are universal.
Suppose now that T |X |. Then for some universal constants κ ± , we have
exp −κ − |X | − |X | log |X | κ − T U N 0,T (0, X ) exp −|X | log |X | κ + T . (4.8)
Proof. See Theorem 5.17 and Theorem 5.25 in [2]; the proofs still apply even with long-range random walks because, for
example, we have α N 1 1 uniformly in N ∈ >0 and {α N k } N ǫ c k=1 have all moments bounded uniformly in N ∈ >0 .
Remark 4.6. Alternatively, the above two-sided estimates hold upon replacing T N ǫ c T , which is, at least potentially, a more straightforward consequence of Theorem 5.17 and Theorem 5.25 in [2]. Indeed, for ǫ c ∈ >0 sufficiently small but still universal, such an estimate would be sufficient for our proof of Proposition 4.2.
Second, in light of the gradient quantities on (2.12) and (4.2), respectively, we require some pointwise estimate for the gradient of the heat kernel. To state this upper bound, it will serve convenient to establish notation for some space-time dilated variation of the heat kernel.
Notation 4.7. For κ P ∈ >0 sufficiently small and κ G ∈ >0 sufficiently large but both universal, we define Before we present the aforementioned pointwise gradient estimate, we first make the following important observation concerning a semigroup-type property for the space-time dilated heat kernels. Lemma 4.9. Consider any L ∈ >0 ; for any sequence of times T 1 < T 2 < . . . < T L ∈ >0 , and any spatial points X , Y ∈ , we have
U N 0,T (X , Y ) • = U N 0,κ P T ( 1 2 X , 1 2 Y ) |X − Y | T, U N 0,κ G T ( 1 2 X , 1 2 Y ) |X − Y | T ;U N 0,T 1 • U N T 1 ,T 2 • . . . • U N T L−1 ,T L (X , Y ) κ L CK,1 U N 0,κ CK,2 T L (X , Y ); (4.10)
above the implied constant is universal, and κ CK,1 , κ CK,2 ∈ >0 are universal as well.
Proof. The estimate follows immediately from the Chapman-Kolmogorov equation for the original heat kernel U N s,T (X , Y ); the inequality follows from possible over-counting within the summation and the estimate U N 0,T (0, X ) U N 0,2T (0, X ± 1) for a universal implied constant, which is direct consequence of the two-sided estimates in Proposition 4.5.
D k U N 0,T (0, X ) T − 1 2 |ℓ| |k| U N 0,T (0, X + ℓ) (4.11)
with a universal implied constant. Moreover, for a universal constant κ 0 ∈ >0 , we have
U N 0,T (0, 1 2 X ) U N 0,T (0, X ) U N 0,κ 0 T (0, 1 2 X );(4.12)
23 the implied constants are both universal.
Proof. The non-gradient estimates follow from a direct computation via the two-sided bounds in Proposition 4.5.
Upon induction in |k| ∈ >0 and the following telescoping identity, we may assume k = ±1: (4.13) where the signs are specified to mutually agree; in particular, there are only two allowed choices of sign above. Moreover, upon symmetry of the following analysis, we assume k = 1.
D ±k ϕ(x) = k−1 ℓ=0 D ±1 ϕ(x + ℓ),
Further, without loss of generality, assume X < Y . Employing the Chapman-Kolmogorov equation for the unperturbed heat kernel U N 0,T (X , Y ), we see To estimate the gradient of the first quantity on the RHS of (4.15), we apply a lattice-type integration-by-parts to obtain w∈ :
U N 0,T (X , Y ) = w∈ U N 0, 1 2 T (X , w) · U N 1 2 T,T (w, Y ) (4.14) = w∈ : |w−X | 1 2 |X −Y | U N 0, 1 2 T (X , w) · U N 1 2 T,T (w, Y ) + w∈ : |w−X |< 1 2 |X −Y | U N 0, 1 2 T (X , w) · U N 1 2 T,T (w, Y ).|w−X | 1 2 |X −Y | D 1 U N 0, 1 2 T (X , w) · U N 1 2 T,T (w, Y ) w∈ : |w−X | 1 2 |X −Y | U N 0, 1 2 T (X , w) · D −1 U N 1 2 T,T (w, Y ) (4.16) + U N 0, 1 2 T (0, 1 2 |X − Y |) · U N 1 2 T,T (w, Y ) + U N 0, 1 2 T (0, − 1 2 |X − Y |) · U N 1 2 T,T (w, Y )
. We provide a bound for each quantity on the RHS of (4.16):
• The first quantity admits a straightforward bound obtained via the on-diagonal gradient estimate from Proposition A.1 in [8] and a straightforward summation employing the two-sided estimates in Proposition 4.5:
w∈ : |w−X | 1 2 |X −Y | U N 0, 1 2 T (X , w) · D −1 U N 1 2 T,T (w, Y ) T −1 w∈ : |w−X | 1 2 |X −Y | U N 0, 1 2 T (X , w) (4.17) T − 1 2 U N 0,T (X , Y ),(4.18)
assuming the constants κ P , κ G ∈ >0 are chosen suitably. • The second and third quantities admit the same upper bound from the on-diagonal estimate from Proposition A.1 in [8] and the non-gradient estimates within the proposition at hand:
U N 0, 1 2 T (0, 1 2 |X − Y |) · U N 1 2 T,T (w, Y ) + U N 0, 1 2 T (0, − 1 2 |X − Y |) · U N 1 2 T,T (w, Y ) T − 1 2 U N 0,T (X , Y ). (4.19)
This provides a final estimate of w∈ :
|w−X | 1 2 |X −Y | D 1 U N 0, 1 2 T (X , w) · U N 1 2 T,T (w, Y ) T − 1 2 U N 0,T (X , Y ).(4.20)
To estimate the second quantity within the RHS of (4.15), we employ the methodology within the first bullet point above;
this completes the proof.
U N 0,T • Z N T 0 +τ, y 2 L 2 ω |k| N 1 2 +ǫ+ǫ c a k U N 0,T +κτ • Z N T 0 , y 2 L 2 ω + e −κN 1 2 +ǫ 0 , (4.21)
where κ ∈ >0 and the implied constant are both universal.
Proof. Employing directly the integral equations (2.12) and (4.2) and applying the Ito isometry for the stochastic integraltype quantity, for |X | N 5 4 + 1 2 ǫ 0 , we have the following estimate consequence of the non-gradient estimate in Proposition 4.10; observe that the second moment of the stochastic-integral type quantity is actually the second moment of an integral, rather than simply a summation over jump-times of the underlying Poisson clocks:
U N 0,T • Z N T 0 +τ, y 2 L 2 ω U N 0,T +τ • Z N T 0 , y 2 L 2 ω + τ 0 ̺ − 1 2 s,τ U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds (4.22) + τ 0 ̺ − 1 2 s,τ U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds + N τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds + N τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds + N 2β X τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds + N 2β X τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds + τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds + τ τ 0 U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds + τ 1 2 N ǫ c k=1 |c k | · U N 0,T • τ 0 D −k U N s,τ • Z N T 0 +s, y 2 L 2 ω ds + τ 1 2 N ǫ c k=1 |c k | · U N 0,T • τ 0 D −k U N s,τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds.
For the final quantity on the RHS of (4.22), we apply both Proposition 4.10 and afterwards Lemma 4.9. Analytically, this transforms the gradient of the heat kernel into a space-time dilated and spatially shifted heat kernel with a time-dependent factor, allowing us to iterate (4.22). In particular, we have
τ 1 2 N ǫ c k=1 |c k | · U N 0,T • τ 0 D −k U N s,τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds N ǫ c τ 1 2 − N ǫ c k=1 τ 0 ̺ − 1 2 s,τ U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds. (4.23)
Analogously, for the second-to-last quantity on the RHS of (4.22), we convert the gradient into a space-time dilated heat kernel and obtain the upper bound
τ 1 2 N ǫ c k=1 |c k | · U N 0,T • τ 0 D −k U N s,τ • Z N T 0 +s, y 2 L 2 ω ds N ǫ c τ 1 2 τ 0 ̺ − 1 2 s,τ − |k| N ǫ c τ k U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds. (4.24) 25
Thus, upon another application of Proposition 4.10, we obtain the following final estimate for |X | N 5 4 + 1
2 ǫ 0 after defining µ N s,τ = N ǫ c ̺ − 1 2 s,τ + N ǫ c N τ: U N 0,T • Z N T 0 +τ, y 2 L 2 ω U N 0,T +τ • Z N T 0 , y 2 L 2 ω + τ 0 µ N s,τ − |k| N ǫ c τ k U N s,T +τ • Z N T 0 +s, y 2 L 2 ω ds (4.25) + τ 0 µ N s,τ − |k| N ǫ c τ k U N s,T +τ • Z N T 0 +s, y 2 L 2 ω 1 C N ds.
We apply N 1 2 +ǫ -many iterations of (4.25). Collecting the resulting quantities, for universal constants C 0 , κ ∈ >0 and for σ N = ⌊N 1 2 +ǫ ⌋, we obtain
U N 0,T • Z N T 0 +τ, y 2 L 2 ω U N 0,T +τ • Z N T 0 , y 2 L 2 ω + N ǫ c τ 0 µ N s,τ − |k| N ǫ c τ k U N s,T +τ • U N 0,s • Z N T 0 +τ, y 2 L 2 ω ds (4.26) + N ǫ c e κN 1 2 τ 0 µ N s,τ − |k| N ǫ c U N s,T +τ • 1 C N ds + σ N j=1 N jǫ c C j 0 I N τ, j + e κN 1 2 σ N j=1 N jǫ c J N τ, j + e κN 1 2 τ 0 . . . s σ N +1 0 σ N +1 ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ ds σ N +2 . . . ds 1 , U N 0,T +τ • Z N T 0 , y 2 L 2 ω + N ǫ c τ 0 µ N s,τ − |k| N ǫ c τ k U N s,T +τ • U N 0,s • Z N T 0 +τ, y 2 L 2 ω ds (4.27) + N ǫ c e κN 1 2 τ 0 µ N s,τ − |k| N ǫ c U N s,T +τ • 1 C N ds + σ N j=1 N jǫ c C j 0 I N τ, j + e κN 1 2 σ N j=1 N jǫ c J N τ, j + e κN 1 2 N −σ N ǫ MS , where I N j,τ • = τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c U N s 1 ,T +τ • U N s 2 ,s 1 . . . • U N s j+1 ,s j • U N 0,s j+1 • Z N T 0 , y 2 L 2 ω ds j+1 . . . ds 1 , (4.28) J N j,τ • = τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c U N s 1 ,T +τ • U N s 2 ,s 1 . . . • U N s j+1 ,s j • 1 C N ds j+1 . . . ds 1 . (4.29)
We first address the quantities of type (4.29). Applying the Chapman-Kolmogorov-type estimate in Lemma 4.9, for some universal constant κ CK ∈ >0 depending only on κ CK,2 ∈ >0 , we see
J N j,τ κ j CK,1 τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ U N 0,T • − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c τ k 1 +...+k j+1 U N κ CK,2 s j+1 ,κ CK,2 τ • 1 T C N ds j+1 . . . ds 1 (4.30) κ j+1 CK,1 τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c τ k 1 +...+k j+1 U N 0,κ CK [T +τ−s] • 1 C N ds j+1 . . . ds 1 ,(4.31)
with the implied constant being universal.
Recall |X | N 5 4 + 1 2 ǫ 0 . This assumption implies that the heat flow operator within this last estimate is actually a summation supported on points in the infinite-volume lattice that are distance at least N 5 4 +ǫ 0 ∈ >0 from X ∈ . Employing the 26 heat-kernel estimate in Proposition 4.5, we see
κ j+1 CK,1 τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c τ k 1 +...+k j+1 U N 0,κ CK [T +τ−s] • 1 C N ds j+1 . . . ds 1 e −N 1 2 +ǫ 0 κ j+1 CK,1 τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c ds j+1 . . . ds 1 (4.32) e −N 1 2 +ǫ 0 κ j+1 CK,1 N − jǫ MS ; (4.33)
above, the implied constants are both universal, and the final estimate is consequence of a straightforward integral calculation identical to that used to obtain (4.27).
Lastly, we address the quantities of type (4.28). To this end, we first proceed in similar fashion as we did for quantities of type (4.29), namely by employing Lemma 4.9; this provides
I N j,τ κ j CK,1 U N 0,T • τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c τ k 1 +...+k j+1 U N 0,κ CK,2 τ • Z N T 0 , y 2 L 2 ω ds j+1 . . . ds 1 . (4.34)
Observe that for τ = N − 1 2 −ǫ MS with ǫ MS ∈ >0 arbitrarily small but universal, for a universal constant κ ∈ >0 we have
τ k 1 +...+k j+1 U N 0,κ CK,2 τ (X , Y ) τ k 1 +...+k j+1 U N 0,κτ (X , Y ) + e − 1 2 |X −Y | 1 |X − Y | N 3 2 −ǫ MS .+ e κ ′ N 1 2 |X −Y | N 3 2 −ǫ MS e − 1 2 |X −Y | · κ j CK,1 τ 0 . . . s j 0 j ℓ=1 µ N s ℓ+1 ,s ℓ · µ N s 1 ,τ ds j+1 . . . ds 1 κ j CK,1 N − jǫ MS · − |k 1 | N ǫ c . . . − |k j+1 | N ǫ c τ k 1 +...+k j+1 U N 0,T +κτ • Z N T 0 , y 2 L 2 ω + κ j CK,1 N − jǫ MS e −N . (4.37)
Finally, a similar calculation shows
U N T +τ • Z N T 0 , y 2 L 2 ω U N T +κτ • Z N T 0 , y 2 L 2 ω + e −N (4.38)
and further Proof of Proposition 4.2. We consider κ = 0 for simplicity, because the following analysis remains valid upon additional κ-dependent factors. In particular, our a priori moment estimates for the initial condition will hold for κ = 0.
N ǫ c τ 0 µ N s,τ − |k| N ǫ c τ k U N s,T +τ • U N 0,s • Z N T 0 +τ, y 2 L 2 ω ds − |k| N ǫ c τ k U N T +κτ • Z N T 0 , y 2 L 2 ω + e −N , (4.39) N ǫ c e κN 1 2 τ 0 µ N s,τ − |k| N ǫ c U N s,T +τ • 1 C N ds e −N
We begin by proving the moment estimate (4.3). To this end, upon inductively applying Lemma 4.11, we obtain the following estimate for a sequence of coefficients { a k } ∞ k=1 with all moments uniformly bounded in N ∈ >0 and for a pair 27 of universal constants κ 1 , κ 2 ∈ >0 :
Z N T,X 2 L 2 ω C N 1 2 +ǫ MS 1 · |k| N 1+ǫ MS +ǫ c a k τ k U N 0,κ 1 T • Z N 0, y 2 L 2 ω + C N 1 2 +ǫ MS 1 e −κ 2 N 1 2 +ǫ 0 . (4.41)
Choosing ǫ MS < 1 2 ǫ 0 , for example, provides a suitable bound for the second quantity on the RHS of (4.41). As for the first quantity on the RHS of (4.41), we first observe that the summation is over lengths bounded above by N 1+ǫ MS +ǫ c for ǫ MS , ǫ c ∈ >0 arbitrarily small but universal. We further split the analysis of the first quantity on the RHS of (4.41) into the situations of near-stationary and narrow-wedge initial data, respectively and in this order.
• By assumption, the first quantity on the RHS of (4.41) is a summation outside the ball of radius N 1+ǫ MS +ǫ c around X ∈ . Uniformly over |X | N 5 4 + 1 2 ǫ 0 , for example, because the initial data have uniformly bounded moments, we have the following estimate courtesy of the heat kernel estimates in Proposition 4.5:
C N 1 2 +ǫ MS 1 · |k| N 1+ǫ MS +ǫ c a k τ k U N 0,κ 1 T • Z N 0, y 2 L 2 ω C N 1 2 +ǫ MS 1 Y ∈ : |Y | N 5 4 + 1 2 ǫ 0 U N 0,κ 1 T (0, Y ) (4.42) C N 1 2 +ǫ MS 1 e −N 1 2 + 1 2 ǫ 0 , (4.43)
from which we may choose ǫ MS < 1 2 ǫ 0 again to obtain the desired estimate in (4.3). • Concerning narrow-wedge initial data, we follow the proof of Lemma 5.1 in [8] with the observation that Z N 0,X is approximately a Dirac point mass supported on nonzero integer multiples of N 5 4 +ǫ 0 ∈ >0 . This provides For D ′ ∈ >0 sufficiently large and ǫ ∈ >0 small, via large-deviations for the Poisson distribution, we have P[Q] e −κN ǫ with κ ∈ >0 universal and ǫ ∈ >0 depending only on ǫ ∈ >0 . Thus, the pathwise probability estimate (4.4) now follows from the probability estimate on Q and taking the union bound over the N D -many events for which (4.3) holds.
C N 1 2 +ǫ MS 1 · |k| N 1+ǫ MS +ǫ c a k τ k U N 0,κ 1 T • Z N 0, y 2 L 2 ω N C N 1 2 +ǫ MS 1 |k| N 1+ǫ MS +ǫ c a k ℓ∈ \{0} U N 0,κ 1 T (X + k, ℓN 5 4 +ǫ 0 ) (4.44) N C N 1 2 +ǫ MS 1 e −N
DYNAMICAL ONE-BLOCK ANALYSIS
We reemphasize that although the particle system evolves on the infinite-volume lattice , the Gartner transform evolves on the compact torus N = −N Before we proceed, we declare the following two assumptions:
• Within this section, any constant ǫ ∈ >0 labeled as "arbitrarily small but universal", with or without subscript or superscript, is allowed to change between different lines an N -independent number of times.
• All particle dynamics within the current section are assumed to allow a maximal jump-length of at most N ǫ c ∈ >0 . This will not affect our results given the moment estimates in Assumption 1.1.
Because this section may be of independent interest, the discussion is held in slight generality. 28 5.1. Dynamical Analysis Ia. The primary objective of the current subsection is establishing one of two crucial dynamical estimates for mesoscopic space-time averages of pseudo-gradient fields.
We first establish the following notation for mesoscopic space-time averages of pseudo-gradient fields with and without a cutoff; this notation will be employed throughout the article. Notation 5.1. Fix some universal constant β X ∈ >0 and an arbitrarily small but universal constant ǫ X ∈ >0 . For any pseudo-gradient field g N T,X , we define the following mesoscopic spatial average and its cutoff:
Av β X g N T,X • = − N β X w=1 τ −w g N T,X , Av β X g N T,X • = Av β X g N T,X 1 Av β X g N T,X N − 1 2 β X +ǫ X . (5.1)
Fix any possibly N -dependent time-scale τ ∈ >0 , and any universal constants β + , β − ∈ >0 . We define the temporal average and its cutoff with any t N ∈ >0 satisfying |t N | N ǫ τ for any constant ǫ ∈ >0 arbitrarily small but universal:
A β X g N T,X (τ) • = − τ 0 τ r Av β X g N T,X dr , A β X g N T,X (τ; β + , β − ) • = A β X g N T,X 1 A β X g N T,X (τ) N −β − 1 τ t N A β X g N T,X (τ) N −β + . (5.2)
Remark 5.2. Briefly, the cutoff incorporated into the mesoscopic spatial average is a natural cutoff coming from martingale LDP-type estimates that pseudo-gradient fields exhibit. The cutoff incorporated into the last temporal average is not natural in the same sense, but arises from a multiscale analysis required in the proof of Theorem 1.6.
Moreover, we will always omit the parameter t N ∈ >0 within our notation for the temporal cut-off since our estimates will be uniform in t N ∈ >0 .
We now present the primary estimate for this subsection and main ingredient towards the proof of Theorem 1.6 consequence of our dynamical analysis.
Proposition 5.3. Suppose g N
T,X is a pseudo-gradient field, and then fix universal constants β X , β + , β − ∈ >0 , and let τ ∈ >0 denote an N -dependent time-scale to be specified. Suppose that all of the following conditions are satisfied:
• We have β X = 1 3 + ǫ 1 for ǫ 1 ∈ >0 an arbitrarily small but universal constant.
• We have β + = [ 1 2 β − + β X − ǫ 2 ] ∧ [ 3 2 β − + β X − 1 3 − ǫ 2 ]
for an arbitrarily small but universal constant ǫ 2 ∈ >0 . • We have τ = N −1−β X − 1 2 β − +β + +ǫ 3 for some arbitrarily small but universal constant ǫ 3 ∈ >0 .
Then for some universal constant β u ∈ >0 , we have, with universal implied constant,
E T 0 U N s,T • e κ, y N 1 2 A β X g N s, y (τ; β + , β − ) ds L ∞ T,X (κ;0) N −β u . (5.3)
As an immediate consequence of Proposition 5.3, we obtain the following multiscale analog which provides the primary utility of the current subsection towards the proof of Theorem 1.6. • We have β 0 = 1 2 β X − ǫ X for ǫ X ∈ >0 the arbitrarily small but universal constant from Notation 5.1.
• For m ∈ 0, M − 1 , we have β m+1 = [ 1 2 β m + β X − ǫ m ] ∧ [β m + ǫ 2 ]
for two arbitrarily small but universal constants ǫ m , ǫ 2 ∈ >0 .
• For each m ∈ 1, M , define τ N } ∞ m=1 remains increasing; for the first bullet point to hold, we must have β m < 1 3 + ǫ ′ m for ǫ ′ m ∈ >0 some arbitrarily small but universal constant, in which case τ (m) N N −1−ǫ ′′ m for ǫ ′′ m ∈ >0 another arbitrarily small but universal constant, so we choose ǫ m+1 ∈ >0 sufficiently small for the next parameter β m+1 ∈ >0 depending on this last ǫ ′′ m ∈ >0 . To complete the proof, it remains to prove that β M > 1 2 for M 1 with a universal implied constant. To this end, we make the following two observations:
• The second constraint within the minimum defining β m+1 indicates that, because β X > 1 3 , we may take β m+1 − β m to be some universal positive constant. • The first constraint within the minimum defining β m+1 does not yield the same indication. However, it yields that we may choose {β m } ∞ m=1 strictly increasing with the limit given by a geometric series; in particular, we have
β ∞ = β X 1 1 − 1 2 (5.5) = 2β X ,(5.6)
which, given β X > 1 3 , is strictly larger than 1 2 . In particular, for a universal constant M ∈ >0 , we have β M > 1 2 . This completes the proof.
The first ingredient towards establishing Proposition 5.3 is some dynamical replacement-type lemma. Roughly speaking, the statement asserts that the expectation of any bounded functional of the path-space that depends only those occupation variables along any given trajectory associated to sites within some sub-lattice Λ ⊆ is approximately equal to the expectation of the same functional, but replacing the original exclusion process with a periodic exclusion process on some neighborhood of Λ ⊆ sufficiently large depending only on the terminal time for the process.
We now make the aforementioned replacement precise. Suppose Λ ⊆ is any connected sub-lattice. For any τ ∈ 0 and κ 1 ∈ >0 , we let Λ N ,τ ⊇ Λ be the neighborhood
Λ N ,τ · = Λ + −Σ N , Σ N , Σ N · = N τ 1 2 log κ 1 N + N 3 2 τ log κ 1 N . (5.7)
Lastly, we define s → η N ,per Lemma 5.5. Fix any n ∈ >0 , and suppose F : Ω n Λ → is any bounded functional. For any κ 2 ∈ >0 , there exists κ 1 ∈ >0 sufficiently large depending only on κ 2 ∈ >0 such that for any t 1 , . . . , t n ∈ [0, τ], where the probability of a discrepancy appearing in Λ ⊆ N by time τ ∈ >0 is exponentially small on the desired scale.
E η N 0 F Π Λ η N • − E Π Λ N,τ η N 0 F Π Λ η N ,per • L ∞ η N 0 F L ∞ e − log κ 2 N , (5.8) where Π Λ η N • • = Π Λ η N t j n j=1 , Π Λ η N ,per • • = Π Λ η N ,per t j n j=1 ,(5.
To this end, we introduce the following coupling from the proof of Lemma 3.6. Provided the current section is possibly of independent interest outside proving Theorem 1.6, we repeat the coupling below.
• Upon realizing the symmetric long-range exclusion process as attaching a Poisson clock to each bond connecting two sites, and the step in the process corresponding to a ringing of this Poisson clock as swapping the occupation variable values at these two sites, for any bonds shared between the global dynamic and the local periodic dynamic we couple the bond-clocks together. We emphasize that this is not the basic coupling. • The remaining Poisson clocks associated to the lower-order totally asymmetric long-range exclusion process are then equipped with the basic coupling, that any of these remaining clocks, if are shared between the two dynamics, are coupled together. Equivalently, any particles that occupy any site shared between the two domains and Λ N ,τ jump together whenever possible.
We make two further observations.
(1) There are not discrepancies between the two initial particle configurations on Λ N ,τ ⊆ by assumption, since the expectations appearing in the statement of this lemma condition on identical initial configurations. Moreover, the coupling constructed above cannot create discrepancies. In particular, any discrepancies that appear in Λ N ,τ ⊆ must be consequence of some interaction that is present in one dynamic and absent in the other. These interactions are those coming from the periodic boundary condition in the process s → η N ,per s or from the interaction of sites within Λ N ,τ ⊆ to sites outside this sub-lattice in the process s → η N s . Ultimately, discrepancies must travel the distance of Σ N ∈ >0 in order to reach Λ ⊆ .
(2) Under the above semi-basic coupling, the dynamics of any tagged trajectory are a free and unsuppressed symmetric long-range random walk plus a randomly suppressed and randomly killed totally asymmetric long-range random walk, where the randomness is a function of the environment of particles. The latter totally asymmetric component occurs with a dampened speed of scale N 3 2 ∈ >0 .
The claimed estimate is thus straightforward consequence of the Azuma-Hoeffding martingale inequality, which estimates from above the maximal distance for any individual tagged discrepancy, combined with the union bound over all tagged discrepancies. As a discrepancy is created only through the ringing of any Poisson clock, the total number of discrepancies is at most N D ∈ >0 with exponentially high probability, where D ∈ >0 is some large but universal constant. Via Lemma 5.5 and a straightforward calculation, we obtain the following result; to state it, we first introduce localized and periodic variants of the mesoscopic space-time averages that were introduced in the beginning of this subsection. Notation 5.6. Fix any possibly N -dependent time-scale τ ∈ >0 , and any pair of universal constants β + , β − ∈ >0 . We define the localized and periodic temporal average and its cutoff:
A β X ,per g N T,X (τ) • = − τ 0 τ r Av β X ,per g N T,X dr , A β X ,per g N T,X (τ; β + , β − ) • = A β X ,per g N T,X 1 N −β + A β X ,per g N T,X (τ) N −β − , (5.10) where τ r Av β X ,per g N T,X • = − N β X w=1 τ −w g N ,per
T +r,X , τ r Av (5.11) and g N ,per
β X ,per g N T,X • = τ r Av β X ,per g N T,X 1 τ r Av β X ,per g N T,X N − 1 2 β X +ǫ , r ∈ [0, τ],
T +r,X denotes the value of the pseudo-gradient field g N when evaluated at the particle configuration obtained by evolving the particle configuration
Π Λ N,τ η N T under the periodic dynamic of generator L ! T Λ N,τ ,per for time r ∈ [0, τ].
Corollary 5.7. For any κ 2 ∈ >0 , there exists some κ 1 ∈ >0 sufficiently large depending only on κ 2 ∈ >0 such that for any
(s, y) ∈ [0, T f ] × N , we have E η N s A β X g N s, y (τ) E Π y+Λ N,τ η N s A β X ,per g N s, y (τ) + e − log κ 2 N , (5.12) E η N s A β X g N s, y (τ; β + , β − ) E Π y+Λ N,τ η N s A β X ,per g N s, y (τ; β + , β − ) + e − log κ 2 N , (5.13)
where κ 1 ∈ >0 is the parameter defining Σ N and the periodic dynamic s → η N ,per s .
Proof. Follows immediately from Lemma 5.5 and the uniform bound g L ∞ 1.
Proof of Proposition 5.3. We demonstrate the analysis for κ = 0; for general κ ∈ >0 the following estimates remain valid up to universal prefactors depending only on κ ∈ >0 .
Defining T N = T − N − 1 2 +β − −ǫ for ǫ ∈ >0 arbitrarily small but universal, we write T 0 U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds = T N 0 U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds + T T N U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds. (5.14)
Proceeding in reverse order, the second quantity on the RHS of (5.14) is straightforward to estimate; because the operator U N s,T is a contraction in L ∞ ( N ), due to the a priori estimate built into the mesoscopic space-time average we observe the bound
T T N U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds N −ǫ . (5.15)
Proceeding with the first quantity on the RHS of (5.14), we employ the on-diagonal estimate for the heat kernel underlying the semigroup operator U N s,T , which we cite from Proposition A.1 in [8], to obtain (5.18) where c N ,β − = N 1+ǫ− 1 2 β − . Thus, it remains to take then analyze the expectation of the integral on the RHS. The first step is invoking Corollary 5.7, which gives
T N 0 U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds N β T N 0 ̺ − 1 2 s,T − y∈ N N 1 2 A β X g N s, y (τ; β + , β − ) ds (5.16) N 1 4 +β + 1 2 ǫ− 1 2 β − T N 0 − y∈ N N 1 2 A β X g N s, y (τ; β + , β − ) ds.T 0 U N s,T * N 1 2 A β X g N s, y (τ; β + , β − ) ds L ∞ T,X N −ǫ + c N ,β ,β − T f 0 − y∈ N A β X g N s, y (τ; β + , β − ) ds,E c N ,β − T f 0 − y∈ N A β X g N s, y (τ; β + , β − ) ds = c N ,β − T f 0 − y∈ N E η N s E η N s A β X g N s, y (τ; β + , β − ) ds (5.19) c N ,β − T f 0 − y∈ N E η N s E Π y+Λ N,τ η N s A β X ,per g N s, y (τ; β + , β − ) ds + c N ,β − e − log κ 2 N (5.20) E c N ,β − T f 0 − y∈ N E Π y+Λ N,τ η N s A β X ,per g N s, y (τ; β + , β − ) ds + e − 1 2 log κ 2 N (5.21)
for κ 2 ∈ >0 large but universal.
Observe the conditional expectation within the RHS of (5.21) is an expectation with respect to the path-space measure induced by the periodic dynamic with initial condition Π y+Λ N,τ η N s = τ y,s Π Λ N,τ η N 0 . In particular, since the periodic dynamic is invariant under space-time translations, this conditional expectation is the space-time translation of a functional of the 32 particle system. We may now proceed with the usual one-block estimate; by Lemma 3.12 applied to κ = N −β − , we obtain the following upon realizing the time-average and its t N -translate are defined on a common lattice of size N ǫ |Λ N ,τ |:
E c N ,β − T f 0 − y∈ N E Π y+Λ N,τ η N s A β X ,per g N s, y (τ; β + , β − ) ds c N ,β − N −2+3ǫ |Λ N ,τ | 3 N −β − (5.22) + c N ,β − N −β − sup ̺∈[−1,1] log E µ can ̺,Λ exp N β − E Π Λ N,τ η N 0 A β X ,per g N 0,0 (τ; β + , β − ) .
We recall |Λ N ,τ | = N τ 1 2 log κ N + N 3 2 τ log κ N + N β X for some arbitrarily large but universal constant κ ∈ >0 . Combine this with a straightforward computation, and we note that the assumptions within Proposition 5.3 imply the first quantity on the RHS of (5.22) is bounded above by N −β u ∈ >0 for some universal constant β u ∈ >0 .
To estimate the second quantity on the RHS of (5.22), we appeal to the following observations:
• The quantity within the exponential is uniformly bounded in N ∈ >0 by definition. Thus, we have the determin-
istic upper bound exp N β − E Π Λ N,τ η N 0 A β X ,per g N 0,0 (τ; β + , β − ) 1 + κN β − E Π Λ N,τ η N 0 A β X ,per g N 0,0 (τ; β + , β − ) (5.23)
for some universal constant κ ∈ >0 . • As consequence, we have
log E µ can ̺,Λ exp N β − E Π Λ N,τ η N 0 A β X ,per g N 0,0 (τ; β + , β − ) log 1 + N β − E µ can ̺,Λ A β X ,per g N 0,0 (τ; β + , β − ) (5.24) N β − E µ can ̺,Λ A β X ,per g N 0,0 (τ; β + , β − ) . (5.25)
• Third, via the Cauchy-Schwarz inequality and Chebyshev inequality, and further applying Lemma 3.18 accompanied by the uniform deterministic bound on g N T,X , we see
E µ can ̺,Λ A β X ,per g N 0,0 (τ; β + , β − ) E µ can ̺,Λ A β X ,per g N 0,0 (τ; β + , β − ) 2 1 2 P µ can ̺,Λ N −β + A β X g N t N ,0 (τ) 1 2 (5.26) N β + E µ can ̺,Λ A β X ,per g N 0,0 (τ; β + , β − ) 2 · E µ can ̺,Λ A β X ,per g N t N ,0 (τ; β + , β − ) 2 1 2 (5.27) N β + E µ can ̺,Λ − τ 0 τ r Av β X g N 0,0 dr 2 · E µ can ̺,Λ − τ 0 τ r Av β X g N t N ,+ N β + E µ can ̺,Λ Av β X g N 0,0 1 Av β X g N 0,0 N − 1 2 β X +ǫ X 2 N β + E µ can ̺,Λ − τ 0 τ r Av β X g N 0,0 dr 2 · E µ can ̺,Λ − τ 0 τ r Av β X g N t N ,0 dr 2 1 2 + e − 1 2 log κ 2 N , (5.29)
with κ 2 ∈ >0 an arbitrarily large but universal constant.
As a final consequence, because the above estimates are uniform in ̺ ∈ [−1, 1], the second quantity on the RHS of (5.22) admits the following bound as consequence of both Lemma 3.15 and Lemma 3.16:
c N ,β − N −β − sup ̺∈[−1,1] log E µ can ̺,Λ exp N β − E Π Λ N,τ A β X ,per g N 0,0 (τ; β + , β − ) c N ,β − N −2−β X +β + τ −1 + + e − 1 2 log κ 2 N . (5.30)
Again, by the assumptions in the statement of Proposition 5.3, this quantity is bounded above by N −β u ∈ >0 for a universal constant β u ∈ >0 . 33 5.2. Dynamical Analysis Ib. Towards the proof for Theorem 1.6, we require a variation of Proposition 5.3 without the additional input of a mesoscopic spatial-average, though with a smaller N -dependent prefactor.
Proposition 5.8. Suppose g N T,X is a pseudo-gradient field, and then fix universal constants β + , β − ∈ >0 , and let τ ∈ >0 denote an N -dependent time-scale to be specified. Suppose that all of the following conditions are satisfied:
• The pseudo-gradient field has a pseudo-gradient factor contained in a sub-lattice of size at most N ǫ c ∈ >0 .
• We have β + = [ 1 12 + β − − ǫ 2 ] ∧ [ 1 4 + 1 2 β − − ǫ 2 ]
for some arbitrarily small but universal constant ǫ 2 ∈ >0 .
• We have τ = N − 5 4 − 1 2 β − +β + −ǫ 3 for some arbitrarily small but universal constant ǫ 3 ∈ >0 .
Then for some universal constant β u ∈ >0 , we have
E T 0 U N s,T • e κ, y N 1 3 +ǫ 1 A 0 g N s, y (τ; β + , β − ) ds L ∞ T,X (κ;0) N −β u , (5.31)
where the implied constant is universal.
Similar to the previous subsection, from Proposition 5.8 we obtain a multiscale estimate.
• Define β 0 = 0. • For m ∈ 0, M − 1 , we define β m+1 = [ 1 4 + 1 2 β m − ǫ m ] ∧ [β m + ǫ 2 ]
for some arbitrarily small but universal constants ǫ m , ǫ 2 ∈ >0 .
• For each m ∈ 1, M , define τ • We apply Lemma 3.17 in addition to Lemma 3.16.
Dynamical Analysis IIa.
We proceed to establish the second of two aforementioned dynamical estimates. Though both the proof and statement of this upcoming second bound resemble those of Proposition 5.3, we separate these two purely for organizational clarity.
Proposition 5.10. Suppose g N T,X is any pseudo-gradient field, consider by β X a universal constant, and let τ ∈ >0 denote an N -dependent time-scale; moreover, suppose the following conditions hold:
• We have β X = 1 3 + ǫ 1 for some arbitrarily small but universal constant
ǫ 1 ∈ >0 . • We have N −2+2β X −2ǫ X +ǫ 2 τ N −1 for some arbitrarily small but universal constant ǫ 2 ∈ >0 .
For any ǫ ∈ >0 sufficiently small but universal, we have
E T 0 U N s,T • e κ, y N 1 2 A β X g N s, y (τ) ds L ∞ T,X (κ;0) ǫ N 5 24 +ǫ + N − 5 24 +ǫ τ − 1 4 . (5.33)
Again, we obtain the following multiscale variation of Proposition 5.10.
Corollary 5.11. Suppose g N T,X is any generic pseudo-gradient field, and consider β X ∈ >0 from Proposition 5.10. Suppose further that {τ m } M m=0 is a sequence of length M ∈ >0 defined as follows: If β u ∈ >0 is sufficiently small, we have M ǫ,β u 1, and
• Define τ 0 = N −2+2β X −2ǫ XM −1 m=0 τ 1 4 m+1 E T 0 U N s,T • e κ, y N 1 2 A β X g N s, y (τ m ) ds L ∞ T,X (κ;0) ǫ,β u N −β u . (5.34)
Proof. The estimate M ǫ,β u 1 is straightforward via τ Proof of Proposition 5.10. Again, we will assume κ = 0 for convenience; the following analysis remains valid for arbitrary κ ∈ >0 up to additional κ-dependent factors.
Applying the Cauchy-Schwarz inequality with respect to the temporal-integration and afterwards the spatial sum, we
have T 0 U N s,T • N 1 2 A β X g N s, y (τ) ds = T 0 ̺ − 1 4 s,T ̺ 1 4 s,T U N s,T • N 1 2 A β X g N s, y (τ) ds (5.35) T 0 ̺ − 1 2 s,T ds 1 2 T 0 ̺ 1 2 s,T U N s,T • N 1 2 A β X g N s, y (τ) 2 ds 1 2 (5.36) T f T 0 ̺ 1 2 s,T U N s,T • N · A β X g N s, y (τ) 2 ds 1 2 (5.37) T f 0 N 5 4 − y∈ N A β X g N s, y (τ) 2 ds 1 2 ; (5.38)
to establish the final inequality, we employed once again the heat kernel estimate from Proposition A.1 in [8]. Therefore, because the final upper bound (5.38) is uniform in space-time, it remains to estimate its expectation. To this end, applying both the Jensen inequality and Corollary 5.7, we obtain for κ 2 ∈ >0 large but universal; this is consequence of the analogous calculation from the proof of Proposition 5.3. Also similarly, by Lemma 3.12 applied to κ = N −β X +2ǫ X , we have
E T f 0 N 5 4 − y∈ N A β X g N s, y (τ) 2 ds 1 2 E T f 0 N 5 4 − y∈ N A β X g N s, y (τ) 2 ds 1 2 (5.39) E T f 0 N 5 4 − y∈ N E Π y+Λ N,τ η N s A β X g N s, y(τ)E T f 0 N 5 4 − y∈ N E Π y+Λ N,τ η N s A β X g N s, y (τ) 2 ds N 5 4 N −2 |Λ N ,τ | 3 N −β X +2ǫ X + N 5 4 N −β X +2ǫ X sup ̺∈[−1,1] log E µ can ̺,Λ exp N β X −2ǫ X E Π Λ N,τ η N 0 A β X g N 0,0 (τ) 2 . (5.41)
Under the assumptions of the current proposition, the first quantity on the RHS of (5.41) is bounded above by N 5 12 +2ǫ X . To estimate the second quantity on the RHS of (5.41), we follow an identical line of reasoning as employed during the proof of Proposition 5.3. Precisely, because the quantity within the exponential is deterministically bounded by a universal constant, we see The following is the corresponding multiscale analog of Proposition 5.12.
N 5 4 N −β X +2ǫ X sup ̺∈[−1,1] log E µ can ̺,Λ exp N β X −2ǫ X E Π Λ N,τ η N 0 A β X g N s, y (τ) 2 N 5 4 E A β X g N 0,0 2 (5.42) N − 3 4 τ −1 N −β X ,(5.
Corollary 5.13. Suppose g N T,X is any pseudo-gradient field satisfying the pseudo-gradient factor constraint from Proposition 5.8.
Suppose further that {τ m } M m=0 is a sequence of length M ∈ >0 defined as follows:
• Define τ 0 = N − 4 3 −ǫ for an arbitrarily small but unviersal constant ǫ ∈ >0 . • For each m ∈ 0, M − 1 , define τ m = τ m−1 N β u , where β u ∈ >0
is an arbitrarily small but universal constant.
• Define M ∈ >0 to be the least positive integer for which τ M = τ The proofs of Proposition 5.12 and Corollary 5.13 follow from the identical updates made in the proofs of Proposition 5.8 and Corollary 5.9 applied to the proof of Proposition 5.8; we thus omit these analyses. 36 5.5. Static Analysis. The current subsection amounts to a technically precise and quantitative refinement of the classical one-block estimate from [14]. Although it serves as a key preliminary step in establishing the framework for the dynamical one-block estimates above, we present it last within this section because previous articles, including [6], have implemented similar strategies.
While we will retain the notation from the previous subsection on the dynamical one-block strategy, we further establish the following notation whose relevance will come directly from the proof of Theorem 1.6.
Notation 5.14. Consider a universal constant β X ∈ >0 , and let ǫ X ∈ >0 denote the arbitrarily small but universal constant from Notation 5.1. For any pseudo-gradient field g N T,X , we define the following mesoscopic spatial average with LDP-type cutoff:
Av β X g N T,X • = Av β X g N T,X 1 Av β X g N T,X N − 1 2 β X +ǫ X (5.46) = Av β X g N T,X − Av β X g N T,X . (5.47)
The static analysis amounts to the following result.
Proposition 5.15. Suppose g N T,X is a pseudo-gradient field with support of size bounded above by N β 0 ∈ >0 . Suppose further that β X = 1 3 + ǫ 1 as in Proposition 5.3. Then for some universal constant β u ∈ >0 , we have
E T 0 U N s,T • e κ, y N 1 2 Av β X g N s, y ds L ∞ T,X (κ;0) N −β u , (5.48)
where the implied constant is universal.
Proof. Again, we will assume κ = 0 just for convenience; the following analysis holds for general κ ∈ >0 up to additional κ-dependent constants.
Define T N = T −N − 1 2 −ǫ , where ǫ ∈ >0 is arbitrarily small but universal. Similarly following the strategy behind proving Proposition 5.3, we have Because this final estimate is uniform in space-time, we may estimate its expectation. Employing Lemma 3.12, we obtain the following estimate for the first quantity, and only remaining interest quantity, on the RHS of (5.51), with κ = κ N to be determined shortly:
E N T f 0 − y∈ N Av β X g N s, y ds N −1 N 3β X κ −1 N + κ −1 N sup ̺∈[−1,1] log E µ can ̺,Λ exp κ N Av β X g N 0,0 . (5.52)
Employing now Corollary 3.19, for κ N = N 1 2 β X −ǫ X , the second quantity within the RHS of (5.52) is bounded above by e − log 100 N , for example. Thus, we have
E N T f 0 − y∈ N Av β X g N s, y ds N −1+3β X − 1 2 β X +ǫ X + e − log 100 N ,(5.53)
which certainly completes the proof given the assumption β X = 1 3 + ǫ 1 with ǫ 1 ∈ >0 arbitrarily small but universal. 37
LATTICE-SHE ANALYSIS I: PRELIMINARY ESTIMATES
We reemphasize that although the particle system evolves on the infinite-volume lattice , the Gartner transform evolves on the compact torus N = −N 5 4 +ǫ , N 5 4 +ǫ , where ǫ ∈ >0 is an arbitrarily small though universal constant. In particular, the semigroup of operators U N s,T denotes the periodic heat flow on this torus. Moreover, all spatial coordinates are implicitly taken modulo this torus; see Corollary 4.3.
6.1. High-Probability Temporal Regularity. The following estimate is a pathwise estimate for the stochastic integral-type quantity within the integral equation (2.12). Proposition 6.1. Consider any set of arbitrarily small but universal constants ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 , ǫ 5 ∈ >0 , any D ∈ >0 , and any τ ∈ >0 . The following estimates hold with probability at least 1 − κ ǫ 1 ,ǫ 2 ,ǫ 3 ,ǫ 4 ,ǫ 5 ,D N −D :
T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y L ∞ T,X (κ;0) ǫ 1 ,...,ǫ 5 ,D N ǫ 1 τ 1 4 −ǫ 2 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) + N −ǫ 4 τ 1 4 −ǫ 2 + N − 1 2 +ǫ 5 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) , (6.1) T 0 U N s,T • Z N s, y dQ N s, y L ∞ T,X (κ;0) ǫ 1 ,...,ǫ 5 ,D N ǫ 1 T 1 4 −ǫ 2 f Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) + N −ǫ 4 T 1 4 −ǫ 2 f + N − 1 2 +ǫ 5 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) . (6.2)
Both estimates in Proposition 6.1 together provide the most nontrivial ingredient towards proving the following regularity estimate.
Proposition 6.2. Suppose τ N −1 . Consider any set of parameters ǫ 1 , ǫ 2 , ǫ 3 ∈ >0 , any D ∈ >0 . We have
P Z N T +τ,X − Z N T,X L ∞ T,X (κ;0) N ǫ 1 τ 1 4 −ǫ 3 + N ǫ 1 τ 1 4 −ǫ 3 Z N T,X 1+ǫ 2 L ∞ T,X (κ;0) ǫ 1 ,ǫ 2 ,ǫ 3 ,D N −D , (6.3)
where the implied constants within the events on the LHS are universal constants.
The first step towards the proofs of Proposition 6.1 and Proposition 6.2 is a pointwise moment estimate.
Lemma 6.3.
Consider any arbitrarily small but universal constants ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 , ǫ 5 ∈ >0 , any D ∈ >0 , and any τ ∈ >0 ; further consider any arbitrarily large but universal D ∈ >0 . For p ǫ 1 ,...,ǫ 5 ,D 1 sufficiently large, we have
P e N X ,−κ T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y N ǫ 1 τ 1 4 −ǫ 2 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) + N −ǫ 4 τ 1 4 −ǫ 2 + N − 1 2 +ǫ 5 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) ǫ 1 ,ǫ 2 ,ǫ 3 ,ǫ 4 ,ǫ 5 ,D N −D , (6.4) P e N X ,−κ T 0 U N s,T • Z N s, y dQ N s, y N ǫ 1 T 1 4 −ǫ 2 f Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) + N −ǫ 4 T 1 4 −ǫ 2 f + N − 1 2 +ǫ 5 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) ǫ 1 ,ǫ 2 ,ǫ 3 ,ǫ 4 ,ǫ 5 ,D N −D . (6.5)
Proof. As before, we prove the estimates for κ = 0, because the following analysis remains valid for κ ∈ >0 up to inserting κ-dependent factors. Moreover, we prove only the estimate (6.4); the proof of (6.5) follows from similar considerations.
Consider first T N −2 . For ǫ 4 ∈ >0 arbitrarily small but universal, we employ the decomposition
T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y = T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0)
N − ǫ 4 (6.6)
+ T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0) < N − ǫ 4 .
We proceed to analyze each quantity on the RHS of (6.6), for which the following notation will serve convenient:
Z N T,X (υ) • = min υe N κ,X , max −υe N κ,X , Z N T,X , υ ∈ >0 . (6.7) 38
Beginning with the second quantity on the RHS of (6.6), we employ the Chebyshev inequality to obtain
P T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0) < N − ǫ 1 N −ǫ 4 τ 1 4 −ǫ 2 N −2pǫ 4 τ − 1 2 p+2pǫ 2 E T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y 2p · 1 Z N T,X L ∞ T,X (κ;0) < N − ǫ 1 (6.8) N −2pǫ 4 τ − 1 2 p+2pǫ 2 E T 0 U N s,T +τ − U N s,T • Z N s, y (N − ǫ 4 ) · dQ N s, y 2p , (6.9)
where the last bound is obtained via the constraint in the indicator function to rewrite the stochastic integrand, and then dropping this indicator. Via the BDG inequality and heat kernel estimates from Proposition A.1 in [8], for p ∈ >1 we have
N −2pǫ 4 τ − 1 2 p+2pǫ 2 E T 0 U N s,T +τ − U N s,T • Z N s, y (N − ǫ 4 ) · dQ N s, y 2p p, ǫ 2 ,T f N −2pǫ 4 +2p ǫ 4 +4p ǫ 2 τ 2pǫ 2 −2p ǫ 2 . (6.10)
Choosing ǫ 2 = ǫ 2 and ǫ 4 = 1 2 ǫ 4 provides the estimate if p ∈ >1 is sufficiently large. It remains to estimate the first quantity on the RHS of (6.6); we further decompose
T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0) N − ǫ 4 = ∞ ℓ=1 T 0 U N s,T +τ − U N s,T • Z N s, y (ℓ + 1) · dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0)
∈ [ℓ, ℓ + 1) (6.11)
+ T 0 U N s,T +τ − U N s,T • Z N s, y (1) · dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0) ∈ [N − ǫ 4 , 1) .
Applying the Chebyshev inequality for p ∈ >1 , we similarly obtain
P T 0 U N s,T +τ − U N s,T • Z N s, y dQ N s, y · 1 Z N T,X L ∞ T,X (κ;0) N − ǫ 4 N ǫ 1 τ 1 4 −ǫ 2 Z N T,X 1+ǫ 3 L ∞ T,X (κ;0) N −2pǫ 1 +2p ǫ 4 τ − 1 2 p+2pǫ 2 E ∞ ℓ=1 ℓ −1−ǫ 5 T 0 U N s,T +τ − U N s,T • Z N s, y (ℓ + 1) · dQ N s, y 2p (6.12) + N −2pǫ 1 +4p ǫ 4 +2p ǫ 4 ǫ 5 τ − 1 2 p+2pǫ 2 E T 0 U N s,T +τ − U N s,T • Z N s, y (1) · dQ N s, y 2p N −2pǫ 1 +2p ǫ 4 τ − 1 2 p+2pǫ 2 ∞ ℓ=1 ℓ −2pǫ 5 E T 0 U N s,T +τ − U N s,T • ℓ −1 Z N s, y (ℓ + 1) · dQ N s, y 2p (6.13) + N −2pǫ 1 +4p ǫ 4 +2p ǫ 4 ǫ 5 τ − 1 2 p+2pǫ 2 E T 0 U N s,T +τ − U N s,T • Z N s, y (1) · dQ N s, y 2p p, ǫ 2 ,T f N −2pǫ 1 +2p ǫ 4 τ 2pǫ 2 −2p ǫ 2 ∞ ℓ=1 ℓ −2pǫ 5 + N −2pǫ 1 +4p ǫ 4 +2p ǫ 4 ǫ 5 τ 2pǫ 2 −2p ǫ 2 .
(6.14)
For any p ǫ 5 1 sufficiently large, the infinite series converges absolutely uniformly in N ∈ >0 . For any given ǫ 1 ∈ >0 , choosing ǫ 4 , ǫ 5 ∈ >0 sufficiently small and ǫ 2 = 1 2 ǫ 2 , we finally choose any exponent p ∈ >1 sufficiently large. 39 Consider now T N −2 . To this end, an identical analysis succeeds upon the adaptation that under this regime, we may directly bound the quadratic variation of the compound Poisson martingale by the number of jumps, which is bounded by N ǫ 5 ∈ >0 with probability at least 1 − e −κN ǫ 5 for κ ∈ >0 a universal constant and ǫ 5 ∈ >0 depending on ǫ 5 ∈ >0 .
Proof of Proposition 6.1. We obtain the space-time uniform estimate (6.1) from (6.4); the mechanism for obtaining (6.2) from (6.5) is identical. Moreover, as earlier we further assume κ = 0 purely for notational simplicity.
We first claim that it suffices to reduce proving the estimate (6.1) to proving the estimate (6.4) for N D ′ -many points for D ′ ∈ >0 an arbitrarily large but universal constant. Indeed, if this reduction were valid, the estimate (6.1) would follow from combining (6.4) with a straightforward union bound.
To establish validity of the reduction, we proceed as in the proof of Proposition 4.2. Suppose D N is any discretization of [0, T f ] × N of N D ′ -many points such that the distance between any two time coordinates is equal to N −D ′′ ∈ >0 with D ′′ ∈ >0 arbitrarily large but universal. For ǫ 6 ∈ >0 arbitrarily small, consider the event
Q • = (T,X )∈D N sup t∈[0,N −D ′′ ] Q N T +t,X − Q N T,X N ǫ 6 .
For D ′′ ∈ >0 sufficiently large and ǫ 6 ∈ >0 sufficiently small, consequence of large-deviations estimates for the Poisson distribution we know P[Q] e −κN ǫ 6 with κ ∈ >0 universal and ǫ 6 ∈ >0 depending only on ǫ 6 ∈ >0 . Upon taking the intersection of Q with the events in (6.4), this completes the proof of the reduction.
Proof of Proposition 6.2. Appealing to the integral equation (2.12), we have
Z N T +τ,X − Z N T,X U N 0,T +τ − U N 0,T * Z N 0, y + T +τ T U N s,T +τ * Z N s, y dQ N s, y + T 0 U N s,T +τ − U N s,T * Z N s, y dQ N s, y (6.15) + N 1 2 T +τ T U N s,T +τ * Z N s, y ds + N 1 2 T +τ T U N s,T +τ − U N s,T * Z N s, y ds + ∞ k=1 |c k | T +τ T D −k U N s,T +τ * Z N s, y ds + ∞ k=1 |c k | T 0 D −k U N s,T +τ − U N s,T * Z N s, y ds.
Concerning the quantities on the RHS of (6.15), we first record the following deterministic estimates consequence of the heat kernel estimates in Proposition A.1 in [8] and straightforward computations, valid for any ǫ ∈ >0 :
N 1 2 T +τ T U N s,T +τ * Z N s, y ds L ∞ T,X (κ;0) κ N 1 2 τ Z N T,X L ∞ T,X (κ;0) , (6.16) N 1 2 T +τ T U N s,T +τ − U N s,T * Z N s, y ds L ∞ T,X (κ;0) κ,ǫ N 1 2 +2ǫ τ 1−ǫ Z N T,X L ∞ T,X (κ;0) , (6.17) ∞ k=1 |c k | T +τ T D −k U N s,T +τ * Z N s, y ds L ∞ T,X (κ;0) κ τ 1 2 Z N T,X L ∞ T,X (κ;0) , (6.18) ∞ k=1 |c k | T 0 D −k U N s,T +τ − U N s,T * Z N s, y ds L ∞ T,X (κ;0) κ,ǫ τ 1 4 −ǫ Z N T,X L ∞ T,X (κ;0)
. (6.19) We further observe that for τ N −1+ǫ 0 with ǫ 0 ∈ >0 sufficiently small but universal, each upper bound above is bounded above by τ
1 4 −ǫ Z N T,X L ∞
T,X (κ;0) for any ǫ ∈ >0 arbitrarily small but universal. It remains to estimate the first three quantities on the RHS of (6.15). The second and third quantities, from Proposition 6.1 provide the appropriate estimates, so it remains to analyze the first quantity on the RHS of (6.15). However, this is a consequence of the assumed temporal regularity of the Gartner transform. The current section is dedicated towards the primary estimate for the nonlinearities:
G N T,X = T 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds, (7.1) G N T,X = T 0 U N s,T • N β X g N s, y Z N s, y ds. (7.2)
We emphasize the following important features concerning the objects above, some of which were previously mentioned.
• We define β X = 1 3 + ǫ 1 , where ǫ 1 ∈ >0 is an arbitrarily small but universal constant. • The pseudo-gradient field g N T,X has support of size at most N ǫ c ∈ >0 with ǫ c ∈ >0 arbitrarily small but universal. • The pseudo-gradient field g N T,X admits a pseudo-gradient factor whose support is contained in X + −2N ǫ c , 0 ⊆ N . Moreover, throughout the current section we continue to follow the notation established in Section 5.
The primary objective for the current section is the following estimate. Proposition 7.1. For a universal constant β u ∈ >0 and for any arbitrarily small but universal constant ǫ ∈ >0 , we have
P G N T,X L ∞ T,X (κ;0) + G N T,X L ∞ T,X (κ;0) ǫ N −β u Z N T,X L ∞ T,X (κ;0) + N −β u Z N T,X 1+ǫ L ∞ T,X (κ;0) ǫ,ǫ 0 N −β u . (7.3)
Upon employing a union bound and the triangle inequality, it suffices to establish the estimate for both G N T,X and G N T,X individually. Moreover, the analysis for G N T,X follows exactly the same procedure as the analysis for G N T,X with exception to one step present for the latter analysis though absent for the former. Therefore, instead of producing identical analysis, we comment on the adjustments needed for G N T,X in Remark 7.
N −β u Z N T,X L ∞ T,X (κ;0) N −β u . (7.4)
Proof. Pointwise in space-time, we have the straightforward estimate
T 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds − T 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds Z N T,X L ∞ T,X (κ;0) · T 0 U N s,T • e κ, y N 1 2 Av β X g N s, y ds. (7.5)
The result now follows from Proposition 5.15. facilitating the statement and proof of the following estimate in Lemma 7.5, we first establish the following notation.
Notation 7.4. Consider some possibly N -dependent time-scale τ ∈ >0 , and define the temporal average:
A β X g N T,X (τ) • = − τ 0 τ r Av β X g N T,X
dr. (7.6) Moreover, consider any sequence {τ j } ∞ j=0 of possibly N -dependent but positive time-scales, and define
A β X g N T,X (τ; j) • = − τ 1 0 . . . − τ j−1 0 τ r 1 +...r j−1 A β X g N s, y (τ j ) dr j−1 . . . dr 1 . (7.7)
Lastly, we define
ζ (1) m • = T 0 − τ m 0 U N s,T − U N s−r,T dr • N 1 2 A β X g N s, y (τ m−1 ; m − 1) · Z N s, y ds, (7.8) ζ (2) m • = − τ m 0 T 0 U N s−r m ,T • N 1 2 A β X g N s, y (τ m−1 ; m − 1) · Z N s, y − Z N s−r m , y ds dr m (7.9)
We briefly mention that A . For a universal constant β u ∈ >0 , for any ǫ ∈ >0 , we have the following estimate with probability at least 1 − N −β u :
β X g N T,X (τ) = A β X g N T,X (τ) toT 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds − T 0 U N s,T • N 1 2 A β X g N s, y (τ M ; M ) · Z N s, y ds L ∞ T,X (κ;0) ǫ N −β u + N −β u Z N T,X 1+ǫ L ∞ T,X (κ;0)
. (7.10)
Proof. We first consider a temporal replacement at some mesoscopic scale. Precisely, we define τ 0 = N −2+2β X −2ǫ X +ǫ for an arbitrarily small though universal constant ǫ ∈ >0 ; we recall ǫ X ∈ >0 from Notation 5.1 is another arbitrarily small but universal constant.
A straightforward computation gives The second and third quantities on the RHS of (7.11) admit almost-deterministic estimates. In particular, we observe the following two upper bounds, consequence of the temporal regularity estimate for the heat kernel from Proposition A.1 in [8] and the temporal regularity estimate for the Gartner transform from Proposition 6.2, valid for any ǫ ′ ∈ >0 :
T 0 − τ 0 0 U N s,T − U N s−r,T dr • N 1 2 Av β X g N s, y Z N s, y ds L ∞ T,X (κ;0) ǫ ′ N 1 2 − 1 2 β X +2ǫ ′ τ 1−ǫ ′ 0 Z N T,X L ∞ T,X (κ;0) (7.12)
and, for any D ∈ >0 ,
P − τ 0 0 T 0 U N s−r m ,T • N 1 2 Av β X g N s, y · Z N s, y − Z N s−r m , y ds L ∞ T,X (κ;0) ǫ ′ N 1 2 − 1 2 β X +ǫ ′ τ 1 4 −ǫ ′ 0 + N 1 2 − 1 2 β X +ǫ ′ τ 1 4 −ǫ ′ 0 Z N T,X 1+ǫ ′ L ∞ T,X (κ;0) ǫ ′ ,D N −D . (7.13)
In particular, because τ 0 = N −2+2β X −2ǫ X +ǫ , considering ǫ X , ǫ ′ ∈ >0 are both sufficiently small but universal, and ǫ ǫ X +ǫ ′ is therefore sufficiently small but universal, for β u ∈ >0 arbitrarily small but universal, we see the following estimate holds with probability at least 1 − κ D N −D for any arbitrarily large but universal D ∈ >0 :
T 0 U N s,T • N 1 2 Av β X g N s, y ds − T 0 U N s,T • N 1 2 A β X g N s, y (τ 0 )Z N s, y ds L ∞ T,X (κ;0) ǫ ′′ N −β u + N −β u Z N T,X 1+ǫ ′ L ∞ T,X (κ;0)
. (7.14) 42 It therefore remains to compare the temporal average on this short, though still mesoscopic, time-scale to that with respect to the desired time-scale. To this end, consider {τ m } M m=0 from Corollary 5.11 and the following multiscale decomposition:
T 0 U N s,T • N 1 2 A β X g N s, y (τ 0 ) · Z N s, y ds = T 0 U N s,T • N 1 2 A β X g N s, y (τ M ; M ) · Z N s, y ds + M −1 m=1 ζ (1) m + ζ (2) m . (7.15)
As M 1, we may estimate each ζ ( j) m -quantity; this is immediate by Corollary 5.11 and the temporal regularity of the heat kernel from Proposition A.1 in [8] and the temporal regularity estimate for the Gartner transform in Proposition 6.2.
Remark 7.6. Concerning the analog of Lemma 7.5 for G N T,X , the same procedure succeeds if, rather than, Corollary 5.11 we employ Corollary 5.13; although there is not an a priori estimate equipped to the spatial average coming from Lemma 7.2 for G N T,X , the initial time-replacement succeeds regardless because the N -dependent prefactor in G N T,X of N
1 − N −β u : T 0 U N s,T • N 1 2 A β X g N s, y (τ M ; M ) · Z N s, y ds L ∞ T,X (κ;0) N −β u Z N T,X L ∞ T,X (κ;0) . (7.16)
It will again serve convenient to establish some notation before we prove Lemma 7.7. Lastly, we now define
ξ m = ξ (1) m + ξ (2) m + ξ (3) m , where ξ (1) m • = T 0 U N s,T • Z N s, y · − J m j=1 τ ι j,m A β X g N s, y (t j,m ; M ) · 1 τ ι j,m m A β X g N s, y (t j,m ; M ) ∈ (N −β m , N −β m−1 ] ds, ξ (2) m • = T 0 U N s,T • Z N s, y · − J m+1 j=1 · − J m+1 J −1 m ℓ=1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) · 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) N −β m 1 E C j,m+1 ds, ξ (3) m • = T 0 U N s,T • Z N s, y · − J m+1 j=1 · − J m+1 J −1 m ℓ=1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) · 1 E C σ j,m,ℓ ,m 1 E j,m+1 ds,
and the events are defined as
1 E j,m+1 • = J m+1 J −1 m ℓ=1 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) N −β m (7.18) 1 A β X g N s, y (t j,m+1 ; M ) N −β m . (7.19)
Proof of Lemma 7.7. We first observe that J m+1 J −1 m N ǫ due to the constraint τ
M . (7.20) Indeed, the decomposition comes from an inductive procedure; to make this estimate an exact identity without the absolute values requires slightly more attention, especially to ξ (1) m -quantities, though this will not be necessary to prove Lemma 7.7. For presentational clarity, we provide the following algorithm which yields (7.20):
• Consider every block B j,1 , and replace the respective A -quantities with time-scales t j,1 ∈ >0 with a cut-off N −β 1 ; this provides the error ξ (1)
.
• Now, group the A -quantities associated to blocks B j,1 according to the larger B j,2 -block that contains each of them.
For each of these larger groups, update these grouped quantities by inserting the factor 1 E j,2 ; the error obtained is bounded by the ξ (2) 1 -quantity.
• What remains is an average of A -quantities grouped according to boxes B j,2 , each of these A -quantities admitting a cut-off N −β 1 and the whole group corresponding to any box B j,2 admitting a factor of 1 E j,2 . For each A -quantity, remove the individual N −β 1 -cutoffs while leaving the 1 E j,2 -factor for each group. This gives the error corresponding to ξ (3) 1 , while what remains is an average of A -quantities, now on larger time-scales t j,2 ∈ >0 , each with a cut-off implemented through 1 E j,2 .
• Repeat this procedure, proceeding in increasing fashion in m ∈ 1, M to larger blocks B j,m and larger time-scales t j,m ∈ >0 and with shaper cut-offs N −β m and 1 E j,m until we finally arrive at the ξ (1) M -quantity, in which case we are left with the first quantity on the RHS of (7.20). To illustrate the inductive procedure, precisely, the next step in the algorithm will be to introduce a sharper cutoff of N −β 2 for each of the A -quantities on scales t j,2 with cutoffs 1 E j,2 , giving the error ξ (2) 1 . Afterwards, we group A -quantities according to their respective B j,3 -block, from where we introduce the cutoff 1 E j,3 on each block and remove the previous 1 E j,2 -factor from each A -quantity, giving the errors ξ
(2) 2 , ξ(3)
2 , respectively. Therefore, it remains to estimate each quantity on the RHS. Concerning the first quantity, because β M > 1 2 is a universal constant, the required upper bound is easy. To estimate the ξ ( j) m -quantities, we observe that the ξ (1) m -quantities are bounded immediately via Corollary 5.4. Second, we observe the next inequality, valid for some positive universal constant ǫ ′ ǫ:
1 E C j,m J m+1 J −1 m ℓ=1 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) > N −β m ∧ 1 A β X g N s, y (t j,m+1 ; M ) N −β m+1 −ǫ ′ . (7.21)
This, along with J m+1 J −1 m N ǫ , provides the following upper bounds which include soon-to-be-introduced events:
ξ (2) m N ǫ T 0 U N s,T • Z N s, y · − J m+1 j=1 · − J m+1 J −1 m ℓ,ℓ ′ =1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) · 1 E j,m,ℓ 1 E C j,m,ℓ ′ ds , (7.22) ξ (3) m T 0 U N s,T • Z N s, y · − J m+1 j=1 · − J m+1 J −1 m ℓ=1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) · 1 H j,m,ℓ ds,(7.23)
where the soon-to-be-introduced events are defined as
1 E j,m,ℓ • = 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) N −β m , (7.24) 1 H j,m,ℓ • = 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) N −β m+1 −ǫ ′ · 1 τ ι σ j,m,ℓ ,m A β X g N s, y (t σ j,m,ℓ ,m ; M ) N −β m . (7.25)
Provided the above bounds, we may now employ Corollary 5.4 again to suitably estimate the ξ (2) m -quantities. Moreover, we would be able to obtain the same for the ξ (3) m -quantities if t σ j,m,ℓ ,m ∈ >0 were replaced by t j,m+1 ∈ >0 for any j ∈ 1, J m+1 . However, because τ (m+1) N τ (m) N N ǫ with ǫ ∈ >0 arbitrarily small but universal, the proof of Corollary 5.4 actually provides the desired estimate; indeed, given the smaller t σ j,m,ℓ ,m time-scale in place of the larger t j,m+1 time-scale, the upper bound of N −β u within Corollary 5.4 thus grows by a factor of N ǫ . This completes the proof. 44 Remark 7.9. Tracking sufficiently carefully the ǫ-type constants in both Corollary 5.4 and the proof of Lemma 7.7 allows one to deduce that the ξ (3) m -quantities are superfluous in our estimation within the proof of Lemma 7.7. However, the proof given above is both more transparent and certainly succeeds without this precise bookkeeping, hence its presentation. Within the current section, we prove Theorem 1.6 for near-stationary initial data. To this end, we require the following. Definition 8.1. Retain the framework of Proposition 2.6. We define Z N T,X to be the unique solution to the integral equation
Z N T,X = U N 0,T • Z N 0, y + T 0 U N s,T • Z N s, y dQ N s, y + T 0 U N s,T • w N s, y Z N s, y ds + ∞ k=1 c k T 0 D −k U N s,T • w N ,k s, y Z N s, y ds. (8.1) Moreover, we define Y N T,X • = Z N T,X − Z N T,X .
The space-time field Z N T,X is a lattice model of the continuum SHE. Precisely, the sequence of random fields {Z N T,X } ∞ N =1 is tight with respect to the Skorokhod topology on D( 0 , C( )); moreover, all limit points concentrate on the unique weak solution to the continuum SHE with appropriate initial data. Thus, it remains to prove the following result.
Proposition 8.2. For some universal constant β u ∈ >0 , we have P Y N T,X L ∞ T,X (κ;0) N −β u N −β u . (8.2)
Roughly speaking, Proposition 8.2 follows courtesy of Proposition 7.1 assuming some a priori estimate on Z N T,X L ∞ T,X (κ;0) . Indeed, if Proposition 8.2 were actually provided, this would follow immediately from moment estimates for Z N T,X combined with the gluing strategy used to prove Proposition 6.2; these moment estimates follow via Proposition 3.2 in [8].
To rigorously execute the heuristic in the above paragraph, we implement a continuity method. Because the initial data Z N 0,X admits its all moments uniformly bounded in N ∈ >0 , we obtain the desired control on Z N T,X at least for the initial data. Our strategy amounts to using Y N T,X to propagate an L ∞ T,X (κ; 0)-estimate, completing the proof. Presenting the proof in technical detail requires the following stopping times.
τ (1) ∞ • = inf T ∈ [0, T f ] : G N T,X L ∞ T,X (κ;0) δ N −β u Z N T,X L ∞ T,X (κ;0) + N −β u Z N T,X 1+δ L ∞ T,X (κ;0) ∧ T f , (8.3a) τ (2) ∞ • = inf T ∈ [0, T f ] : Z N T,X L ∞ T,X (κ;0) N 2ǫ ST δ ∧ T f , (8.3b) τ (3) ∞ • = inf T ∈ [0, T f ] : Z N T,X L ∞ T,X (κ;0) N ǫ ST δ ∧ T f , (8.3c) τ ∞ • = τ (1) ∞ ∧ τ (2) ∞ ∧ τ (3) ∞ ; (8.3d)
above, the implied δ-dependent constant defining τ (1) ∞ ∈ 0 is chosen so that τ (1) ∞ = T f with probability at least 1 − N −β u for a universal constant β u ∈ >0 , courtesy of Proposition 7.1.
Next, it will be convenient to introduce the following "pathwise model" for Y N T,X .
+ T 0 U N s,T • w N s, y Y N s, y ds + ∞ k=1 c k T 0 ∇ ! X k U N s,T • w N ,k s, y Y N s, y ds (8.4) + T ∧τ ∞ 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds + T ∧τ ∞ 0 U N s,T • N β X g N s, y Z N s, y ds.
Remark 8.5. Uniqueness for the above integral equation follows from almost an identical argument for uniqueness of the equation defining Z N T,X , but the stopping time must be addressed; this can be done through analyzing the path before and after τ ∞ ∈ >0 .
To justify referring to Y N T,X as a model for Y N T,X , we appeal to the following lemma which identifies these two space-time random fields until the stopping time τ ∞ ∈ >0 . Lemma 8.6. For any T 0 ∈ [0, T f ], we have the containment of path-space events
{τ ∞ T 0 } ⊆ (T,X )∈[0,T 0 )× N Y N T,X = Y N T,X . (8.5)
Proof. This is an immediate consequence of the observation Y N
T ∧τ − ∞ ,X = Y N T ∧τ −
∞ ,X . The next lemma, which justifies studying Y N T,X as opposed to Y N T,X itself, is a collection of moment estimates.
Lemma 8.7.
There exists a universal constant β u ∈ >0 such that for any p ∈ >1 , we have
sup (T,X )∈[0,T f ]× N e −κ,X E Y N T,X 2p p,T f N −2pβ u . (8.6)
Moreover, for any D ∈ >0 ,
P Y N T,X L ∞ T,X (κ;0) N − 1 2 β u T f ,D N −D . (8.7)
Proof. We first observe that, by definition of τ ∞ ∈ >0 , for a universal constant β ′ u ∈ >0 , since U N s,T is a contraction on L ∞ ( ), we have
T ∧τ ∞ 0 U N s,T • N 1 2 Av β X g N s, y Z N s, y ds + T ∧τ ∞ 0 U N s,T • N β X g N s, y Z N s, y ds L ∞ T,X (κ;0) T ∧τ ∞ 0 U N s,T ∧τ ∞ • N 1 2 Av β X g N s, y Z N s, y ds + T ∧τ ∞ 0 U N s,T ∧τ ∞ • N β X g N s, y Z N s, y ds L ∞ T,X (κ;0) (8.8) N −β ′ u . (8.9)
The desired moment estimate now follows immediately from the proof under both Proposition 3.2 and Lemma 5.2 in [8].
To obtain the high-probability space-time uniform pathwise estimate, similarly to the proof for Proposition 6.2 it suffices to apply the a priori moment estimate for sufficiently large but universal exponent p ∈ >1 , combined with the trivial union bound over any suitable discretization of space-time by N D -many points, where D ∈ >0 is arbitrarily large but universal.
This completes the proof upon possibly updating β u ∈ >0 .
8.1. Proof of Proposition 8.2. Via Lemma 8.6 and Lemma 8.7, it remains to prove that τ ∞ = T f with probability at least 1− N −β ′ u , for β ′ u ∈ >0 some universal constant. To this end, for any ǫ ∈ >0 arbitrarily small but universal, define another set of auxiliary stopping times
τ (4) ∞ • = inf T ∈ [0, T f ] : Y N T,X L ∞ T,X (κ;0) N − 1 2 β u ∧ T f , (8.10a) τ (5) ∞ • = inf T ∈ [0, T f ] : sup (T 0 ,X )∈[0,T ]× N Q N T 0 +N −10 ,X − Q N T 0 ,X N ǫ ∧ T f . (8.10b) 46
Finally, our last preliminary observation is the following estimate for β u ∈ >0 a universal constant:
P j =2 τ ( j) ∞ = T f N −β u . (8.11)
Indeed, this is consequence of Proposition 7.1, Proposition 3.2 in [8], Lemma 8.7, and large-deviations principles for the Poisson distribution all combined with a union bound over N D -many space-time points with D ∈ >0 arbitrarily large but universal. Thus, it remains to establish the following bound on conditional probability for any D ∈ >0 :
P τ (3) ∞ = T f j =2 τ ( j) ∞ = T f D N −D . (8.12)
Indeed, this estimate is consequence of the following observations, in which 100 plays no particular importance.
• If τ (3) ∞ < T f , consider the time τ (3.1) ∞ = τ (3) ∞ − N −100 ∈ >0
; indeed, courtesy of the a priori estimate in Lemma 4.4 and the union bound argument within the proof of Proposition 6.2, we know τ (3) ∞ N −1 with probability at least
1 − κ D N −D for any D ∈ >0 . • Courtesy of Lemma 8.6, we know Y N τ (3.1) ∞ ,X = Y N τ (3.1) ∞ ,X
, and thus we have
Z N T,X L ∞ τ (3.1) ∞ ,X (κ;0) Y N T,X L ∞ T,X (κ;0) + Z N T,X L ∞ τ (3.1) ∞ ,X (κ;0) . (8.13)
• Consequence of conditioning on τ (4)
∞ = T f and τ (3) ∞ = T f , from (8.13) we have Z N T,X L ∞ τ (3.1) ∞ ,X (κ;0) N − 1 2 β u + N ǫ ST δ (8.14) N ǫ ST δ ,(8.15)
where the implied constants are universal. • Moving forward in time by an increment of N −99 ∈ >0 from τ (3.1) ∞ to τ (3.2) ∞ = τ (3.1) ∞ + N −99 , because we have conditioned on τ (5) ∞ = T f , by definition of the Gartner transform we see
Z N T,X L ∞ τ (3.2) ∞ ,X (κ;0) Z N T,X L ∞ τ (3.1) ∞ ,X (κ;0) (8.16)
again with another universal implied constant. • Lastly, combining (8.15) and (8.16), we contradict the condition defining τ (3) ∞ ∈ >0 at the time τ (3.2)
∞ > τ (3) ∞ . Precisely, we deduce P τ (3) ∞ = T f j =2 τ ( j) ∞ = T f P τ (3) ∞ N −1 j =2 τ ( j) ∞ = T f (8.17) D N −D (8.18)
with the final inequality a consequence of Lemma 4.4 for time-scale N −1 ∈ >0 , which completes the proof.
LATTICE SHE ANALYSIS IIIB: NARROW-WEDGE INITIAL DATA
We reemphasize that although the particle system evolves on the infinite-volume lattice , the Gartner transform evolves on the compact torus N = −N In the current section, we address the necessary updates to the previous results and steps within the article in order to establish Theorem 1.6 for narrow-wedge initial data. To avoid repeating technical details, we comment on the differences without providing complete proofs as before. 47 Before we begin, however, we recall the following space-time norm, which is an adaptation of the previous L ∞ T,X (κ; 0)topology addressing the singular nature with respect to time of the heat kernel:
F (T, X ) L ∞ T,X (κ;δ) • = sup T ∈(0,T f ] sup X ∈ N T 1 2 −δ e N −κ,X |F (T, X )|. (9.1)
For the remainder of the article, we choose δ ∈ >0 arbitrarily small but universal. Lastly, we recall the normalized Gartner transform Z N T,X from (1.15).
9.1. Preliminary ℓ 1 X -Estimate. Unlike the situation for near-stationary initial data, we first need an auxiliary estimate for the lattice-analog of the L 1 X -norm for the Gartner transform. Before we state the result, we first define the norm precisely:
F ℓ 1 X • = N −1 X ∈ N |F (X )|. (9.2)
Lemma 9.1. For some universal constant β u ∈ >0 and for any ǫ ∈ >0 , we have
P sup T ∈[0,T f ] Z N T,X ℓ 1 X N 1 8 + N 1 8 +ǫ Z N T,X 1+ǫ L ∞ T,X (κ;δ) ǫ N −β u . (9.3)
Throughout the proof for Lemma 9.1 to follow, we mention adaptations of necessary results established previously in this article; we do not explicitly write details because more difficult versions of these details will be addressed later within the current subsection.
Proof. We first assume T f = 1 2 ; for general T f ∈ >0 , the result follows from iterating the following analysis for O(T f )-many instances.
From (2.12), positivity of the Gartner transform, and the unit ℓ 1 X -norm of the heat kernel, we have
Z N T,X ℓ 1 X = Z N 0,X ℓ 1 X + T 0 N −1 y∈ N Z N s, y dQ N s, y + T 0 N −1 y∈ N N 1 2 · − N β X w=1 τ −w g N s, y · Z N s, y ds (9.4) + T 0 N −1 y∈ N N β X g N s, y Z N s, y ds + T 0 N −1 y∈ N w N s, y Z N s, y ds.
The first quantity on the RHS is bounded above by a universal constant as noted in the proof of Lemma 5.1 in [8].
As for the third quantity on the RHS of (9.4), we proceed like in the proof of Proposition 5.3; define T N = N − 1 2 −ǫ for ǫ ∈ >0 arbitrarily small but universal, and consider
T 0 N −1 y∈ N N 1 2 · − N β X w=1 τ −w g N s, y · Z N s, y ds T T N N −1 y∈ N N 1 2 · − N β X w=1 τ −w g N s, y · Z N s, y ds + N −ǫ sup T ∈[0,T f ] Z N T,X ℓ 1 X . (9.5)
Proceeding like in the respective proofs of Proposition 7.1, Proposition 5.3, Proposition 5.10, and Proposition 5.15, we obtain the following estimate for a universal constant β u ∈ >0 and arbitrarily small but universal constant ǫ ∈ >0 , with probability at least 1 − N −β u :
T T N N −1 y∈ N N 1 2 · − N β X w=1 τ −w g N s, y · Z N s, y ds ǫ N −β u Z N T,X L ∞ T,X (κ;δ) + N −β u Z N T,X 1+ǫ L ∞ T,X (κ;δ)
. (9.6) Indeed, we outline the differences below.
• Employing the temporal regularity of the heat kernel from Proposition A.1 in [8], the temporal regularity for the Gartner transform obtained in Proposition 6.2 applies to the narrow-wedge initial data upon replacing L ∞ T,X (κ; 0)norms with L ∞ T,X (κ; δ)-norms. To justify this, we observe the stochastic integral-type estimates in Lemma 6.3 hold for the L ∞ T,X (κ; δ)-norms with implied constant depending on δ ∈ >0 by means of writing Z N s, y = s − 1 2 +δ s 1 2 −δ Z s, y within the stochastic-integrand and integrability of the function s −1+2δ for s ∈ [0, T f ]. Moreover, separating the initial data quantity U N 0,T • Z N 0, y quantity from the remainder of the integral equation (2.12), the proof of Proposition 6.2 remains valid for the L ∞ T,X (κ; δ)-norms; the temporal regularity of U N 0,T • Z N 0, y matches that of the heat kernel itself. 48 • The previous bullet point indicates that the multiscale analysis in Lemma 7.5 remains valid for the narrow-wedge initial data and for the L ∞ T,X (κ; δ)-norms. Thus, to complete the remainder of the steps in establishing Proposition 7.1, it suffices to establish analogs of Proposition 5.3 and Proposition 5.10 for the L ∞ T,X (κ; δ)-norms for the third quantity on the RHS of (9.4). However, this is actually a simple adaptation of these aforementioned propositions upon writing Z N s, y = s − 1 2 +δ s 1 2 −δ Z s, y inside the first quantity within the RHS of (9.5) and proceeding as in Proposition 5.3 and Proposition 5.10, given that within (9.5) we have already introduced the time-scale T N ∈ >0 .
Similarly, following similar adaptations of Proposition 7.1, Proposition 5.8, and Proposition 5.12, we deduce the following for any ǫ ∈ >0 arbitrarily small but universal and possibly updated universal constant β u ∈ >0 , again with probability at least 1 − N −β u : . (9.11) This completes the proof.
9.2. Proof of Theorem 1.6. Towards establishing Theorem 1.6 with narrow-wedge initial data, the strategy we adopt is exactly that presented in Section 8. However, to this end, we need an analog of Proposition 7.1 with respect to L ∞ T,X (κ; δ)norms. We emphasize that this differs from the adjustments made in the proof of Lemma 9.1 because within this previous lemma, the presence of the heat kernel is eliminated by taking the ℓ 1 X -norm. Actually, as mentioned just prior to the proof of Lemma 9.1, the presence of the heat kernel renders the estimates more challenging to obtain.
To begin, we first establish the appropriate analogs of Corollary 5.4, Corollary 5.9, Corollary 5.11, Corollary 5.13, and Proposition 5.15. We begin with the first of these results. , (9.12) where the implied constant is universal outside of its dependence on the terminal index M ∈ >0 , which itself is universal.
Lastly, the sequence {τ from which the proof of Proposition 5.3 yields the desired estimate if δ ∈ >0 is sufficiently small.
We return to the first quantity on the RHS of (9.13). To this end, we note Employing similar adjustments, we obtain the following analog of Corollary 5.9. • For each m ∈ 1, M , define τ (m) N = N − 5 4 +β m −ǫ 3 for some arbitrarily small but universal constant ǫ 3 ∈ >0 . • The terminal index M ∈ >0 is the smallest positive integer for which β M > 1 3 + ǫ 4 , where ǫ 4 ∈ >0 is an arbitrarily small but universal constant.
For ǫ 1 ∈ >0 arbitrarily small but universal, the terminal index satisfies M ǫ 1 ,...,ǫ 4 1. Moreover, for some universal constant β u ∈ >0 , we have the following estimate with probability at least 1 − N −β u and for any ǫ ∈ >0 : Because the proof consists of the same adjustments made in the proof of Proposition 9.2, we omit the details.
The next step is an analog of Corollary 5.11. Proposition 9.4. Suppose g N T,X is any generic pseudo-gradient field, and consider β X ∈ >0 from Proposition 5.10. Suppose further that {τ m } M m=0 is a sequence of length M ∈ >0 defined as follows: • Define τ 0 = N −2+2β X −2ǫ X +ǫ for an arbitrarily small but universal constant ǫ ∈ >0 , and where ǫ X ∈ >0 denotes the constant from Notation 5.1.
• For each m ∈ 0, M − 1 , define τ m = τ m−1 N β u , where β u ∈ >0 any arbitrarily small but universal constant.
• Define M ∈ >0 to be the least positive integer for which τ M = τ We then proceed exactly as in the proof of Proposition 5.10.
Once again, we obtain the following analog of Corollary 5.13 via the same adjustments. If β u ∈ >0 is sufficiently small, we have M ǫ,β u 1, and the following estimate holds with probability at least 1 − N −β u : We conclude our discussion concerning adaptations of estimates within Section 5 with the following analog of Proposition 5.15.
Proposition 9.6. Suppose g N T,X is a pseudo-gradient field with support of size bounded above by N β 0 ∈ >0 . Suppose further that β X = 1 3 + ǫ 1 as in Proposition 5.3. Then for some universal constant β u ∈ >0 , we have the following with probability at least 1 − N −β u : Proof. The adjustments are the same as in Proposition 9.2; we are then left with the constraint that − 15 16 + 5 2 β X < 0 if we otherwise follow the proof of Proposition 5.15. However, β X = 1 3 + ǫ 1 with ǫ 1 ∈ >0 arbitrarily small but universal certain satisfies this constraint, which completes the proof.
The final ingredient towards proving Proposition 7.1 for the L ∞ T,X (κ; δ)-topology is an analog of the temporal regularity estimate in Proposition 6.2. Indeed, this would provide the temporal bootstrapping result in Lemma 7.5. Proof. The estimate follows from the same considerations applied to all but the first quantity on the RHS of (6.15). As for the first quantity, we simply apply the temporal regularity estimate for the heat kernel from Proposition A.1 in [8], which gives the additional quantity τ 1−ǫ 0 T −1+ǫ 0 .
Accumulating the estimates within this subsection, we obtain an analog for Proposition 7.1 with respect to the L ∞ T,X (κ; δ)topology. Considering the auxiliary solution Z N T,X with the narrow-wedge initial data and scaled appropriately in the same fashion as Z N T,X , the proof behind Proposition 8.2 for the narrow-wedge initial data thus holds if we implement the moment estimates from Lemma 5.1 in [8] rather than those from Proposition 3.2 in [8]. In particular, we deduce that the following event holds with probability at least 1 − N −β u , with β u ∈ >0 some universal constant.
• For all T 0 ∈ >0 , under the narrow-wedge initial data, we have , we know the sequence of random space-time fields {Z N T,N X } ∞ N =1 is tight with respect to the Skorokhod topology on D( >0 , C( )) and that all limit points concentrate on the weak solution to the continuum SHE (1.2) with initial data given by the Dirac point mass δ 0 supported at the origin. Via these last two pieces of information, we deduce Theorem 1.6 for narrow-wedge initial data. 52
Definition 1. 4 .
4For T ∈ >0 , we define h N T,0 as the net flux of particles across the origin, adopting the convention counting leftward moving particles contribute positive flux.
−κ,X Z N 0,X − Z N 0, y L 2p ω κ,p,u N −u |x − y| u ; (1.13)
•
The fields w N s, y , w N ,1 s, y , . . . , w N ,N β 0 s, y weakly vanishing quantities. • The coefficients {c k } ∞ k=1 admit all moments.
g N ,k s, y · Z N s, y ds, (2.16) 9
Lemma 3. 6 .
6Retain the framework within Proposition 3.5, and for any arbitrarily small but universal constant ǫ ′′ ∈ >0 , we define the sub-lattice N = −N 3 2 +ǫ ′′ , N 3 2 +ǫ ′′ along with the probability measures
Notation 3. 9 .
9Consider any ǫ ǫ with the implied constant universal to be determined shortly. Omitting the time-variable from notation without risk of confusion, we define both the sub-lattice ℓ N = −N 5 4 +ǫ −ℓN ǫ , N 5 4 +ǫ +ℓN ǫ and, for any x, y ∈ ,
Notation 3. 10 .. 11 .
1011We define the space-Consider any sub-lattice Λ ⊆ and any ̺ ∈ [−1, 1]. Define
Lemma 3. 16 .
16Consider any sub-lattice Λ ⊆ and any parameter ̺ ∈ [−1, 1]. Furthermore, suppose we have a collection of bounded functions ϕ 1 , . . . , ϕ J : Ω Λ → satisfying the following constraints:
Lemma 3. 18 .
18Consider any sub-lattice Λ ⊆ and any parameter ̺ ∈ [−1, 1]. Furthermore, suppose we have a collection of bounded functions ϕ 1 , . . . , ϕ J : Ω Λ → satisfying the following constraints:
Lemma 3 .
318 provides a fundamental estimate for our hydrodynamical-type analysis of pseudo-gradient fields in Section 5. For instance, from the sub-Gaussian estimate in Lemma 3.18 we obtain the following Laplace transform estimate.
Corollary 3. 19 .
19Retain the setting within Lemma 3.18 and further define
Notation 4. 1 .
1Define
Proposition 4. 2 .
2Consider any constant ǫ 0 ∈ (0, ǫ 0 ). For any T ∈ >0 and any X ∈ ǫ 0 , the following estimate holds with a universal constant κ 0 ∈ >0 :
Corollary 4. 3 .
3To prove Theorem 1.6, for both the near-stationary and narrow-wedge initial data it suffices to prove the result therein upon replacing Z N T,X by Z N ,p T,X .
4. 1 .
1A Priori Moment Estimates. The proof of Proposition 4.2 requires first the following pointwise moment estimate derived in exactly the fashion that Proposition 3.2 in [8] is obtained, with minor modifications. Moreover, upon presenting these pointwise moment bounds the validity of Proposition 4.2 is elucidated.
Lemma 4. 4 .
4Consider any p ∈ >1 . We have the following estimate for a universal constant κ 0 ∈ >0 : Moreover, the same result holds for Z N ,p T,X in place of Z N T,X .
(4. 9 )
9the above implied constants are also universal. The values of the heat kernel are extended to the half-integer lattice1 2 through piecewise linear interpolation of the values on the integer lattice .Lastly, by U N s,T we denote the operator corresponding to convolving by U N s,T (X , Y ).
Remark 4. 8 .
8The dilation constants κ P , κ G ∈ >0 correspond to the distinct Poisson-type and Gaussian-type regimes for the heat kernel U N s,T (x, y).
Proposition 4. 10 .
10Consider any k ∈ . Uniformly in (T, X ) ∈ >0 × , we have
of these two aforementioned estimates is a short-time estimate similar to the short-time analysis employed in proving Lemma 4.4. To state the estimate, we first establish notation and define Z N T,X = Z N T,X − Z N ,p T,X .Lemma 4.11. Consider T, T 0 ∈ [0, T f ], and consider τ = N − 1 2 −ǫ MS for a ǫ MS ∈ >0 arbitrarily small but universal depending only on ǫ c , ǫ 0 ∈ >0 . Then there exists some universal constant κ ∈ >0 and some set of coefficients {a k } ∞ k=1 with all moments uniformly bounded in N ∈ >0 such that for any |X | N
the first quantity within the RHS suffices within the Gaussian regime in Proposition 4.5, and the second quantity suffices within the Poisson regime, as j σ N and in this regime, we have the lower bound |X − Y | N 2 τ N 3 2 −ǫ MS . Thus, upon possibly updating the constant κ ∈ >0 , courtesy of Lemma 4.| N ǫ c τ k 1 +...+k j+1 U N 0,T +κτ • Z
), and (4.40) gives the result upon straightforward moment bounds for {a k } ∞ k=1 .
ǫ 0 .
0Provided ǫ MS < 1 2 ǫ 0 , this establishes (4.3) for narrow-wedge initial data. We now conclude the proof of Proposition 4.2 by establishing the pathwise probability estimate (4.4); for this, we simply partition [0, T f ] × N into any lattice of N −D -many evenly-spaced space-time points with D ∈ >0 arbitrarily large but universal. For ǫ ∈ >0 arbitrarily small, consider the event, for D ′ ∈ >0 arbitrarily large but universal,
, where ǫ ∈ >0 is an arbitrarily small though universal constant. In particular, the semigroup of operators U N s,T denotes the periodic heat flow on this torus. Moreover, all spatial coordinates are implicitly taken modulo this torus; see Corollary 4.3.
Corollary 5. 4 .
4Suppose g N T,X is a pseudo-gradient field, and consider the constant β X ∈ >0 from Proposition 5.3. Suppose further that for some universal constant M ∈ >0 and two sequences {β m } M m=0 and {τ (m) N } M m=1 the following conditions are satisfied:
N(
= N −1−β X − 1 2 β m−1 +β m +ǫ 3 for an arbitrarily small but universal constant ǫ 3 ∈ >0 .• The terminal index M ∈ >0 is the smallest positive integer for which β M > 1 2 . Then for some universal constant β u ∈ >0 , we have s,T • e κ, τ m ; β m , β m−1 ) dsL ∞ T,X (κ;0) M N −β u ,(5.4)where the implied constant is universal outside of its dependence on the terminal index M ∈ >0 , which itself is universal.Finally, the sequence {τ (m) N } M m=1is increasing in m ∈ 1, M , and for some arbitrarily small but universal constant ǫ ∈ >0 , we have τ follows from the following three observations:• Suppose β m+1 = β m + ǫ 2 . In this situation, we have τ (m) N = N −1−β X + 1 2 β m−1 +ǫ 2 +ǫ 3 , which is certainly increasing since {β m } ∞ m=1 is increasing. Moreover, it is clear that τ for ǫ ∈ >0 arbitrarily small but universal withinthis regime for β m+1 , β m ∈ >0 . • Suppose β m+1 = 1 2 β m + β X − ǫ m . In this case, we have τ (m) N = N −1−ǫ m , which is certainly increasing if we choose {ǫ m } ∞ m=1decreasing; the property τ for ǫ ∈ >0 arbitrarily small but universal is also immediate. • Considering the index m ∈ >0 from which we transition from the first bullet point to the second bullet point, the sequence {τ (m)
s to be the exclusion process with on Λ N ,τ with periodic boundary conditions on the time-interval s ∈ [0, τ], or equivalently with the generator L ! T Λ N,τ ,per .
cost of a sub-optimal upper bound, we may extend the time-integral to the domain [0, T f ], providing the space-time uniform estimate upon recalling (5.14) and(5.15)
Corollary 5. 9 .
9Suppose g N T,X is a pseudo-gradient field satisfying the constraint within Proposition 5.8. Suppose further that for some universal constant M ∈ >0 and two sequences {β m } M m=0 and {τ (m) N } M m=1 the following conditions are satisfied:
NN
= N − 5 4 +β m −ǫ 3 for some arbitrarily small but universal constant ǫ 3 ∈ >0 .• The terminal index M ∈ >0 is the smallest positive integer for which β M > 1 3 + ǫ 4 , where ǫ 4 ∈ >0 is an arbitrarily small but universal constant.For ǫ 1 ∈ >0 arbitrarily small but universal, we have M ǫ 1 ,...,ǫ 4 1. Moreover, for some universal constant β u ∈ >0 , } M m=1 is increasing in m ∈ 1, M , and for some arbitrarily small but universal constant ǫ ∈ >0 , we have τThe respective proofs of Proposition 5.8 and Corollary 5.9 follow from exactly those of Proposition 5.3 and Corollary 5.4, respectively, with the following changes:• We update the constant c N ,β − = N3 4 +ǫ− 1 2 β − within the proof of Proposition 5.3. • We update β X = 0 throughout the proof of Proposition 5.3.
N
+ǫ for an arbitrarily small but universal constant ǫ ∈ >0 , and where ǫ X ∈ >0 denotes the constant from Notation 5.1.• For each m ∈ 0, M − 1 , define τ m = τ m−1 N β u , where β u ∈ >0any arbitrarily small but universal constant. • Define M ∈ >0 to be the least positive integer for which τ M = τ ∈ >0 denotes the terminal time-scale within Corollary 5.4 with possibly different terminal index M ∈ >0 .
N
−1 coupled with a short calculation. The subsequent estimate follows from a repeated application of Proposition 5.10 combined with the observation that τ
final estimate consequence of Lemma 3.15 and Lemma 3.16. Because of the prior assumptionτ N −2+2β X −2ǫ X +ǫ 2 , we obtain our final upper bound N − 3 4 τ −1 N −β X N 1 4 −2β X +ǫ X −ǫ 2 τ − 1 2 ,which completes the proof. 5.4. Dynamical Analysis IIb. Again, we require some variation of Proposition 5.10 and Corollary 5.11 without a spatial average but with a smaller N -dependent prefactor. Proposition 5.12. Suppose g N T,X is any pseudo-gradient field satisfying the pseudo-gradient factor constraint from Proposition 5.8, and let τ ∈ >0 an N -dependent time-scale satisfying N − 4 3 −ǫ τ N −1 for some arbitrarily small but universal constant ǫ ∈ >0 .
NU
∈ >0 denotes the terminal time-scale from Corollary 5.9 with a possibly different terminal index M ∈ >0 . If β u ∈ >0 is sufficiently small, we have M ǫ,β N s,T • e κ, y N 1 3 +ǫ 1 A 0 g N s, y (τ m ) ds L ∞ T,X (κ;0) ǫ,β u N − 1 24 +ǫ + N −β u . (5.45)
, where ǫ ∈ >0 is an arbitrarily small though universal constant. In particular, the semigroup of operators U N s,T denotes the periodic heat flow on this torus. Moreover, all spatial coordinates are implicitly taken modulo this torus; see Corollary 4.3.
Remark 7. 3 .
3Our analysis for G N T,X will not require an analog of Lemma 7.2. 7.2. Dynamical Multiscale Analysis I. The second step towards the proof behind Proposition 7.1 is a multiscale analysis implemented to replace the space-average with cutoff by its temporal average on a suitable mesoscopic time-scale. Towards
possibly clear any confusion. Moreover, we remark that the conclusions of Proposition 5.3, Corollary 5.4, Proposition 5.10, and Corollary 5.13 all remain valid upon the replacement of the space-; j) for any j ∈ 0 , upon taking the time-average of the expectations. Lemma 7.5. Consider the set of times {τ m } M m=0 from Corollary 5.11
Notation 7. 8 .
8Recall the sequences {β m } M m=0 and {τ(m) N } N m=1 from Corollary 5.4. For m ∈ 1, M , we further let {B j,m } J m j=1 denote some decomposition of the time-block [0, τ (M ) N ] into disjoint and connected sub-intervals whose lengths satisfy |B j,m | ≍ τ (m) N , and J m ∈ >0 denotes the number of such sub-intervals. Moreover, we define t j,m • = |B j,m |, ι j,m • = inf B j,m , σ j,
N
N ǫ from Corollary 5.4. Second, observe the following decomposition with β M > 1 2 as in Corollary 5.4: T 0 U N s,T • N 1 2 A g N s, y (τ M ; M ) · Z N s, y ds M ; M ) · 1 E M Z N s, y ds + N
Notation 8. 3 .
3Retain the setting in Proposition 7.1. For ǫ ST ∈ >0 arbitrarily small but universal, define the stopping times
, where ǫ ∈ >0 is an arbitrarily small though universal constant. In particular, the semigroup of operators U N s,T denotes the periodic heat flow on this torus. Moreover, all spatial coordinates are implicitly taken modulo this torus; see Corollary 4.3.
to analyze the stochastic integral-type quantity on the RHS of (9.4). To this end, we observe the proof of Lemma 6.3 gives the following estimate for any ǫ ∈ >0 with probability at least 1 − κ D N −D for any D ∈ >0 : this estimate is actually simpler to obtain because the quantity on the LHS is an honest martingale. Nevertheless, we finally obtain the following estimate with probability at least 1 − N −β
Proposition 9. 2 .
2Suppose g N T,X is a pseudo-gradient field, and consider the constant β X ∈ >0 from Proposition 5.3. Suppose further that for some universal constant M ∈ >0 and two sequences {β m } M m=0 and {τ (m) N } M m=1 the following conditions are satisfied:• We have β 0 = 1 2 β X − ǫ X for ǫ X ∈ >0the arbitrarily small but universal constant from Notation 5.1; • For m ∈ 0, M −1 , we have β m+1 = [ 1 2 β m +β X − 1 16 −ǫ m ]∧[β m +ǫ 2 ] for two arbitrarily small but universal constants ǫ m , ǫ 2 ∈ >0 . • For each m ∈ 1, M , define τ (m) N = N −1−β X − 1 2 β m−1 +β m +ǫ 3 for an arbitrarily small but universal constant ǫ 3 ∈ >0 . • The terminal index M ∈ >0 is the smallest positive integer for which β M > 1 2 .49 Then for some universal constant β u ∈ >0 , the following estimate holds with probability at least 1 − N −β u for any ǫ ∈ >0 :
(τ m ; β m , β m− 1 )(τ m ; β m , β m− 1 )
11Z Z m ; β m , β m−1 ) Z N s, y ds.Concerning the second quantity on the RHS,m ; β m , β m−1 ) Z N s, y ds N 2δ Z N T,X L ∞ T,X m ; β m , β m−1 ) ds,(9.14)
(τ m ; β m , β m− 1 )(τ m ; β m , β m− 1
11m ; β m , β m−1 ) m ; β m , β m−1 ) Z N s, y ds. m ; β m , β m−1 ) Z N s, y ds = m ; β m , β m−1 ) Z N s, y ds (m ; β m , β m−1 ) Z N s, y ds.We estimate the first quantity within the RHS via Lemma 9.1 combined with the a priori estimate within | A m ; β m , β m−1 )|. Z m ; β m , β m−1 ) ds,(9.18) from which we obtained the desired estimate from the proof of Proposition 5.3. This completes the proof.
Proposition 9. 3 . 50 •
350Suppose g N T,X is a pseudo-gradient field satisfying the constraint within Proposition 5.8. Suppose further that for some universal constant M ∈ >0 and two sequences {β m } M m=0 and {τ (m) N } M m=1 the following conditions are satisfied:• Define β 0 = 0. For m ∈ 0, M − 1 , we define β m+1 = [ 3 16 + 1 2 β m − ǫ m ] ∧ [β m + ǫ 2 ]for some arbitrarily small but universal constants ǫ m , ǫ 2 ∈ >0 .
N
; β m ; β m−1 ) Z N s, y ds L ∞ T,X (κ;δ) M ,ǫ N −β u + N −β u Z } M m=1 is strictly increasing in m ∈ 1, M .
N
∈ >0 denotes the terminal time-scale within Proposition 9.2 with possibly different terminal index M ∈ >0 .If β u ∈ >0 is sufficiently small, we have M ǫ,β u 1, and the following estimate holds with probability at least 1 − N −β u :Proof. The necessary adjustments within the proof of Corollary 5.11 are simpler. Indeed, following the proof of Corollary 5.11, for any τ ∈ >0 we have
Proposition 9. 5 .
5Suppose g N T,X is any pseudo-gradient field satisfying the pseudo-gradient factor constraint from Proposition 5.8.Suppose further that {τ m } M m=0 is a sequence of length M ∈ >0 defined as follows: • Define τ 0 = N − 4 3 −ǫ for an arbitrarily small but universal constant ǫ ∈ >0 . • For each m ∈ 0, M − 1 , define τ m = τ m−1 N β u , where β u ∈ >0 is an arbitrarily small but universal constant. • Define M ∈ >0 to be the least positive integer for which τ M = τ (M ) N , where τ (M ) N ∈ >0 denotes the terminal time-scale from Proposition 9.3 with a possibly different terminal index M ∈ >0 .
the implied constant is universal, and where we recall the definition of Av
Proposition 9. 7 .
7Suppose τ N −1 . Consider any set of parameters ǫ 0 , ǫ 1 , ǫ 2 , ǫ 3 ∈ >0 , any D ∈ >0 . We haveP Z N T +τ,X − Z N T,X L ∞ T,X (κ;δ) τ 1−ǫ 0 T −1+ǫ 0 + N ǫ the impliedconstants within the events on the LHS are universal constants. Indeed, with Proposition 9.7, similar considerations as in the proofs of Proposition 9.2 and Proposition 9.4 provide the analog of Lemma 7.5 with respect to the L ∞ T,X (κ; δ)-topology.
1.3 in [8]
• The moment estimates for {c k } ∞ k=1 follow from the a priori moment bounds for the coefficients {α k } ∞ k=1 and {γ k } ∞k=1
from Assumption 1.1.
This completes the proof.
2.4. Strategy. As suggested by Proposition 2.6, the primary challenge in proving Theorem 1.6 from the integral equation
(2.12) is a suitable analysis of the quantities containing pseudo-gradient fields therein. To elucidate the difficulties behind
Proof. Because the desired estimate concerns only the path-space law of the trajectories s → η N s and s → η N ,per projected onto Λ ⊆ N , it suffices to construct a coupling of these trajectories with the same initial particle configuration on Λ N ,τ ⊆ N9)
30
and where κ 1 ∈ >0 is the parameter defining Σ N and the periodic dynamic s → η N ,per
s
.
s
Remark 7.10. The same procedure succeeds for G N T,X if we instead apply Corollary 5.9 instead of Corollary 5.4.8. LATTICE SHE ANALYSIS IIIA: NEAR-STATIONARY INITIAL DATAWe reemphasize that although the particle system evolves on the infinite-volume lattice , the Gartner transform evolves on the compact torus N = −N +ǫ , where ǫ ∈ >0 is an arbitrarily small though universal constant. In particular, the semigroup of operators U N s,T denotes the periodic heat flow on this torus. Moreover, all spatial coordinates are implicitly taken modulo this torus; see Corollary 4.3.5
4 +ǫ , N
5
4
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| []
|
[
"Using Hand Pose Estimation To Automate Open Surgery Training Feedback",
"Using Hand Pose Estimation To Automate Open Surgery Training Feedback"
]
| [
"Eddie Bkheet \nData and Decision Sciences\nTechnion Institute of Technology\nHaifaIsrael\n",
"Anne-Lise D'angelo [email protected] \nSurgery\nMayo Clinic\nRochesterMinnesotaUnited States\n",
"Adam Goldbraikh [email protected] \nApplied Mathematics\nTechnion Institute of Technology\nHaifaIsrael\n",
"Shlomi Laufer [email protected] \nData and Decision Sciences\nTechnion Institute of Technology\nHaifaIsrael\n"
]
| [
"Data and Decision Sciences\nTechnion Institute of Technology\nHaifaIsrael",
"Surgery\nMayo Clinic\nRochesterMinnesotaUnited States",
"Applied Mathematics\nTechnion Institute of Technology\nHaifaIsrael",
"Data and Decision Sciences\nTechnion Institute of Technology\nHaifaIsrael"
]
| []
| Purpose: This research aims to facilitate the use of state-of-the-art computer vision algorithms for the automated training of surgeons and the analysis of surgical footage. By estimating 2D hand poses, we model the movement of the practitioner's hands, and their interaction with surgical instruments, to study their potential benefit for surgical training. Methods: We leverage pre-trained models on a publicly-available hands dataset to create our own in-house dataset of 100 open surgery simulation videos with 2D hand poses. We also assess the ability of pose estimations to segment surgical videos into gestures and toolusage segments and compare them to kinematic sensors and I3D features. Furthermore, we introduce 6 novel surgical dexterity proxies stemming from domain experts' training advice, all of which our framework can automatically detect given raw video footage. Results: State-of-the-art gesture segmentation accuracy of 88.35% on the Open Surgery Simulation dataset is achieved with the fusion of 2D poses and I3D features from multiple angles. The introduced surgical skill proxies presented significant differences for novices compared to experts and produced actionable feedback for improvement. Conclusion: This research demonstrates the benefit of pose estimations for open surgery by analyzing their effectiveness in gesture Using Hand Pose Estimation To Automate Open Surgery Training Feedback segmentation and skill assessment. Gesture segmentation using pose estimations achieved comparable results to physical sensors while being remote and markerless. Surgical dexterity proxies that rely on pose estimation proved they can be used to work towards automated training feedback. We hope our findings encourage additional collaboration on novel skill proxies to make surgical training more efficient. | 10.1007/s11548-023-02947-6 | [
"https://export.arxiv.org/pdf/2211.07021v2.pdf"
]
| 257,901,221 | 2211.07021 | 80253ce32712c08c91f62d37a524de0577c4e4a1 |
Using Hand Pose Estimation To Automate Open Surgery Training Feedback
Eddie Bkheet
Data and Decision Sciences
Technion Institute of Technology
HaifaIsrael
Anne-Lise D'angelo [email protected]
Surgery
Mayo Clinic
RochesterMinnesotaUnited States
Adam Goldbraikh [email protected]
Applied Mathematics
Technion Institute of Technology
HaifaIsrael
Shlomi Laufer [email protected]
Data and Decision Sciences
Technion Institute of Technology
HaifaIsrael
Using Hand Pose Estimation To Automate Open Surgery Training Feedback
Springer Nature 2021 L A T E X templateMachine LearningComputer VisionGesture RecognitionSurgical Skill AssessmentPose EstimationSurgical Training
Purpose: This research aims to facilitate the use of state-of-the-art computer vision algorithms for the automated training of surgeons and the analysis of surgical footage. By estimating 2D hand poses, we model the movement of the practitioner's hands, and their interaction with surgical instruments, to study their potential benefit for surgical training. Methods: We leverage pre-trained models on a publicly-available hands dataset to create our own in-house dataset of 100 open surgery simulation videos with 2D hand poses. We also assess the ability of pose estimations to segment surgical videos into gestures and toolusage segments and compare them to kinematic sensors and I3D features. Furthermore, we introduce 6 novel surgical dexterity proxies stemming from domain experts' training advice, all of which our framework can automatically detect given raw video footage. Results: State-of-the-art gesture segmentation accuracy of 88.35% on the Open Surgery Simulation dataset is achieved with the fusion of 2D poses and I3D features from multiple angles. The introduced surgical skill proxies presented significant differences for novices compared to experts and produced actionable feedback for improvement. Conclusion: This research demonstrates the benefit of pose estimations for open surgery by analyzing their effectiveness in gesture Using Hand Pose Estimation To Automate Open Surgery Training Feedback segmentation and skill assessment. Gesture segmentation using pose estimations achieved comparable results to physical sensors while being remote and markerless. Surgical dexterity proxies that rely on pose estimation proved they can be used to work towards automated training feedback. We hope our findings encourage additional collaboration on novel skill proxies to make surgical training more efficient.
Introduction
Until this day, the traditional surgical training methodology of "see one, do one, teach one" by Dr. William Halsted [1] remains the most practiced method among medical practitioners. A major drawback of this methodology is the limited availability of expert surgeons. To make the training process more efficient, several works have been done that aim to recognize gestures and estimate surgical skill levels using different input modalities, including kinematic sensory data [2,3], optical flow [4], and RGB videos [5][6][7][8][9][10].
While sensor-based models [2,3] provide accurate spatial coordinates of the hands and surgical instruments, they can be challenging to implement due to the expensive hardware and setup required. In contrast, computer vision models that utilize RGB videos as input present a more viable solution, as they are less intrusive to the surgical workflow and are easier to set up.
Some of the popular datasets in this field are the JIGSAWS dataset [11] and the more recent RARP-45 dataset [12]. Existing studies focus on hand detection [13][14][15], surgical instrument detection [6,16], tool usage [6,17], surgical gesture recognition [3][4][5]10], and surgical skill classification [2,[7][8][9][10]. Previous studies [15,18] have also demonstrated the effectiveness of pose estimations for the task of gesture recognition.
Existing work on skill assessment focuses on classifying the performance using categorical labels or a numeric score and statistically comparing the performance of novices to experts. Existing frameworks offer feedback that includes (1) highlighting the segments that contributed to the skill classification [2] and (2) providing global motion metrics and statistics for novice compared to experts [3,6,7]. To the best of our knowledge, a fully automated framework that performs explainable skill assessment with task-specific actionable feedback hasn't been released yet.
This research aims to bridge the gap in the implementation of computer vision models in the surgical training environment by exploring surgical dexterity proxies based on 2D hand pose estimations, that automate the expert's advice. The proposed proxies produce task-specific feedback focused on surgical tool utilization for scissors, needle drivers, and forceps, as well as suture holding, and cutting positions. This paper's contributions are as follows:
1. A fully automated modular surgical dexterity assessment framework with actionable and explainable feedback. 2D hand pose dataset of open surgery simulation with annotations of instruments, hands, tool usage, gestures, and skill levels. 3. State-of-the-art results on the multi-task gesture segmentation problem using the predicted skeletons.
A novel
The Dataset
The dataset used in this research is the Open Surgery Simulation dataset [3]. It contains 100 videos of 25 clinicians with varying levels of expertise. The dataset contains temporal segmentation annotations for tool utilization and gesture recognition. The tools are needle driver, scissors, and forceps. The gestures are "No Gesture", "Needle Passing", "Pull The Suture", "Instrumental Tie", "Lay The Knot", and "Cut The Suture". The dataset was split into 5 folds using cross-user validation (Leave One User Out [11] modified for groups of users). In order to evaluate the added value of pose estimations to surgical skill assessment, 925 frames (0.01% of the dataset) sampled from 15 videos were annotated with 2D hand poses that include bounding boxes and 21 key-points. This results in a total of 1987 annotated hand instances.
Methods
Object Detection
For the task of hands and tools detection, a real-time YOLO-X [19] object detection model was used. The model configuration was according to the YOLOX-S version as in the official release. The model was trained using the "mmdetection" framework [20] for 400 epochs, each epoch took about 79 seconds on our hardware. The weights were initialized using a pre-trained model on the COCO [21] dataset. The prediction head was modified to detect 2 hands (left and right) and 4 surgical tools (needle driver, scissors, forceps, forcepsnot-used). The model was evaluated using the mean average precision (mAP) of intersection over union (IoU). Finally, the trained model was used to extract per-frame bounding boxes using a confidence threshold of 0.5 where a single bounding box with the highest confidence threshold was kept for each class. The model runs inference at 38 FPS on our hardware.
2D Pose Estimation
The pose estimation model was trained using the "mmpose" framework [22]. HRNetV2-W18 [23] was compared to Simple-Baseline [24] with Resnet 50 backbone, and the latter was used during the succeeding stages due to its speed of 43 items/second compared to HRNet's 9 items/second on our hardware. These models were evaluated using the probability of correct keypoint (PCK), area under the curve (AUC), and endpoint error (EPE) metrics. The models were pre-trained on the OneHand [25] dataset and fine-tuned on our dataset for 85 additional epochs. The trained model was used to extract per-frame pose estimations using a key point confidence threshold of 0.3. The missing keypoints were imputed using the last observation carried forward (LOCF) method. Finally, the Savitzky-Golay [26] signal smoothing algorithm was used to minimize the jittering of key-points across time caused by the frame-wise inference.
YOLOX
Multi-task Temporal Activity Segmentation
The extracted keypoints, and the bounding box centers are concatenated to form the input features of our spatio-temporal deep learning architecture as seen in Fig. 1. The architecture uses MSTCN++ [27] to predict the performed gesture and tools utilized for each frame by modifying the prediction head to produce multiple predictions. Our input features were compared to I3D [28] as a baseline feature extraction method for video processing, and to the combination of our input and I3D features. The I3D model was pre-trained on kinetics-400 [29] and fine-tuned on our dataset for 100 epochs taking 265 seconds per epoch. The features were extracted with a sliding window of 32 frames and a stride of 16 at 127 FPS. Our results were also compared to the current state-of-the-art benchmark on this dataset which uses kinematic sensor data [3]. The hyper-parameter space of MSTCN++ was explored using the random search strategy with 100 trials.
Surgical Skill Assessment
For the purpose of surgical skill assessment, we propose an approach that relies on the building block of a Surgical Skill Proxy [8]. In our case, a surgical skill proxy is defined as a metric that can be calculated using the tools and skeletons in the video, and that can be explained to the performer in simple words. In order to measure the statistical significance of the proxy differences between novices and experts, a 2-sided t-test was used which assumes a normal distribution of the proxy means across participants per task occurrence. In order to come up with meaningful proxies and provide evidence of validity, a domain expert (Author AD) was consulted during the proxy engineering process to provide face validity. The initial proxies were created based on known techniques, and later on, new ones were identified based on the data in an iterative process. Following are the proxies that showed promising feedback and how they model the practitioner's performance.
Gesture Duration
Proxy measurement: This proxy measures the time spent on a particular portion of the operation.
Clinical relevance: Previous studies [3] have shown differences in duration between novice and expert when performing different gestures. Furthermore, the background gesture which indicates the time spent on setup and non-specific movement between gestures tends to be longer for residents as observed in figure 3.
Hand Orientation
Proxy measurement: This proxy measures the level of pronation of the hand as seen in figure 4. It's calculated as x index − x pinky where x i is the x coordinate of the key point corresponding to the Metacarpophalangeal (MCP) joints of the i'th finger. The proxy produces high positive values when the hand is in a completely pronated position (palm rotated down), values around zero when the hand is directed to the sides (palm to the side and thumb up or down), and low negative values when the hand is fully supinated (palm rotated up).
Clinical relevance: Research has shown that pronation and supination of the hand are considered key skills when learning to perform surgical sutures. For example, when cutting the suture, full supination of the hand leads to the scissors forming a 90-degree angle with the suture, increasing the chance of cutting the knot. Therefore, slight supination of the hand is encouraged to form a 45-degree angle instead [30]. When holding forceps, full pronation of the hand indicates an incorrect holding position, whereas slight supination leads to a pencil-like holding position, as is taught by surgical experts [30]. Another example is needle passing, where the literature instructs starting with a pronated position, and ending the gesture with a slightly supinated position [30]. Beginning the gesture in a supinated position in this case limits the freedom of hand movement, resulting in awkward hand positions with less granularity of control.
Distance Between Thumb and Index Fingers
Proxy measurement: This proxy measures the distance in pixels between the tip of the thumb and index fingers as seen in figure 4.
Clinical relevance: The basic method of holding a needle driver is to straighten the index finger while slightly inserting the thumb through one of the handles. This way, the index finger guides and stabilizes the needle driver while the thumb is used to open and close it [31]. A more advanced technique is to palm the needle driver, by resting one side of the handle on the thenar eminence rather than inserting the thumb into the handle and using the thenar eminence to open and close it. This has the advantage of allowing a wider range of rotational motion of the instrument within the hand [31]. Another example is the suture holding position. When holding the suture, we observed that experienced surgeons tend to hold the suture at the edge of their fingers, allowing them a stronger grip on the suture, whereas novice surgeons end up holding the suture with different parts of their thumb, perhaps due to the larger surface area, allowing easier yet less granular grip of the suture.
Fingers to Tissue Distance
Proxy measurement: This proxy measures the distance in pixels between the tissue and the fingers holding the suture as seen in figure 4.
Clinical relevance: When cutting the suture, the proximity of the hand to the tissue impacts the surgeon's field of view. A small distance could lead to obscuring the suture, making the cut prone to errors. Oftentimes, assistants cut sutures for the leading surgeon, therefore, holding the suture at distance is required for better surgical performance.
Hand Velocity
Proxy measurement: This proxy measures the velocity of the hand when pulling the suture using a needle driver. The velocity of the hand is measured from multiple key points to account for subtle rotations of the hand. Such movements might be undetectable solely using object detection methods.
Clinical relevance: When pulling the suture, the surgeon needs to leave just the right amount of tail to allow for efficient performance during instrument ties [32]. Too short of a tail does not allow enough suture to complete the knot and too long of a tail makes it cumbersome to pull through the loop to form the knot. Novice surgeons who aren't familiar with the gesture, need to move their hand slowly while keeping an eye on the tail to leave the right amount, whereas expert surgeons who are familiar with the gesture move, their hand faster consistently leaving an accurate tail length. Table 1 shows the pose estimation results of HRNet compared to Simple Baseline. Simple Baseline shows comparable accuracy but offers a significant advantage in speed. The gesture segmentation results of our multi-task network are presented in table 2. The highest accuracy of 88.35% was achieved from the fusion of 2D poses and I3D visual features from multiple angles as seen in figure 2. Figure 5 shows a box plot of mean proxy values of novices compared to experts for the presented proxies during the relevant gestures. All experiments were conducted on a Tesla V100 GPU with 32GB memory.
Experiments and Results
Conclusion and Discussion
When it comes to gesture segmentation, remote pose estimations showed comparable accuracy to sensors on the frontal view (81.81% vs 82.40%), and better accuracy on the close-up and mixed views (84.16% and 85.39%). Despite being less accurate than I3D features, they offer the advantage of context isolation, and support for concrete feedback through skill proxies. Furthermore, concatenating key points from multiple views leads to a 1.23% increase in accuracy which leads us to believe that 3D pose estimations could outperform current results. Finally, we see that combining pose estimations and I3D features results in up to 0.53% added accuracy. As for skill assessment and training feedback, the proxies presented in section 3.4 highlight the notable distinctions between novices and experts. This allows the system to automatically offer task-specific feedback, such as instructing the user to keep the suture away from the tissue while cutting or to straighten the index finger while holding the needle driver. This is accomplished by comparing the proxy values of new samples to the average of the experts during the specific task, and providing feedback if the difference surpasses a pre-determined threshold that is independently adjusted for each proxy. This is a significant improvement over prior methods such as those used by Liu et al. [8], who relied on semantic visual features and tool trajectory, and Goldbraikh et al. [6], who used global metrics.
An important limitation to note is our use of 2D pose estimations instead of 3D. This limits some of our proxies such as hand orientation 3.4.2 to a single camera angle. Given that we used the videos of the frontal view in our dataset, applying this proxy was successful. When applying it using different camera angles, a 3D pose would provide a more accurate result.
To conclude, this research aims to bridge the gap in the application of novel computer vision algorithms to the domain of surgical training and monitoring. We demonstrate how 2D pose estimation can be applied to new open surgery datasets, and how it can be utilized for gesture segmentation and skill assessment. By creating the proxies depicted in section 3.4, and assessing known groups' validity evidence with our dataset, we demonstrate how the expert's advice can be automated through the surgical skill proxy methodology. This paves the way to work towards a fully automated surgical training framework requiring only a performance video.
Declarations
Fig. 4 :
4Surgical Proxy Visualization
Fig. 5 :
5Surgical Dexterity Proxy Novice Vs. Expert (*) for p-value < 0.05, (**) for p-value < 0.005
No Gesture Cut The Suture Instrumental Tie Lay The Knot Needle Passing Pull The Suture Average Gesture Duration in SecondsThe range for learning rate was
[0.0001, 0.0005, 0.005, 0.001], for feature maps it was [32, 64, 128, 256], for
prediction generation layers it was [3, 7, 11, 15], for refinement layers it was
[6, 10, 14], and for refinement stages it was [1, 3, 6, 12].
Predicted:
Actual:
No Gesture
Needle Passing
Pull The Suture
Instrumental Tie
Cut The Suture
Lay The Knot
Fig. 2: Gesture Segmentation on
Test Sample Frontal View
4.48
0.90
1.59
1.83
5.60
2.79
3.00
0.90
1.13
1.16
3.56
2.16
0.00
1.00
2.00
3.00
4.00
5.00
6.00
Novice
Expert
Fig. 3: Duration Per Gesture In
Seconds
Table 1 :
12D Hand Pose Estimation ResultsModel Name
PCK
AUC
EPE
ETT 1
Item/s
Resnet Simple Baseline Pre-trained 0.729
0.567
15.107 -
43.47
HRNet Pre-trained
0.771
0.603
13.279 -
9.52
Resnet Simple Baseline Fine-tuned
0.949
0.774
7.178
17
43.47
HRNet Fine-tuned
0.951 0.776 7.091
29
9.52
1 ETT: Epoch training time in seconds.
Table 2 :
2Multi-task Network Gesture Segmentation ResultsAccuracy
Edit
[email protected] [email protected] [email protected] ETT 1
Sensors [3]
82.40 ± 6.58
85.99
89.09
85.80
71.41
-
RGB Frontal View
Keypoints
81.22 ± 5.61
83.61
86.39
82.62
67.41
9.21
I3D
83.11 ± 5.84
86.35 89.10
86.18
73.76
14.49
I3D + Keypoints 83.24 ± 6.11 85.60
88.33
84.90
71.85
14.75
RGB Closeup View
Keypoints
84.16 ± 5.37
79.95
84.25
82.02
72.63
9.21
I3D
87.16 ± 4.72
84.66 89.70
88.33
82.10
14.49
I3D + Keypoints 87.69 ± 4.40 83.32
88.16
86.72
79.99
14.75
RGB Multi-view
Keypoints
85.39 ± 4.35
81.63
85.76
84.16
75.83
10.18
I3D
87.89 ± 4.13
85.76 90.61
89.08
82.82
20.29
I3D + Keypoints 88.35 ± 4.15 85.32
89.68
88.28
82.32
20.80
The chosen MSTCN++ hyperparameters are learning rate = 0.001, number of feature maps
= 64, prediction generation layers = 11, refinement layers = 10, refinement stages = 1.
1 Epoch training time in seconds for MSTCN++.
Acknowledgments This research was partially funded by the Technion Center for Machine Learning and Intelligent Systems and by Israel PBC-VATAT. Conflict of interest The authors declare that they have no conflict of interest. Ethical approval All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki Declaration and its later amendments or comparable ethical standards. Informed consent Informed consent was obtained from all individual participants included in the study. Code availability The code will be released after publication at https://github.com/edybk/Hand-Pose-Estimation-For-Surgical-Training
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| A linear Breit-Wheeler module for the code Geant4 has been developed. This allows signal-tonoise ratio calculations of linear Breit-Wheeler detection experiments to be performed within a single framework. The interaction between two photon sources is modelled by treating one as a static field, then photons from the second source are sampled and tracked through the field. To increase the efficiency of the module, we have used a Gaussian process regression, which can lead to an increase in the calculation rate by a factor of up to 1000.To demonstrate the capabilities of this module, we use it to perform a parameter scan, modelling an experiment based on that recently reported by Kettle et al.[1]. We show that colliding 50 fs duration γ-rays, produced through bremsstrahlung emission of a 100 pC, 2 GeV laser wakefield accelerator beam, with a 50 ps X-ray field, generated by a germanium burn-through foil heated to temperatures > 150 eV, this experiment is capable of producing > 1 Breit-Wheeler pair per shot.I. | 10.1103/physrevaccelbeams.26.054601 | [
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Monte Carlo modelling of the linear Breit-Wheeler process within the GEANT4 framework
R A Watt
The John Adams Institute for Accelerator Science
Imperial College London
SW7 2AZLondonUnited Kingdom
S J Rose
The John Adams Institute for Accelerator Science
Imperial College London
SW7 2AZLondonUnited Kingdom
B Kettle
The John Adams Institute for Accelerator Science
Imperial College London
SW7 2AZLondonUnited Kingdom
S P D Mangles
The John Adams Institute for Accelerator Science
Imperial College London
SW7 2AZLondonUnited Kingdom
Monte Carlo modelling of the linear Breit-Wheeler process within the GEANT4 framework
(Dated: February 9, 2023)
A linear Breit-Wheeler module for the code Geant4 has been developed. This allows signal-tonoise ratio calculations of linear Breit-Wheeler detection experiments to be performed within a single framework. The interaction between two photon sources is modelled by treating one as a static field, then photons from the second source are sampled and tracked through the field. To increase the efficiency of the module, we have used a Gaussian process regression, which can lead to an increase in the calculation rate by a factor of up to 1000.To demonstrate the capabilities of this module, we use it to perform a parameter scan, modelling an experiment based on that recently reported by Kettle et al.[1]. We show that colliding 50 fs duration γ-rays, produced through bremsstrahlung emission of a 100 pC, 2 GeV laser wakefield accelerator beam, with a 50 ps X-ray field, generated by a germanium burn-through foil heated to temperatures > 150 eV, this experiment is capable of producing > 1 Breit-Wheeler pair per shot.I.
INTRODUCTION
The linear Breit-Wheeler (BW) process is the annihilation of two photons to produce an electron positron pair (γγ → e + e − ) and is the simplest mechanism by which matter can be generated from light [2]. The process was first predicted in 1934, however, despite the long time that has passed, the annihilation of real photons has never been directly observed in the laboratory. This is due to the difficulty in generating the high energydensity photon sources required to overcome the relatively small cross-section. However, such photon sources are routinely found in astrophysical environments, and the BW process is predicted to play an important role in a range of phenomena, with examples including gamma ray bursts and the emission from quasars [3,4]. Also for photons propagating through the intergalactic medium there is a high energy (< 100 GeV) cut-off in the cosmic gamma ray spectrum observed at Earth due to BW annihilation with the cosmic microwave background [5,6]. This mechanism was thought to be well understood. However, recent observations have found a larger number of high energy photons from quasar 3C 279 reaching Earth than expected [7]. This demonstrates the need for more experimental work, to better understand these astrophysical environments.
Recently, the STAR collaboration [8] have reported the observation of the BW process from quasi-real photons in the peripheral collisions of high energy ions. However, observing the annihilation of real photons remains elusive. With the advance of high power laser systems, generating high energy-density photon sources in the laboratory has become possible. This has led to the proposal of several laser based BW detection experimental schemes utilising current and future facilities. Pike et al. [9] have suggested a scheme involving the generation of a thermal X-ray field with a temperature of ∼ 300 eV, using a laser heated hohlraum. A beam of GeV gamma rays, produced by bremsstrahlung emission of a laser wakefield accelerated (LWFA) electron beam in a high Z material target, then interacts with the X-ray photons. To produce the thermal X-ray field requires a facility such as the National Ignition Facility (NIF). Pike et al. predict up to 10 5 pairs can be produced in a single shot. A second scheme involving the interaction between two symmetrical photon sources has been proposed by Ribeyre et al. [10]. With symmetrical sources, the photon energy required to produces the rest mass of the e + e − pair is reduced to the MeV scale. Ribeyre et al. have compared multiple sources and found synchrotron emission from a highly energetic electron beam in an intense laser field and bremsstrahlung emission to be the most efficient methods. A pair yield of ∼ 10 4 per shot is predicted with a relatively modest laser energy of 100 J. Finally, building on this symmetric setup, Drebot et al. [11] have investigated a scheme involving the interaction of Compton sources produced by ∼ 260 MeV electron beams and joule class lasers. The predicted yield of ∼ 10 −4 pairs per shot is far lower than other experimental schemes, however, due to the low laser energy required, a high repetition rate system can be used. The BW process would then be detected over many shots.
In the Spring of 2018 a BW detection campaign took place at the Gemini laser of the Central Laser Facility at the Rutherford Appleton Laboratory, in the UK. The experiment was based on the scheme proposed by Pike et al. [9] and a diagram of the experimental setup is shown in figure 1. A more in depth review of this experiment can be found in Kettle et al. [1]. An electron beam generated by a LWFA interacted with a 1 mm bismuth target, producing a ∼ 50 fs pulse of gamma rays with a spectrum extending up to 100s of MeV. This was directed through a ∼ 50 ps duration X-ray field, generated by the thermal emission of a laser heated germanium foil which emits strongly between 1 keV and 2 keV [12]. Both beams of the Gemini laser are capable of providing 15 J, which is low compared to a high energy facility such as NIF at 2 MJ. However, the Gemini laser has a much higher repetition rate of 0.05 Hz compared to 1 shot per day. With a large number of shots a better characterisation of the background can be performed as well as a statistical analysis of the interaction.
All linear BW detection experiments require the generation of high energy (> 0.511 MeV) photons. As a result, the environment is inherently noisy. There are two main sources of noise: the direct detection of photons by the particle detectors, and photons interacting with the experimental setup, producing background e + e − pairs through the Bethe-Heitler (BH) process [13] (the annihilation of a photon with a virtual photon in the nuclear field of an atom). For a successful experiment, the ratio of signal BW pairs to the background noise should be maximised. An estimation of the background noise can be obtained by performing full scale simulations of the passage of high energy particles through the experimental setup. Several publicly available Monte Carlo particle tracking codes exist for this purpose, including Geant4, Fluka and MCNP [14][15][16]. For the 2018 Gemini campaign Geant4 was used to analyse the background noise [1]. However, the linear BW process is not included within the standard Geant4 physics package. It is advantageous to perform both signal and noise calculations within a single framework, and the aim of this work has been to develop a linear BW module for Geant4 to enable this.
Geant4 models the passage of individual particles through matter. To include the linear BW process within Geant4, it must be modelled within this same framework. This has been achieved by treating one photon source as a static photon field. Individual photons from the second source are then sampled and tracked through this field. In this work, we will refer to a photon from the field as static and a photon tracked through the field as dynamic. Using this method, the temporal evolution of the static photon source cannot be accounted for. This module is therefore suited for modelling experiments with asymmetric photon sources, in which one is constant in time, to a good approximation, over the full interaction. Figure 1 shows an example of such an experiment.
II. MODULE OVERVIEW
As discussed in section I, this module involves tracking dynamic photons through a static photon field. The static photon field is represented by a physical volume within the Geant4 computational domain. It is fully defined by n( , θ, φ), the number of photons per unit volume with energy between and + d , and travelling at an angle between θ + dθ and φ + dφ. For a dynamic photon which enters a static photon field the calculation of the interaction involves two steps:
1. Calculating the probability of the interaction occurring.
Calculating the dynamics of the interaction.
A dynamic photon enters the static photon field at some arbitrary angle in the simulation frame. To perform the calculation of step 1, we transform n( , θ, φ) into a frame where the z-axis is the direction of propagation of the dynamic photon. We will refer to this frame as the dynamic photon frame. For the linear BW process to occur, the total energy in the centre-of-mass (c.m.) frame must be larger than twice the rest mass energy of an electron. This sets the following interaction threshold
s = 2 E(1 − cos θ) > 4m 2 c 4 (1)
where E is the energy of the dynamic photon. If n( , θ, φ) = 0 for all , θ and φ satisfying equation 1, the threshold is not met and the probability of the dynamic photon interacting is 0. In this case the dynamic photon will propagate through the static field unaffected. However, if n( , θ, φ) = 0 there is a finite probability of the dynamic photon interacting. To obtain this probability requires the mean free path of the interaction, λ, which is obtained through the following [17]
1 λ = 2π 0 dφ π 0 dθ ∞ 0 d σ BW (s) n(φ, θ, ) (1 − cos θ)(2)
where the factor of (1 − cos(θ)) accounts for the relative velocity between photons and σ BW is the polarisation averaged, total BW cross-section [18]:
σ BW = π 2 r 2 e (1−β 2 ) −2β(2−β 2 )+(3−β 4 ) ln 1 + β 1 − β(3)
where β = (1 − 4s −1 ) and r e is the classical electron radius. The probability that a dynamic photon propagates a distance l is
P (l) = λ −1 e −l/λ .(4)
For each dynamic photon, a value of l is sampled from this distribution. If l is longer than the static photon field, the dynamic photon will propagate through unaffected. If l is shorter, the dynamic photon will propagate l and annihilate. When a dynamic photon annihilates it is removed from the simulation. We then move onto step 2, determining the dynamics of the interaction. First the properties of the annihilating static photon ( , θ, φ) must be calculated. These are obtained by sampling from the integrand in equation 2. In the c.m. frame, both the electron and positron receive an energy, E +,− , of √ s/2. The c.m. frame positron polar scattering angle, θ + , is obtained by sampling from the BW differential cross-section [18]
dσ BW dΩ = r 2 e β s 1 + 2βsin 2 θ − β 4 − β 4 sin 4 θ (1 − β 2 cos 2 θ) 2(5)
which is plotted in figure 2. The positron azimuthal scattering angle, φ + , is sampled from the uniform distribution U(0, 2π). Using conservation of momentum, the electron polar and azimuthal scattering angles are given by A tungsten collimator and block are placed in the beam path to remove highly divergent gamma rays and those directed towards the X-ray foil respectively. A large number of BH pairs are produced in the converter foil, collimator and block. These are removed with an on-axis magnet before the interaction zone. The gamma rays interact with an X-ray field, generated by a laser heated germanium foil, producing BW pairs. The residual gamma rays continue on axis to a spectrometer. The BW pairs pass through a magnetic chicane to single particle detectors, situated behind lead shielding (not shown). Diagram provided by E. Gerstmayr. θ − = π − θ + and φ − = φ + − π. From E +,− , θ +,− and φ +,− the momentum in the c.m. frame, P +,− , is calculated. Finally a Lorentz boost is performed into the laboratory frame and an electron and positron are added with the calculated momentum.
III. INCREASING EFFICIENCY WITH GAUSSIAN PROCESS REGRESSION
Due to the low photon density, length of field and value of σ BW , the probability that an individual photon will undergo the linear BW process is small. To analyse the interaction, many dynamic photons must be simulated. For each dynamic photon λ is calculated, involving a triple integral over the static photon variables (equation 2). This makes the method outlined in section II computationally expensive.
A common solution to this problem is to replace the direct, expensive calculation of λ with a lookup table. λ is a function of three variables of the dynamic photon, its energy, E, and direction, θ d and φ d , through the rotation to the dynamic photon frame. A lookup table would involve discretizing E, θ d and φ d to form a three dimensional grid. At each grid point λ is calculated and values are obtained at runtime by interpolating from the grid. However, a lookup table is not appropriate in this case for the following reasons 1. λ varies rapidly with E, θ d and φ d . Therefore, to avoid numerical errors, a fine grid resolution is required. The generation of this lookup table is then computationally expensive.
2. To avoid numerical errors through extrapolation, the limits of the table must be known prior to the simulation runtime. This can be challenging if the dynamic photons are generated through a multistep process.
3. The generation of a uniform dense lookup table does not represent the distribution of the dynamic photon parameters, making it inefficient.
We have used an alternative approach where a Gaussian process regression (GPR) model is trained dynamically and used to quickly calculate λ when required. GPR is a Bayesian method for approximating predictions from an expensive physical model. Here, we will only discuss our implementation and not the concepts of GPR. For a full review of GPR refer to Rasmussen and Williams (2006) [19].
We can consider equation 2 as an expensive function mapping the dynamic photon parameters x ≡ (E, θ d , φ d ) to λ. The goal is to replace equation 2 with a cheaper function, f . However, λ varies rapidly with x, therefore, it is more numerically accurate to find a function which returns the natural logarithm of the mean free path estimate
λ a = e f (x) (6)
where f (x) is provided by a GPR. By using a GPR we have assumed that f (x) is Gaussian distributed. Therefore, λ a is log-normal distributed with mean and variance given by
µ λa = exp µ f − σ 2 f 2 σ 2 λa = exp(σ 2 f ) − 1 exp(2µ f + σ 2 f )(7)
where µ f and σ 2 f are the mean and variance of f (x). To train the GPR a data set, D = {x i , log(λ i )}, is required. This is generated by calculating equation 2 for a limited number of events. In the units system of Geant4, the dimensions of x vary over vastly different scales. Therefore, if the regression model is trained directly on D it would perform poorly. To solve this problem x is normalised using min-max feature scaling.
In this project the open source library libgp [20], was used for the GPR and the implementation can be broken down into the following three stages:
1. Data accumulation stage: Equation 2 is solved in full for n d dynamic photons and the result saved, generating D.
2.
Training stage: After simulating n d dynamic photons the GPR is trained on D by optimising the model hyperparameters.
3. Acceleration stage: For subsequent events, σ λa is calculated using equation 7 if σ λa < σ max where σ max is a user defined limit, λ is given by µ λa . If σ λa > σ max equation 2 is solved and the result is appended to D. After another n d points are added to D the hyperparameters are again optimised.
As the simulation progresses D grows. This causes the confidence of the GPR to increase, so it is used more often than the full calculation of equation 2. The rate at which dynamic photons are simulated then increases. The values of n d and σ max can be tuned to optimise the efficiency for an acceptable level of error. This GPR scheme overcomes all the shortfalls of a lookup table listed above. As the GPR is trained dynamically, no expensive calculations prior to runtime are required. Also, no limits on the dynamic photon parameters are set prior to runtime. This avoids extrapolation as λ is calculated in full when the dynamic photon parameters fall well outside the data set. Finally, as the GPR is trained on a sample from the dynamic photon population, it is fully representative of the dynamic photon parameters distribution.
To demonstrate the increase in computational efficiency a test simulation has been carried out for both the full calculation of equation 2 and the GPR scheme. This test simulation involves a gamma ray beam with a divergence of 10 mrad and a Gaussian energy spectrum with mean 1 GeV and standard deviation 0.25 GeV, interacting with a 1 cm long, 300 eV isotropic black body radiation field
n( , θ, φ) = 2 π 2 1 e /T − 1 sin θ 4π .(8)
For the GPR scheme the parameters n d and σ max are set to 100 and 10 −2 respectively. Figure 3 shows the number of dynamic photons simulated against runtime. The red curve corresponds to the full calculation of equation 2 whereas the blue curve corresponds to the GPR scheme. At the start of the simulation figure 3 shows both schemes have the same rate of dynamic photon calculation. This is during the data accumulation stage. Just after 500 s, the GPR enters the training stage. Beyond this point the GPR model dynamic photon computation rate increases rapidly. After 2000 s the GPR scheme has calculated 100 times more dynamic photons. When performing experimental parameter optimisation, the same static photon field is often used in multiple simulations. In this case the GPR model can be saved after the end of a simulation and reused in a later one. This avoids the slow data accumulation stage and provides an even greater increase in efficiency as shown by the black curve in figure 3.
To test if the GPR scheme is accurately emulating the full calculation of equation 2, simulations can be carried out for both algorithms and the number of pairs generated, N BW , compared. However, both algorithms include uncertainty in N BW due to the finite number of events simulated. Therefore, an ensemble of simulations should be carried out and the distributions over N BW compared. The result of this is shown in figure 4 for two different values of σ max . When σ max = 0.01 there is little difference between the distributions of the two algorithms, suggesting that the GPR scheme is accurately emulating 2. However, figure 4 also shows that if σ max is set too large, this is not the case and the GPR algorithm introduces errors to the estimate.
IV. MODULE DISCUSSION
Fully modelling the interaction between two photon sources where both are treated dynamically is an N-body problem with complexity of O(N 2 ). This is too computationally expensive to directly solve for the large number of photons required to statistically analyse the interaction. Through the use of a tree code, as demonstrated by Jansen et al. [21], the problem complexity can be re-FIG. 5: Geant4 visualisation of basic Breit-Wheeler experiment. An electron (red) produces a high energy gamma ray (green) in a bismuth target. The particles propagate through an on-axis magnet, diverting seed electron. The gamma ray propagates into an X-ray field annihilating and forming an e + e − pair. duced to O (N log N). Here, by treating one source as a static field, the complexity is reduced further to O(N). However, the type of photon-photon interactions which can be modelled are also constrained. By defining the static photon field with n( , θ, φ) neither spatial or temporal gradients are accounted for. It is trivial to redefine the static field also as a function of position, n(r, , θ, φ), and account for spatial gradients by constructing the full field through multiple sub-fields with different n( , θ, φ). But it is not possible to relax the temporal gradient constraint due to the static nature of the Geant4 computational domain. Therefore, the module can only be used when one photon source is effectively stationary over the full duration of the interaction.
Although it constrains the type of experiment which can be modelled, developing the module within Geant4 allows us to make use of its extensive toolkit. It has the capability to model particle tracking, the geometry of experiments and the response of detectors to energetic particles. Combining these capabilities with the module discussed here, start-to-end simulations can be performed. This includes: the generation of the gamma ray beam in the bremsstrahlung target; the background noise; the tracking of pairs through the analyser magnets and the particle detector interaction. This can be used for experimental parameter sensitivity analysis and optimisation. The ability to fully visualise the setup with Geant4 (see figure 5) also makes experimental design much quicker [1].
V. MODULE DEMONSTRATION
Here we demonstrate how the module can be used to analyse a BW detection experiment similar to the Kettle et al. experiment discussed in section I but with a simplified geometry and representation of the X-ray source. The setup is shown in figure 5. An electron beam is incident onto a 1 mm bismuth converter foil. This produces gamma rays through the bremsstrahlung process. The particles then pass through an on-axis magnet, removing the seed electrons and BH pairs generated in the converter foil. The gamma rays propagate into a 1 mm long X-ray field, generated by a laser heated germanium foil, and annihilate. The X-ray field has spatial gradients, is anisotropic and has a non-blackbody energy spectrum, making this an ideal test for the module. To model the spatial gradients, the field is split up into an array of 10 × 10 × 10 photon volumes, each with a different n( , θ, φ). We assume that n( , θ, φ) is not correlated in energy and angle. This allows us to separate the energy and angular dependant parts, n( , θ, φ) = f ( )Φ(θ, φ). f ( ) is then modelled using the atomic physics code Flychk [22] and an example is shown in figure 6 with a foil temperature of 350 eV. Φ(θ, φ) is modelled using a Monte-Carlo X-ray photon tracking code. Here, photons are launched from an emitting plane and tracked through the photon volume array. By launching many photons, Φ(θ, φ) can be estimated. Figure 7 shows the angle and energy probability distribution for BW positrons produced with a 1 GeV electron beam and a burn-through foil electron temperature of 350 eV. From figure 7 we can see the positron beam is well collimated with a divergence of ∼ 10 mrad. The divergence of the positrons decreases with increasing energy. The particle beaming is a result of the gamma ray carrying almost all the momentum of the interaction. This is a key advantage of the Pike et al. detection scheme. If the positron beam had a large divergence, which is the case for two symmetric photon sources, to prevent a low detection efficiency the interaction region would have to be surrounded by detectors. A large number of high divergence background BH pairs would also be detected. However, with a highly beamed source, an analyser chicane can be used to select positrons within a specific divergence and energy band and transport them to an area with low noise to be detected. With the increased computational efficiency, as a result of implementing a GPR, the module can be used to perform scans over experimental parameters. This is demonstrated in figure 8 where we can see how the yield of BW pairs is affected by the burn-through foil temperature and the LWFA electron beam energy. Each simulation consisted of 10 9 electrons (160 pC) and the electron beam energy was varied from 500 MeV to 3000 MeV and foil temperature from 50 eV to 500 eV. The Gemini laser facility is capable of producing 2 GeV electron beams with a charge of < 100 pC. Therefore, these results suggest that with a foil temperature of > 150 eV, it is possible to generate > 1 BW pair per shot. Utilising the relatively high repetition rate of the Gemini laser facility, over multiple shots a detectable number of BW pairs above background could be produced.
VI. CONCLUSION
As the power of laser facilities has increased in recent years, it is now theoretically possible to produce a detectable number of BW pairs in the laboratory. However, the expected signal-to-noise ratio of such an experiment is low, making detailed numerical modelling vital. Here, we have presented the development of a new linear BW module for Geant4. This will allow signal-to-noise ratio calculations to be performed within a single framework. We have shown how a Gaussian process regression can be used to greatly increase the rate at which photons are simulated without a loss in the accuracy of the module. We have used this increase in efficiency to perform a parameter scan which suggests that > 1 BW pair per shot can be produced using the Gemini laser facility.
The module presented here can be readily extended to include other particle-photon interactions such as Compton scattering (e − γ → e − γ) and photon-photon scattering (γγ → γγ). Photon-photon scattering from two real photons is another QED process with astrophysical relevance that has never been directly observed in the laboratory. The difficulties of observing the process arise due to the relatively small cross-section (∼ α 2 times smaller than σ BW ) and the incoming and outgoing particles all being photons, making background subtraction challenging. The scattering of quasi-real photons in the collision of heavy ion beams has recently been observed by the AT-LAS detector at the Large Hadron Collider [23]. With improvements to the experimental scheme from section I, it may also be possible to study photon-photon scattering of two real photons, which could be modelled using this module.
FIG. 1 :
1Schematic of Gemini linear BW detection experiment. Starting from the left: a LWFA generates an electron beam which is converted into a gamma ray beam through bremsstrahlung emission in a thin bismuth converter foil.
FIG. 2 :
2Polarisation averaged BW differential cross-section.
FIG. 3 :
3Number of simulated dynamic photons against runtime for the full calculation of equation 2 (red), the accelerated GPR scheme (blue) and the pre-trained GPR scheme (black).
FIG. 4 :
4Histograms of number of positrons generated per simulation. The red curve shown the full calculation of equation 2 and the blue and black curves show calculations performed with the GPR scheme with a σ max of 0.01 and 1.0 respectively.
FIG. 6 :
6Radiation spectrum emitted from a static, solid density germanium plasma with an electron temperature of 350 eV. This was generated using the atomic physics code Flychk[22].
FIG. 8 :
8Breit-Wheeler pair yield against X-ray foil temperature and electron beam energy.
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| [
"https://github.com/mblum/libgp"
]
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[
"arXiv:math/0309206v1 [math.SP] 12 Sep 2003 NECESSARY AND SUFFICIENT CONDITIONS IN THE SPECTRAL THEORY OF JACOBI MATRICES AND SCHRÖDINGER OPERATORS",
"arXiv:math/0309206v1 [math.SP] 12 Sep 2003 NECESSARY AND SUFFICIENT CONDITIONS IN THE SPECTRAL THEORY OF JACOBI MATRICES AND SCHRÖDINGER OPERATORS"
]
| [
"David Damanik ",
"Rowan Killip ",
"Barry Simon "
]
| []
| []
| We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator − d 2 dx 2 + V (x) on L 2 (0, ∞) with V ∈ L 2 (0, ∞) and u(0) = 0 boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate. | null | [
"https://export.arxiv.org/pdf/math/0309206v1.pdf"
]
| 18,054,461 | math/0309206 | bb23cbf139b14eb10bb2bdc13444f861115e8358 |
arXiv:math/0309206v1 [math.SP] 12 Sep 2003 NECESSARY AND SUFFICIENT CONDITIONS IN THE SPECTRAL THEORY OF JACOBI MATRICES AND SCHRÖDINGER OPERATORS
David Damanik
Rowan Killip
Barry Simon
arXiv:math/0309206v1 [math.SP] 12 Sep 2003 NECESSARY AND SUFFICIENT CONDITIONS IN THE SPECTRAL THEORY OF JACOBI MATRICES AND SCHRÖDINGER OPERATORS
We announce three results in the theory of Jacobi matrices and Schrödinger operators. First, we give necessary and sufficient conditions for a measure to be the spectral measure of a Schrödinger operator − d 2 dx 2 + V (x) on L 2 (0, ∞) with V ∈ L 2 (0, ∞) and u(0) = 0 boundary condition. Second, we give necessary and sufficient conditions on the Jacobi parameters for the associated orthogonal polynomials to have Szegő asymptotics. Finally, we provide necessary and sufficient conditions on a measure to be the spectral measure of a Jacobi matrix with exponential decay at a given rate.
Introduction
In this note, we want to describe some new results in the spectral and inverse spectral theory of half-line Schrödinger operators and Jacobi matrices. Given V ∈ L 1 loc (0, ∞) with a mild regularity condition at infinity (ensuring limit-point case there, cf. [20]), one can define a unique selfadjoint operator which is formally
H = − d 2 dx 2 + V (x) (1.1)
with u(0) = 0 boundary condition (see, e.g., [20]). For any z / ∈ R, there is a solution u(x; z) of −u ′′ + V u = zu which is L 2 at infinity and unique up to a constant. The Weyl m-function is then defined by
m(z) = u ′ (0; z) u(0; z) (1.2)
It obeys Im m(z) > 0 when Im z > 0, which implies that Im m(E + iε) has a boundary value as ε ↓ 0 in distributional sense:
dρ(E) = w-lim ε↓0 1 π Im m(E + iε) dE (1.3)
dρ is called the spectral measure. In this way, each V gives rise to a spectral measure dρ. In fact, the correspondence is one-to-one: Gel'fand-Levitan [10] (see also Simon [26]) found an inverse procedure to go from dρ to V.
Similarly, given a Jacobi matrix, a n > 0, b n ∈ R:
J = b 1 a 1 0 0 · · · a 1 b 2 a 2 0 · · · 0 a 2 b 3 a 3 · · · . . . . . . . . . . . . . . . (1.4)
on ℓ 2 (Z + ), we define dµ to be the measure associated to the vector δ 1 by the spectral theorem. That is,
m(z) ≡ δ 1 , (J − z) −1 δ 1 = dµ(E) E − z (1.5)
In this setting, the inverse procedure dates back to Jacobi, Chebychev, Markov, and Stieltjes. It is easy to describe: By applying Gram-Schmidt to {1, E, E 2 , . . .} in L 2 (dµ), we obtain the orthonormal polynomials p n (E). These obey the three-term recursion relation Ep n (E) = a n+1 p n+1 (E) + b n+1 p n (E) + a n p n−1 (E) (1.6) Alternatively, one can obtain a n , b n from a continued fraction expansion of m ( [30,37]). The main subject of spectral theory is to find relations between general properties of the spectral measures dρ or dµ and of the differential/difference equation parameters V and a n , b n . Clearly, the gems of the subject are ones that provide necessary and sufficient conditions, that is, a one-to-one correspondence between some explicit family of measures and some explicit set of parameters. In this note, we announce three such results (one involving asymptotics of orthogonal polynomials rather than the measures) whose details will appear elsewhere [19,4,5].
In the context of orthogonal polynomials on the unit circle [28], Verblunsky's form [36] of Szegő's theorem [32,33,34] is such a one-toone correspondence between a measure and the recurrence coefficients for its orthogonal polynomials. Baxter's theorem [1,2] and Ibragimov's theorem [17,13] can be viewed as other examples.
Our work here is related to and motivated by the more recent result of Killip-Simon [18]:
Theorem 1.1 ([18]). J − J 0 is Hilbert-Schmidt, that is ∞ n=1 (a n − 1) 2 + b 2 n < ∞ (1.7)
if and only if the spectral measure dµ obeys
(i) (Blumenthal-Weyl) supp(dµ) = [−2, 2] ∪ {E + j } N + j=1 ∪ {E − j } N − j=1 with E + 1 > E + 2 > · · · > 2 and E − 1 < E − 2 < · · · < −2 with lim j→∞ E ± j = ±2 if N ± = ∞. (ii) (Normalization) µ is a probability measure. (iii) (Lieb-Thirring Bound) ±,j (|E ± j | − 2) 3/2 < ∞ (1.8) (iv) (Quasi-Szegő Condition) Let dµ ac (E) = f (E) dE. Then 2 −2 log[f (E)] √ 4 − E 2 dE > −∞ (1.9)
Our first result is the analog of this theorem for Schrödinger operators. This is discussed in Section 2.
Our second result concerns Szegő asymptotics for orthogonal polynomials. In 1922, Szegő [35]
proved that if dµ = f (E) dE where f is supported on [−2, 2] and log[f (E)] dE √ 4 − E 2 > −∞ (1.10) then lim n→∞ z n p n (z + z −1 ) (1.11)
exists and is nonzero (and finite) for all z ∈ D. There is work by Gončar [14], Nevai [22], and Nikishin [24] that allow point masses outside [−2, 2]. The following summarizes more recent results on this subject by Peherstorfer-Yuditskii [25], Killip-Simon [18], and Simon-Zlatoš [29]:
Theorem 1.2. Suppose dµ = f (E) dE + dµ s with supp(dµ sc ) ∪ supp(f ) ⊂ [−2, 2] and j,± (|E ± j | − 2) 1/2 < ∞ (1.12)
Then the following are equivalent:
(i) inf(a 1 .
. . a n ) > 0 (ii) All of the following:
(a) ∞ n=1 |a n − 1| 2 + |b n | 2 < ∞ (1.13) (b)
lim n→∞ a n . . . a 1 exists and is finite and nonzero.
(c) lim n→∞ n j=1 b j exists. (iii) 2 −2 log[f (E)] dE √ 4 − E 2 > −∞ (1.14)
Moreover, if these hold, then the limit (1.11) exists and is finite for all z ∈ D and is nonzero if .12) is required a priori here, this result is not a necessary and sufficient condition with only parameter information on one side and only spectral on the other. In Section 3, we will discuss a necessary and sufficient condition for the asymptotics (1.11) to hold, thereby closing a chapter that began in 1922.
z + z −1 / ∈ {E ± j }. Because (1
Finally, in Section 4, we discuss necessary and sufficient conditions on the measure for the a's and b's to obey
lim sup (|a n − 1| + |b n |) 1/2n ≤ R −1 (1.15)
for some R > 1. Namely, dµ must give specified weight to those eigenvalues E j with |E j | < R + R −1 and the Jost function must admit an analytic continuation to the disk {z : |z| < R}. The Jost function is naturally defined in terms of scattering; however, there is a simple procedure for determining it from the measure and vice versa. See (4.2).
Acknowledgment. We would like to thank Roman Romanov for drawing our attention to [8].
Schrödinger Operators with L 2 Potential
The proofs of the results in this section will appear in [19]. Given a measure dρ on R, defineσ on [0, ∞) by
∞ 0 g √ E dρ(E) = ∞ 0 g(k) dσ(k) (2.1) that is, formally dσ(k) = χ (0,∞) (k 2 ) dρ(k 2 )
. For the Schrödinger operator with V = 0,
dρ 0 (E) = π −1 χ [0,∞) (E) √ E dE (2.2) and dσ 0 (p) = 2π −1 χ [0,∞) (p)p 2 dp (2.3)
Given ρ, defineF bỹ
F (q) = π −1/2 p≥1 p −1 e −(q−p) 2 [dσ(p) − dσ 0 (p)] (2.4) Since dρ obeys dρ(E) 1 + E 2 < ∞ the integral in (2.4) is convergent.
Define M(k) by M(k) = m(k 2 ) with m given by (1.2). Here is our main result on L 2 potentials: Theorem 2.1. Let dρ be the spectral measure associated to a potential,
V. Then V ∈ L 2 ([0, ∞)) if and only if (i) (Weyl) supp(dρ) = [0, ∞) ∪ {E j } N j=1 with E 1 < E 2 < · · · < 0 and lim j E j = 0 if N = ∞. (ii) (Local Solubility) ∞ 0 |F (q)| 2 dq < ∞ (2.5) (iii) (Lieb-Thirring) j |E j | 3/2 < ∞ (2.6) (iv) (Quasi-Szegő) log |M(k; 0) + ik| 2 4k Im M(k; 0) k 2 dk < ∞ (2.7)
Remarks. 1. While there is a parallelism with Theorem 1.1, there are two significant differences. First, the innocuous normalization condition is replaced with (2.5) and, second, (2.7) involves M and not just µ.
2. Equation (2.5) (assuming (2.7) holds) is an expression of the fact that dρ is the spectral measure of an L 2 loc potential essentially because it implies (by [12]) that the A-function of [26] is in L 2 loc . 3. The integrand in (2.7) is − log(1 − |R(k)| 2 ) where R is a reflection coefficient. Weak lower semicontinuity of the negative of the entropy used in [18] is replaced by lower semicontinuity of the L 2n norm. 4. The key to the proof of Theorem 2.1 is a strong version of the Zaharov-Faddeev [38] sum rules; essentially following [18,29,27], we provide a step-by-step sum rule for V ∈ L 2 loc and take suitable limits. What is interesting is that we use a whole-line, not half-line, sum rule.
We note that prior to our work, V ∈ L 2 ⇒ (2.6) was proved by Gardner et al. in [9]. Bounds of this type are often called Lieb-Thirring inequalities after their work on moments of eigenvalues for V ∈ L p (R d ); see [21]. [6], proved that V ∈ L 2 implies f (E) > 0 for a.e. E > 0. There is related work when − d 2 dx 2 + V ≥ 0 in Sylvester-Winebrenner [31] and Denisov [7].
Szegő Asymptotics
The proofs of the results in this section will appear in [4]. For the study of Szegő asymptotics, it is useful to map D = {z : |z| < 1} to C\[−2, 2] by z → E = z + z −1 . Our main result on this issue uses the following conditions:
∞ n=1 |a n − 1| 2 + |b n | 2 < ∞ (3.1) lim N →∞ N n=1
log(a n ) exists (and is finite) 3) hold, then z n p n (z + z −1 ) converges uniformly on compact subsets of D and has a non-zero limit for those z = 0 where z + z −1 is not an eigenvalue of J.
Remarks. 1. By Theorem 1.1, (3.1) implies only the quasi-Szegő condition (1.9) whereas all prior discussions of Szegő asymptotics have assumed the stronger Szegő condition (1.14). We have examples in [4] where (3.1)-(3.3) hold and (|E ± n | − 2) 1/2 = ∞ which, by [29], implies that (1.14) fails, so we have examples where Szegő asymptotics hold, although the Szegő condition fails.
2. The first step in the proof is to show that for fixed z ∈ D, Szegő asymptotics hold if and only if there is a solution with Jost asymptotics, that is, for which lim z −n u n (z) (3.4) exists and is non-zero.
3. We have two constructions of the Jost solution when (3.1)-(3.3) hold: one using the nonlocal step-by-step sum rule of [27] and the other using perturbation determinants [18]. In either case, one makes a renormalization: In the first approach, one renormalizes Blaschke products and Poisson-Fatou representations, and, in the second case, one uses renormalized determinants for Hilbert-Schmidt operators.
While these are the first results we know for Szegő/Jost asymptotics for Jacobi matrices with only L 2 conditions, Hartman [15] and Hartman-Wintner [16] (see also Eastham [8,Ch. 1]) have found Jost asymptotics for Schrödinger operators with V ∈ L 2 with conditionally convergent integral.
Jacobi Parameters With Exponential Decay
The proofs of the results in this section will appear in [5]. If m is given by (1.5), we define M(z) by
M(z) = −m(z + z −1 ) (4.1)
Suppose M(z) is the M-function of a Jacobi matrix and that M(z) has an analytic continuation to a neighborhood ofD with the only poles in D lying inD ∩ R and all such poles are simple. Then we can define
u(z) = N k=1 b(z, z k ) exp e iθ + z e iθ − z log sin θ Im M(e iθ ) dθ 4π (4.2)
where {z k } ∞ k=1 are the poles of M in D. This agrees with the Jost function from scattering theory (see [18]), so we call it by this name.
Given M and the Jost function, u, suppose u is analytic in {z : |z| < R} and z 0 is a zero of u (pole of M) with R −1 < |z 0 | < 1. We say M has a canonical weight at z 0 if Given u and not m, there is a normalization issue, so it is easier to discuss the perturbation determinant [18] which obeys (2) The only zeros of L inD lie onD ∩ R and are simple.
lim z→z 0 z =z 0 (z − z 0 )M(z) = (z 0 − z −1 0 )[u ′ (z 0 )u(z −1 0 )] −1 (4.3)L(z) = u(z) u(0) = ∞ n=1 a n u(z) (4.4)
In that case, there is a unique J with J − J 0 finite rank, so L is its perturbation determinant.
Remarks. 1. While there is a unique J with J − J 0 finite rank, if L has k zeros in D, there is a k-parameter family of other J's with L as their perturbation determinant (each such J has |a n −1|+|b n | decaying exponentially, but only one has J − J 0 finite rank).
2. There is an analog of Theorem 4.3 if L is only analytic in {z : |z| < R}.
3. The proofs of these results depend on analyzing the map (u, M) → (u (1) , M (1) ) where u (1) , M (1) are the Jost function and M-function for J (1) , the Jacobi matrix with the top row and leftmost column removed. We control |||u (1) ||| in terms of |||u||| where |||u||| 2 = |u(R 1 e iθ ) − u(0)| 2 dθ 2π with R 1 < R, and are thereby able to show |||u (n) ||| goes to zero exponentially. While we are aware of no prior work on the direct subject of this section, we note that Nevai-Totik [23] solved the analogous problem for orthogonal polynomials on the unit circle, that Geronimo [11] has related results for Jacobi matrices (but only under an a priori hypothesis on M), and that there are related results in the Schrödinger operator literature (see, e.g., Chadan-Sabatier [3]).
. 1 .
1If for some ε > 0, z n p n (z + z −1 ) converges uniformly on compact subsets of {z : 0 < |z| < ε} to a non-zero value, then(3.1)-(3.3) hold.Conversely, if (3.1)-(3.
Theorem 4. 1 .
1Let M be the M-function of a Jacobi matrix, J. Then J − J 0 is finite rank if and only if (i) M is rational and has only simple poles inD with all such poles in R. (ii) u is a polynomial. (iii) M has canonical weight at each z ∈ D which is a pole of M.
Theorem 4 . 2 .
42Let M be the M function of a Jacobi matrix, J. Then the parameters of J obey (|a n − 1| + |b n |) ≤ C ε (R −1 + ε) 2n for some R > 1 and all ε > 0 if and only if (i) M is meromorphic on {z : |z| < R} with only simple poles insidē D with all such poles in R. (ii) u is analytic in {z : |z| < R}. (iii) M has canonical weight at each pole of M, z 0 ∈ D, with R −1 < |z 0 | < 1.
Theorem 4. 3 .
3Let L be a polynomial with L(0) = 1. Then L is a perturbation determinant of a Jacobi matrix, J, with J − J 0 finite rank if and only if (1) L(z) is real if z ∈ R.
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Discrete Sturm-Liouville operators and some problems of function theory. E M Nikishin, J. Sov. Math. 35E.M. Nikishin, Discrete Sturm-Liouville operators and some problems of func- tion theory, J. Sov. Math. 35 (1986), 2679-2744.
Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points. F Peherstorfer, P Yuditskii, Proc. Amer. Math. Soc. 129F. Peherstorfer and P. Yuditskii, Asymptotics of orthonormal polynomials in the presence of a denumerable set of mass points, Proc. Amer. Math. Soc. 129 (2001), 3213-3220.
A new approach to inverse spectral theory. I. Fundamental formalism. B Simon, Ann. of Math. 2B. Simon, A new approach to inverse spectral theory. I. Fundamental formal- ism, Ann. of Math. (2) 150 (1999), 1029-1057.
A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices. B Simon, preprintB. Simon, A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices, preprint.
. B Simon, Orthogonal Polynomials on the Unit Circle. AMS Colloquium Publications SeriesB. Simon, Orthogonal Polynomials on the Unit Circle, AMS Colloquium Pub- lications Series, expected 2004.
Sum rules and the Szegő condition for orthogonal polynomials on the real line. B Simon, A Zlatoš, to appear in Comm. Math. PhysB. Simon and A. Zlatoš, Sum rules and the Szegő condition for orthogonal polynomials on the real line, to appear in Comm. Math. Phys.
Recherches sur les fractions continues. T Stieltjes, Ann. Fac. Sci. Univ. Toulouse. 8T. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Univ. Toulouse 8 (1894-1895), J1-J122; 9, A5-A47.
Linear and nonlinear inverse scattering. J Sylvester, D P Winebrenner, SIAM J. Appl. Math. 59J. Sylvester and D.P. Winebrenner, Linear and nonlinear inverse scattering, SIAM J. Appl. Math. 59 (1999), 669-699. [electronic]
Ein Grenzwertsatzüber die Toeplitzschen Determinanten einer reellen positiven Funktion. G Szegő, Math. Ann. 76G. Szegő, Ein Grenzwertsatzüber die Toeplitzschen Determinanten einer reellen positiven Funktion, Math. Ann. 76 (1915), 490-503.
Beiträge zur Theorie der Toeplitzschen Formen, I, Math. G Szegő, Z. 6G. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, I, Math. Z. 6 (1920), 167-202.
Beiträge zur Theorie der Toeplitzschen Formen. G Szegő, Math. Z. IIG. Szegő, Beiträge zur Theorie der Toeplitzschen Formen, II, Math. Z. 9 (1921), 167-190.
Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind. G Szegő, Math. Ann. 85G. Szegő,Über den asymptotischen Ausdruck von Polynomen, die durch eine Orthogonalitätseigenschaft definiert sind, Math. Ann. 85 (1922), 114-139.
On positive harmonic functions (second part ). S Verblunsky, Proc. London Math. Soc. 40S. Verblunsky, On positive harmonic functions (second part ), Proc. London Math. Soc. 40 (1936), 290-320.
Analytic Theory of Continued Fractions. H S Wall, AMS ChelseaNew York; Providence, R.I.H.S. Wall, Analytic Theory of Continued Fractions, Van Nostrand, New York, 1948; AMS Chelsea, Providence, R.I., 2000.
The Korteweg-de Vries equation is a fully integrable Hamiltonian system. V E Zaharov, L D Faddeev, Funkcional. Anal. i Priložen. 5RussianV.E. Zaharov and L.D. Faddeev, The Korteweg-de Vries equation is a fully integrable Hamiltonian system, Funkcional. Anal. i Priložen. 5 (1971), 18-27. [Russian]
| []
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[]
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"Liyong Zhang \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Jianjun He \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Richard J Deboer \nNuclear Science Laboratory\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n",
"Michael Wiescher [email protected] \nNuclear Science Laboratory\nUniversity of Notre Dame\nNotre Dame\n46556IndianaUSA\n",
"Alexander Heger \nSchool of Physics and Astronomy\nMonash University\n3800VictoriaAustralia\n",
"Daid Kahl \nExtreme Light Infrastructure -Nuclear Physics\nHoria Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH)\n077125Bucharest-MȃgureleRomania\n",
"Jun Su \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Daniel Odell \nInstitute of Nuclear and Particle Physics\nDepartment of Physics and Astronomy\nOhio University\n45701AthensOhioUSA\n",
"Yinji Chen \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Xinyue Li \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Jianguo Wang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Long Zhang \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Fuqiang Cao \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Hao Zhang \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Zhicheng Zhang \nCollege of Physics and Optoelectronic Engineering\nShenzhen University\n518060ShenzhenChina\n",
"Xinzhi Jiang \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Luohuan Wang \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Ziming Li \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Luyang Song \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Hongwei Zhao \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Liangting Sun \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Qi Wu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Jiaqing Li ",
"Baoqun Cui \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Lihua Chen \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Ruigang Ma \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Ertao Li \nCollege of Physics and Optoelectronic Engineering\nShenzhen University\n518060ShenzhenChina\n",
"Gang Lian \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Yaode Sheng \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Zhihong Li \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Bing Guo \nChina Institute of Atomic Energy\nP. O. Box 275(\n",
"Xiaohong Zhou \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Yuhu Zhang \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Hushan Xu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n",
"Jianping Cheng \nCollege of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina\n\n102413BeijingChina\n",
"Weiping Liu [email protected] \nChina Institute of Atomic Energy\nP. O. Box 275(\n"
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"College of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina",
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"Extreme Light Infrastructure -Nuclear Physics\nHoria Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH)\n077125Bucharest-MȃgureleRomania",
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"102413BeijingChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"China Institute of Atomic Energy\nP. O. Box 275(",
"China Institute of Atomic Energy\nP. O. Box 275(",
"College of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina",
"102413BeijingChina",
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"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"China Institute of Atomic Energy\nP. O. Box 275(",
"China Institute of Atomic Energy\nP. O. Box 275(",
"China Institute of Atomic Energy\nP. O. Box 275(",
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"China Institute of Atomic Energy\nP. O. Box 275(",
"College of Nuclear Science and Technology\nKey Laboratory of Beam Technology of Ministry of Education\nBeijing Normal University\n100875BeijingChina",
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"China Institute of Atomic Energy\nP. O. Box 275(",
"China Institute of Atomic Energy\nP. O. Box 275(",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
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| The origin of calcium production in the first stars (Pop III stars), which formed out of the primordial matter of the Big Bang, and their fates, remain most fascinating mysteries in astrophysics. Advanced nuclear burning and supernovae were thought to be the dominant source of the Ca production seen in all stars. | 10.1038/s41586-022-05230-x | [
"https://export.arxiv.org/pdf/2302.09802v1.pdf"
]
| 253,155,735 | 2302.09802 | b64b2db7e43cebc9ebf80a0494bf2ada782e8972 |
20 Feb 2023
Liyong Zhang
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Jianjun He
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Richard J Deboer
Nuclear Science Laboratory
University of Notre Dame
Notre Dame
46556IndianaUSA
Michael Wiescher [email protected]
Nuclear Science Laboratory
University of Notre Dame
Notre Dame
46556IndianaUSA
Alexander Heger
School of Physics and Astronomy
Monash University
3800VictoriaAustralia
Daid Kahl
Extreme Light Infrastructure -Nuclear Physics
Horia Hulubei National Institute for R&D in Physics and Nuclear Engineering (IFIN-HH)
077125Bucharest-MȃgureleRomania
Jun Su
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Daniel Odell
Institute of Nuclear and Particle Physics
Department of Physics and Astronomy
Ohio University
45701AthensOhioUSA
Yinji Chen
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Xinyue Li
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Jianguo Wang
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Long Zhang
China Institute of Atomic Energy
P. O. Box 275(
Fuqiang Cao
China Institute of Atomic Energy
P. O. Box 275(
Hao Zhang
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Zhicheng Zhang
College of Physics and Optoelectronic Engineering
Shenzhen University
518060ShenzhenChina
Xinzhi Jiang
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Luohuan Wang
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Ziming Li
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Luyang Song
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Hongwei Zhao
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Liangting Sun
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Qi Wu
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Jiaqing Li
Baoqun Cui
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
China Institute of Atomic Energy
P. O. Box 275(
Lihua Chen
China Institute of Atomic Energy
P. O. Box 275(
Ruigang Ma
China Institute of Atomic Energy
P. O. Box 275(
Ertao Li
College of Physics and Optoelectronic Engineering
Shenzhen University
518060ShenzhenChina
Gang Lian
China Institute of Atomic Energy
P. O. Box 275(
Yaode Sheng
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Zhihong Li
China Institute of Atomic Energy
P. O. Box 275(
Bing Guo
China Institute of Atomic Energy
P. O. Box 275(
Xiaohong Zhou
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Yuhu Zhang
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Hushan Xu
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
Jianping Cheng
College of Nuclear Science and Technology
Key Laboratory of Beam Technology of Ministry of Education
Beijing Normal University
100875BeijingChina
102413BeijingChina
Weiping Liu [email protected]
China Institute of Atomic Energy
P. O. Box 275(
20 Feb 2023Measurement of 19 F(p, γ) 20 Ne reaction suggests CNO break-out in first stars
The origin of calcium production in the first stars (Pop III stars), which formed out of the primordial matter of the Big Bang, and their fates, remain most fascinating mysteries in astrophysics. Advanced nuclear burning and supernovae were thought to be the dominant source of the Ca production seen in all stars.
Here we report on a qualitatively different path to Ca production through break-out from the "warm" carbon-nitrogen-oxygen (CNO) cycle. We extend direct measurement of the 19 F(p, γ) 20 Ne break-out reaction down to an unprecedentedly low energy point of 186 keV and discover a key resonance at 225 keV. In the domain of astrophysical interest, at around 0.1 giga kelvin, this thermonuclear 19 F(p, γ) 20 Ne rate is up to a factor of 7.4 larger than the previous recommended rate. Our stellar models show a stronger break-out during stellar hydrogen burning than thought before, and may reveal the nature of Ca production in Pop III stars imprinted on the oldest known ultra-iron poor star, SMSS0313-6708. This result from the China Jinping Underground Laboratory, the deepest laboratory in the world, offering an environment with extremely low cosmic-ray induced background, has far-reaching implications on our understanding of how the first stars evolve and die. Our rate showcases the impact that faint Pop III star supernovae can have on the nucleosynthesis observed in the oldest known stars and first galaxies, key mission targets of the James Webb Space Telescope.
Stars are the nuclear forges of the cosmos, responsible for the creation of most elements heavier than helium in the Universe. Some of these elements are created in the hearts of stars over the course of billions of years, whereas others are formed in just a few seconds during the explosive deaths of massive stars. These heavy elements play an important role in the universe, allowing for the formation of complex molecules and dust which facilitate the cooling and condensation of molecular clouds, aiding the formation of new stars like our Sun. The first generation of stars, called Population III, Pop III stars, or primordial stars, formed from the pristine matter left by the Big Bang, thus play a special role in seeding the universe with the first heavy elements and creating suitable conditions for future generations of stars and galaxies.
Every star, regardless of its mass, spends the majority of its life quiescently fusing hydrogen into helium in its core through two primary mechanisms: the p-p chains and the catalytic carbon-nitrogen-oxygen (CNO) cycles (1)(2)(3). Which mechanism dominates hydrogen burning is determined by the temperature in the core of a star. In stars with initial masses less than ∼1.2 solar masses (M ), with relatively cool cores (T ≤ 0.02 GK), the p-p chains dominate the hydrogen fusion, whereas in stars with higher initial masses and hotter cores, the CNO cycles take over. As a catalytic reaction, the total number of CNO nuclei remains constant, unless a breakout reaction sequence causes a leakage toward the NeNa mass region, or if temperature and density are high enough to forge new carbon by the triple-alpha (3α) process. The latter two occur in primordial massive stars. The only reaction that can potentially remove the catalytic material from the cycle at lower temperatures is the fusion of 19 F with a proton to form 20 Ne, denoted 19 F(p, γ) 20 Ne (4). Previously, this reaction was thought to be rather weak compared to the competing 19 F(p, α) 16 O reaction, so most of the 19 F produced by the CNO cycle would be recycled back into 16 O, with no significant chemical abundance changes (5).
The most metal-poor stars observed in our Milky Way's halo today display the diluted nucleosynthetic signatures resulting from Pop III stars that preceded them (6). Keller et al. (7) discovered one of the oldest known stars in the Universe, SMSS0313-6708, and, based on the stellar models by Heger & Woosely (8), suggested that a CNO breakout during hydrogen burning is the source of calcium (Ca) production, reporting [Ca/H] = -7.2 (7). Takahashi et al. (9) also cited such a breakout as the Ca production mechanism for the stars HE 1327-2326 and HE 0107-5240, with [Ca/H] = -5.3 and -5.13, respectively. Pop III stars begin their lives with primordial Big-Bang composition and contract until the central temperature is high enough (∼0.1 GK) to ignite the 3α-process, creating a small abundance of carbon (10), e.g., X12 C ∼ 10 −9 to serve as a catalyst and initiate the CNO cycles. The stellar evolution simulations of Clarkson & Herwig (11) using the NACRE rate set (12) that supersedes the rates used by Heger & Woosely (8) confirmed that the CNO cycling takes place at a core H-burning temperature of up to ∼0.12 GK. Their nucleosynthesis calculations found that it was unlikely that large amounts of Ca could be produced by hot CNO breakout. Their predicted Ca abundance was between ∼0.8 and nearly 2 dex lower than required by observations of the most metal-poor stars. If, however, ratio of the 19 F(p, γ) 20 Ne and the 19 F(p, α) 16 O reaction rates were a factor of ∼ 6 higher than that reported in the NACRE compilation (12), their models could produce Ca at the level observed in ultra-metal poor stars such as SMSS0313-6708.
SMSS0313-6708 is an ultra-metal poor (UMP) star that is speculated to be a direct decedent of the first generation of stars in the universe that formed after the Big Bang. The observable composition of an UMP star is a time capsule to the environment before the first galaxies formed -complementing the exiting upcoming observations of the James Webb telescope (13), which is now aiming to give a first look at the earliest stars and galaxies.
Here, the (p, γ)/(p, α) rate ratio can provide an invaluable tool to diagnose how the first stars evolved and died, and has far-reaching implications on the stellar modeling. If Ca were produced from such hot hydrogen burning, the Ca produced in the later Si-burning phases can fall back onto a central black hole during the supernova (14), which is a key ingredient in the prevailing faint supernova with efficient fallback scenario. Otherwise, such a scenario has to be revised, or an alternative source must be validated. Other potential sources include a convectivereactive light Pop III i-process (15) or Ca synthesis from explosive burning (16). Therefore, an accurate determination of the 19 F(p, γ) 20 Ne rate around 0.1 GK is extremely important to pin down the origin of Ca made by Pop III stars, as well as validating the stellar evolution models.
In the center-of-mass energy region of primary astrophysical interest (E c.m. < 1 MeV), very limited experimental data are available for the 19 F(p, γ) 20 Ne reaction due to the very strong 6.130-MeV γ-ray background from the competing 19 F(p, αγ) 16 16 O experimental data in the R-matrix framework, and estimated the corresponding rates for these two reactions. The Pop III star Ca production problem, however, was even intensified with their estimated ratio of the 19 F(p, γ) 20 Ne and 19 F(p, α) 16 O rates, where the ratio was about a factor of 4 lower than that of NACRE.
To date, there is a scarcity of experimental data in the energy region below E c.m. ≈ 0.35 MeV.
To provide an accurate thermonuclear rate, it is of paramount importance to directly measure the 19 F(p, γ) 20 Ne reaction cross section in this region. Since the cosmic-ray background radiation is very strong on the Earth's surface, i.e., above-ground laboratories, direct measurements of such small cross sections are extremely challenging. The China Jinping underground laboratory (CJPL) is located in a traffic tunnel of a hydropower station under Jinping Mountain, in the southwest of China (26) with about 2400 m of vertical rock overburden. By this measure, it is the deepest operational underground laboratory for particle and nuclear physics experiments in the world. It offers a great reduction in the muon and neutron fluxes by six and four orders of magnitude, respectively, compared to those at the Earth's surface. The cosmic-ray induced background measured at CJPL (27) is about two orders of magnitude lower than that in LUNA (1400 m thick dolomite rocks) (28). With such a unique ultra-low-background environment, the Jinping Underground Nuclear Astrophysics Experiment (JUNA) (29) was initiated, and we have performed a 19 F(p, γ) 20 Ne direct measurement campaign as one of the Day-one experiments.
The experiment was performed in normal kinematics using a high-current 400 kV electrostatic accelerator (30) 20 Ne experiments. The γ rays were detected using a nearly 4π BGO detector array that was also employed in the preceding JUNA experiments (32,33). The typical γ-ray spectra are shown in Extended Data Fig. 2. Owing to the different detection efficiency, the contribution of the summing γ sum rays (at ∼ 13 MeV) from 19 F(p, γ) 20 Ne reaction has been separated into two components: one involves only the transition to the ground state (g.s.) in 20 Ne, hereafter referred to as (p, γ 0 ); another involves all transitions through the 1.634 MeV first excited state to the g.s. in 20 Ne, hereafter referred to as (p, γ 1 ). This way, the (p, γ 1 ) component can be determined precisely based on the coincident technique described here, because the nearby heavy summing signal (at ∼12 MeV) induced by the 6.130-MeV γ rays interferes with the total counts of the summed γ rays, as do those γ rays from the 11 B(p, γ) 12 C contamination reaction at lower proton energies. As shown in the inset of Extended Data Fig. 2, the 1.634→g.s. transition can be clearly observed by gating on the summing γ sum rays, which correspond to the 19 F(p, γ 1 ) 20 Ne component. Figure 1(a) shows the resulting 1.634 MeV γ-ray yields obtained using this coincidence technique.
A new resonance has been discovered at E c.m. = 225 keV for the first time, well below the well-known resonance at E c.m. = 323 keV. For the known 323 keV resonance, the γ-yield Figure 1: (a) Experimental yields of the 19 F(p, γ 0,1 ) 20 Ne reaction measured at JUNA. Previous experimental (p, γ 1 ) data (22), which overlap with the present energy regime, are shown for comparison. The Geant4 simulated yield curve is depicted using the R-matrix fit ("Fit1").
Here, E p denotes the proton beam energy delivered from the accelerator. (b) Three probable astrophysical S-factor curves for the 19 F(p, γ 1 ) 20 Ne reaction fitted by the R-matrix calculations. Six data points are derived from the present JUNA experiment. The uncertainties are purely statistical. The error bars are invisible where they are smaller than the data-point size. See Extended Data Fig. 4 for fitting covariance matrix. ratios between the (p, αγ) and (p, γ) channels obtained are shown in Extended Data Fig. 3. We determined partial strengths of ωγ (p,γ 1 ) = 2.09 ± 0.21 meV and ωγ (p,γ 0 ) = 1.07 ± 0.21 meV, respectively. Thus, its total strength is determined to be ωγ (p,γtot) = 3.16 ± 0.33 meV, where the statistical and systematical errors are 0.23 meV and 0.24 meV, respectively. The present ωγ (p, γ 1 ) 20 Ne reaction, with total errors listed in the parentheses. R-matrix fit parameters are tabulated, including a sub-threshold and near-threshold 11 keV resonances as shown in Figure 1(b). The fit includes the additional levels and from Ref. (25) as fixed background terms. See Method for details.
(keV) (MeV) ωγ tot (meV) (fm −1/2 ) (eV) E c.m. E x J π Present NACRE (12) ANC Γ α2 Γ γ1 −448 12.396(4) a 1 + 15 b 60 +40 −30 <3.4 11 12.855(4) a 1 + 1.14×10 −28 c −590 +230 −290 <4.8 212.7(10) 13.057 2 − <4.2×10 −3 <1.3(13)×10 −3 225.2(10) 13.069 3 − 4.19(33)×10 −2 323.9 13.168(2) b 1 + 3.16(33) 5(3)
value is about a factor of 1.5 larger than the previous value of 1.38±0.44 meV (22). Both values agree within a 2-σ uncertainty, but our value has much improved precision. In addition, our total value of ωγ (p, γtot) is consistent with the NACRE adopted value of 5 ± 3 meV, as well as with the recently reported value of 3.3 +1.1 −0.9 meV (24) that was also sensitive to the direct capture to g.s., but with 4-7 times better precision. For the newly observed 225 keV resonance, its strength is determined to be ωγ (p, γ 1 ) = (4.19 ± 0.33)×10 −2 meV based on the yield ratio between the 323 keV (p, αγ) and 225 keV (p, γ 1 ) resonances. Our resonance strengths were all determined relatively to the well-known (p, αγ) strength of the E c.m. = 323 keV resonance. Here, the yield corresponds to the integrated γ-ray counts (corrected for efficiency) under the yield curve over the resonance. We find that the (p, γ 0 ) contribution is negligibly small in the energy region below E c.m. ≈ 322 keV (see Figure 1(a)), and hence ωγ (p, γtot) ≈ ωγ (p, γ 1 ) for this resonance.
For the previously theorized E c.m. = 213 keV resonance, estimates placed upper limits on the strength at 1.3 × 10 −3 meV (12) and 9.3 × 10 −4 meV (22); we now firmly constrain its strength to be < 4.2 × 10 −3 meV (i.e., less than 10% of that of the 225 keV resonance) based on the present experimental data. Table 1 summarizes the resonance properties.
A multilevel, multichannel R-matrix analysis, using the code AZURE2 (34,35), was used to fit the data. The R-matrix analysis is an extension of that presented in Ref. (25), and includes all those data together with the new CJPL (p, γ 1 ) and (p, αγ) data. With this method, various possible contributions can be strictly constrained. The curve shown in Figure 1(a) represents the Geant4 (36) simulated results by using one of the lowest χ 2 R-matrix fits ("Fit1") to the S-factor data. Figure 1(b) shows six off-resonance data points derived from the present JUNA experiment. Numeric samples of the S-factors and the associated uncertainties in the offresonance region are tabulated in Extended Data Table 1. We present the three best R-matrix fits. Here, "Sub" denotes the 1 + sub-threshold state at E x = 12.396 MeV, and "11 keV"denotes the 11 keV 1 + resonance at E x = 12.855 MeV. For example, the label "Fit1: Sub, 11 keV"
indicates the R-matrix fit including both the sub-threshold state and the 11 keV resonance. The nuclear level properties in the R-matrix fits were co-varied over a large parameter space. The resultant resonance properties deduced from the R-matrix fits are listed in Table 1 (25) is also shown for comparison. The associated uncertainties are shown as the colored bands. The inset shows the ratios at a temperature of 0.14 GK. ment factor of 5.4-7.4 for the (p, γ)/(p, α) rate ratio relative to that of NACRE (12) at around 0.1 GK. We find an even larger enhancement below ∼ 0.08 GK. By a simple scaling argument to the model calculations in Refs. (11,25), the observed Ca abundances in the oldest known SMSS0313-6708 star can now be reproduced reasonably with our new 19 F(p, γ) 20 Ne rate.
0 . 0 1 0 . 1 1 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 R a t e / N A C R E T 9 J U N A d e B o e r N A C R E N A C R E d e B o
We have investigated the impact of the thermonuclear 19 F(p, γ) 20 Ne rate on a range of nucleosynthesis modelling techniques, and the calcium production is summarized in Figure 3 (Left Panel ). Our studies comprise simple trajectories (see Extended Data Table 3), new mix models (see Extended Data Table 4), and full stellar models (see Extended Data Table 5) calculations.
We find that all our nucleosynthesis models can reproduce the observed calcium production. We conclude that the 40 Ca observed in the oldest known ultra-iron poor stars (e.g., SMSS0313-6708) may indeed originate in hydrostatic burning in Pop III stars, requiring only the supernova ejection of their outer layers, whereas the metal-rich core may collapse to a black hole. Previously, the ejection of the metal-rich core was required as the source of calcium abundance observed in the oldest stars. On the contrary, here we show a much stronger breakout from a "warm" CNO cycle scenario via 19 F(p, γ) 20 Ne, which significantly increases the production of Ne -Ca. Figure 4 shows the ratio of final abundances of using our JUNA mean 19 F(p, γ) rate compared to using the NACRE mean rate. The production of all elements beyond Z = 9
is shifted by a constant factor and hence can be well represented by single species, e.g., the double-magic nucleus 40 Ca that was observed in ultra-metal poor stars. This clearly shows the bottle-neck nature of the 19 F(p, γ)-19 F(p, α) branching point. See Supplemental Information for more details.
To conclude, we have directly measured the 19 F(p, γ) 20 Ne reaction down to the unprecedentedly low energy point of E c.m. ≈ 186 keV by exploiting the extremely low background environment deep underground, high-intensity beam and newly developed durable target(s).
All these unique and featured conditions allowed us to measure this crucial reaction at the stellar energy region, which is inaccessible in the above-ground laboratories for decades. We have discovered a new key resonance and determined for the first time a precise thermonuclear rate over the temperature region of astrophysical importance. Our enhanced new rate leads to a stronger breakout in a "warm" CNO scenario as the origin of the calcium discovered in the oldest, ultra-iron poor stars. Our results provide a strong experimental foundation to the faint supernova model of first-generation primordial stars as source for the observed chemical abundance signature. The astrophysical implications of our new rate on novae, X-ray bursts, AGB stars, and other star sites are still subject to future detailed investigations. By adjusting the beam intensity in each run, the counting rate of the BGO array was limited to about 10 kHz to prevent the signal pileups and reduce the dead-time of the DAQ system. In addition, the waveforms of pulses were recorded in the DAQ system to monitor the pileup events during the experiment. We found that the pileup events are very rare, and can be completely ignored.
Acknowledgments
Extended Data Fig. 2 shows the typical γ-ray spectra taken for two typical energy points, (a) at E p = 356 keV and (b) at E p = 250 keV. Here, E p denotes the proton beam energy delivered from the accelerator, and the real bombarding energy on the fluorine atoms is reconstructed by taking into account the energy loss through the Cr protective layer with a Geant4 simulation (36). It shows that the 6.130-MeV γ rays (from the αγ 2 channel) dominate the whole spectra, while the 6.917-MeV (from the αγ 3 channel) and 7.117-MeV (from the αγ 4 channel) γ rays observed at certain proton energies, only make a maximum contribution of ≈2.4% in the energy region studied in this work. Here, we are mainly concerned with the summing γ-ray peak for the targeted 19 F(p, γ) 20 Ne channel around 13 MeV. The γ rays induced by the 11 B, 12 C and 13 C contaminants were observed at certain energies, and their origins were clearly identified (31,32). The 19 F target material loss was monitored and found to be negligible since the total beam dose utilized in this measurement was only about 41 C, which was consistent with prior expectations (31).
A precise determination of the absolute 19 F number density is challenging because of the complicated target structure and the unknown self-sputtering rate during the implantation procedure. Similar to previous work (32), we derived the (p, γ) strengths of the 225 keV and 323 keV resonances relative to the well-known (p, αγ) strength of the 323-keV resonance. Its strength was evaluated as ωγ (p, αγ) = 23.1 ± 0.9 eV in NACRE, i.e., with an uncertainty of about 4%.
For the 323-keV resonance, the ratios between (p, αγ) and (p, γ) yields are obtained at five energy points over the resonance, by comparing the corresponding γ-ray counts corrected by the efficiency. The corresponding ratios are shown in Extended Data Fig. 3. The weighted average ratios and the associated uncertainties are plotted as the solid and dashed lines, respectively.
We find weighted average ratios of (p, αγ)/(p, γ 1 ) and (p, αγ)/(p, γ 0 ) of (1.11±0.07)×10 4 and (2.15±0.39)×10 4 , respectively.
Astrophysical S factors. Selected astrophysical S factors derived for the 19 F(p, γ) 20 Ne reaction in the non-resonance region are listed in Extended Data Table 1, which are shown in Figure 1(b) (with statistical uncertainties shown only). Here, the statistical uncertainties range from 8.3% to 27.4% as listed in the last column of Extended Data Table 1. The systematic uncertainties mainly include the following contributions: 1) a 5% uncertainty estimated for the Geant4 simulation by assuming a 0.5 keV uncertainty in the reconstructed E c.m. energy; 2) a 3.9% uncertainty of the 323-keV resonance strength (from the normalization); and 3) a 5-10% uncertainty of the 1634-keV γ-ray coincidence efficiency. From this, conservatively, we estimate an overall systematic uncertainty of 12%.
R-matrix fit. The temperature relevant to the Population III stars is about 0.1-0.12 GK, corresponding to an energy range around 100 keV. At such low energies, the Coulomb repulsion between the two interacting particles -proton and 19 F -makes the cross sections so small that their laboratory measurement is very challenging due to the low event rate. Therefore, mea-surements are typically made at higher energies, and then a model with underlying physical motivation is used to extrapolate to the low energies of interest. In low energy nuclear physics, R-matrix analysis is one of the most successful of these phenomenological reaction models.
The model is both very flexible, applicable to a wide verity of different reactions, yet still has fundamental physical constraints.
At JUNA, we have obtained both 19 F(p, γ) 20 Ne and 19 F(p, αγ) 16 O cross section data. The latter was already described elsewhere (32). Whereas the new 19 F(p, γ) 20 Ne cross section measurements extend to lower energies than any previous measurement, they are still higher in energy than the energy range of interest to astrophysics. Therefore the AZURE2 (34,35) R-matrix code has been used to simultaneously fit both reactions using our new data. This R-matrix analysis is an extension of earlier work presented in Ref. (25).
It still remains unknown which resonance contributions dominate at very low energies, thus several R-matrix fits were attempted, taking into account different contributions from either a subthreshold resonance ("Sub") or a near-threshold resonance at 11 keV ("11 keV"). The three most probable fit solutions are shown in Figure 1(b). To quantify the uncertainty stemming from the experimental data and the ambiguity in the low energy resonance structure, a Bayesian uncertainty analysis has been performed (38). Extended Data Fig. 4 shows the covariance matrix from an MCMC analysis that includes Γ γ 1 for both the sub-threshold and near-threshold resonances. The data indicate that at least one of these components is needed. When both are included, the MCMC analysis indicates a non-zero contribution from the sub-threshold resonance contribution and a value that is consistent with zero for the near-threshold resonance. This analysis quantifies an upper limit for the S-factor extrapolation as indicated in Figure 1(b) at the 68% level. A lower limit is calculated using an analysis that only includes the sub-threshold resonance, resulting in a nearly constant low energy S-factor. In main Table 1 Reaction Rates. At each temperature point in Figure 2, three reaction rates were calculated based on the three S-factor curves shown in Figure 1(b). The maximum and minimum of the three rates were adopted as the high and low limits, and the average of the maximum and minimum was adopted as the recommended median rate. Where the rate errors (i.e., low and high limits) are smaller than those caused by the JUNA S-factor errors, the S-factor errors were adopted accordingly as the total reaction rate error (i.e., low and high limits). In this way, the present median rate and the associated uncertainties are obtained in a temperature region of 0.01-1 GK. Beyond 1 GK, the NACRE rates can be used. Extended Data Table 2 lists the presently recommended thermonuclear 19 F(p, γ) 20 Ne rates, and the associated uncertainties (low and high values).
The present mean rate can be parameterized by the standard format of (40),
N A σv = exp(−8.41786 − 3.8921 T 9 + 30.2621 T 1/3 9 − 1.33705 ln T 9 )
with a fitting error of less than 1% over the temperature region of 0.01-1 GK.
Astrophysical calculations. We have investigated the impact of thermonuclear 19 F(p, γ) 20 Ne rate on a range of nucleosynthesis modelling techniques. We have performed a range of full stellar model calculations for a 40 M star of initially primordial composition with the KE-PLER code (41). The calcium production is briefly summarized in Figure 3 1.98×10 −01 0.5 1.31×10 +00 1.20×10 +00 1.45×10 +00 1.63×10 +00 6.04×10 −01 1.08×10 +00 0.6 4.81×10 +00 4.39×10 +00 5.30×10 +00 6.19×10 +00 2.90×10 +00 4.04×10 +00 0.7 1.
46×10 +01 1.32×10 +01 1.60×10 +01 1.92×10 +01 1.04×10 +01 1.22×10 +01 0.8 3.73×10 +01 3.38×10 +01 4.09×10 +01 4.88×10 +01 2.88×10 +01
3.03×10 +01 0.9 8.13×10 +01 7.41×10 +01 8.95×10 +01 1.04×10 +02 6.41×10 +01 6.36×10 +01 1.0 1.56×10 +02 1.42×10 +02 1.72×10 +02 1.93×10 +02 1.21×10 +02 1.17×10 +02 Extended Data Table 3. Calcium yields (logarithm base 10 values) for fixed trajectory with constant ρ = 39.8 g cm −1 , T = 1.19 × 10 8 K and primordial initial composition (42). Williams Low -11.74 -11.88 -11.99 -11.81 -11.85 -12.11 Mean -11.66 -11.80 -11.91 -11.73 -11.77 -12.03 High -11.57 -11.71 -11.82 -11.64 -11.69 -11.94
Extended Data Table 5. Calcium yields (logarithm base 10 values) for full stellar models.
The first data column gives the average abundance over the entire star at the terminal-age main-sequence (TAMS), defined by a core hydrogen mass fraction of 0.01, consistent with what we use elsewhere. The last three columns list the average calcium mass fraction at the pre-supernova stage: in the hydrogen envelope (hydrogen mass fraction ≥ 0.01), in the helium shell (helium mass fraction ≥ 0.01 and hydrogen mass fraction < 0.01), and in the combination of both (helium mass fraction ≥ 0.01), respectively.
of the R-matrix calculations). The thermonuclear 19 F(p, γ) 20 Ne reaction rate as a function of temperature is calculated by numerical integration of the S-factors shown in Figure 1(b) (2). The mean rate and the associated uncertainties (Low and High limits) are obtained in a temperature region of 0.01-1 GK and presented in Extended Data
Figure 2 :
2Ratio of the present (labelled as JUNA) relative to NACRE's rate(12). The corresponding ratio for deBoer et al.'s rate
Figure 3 :
3(Left Panel) Range of results for different rate combinations and different modelling techniques. (I) is for trajectories of fixed temperature and density; (II) is for time-dependent trajectories (Figure S2) that include the effect of mixing due to convection; (III) is for yields from full stellar models. (Right Panel) The four classical CNO cycles (4) (solid lines) and the hot CNO cycle shortcut (black dotted lines). The breakout 19 F(p, γ) 20 Ne reaction route is indicated as red dotted arrow. See Supplemental Information for more details.
Figure 4 :
4Ratio of final abundances of using our JUNA mean 19 F(p, γ) rate compared to using the NACRE (12) mean rate. Both use the NACRE mean 19 F(p, α) rate. The ratio is shown at core hydrogen depletion, at a hydrogen mass fraction of 0.01 and using the mixing model. The linear size of the symbol indicates the logarithm of the absolute mass fraction (see legend ). Their numerical values are given above/below the symbols.
We thank the staff of the CJPL and Yalong River Hydropower Development Company (N.C. Qi, W.L. Sun, X.Y. Guo, P. Zhang, Y.H. Chen, Y. Zhou, J.F. Zhou, J.R. He, C.S. Shang, M.C. Li) for logistics support. We thank F. Herwig, Y. Sun and S.E. Malek for discussions. We acknowledge support from the National Natural Science Foundation of China (Nos. 11825504, 11490560, 12075027, 12125509). R.D. and M.W. were supported by the NSF through Grant No. Phys-2011890. R.D., M.W., and A.H. were supported by the Joint Institute for Nuclear Astrophysics through Grant No. PHY-1430152 (JINA Center for the Evolution of the Elements). A.H. was supported by the Australian Research Council (ARC) Centre of Excellence (CoE) for Gravitational Wave Discovery (OzGrav) through project number CE170100004, by the ARC CoE for All Sky Astrophysics in 3 Dimensions (ASTRO 3D) through project number CE170100013. D.K. acknowledges the support of the Romanian Ministry of Research and Innovation under research contract 10N/PN 19 06 01 05. Author contributions: M.W. proposed the original idea of this research. J.H. and W.L. proposed this JUNA experiment. L.Z., J.H. designed the experimental setup and led all the tests and experiments, and performed the data reduction and analysis. R.D., and M.W. performed the R-matrix analysis. A.H. made the astrophysical model calculation and interpretation. J.S., Y.C., X.L., H.Z., X.J., L.W., Z.L., and L.S. participated in the experiment. J.H., A.H., D.K., R.D., M.W., W.L. prepared the draft of the manuscript. D.K. made major contributions to the manuscript polishing. All authors read the manuscript, gave comments, suggested changes, and agreed with the final version. L.Z., F.C., Y.C., and Z.Z. took main responsibility for the operation of the JUNA accelerator. J.S., and Z.L. developed the 4π BGO detector array, and J.W. developed the DAQ system. L.S., Q.W., J.L., and H.Z. designed and constructed the ECR ion source. B.C., L.C., R.M., and G.L. designed and constructed the 400-kV accelerator. J.H. supervised the experiment and verified that the data were acquired correctly as a P.I. of this sub-project. W.L. leads the JUNA project, and J.C. leads the CJPL. Competing interests: The authors declare no competing interests. Data availability Experimental data taken at JUNA are proprietary to the collaboration but can be made available from the corresponding authors upon reasonable request. Code availability The R-matrix code can be made available upon request to R.D. (e-mail: [email protected]). Extended data is available for this paper at https://doi.org/10.1038/s41586-022-05230-x Correspondence and requests for materials should be addressed to Jianjun He, Michael Wiescher or Weiping Liu. 35. Uberseder, E. & deBoer, R. J. AZURE2 User Manual (2015). 36. Agostinelli, S. et al. Geant4--a simulation toolkit. Nucl. Instrum. Meth. A 506, . The Jinping Underground Nuclear Astrophysics Experiment (JUNA) (29) was initiated in 2015. One of the Day-one goals (37) was to directly measure the 19 F(p, αγ) 16 O reaction at Gamow energies. The measurement was accomplished and results were published elsewhere (32). The present 19 F(p, γ) 20 Ne experiment was immediately followed that (p, αγ) run with the same experiment setup, acting as one of the Day-one campaigns. In combination with the ultra-low background environment, the strong beam intensity, the durable target, as well as the coincidence technique, it ultimately makes this direct 19 F(p, γ) 20 Ne measurement possible. The schematic view of the experimental setup is shown in Extended Data Fig. 1. A proton beam from the accelerator was undulated over a rectangular area of about 4×4 cm 2 by oscillating the magnetic field of the beam deflector. A well-focused, intense beam was uniformly spread across the target, mitigating damage to the target. The scanning proton beam was collimated by two apertures (φ15 upstream and φ12 mm downstream) and then impinged on a water-cooled target, where the beam current reached up to 1 mA, with a spot size of about φ10 mm. An inline Cu shroud cooled to LN 2 temperature extended close to the target to minimize carbon build-up on the target surface. Together with the target, the Cu shroud constituted the Faraday cup for beam integration. A negative voltage of 300 V was applied to the shroud to suppress secondary electrons from the target. A very strong and durable implanted 19 F target (31) was utilized in this work. The optimum scheme for target production is: first, implanting 19 F ions into the pure Fe backings with an implantation energy of 40 keV, and then sputtering a 50-nm thick Cr layer to further prevent the fluorine material loss. The 4π Bi 4 Ge 3 O 12 (BGO) detector array specially designed for the JUNA project is composed of eight identical segments with a length of 250 mm and a radial thickness of 63 mm, each covering a 45 • azimuthal angle. For the 6.130-MeV γ rays, the total absolute detection efficiency was ≈58%, with a ≈6% energy resolution achieved by alcohol-cooling the BGO crystals (≈-5 • C). To further suppress the natural background emitted from the rocks and induced by neutron capture reactions, the BGO array was passively shielded by 5-mm copper, 100-mm lead and 1-mm cadmium, respectively.
, E x values fixed to those determined in previous analyses are indicated by a (39), b (25) and c (23), where the corresponding uncertainties are adopted. While for the present results, an assumed uncertainty of 1.0 keV are quoted on the resonance energies. The sign of a partial width indicates the sign of the corresponding reduced width amplitude. No uncertainties are quoted for the ANCs of both the bound and near threshold states as none are available from previous literature. Further, the present data only constrain the product of the ANC and the γ-width and we have chosen to indicate this uncertainty on the γ-width. The details will be published elsewhere.
Fig. 3 :
3Fig. 2: (Left Panel) Typical γ-ray spectra taken with a 4π BGO array at JUNA during proton bombardment of an implanted 19 F target, at proton energies of (a) 356 keV and (b) 250 keV. The heavy γ-ray background from the competing 19 F(p, αγ)16 O channel and their summing signals are indicated. The summing γ-ray peak for the target 19 F(p, γ) 20 Ne channel is indicated by red arrows. The inset shows the coincident γ-ray spectrum gated on the summing peak located in the shaded region, where several γ-ray transitions are observed, and locations are illustrated in the corresponding level scheme (Right Panel) . Yield ratio of (p, αγ)/(p, γ 1 ) (Upper Panel ) and that of (p, αγ)/(p, γ 0 ) (Lower Panel ) over the 323-keV resonance (statistical error only). The weighted average ratios and the associated uncertainties are plotted as solid and dashed lines, respectively.
.03×10 −25 2.91×10 −27 2.05×10 −25 4.25×10 −28 1.10×10 −28 8.17×10 −28 0.015 1.30×10 −21 8.78×10 −23 2.52×10 −21 1.37×10 −23 3.50×10 −24 2.61×10 −23 0.02 5.16×10 −19 5.80×10 −20 9.73×10 −19 9.45×10 −21 2.38×10 −21 1.79×10 −20 0.03 9.45×10 −16 1.94×10 −16 1.70×10 −15 3.41×10 −17 8.44×10 −18 6.45×10 −17 0.04 1.08×10 −13 3.15×10 −14 1.84×10 −13 5.95×10 −15 1.44×10 −15 1.11×10 −14 0.05 3.13×10 −12 1.16×10 −12 5.09×10 −12 2.34×10 −13 5.57×10 −14 4.38×10 −13 0.06 4.07×10 −11 1.79×10 −11 6.33×10 −11 3.84×10 −12 8.98×10 −13 7.19×10 −12 0.07 3.15×10 −10 1.59×10 −10 4.70×10 −10 3.60×10 −11 8.25×10 −12 6.65×10 −11 0.08 1.70×10 −09 9.57×10 −10 2.45×10 −09 2.28×10 −10 5.15×10 −11 4.21×10 −10 0.09 7.15×10 −09 4.39×10 −09 9.91×10 −09 1.10×10 −09 2.46×10 −10 2.02×10 −09 0.1 2.53×10 −08 1.69×10 −08 3.39×10 −08 4.31×10 −09 9.93×10 −10 7.76×10 −09 0.15 5.68×10 −06 4.86×10 −06 6.51×10 −06 9.55×10 −07 3.36×10 −07 1.18×10 −06 0.2 2.58×10 −04 2.35×10 −04 2.84×10 −04 8.67×10 −05 2.78×10 −05 6.92×10 −05 0.3 2.18×10 −02 1.98×10 −02 2.40×10 −02 1.93×10 −02 5.63×10 −03 1.33×10 −02 0.4 2.60×10 −01 2.37×10 −01 2.87×10 −01 2.93×10 −01 8.99×10 −02
O channel. This makes measurements of such small 19 F(p, γ) 20 Ne cross section extremely difficult. Most of the previous exper-iments detected the >11 MeV primary transition to the first excited state of 20 Ne (17-21) using small-volume NaI(Tl) detectors with low resolution and efficiency. The earlier measurements also suffered from pileup from the 6.130-MeV γ rays because of insufficient energy resolution to separate the two components. Later, Couture et al. (22) developed a coincident detection technique (between HPGe and NaI detectors) to measure the 19 F(p, γ) 20 Ne and 19 F(p, αγ) 16 O reactions over an energy range of E c.m. = 200-760 keV. Due to their limited sensitivity, only an upper-limit for the strength of the E c.m. = 213 keV resonance was given and no estimate was made for the 225 keV resonance, although it had been observed as a resonance in the 19 F(p, αγ) 16 O reaction by Spyrou et al. (23). Williams et al. (24) measured a factor of 2 larger strength value than that of Couture et al. for the 323-keV resonance by using the inverse kinematics method, because their measurement also included contribution owing to the ground-state transition. Recently, deBoer et al. (25) reanalyzed the available 19 F(p, γ) 20 Ne and 19 F(p, α)
at CJPL. A well-focused, high intensity proton beam uniformly impinged on a 19 F water-cooled target with a current up to ≈1 mA. The experimental setup is shown in Extended Data Fig. 1. Durable implanted 19 F targets (31) were used in both 19 F(p, αγ) 16 O (32) and 19 F(p, γ)
Table 1 :
1Relevant resonance strengths ωγ tot determined for the 19 F(p, γ)
Table 2 .
2The ratios between the present rate and the NACRE recommended rate are shown in Figure 2. It shows that our new rate is enhanced by a factor of 5.4-7.4 at the temperature around 0.1 GK. This enhancement is attributed to the newly observed 225 keV resonance. In addition, our new rate is about 200 times larger at temperatures around 0.01 GK mainly because of the 11 keV resonance (32). The uncertainty of the present rate is drawn as a colored band, which we estimate based on the uncertainties of the resonance strengths and R-matrix calculations. The uncertainties in the present S-factor and rate over the range of astrophysical interest have been significantly reduced compared to previous estimates (25). deBoer et al. (25) recommended a 19 F(p, α) 16 O mean rate quite similar to that of NACRE (12).Thus, we adopt NACRE's 19 F(p, α)16 O rate as our reference, and hence obtain an enhance-
Extended DataTable 1. Selected astrophysical S factors for 19 F(p, γ) 20 Ne derived in this work. The total uncertainties are listed in the parentheses, and the statistical uncertainties are listed in the last column. Conservatively, we estimate an overall systematical uncertainty of 12%.E COM (keV) S-factor (MeV·b) Statistical uncertainty (%)
186.4
0.0140(0.0040)
26.1
195.5
0.0132(0.0040)
27.4
252.7
0.0125(0.0018)
8.3
273.8
0.0129(0.0028)
18.3
283.1
0.0262(0.0050)
14.9
291.5
0.0304(0.0047)
9.7
61.23 +40.19 31.14
1 .
Extended DataFig. 4. Corner plot of the covariance matrix for an MCMC analysis of the R-matrix fit. The vertical dashed lines indicate 16%, 50%, and 84% quantiles. Here 'sub' refers to the subthreshold state at E x = 12.396 MeV, 'thresh' the near threshold state at E x = 12.855 MeV, and 'n pα ' and 'n pγ ' are the normalization factors for the (p, γ) and (p, α) data sets, respectively. Uniform priors were taken for all parameters of the analysis.Extended DataTable 2. Thermonuclear reaction rates of 19 F(p, γ) 20 Ne in units of cm 3 s −1 mol −1 . The rates are for the bare 19 F nuclei in the laboratory, i.e., no thermally excited target states are considered.1.03 +0.06 0.06
Present rate
NACRE (3) deBoer21 (5) Williams21 (27)
The trajectories were run until the hydrogen mass fraction dropped below 0.01. Other nondegenerate binary reactions are taken from REACLIB v2.2. The reference rate for just using the original REACLIB rates for all binary reactions gives a mass fraction of log( 40 Ca) = −12.33. 19 F(p, γ) rate NACRE 19 F(p, α) rate deBoer 19 F(p, α) rate Low Mean High Low Mean High JUNA Low -11.59 -11.72 -11.83 -11.70 -11.68 -11.90 Mean -11.47 -11.60 -11.70 -11.57 -11.55 -11.77 High -11.37 -11.50 -11.60 -11.48 -11.46 -11.67 Low -13.25 -13.38 -13.48 -13.36 -13.34 -13.55 Mean -12.81 -12.94 -13.05 -12.92 -12.90 -13.12 High -12.05 -12.18 -12.28 -12.16 -12.13 -12.35 Extended Data Table 4. Similar to Extended Data Table 3 but uses the actual central temperature trajectory of a 40 M Population III star model (8) and using a mixing model to emulate convection. The trajectory was run until a core hydrogen mass fraction of 0.01. The mixing model assumes that the trajectory represents a fraction of τ mix = 0.0595 of the total reservoir such that burning reduced the H mass fraction in the trajectory to 0.01, consistent with the stellar model. The reference rate for just using the original REACLIB rates for all binary reactions is log( 40 Ca) = −12.02. 19 F(p, γ) rate NACRE 19 F(p, α) rate deBoer 19 F(p, α) rate Low Mean High Low Mean High JUNA Low -11.29 -11.44 -11.54 -11.36 -11.41 -11.66 Mean -11.14 -11.29 -11.39 -11.22 -11.26 -11.51 High -11.03 -11.18 -11.29 -11.11 -11.15 -11.41 Low -13.13 -13.27 -13.38 -13.20 -13.24 -13.49 Mean -12.52 -12.66 -12.77 -12.59 -12.63 -12.89 High -11.72 -11.86 -11.97 -11.79 -11.83 -12.09NACRE
Low
-12.51 -12.64 -12.75
-12.62 -12.60 -12.82
Mean
-12.21 -12.35 -12.45
-12.32 -12.30 -12.52
High
-12.04 -12.17 -12.27
-12.15 -12.13 -12.34
deBoer
Williams
Low
-12.07 -12.21 -12.31
-12.18 -12.16 -12.38
Mean
-11.99 -12.13 -12.23
-12.10 -12.08 -12.30
High
-11.91 -12.04 -12.15
-12.02 -12.00 -12.22
NACRE
Low
-12.19 -12.33 -12.44
-12.26 -12.31 -12.56
Mean
-11.89 -12.04 -12.14
-11.97 -12.01 -12.26
High
-11.72 -11.86 -11.97
-11.79 -11.83 -12.09
deBoer
19 F(p, γ) rate 19 F(p, α) rate TAMS Pre-supernova Star H envel. He shell H+He envel.JUNA
Low
NACRE
High
-10.81
-11.18
-10.74
-11.13
Mean
Mean
-11.10
-11.47
-11.05
-11.41
High
Low
-11.41
-11.77
-11.39
-11.73
NACRE
Low
NACRE
High
-11.56
-11.93
-11.51
-11.88
Mean
Mean
-11.90
-12.27
-11.86
-12.21
High
Low
-12.31
-12.68
-12.25
-12.63
deBoer
Low
NACRE
High
-11.56
-11.92
-11.49
-11.88
Mean
Mean
-12.51
-12.88
-12.46
-12.85
High
Low
-13.62
-13.98
-13.57
-13.94
Williams
Low
NACRE
High
-11.40
-11.77
-11.39
-11.73
Mean
Mean
-11.64
-12.01
-11.62
-11.96
High
Low
-11.83
-12.20
-11.83
-12.16
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| []
|
[
"Transforming Feature Space to Interpret Machine Learning Models",
"Transforming Feature Space to Interpret Machine Learning Models",
"Transforming Feature Space to Interpret Machine Learning Models",
"Transforming Feature Space to Interpret Machine Learning Models"
]
| [
"Alexander Brenning \nMichael Stifel Center Jena for Data-Driven and Simulation Science (MSCJ)\nGeographic Information Science group\nDepartment of Geography\nFriedrich Schiller University Jena\nGermany\n",
"Alexander Brenning \nMichael Stifel Center Jena for Data-Driven and Simulation Science (MSCJ)\nGeographic Information Science group\nDepartment of Geography\nFriedrich Schiller University Jena\nGermany\n"
]
| [
"Michael Stifel Center Jena for Data-Driven and Simulation Science (MSCJ)\nGeographic Information Science group\nDepartment of Geography\nFriedrich Schiller University Jena\nGermany",
"Michael Stifel Center Jena for Data-Driven and Simulation Science (MSCJ)\nGeographic Information Science group\nDepartment of Geography\nFriedrich Schiller University Jena\nGermany"
]
| []
| Model-agnostic tools for interpreting machine-learning models struggle to summarize the joint effects of strongly dependent features in high-dimensional feature spaces, which play an important role in pattern recognition, for example in remote sensing of landcover. This contribution proposes a novel approach that interprets machine-learning models through the lens of feature space transformations. It can be used to enhance unconditional as well as conditional post-hoc diagnostic tools including partial dependence plots, accumulated local effects plots, or permutation feature importance assessments. While the approach can also be applied to nonlinear transformations, we focus on linear ones, including principal component analysis (PCA) and a partial orthogonalization technique. Structured PCA and diagnostics along paths offer opportunities for representing domain knowledge. The new approach is implemented in the R package wiml, which can be combined with existing explainable machine-learning packages. A case study on remote-sensing landcover classification with 46 features is used to demonstrate the potential of the proposed approach for model interpretation by domain experts. | 10.1007/s10994-023-06327-8 | [
"https://export.arxiv.org/pdf/2104.04295v1.pdf"
]
| 233,204,490 | 2104.04295 | 9908b289989061e0d6fc65063b323e17be8be19d |
Transforming Feature Space to Interpret Machine Learning Models
Alexander Brenning
Michael Stifel Center Jena for Data-Driven and Simulation Science (MSCJ)
Geographic Information Science group
Department of Geography
Friedrich Schiller University Jena
Germany
Transforming Feature Space to Interpret Machine Learning Models
Model-agnostic tools for interpreting machine-learning models struggle to summarize the joint effects of strongly dependent features in high-dimensional feature spaces, which play an important role in pattern recognition, for example in remote sensing of landcover. This contribution proposes a novel approach that interprets machine-learning models through the lens of feature space transformations. It can be used to enhance unconditional as well as conditional post-hoc diagnostic tools including partial dependence plots, accumulated local effects plots, or permutation feature importance assessments. While the approach can also be applied to nonlinear transformations, we focus on linear ones, including principal component analysis (PCA) and a partial orthogonalization technique. Structured PCA and diagnostics along paths offer opportunities for representing domain knowledge. The new approach is implemented in the R package wiml, which can be combined with existing explainable machine-learning packages. A case study on remote-sensing landcover classification with 46 features is used to demonstrate the potential of the proposed approach for model interpretation by domain experts.
Introduction
Interpreting complex nonlinear machine-learning models is an inherently difficult task. A common approach is the post-hoc analysis of black-box models for dataset-level interpretation (Murdoch et al. 2019) using model-agnostic techniques such as the permutation-based variable importance, and graphical displays such as partial dependence plots that visualize main effects while integrating over the remaining dimensions .
These tools are so far limited to displaying the relationship between the response and one (or sometimes two) predictor(s), while attempting to control for the influence of the other predictors. This can be rather unsatisfactory when dealing with a large number of highly correlated predictors, which are often semantically grouped. While the literature on explainable machine learning has often focused on dealing with dependencies affecting individual features, e.g. by introducing conditional diagnostics (Strobl et al. 2008;, no practical solutions are available yet for dealing with model interpretation in highdimensional feature spaces with strongly dependent features Molnar, König, Herbinger, et al. 2020).
These situations routinely occur in environmental remote sensing and other geographical and ecological analyses (Landgrebe 2002;Zortea, Haertel, and Clarke 2007), which motivated the present proposal to enhance existing model interpretation tools by offering a new, transformed perspective. For example, vegetation 'greenness' as a measure of photosynthetic activity is often used to classify landcover or land use from satellite imagery acquired at multiple time points throughout the growing season (Peña and Brenning 2015;Peña, Liao, and Brenning 2017). Spectral reflectances of equivalent spectral bands (the features) are usually strongly correlated within the same phenological stage since vegetation characteristics vary gradually. Similarly. when using texture features to characterize image structure based on a filter bank, features with similar filter settings can be strongly correlated, as is in the case in our case study (Brenning, Long, and Fieguth 2012).
Although it may be tempting in these situations to use feature engineering or feature selection techniques to reduce the complexity of feature space, experience shows that this may lead to a decline in predictive performance. Also, re-training a model using modified features is not normally an option in post-hoc analyses of machine-learning models.
Considering these challenges, and the inherent need to reduce the complexity at the time of interpreting an already trained model, a novel strategy to visualize machine-learning models along cross-section through feature space is proposed in this paper. In many situations, principal components (PCs) offer a particularly appealing perspective onto feature space from a practitioner's point of view, although the proposed approach is not limited to this transformation. In addition, a modification is proposed that focuses on subgroups of features and their principal axes in order to allow for a more structured approach to model interpretation that more consistent with the data analyst's domain knowledge.
Turning our attention back to the importance of individual features, an orthogonalization technique that can be used to single out the effect of individual features on model predictions, avoiding the sometimes complex structure of PCs. This approach can, in principle, be applied to arbitrary paths through feature space, such as nonlinear curves defined by domain-specific perspectives, or data-driven transitions between clusters of observations.
The proposed approaches can be combined with commonly used plot types and diagnostics including partial dependence plots, accumulated local effects (ALE) plots, and permutation-based variable importance measures, among other model-agnostic techniques that only have access to the trained model (Apley and Zhu 2020;Molnar 2019). While the focus of this contribution is on visualizing main effects, analyses of conditional relationships may also benefit from this perspective (Strobl et al. 2008;).
Proposed Method
Let's consider a regression or classification model
f : x →f (x) ∈ R
that was fitted to a training sample L in the (original, untransformed) p-dimensional feature space X ⊂ R p . I will assumef (x) ∈ R; in the case of classification problems,f (x) shall therefore represent predictions of some real-valued quantity such as the probability or logit of a selected target class. One of the features, referred to as x s , is selected as the feature of interest, and the remaining features are denoted by x C .
Example: partial dependence plots
In this situation, the partial dependence plot off with respect to x S can formally be defined aŝ
f x S ,P DP (x S ) = E X C f (x S , X C ) = x Cf (x S , x C ) dP (x C ) (Molnar 2019
). This plot, which can be generalized to more than one x s dimension, was introduced by Friedman (2001) to visualize main effects of predictors in machine-learning models.
Partial dependence plots have some disadvantages such as the extrapolation off beyond the region in X for which training data is available (Apley and Zhu 2020;Molnar, König, Herbinger, et al. 2020). This is especially the case when predictors are strongly correlated, as in our case study. Nevertheless, without loss of generality, this simple plot type helps to illustrate the proposed approach.
Transformed feature space
When several predictors are strongly correlated and/or express the same domain-specific concept such as 'early-season vegetation vigour' in vegetation remote-sensing, we may be more interested in exploring the overall effect of these predictors. Principal component analysis (PCA) and related data transformation techniques such as factor analysis are tools that are often used by practitioners to synthesize and interpret multivariate data Basille et al. (2008).
More generally speaking, we could think of bijective (invertible) transformation function
T : X → W ⊂ R p , w = T(x)
that can be used to re-express the features in our data set. We will assume that T is continuous and differentiable. PCA is one such example.
Through the composition of the back transformation T −1 and the model functionf , we can now formally define a modelĝ on W ,ĝ :=f • T −1 which predicts the real-valued response based on 'data' in W although it was trained using a learning sample L ⊂ X in the untransformed space.
We can use this to formally re-express the partial dependence plot as a function of w s :
f w S ,P DP (w S ) = E w C (f • T −1 )(w S , w C ) = w C (f • T −1 )(w S , w C ) dP w C
Note that T −1 , when used only on data in Im T (X) := T(X), does not create x values outside the datasupported region X, and it therefore avoids extrapolation off .
Also, when choosing PCA for T the w variables in T(L) are linearly independent, and statistically independent if L arises from a multivariate normal distribution. Thus, the PCA approach overcomes one of the limitations of partial dependence plots and broadens their applicability.
Orthogonalization approaches
In some instances, PCs (and other multivariate transformations) of large and complex feature sets can be difficult to interpret, and analysts would therefore like to focus on individual features that are perhaps 'representative' of a larger group of features -for example, vegetation greenness in mid-June may be a good proxy for vegetation greenness a few weeks earlier and later, as expressed by other features in the feature set (Peña and Brenning 2015).
This can be addressed by proposing a transformation of X in which w s := x s is retained, while making the remaining base vectors linearly independent of x s . This can be achieved through orthogonalization:
w i := x i − b i x s
where b i equals Pearson's correlation coefficient of x s and x i , and where we assume for simplicity of notation that all features are zero-mean with a unit standard deviation.
This defines a linear transformation T : X → W , which can be represented by its coefficient matrix. Note that T can be inverted using
x i = w i + b i w s
since x s = w s , and assuming that all b i < 1. A related iterative orthogonalization approach has previously been proposed in the context of feature ranking (Adebayo and Kagal 2016).
Dependence plots along paths
Data analysts may more generally want to visualize the effect of a real-valued function of multiple features.
As an example, knowing that several features are strongly correlated, how does the response vary with the mean value of these features, or more generally a linear combination? This information is sometimes hidden in an ocean of individual main effects plots or variable importance measures.
In other situations, there may be simple process-based models that have the potential to provide deeper insights into black-box models. These models may be candidates for an enhancement of feature space, or they might express specific theories or hypotheses.
Any of these transformations can be thought of as a function
p : P ⊂ R → X ⊂ R p , t → p(t)
that defines a one-dimensional path in feature space.
In different use cases there may be different ways of constructing paths of interest to the data analyst:
• A group of strongly positively correlated features could be averaged to obtain an overall signal, or contrasts between groups of features could be calculated. • A linear path can be drawn from one cluster centre to another, where cluster centres c 1 , . . . , c k ∈ X are obtained by unsupervised clustering in feature space (e.g., k-means). The path between clusters 1 and 2 is simply defined as c 1 + tc 2 , etc. • A linear path between user-defined points in feature space, e.g. in remote sensing so-called endmembers representing spectral characteristics of 'pure' surface types such as asphalt or water (Somers et al., 2016).
Evidently, creating a dependence plot for a feature x s is just a special case that follows one of the base vectors of feature space. In other words, without loss of generality we can formally include t as an additional feature in our feature set and denote it with x s . This is just a formal inclusion, without actually offering this new variable to the model for training.
For a formally more accurate treatment, and keeping t out of the feature space X ⊂ R p , we write w s := t and construct the w i 's as in the previous section by orthogonalization, however this time for all i = 1, . . . , p.
Thus, w s comes on top of the other p dimensions, and thus W ⊂ R (p+1) .
Due to the redundancy of w s , the transformation function can now be defined as a mapping T : W → X from (p + 1)-to p-dimensional space, with the x i 's being recovered from the w i 's as in the previous section.
Other model-agnostic plots
The same principles outlined in the previous section can be applied to ALE plots and related model-agnostic tools, including permutation-based variable importance and their conditional modifications (see reviews by Molnar et al., 2020a andMolnar, 2021). Also, this is not limited to x s ∈ R -this principle equally applies to bivariate x s ∈ R 2 relationships, which can be used to display pairwise interactions. Clearly, in a high-dimensional situation, the need to reduce dimensionality in post-hoc model interpretation is even more pressing when interpreting up to p * (p − 1)/2 pairwise interactions, and the proposed approach offers a practical tool to address this in situations where dimension reduction is viable.
Implementation
The proposed methods are provided as an R package wiml (https://github.com/alexanderbrenning/wiml). It implements transformation functions called 'warpers' based on PCA (of all features or a subset of features), structured PCA (for multiple groups of features), and feature orthogonalization, all of which are based on rotation matrices and therefore share a common core. Due to the modular and object-oriented structure, users can easily implement their own transformations without requiring changes to the package.
These warpers can be used to implement the compositionf •T −1 by 'warping' a fitted machine-learning model. The resulting object behaves like a regular fitted machine-learning model in R, offering an adapted predict method. From a user's perspective, the resulting model feels like it had been fitted to the transformed data T −1 (L), except that it hasn't. This 'warped' fitted model can, in principle, be used with any model-agnostic tool that doesn't require refitting. An implementation of the composition f • T −1 involving the untrained model f is also available; this can be used for drop and re-learn or permute and re-learn techniques (Hooker and Mentch 2019).
The package has been tested and works well with the iml package for interpretable machine learning (Molnar, Bischl, and Casalicchio 2018), but it can also be combined with other frameworks since it only builds thin wrappers around standard R model functions. Initial tests with the DALEX framework for explainable machine-learning (Biecek 2018) and its interactive environment modelStudio (Baniecki and Biecek 2019) have been successful, as have been tests with the pdp package (Greenwell 2017).
Case Study
The potential of the proposed methods will be demonstrated in a case study from land cover classification, which is a common machine-learning task in environmental remote sensing (e.g., Mountrakis, Im, and Ogole 2011;Peña and Brenning 2015). One particularly challenging task is the detection of rock glaciers, which, unlike 'true' glaciers, do not present visible ice on their surface; they are rather the visible expression of creeping ice-rich mountain permafrost. In the present case study, we look at a subset of a well-documented data set consisting of a sample of 1000 labelled point locations (500 presence and 500 absence locations of flow structures on rock glaciers) in the Andes of central Chile (Brenning, Long, and Fieguth 2012).
There are 46 features in total, which are divided into two unequal subsets: Six features are terrain attributes (local slope angle, potential incoming solar radiation, mean slope angle of the catchment area, and logarithm of catchment height and catchment area), which are proxies for processes related to rock glacier formation. The other 40 features are Gabor texture features (Clausi and Jernigan 2000), which are designed to detect the furrow-and-ridge structure in high-resolution (1 m × 1 m) satellite imagery, in this case panchromatic IKONOS imagery (see Brenning, Long, and Fieguth 2012 for details). The 40 Gabor features correspond to different filter bandwidths (5, 10, 20, 30 and 50 m), anisotropy factors (1 or 2), and types of aggregation over different filter orientations (minimum, median, maximum, and range).
Texture features with similar filter settings are often strongly correlated with each other. This is especially true for minimum and median aggregation with otherwise equal settings, and for maximum and range aggregation. Overall, the median of each feature's strongest Pearson correlation is 0.92 (minimum: 0.80). Correlations among terrain attributes are much smaller (median strongest correlation: 0.60). Terrain attributes and texture features are weakly correlated (maximum correlation: 0.32). Correlation statistics are very similar for Spearman's rank-based correlation. To explore the feature sets, PCAs were performed for the entire set of 46 feature and for the subset of 40 Gabor features (Figure 1). In the entire feature set, 63.6% of the variance is concentrated in the first two PCs (first six PCs: 83.7%). In the more strongly correlated Gabor feature set, in contrast, the first two PCs make up 72.2% of the variance (first six PCs: 89.5%). The main PCs turned out to be interpretable by domain experts. PC #1 of the Gabor feature set ('Gabor1,' in the figures) is basically an overall average of all texture features, meaning that it expresses the overall presence of striped patterns of any characteristics. Gabor PC #2 represents the contrast between minimum and median aggregated anisotropic Gabor features and the rest; large values are interpreted as incoherent patterns with no distinct, repeated stripe pattern. Gabor PC #3 expresses differences between large-wavelength range or maximum-aggregated features versus the short-wavelength features, which represents the heterogeneity in the width of stripes, and thus the size of linear surface structures. Very large values correspond to distinct patterns of large amplitude.
A random forest classifier is used for the classification of rock glaciers based on the features introduced above. Its overall accuracy, estimated by spatial cross-validation between the two sub-regions (Brenning 2012), is 80.1%. Omitting terrain attributes from the feature set has a greater impact on performance than omitting the texture features (Table 1).
Results
With 46 features that are grouped into two semantic feature types (terrain attributes, texture features), it can be challenging to interpret the patterns represented by marginal effects plots (Figure 2). Although there appears to be some consistency in direction among many of the texture features, it is difficult to identify an overall pattern that can be summarized verbally, and it would be unreasonable to present such detailed visual information to a conference audience that is expecting a concise and coherent narrative.
The ALE plots along principal axes distill 71.6 percent of the feature variance into only three plots ( Figure 3). Nevertheless, considering the semantic differences and weak correlations between texture features and terrain attributes, it seems unnecessary to combine all features in a joint PCA, which results in PCs with an at least slightly mixed meaning.
The structured PCA approach, in contrast, allows us to explicitly separate the model's representation of effects of texture features and terrain attributes, which is desitable from a domain expert's perspective and statistically justifiables based on the weak correlations between these feature groups. Larger overall texture signals (Gabor PC #1) are associated with higher predicted rock glacier probability (Figure 4). However, a large contrast between minimum/median anisotropic texture features and the remaining texture features, as expressed by a high Gabor PC #2 value, is more often associated with an absence of rock glaciers. In other words, the absence of coherently oriented, repeated stripes is not typical of rock glaciers -these may be more typical of non-repeated stripes (e.g., erosion gullies, jagged rock slopes).
The permutation-based assessment of the importance of texture PCs and terrain attributes shows that subsequent PCs contribute much less to the predictive performance, and that slope angle is the most salient feature overall ( Figure 5). Clearly, the combined importance of Gabor features as summarized by Gabor PCs #1 and #2 provides a more comprehensible summary than an incoherent litany of individual feature importances of strongly correlated features, which should not be permuted independently of each other.
Discussion
Overall, interpretation plots along the principal axes are capable of distilling complex high-dimensional relationships into low-dimensional summaries, thus providing a tidier, better structured and more focused approach to model interpretation than traditional tools that focus on individual predictors in an ocean of highly correlated features. This behaviour is highly desirable from a domain expert's perspective, and applying it in a structured manner allows the analyst to honour domain knowledge and feature semantics.
Of course fitting the classifier to PCA-transformed data as input features could have provided direct access to ALE plots along principal axes. However, we would want our feature engineering decisions to be directed towards improving predictive performance, and we would therefore prefer not to risk compromising an optimal performance to satisfy our desire to interpret our model. While this is not an issue in the present case study (overall accuracy 0.791 with PC features versus 0.801 with the original predictors), our experience shows that PCA-transformed predictors can worsen the predictive performance. Also, model-agnostic post-hoc analysis tools are precisely meant to be applicable to black-box models that are provided 'as is,' without the possibility of altering their input features, in which case the proposed 'hands-off' access to transformed perspectives is particularly valuable. The proposed use of PCA and related linear transformation technique appears to be in contradiction to the use of complex nonlinear machine-learning models. Nevertheless, it could be argued that linear cross-sections of feature space along the original feature axes are no less arbitrary and limiting, considering the often strong correlations with other features that would have to be interpreted simultaneously. From that perspective, principal axes provide a 'tidier' perspective and smarter peek into feature space than traditional ALE or partial dependence plots. Linear transformations similar to PCA may further enhance interpretability by offering a more structured or target oriented perspective based on simple components (Rousson and Gasser 2004) or discriminant functions (Cunningham and Ghahramani 2015).
One could even argue that ALE plots can be misleading for highly correlated features as they look at the often tiny contributions of individual features in an isolated way, while the proposed approach focuses on the bigger picture and captures the combined effect of a bundle of features. This also becomes evident in permutation-based feature importance assessments, where individual texture features consistently achieve discouragingly low importances, while the first two PCs of the texture features are ranked very highly. For the specific case of permutation assessments, it has also been proposed to jointly permute groups of features (Molnar 2019); unlike the techniques proposed here, this approach is not transferable to other interpretation tools that are not based on permutations.
Beyond linear transformations, the proposed approach provides a general framework even for nonlinear perspectives on feature space and model functions. In particular, paths proposed in section 2.4 may well be nonlinear, as e.g. defined by a physical model that could be used by domain experts to check model plausibility. Also, curvilinear component analyses (CCA) or autoencoders as state-of-the-art multivariate nonlinear transformation methods provide a logical extension of PCA and highlight the link between explainable machine-learning and projection-based visual analytics (Schaefer et al. 2013).
Finally, due to the orthogonality and thus linear independence of PCs, the more naturally interpretable partial dependence plots become a more viable option for the interpretation of black-box machine-learning models. In original feature space, in contrast, the less intuitive and sometimes rather coarse ALE plots should usually be preferred despite their limitations (compare Figures 6 and 7).
Conclusions
Despite the inherent limitations of post-hoc machine-learning model interpretation, feature space transformations, and structured PCA transformations in particular, are a powerful tool that allows us to distill complex nonlinear relationships into an even smaller number of univariate plots than previously possible, representing perspectives that are informed by domain knowledge. These transformations provide an intuitive access to feature space, which can be easily wrapped around existing model implementations. Model interpretation through the lens of feature transformation and dimension reduction allows us to peek into the feature space at an oblique angle -a strategy that many of us have have successfully applied when checking if our kids are asleep in their beds, and a much more successful strategy than staring along the walls, i.e. the original feature axes, especially when these are nearly parallel.
Figure 1 :
1Feature (sub)space diagrams for the first PCs of the entire feature set.
Figure 2 :Figure 3 :
23Ordinary ALE plots for all 46 features. ALE plots along the first six principal axes, applying PCA to the entire feature set.
Figure 4 :Figure 5 :Figure 6 :Figure 7 :
4567ALE plots along the first principal axes of texture features, and for the most important terrain attributes. Permutation feature importances of the 10 top-ranked texture principal components and terrain attributes. Two-dimensional ALE plot with respect to the first and second principal axes of the texture features. Two-dimensional partial dependence plot with respect to the first and second principal axes of the texture features.
Table 1 :
1Overall accuracies of random forest classification of rock glaciers using the entire feature set, and omitting either the terrain attributes or the texture features.All features Only terrain attributes Only texture features
Accuracy
0.801
0.759
0.696
Decrease in accuracy 0.000
0.042
0.105
AcknowledgementsI would like to thank Philipp Lucas (DLR Institute for Data Science) for encouraging discussions on an early version of the manuscript.
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"Novel critical phenomena in compressible polar active fluids: Dynamical and Functional Renormalization Group Studies",
"Novel critical phenomena in compressible polar active fluids: Dynamical and Functional Renormalization Group Studies"
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| [
"Patrick Jentsch \nDepartment of Bioengineering\nImperial College London\nSouth Kensington CampusSW7 2AZLondonU.K\n",
"Chiu Fan Lee \nDepartment of Bioengineering\nImperial College London\nSouth Kensington CampusSW7 2AZLondonU.K\n"
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| Active matter is not only relevant to living matter and diverse nonequilibrium systems, but also constitutes a fertile ground for novel physics. Indeed, dynamic renormalization group (DRG) analyses have uncovered many new universality classes (UCs) in polar active fluids (PAFs) -an archetype of active matter systems. However, due to the inherent technical difficulties in the DRG methodology, almost all previous studies have been restricted to polar active fluids in the incompressible or infinitely compressible (i.e., Malthusian) limits, and, when the -expansion was used in conjunction, to the one-loop level. Here, we use functional renormalization group (FRG) methods to bypass some of these difficulties and unveil for the first time novel critical behavior in compressible polar active fluids, and calculate the corresponding critical exponents beyond the one-loop level. Specifically, we investigate the multicritical point of compressible PAFs, where the critical order-disorder transition coincides with critical phase separation. We first study the critical phenomenon using a DRG analysis and find that it is insufficient since two-loop effects are important to obtain a nontrivial correction to the scaling exponents. We then remedy this defect by using a FRG analysis. We find three novel universality classes and obtain their critical exponents, which we then use to show that at least two of these universality classes are out of equilibrium because they violate the fluctuation-dissipation relation. | 10.1103/physrevresearch.5.023061 | [
"https://export.arxiv.org/pdf/2210.03830v2.pdf"
]
| 258,060,283 | 2210.03830 | 8a7f3252c5169e6171829321934f388f00e5ba97 |
Novel critical phenomena in compressible polar active fluids: Dynamical and Functional Renormalization Group Studies
Patrick Jentsch
Department of Bioengineering
Imperial College London
South Kensington CampusSW7 2AZLondonU.K
Chiu Fan Lee
Department of Bioengineering
Imperial College London
South Kensington CampusSW7 2AZLondonU.K
Novel critical phenomena in compressible polar active fluids: Dynamical and Functional Renormalization Group Studies
(Dated: April 12, 2023)
Active matter is not only relevant to living matter and diverse nonequilibrium systems, but also constitutes a fertile ground for novel physics. Indeed, dynamic renormalization group (DRG) analyses have uncovered many new universality classes (UCs) in polar active fluids (PAFs) -an archetype of active matter systems. However, due to the inherent technical difficulties in the DRG methodology, almost all previous studies have been restricted to polar active fluids in the incompressible or infinitely compressible (i.e., Malthusian) limits, and, when the -expansion was used in conjunction, to the one-loop level. Here, we use functional renormalization group (FRG) methods to bypass some of these difficulties and unveil for the first time novel critical behavior in compressible polar active fluids, and calculate the corresponding critical exponents beyond the one-loop level. Specifically, we investigate the multicritical point of compressible PAFs, where the critical order-disorder transition coincides with critical phase separation. We first study the critical phenomenon using a DRG analysis and find that it is insufficient since two-loop effects are important to obtain a nontrivial correction to the scaling exponents. We then remedy this defect by using a FRG analysis. We find three novel universality classes and obtain their critical exponents, which we then use to show that at least two of these universality classes are out of equilibrium because they violate the fluctuation-dissipation relation.
I. INTRODUCTION
Active matter refers to many-body systems in which the microscopic constituents can exert forces or stresses on their surroundings and, as such, detailed balance is broken at the microscopic level [1,2]. However, even if the microscopic dynamics are fundamentally different from more traditional systems considered in physics, it remains unclear whether novel behavior will emerge in the hydrodynamic limits (i.e., the long time and large distance limits [3]). One unambiguous way to settle this question is to identify whether the system's dynamical and temporal statistics are governed by a new universality class (UC), typically characterized by a set of scaling exponents [4][5][6]. These exponents can in principle be determined using either simulation or renormalization group (RG) methods. However, simulation studies can be severely plagued by finite-size effects (e.g., two recent controversies concern the scaling behavior of active polymer networks [7,8] and critical motility-induced phase separation [9][10][11]). Therefore, RG analyses remain as of today the gold standard in the categorization of dynamical systems into distinct UCs. This perspective has been particularly fruitful in biological physics, where many new nonequilibrium universality classes have been discovered in biology inspired systems [12][13][14][15]. Specifically, for polar active fluids (PAFs) [16][17][18], an archetype of active matter systems, the use of dynamic renormalization group (DRG) [19] analyses have led to, on one hand, surprising realizations that certain types of PAFs are no different from thermal systems in the hydrodynamic limit [20,21], and on the other hand discoveries of diverse novel phases [17,18,[22][23][24][25][26][27][28][29][30], critical phenomena [31][32][33] and discontinuous phase transitions [34]. However, due to the inherent technical difficulties in DRG methods, all of these studies have been restricted to PAFs in the incompressible or infinitely compressible (i.e., Malthusian) limits except for rare exceptions [23,24]. Further, when a DRG analysis was used in conjunction with the -expansion method, which was typically the case, it has always been restricted to the one-loop level.
In this work, we apply for the first time functional RG (FRG) methods on compressible PAFs and overcome some of these technical challenges. Specifically, we investigate a multicritical region of dry compressible PAFs. Although experimentally less accessible than simple critical points, multicritical points (MCPs) can offer surprising new physics, even in models that are thought to be well understood. For instance, nonperturbative fixed points have been discovered in the extensively studied O(N ) model [35], and in systems where two order parameters compete, whose individual critical points belong to equilibrium universality classes, the multicritical region where both critical points coincide can be manifestly out of equilibrium and demonstrate very interesting, spiral phase diagrams [36].
We will first apply a traditional one-loop DRG approach to the MCP of our interest, demonstrating how it is insufficient to capture its universal physics and then, for the first time for PAFs, apply a FRG [37][38][39][40][41][42][43][44] analysis that goes beyond the equivalent perturbative one-loop level.
FRG analyses are intrinsically non-perturbative and are based on an exact RG flow equation to which ap-proximate solutions can be readily obtained numerically. Recent successes in the applications of FRG include the elucidation of scaling behavior in, e.g., critical and multicritical N -component ferromagnets [35,[45][46][47], reactiondiffusion systems [48][49][50][51][52], the Kardar-Parisi-Zhang model [53][54][55], and turbulence [56][57][58][59][60], as well as non-universal observables far from scaling regimes [61,62]. Using FRG, we uncover here three novel nonequilibrium UCs by studying a multicritical region of dry compressible PAFs and quantify the associate scaling behaviors beyond the one-loop level.
The outline of this paper is as follows. In Sec. II, we introduce the hydrodynamic theory of compressible polar active matter and discuss salient features in its phase diagram, which enables us to define the multicritical point of interest. We then show how general scaling invariance of the equations of motion leads to powerlaw behavior in the correlation functions in Sec. III. For the multicritical point, we show this first in the linear regime in Sec. IV and then turn to the nonlinear regime in Sec. V. In Sec. VI, we perform the one-loop DRG calculation and argue why it is not sufficient to take into account the nonlinearities and then present the FRG approach in Sec. VII that we use instead. Using this method we find three RG fixed points which represent three novel nonequilibrium UCs. We discuss them and their scaling behavior in Sec. VIII. Finally, we summarize our findings and give an outlook on future work in Sec. IX.
II. COMPRESSIBLE POLAR ACTIVE FLUIDS
A. Equations of motion from symmetry and conservation laws
Polar active matter aims to describe the collective behavior of swarming animals, e.g., flocks of birds, schools of fish or bacterial swarms. In the fluid state, as in passive fluids that are describable by the Navier-Stokes equations, the relevant dynamical variables are the momentum density and density fields, denoted by g and ρ respectively. Without any assumptions about the microscopic realization, one can then, based on symmetry and conservation laws, construct a generic set of hydrodynamic equations of motion (EOM) for these variables.
Here, we assume that the particle number is conserved (as opposed to, e.g., a Malthusian system in which birth and death of particles can occur [22,26,27]). We thus arrive at a continuity equation as the EOM of the density field ρ:
∂ t ρ + ∇ · g = 0 .
(1)
For the momentum density field, we assume temporal, translational, rotational and chiral invariance. In addition, we focused on active systems in which the constituents move on a fixed frictional substrate, i.e., dry active matter systems (as opposed to wet active systems such as active suspensions) [2]. This symmetry consideration leads to the following generic hydrodynamic EOM of g [17,18,63]: ∂ t g + λ 1 ∇(|g| 2 ) + λ 2 (g · ∇)g + λ 3 g(∇ · g) = µ 1 ∇ 2 g + µ 2 ∇(∇ · g) − αg − β|g| 2 g − κ∇ρ + ... + f ,
where the ellipsis represents the omitted higher-order terms (i.e. terms of higher order in spatial derivatives and the momentum density field).
In the EOM above, all coefficients are functions of ρ, and the noise term f (r, t) is a zero mean Gaussian white noise of the form
f i (r, t)f j (r , t ) = 2Dδ ij δ d (r − r )δ(t − t ) . (3)
The above hydrodynamic EOM (1,2) are termed the Toner-Tu EOM [17,18]. However, in contrast to the original Toner-Tu formulation, we have chosen to use the momentum field as the hydrodynamic variable instead of the velocity field so that there is a linear relationship between ρ and g, which facilitates our discussion later.
B. Mean-field theory and homogeneous phases
Given the hydrodynamic EOM, one of the first, and simplest, question to ask is: what are the mean-field homogeneous solutions to the EOM? Answering this question amounts to focusing on temporally invariant, spatially homogeneous, and noise-free solutions to the EOM. These solutions are readily seen to be
|g| = |α| β , if α < 0 0 , otherwise ,(4)
where β is taken to be always positive for reasons of stability. Since g can point in any direction, the |g| > 0 state implies a spontaneous symmetry breaking of the rotational symmetry, and corresponds to the homogeneous ordered phase of polar active matter where collective motion emerges. In contrast, the |g| = 0 state corresponds to the homogeneous disordered phase, i.e., there is no collective motion.
C. Phase diagram: phase separations, critical and multicritical phenomena
The homogeneous phases discerned from the previous mean-field analysis are, however, not always stable, even in the absence of the noise term f (3). The standard way to ascertain the in/stability of the homogeneous phases in this noiseless regime is to use a linear stability analysis. Here, the temporal evolution of an initially small perturbation to a homogeneous solution is studied and the growth or decay of its amplitude signifies whether the homogeneous state is unstable or stable, respectively. Since the nature of the inhomogeneous states does not follow from linear stability analysis alone and inhomogoneous, analytic solutions of the noiseless mean-field equations are most often very difficult, it is typically explored via simulations. Using this method, complex phase diagrams of polar active fluids have been uncovered [64][65][66]. In particular, distinct types of bulk phase separations, i.e., an inhomogeneous state where two different phases co-exist, have been demonstrated. As the large length and timescale-limit of all these different models, it is expected that the hydrodynamic EOM captures the same phenomenology in an encompassing phase diagram, schematically depicted in Fig. 1.
In particular, expressing α and κ in Eq. (2) as
α = n≥0 α n δρ n , κ = n≥0 κ n δρ n ,(5)
where δρ = ρ − ρ 0 with ρ 0 being the average particle density in the system, two disordered phases (with distinct densities) can co-exist if α 0 > 0 and κ 0 < 0 (blue region in Fig. 1(a)) [10], while an ordered phase can coexist with a disordered phase if κ 0 > 0 and α 0 < 0 (green region) [65]. Further, the system can become critical upon fine-tuning: if α 0 > 0 and κ 0 = κ 1 = 0, the resulting critical behavior belongs to the Ising universality class (UC) (blue triangle) [10], while if α 0 = α 1 = 0 and κ 0 > 0, the associate critical behavior corresponds to a yet to be characterized UC (yellow inverted triangle) [65]. Recently, a third type of critical behavior was identified [66], which corresponds to the merging of these two distinct critical points by simultaneously fine-tuning α 0 , α 1 , κ 0 and κ 1 to zero (red circle in Fig. 1(b)). The universal behavior of this new multicritical point is the focus of this paper.
It is interesting to note that apart from these, either homogeneous or bulk phase-separated, states that join at the MCP, different states of microphase separation have been observed as well [67][68][69]. [66]. (a) Depending on the model parameter, e.g., α0, and the average density ρ0, the system can be in the homogeneous disordered phase (white region denoted by D) or the polar ordered phase (yellow region denoted by O); It can also phase separate into two disordered phases with different densities (blue region), or into one ordered phase and one disordered phase, again with different densities (green region flanking the homogeneous ordered phase). The critical behavior associated with the first type of phase separation is generically described by the Ising UC (blue triangle) [10], and that associated with the second type is described by a putatively novel UC yet to be described (yellow inverted triangle) [65]. (b) Upon further fine-tuning, these two critical points can coincide (red circle) [66], and the resulting critical point is described by a novel UC uncovered in the present work.
III. SCALE INVARIANT EQUATIONS OF MOTION
It is generally expected that the EOM of general systems at critical points become invariant under rescaling of lengths, time and fields [4][5][6]. Hence, at the multicritical point (MCP), we expect that the EOM (1,2) are invariant under the rescaling r → re , t → te z , ρ → ρe χρ , g → ge χg , (6) for some exponents z, χ ρ and χ g , that are a priori not known. If this is the case, however, this defines a rescaling symmetry of the theory which the correlation functions have to obey as well. Take for example the densitydensity correlation function: C ρ (r, t) = ρ(r, t)ρ(0, 0) = e −2χρ ρ(re , te z )ρ(0, 0) .
(7) Choosing = − ln r, r being measured against some reference scale, we see immediately that
C ρ (r, t) = r 2χρ S ρρ t r z ,(8)
where S ρρ (.) is a scaling function that only depends on the ratio t/r z which is invariant under rescaling (6). So we immediately see that, if the EOM are invariant under a rescaling transformation, the correlation functions will generally express powerlaw behavior. Likewise, this argument can be applied to the momentum-momentum correlation function C g (r, t) = g(r, t)g(0, 0) = r 2χg S gg t r z , (9) where S gg is again a scaling function with similar properties as S ρρ . Ultimately, we will demonstrate that the EOM does become scale invariant and determine the scaling exponents using a FRG analysis, but first, we will illustrate the scale invariance discussed here using the simple, but quantitatively incorrect, linear theory.
IV. LINEAR REGIME
In the linear regime, i.e., when the non-linear terms in Eq. (2) are neglected, the scaling behavior discussed above can readily be seen. Around the MCP at which critical disordered phase separation (blue triangle in Fig. 1a)) merges with critical disorder-order of a generic compressible PAF (yellow inverted triangle), |g| ≈ 0, such that the linearized EOM are
∂ t ρ = −∇ · g ,(10a)∂ t g = µ 1 ∇ 2 g + µ 2 ∇(∇ · g) + ζ∇ 2 ∇ρ + f ,(10b)
where we have introduced the term characterized by ζ since, when κ 0 is fine-tuned to zero, this term is now the leading order term linear in ρ. In Eq. (10), we have redefined ρ to be δρ to ease notation, and we will continue to do so from now on.
A. Scaling exponents
Correlation functions
Upon rescaling time, lengths, and fields according to Eq. (6), the linearized EOM (10) become
e (χρ−z) ∂ t ρ = −e (χg−1) ∇ · g ,(11a)e (χg−z) ∂ t g = e (χg−2) µ 1 ∇ 2 g + µ 2 ∇(∇ · g) + e (χρ−3) ζ∇ 2 ∇ρ + e −(z+d) /2 f . (11b)
They thus remain unchanged if
z lin = 2, χ lin ρ = 4 − d 2 , χ lin g = 2 − d 2 .(12)
At the linear level we can therefore directly conclude that
C ρ (r, t) = r 2χ lin ρ S lin ρρ t r z lin ,(13a)C g (r, t) = r 2χ lin g S lin gg t r z lin ,(13b)
using the argument from Sec. III. Since the linearized EOM (10) are solvable analytically by performing a spatio-temporal Fourier transform, the scaling behavior of the correlation functions can in fact be demonstrated explicitly. This has the added advantage that the expressions of the aforementioned scaling functions, S ρρ and S gg , can be obtained in the form of integrals.
Specifically, by performing a spatiotemporal Fourier transform, the linear EOM can be written as
ρ(q) = q ω G (q)f (q) ,(14a)g (q) = G (q)f (q) ,(14b)g ⊥ (q) = G ⊥ (q)f ⊥ (q) ,(14c)
where g (q) = g(q) ·q, g ⊥ = g − g q, withq being the unit vector in the direction of q, q = |q|,q = (q, ω), and the G's in (14), or the "propagators", are:
G (q) = ω −i(ω 2 − ζq 4 ) + ωµ q 2 ,(15a)G ⊥ (q) = 1 −iω + µ 1 q 2 ,(15b)
where µ ≡ µ 1 + µ 2 . Given the above expressions, the correlation functions are now obtained straightforwardly:
C ρ (r, t) = q e iq·r 2Dq 2 (ω 2 − ζq 4 ) 2 + ω 2 µ 2 q 4 ,(16a)C g = C ⊥ g + C g ,(16b)C ⊥ g (r, t) = q e iq·r 2DP ⊥ (q) ω 2 + µ 2 1 q 4 ,(16c)C g (r, t) = q e iq·r 2Dω 2 P (q) (ω 2 − ζq 4 ) 2 + ω 2 µ 2 q 4 ,(16d)
where q ≡ d d qdω/(2π) (d+1) ,q ·r = q · r − ωt, and P ij (q) ≡ q i q j /q 2 and P ⊥ ij (q) ≡ δ ij − q i q j /q 2 are the projectors parallel and transverse to q respectively.
Focusing on C ρ (r, t) (16a) as an example, the substitutions ω = Ω/r 2 and q = Q/r in the integral lead to
C ρ (r, t) = r 4−d d d QdΩ (2π) (d+1) 2DQ 2 exp i Q ·r − Ωt r 2 (Ω 2 − ζQ 4 ) 2 + Ω 2 µ 2 Q 4 ,(17)
which demonstrates the scaling form (8) with the scaling exponents from the linear theory (12), and
S lin ρρ (y) = d d QdΩ (2π) (d+1) 2DQ 2 exp [i (Q ·r − Ωy)] (Ω 2 − ζQ 4 ) 2 + Ω 2 µ 2 Q 4 .(18)
As we will show later, all the scaling exponents from the linear theory (12) are in fact incorrect for describing the hydrodynamic behavior around the MCP due to the nonlinearities in the EOM.
2.
Divergence of correlation length
Besides the scaling exponents in the correlation functions right at the MCP, the divergence of the correlation length, as one approaches the MCP, is also governed by another set of scaling exponents. For the Ising model, this is the temperature. For the present MCP however, this divergence is associated to two parameters α 0 , κ 0 . The other two relevant parameters, α 1 and κ 1 , take a role akin to the magnetic field in the Ising model.
Since κ 0 appears in the EOM as a speed of sound for density wave, it is not immediately clear how it might be related to the correlation length. However one can show by rederiving the correlation functions (16) in the presence of these two couplings, that the equal-time correlation functions are
C ρ (r, 0) = q e iq·r D (α 0 + µ q 2 )(κ 0 + ζq 2 ) ,(19a)C ⊥ g (r, 0) = q e iq·r DP ⊥ (q) α 0 + µ 1 q 2 , (19b) C g (r, 0) = q e iq·r Dω 2 P (q) α 0 + µ q 2 .(19c)
From this standard form it is clear that, in the linear theory, both α −1/2 0 and κ −1/2 0 define a crossover scale, and the larger of the two a correlation length for density correlations, while α −1/2 0 is always the correlation length for momentum correlations.
As the divergence of the correlation length is described by these two parameters, α 0 and κ 0 , there are also two exponents which we call y 1 and y 2 . At the linear level, these exponents correspond expectedly to their meanfield values:
y lin 1 = y lin 2 = 2 .(20)
We note that these scaling exponents are again expected to be modified by the nonlinearities, as we shall see in the next section. Experimentally, these exponents define a relationship between the correlation length ξ and the distances t 1 and t 2 in the phase diagram (see the inset of Fig. 2) from the critical points of disordered phase separation (blue upwards triangle in Fig. 1 and blue line in the inset of Fig. 2) and the critical order-disorder transition (yellow downwards triangle in Fig. 1 and yellow line in in the inset of Fig. 2),
ξ ∼ t − 1 y 1 1 ∼ t − 1 y 2 2 .(21)
This means that, upon approaching the MCP, one has to enforce the relationship given by the right proportionality in Eq. (21) to see a clear scaling behavior in the correlation length. This relationship is also visualized in shows the critical scaling behavior characterized by the scaling exponent χρ (slope triangle). In reality though, the system will never be exactly at the critical point. This can be characterized by the distances t1 and t2 (gray lines in inset) from the critical point of disordered phase separation (blue line in inset) and from the critical order-disorder transition (yellow line in inset) respectively. This manifests in a finite correlation length ξ above which the scaling behavior breaks down (gray lines in main figure). As one approaches the fixed point, by decreasing t1 and t2, the correlation length diverges. If this is done carefully, such that the second relation in Eq. (21) remains unchanged, for example by rescaling t1 and t2 by a factor s y 1 and s y 2 respectively (gray arrow in inset), the correlation length rescales according to Eq. (21), i.e. by a factor s −1 (gray arrow in main figure).
V. NONLINEAR REGIME
While the scaling behavior described in Sec. IV is qualitatively expected generally, the critical exponents (12,20) obtained in the linear theory are only expected to be exact when the spatial dimension d is high enough. Below a certain upper critical dimension d c , nonlinear terms become important which modify the scaling and correlation length exponents. The linear exponents (12) can however be used to gauge the importance of various nonlinearities in the EOM (2) as d is lowered.
We now turn to the full EOM of g (2), and perform a rescaling (6) with the linear exponents (12). If the spatial dimension d is large enough, all nonlinear terms are irrelevant, i.e., they vanish as → ∞. As d decreases from, say infinity, the nonlinear terms that first become relevant (and are not fine-tuned to zero), i.e., terms that diverge as → ∞, are
α 2 ρ 2 g and κ 2 ρ 2 ∇ρ ,(22)
which happens at the upper critical dimension d c = 6. These non-linear terms, together with the linear terms, support the following symmetry:
ρ → −ρ and g → −g .(23)
One can therefore simplify the consideration by restricting to the subspace of EOM compatible with this symmetry, in which α 1 and κ 1 are vanishing and are not generated under RG transformations. Higher-order terms breaking this symmetry are irrelevant close to the upper critical dimension, modifying our results only beyond the order considered in this work. Therefore, close to the upper critical dimension, the symmetry (23) is a property of the MCP. A physical interpretation of this emergent symmetry (23) corresponds to the equivalence between a highdensity band traveling in directionn and the corresponding low-density band (with the same profile but inverted) traveling in the −n direction.
Just below six dimensions, the universal hydrodynamic EOM (2) is therefore
γ∂ t g = µ 1 ∇ 2 g + µ 2 ∇(∇ · g) − α 0 g − κ 0 ∇ρ + f −α 2 ρ 2 g − κ 2 3 ∇ ρ 3 + ζ∇ 2 ∇ρ ,(24)
where γ is a dimensionless coefficient introduced to allow for renormalization of the temporal derivative term associated to g. Note that such a RG correction to the temporal derivative is also present in a recent study of incompressible active fluids with quenched disorder [30]. Further, we note that the signs of the nonlinear terms (with α 2 , κ 2 > 0) are chosen for the sake of stability. By the same token, the term ζ∇ 2 ∇ρ stabilizes the system in the case of κ 0 < 0. In fact, this term is marginal according to our linear theory and is therefore required in our discussion. It can be interpreted as an effective "pressure" or compressibility term for the momentum density field which, in the limit of small ζ manifests as an effective diffusion for the density mode. Its dispersion relation is given as
ω ρ = µ 2γ i − 4γζ µ 2 − 1 q 2 ≈ i ζ γ q 2 ,(25)
where the last approximation is valid in the limit of ζ µ 2 /(4γ) such that the diffusion constant of the density mode is given by ζ/γ.
VI. DRG ANALYSIS
Traditionally, a DRG analysis together with theexpansion method is now applied. As we will demonstrate now however, a one-loop calculation as is usually performed, will not be sufficient.
The DRG [19] is usually performed by first transforming the EOM (1,24) to Fourier-space and then splitting the fields into small and large scale modes, arbitrarily split at an inverse length-scale Λ = Λe − , which is a fraction of the physical cutoff scale Λ that defines the smallest length scale of the system, e.g., the average distance between individual particles:
ρ(q, ω) = ρ > (q, ω) + ρ < (q, ω) , (26a) g(q, ω) = g > (q, ω) + g < (q, ω) , (26b) such that ρ > (q, ω) = ρ(q, ω) if |q| > Λ and ρ > (q, ω) = 0 otherwise, etc.
One can then eliminate the small scale modes ρ > and g > from the EOM by recursively reinserting the formal solution for the small scale modes in terms of the large scale modes provided by the EOM. This generates a hierarchy of terms, which can only be truncated by assuming the interaction terms are small, i.e., by going to the perturbative limit. Averaging this expression over the small scale noise terms f > that remained so far in the equation, then generates effective contributions to the coefficients of the EOM (24) that depend on the coarse-graining scale Λ . These terms can be represented diagrammatically through Feynman diagrams which can be classified by their number of loops, i.e., number of integrals one needs to solve to determine the correction.
The conventions we will be using in this paper arẽ
q = G(q) , (27a) q = 1 −iω q , (27b) q = −iq ,(27c)= 2D id , (27d) = α 2 id , (27e) q = iqκ 2 ,(27f)
where id is the identity matrix and G is the propagator of the momentum density fields
G(q) = G (q)P (q) + G ⊥ (q)P ⊥ (q) ,(28a)G (q) = −iω −iω(−iγω + α 0 + µ q 2 ) + κ 0 q 2 + ζq 4 , (28b) G ⊥ (q) = 1 −iγω + α 0 + µ 1 q 2 .(28c)
In general, each diagram expression is of the tensorial rank equal to the number of open unbroken lines. These, so-called, graphical corrections are
idGr DRG α2 = − 2 − 4 − 4 − 4 − 2 ,(29)iqGr DRG κ2 = − 4 q q − 4 q q − 6 q q − 4 q q (30) − 4 q q − 6 q q , idGr DRG α0 = , iqGr DRG κ0 = q q ,(31)
where outgoing lines without a wave vector imply that the corresponding wavevector has been set to zero. The loop wavevector has not been written explicitly and its integral is implied. Whether the external wavenumber q is routed through the upper or the lower part of the loop is irrelevant up to order . Also, the tildes are omitted since the external frequency ω q is set to 0 for every diagram in our approximation. Only internal lines correspond to a propagator according to Eq. (27). External lines only mark open vector indices and external wavevectors.
We further note that the different prefactors in Eq. (30) stem from the fact that the right-hand side has to be expanded to linear order in q. For diagrams whose first vertex has three outgoing density propagators (dotted lines), i.e. the vertex in Eq. (27f), it is immaterial at this order whether the external wave vector leaves before or after the loop part, e.g.,
q q = q q + O(q 2 ). (32)
For diagrams where the first vertex has two outgoing density fields and one outgoing momentum field (two dotted and one unbroken line), i.e. the vertex in Eq. (27e), contributions where the external wave vector leaves before passing through the loop part vanish identically since they are wave vector independent and the wave vector independent part vanishes due to antisymmetry of the integrand, e.g.,
q q = 0.(33)
This attributes the relative factor 2/3 between these two types of diagrams since all permutations of the outgoing wavevector need to be considered.
At the 1-loop level, we can set κ 0 and α 0 to zero on the right-hand-side of Eqns. (29) and (30). For Eq. (31), the expansion must be carried out explicitly to second order in κ 0 and α 0 however, which is most conveniently done after the frequency integration. Further details of the evaluation of the diagrams can be found in App. C.
As an example, for one of the specific 1-loop diagrams from Eq. (29) that provides a RG correction to α 2 the analytical expression is,
= α 2 2 Λ Λe − d d p (2π) d ∞ −∞ dω p 2π × 2D p ⊗ p (γω 2 p − ζp 4 ) 2 + ω 2 p µ 2 p 4 ω p i(γω 2 p − ζp 4 ) + ω p µ p 2 .
We note here that the ζ term is indeed crucial to regularize the RG calculation, as without it the above 1-loop frequency integral is clearly divergent.
Now together with these corrections, stemming from the nonlinear terms, we can reattempt the rescaling in Eq. (6) which we include as an effective rescaling of the couplings. In addition, the EOM (24) is divided by γ, to fix the time-derivative coefficient to unity:
µ 1 → µ 1 e (z−2) γ , (34a) µ → µ e (z−2) γ , (34b) ζ → ζ e (z−3+χρ−χg) γ , (34c) D → D e (−2χg+z−d) γ 2 , (34d) α 0 → α 0 e z γ ,(34e)κ 0 → κ 0 e (z+χρ−χg) γ , (34f) α 2 → α 2 e (z+2χρ) γ ,(34g)κ 2 → κ 2 e (z+3χρ−χg) γ . (34h)
If we now consider that is an infinitesimal number → d , defining the so-called Wilsonian momentum shell, we can write down the DRG flow equations:
∂ µ 1 = (z − 2 − Gr DRG γ /γ)µ 1 + Gr DRG µ1 , (35a) ∂ µ = (z − 2 − Gr DRG γ /γ)µ + Gr DRG µ , (35b) ∂ ζ = (z − 3 + χ ρ − χ g − Gr DRG γ /γ)ζ + Gr DRG ζ ,(35c)∂ D = (−2χ g + z − d − 2Gr DRG γ /γ)D + Gr DRG D ,(35d)∂ α 0 = (z − Gr DRG γ /γ)α 0 + Gr DRG α0 ,(35e)∂ κ 0 = (z + χ ρ − χ g − Gr DRG γ /γ)κ 0 + Gr DRG κ0 , (35f) ∂ α 2 = (z + 2χ ρ − Gr DRG γ /γ)α 2 + Gr DRG α2 , (35g) ∂ κ 2 = (z + 3χ ρ − χ g − Gr DRG γ /γ)κ 2 + Gr DRG κ2 . (35h)
This means that there are three parts that contribute to the flow equation of each coupling: the rescaling of fields, lengths and time, the graphical correction of each term, which have been rescaled by the same factor as the respective coupling, and finally the graphical correction of γ from dividing the total EOM by this factor. The coupling γ itself is not rescaled
∂ γ = γ + Gr DRG γ(36)
and will in fact not approach a fixed point. But this does not matter since the rescaled EOM no longer depend on γ.
If the flow equations (35) are vanishing, this means that the EOM are invariant under this rescaling transformation in the presence of nonlinearities, which implies power-law correlations as discussed in Sec. III.
To facilitate the comparison between our DRG calculation and our FRG analysis to be presented later, we further define the dimensionless couplings through which all the flow equations can be expressed,
µ = µ 1 µ (37a) ζ = γζ µ 2 (37b) α 0 = α 0 µ Λ 2 ,(37c)κ 0 = κ 0 ζΛ 2 ,(37d)α 2 = α 2 Λ d−6 DS d µ 2 ζ(2π) d ,(37e)κ 2 = κ 2 Λ d−6 DS d µ ζ 2 (2π) d ,(37f)
where the geometric factor S d /(2π) d , with the surface area of a d-dimensional unit sphere S d = 2π d/2 /Γ(d/2) and the Euler gamma function Γ, was introduced for convenience. In this way, the rescaling introduced earlier is removed again from the flow equations, which might seem surprising. However, engineering and scaling dimension are actually very closely related. Any anomalous scaling exponent is generated by the renormalization of couplings that relate different units. For example, the diffusion constant µ /γ relates time and length scales. The four couplings carrying that information in this system are µ , ζ, γ and D which are all removed from appearing explicitly in the flow equations due to making them dimensionless (37). What remains in the flow equations, however, is their graphical corrections
η µ = Gr DRG µ µ , (38a) η ρ = Gr DRG ζ ζ + η γ − 2η µ ,(38b)η γ = Gr DRG γ γ , (38c) η D = Gr DRG D D . (38d)
The dimensionless flow equations can then be written as
∂ μ = −η µμ +Ḡr DRG µ1 , (39a) ∂ ζ = η ρζ ,(39b)∂ ᾱ 0 = (2 − η µ )ᾱ 0 +Ḡr DRG α0 ,(39c)∂ κ 0 = (2 − 2η µ + η γ − η ρ )κ 0 +Ḡr DRG κ0 ,(39d)∂ ᾱ 2 = (6 − d − 4η µ + η γ + η D − η ρ )ᾱ 2 +Ḡr DRG α2 ,(39e)∂ κ 2 = (6 − d − 5η µ + 2η γ + η D − 2η ρ )κ 2 +Ḡr DRG κ2 . (39f)
The dimensionless graphical correctionsḠr DRG are defined analogously to their respective couplings. Now, since all nonlinearities in the EOM are cubic in nature (the α 2 and κ 2 terms), the only coefficients that receive graphical corrections are α 0,2 and κ 0,2 at the 1loop level, which are shown in Eq. (29)(30)(31). The other graphical corrections are all zero, i.e.,
Gr DRG µ1 = Gr DRG µ = Gr DRG ζ = Gr DRG D = Gr DRG γ = 0 ,(40)
and as a result,
η µ = η ρ = η γ = η D = 0.(41)
Therefore, we can directly infer that the scaling exponents are unchanged from the linear theory
z DRG = 2, χ DRG ρ = 4 − d 2 , χ DRG g = 2 − d 2 .(42)
The remaining flow equations, perturbatively expanded to second order inᾱ 0 ,κ 0 ,ᾱ 2 andκ 2 are This suggests that there are potentially three novel universality classes, since for each of them a different rescaling transformation exists, implying different critical exponents. However, since two of the FP locations, especially that of the attractive one, depend onμ andζ, so do the critical exponents. Since the one-loop DRG calculation predicts no renormalization of these couplings, they can take arbitrary values, according to whichever microscopic model is realized, suggesting that the critical exponents are not universal, which is indicative of the unreliability of this 1-loop calculation.
∂ ᾱ 0 = 2ᾱ 0 +ᾱ 2 (1 −ᾱ 0 −κ 0 ) ,(43a)∂ κ 0 = 2κ 0 +κ 2 (1 −ᾱ 0 −κ 0 ) , (43b) ∂ ᾱ 2 = ᾱ 2 −ᾱ 2κ2 − 5 3 2 + 3μ +μ 2 +ζ µ +μ 2 +ζᾱ 2 2 , (43c) ∂ κ 2 = κ 2 − 3κ 2 2 − 11 3ᾱ 2κ2 + 2 3 5 + 5μ + 2μ 2 + 2μ 3 + 2μζ µ 2 +μ 3 +μζᾱ
This defect can potentially be cured by taking into account two-loop effects, where O( 2 ) corrections to the flow equations may lead to nontrivial and universal FP values of the couplingsμ andζ. As far as we are aware, such a two-loop calculation has never been done explicitly for active matter systems using the DRG formalism. This may be due to the use of the sharp wavenumber cutoff in the Wilsonian momentum shell regularization that renders a two-loop calculation difficult. For similar problems in the literature, one has usually resorted to a field-theoretic approach, where the large scale regularization can be made smooth, see e.g., [70]. In this work, we will, however, pursue a functional renormalization group approach instead.
VII. FRG ANALYSIS
Our FRG analysis is based on the so-called Wetterich equation [37][38][39]:
∂ k Γ k = 1 2 Tr Γ (2) k + R k −1 ∂ k R k ,(44)
where Γ k is the k-dependent effective average action, with k being the inverse length scale up to which fluctuations have been averaged out (i.e., k plays the equivalent role of Λ = Λe − in the previous DRG analysis.) The functional Γ k interpolates from the microscopic action Γ Λ to the macroscopic effective average action Γ 0 .
As the Legendre transform of the logarithm of the partition function, it contains all statistics of the many-body problem and can therefore be regarded as its full solution. It can further be regarded as the classical action for the average of the fields. The gradual incorporation of fluctuations as k → 0 is facilitated by the "regulator" R k , which serves to suppress fluctuations of length scales greater than k −1 . The regulator can be chosen arbitrarily as long as R Λ ≈ ∞ and R 0 = 0 to ensure the correct boundary conditions for Γ k . Further, Γ
k in Eq. (44) denotes the field dependent matrix of the second functional derivatives of Γ k (i.e., entries are of the form δ 2 Γ k /(δgδρ), etc), and Tr stands for the matrix trace over internal indices and integration over the internal wave vector and frequency.
A. Choice of functional in the Wetterich equation
While the Wetterich equation (44) is in principle exact, the actual implementation of the RG flow relies on restricting the functional Γ k to a manageable form. Here, we will take Γ k to be the functional obtained from the EOM (1,24) via the Martin-Siggia-Rose-de Dominicis-Janssen formalism [44,[71][72][73]:
Γ k [ḡ, g,ρ, ρ] = r ρ (∂ t ρ + ∇ · g) − D|ḡ| 2 +ḡ · γ∂ t g − µ 1 ∇ 2 g − µ 2 ∇(∇ · g) + α 0 g +κ 0 ∇ρ + α 2 ρ 2 g + κ 2 ρ 2 ∇ρ − ζ∇ 2 ∇ρ ,(45)
where r ≡ d d rdt, and all coefficients above (µ 1 , µ 2 , α 0 , etc.) are now k dependent. The response fields introduced by the formalism are denoted byḡ andρ. Note the absence of any coefficients in the density 'sector' of Γ k , i.e., terms proportional toρ in (45). This is due to the fact that it does not renormalize because of an extended symmetry as defined in Ref. [74]. Specifically, this extended symmetry stems from the linear nature of the continuity equation (1), which implies that, under the transformation ρ(r, t) →ρ(r, t) + ε(r, t), with an arbitrary field ε, the microscopic action Γ Λ transforms linearly in the fields,
δΓ Λ = r ε(∂ t ρ + ∇ · g).(47)
Since the transformation (46) is scale-independent it commutes with the scale derivative ∂ k and we can use the Wetterich equation (44) to see how this relation changes under RG transformations in the case of infinitesimal ε:
∂ k δΓ k = (48) − 1 2 Tr Γ (2) k + R k −1 δΓ (2) k Γ (2) k + R k −1 ∂ k R k .
As δΓ k is linear initially at the scale k = Λ, δΓ
k vanishes, so δΓ k remains unchanged at the infinitesimally larger RG scale k = Λ + dk. This argument can be repeated at this scale and so on, showing that ∂ k δΓ k = 0 and δΓ Λ = δΓ k = δΓ 0 at all scales. Thus, the density sector does not renormalize, which is why, in Eq. (45), we have set the coefficients characterizing it to unity, fixing the engineering dimensions of the fields. This nonrenormalization further implies the hyperscaling relationship between the density and momentum density field,
χ ρ − χ g − 1 = z − 2 .(49)
For the momentum density 'sector' of Γ k , i.e., terms proportional toḡ in (45), we know from our linear theory that this form of Γ k is sufficient only around the critical dimension d c = 6. As a result, we expect that the validity of our quantitative predictions is limited to around d c . Therefore, we will express our results as corrections to the linear theory in terms of = d c − d. In particular, our results for universal exponents will coincide with the perturbative DRG results to order , if the fixed point values forμ andζ are put into the DRG calculation by hand. At the same time, corrections of order O( 2 ) are expected to differ from the DRG results at the same order, since the FRG analysis is nonperturbative in nature. Even though our approach, therefore, becomes perturbative in the couplings α 0 , κ 0 , α 2 and κ 2 , since their FP values are controlled by , our approach is not fully perturbative, since we take the full dependence of the flow equations onμ andζ into account, whose FP values are not controlled by .
B. Regulator
Besides the form of the average action, the regulator R k needs to be specified, which we choose to be, in spatiotemporally Fourier transformed space,
R k (q,p) = (2π) d+1 δ d+1 (q +p) × 0 idA k (q 2 ) 0 iqB k (q 2 ) idA k (q 2 ) 0 0 0 0 0 0 0 −iqB k (q 2 ) 0 0 0 ,(50)
where the ordering of the matrix entries is: (ḡ, g,ρ, ρ). The choice of a time-independent regulator is common for dynamical systems [44]. Also, this matrix form does not regulate the density sector directly but rather introduces a k dependent "pressure term" in the momentum field sector. This regularization sufficiently cuts off large and small scale fluctuations (for appropriate choices of A k and B k ) while following the overall structure of the EOM. In particular, it leaves the extended symmetry unmodified, implying that density remains conserved, even in the regulated theory, which we believe to be crucial to obtain the correct scaling behavior (e.g., the value of the dynamic exponents in dynamic Ising models depends on whether the dynamics are conservative or not [4]). We also define the following in Eq. (50):
A k (q 2 ) = µ ,k k 2 m(q 2 /k 2 ) , B k (q 2 ) = ζ k k 2 m(q 2 /k 2 ) , (51a) m(y) = a/y ,(51b)
where we write the k-dependence of the couplings explicitly and a is an arbitrary positive constant. In principle, all results obtained should be independent of the regulator choice, however, truncating the form of Γ k usually introduces some form of regulator dependence. This dependence can be judged by the a-dependence of the critical exponents. It turns out that for an algebraic regulator as in Eq. (51b), the critical exponents are independent of a. This is shown numerically below, but an analytical argument has been given in Ref. [75] as well.
We further verify our results using also another class of regulator, a generalization of the Litim regulator [76]
m(y) = a(1 − y) 4 Θ(1 − y) .(52)
The fourth order is required to ensure continuous integrands in the RG flow equations, as derivatives of m(y) up to fourth order appear.
C. FRG flow equations
With the forms of Γ k and R k defined, one can then use the Wetterich equation (44) to project a set of coupled ordinary differential equations (ODEs), one for each coefficient in the functional (45). For instance, since
α 0,k = 1 V T 1 d Tr δ 2 Γ k δḡ(q)δg(q) ρ=0 ,(53)
where V T = (2π) d+1 δ d+1 (0) is the spatio-temporal volume, we obtain from the Wetterich equation (44) that
∂ α 0,k = −α 2,k Λ 0 d d p (2π) d ∞ −∞ dω 2π 4Dp 2 ([κ 0 + B k ]p 2 + ζp 4 − γω 2 )p 2 ∂ B k (p 2 ) + (µ p 2 + α 0 + A k (p 2 ))ω 2 ∂ A k (p 2 ) ([κ 0 + B k (p 2 )]p 2 + ζp 4 − γω 2 ) 2 + ω 2 (µ p 2 + α 0 + A k (p 2 )) 2 2(54)
where ≡ − ln(k/Λ). The full set of such FRG flow equations correspond to the graphical corrections to the RG flow equations in our previous DRG analysis (29)(30)(31). At the same time, as in our DRG analysis (37), it is convenient to introduce dimensionless couplings, for the determination of potential RG fixed points (FPs). In the FRG formalism, the non-dimensionalization and rescaling are performed in a single step, where the inverse scale k takes the role of e − Λ. This essentially skips the step where the flow equations are written as in Eq. (35). Specifically, we define the following:
µ = µ 1 µ , (55a) ζ = γζ µ 2 ,(55b)α 0 = α 0 µ k 2 ,(55c)κ 0 = κ 0 ζk 2 ,(55d)α 2 = α 2 k d−6 DS d µ 2 ζ(2π) d ,(55e)κ 2 = κ 2 k d−6 DS d µ ζ 2 (2π) d .(55f)
The rescaling is mostly prescribed by the dimensionality of the couplings. However, we have taken the liberty to rescaleκ 0 andκ 2 with a factor that contains the dimensionless couplingζ. This particular choice ensures that the flow equations remain regular in the case thatζ → 0.
Like in the DRG calculation, we can now again define the anomalous scaling dimensions
∂ µ = η µ µ = Gr FRG µ , (56a) ∂ D = η D D = Gr FRG D ,(56b)∂ γ = η γ γ = Gr FRG γ ,(56c)
which enter the flow equations of the other couplings through the nondimensioning. The FRG flow equations can thus be written in a similar fashion as in Eq. (39):
∂ μ = −η µμ +Ḡr FRG µ1 , (57a) ∂ ζ = (η γ − 2η µ +Ḡr FRG ζ )ζ = η ρζ ,(57b)∂ ᾱ 0 = (2 − η µ )ᾱ 0 +Ḡr FRG α0 ,(57c)∂ κ 0 = (2 − 2η µ + η γ − η ρ )κ 0 +Ḡr FRG κ0 , (57d) ∂ ᾱ 2 = (6 − d − 4η µ + η γ + η D − η ρ )ᾱ 2 +Ḡr FRG α2 ,(57e)∂ κ 2 = (6 − d − 5η µ + 2η γ + η D − 2η ρ )κ 2 +Ḡr FRG κ2 . (57f)
Note that the flow equation ofζ admits two different kinds of FPs: either η ρ = 0 andζ = 0 or η ρ = 0 andζ = 0. This motivates our choice of dimensionless couplings.
The nondimensional graphical corrections to the FRG flow equations can be written in terms of the second-order functional derivatives:
F g = 1 V T −k µ k 2 δ 2 ∂ k Γ k δḡ(q)δg(−q) ρ=ρ unif ,(58a)F ρ = 1 V T −k ζk 2 δ 2 ∂ k Γ k δḡ(q)δρ(−q) ρ=ρ unif ,(58b)F D = 1 V T −k D δ 2 ∂ k Γ k δḡ(0)δḡ(0) ρ=ρ unif ,(58c)
evaluated at vanishing fieldsḡ = g =ρ = 0, except for the density which is set to a value ρ unif , uniform in space and time, or their dimensionful equivalents, F g = µ k 2F g , F ρ = ζk 2F ρ and F D = DF D . Akin to the DRG, the F terms can be represented diagrammatically, hence we call them graphical corrections too. Together with the detailed analytical expressions, they are derived in App. A.
Before we proceed to detail what these graphical corrections are exactly, we will first discuss how FRG enables us to go beyond the 1-loop calculation in the previous DRG calculation. The strategy that we use here follows from Ref. [42], which is to evaluate the aboveF's at a non-vanishing (i.e., off-critical) density. This procedure is supported by the following physical argument: The effective average action at a nonzero scale k, Γ k , serves as an effective theory that describes subsystems of size k −1 . Within this subsystem, the mean density background can be different from the total density background which is vanishing at the MCP, ρ k = ρ 0 = 0, as mass can be exchanged between the subsystems. This is the case when a linear stability analysis on Γ k , which includes nonlinear effects on scales smaller than k −1 , predicts that the homogeneous state ρ = 0 is unstable, which happens when α 0,k < 0 and/or κ 0,k < 0 (the scale dependence has been made explicit here to emphasize that the instability is scale dependent). Then, induced by fluctuations, the system will locally phase separate and spontaneously select a new local density until local stability is reached again. The new local equilibrium is reached when α(ρ) = α 0 + α 2 ρ 2 = 0 and κ(ρ) = κ 0 + κ 2 ρ 2 > 0 or κ(ρ) = 0 and α(ρ) > 0, whichever happens first. Since this is the physical state of the system, we choose it as the constant background field value for ρ unif when evaluating the second-order functional derivatives at scale k (58) for the flow equations for µ 1 , µ and ζ, i.e.,
ρ unif = max α0 α2 , κ0 κ2 if α0 α2 < 0 and κ0 κ2 < 0 α0 α2 if α0 α2 < 0 and κ0 κ2 > 0 κ0 κ2
if α0 α2 > 0 and κ0 κ2 < 0 0 otherwise . (59) For this definition to work, the flow equations for α 0 , κ 0 , α 2 and κ 2 must be evaluated at ρ unif = 0 though, e.g.,
α 0 = Tr δ 2 Γ k δḡ(q)δg(−q) ρ=0,q=0 ,(60)
as Tr
δ 2 Γ k δḡ(q)δg(−q) ρ=ρ unif ,q=0 = 0 ,(61)
by definition. The procedure seemingly reintroduces the interaction terms that would also be introduced by the couplings α 1 and κ 1 , which we excluded above since we were restricting ourselves to the theory obeying the symmetry (23). This is however not the case and is instead solely an effect of the projection. The interaction terms are not free variables of the RG-flow but are in fact always fixed by the relationship (59). The same effect takes place in Ref. [42] where new effective interaction terms appear that seemingly break the Ising or O(N ) symmetry and ultimately yield a nontrivial anomalous dimension.
Note also that distinct from the procedure applied to the Ising model in Ref. [42], we do not make the change of variables: α 0 → ρ unif , in the flow equations. Instead, we treat ρ unif as an auxiliary variable that is determined from Eq. (59) at each RG step. Our procedure is more advantageous here because of the following: since α 0 , κ 0 , α 2 and κ 2 are always changing smoothly in k, the flow equations never become singular, whereas, since ρ unif can have cusps, e.g., if α 0 flips its sign while κ 0 > 0, the flow equation for ρ unif has singular behavior at ρ unif = 0.
Finally, we can then write the graphical corrections to Eq. (57) as
Gr FRG α0 = 1 d TrF g ρ unif =0 q=0 , (62a) Gr FRG κ0 = q iq 2 ·F ρ ρ unif =0 q=0 ,(62b)Gr FRG α2 = 1 2d ∂ 2 ∂ρ 2 unif TrF g ρ unif =0 q=0 ,(62c)Gr FRG κ2 = 1 2 ∂ 2 ∂ρ 2 unif q iq 2 ·F ρ ρ unif =0 q=0 ,(62d)Gr FRG µ1 = 1 d − 1 1 2 ∂ 2 ∂q 2 TrP ⊥ (q)F g q=0 ,(62e)Gr FRG ζ = 1 2 ∂ 2 ∂q 2 q iq 2 ·F ρ q=0 , Gr FRG µ = 1 2 ∂ 2 ∂q 2 1 q 2q ·F g ·q q=0 , (62f) Gr FRG γ = i d ∂ ∂ω TrF g q=0 ,(62g)Gr FRG D = − 1 2d TrF D ,(62h)
where theF's are given in Eq. (58), and we have introduced the dimensionless wavevectorq = q/k,q = |q|, and frequencyω = ωµ /(ζk 2 ). As described above, the couplings α 0 , κ 0 , α 2 and κ 2 are evaluated at vanishing background density fluctuation, while the remaining couplings and anomalous dimensions are evaluated at the value shown in Eq. (59).
We now impose the small expansion by neglecting all but the terms from leading order in by usinḡ
µ ∼ 0 ,ζ ∼ 0 , (63a) α 0 ∼ ,κ 0 ∼ ,(63b)α 2 ∼ ,κ 2 ∼ , (63c) η γ ∼ 2 , η µ ∼ 2 , (63d) η D ∼ 2 , η ρ ∼ 2 ,(63e)
which we know already from our DRG analysis. The flow equations, akin to Eq. (43) in the DRG analysis, can then be written as
∂ ᾱ 0 = 2ᾱ 0 + 1 d TrF g ρ unif =0 q=0 + O( 2 ) ,(64a)∂ κ 0 = 2κ 0 + q iq 2 ·F ρ ρ unif =0 q=0 + O( 2 ) , (64b) ∂ ᾱ 2 = ᾱ 2 + 1 2d ∂ 2 ∂ρ 2 unif TrF g ρ unif =0 q=0 + O( 3 ) , (64c) ∂ κ 2 = κ 2 + 1 2 ∂ 2 ∂ρ 2 unif q iq 2 ·F ρ ρ unif =0 q=0 + O( 3 ) , (64d) ∂ μ = −η µμ + 1 d − 1 1 2 ∂ 2 ∂q 2 TrP ⊥ (q)F g q=0 + O( 3 ) , (64e) ∂ ζ = η ρζ = (η γ − 2η µ )ζ + 1 2 ∂ 2 ∂q 2 q iq 2 ·F ρ q=0 + O( 3 ) ,(64f)
where the anomalous dimensions are
η µ = 1 2 ∂ 2 ∂q 2 1 q 2q ·F g ·q q=0 + O( 3 ) , (65a) η γ = i d ∂ ∂ω TrF g q=0 + O( 3 ) ,(65b)η D = − 1 2d TrF D q=0 + O( 3 ) . (65c)
The evaluation of theF's at a nonvanishing density can now be seen to serve two purposes: First, it enables the flow equations for the nonlinear couplingsᾱ 2 and κ 2 to be projected fromF g andF ρ by taking a secondorder derivative with respect to ρ unif . Secondly, if we were to set ρ = 0 for the evaluation of the flow equations ofμ andζ as well as the anomalous dimensions, they would be vanishing. Then the flow equations would be equivalent to the one-loop DRG result. (In fact, we will show in App. C that one can obtain the one-loop DRG equations exactly from the FRG formalism by using a specific "sharp" regulator, as in Ref. [77].) Choosing a nonvanishing ρ unif , therefore, allows us to incorporate effects that go beyond the one-loop level.
VIII. NOVEL RG FIXED POINTS
The fixed points of the FRG flow equations (64,65) determine the universality classes of the system and their associated scaling behavior. While the flow equations (64,65) can in principle be expressed analytically, the number of terms involved renders them unilluminating. Further, we are unable to analytically solve some of the integrals buried in the definitions of the F's (A10). We, therefore, use a combination of computer algebra and numerical methods to solve the FRG equations and thus discern the flow of these couplings upon decreasing the inverse length scale k. Details of the implementation are given in App. B.
In a typical perturbative DRG calculation to one-loop order, one would find that the flow equations for the nonlinear couplings κ 2 and α 2 decouple from the relevant couplings κ 0 and α 0 . Their FP values can therefore be easily obtained even in a numerical calculation since usually at least one FP in this subspace is attractive. In our FRG approach, however, the flow equations for the amplitude ratiosμ andζ, which the non-linear couplings depend on, are directly proportional to the relevant couplingsᾱ 0 andκ 0 through Eq. (59). Therefore, one has to solve all flow equations simultaneously. This is problematic since the relevant couplings diverge from the FP. To tackle this problem in an FRG calculation, one typically invokes the shooting method [40,42] to fine-tune the relevant parameters, which however becomes difficult when there are many parameters to fine-tune. Here we have developed the following simple method to tackle this problem.
A. Fine-tuning by reversing RG flows
To steer the couplings towards the fixed points, we invert the sign of the relevant flow equations. This operation manifestly leaves the locations of the FPs invariant, but changes their stability. The flow equations, therefore, fine-tune themselves. Once the fixed point solution is found, the original signs can be restored to obtain the critical exponents. This method can also be extended to explore other unstable FPs by inverting additional flow equations.
With the help of this simple trick, we find a total of four FPs (Fig. 4 and Tab. I). One, FP3, is stable and therefore governs generically the universal critical behavior of the MCP under consideration. It is denoted by the red circle in Fig. 4 and reached by performing the following inversions
∂ ᾱ 0 → −∂ ᾱ 0 , ∂ κ 0 → −∂ κ 0 .(66)
Here, "stability" refers to the stability within the "critical manifold". We also obtain two other unstable nontrivial FPs: FP2, the green square in Fig. 4 reached by, additionally to (66), inverting
∂ ᾱ 2 → −∂ ᾱ 2 ,(67)
and FP4, the blue diamond in Fig. 4 reached by, additionally to (66), inverting
∂ κ 2 → −∂ κ 2 .(68)
Finally, there is the trivial Gaussian FP, FP1, denoted by the yellow pentagon.
To the best of our knowledge, the universality classes associated to all FPs are novel, except for the Gaussian FP (yellow pentagon).
FIG. 4.
A projection of the RG flow diagram on the "critical manifold" to the space spanned byᾱ2,κ2 andμ. Our FRG analysis enables us to find four fixed points (FPs): one is stable (FP3, denoted by the red circle) and three are unstable (FP1, 2, and 4, denoted by the yellow pentagon, green square, and blue diamond, respectively). In this projection, FP1 and FP2 constitute lines of fixed points (yellow and green, respectively). The marked yellow pentagon and green square show the specific FPs reached corresponding to our choice of initial conditions (μ = 1 andζ = 1, see App. B).
B. Genuine nonequilibrium UCs
In equilibrium the fluctuation dissipation theorem implies η γ = η D . Since this is clearly broken for FP3 and FP4 (see Tab. I), we can conclude that FP3 and FP4 are novel nonequilibrium universality classes. While the fluctuation dissipation theorem does not seem to be broken for FP2, this does not necessarily imply that FP2 describes the critical phenomenon of an equilibrium system. We discuss this further in Sec. VIII E.
C. Nonlinear scaling
Now that we have actually found fixed points through our RG analysis, we will revisit the scaling behavior of the theory as well as the correlation functions.
At the FPs, the anomalous dimensions, i.e., the η's (56), take on universal FP values. Therefore, under a RG transformation from a reference scale k , where the system is already sufficiently close to the FP, to the scale k, the dimensionful EOM (24) in our truncation trans- form as
γ e ηγ ∂ t g = µ e ηµ μ * ∇ 2 g + (1 −μ * )∇(∇ · g) −ᾱ * 0 e −2 g − α * 2 e (d−6+3ηµ−η D −ηγ +ηρ) ρ 2 g +ζ e (2ηµ+ηρ−ηγ ) ∇ 2 ∇ρ − κ * 0 e −2 ∇ρ −κ * 2 e (d−6+3ηµ−η D −ηγ +ηρ) ρ 2 ∇ρ + e η D 2 f ,(69)
where = log(k /k), primed couplings denote couplings at the reference scale k and starred couplings denote the FP value of the couplings. This EOM together with the continuity equation (1) is scale-invariant if we rescale lengths, time and fields,
r → re , t → te z , ρ → ρe χρ , g → ge χg ,(70)
with the nonlinear scaling exponents:
z = 2 − η µ + η γ ,(71a)χ g = 2 − d − η µ − η γ + η D 2 ,(71b)χ ρ = 4 − d − 3η µ + η γ + η D − η ρ 2 .(71c)
Note that this only works if η ρ = 0, which, for the nontrivial FPs, is only the case for FP2 and FP3, but not FP4. We will discuss the case η ρ = 0 for FP4 in Sec. VIII F. If η ρ = 0, we have found a rescaling transformation under which the EOM are invariant, and can, therefore, apply the argument from Sec. III to deduce the scaling of the correlation functions (8,9). But why is it even possible to extract the scaling behavior from the coefficients at finite k, i.e. in a regulated theory with a finite IR cutoff? To see this, consider the following argument (compare also to [27,78]). Suppose, we have the inverse propagator in Fourier-space sufficiently close to the FP, i.e. Γ (1,1,0,0) k (ω, q) scales homogeneously under an RG transformation, k → sk, and simultaneous rescaling of q → sq and ω → s z ω (the scaling behavior of Γ (1,1,0,0) is inverse to that of G), i.e.
for some χ. Now we can consider two equivalent cases. First we set s = k /q with a constant scale k
Γ (1,1,0,0) k (ω, q) = k q χ Γ (1,1,0,0) k k/q k q z ω, k ,(73)
where we can now safely take the limit k → 0, showing that the propagator of the effective action, which is the full solution to the many-body problem, all nonlinear fluctuations included, follows a powerlaw in q with the exponent χ. Secondly we set s = k /k
Γ (1,1,0,0) k (ω, q)= k k χ Γ (1,1,0,0) k k k z ω, k q k ≈ k k χ α 0 ,(74)
where in the last line, we developed Γ (1,1,0,0) k to zeroth order in q and ω, corresponding to our truncation which, by neglecting higher order derivative terms, also assumes that q
k. This last result shows, that the constant part of the inverse propagator scales in k with the same exponent χ. And this exponent has already been obtained through our FRG analysis, i.e., χ = η µ − 2.
A similar argument can be made for the rest of the entries is Γ (2) k , showing the scaling behavior (8,9) for the realspace propagators, obtained from inverting and then Fourier-transforming Γ (2) k .
D. Universal critical exponents and amplitude ratios
The resulting values for the critical exponents at these FPs, depending on the regulator parameter a are plotted in Fig. 5 for both regulators, Eq. (51b) and (52), exemplarily at = 0.1. This clearly shows that, for the algebraic regulator (51b), the critical exponents are independent of the parameter a. While we have shown this numerically, an analytical argument for this is given in [75]. In contrast, the results using the Litim regulator of minimal sensitivity [79], we, therefore, chose the result of the algebraic regulator as our main results which, expressed in terms of = (d c − d), are shown in Table II.
In addition to these critical exponents, we can also provide quantitative predictions on the two universal amplitude ratios:μ andζ, shown in Tab. I. These could in principle be measured experimentally from diffusion constants and the wavelength of density waves.
Finally, we can also determine the exponents y 1 and y 2 , describing the divergence of the correlation length according to Eq. (21), which are also shown in Tab. II.
The dependence of the amplitude ratiosμ andζ, as well as the correlation length exponents y 1 and y 2 on the regulator parameter a is similar to the other exponents in Fig. 5, i.e. they are independent of a for the algebraic regulator (51b) and the results obtained with the Litim regulator (52), while varying in a, are compatible with those of the algebraic regulator up to a correction of nextto-leading order in . For the second fixed point, we observe that the inverse correlation length exponent y 1 agrees with the correla-tion length exponent of the Ising universality class in 4 − dimensions. Though the difference in upper critical dimension between the two universality classes implies their distinctness, it is nevertheless interesting to explore how this relationship arises, which we will do in this subsection.
At FP2, the FP values of all α-couplings vanish, which implies that the EOM reduce to
∂ t ρ = − ∇ · g (75a) γ∂ t g = µ 1 ∇ 2 g + µ 2 ∇(∇ · g) + f − ∇ κ 0 ρ + κ 2 3 ρ 3 − ζ∇ 2 ρ ,(75b)
which is linear in g. We can immediately see, that the transverse component decouples from both the density and longitudinal momentum field and that its dynamics are given by the mean field critical O(d − 1) EOM
∂ t g ⊥ = µ 1 ∇ 2 g ⊥ + f ⊥ .(76)
In the parallel sector, we can eliminate g from the EOM to obtain
(γ∂ t − µ ∇ 2 )∂ t ρ = ∇ 2 κ 0 ρ + κ 2 3 ρ 3 − ζ∇ 2 ρ − ∇ · f .(77)
This equation is again reminiscent of Model B dynamics, except that the time-derivative term is heavily modified. The linear mode of the momentum field, though eliminated from the equation, manifests now in the second order time-derivative. Here, all anomalous dimensions are zero. As a result, the fluctuation-dissipation relation is not explicitly broken and it remains to be seen whether the model equation (77) corresponds to an equilibrium system or not.
This EOM can therefore be seen as an Ising model with exotic two-mode dynamics, that rises the scaling dimension of the field, and therefore also the upper critical dimension.
F. FP4: Emergence of two time-scales
So far, we have discussed the FPs, where η ρ = 0. Now we turn to the case η ρ = 0, which is the case for FP4. In this case, it is impossible to choose rescaling exponents z, χ g and χ ρ such that all the terms in the EOM (69) rescale homogeneously. For instance, if we were to choose the same exponents as for FP2 and 3 (71), in the large , i.e. hydrodynamic limit, the continuity equation (1) would reduce to a statement of staticality,
∂ t ρ = 0 ,(78)
and all the "pressure-terms", proportional to ∇ρ in (69), would vanish, decoupling the momentum density field from the density field at the linear level. Since the continuity equation is modified, this is the only FP where the hyperscaling relation by the extended symmetry of the continuity equation, is broken.
χ ρ − χ g − 1 = z − 2, enforced FP z − 2 χg + (d − 2)/2 χρ + (d − 4)/2 y1 − 2 y2 − 2 1 0 0 0 0 0 2 0 0 0 −0.FP z − 2 χg + (d − 2)/2 χρ + (d − 4)/2 y1 − 2 y2 − 2 2' 0 0 −0.022 2 −0.33
If we interpret this as the momentum-densitycouplings becoming irrelevant at this fixed point, we can simply omit these and obtain a scale invariance on the remaining terms. For the momentum density correlation function, we can therefore conclude that,
C g (r, t) = r 2χg S gg t r z .(79)
At the linear level, they would look just like in Eq. (16d) with ζ = 0. In this decoupled limit however, the density correlation function cannot be determined since the density field seemingly decouples from the noise term f . This suggests, that this choice of rescaling is not the correct one when looking at density-density correlations. If we were to choose instead a different time and momentum field rescaling,
z alt = 2 − η µ + η γ − η ρ ,(80a)χ alt g = 2 − d − η µ − η γ + η D + η ρ 2 ,(80b)χ alt ρ = χ ρ = 4 − d − 3η µ + η γ + η D − η ρ 2 . (80c)
the continuity equation remains scale invariant and the "pressure-terms" stay relevant. However, now the timederivative term on the left-hand-side of Eq. (69) vanishes in the large , i.e., hydrodynamic, limit. Now, again regarding this term as irrelevant, the remaining terms support a scale invariance, from which we can conclude that the density correlation function scales as
C ρ (r, t) = r 2χ alt ρ S ρρ t r z alt .(81)
At the linear level, with this rescaling, the correlation functions would be
C ρ (r, t) = 1 ζ q e iq·r 2Dq 2 q 8 + ω 2 µ 2 q 4 ,(82a)
C g (r, t) = ζ q e iq·r 2Dω 2 P (q)
q 8 + ω 2 µ 2 q 4 .(82b)
The transverse components of the momentum density remain unmodified. This clearly shows that at FP4 two separate time-scales are emerging; one fast time-scale at which the momentum density can react quickly to perturbations in the presence of a density field frozen in time, and a slow time-scale over which the density relaxes once momentum fluctuations have already dissipated long ago. In other words, the momentum density ceases to be a hydrodynamic variable at this fixed point, since it becomes a "fast" mode.
Even though our analysis has shown that in the vicinity of FP4 the "pressure-terms" proportional to ζ are irrelevant, they cannot be neglected in the RG analysis. First, as discussed in Sec. VI if ζ is naively set to zero, the frequency integrals are clearly divergent. Secondly, the FP value of κ 2 plays an important role in determining the critical exponents of FP4. This coupling must therefore be included necessarily in the discussion.
The right coupling to neglect instead is therefore the time-derivative term of the momentum density. Then, its EOM becomes an exact, time-invariant identity enslaving the momentum field to the density field. It can be used to eliminate the momentum field from the continuity equation to obtain, for simplicity at the linear level,
that α 0 − µ ∇ 2 ∂ t ρ = ∇ 2 κ 0 ρ − ζ∇ 2 ρ − ∇ · f , (83)
which, except for the µ term, is exactly the general equation of model B dynamics one would write down for a conserved, scalar quantity [4]. Fine-tuning κ 0 → 0 alone yields the critical model B, Ising universality class. We can therefore interpret FP4 as a genuine nonequilibrium multicritical point of model B, where, in addition to κ 0 , also the time-derivative term, characterized by α 0 , is finetuned to zero. For reasons of stability, the higher-order derivative term characterized by µ , is needed.
The ζ = 0 manifold, therefore, describes the model B theory subspace. Within it, in addition to FP4, we find two additional FPs which are related to FP2 and FP3, so we name them FP2' and FP3' accordingly. Since they are unstable in the ζ direction and already described by the model B EOM, we will not discuss them in detail. Instead, we will simply report here the FP values and critical exponents in Tab. III.
IX. SUMMARY & OUTLOOK
We have demonstrated in this work the whole process from first formulating the large scale hydrodynamic equations to discovering novel universality classes on the particular problem of the multicritical point in the phase diagram of compressible active matter, where the critical point of the flocking transition coincides with the critical point of disordered phase-separation.
We started with the Toner-Tu equations, derived purely from symmetry arguments and conservation laws, which therefore describe general compressible polar active fluids, i.e. systems of self-propelling and aligning particles at the hydrodynamic level. Using mean-fieldtheory and linear stability analysis one can show, first, the existence of two homogeneous phases and, secondly, the existence of regions where the homogeneous state is unstable indicating phase-separation, revealing the phase diagram of compressible polar active matter. Linear stability analysis further reveals the existence of two critical points, the critical order-disorder transition and the critical point of disordered phase-separation. Further, there is a multicritical point in the phase diagram where both these critical points overlap.
Analyzing the linearized equations of motion around this multicritical point, then revealed the scale invariance of the theory at the linear level, which results in powerlaw scaling of the correlation functions. We elucidated how these functions, and thereby the scaling exponents as well as the scaling behavior of the correlation length, could be measured in principle.
The scaling behavior of the linear theory further informed us about the relevance of the possible nonlinear terms and the critical dimension, above which the linear theory is expected to be exact. While in the nonlinear regime, the correlation functions could no longer be calculated explicitly, scale invariance will in general still lead to powerlaw correlation functions, though their scaling exponents are no longer trivially determined.
To determine these nontrivial exponents, we attempted a one-loop dynamic renormalization group (DRG) approach within the -expansion, which ultimately failed, since two-loop effects are necessary to capture the universal physics of this problem.
To take these two-loop effects into account, we set up a functional renormalization group (FRG) ansatz which describes effective theories at intermediate scales and whose scale dependence is described by the Wetterich equation. A key ingredient to this approach was the physical insight that subsystems at an intermediate scale are locally in a homogeneous state even if the total system is phase separating. This enabled our FRG approach to go beyond the perturbative DRG approach.
Using various computer algebra and numerical methods developed for this work, the renormalization group flow equations could then be evaluated. Further, by inverting the sign of the flow equation for the relevant couplings, we found three renormalization group fixed points. Finding them proves the existence of three novel universality classes, at least two of them being demonstrated to be genuinely out of equilibrium.
In summery, our achievements are three folds: (1) the discovery of three novel universality classes, two of them being demonstrably out of equilibrium [80], (2) the first analytical elucidation of critical behavior for compressible active fluids, and (3) the first application of FRG on active matter systems beyond the equivalence of the perturbative one-loop level.
Interesting future directions include the applications of FRG to explore open questions in compressible active matter such as: what are the universality classes of the critical order-disorder transition [65] and of the ordered phase [63,81].
The propagator G is essentially the same as in the DRG calculation (28), except that it now contains the terms introduced by the regulator, which make the flow equations IR convergent, where in the DRG calculation this is taken care of by the integral boundaries.
Similarly to the DRG analysis (27), we introduce a graphical notatioñ
q = G(q) ,q = 1 −iω q ,q = −iq , = 2D id , (A6a) = α 2 id , q = iqκ 2 , = α 2 ρ unif id , q = iqκ 2 ρ unif ,(A6b)
which differs from the DRG notation only by the redefinition of the propagators (A5) and the the three-point-vertices being nonzero.
If we regard the individual entries of Eq. (A2), especially those corresponding to the F's in Eq. (58), we can arrange the vertices, Eqns. (A3f-A3i), in matrices similar to that in Eq. (A4), such that the matrix products in Eq. (A2) can be carried out and traced over. Then the F's can be decomposed into individual terms,
F g = idF g + P (q)F g + P ⊥ (q)F ⊥ g ,(A7a)F g = F ,a g + F ,b g + F ,c g ,(A7b)F ⊥ g = F ⊥,a g + F ⊥,b g ,(A7c)F ρ = iq F a ρ + F b ρ + F c ρ + F d ρ + F e ρ + F f ρ + F g ρ ,(A7d)F D = id F a D + F b D ,(A7e)
which can then be represented diagrammatically,
idF g = , F ,a g + F ⊥,a g = −4qq , F ,b g + F ⊥,b g = −4qq , (A8a) F ,c g = −4qq , iqF a ρ = q q , iqF b ρ = −4 q q ,(A8b)iqF c ρ = −4 q q , iqF d ρ = −4 q q , iqF e ρ = −4 q q ,(A8c)iqF f ρ = −4 q q , iqF g ρ = −4 q q , idF a D = −4 ,(A8d)idF b D = −4 .(A8e)
In dimensionless units,Ḡ = µ k 2 G , (64) is then solved using a fourth order adaptative Runge-Kutta-Fehlberg (4,5) algorithm provided by the GNU Scientific Library [82], where at each RG "time-step", the wave number integral (B6) in (A10) is solved numerically using an adaptative quadrature routine, with the 15 point Gauss-Kronrod rule for infinite boundary integrals in case of the algebraic regulator and the 61 point Gauss-Kronrod rule for finite boundary integrals in case of the Litim regulator, again provided by the GNU Scientific Library [82]. For the adaptative ODE-solver we set a maximum relative and absolute error of e r = 10 −11 and e a = 10 −13 respectively. For the integration they are set to e r = 10 −12 and e a = 10 −10 . Since, we are modifying the flow equations such that each fixed point we would like to investigate is attractive, the initial conditions do not matter too much (as long as they are within the attractive basin). In our analysis we set them close to the Gaussian FP. Since we obtained all our results in Tab. II at = 0.1, the following initial conditions are sufficiently close to the Gaussian FP:μ =ζ = 1,ᾱ 2 =κ 2 = 10 −3 andᾱ 0 =κ 0 = 0. We use these initial conditions to analyze all FPs, except for the following modifications: if we are interested in FP2, we setᾱ 2 = 0 (with this condition it is not necessary to invert the flow equation forᾱ 2 , as it is not being generated if it is vanishing initially), and when we investigate FP4, we setκ 2 = −10 −3 .
G ⊥ = µ k 2 G ⊥ ,(A9a)
When solving the ODEs we set an initial "time-step" of d = 5 × 10 −4 initially, but it is being quickly changed by the adaptative algorithm. The adaptative nature of the algorithm is key to solve this problem, since there are three vastly different RG "time-scales", given by the critical exponents, to resolve. First, the relevant couplings α 0 and κ 0 converge very quickly with a time-scale of ∼ 0.5 to the critical surface. The next time-scale is that of nonlinear couplings α 2 and κ 2 and of order ∼ 1/ . Then the final and longest time-scale sets in, wherein the amplitude ratiosμ andζ converge. This time-scale is approximately of the size of the anomalous dimensions, i.e., of order ∼ 1/ 2 . However, since the prefactor of the O( 2 ) critical exponents is typically a lot smaller than unity, we find total convergence of the fixed points only at about ∼ 5 × 10 4 , though FP4 converges somewhat quicker, since the scaling exponents are generally larger, compare Fig. 5 and Tab. II.
Finally, once the ODEs have converged to the desired FP, the correlation length exponents can be obtained by first restoring the original signs of the flow equations, and then taking discrete derivatives with a finite difference of dc = 10 −7 with respect to the couplings. The largest two eigenvalues of the so obtained matrix characterizing the linearized flow equations around the FP give the correlation length exponents y 1 and y 2 .
The complete algorithm is summarized again in Alg. 1.
FIG. 1 .
1Polar active fluids admit diverse phase transitions and phase separations. These figures show qualitatively two possible instances already discussed in
Fig. 2 .
2
FIG. 2 .
2Scaling behavior of the correlation length when approaching the MCP. The inset shows the phase diagram in terms of the couplings α0 and κ0 under the assumption that α1 = κ1 = 0, and the main figure shows the equal-time density correlation function Cρ(r, 0) in log-log scale (black lines) which, if sufficiently close to the MCP (red circle in the inset)
equations have four different fixed point solutions, three of which are nontrivial and depend on the initial values ofμ andζ. In Fig. VI, we show two examples of flow diagrams for different values ofμ and ζ.
FIG. 3 .
3It is a priori not clear which values to choose forμ andζ and the flow diagram and the location of the four fixed points, indicated by the yellow pentagon (1), the green square (2), the red circle (3) and the blue diamond (4), depend on this choice. To show this, we plot here two examples of the flow diagram obtained with the different values forμ andζ shown, yielding different results. In both cases we chose = 0.1.
. Fixed point values for all four fixed points, expressed as an -expansion from the upper dimension dc = 6. The fixed point values of α0 and κ0 are normalized such that they are independent of the regulator parameter a. The universal amplitude ratiosμ andη as well as the anomalous dimensions, the η's, are universal. When no value forμ orζ is given, they can take any arbitrary value and are not universal in this case.
, q) = s χ Γ (1,1,0,0) sk (s z ω, sq),
=
(52) do depend on a, however, the estimated values for the critical exponents are compatible with those of the algebraic regulator, i.e. the deviations are all smaller than 3 . Due to the expansion accuracy can anyways only be expected to this degree. By virtue of the principle χg + (d − 2)/2 and χ anom ρ = χρ + (d − 4)/2, on the regulator parameter a obtained at = 0.1. The different colors label different fixed points, corresponding to Fig. 4, and are annotated by brackets. The different line styles denote the different exponents. For the algebraic regulator (51b), the exponents are completely independent of a (saturated lines), which is why we accept them as our final results, see Tab. II, according to the principle of minimal sensitivity. While the results obtained with the Litim regulator (52) do depend on a (faint lines), they are compatible with those of the algebraic regulator and the deviations are always smaller than what is expected of next order corrections, i.e. smaller than 3 = 10 −3 .
E. FP2: a nonequilibrium version of the Ising universality class?
. Fixed point values and universal critical exponents for the two additional fixed points in theζ = 0 plane, expressed as an -expansion from the upper dimension dc = 6. The fixed point values of α0 and κ0 are again normalized such that they are independent of the regulator parameter a. When no value forμ is given, it can take any arbitrary value and is not universal in this case.
25
d ← 5 × 10 −4 14 Initialize GSL ODE solver environment with standard control, rfk45 step type and errors of epsabs ← 10 −11 , epsrel ← 10 −13 15 while < 500/ 2 do 16 Calculate flow equation (μ,ζ,ᾱ2,κ2,ᾱ0,κ0): 17 if algebraic regulator then 18 Initialize GSL QAG integral environment with infinite boundaries and errors of epsabs ← 10 −12 , epsrel ← 10 −10 19 else if Litim regulator then 20 Initialize GSL QAG integral environment with finite boundaries, 61 point Gauss-Kronrod rule and errors of epsabs ← 10 −12 , epsrel ← 10 Restore original signs of flow equations 26 Take a discrete derivative with finite difference dc = 10 −7 of flow equation with respect to all couplings 27 Calculate 2 largest eigenvalues of the derivative matrix 28 Save FP values and 2 largest eigenvalues RG flow equations
Algorithm 1: The numerical procedure to obtain the FP solutions and critical exponents Select FP 2 Select Regulator type and parameter 3μ ←ζ ← 1 4ᾱ0 ←κ0 ← 0 5ᾱ2 ←κ2 ← 10 −2 6 Invert sign of α0 and κ0 flow equation7 if FP2 then1 8
α2 ← 0
9 else if FP3 then
Appendix A: FRG Flow equationsIn the following we give details on how the right-hand-side of Eqns.(58)can be obtained. As a first step, to reduce some of the complexity of the following calculation, we use a trick commonly used in FRG. We reformulate the Wetterich equation(44)to the formwhere the derivative ∂ = k ∂ k on the right-hand-side, only acts on the k-dependence of the regulator. This reduces the number of propagators (Γ(2)k + R k ) −1 in the following expressions by one. As the projections in Eqns.(58)involve two functional derivatives of Eq. (A1), the right-hand-side of Eqns.(58)after derivation will involve, in addition to Γ(2)k also the third and fourth order functional derivatives of Γ k :where we didn't write explicitly the dependence of Γ(3)k and Γ(4)k , on the external wavevectors and frequencies that come from the functional derivatives. As written in Eqns.(58), the expressions are then to be evaluated at the constant field ρ(r, t) = ρ unif . The expressions for Γ(2)k , Γk and Γ(4)k with this background field are straightforwardly determined from our ansatz, Eq.(45):where we introduced the short-hand notationδ qp... = (2π) d+1 δ d+1 (q +p + . . . ). All other functional derivatives of Γ k , up to fourth order, evaluated at these background fields and not mentioned here are vanishing. From these equations (A3) we can determine the regulated propagator by inverting the 4 × 4 matrix Γwhere we have defined,their analytical expressions are given bȳwhere we introduced the cosine between the internal and external wave vector z =q ·p/qp and the "primed integral",Other terms, not listed in Eq. (A10), one would naively expect to appear, but vanish for reasons of causility (see e.g.[44]), i.e., terms that have poles only in the lower/upper half of the complex plane in the variableω p . Notice that the limit1has not yet been applied yet.From this point on, further calculations by hand quickly become unmanagable, even though analytical treatment is still possible for a few more steps. Going from here to integrating the flow equations (64) is described in App. B.Appendix B: Computer algebra and numerical methodsIn the following, we describe how, from the equations above (64, A10), one can use computer algebra and numerical methods to solve the flow equations to find FP solutions and obtain the corresponding critical exponents characterizing their universality class.For each F in Eq. (A10), one has to solve the three different integrals as defined in Eq. (A11): the frequency integral overω p , the angular integral over the directional degrees of freedom ofp and the integral over the wavenumberp. The frequency integrals can be solved exactly using Cauchy's integral formula and the angular integral can be solved analytically in the limit ofq 1, which fortunately is the limit we are interested in. However, the remaining integral overp must be solved numerically sinceμ andζ are nonnegligible, scale-dependent numbers.Additionally, one has to perform the ∂ derivative in (A11), the expansion into small and the expansion into small external wave numbers and frequencies,q andω q , (depending on which couplings, one either needs the zeroth or second order Taylor coefficient inq and zeroth or first order Taylor coefficient inω q according to Eq. (64)). All three of these operations are applied after the frequency integral, where it is most convenient.As the pole-structure in any of the integrands of Eqns. (A10) is simple enough -one can chose to close the contour such that the only possible poles appearing arēi.e., independent of the external wave vectorq, and where we have definedĀ k (p 2 ) = A k (p 2 )/[µ k 2 ] andB k (p 2 ) = B k (p 2 )/[ζk 2 ] -one can apply Cauchy's integral formula to solve the frequency integrals analytically. Then the ∂ derivative can be carried out, which is straightforward since onlyĀ k andB k depend on k and their derivatives evaluate toĀKeep in mind thatĀ k andB k can appear withq 2 or |p −q| 2 as arguments. Since the anomalous dimensions η µ , η ρ and η γ are of O( 2 ), they can be neglected in Eqns. (B3). This is for example a convenient point to perform the expansion in small , noting that the couplingsᾱ 0 ,κ 0 ,ᾱ 2 andκ 2 are of order O( ). Finally, as a preparatory step for the angular integrals, the resulting integrands are developed in smallq (to second order) andω q (to first order). This produces derivatives of m up to fourth order.The only angular dependence of the integrals comes from the dependence of z which, due to Taylor expansion into smallq, is polynomial up to order z 4 . Using the Jacobian for d-dimensional spherical coordinates, one can show that d dpAn integral over any power of z and, therefore, also any polynomial of z, can be performed using the formulafor even n. For odd n the angular integral vanishes. Finally, the wave number integral can be reformulated in terms of y =p 2 ,All steps in App. B so far have been performed using computer algebra. The resulting expressions have then been converted into C++ code such that the final wave number integral can be solved numerically. The complete set of Appendix C: DRG Flow equations Instead of going through the usual procedure of obtaining the graphical corrections to the DRG flow equations, we instead obtain them from the FRG calculation. The flow equations are still described by Eqns.(64,65), but the F's in Eq. (A10) need to be adapted. While there is no finite background field in the DRG formalism, it is nevertheless important to realize, that the derivatives with respect to ρ unif in Eq. (62) essentially represent attachments of external legs with vanishing wavevector to the diagrams, e.g.,This important fact creates the relationship between FRG diagrams and the corresponding DRG diagrams. Only after these derivatives have been performed, the background field is set to ρ unif = 0 in all expressions, since in the perturbative DRG the background field is vanishing. Since no derivatives with respect to ρ unif appear in the projections for the higher order derivative term couplings, i.e. µ 1 , µ and ζ, their graphical corrections are therefore directly set to zero since they are proportional to ρ unif .Secondly, the regulator needs to be redefined to a sharp cutoff. The technical aspects of this cutoff are well described in[77], but here we will describe its practical implications: 1) Any regulator functions in the integrands can directly be set to zero,Ā k =B k = 0. 2) The frequency integral remains unmodified. 3) In diagrams where the external wavevector is nonzero, the integrand needs to be averaged over the two possible paths the external wavevector can take through the diagram, i.e. 1/2 the integrands from Eqn. (A10) plus 1/2 the same integrand with the replacement p →p +q. This step can be skipped in the small expansion since the graphical corrections are independent of the wavevector routing to linear order in . 4) The integral over the magnitude of the loop wavevector and the ∂ derivative are removed, i.e. ∞ 0 dp∂ → 1.(C2)5) The magnitude of the dimensionless wavenumber is set to unityp → 1.These five rules can formally be derived by realizing that the sharp regulator amounts to a change in regulator:The theta and delta functions appearing after taking the ∂ derivative realize the rules described above. The DRG flow equations obtained this way have been displayed in Eq.(43).As an example consider the term F d ρ . In the full FRG formalism, it would contribute to the graphical corrections of κ 2 and ζ. But because it is proportional to ρ 2 unif , it can only contribute to κ 2 in the adapted DRG equations, since it is the only coupling requiring two derivatives with respect to ρ unif in its projection according to Eq.(62). 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| []
|
[
"VAIN: Attentional Multi-agent Predictive Modeling",
"VAIN: Attentional Multi-agent Predictive Modeling"
]
| [
"Yedid Hoshen [email protected] \nFacebook AI Research\nNYC\n"
]
| [
"Facebook AI Research\nNYC"
]
| []
| Multi-agent predictive modeling is an essential step for understanding physical, social and team-play systems. Recently, Interaction Networks (INs) were proposed for the task of modeling multi-agent physical systems, INs scale with the number of interactions in the system (typically quadratic or higher order in the number of agents). In this paper we introduce VAIN, a novel attentional architecture for multi-agent predictive modeling that scales linearly with the number of agents. We show that VAIN is effective for multi-agent predictive modeling. Our method is evaluated on tasks from challenging multi-agent prediction domains: chess and soccer, and outperforms competing multi-agent approaches. | null | [
"https://arxiv.org/pdf/1706.06122v1.pdf"
]
| 347,868 | 1706.06122 | cab19139d9f2c8479766db0bb39195db7701d984 |
VAIN: Attentional Multi-agent Predictive Modeling
Yedid Hoshen [email protected]
Facebook AI Research
NYC
VAIN: Attentional Multi-agent Predictive Modeling
Multi-agent predictive modeling is an essential step for understanding physical, social and team-play systems. Recently, Interaction Networks (INs) were proposed for the task of modeling multi-agent physical systems, INs scale with the number of interactions in the system (typically quadratic or higher order in the number of agents). In this paper we introduce VAIN, a novel attentional architecture for multi-agent predictive modeling that scales linearly with the number of agents. We show that VAIN is effective for multi-agent predictive modeling. Our method is evaluated on tasks from challenging multi-agent prediction domains: chess and soccer, and outperforms competing multi-agent approaches.
Introduction
Modeling multi-agent interactions is essential for understanding the world. The physical world is governed by (relatively) well-understood multi-agent interactions including fundamental forces (e.g. gravitational attraction, electrostatic interactions) as well as more macroscopic phenomena (electrical conductors and insulators, astrophysics). The social world is also governed by multi-agent interactions (e.g. psychology and economics) which are often imperfectly understood. Games such as Chess or Go have simple and well defined rules but move dynamics are governed by very complex policies. Modeling and inference of multi-agent interaction from observational data is therefore an important step towards machine intelligence.
Deep Neural Networks (DNNs) have had much success in machine perception e.g. Computer Vision [1,2,3], Natural Language Processing [4] and Speech Recognition [5,6]. These problems usually have temporal and/or spatial structure, which makes them amenable to particular neural architectures -Convolutional and Recurrent Neural Networks (CNN [7] and RNN [8]). Multi-agent interactions are different from machine perception in several ways:
• The data is no longer sampled on a spatial or temporal grid. • The number of agents changes frequently.
• Systems are quite heterogeneous, there is not a canonical large network that can be used for finetuning. • Multi-agent systems have an obvious factorization (into point agents), whereas signals such as images and speech do not.
To model simple interactions in a physics simulation context, Interaction Networks (INs) were proposed by Battaglia et al. [9]. Interaction networks model each interaction in the physical interaction graph (e.g. force between every two gravitating bodies) by a neural network. By the additive sum of the vector outputs of all the interactions, a global interaction vector is obtained. The global interaction alongside object features are then used to predict the future velocity of the object. It was shown that Interaction Networks can be trained for different numbers of physical agents and generate accurate results for simple physical scenarios in which the nature of the interaction is additive and binary (i.e. pairwise interaction between two agents) and while the number of agents is small.
Although Interaction Networks are suitable for the physical domain for which they were introduced, they have significant drawbacks that prevent them from being efficiently extensible to general multiagent interaction scenarios. The network complexity is O(N d ) where N is the number of objects and d is the typical interaction clique size. Fundamental physics interactions simulated by the method have d = 2, resulting in a quadratic dependence and higher order interactions become completely unmanageable. In Social LSTM [10], this was remedied by pooling a local neighborhood of interactions. The solution however cannot work for scenarios with long-range interactions. Another solution offered by Battaglia et al. [9] is to add several fully connected layers modeling the high-order interactions. This approach struggles when the objective is to select one of the agents (e.g. which agent will move), as it results in a distributed representation and loses the structure of the problem.
In this work we present VAIN (Vertex Attention Interaction Network), a novel multi-agent attentional neural network for predictive modeling. VAIN's attention mechanism helps with modeling the locality of interactions and improves performance by determining which agents will share information. VAIN can be said to be a CommNet [11] with a novel attention mechanism or a factorized Interaction Network [9]. This will be made more concrete in Sec. 2. We show that VAIN can model high-order interactions with linear complexity in the number of vertexes while preserving the structure of the problem, this has lower complexity than IN in cases where there are many fewer vertexes than edges (in many cases linear vs quadratic in the number of agents).
For evaluation we introduce two non-physical tasks which more closely resemble real-world and game-playing multi-agent predictive modeling, as well as a physical Bouncing Balls task. Our non-physical tasks are taken from Chess and Soccer and contain different types of interactions and different data regimes. The interaction graph on these tasks is not known apriori, as is typical in nature.
An informal analysis of our architecture is presented in Sec. 2. Our method is presented in Sec. 3. Description of our experimental evaluation scenarios are presented in Sec. 4. The results are provided in Sec. 5. Conclusion and future work are presented in Sec. 6.
Related Work
This work is primarily concerned with learning multi-agent interactions with graph structures. The seminal works in graph neural networks were presented by Scarselli et al. [12,13] and Li et al. [14]. Another notable iterative graph-like neural algorithm is the Neural-GPU [15]. Notable works in graph NNs includes Spectral Networks [16] and work by Duvenaud et al. [17] for fingerprinting of chemical molecules.
Two related approaches that learn multi-agent interactions on a graph structure are: Interaction Networks [9] which learn a physical simulation of objects that exhibit binary relations and Communication Networks (CommNets) [11], presented for learning optimal communications between agents. The differences between our approach VAIN and previous approaches INs and CommNets are analyzed in detail in Sec. 2.
Another recent approach is PointNet [18] where every point in a point cloud is embedded by a deep neural net, and all embeddings are pooled globally. The resulting descriptor is used for classification and segmentation. Although a related approach, the paper is focused on 3D point clouds rather than multi-agent systems. A different approach is presented by Social LSTM [10] which learns social interaction by jointly training multiple interacting LSTMs. The complexity of that approach is quadratic in the number of agents requiring the use of local pooling that only deals with short range interactions to limit the number of interacting bodies.
The attentional mechanism in VAIN has some connection to Memory Networks [19,20] and Neural Turning Machines [21]. Other works dealing with multi-agent reinforcement learning include [22] and [23].
There has been much work on board game bots (although the approach of modeling board games as interactions in a multi agent system is new). Approaches include [24,25] for Chess, [26,27,28] for Backgammons [29] for Go.
the-art performance for machine translation. The differences between our work and Vaswani et al.'s concurrent work are substantial in application and precise details.
Factorizing Multi-Agent Interactions
In this section we give an informal analysis of the multi-agent interaction architectures presented by Interaction Networks [9], CommNets [11] and VAIN.
Interaction Networks model each interaction by a neural network. For simplicity of analysis let us restrict the interactions to be of 2nd order. Let ψ int (x i , x j ) be the interaction between agents A i and A j , and φ(x i ) be the non-interacting features of agent A i . The output is given by a function θ() of the sum of all of the interactions of A i , j ψ int (x i , x j ) and of the non-interacting features φ(x i ).
o i = θ( j =i ψ int (x i , x j ), φ(x i ))(1)
A single step evaluation of the output for the entire system requires O(N 2 ) evaluations of ψ int ().
An alternative architecture is presented by CommNets, where interactions are not modeled explicitly.
Instead an interaction vector is computed for each agent ψ com (x i ). The output is computed by: . The interaction is approximated by:
o i = θ( j =i ψ com (x j ), φ(x i ))(2)ψ int (x i , x j ) = e |ai−aj | 2 ψ vain (x j )(3)
The output is given by:
o i = θ( j =i e |ai−aj | 2 ψ vain (x j ), φ(x i ))(4)
In cases where the kernel function is a good approximation for the relative strength of interaction (in some high-dimensional linear space), VAIN presents an efficient linear approximation for IN which preserves CommNet's complexity in ψ().
Although physical interactions are often additive, many other interesting cases (Games, Social, Team Play) are not additive. In such cases the average instead the sum of ψ should be used (in [9] only physical scenarios were presented and therefore the sum was always used, whereas in [11] only non-physical cases were considered and therefore only averaging was used). In non-additive cases VAIN uses a softmax:
K i,j = e |ai−aj | 2 / j e |ai−aj | 2(5)
Model Architecture
In this section we model the interaction between N agents denoted by A 1 ...A N . The output can be either be a prediction for every agent or a system-level prediction (e.g. predict which agent will act next). Although it is possible to use multiple hops, our presentation here only uses a single hop (and they did not help in our experiments).
Features are extracted for every agent A i and we denote the features by F i . The features are guided by basic domain knowledge (such as agent type or position). We define the communication encoding function E c (). The encoding function is applied to all agent features F i to yield both encoding e c i and attention vector a i . The attention vector is used for addressing the agents with whom information exchange is sought. E c () is implemented by fully connected neural networks (from now FCNs).
E c (F i ) = (e c i , a i )(7)
For each agent we compute the pooled feature P i , the interaction vectors from other agents weighted by attention. We exclude self-interactions by setting the self-interaction weight to 0:
P i = j e j * Sof tmax(−||a i − a j || 2 ) * (1 − δ j=i )(8)
This is in contrast to the average pooling mechanism used in CommNets and we show that it yields better results. The motivation is to average only information from relevant agents (e.g. nearby or particularly influential agents). The weights
w i,j = Sof tmax j (−||a i − a j || 2 )
give a measure of the interaction between agents. Although naively this operation scales quadratically in the number of agents, it is multiplied by the feature dimension rather by a full E() evaluation and is therefore significantly smaller than the cost of the (linear number) of E() calculations carried out by the algorithm. In case the number of agents is very large (>1000) the cost can still be mitigated: The Softmax operation often yields a sparse matrix, in such cases the interaction can be modeled by the K-Nearest neighbors (measured by attention). The calculation is far cheaper than evaluating E c () O(N 2 ) times as in IN. In cases where even this cheap operation is too expensive we recommend to default to CommNets which truly have an O(N) complexity.
The pooled-feature P i is concatenated to the original features F i to form intermediate features C i :
C i = (P i , e i )(9)
The features C i are passed through decoding function D() which is also implemented by FCNs. The result is denoted by o i :
o i = D(C i )(10)
For regression problems, o i is the per-agent output of VAIN. For classification problems, D() is designed to give scalar outputs. The result is passed through a softmax layer yielding agent probabilities:
P rob(i) = Sof tmax(o i )(11)
Several advantages of VAIN over Interaction Networks [9] are apparent:
Representational Power: VAIN does not assume that the interaction graph is pre-specified (in fact the attention weights w i,j learn the graph). Pre-specifying the graph structure is advantageous when it is clearly known e.g. spring-systems where locality makes a significant difference. In many multi-agent scenarios the graph structure is not known apriori. Multiple-hops can give VAIN the potential to model higher-order interactions than IN, although this was not found to be advantageous in our experiments.
Complexity: As explained in Sec. 2, VAIN features better complexity than INs. The complexity advantage increases with the order of interaction.
Experiments
We presented VAIN, an efficient attentional model for predictive modeling of multi-agent interactions.
In this section we show that our model achieves better results than competing methods while having a lower computational complexity.
We perform experiments on tasks from two different multi-agent domains to highlight the utility and generality of VAIN: chess move and soccer player prediction.
Chess Piece Prediction
Chess is a board game involving complex multi-agent interactions. There are several properties of chess that make it particularly difficult from a multi-agent perspective:
• There are 12 different types of agents with distinct behaviors.
• It has a well defined goal and near-optimal policies in professional games.
• Many of the interactions are non-local and very long ranged.
• At any given time there are multiple pieces interacting in a high-order clique (e.g. blocks, multiple defenders and attackers).
In this experiment we do not attempt to create an optimal chess player. Rather, we are given a board position from a professional game. Our task is to to identify the piece that will move next (MPP). Although we envisage that deep CNNs will achieve the best performance on this task, our objective here is to use chess as a test-bed for multi-agent interactive system predictors using only simple features for every agent. For recent attempts at building an optimal neural chess player please refer to [27,28]. The position illustrates the challenges of chess: non-local interactions, large variety of agents, blockers, hidden and implied threats, very high order interactions (here there's a clique between pawn, rook, queen, bishop etc.).
There are 12 categories of piece types in chess, where the category is formed of the combination of piece type and color. There are 6 types: Pawn, Rook, Knight, Bishop, Queen and King and two colors: Black and White. A chess board consists of 64 squares (organized in 8 rows and 8 columns). Every piece is of one category p i and is situated at a particular board square (x i , y i ). All methods evaluated on this task use the features: (p i , x i , y i for all pieces on the board I). The output is the piece position in the input (so if the input is (12,7,7), (11,6,5), ... output label 2 would mean that piece with features (11,6,5) will move next). There are 32 possible input pieces, in the case that fewer than 32 pieces are present, the missing pieces are given feature values (0, 0, 0).
For training and evaluation of this task we downloaded 10k games from the FICS Games Dataset, an on-line repository of chess games. All the games used are standard games between professionally ranked players. 9k randomly sampled games were used for training, and the remaining 1k games for evaluation. Moves later in the game than 100 (i.e. 50 Black and 50 White moves), were dropped from the dataset so as not to bias it towards particularly long games. The total number of examples is around 600k.
We use the following methods for evaluation:
• Rand: Random piece selection.
• F C: A standard FCN with three hidden layers (Node numbers: Input -32 * (13 + 16), 64, 64, 32). The input is the one-hot encoding of the features of each of the 32 pieces, the output is the index of the output agent. This method requires indexing to be learned.
• SM ax: Per-piece embedding neural network with scalar output. The outputs from all input pieces are fed to a softmax classifier predicting the output label. Note that this method preserves the structure of the problem, but does not model high-order interactions.
• 1hop − F C: A one-hop network followed by a deep (3 layers) classifier. The classifier predicts the label of the next moving piece . Note that the deep classifier removes the structure of the problem. The classifier therefore has to learn to index.
• CommN et: A standard CommNet (no attention) [11]. The protocol for CommNet is the same as VAIN.
• IN : An Interaction Network followed by Softmax (as for VAIN). Inference for this IN required around 8 times more computation than VAIN and CommNet.
• ours − V AIN .
Soccer Players
Team-player interaction is a promising application area for end-to-end multi-agent modeling as the rules of sports interaction are quite complex and not easily formulated by hand-coded rules. An additional advantage is that predictive modeling can be self-supervised and no labeled data is necessary. In team-play situations many agents may be present and interacting at the same time making the complexity of the method critical for its application.
In order to evaluate the performance of VAIN on team-play interactions, we use the Soccer Video and Player Position Dataset (SVPP) [33]. The SVPP dataset contains the parameters of soccer players tracked during two home matches played by Tromsø IL, a Norwegian soccer team. The sensors were positioned on each home team player, and recorded the player's location, heading direction and movement velocity (as well as other parameters that we did not use in this work). The data was re-sampled by [33] to occur at regular 20 Hz intervals. We further subsampled the data to 2 Hz. We only use sensor data rather than raw-pixels. End-to-end inference from raw-pixel data is left to future work.
The task that we use for evaluation is predicting from the current state of all players, the position of each player for each time-step during the next 4 seconds (i.e. at T + 0.5, T + 1.0 ... T + 4.0). Note that for this task, we just use a single frame rather than several previous frames, and therefore do not use RNN encoders for this task.
We evaluated several methods on this task:
• Static: trivial prediction of 0-motion.
• P ALV : Linearly extrapolating the agent displacement by the current linear velocity.
• P ALAF : A linear regressor predicting the agent's velocity using all features including the velocity, but also the agent's heading direction and most significantly the agent's current field position.
• P AD: a predictive model using all the above features but using three fully-connected layers (with 256, 256 and 16 nodes).
• CommN et: A standard CommNet (no attention) [11]. The protocol for CommNet is the same as VAIN.
• IN : An Interaction Network [9]. This results in O(N 2 ) network evaluations.
• ours − V AIN . Figure 2: a) A soccer match used for the Soccer task. b) A chess position illustrating the high-order nature of the interactions in next move prediction. Note that in both cases, VAIN uses agent positional and sensor data rather than raw-pixels.
We excluded the second half of the Anzhi match due to large sensor errors for some of the players (occasional 60m position changes in 1-2 seconds).
Bouncing Balls
Following Battaglia et al. The task which we evaluate is the prediction of the displacement and change in velocity of each ball in the next time step. We evaluate the prediction accuracy of our method V AIN as well as Interaction Networks [9] and CommNets [11]. We found it useful to replace VAIN's attention mechanism by an unnormalized attention function due to the additive nature of physical forces: In all scenarios the attention vector a i is of dimension 10 and shared features with the encoding vectors e i . Regression problems were trained with L2 loss, and classification problems were trained with cross-entropy loss. All methods were implemented in PyTorch [34] in a Linux environment. End-to-end optimization was carried out using ADAM [35] with α = 1e − 3 and no L2 regularization was used. The learning rate was halved every 10 epochs. The chess prediction training for the MPP took several hours on a K80 GPU, other tasks had shorter training times due to smaller datasets.
p i,j = e −||ai−aj || 2 − δ i,j(12)
Results
Qualitative Visualization
Let us first look at the attention maps generated by VAIN for our experimental scenarios. This visualization serves as a tool for understanding the nature of interactions between the agents. Note that VAIN only receives feedback on its future prediction but never receives explicit supervision on the nature of interaction between the agents.
Bouncing Balls: In Fig. 3 we can observe the attention maps for two different balls in the Bouncing Balls scenario. The position of the ball is represented by a circle. The velocity of each ball is indicated In the first scenario we observe that the two balls near the target receive attention whereas other balls are suppressed. This shows that the system exploits the sparsity due to locality inherent in this multi-agent system. In the second scenario we observe, that the ball on collision course with the target receives much stronger attention, relative to a ball that it much closer to the target but is not likely to collide with it. This indicates VAIN learns important attention features beyond the simple positional hand-crafted features typically used.
Soccer: A few visualizations of the Soccer scenario can be seen in Fig. 4. The positions of the players are indicated by green circles, apart from a target player (chosen by us), that is indicated by a blue circle. The brightness of each circle is chosen to be proportional to the strength of attention between each player and the target player. Arrows are proportional to player velocity. We can see in this scenario that the attention to nearest players (attackers to attackers, midfielder to midfielders) is strongest, but attention is given to all field players. The goal keeper normally receives no attention (due to being far away, and in normal situations not affecting play). This is an example of mean-field rather than sparse attention.
Chess: For the Chess scenario, the attention maps were not easily interpretable. We think this is due to the interactions in Chess being complex and high-order. The main visible trend was stronger attention to important and nearby pieces.
Chess MPP The results for next moving chess piece prediction can be seen in Table. 1. Our method clearly outperforms the competing baselines illustrating that VAIN is effective at selection type problems -i.e. selecting 1 -of-N agents according to some criterion (in this case likelihood to move). The non-interactive method SM ax performs much better than Rand (+9%) due to use of statistics of moves. Interactive methods (F C, 1hot − F C, CommN et, IN and V AIN ) naturally perform better as the interactions between pieces are important for deciding the next mover. It is interesting that the simple F C method performs better than 1hop − F C (+3%), we think this is because the
Bouncing Balls
The results of our bouncing balls experiments can be seen in Tab. 3. We see that in this physical scenario VAIN significantly outperformed CommNets, and achieves better performance than Interaction Networks for similar computation budgets. In Fig. 5 we see that the difference increases for small computation budgets. The attention mechanism is shown to be critical to the success of the method. Table 3: RMS accuracy of bouncing ball next step prediction. We observe that CommNet does a little better than the no-interaction baseline due to its noisy interaction vector. Our method VAIN improves over Interaction Networks and both are far better than CommNet. Figure 5: Accuracy differences between VAIN and IN for different computation budgets: VAIN outperforms IN by spending its computation budget on a few larger networks (one for each agent) rather than many small networks (one for every pair of agents). This is even more significant for small computation budgets.
Analysis and Limitations
Our experiments showed that VAIN achieves better performance than other architectures with similar complexity and equivalent performance to higher complexity architectures, mainly due to its attention mechanism. There are two ways in which the attention mechanism implicitly encodes the interactions of the system: i) Sparse: if only a few agents significantly interact with agent a o , the attention mechanism will highlight these agents (finding K spatial nearest neighbors is a special case of such attention). In this case CommNets will fail. ii) Mean-field: if a space can be found where the important interactions act in an additive way, (e.g. in soccer team dynamics scenario), attention would find the correct weights for the mean field. In this case CommNets would work, but VAIN can still improve on them.
VAIN is less well-suited for cases where both: interactions are not sparse such that the K most important interactions will not give a good representation and where the interactions are strong and highly non-linear so that a mean-field approximation is non-trivial. One such scenario is the M body gravitation problem. Interaction Networks are particularly well suited for this scenario and VAIN's factorization will not yield an advantage.
Conclusion and Future Work
We have shown that VAIN, a novel architecture for factorizing interaction graphs, is effective for predictive modeling of multi-agent systems with a linear number of neural network encoder evaluations. We analyzed how our architecture relates to Interaction Networks and CommNets. Examples were shown where our approach learned some of the rules of the multi-agent system. An interesting future direction to pursue is interpreting the rules of the game in symbolic form, from VAIN's attention maps w i,j . Initial experiments that we performed have shown that some chess rules can be learned (movement of pieces, relative values of pieces), but further research is required.
A
single step evaluation of the CommNet architecture requires O(N ) evaluations of ψ com (). A significant drawback of this representation is not explicitly modeling the interactions and putting the whole burden of modeling on θ. This can often result in weaker performance (as shown in our experiments). VAIN's architecture preserves the complexity advantages of CommNet while addressing its limitations in comparison to IN. Instead of requiring a full network evaluation for every interaction pair ψ int (x i , x j ) it learns a communication vector ψ c vain (x i ) for each agent and additionally an attention vector a i = ψ a vain (x i ). The strength of interaction between agents is modulated by kernel function e |ai−aj | 2
Figure 1 :
1A schematic of a single-hop VAIN: i) The agent features F i are embedded by singleton encoder E s () to yield encoding e s i and communications encoder E c () to yield vector e c i and attention vector a i ii) For each agent an attention-weighted sum of all embeddings e c i is computed P i = j w i,j * e c j . The attention weights w i,j are computed by a Softmax over −||a i − a j || 2 . The diagonal w i,i is set to zero to exclude self-interactions. iii) The singleton codes e s i are concatenated with the pooled feature P i to yield intermediate feature C i iv) The feature is passed through decoding network D() to yield per-agent vector o i . For Regression: o i is the final output of the network. vii) For Classification: o i is scalar and is passed through a Softmax.We use two agent encoding functions: i) a singleton encoder for single-agent features E s () ii) A communication encoder for interaction with other agents E c (). The singleton encoding function E s () is applied on all agent features F i to yield singleton encoding e
[ 9 ]
9, we present a simple physics-based experiment. In this scenario, balls are bouncing inside a 2D square container of size L. There are N identical balls (we use N = 50) which are of constant size and are perfectly elastic. The balls are initialized at random positions and with initial velocities sampled at random from [−v 0 ..v 0 ] (we use v 0 = 3ms −1 ). The balls collide with other balls and with the walls, where the collisions are governed by the laws of elastic collisions.
Implementation
Soccer: The encoding and decoding functions E c (), E s () and D() were implemented by fullyconnected neural networks with two layers, each of 256 hidden units and with ReLU activations. The encoder outputs had 128 units. For IN each layer was followed by a BatchNorm layer (otherwise the system converged slowly to a worse minimum). For VAIN no BatchNorm layers were used. Chess: The encoding and decoding functions E() and D() were implemented by fully-connected neural networks with three layers, each of width 64 and with ReLU activations. They were followed by BatchNorm layers for both IN and VAIN. Bouncing Balls: The encoding and decoding function E c (), E s () and D() were implemented with FCNs with 256 hidden units and three layer. The encoder outputs had 128 units. No BatchNorm units were used. For Soccer, E c () and D() architectures for VAIN and IN was the same. For Chess we evaluate INs with E c () being 4 times smaller than for VAIN, this still takes 8 times as much computation as used by VAIN. For Bouncing Balls the computation budget was balanced between VAIN and IN by decreasing the number of hidden units in E c () for IN by a constant factor.
Figure 3 :
3A visualization of attention in the Bouncing Balls scenario. The target ball is blue, and others are green. The brightness of each ball indicates the strength of attention with respect to the (blue) target ball. The arrows indicate direction of motion. Left image: The ball nearer to target ball receives stronger attention. Right image: The ball on collision course with the target ball receives much stronger attention than the nearest neighbor of the target ball.
Figure 4 :
4A visualization of attention in the Soccer scenario. The target ball is blue, and others are green. The brightness of each ball indicates the strength of attention with respect to the (blue) target ball. The arrows indicate direction of motion. This is an example of mean-field type attention, where the nearest-neighbors receive privileged attention, but also all other field players receive roughly equal attention. The goal keeper typically receives no attention due to being far away. by a line extending from the center of the circle, the length of the line is proportional to the speed of the ball. For each figure we choose a target ball A i , and paint it blue. The attention strength of each agent A j with respect to A i is indicated by the shade of the circle. The brighter the circle, the stronger the attention.
Table 2 :
2Soccer Prediction errors (in meters) on the three datasets evaluated with the leave-one-out protocol. This is evaluated for 7 methods. Non-interactive methods: no motion, linear velocity, linear all features, DNN all features and interactive methods: Interaction Networks[9], CommNet and VAIN (ours). All methods performed better than the trivial no-motion. Methods using all features performed better than velocity only, and DNN did better than linear-only. The interactive methods significantly outperformed the non-interactive methods, with VAIN outperforming other methods. We conclude that on this task VAIN could capture the multi-agent interaction without modeling each interaction individually, thus being only 4% of the size of IN. classifier in 1hop−F C finds it hard to recover the indexes after the average pooling layer. This shows that one-hop networks followed by fully connected classifiers (such as the original formulation of Interaction Networks) struggle at selection-type problems. Our method V AIN performs much better than 1hop − IN (11.5%) due to the per-vertex outputs o i , and coupling between agents. V AIN also performs significantly better than F C (+8.5%) as it does not have to learn indexing. It outperforms vanilla CommNet by 2.9%, showing the advantages of our attentional mechanism. It also outperforms INs followed by a per-agent Softmax (similarly to the formulation for VAIN) by 1.8% even though the IN performs around 8 times more computation than VAIN.Soccer We evaluated our methods on the SVPP dataset. The prediction errors inTable.2 are broken down for different time-steps and for different train / test datasets splits. It can be seen that the non-interactive baselines generally fare poorly on this task as the general configuration of agents is informative for the motion of agents beyond a simple extrapolation of motion. Examples of patterns than can be picked up include: running back to the goal to help the defenders, running up to the other team's goal area to join an attack. A linear model including all the features performs better than a velocity only model (as position is very informative). A non-linear per-player model with all features improves on the linear models. Both the interaction network, CommNet and VAIN significantly outperform the non-interactive methods. VAIN outperformed CommNet and IN, achieving this with only 4% of the number of encoder evaluations performed by IN. This validates our premise that VAIN's architecture can model object interactions without modeling each interaction explicitly.
Concurrent work: We found on Arxiv two concurrent submissions which are relevant to this work. Santoro et al.[30]discovered that an architecture nearly identical to Interaction Nets achieves excellent performance on the CLEVR dataset[31]. We leave a comparison on CLEVR for future work. Vaswani et al.[32]use an architecture that bears similarity to VAIN for achieving state-of-AcknowledgementWe thank Rob Fergus for significant contributions to this work. We also thank Gabriel Synnaeve and Arthur Szlam for fruitful comments on the manuscript.
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"Maximising the Utility of Validation Sets for Imbalanced Noisy-label Meta-learning",
"Maximising the Utility of Validation Sets for Imbalanced Noisy-label Meta-learning"
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| [
"Anh Hoang \nVasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n\n",
"Dung \nVasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n\n",
"Cuong Nguyen [email protected] \nVasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n\n",
"Gustavo Carneiro [email protected] \nVasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n\n"
]
| [
"Vasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n",
"Vasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n",
"Vasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n",
"Vasileios Belagiannis Otto von Guericke University Magdeburg\nUniversity of Adelaide\nUniversity of Adelaide\nUniversity of Adelaide\n"
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| Meta-learning is an effective method to handle imbalanced and noisy-label learning, but it depends on a validation set containing randomly selected, manually labelled and balanced distributed samples. The random selection and manual labelling and balancing of this validation set is not only sub-optimal for meta-learning, but it also scales poorly with the number of classes. Hence, recent meta-learning papers have proposed ad-hoc heuristics to automatically build and label this validation set, but these heuristics are still sub-optimal for meta-learning. In this paper, we analyse the meta-learning algorithm and propose new criteria to characterise the utility of the validation set, based on: 1) the informativeness of the validation set; 2) the class distribution balance of the set; and 3) the correctness of the labels of the set. Furthermore, we propose a new imbalanced noisy-label meta-learning (INOLML) algorithm that automatically builds a validation set by maximising its utility using the criteria above. Our method shows significant improvements over previous meta-learning approaches and sets the new state-ofthe-art on several benchmarks. | 10.48550/arxiv.2208.08132 | [
"https://export.arxiv.org/pdf/2208.08132v3.pdf"
]
| 251,622,505 | 2208.08132 | 72ed025b16c250922d260e2d9a23327c381d218a |
Maximising the Utility of Validation Sets for Imbalanced Noisy-label Meta-learning
Anh Hoang
Vasileios Belagiannis Otto von Guericke University Magdeburg
University of Adelaide
University of Adelaide
University of Adelaide
Dung
Vasileios Belagiannis Otto von Guericke University Magdeburg
University of Adelaide
University of Adelaide
University of Adelaide
Cuong Nguyen [email protected]
Vasileios Belagiannis Otto von Guericke University Magdeburg
University of Adelaide
University of Adelaide
University of Adelaide
Gustavo Carneiro [email protected]
Vasileios Belagiannis Otto von Guericke University Magdeburg
University of Adelaide
University of Adelaide
University of Adelaide
Maximising the Utility of Validation Sets for Imbalanced Noisy-label Meta-learning
Meta-learning is an effective method to handle imbalanced and noisy-label learning, but it depends on a validation set containing randomly selected, manually labelled and balanced distributed samples. The random selection and manual labelling and balancing of this validation set is not only sub-optimal for meta-learning, but it also scales poorly with the number of classes. Hence, recent meta-learning papers have proposed ad-hoc heuristics to automatically build and label this validation set, but these heuristics are still sub-optimal for meta-learning. In this paper, we analyse the meta-learning algorithm and propose new criteria to characterise the utility of the validation set, based on: 1) the informativeness of the validation set; 2) the class distribution balance of the set; and 3) the correctness of the labels of the set. Furthermore, we propose a new imbalanced noisy-label meta-learning (INOLML) algorithm that automatically builds a validation set by maximising its utility using the criteria above. Our method shows significant improvements over previous meta-learning approaches and sets the new state-ofthe-art on several benchmarks.
Introduction
Within the past decade, there have been great advancements in visual classification [24,19,61], object detection [73,51,36] and segmentation [7,32] thanks in part to deep learning models. The functionality of these models partly depends on large training sets containing samples that have been correctly labelled and that are wellbalanced among the classes. The difficulty in obtaining such training sets is motivating researchers to develop methods that can work with less well curated data sets [65,35]. Unfortunately, such poorly curated datasets are more likely to contain label noise and imbalanced class distribution.
In the literature, the problems of imbalanced learning and noisy-label learning are generally treated separately. While noisy label methods are based on robust loss functions [58,62], label cleaning [22,70], meta-learning [50,18], ensemble learning [43], and other methods [28,74], imbalanced learning approaches are based on metalearning [50,18,79], transfer learning [10,60], classifier design [64,37], re-sampling [57], and etc. Among those approaches, meta-learning based methods [24,50,80,79,66,52,2,1,56] can address both noisy-label and imbalanced learning problems.
Meta-learning is often formulated as a bi-level optimisation, where the upper level estimates the meta parameters using the validation set, and the lower level trains a classifier using the training set and the estimated metaparameters, where the validation set is commonly built by randomly selecting and manually labelling training samples. However, the process of building these validation sets scales poorly with the number of classes, and the random selection may not pick the most informative samples.
These issues have motivated the design of ad-hoc methods to build the validation set [79,66]. Unfortunately, their results are not as competitive as approaches that rely on manually-curated validation sets. This issue may be due to a shortcoming in their proposed heuristics [79,66], which characterises balanced distribution and label cleanliness but ignores the informativeness for the metalearning algorithm.
In this paper, we propose a new imbalanced noisylabel meta-learning (INOLML) method that automatically builds a validation set by maximising its utility in terms of sample informativeness, class distribution balance, and label correctness. The central contribution of the paper is the definition of the validation set utility criteria, which is motivated by the bi-level optimisation meta-learning algorithm. The proposed method, depicted in Fig. 1, consists of an iterative 3-step approach, namely: 1) pseudoclean sample detection and robust labelling from the noisy training set; 2) validation set formation from the robustly labelled pseudo-clean set in step (1), using the proposed utility criteria; and 3) meta learning using the validation set from step (2). The main contributions of our paper can be summarised as follows:
• A new method to build the meta-learning validation set by maximising its utility for sample informativeness, class distribution balance, and label correctness;
• An innovative meta-learning algorithm ( Fig. 1), comprising the steps: 1) detection and robust labelling of pseudo-clean samples from the noisy training set; 2) formation of the validation set using the proposed utility criteria; and 3) meta learning using the validation set from step (2).
With the two technical contributions above, our method shows improvements over previous meta-learning approaches on imbalanced noisy-label learning benchmarks. In balanced noisy-label benchmarks, our method is competitive or better than the state-of-the-art.
Related Work
We review methods that can deal with imbalanced noisylabel learning, focusing on meta-learning approaches.
Noisy-label Learning
Current noisy-label learning methods can rely on many strategies, such as: robust loss functions [59,62,38], ensemble learning [43], student-teacher model [55], label cleaning [70,22], co-teaching [33,24,41,19,68], dimensionality reduction [40], iterative label correction [77], semi-supervised learning [46,33,47], meta-learning [18,53,24,50,80,79,66,52,2,1,56], and hybrid methods [69,28,74,45,23]. Usually, most of the methods above assume that the training set has a balanced distribution of samples per class, except for the meta-learning approaches [24,50,80,79,66,52,2,1,56] that not only address the noisy-label problem, but also the learning with an imbalanced training set. Meta learning is a versatile solution for many problems (few-shot learning, reinforcement learning, etc.) that optimises meta-parameters in order to benefit the training process. In noisy-label meta learning papers [24,50,80,79,66,52,2,1,56], the meta parameters consist of a weight for each training sample [79,80], and the meta learning methods optimise the model based on a weighted cross entropy loss that automatically downweights noisy samples and upweights clean samples. For example, L2LWS [13] and CWS [12] comprise a target deep neural network (DNN) and a meta-DNN that is pretrained on a small clean validation dataset to re-weight the training samples to model the target DNN. Automatic reweighting [50] weights training samples based on the performance of one-step-ahead model on the validation set. Except for recent methods [79,66], meta-learning approaches require a clean validation set that can be expensive to acquire or unavailable in real world scenarios. Therefore, similarly to [79,66], we focus on the development of an approach that can automatically build a clean validation set, but unlike them, we propose an approach that is motivated by the meta-learning algorithm.
Imbalanced Learning
Imbalance learning is another challenging classification problem that is commonly present in real-world datasets, where a small portion of majority classes have a massive amount of training samples, and minority classes only have a few training samples [76]. This can easily result in a biased model that shows good accuracy Figure 1: Main stages of INOLML: 1) classify the noisy-label samples from D into D (c) (with samples that are likely to have clean labels) and D (n) (samples likely to have noisy labels); 2) build a validation set D (v) containing samples that are informative (from a meta-learning perspective), balanced and with a high likelihood of containing clean labels tested with the moving average robust labeller, where the training set D (t) = D (c) \D (v) ; and 3) train the meta-learning classifier with D (t) and D (v) . These three steps are iterated during training.
for majority classes, but poor performance for the minority ones. To address this problem, many imbalanced learning methods have been proposed [60,10,37,64,57], where the main techniques are [76]: transfer learning [10,60], classifier design [37,64], re-sampling (e.g., meta-learning) [57], decoupled training [26,25], ensemble learning [81,17], cost-sensitive learning [82,54,15], data augmentation [71,9], logit adjustment [42,49] and representation learning [75,21]. Unfortunately, existing methods designed to learn from long-tailed class distributions assume the labels to be clean, making their performance unclear in a more realistic scenario where the datasets are also noisy.
Noisy-label and Imbalanced Learning
Most of the papers listed in Sections 2.1 and 2.2 studied noisy label and imbalance learning problems as separate problems, except FSR [79] -a recent meta-learning approach that aims to solve both problems with metalearning. The presence of label noise in imbalanced datasets has also been considered by non meta-learning approaches [4,63,27], but they either have different setups or achieve inferior results compared with recently proposed meta-learning approaches. As mentioned in Sec. 1, the validation set plays a central role in metalearning, but we are not aware of papers that study how to maximise its utility during optimisation. Classic metalearning approaches [50,52] relies on a random sample selection and manually labelling approach that will likely result in a sub-optimal validation set. In addition, the fact that such manually-curated validation set is fixed for the whole training process may hinder the generalisation of the model. Recent meta-learning approaches try to automatically build a validation set that varies throughout the training process. For instance, FaMUS [66] selects training samples with low losses to form the validation set, while FSR [79] chooses samples that can be well optimized after a training iteration to build the validation set. Such methods, however, form a validation set based on heuristics that are not directly related to the meta-learning optimisation, which is the problem being studied in this paper.
Method
The initial training set is defined as
D = {(x i , y i )} |D| i=1
, with x i ∈ X ⊆ R H×W ×R representing an image of size H × W pixels and R colour channels, and y i ∈ Y = {v : v ∈ {0, 1} C and C k=1 v(k) = 1} being the noisy one-hot and C denoting the number of classes [20]. The classification model is represented by f θ : X → ∆ C−1 parameterised by θ ∈ Θ, with the C − 1 probability simplex ∆ C−1 = {p : p ∈ [0, 1] C and C k=1 p(k) = 1}. The proposed INOLML follows a bi-level optimisation [50,80] that relies on the meta-parameter ω = {ω i }
|D (t) | i=1
(ω i ≥ 0) that weights the samples in the training set D (t) based on their utility regarding informativeness and label cleanliness, and λ = {λ i }
|D (t) | i=1 (λ i ∈ [0, 1]) that weights the contribution of model pre- diction in the pseudo-label estimation, as inŷ i (λ i ) = λ i y i + (1 − λ i )f θ (x i ).
The meta-learning optimisation is defined by:
ω * , λ * = arg min ω,λ 1 |D (v) | (xj ,yj )∈D (v) (v) (x j , y j ; θ * (ω, λ)) s.t.: θ * (ω, λ) = arg min θ 1 D (t) (xi,yi)∈D (t) ω i (t) (x i ,ŷ(λ i ); θ) ,(1)where: (v) (x j , y j ; θ * (ω, λ)) = CE (y j , f θ * (ω,λ) (x j ))
is the cross-entropy (CE) loss between the label y j and model prediction, (t) (x i ,ŷ(λ i ); θ) is defined below in (12). The validation set D (v) is obtained by:
D (v) = MaxUtility D (c) ,(2)
which depends on the pseudo-clean set D (c) obtained from:
D (c) = PseudoCleanDetector (D) .(3)
Our main contribution is the definition of the utility criteria in MaxUtility(D (c) ) in (2) to select the validation set D (v) and training set
D (t) , where D (v) ⊂ D (c) , with D (t) ∩ D (v) = ∅ and D (t) ∪ D (v) = D.
The validation set D (v) has a balanced distribution of samples per class, contains samples that are informative for the metalearning optimisation in (1) and are likely to have clean labels.
The pseudo-clean sample set D (c) is estimated with a noisy-label classifier, represented by PseudoCleanDetector(.) shown in (3). Such classifier selects pseudo-clean samples based on the small CE loss hypothesis [19,33] where the loss is computed between the labels in D and the predictions by f θ (.). The remaining samples form D (n) , with D (c) ∪ D (n) = D and D (c) ∩ D (n) = ∅. These sets are regularly updated during training. The initial pseudo-clean set at the first training iteration is estimated from the model f θ (.) trained with early-stopping. In the following subsections, we describe how to select a validation set that maximises its utility in terms of informativeness, label cleanliness and class distribution balance.
Maximising the Utility of the Validation Set
The maximisation of the utility of the validation set is motivated by the bi-level optimisation in (1), where we focus on the weighting of each training sample, represented by ω i , which estimates the importance of that sample in the training process. The optimisation in (1) is solved by iterating the following 2 steps. In the first step, the locallyoptimal model parameter θ * (ω, λ) in the lower-level is obtained by applying stochastic gradient descent (SGD) on the training set D (t) with each step defined by:
θ(ω, λ) =θ(ω, λ)− η θ ∇ θ 1 |D (t) | (xi,yi)∈D (t) ω i (t) (x i ,ŷ i (λ i ); θ) θ=θ(ω,λ)
.
(4) In the second step, the meta-parameters, ω and λ, in the upper-level, are updated by applying one SGD step on the validation set D (v) . For ω, the update is defined as:
ω * i = max 0, − η ω |D (v) | (xj ,yj )∈D (v) ∂ ∂ω i (v) (x j , y j ; θ * (ω, λ)) ωi=0 ,(5)
and the update for λ is defined below in (11). The obtained meta-parameters are then used in the next bi-level optimisation iteration.
According to [50], the gradient w.r.t. ω is expressed as:
(xj ,yj )∈D (v) ∂ ∂ω i (v) (x j , y j ; θ * (ω, λ)) ωi=0 ∝ − (xj ,yj )∈D (v) L l=1 (z (v) j,l−1 z (t) i,l−1 )(g (v) j,l g (t) i,l ),(6)where z (v)
j,l−1 denotes the feature from validation image x j to be processed by layer l of the model (similarly for the training image feature z
(t) i,l−1 ), and g (v)
j,l represents gradient from layer l for the validation image x j (similarly for the training image gradient g (t)
i,l ). Hence, the weight of a training sample is high if both its feature and gradient are similar to the feature and gradient of at least one of the validation samples; otherwise, the weight is low.
Therefore, a validation set that maximises the weight of samples in the training set maximises its utility for the meta-learning optimisation. This observation is at the crux of our validation sample selection approach, where we first form a pseudo-clean set from the training set and then search within that pseudo-clean set to form a validation set that is balanced and maximises the sum in (6). (2) with the following bi-level optimisation:
The validation set D (v) ⊂ D (c) is built with the function MaxUtility(.) fromD (v) = arg max D (v) ⊂ D (v) | D (v) |=M ×C Clean D (v) , D (c) s.t.: D (v) = arg max D (v) ⊂D (c) |D (v) |=K×C Info D (v) , D (c) . (7)
The function Info(.) in the lower-level of (7) is defined as:
Info(D (v) , D (c) ) = (xi,yi)∈D (c) \D (v) max (xj ,yj )∈D (v) yj =yi ι(x i , x j ),(8)with ι(x i , x j ) = L l=1 (z j,l−1 z i,l−1 )(g j,l g i,l ),(9)
where, similarly to (6), z j,l−1 is the image feature input to layer l from x j (same for z i,l−1 from x i ), and g j,l denotes the validation image gradient of layer l from x j (same for g i,l from x i ). Note that the function ι(.) defined in (9) is the weight defined in (6) between the training sample (x i , y i ) and the validation sample (x j , y j ), or the "information" that (x i , y i ) can get from (x j , y j ). Intuitively, the lower-level summation in (7) (and in particular (8)) is designed to form a candidate balanced set D (v) by maximising the maximum "information content" that the pseudo-clean samples from D (c) \ D (v) can get from the samples in D (v) . The reason we maximise the maximum instead of the average "information content" is to guarantee that each clean training sample get upweighted by at least one clean validation sample. Unfortunately, the samples selected to be in D (v) can still have noisy labels since D (c) is not completely clean and the function Info(.) tends to return high values if samples in D (v) have low confidence logit scores and high gradient values. Simply filtering out samples with higher gradient will force the validation set to contain samples that are more likely to be clean, but less likely to be informative. Therefore, we aim to identify samples that are likely to have clean labels without relying on their prediction logits.
To search for clean samples in D (v) , we observe that the samples from this set are more likely to be clean when they have higher similarity with other samples belonging to the same class. Given this observation, we therefore propose a heuristic based on the cosine similarity between the sample of interest and other samples of the same class in the pseudo-clean set D (c) . The heuristic is represented by the function Clean(.), which is defined as follows:
Clean D (v) , D (c) = (xj ,yj )∈ D (v) (xi,yi)∈D (c) \ D (v) yi=yj L l=1 z j,l−1 z i,l−1 .
(10) We also impose a constraint that selects M samples for each of the C classes with M K as shown in the upperlevel of (7) to obtain a balanced validation subset D (v) .
Given that both optimisations in (7) consist of combinatorial problems, we resort to a greedy approach that loops through the classes and sequentially selects M and K samples for each class (for the upper and lower optimisation, respectively) that maximise the respective objective function. As the solution needs to iterate through all layers of a neural network of interest, the calculation of gradient in (6) and the optimisation in (7) might be expensive, especially for large-scale deep neural networks. However, according to Zhang et al. [79], the weights of training samples in meta-learning mostly depend on the last layer of the model. Hence, we use only on the last layer L of the model for (6) and (7) to reduce the computational cost.
Training Procedure
Our training procedure follows the 3-step iterative approach depicted in Fig. 1, where step 1 (pseudo-clean label detector) and step 2 (maximise the utility of the validation set) have been explained in Section 3.1.
Step 3 (meta-learning) is based on the optimisation in (1), where our training loss (.) follows the one defined in [80].
To optimise (1), we first estimate ω * with (5) and λ * (i.e., the pseudo-labelling parameter defined in (1)) with [80]:
λ * i = sign (xj ,yj )∈D (v) ∂ ∂λ i (v) (x j , y j ; θ * (ω, λ)) + ,(11)where (v) (x j , y j ; θ * (ω, λ)) is defined in (1).
After estimating ω * and λ * , we optimise the lower level of (1) to estimate the model parameter using the following loss function [80]:
(t) (x i , y i ; θ) =ω * i CE ( y i (λ 0 ), f θ (x i )) + 1 B CE (y * i (λ * i ), f θ (x i ))+ p × CE (y β i , f θ (x β i )) + k × KL (f θ (x i ), f θ (x i )),(12)
where y i (λ 0 ) is a pseudo-label, defined as in (1), with a fixed weight λ 0 = 0.9,
y * i (λ * i ) = y i if λ * i > 0, y * i = f θ (x i ) if λ * i ≤ 0, y β
i and x β i are obtained via the mixup operator [72] using the training and validation sets, KL (., .) represents the Kullback-Leibler (KL) divergence [30] between the model response for training image x i and its data augmented versionx i , p and k are hyperparameters, and B is the batch size.
The effectiveness of the optimisation in (7)
Experiment and Analysis
We evaluate our method INOLML on four datasets: CI-FAR10, CIFAR100 [29], WebVision [35] and Controlled Noisy Web Labels (CNWL) [66] with different noise settings, including symmetric, asymmetric, openset [3,61], and long-tailed imbalance with and without symmetric noise [79]. For each type of experiment, we keep the noisy training set the same across all models for a fair comparison.
Datasets
CIFAR10 and CIFAR100 datasets [29] contain 50k and 10k images used for training and testing, respectively. Each image has size 32 × 32 pixels and is labelled as one of 10 or 100 classes. WebVision [35] is a dataset of 2.4 million images crawled from Google and Flickr based on the 1,000 ImageNet classes [14]. The dataset is more challenging than CIFAR since it is class-imbalanced and contains real-world noisy labels. Following [79], we extract a subset that contains the first 50 classes to create the WebVision mini dataset [24]. CNWL [23] is a new benchmark of controlled real-world label noise from the web that contains various noise rates ranging from 0 to 0.8. Following FaMUS [66], we evaluate the proposed method on Red Mini-ImageNet dataset that consists of 50k training images from 100 classes for training and 5k images for testing.
Implementation Details
For all experiments on CIFAR datasets, except longtail imbalance, we use the same hyperparameters and network architectures as the Distill model [80]. We adopt the cosine learning rate decay with warm restarting [39] and SGD optimiser. For CIFAR datasets, we train WideResnet28-10 with 100k iterations and a batch size of 100. We also train a smaller network (Resnet29) to fairly compare with [80]. For WebVision, we follow FSR [79] and train a single Resnet50 network with 1 million iterations and a batch size of 16. For Red mini-ImageNet, we run experiments with 150k iterations and a batch size of 100. For CNWL, we use a single PreAct Resnet18 network that is similar to previous works [11,47] on this benchmark. For the class imbalance problems, we use the popular Resnet32 model to fairly compare with Fa-MUS [66] and FSR [79]. We report the prediction accuracy of each experiment on their corresponding testing sets. Please refer to Appendix B for implementation details and hyper-parameters values. Table 1 shows the test accuracy of many methods, including meta-learning based approaches that require a clean validation set (indicated with T ) and others that automatically build their validation sets, at various level of noise rates ranging from 20% to 80%. In general, the proposed method outperforms most of the previous methods, even though we do not require a clean validation set. The slightly lower performance than Distill on CIFAR100 at 80% noise rate can be explained by Distill's large manually curated clean validation set with 10 clean samples per class. In addition, as shown in Fig. 2a, at 80% symmetric noise rate, a significant portion (20% to 45%) of our clean validation set contains noisy samples at the final training stages, which deteriorates the efficacy of our approach. We also carry out additional experiments with different validation set sizes to fairly compare with Distill in Appendix C, in which our method outperforms Distill by 1 to 3% in majority of scenarios. Overall, these results show that a pseudo-clean, balanced, and informative validation set, can outperform a randomly-collected clean validation set in most symmetric noise scenarios. Our results also set new state-of-the-art (SOTA) results on the symmetric label noise benchmarks for methods that do not require clean validation set.
Symmetric Noise
Asymmetric Noise
We compare our algorithm with Distill [80], FSR [79] and other approaches on CIFAR10 at 40% asymmetric noise rate. Similarly to the symmetric noise cases, we also use (pseudo-)clean validation sets of sizes 1, 5 and 10 samples per class and show the results of Distill and our method in Table 2 (table on left). Although our proposed method does not rely on a manually-labelled validation set, it performs better than Distill, especially with small model architectures (Resnet29) and small validation sets (1 sample per class). Our active selection strategy has slightly lower accuracy with larger clean validation set sizes (larger than or equal 5 random clean samples per classes) on larger model architectures (WideRes-net28). This might be caused by the confirmation bias of asymmetric noise in our selected pseudo-clean validation subset and the high capacity of larger models, such as WideResnet28-10, which are more prone to overfit label noise, especially when being trained on a small dataset with only 10 classes. We further evaluate the proposed method and show the higher performance of our method compared to other methods, such as FSR and L2R metalearning methods, in Table 2 (table on right).
Imbalanced Learning
We evaluate our INOLML on the imbalanced (longtailed) CIFAR datasets following the same setting as [79]. The prediction accuracy in Table 3 shows that INOLML considerably surpasses all previous meta-learning ap- proaches.
Imbalanced Noisy-label Learning
We evaluate our proposed method in the setting that combines class imbalance and label noise, which has been proposed in [79]. We follow the same experimental configuration by adding 20% and 40% symmetric noise to the CIFAR10 imbalanced datasets with different imbalance ratios (10, 50 and 200). The results in Table 4 show that our proposed method outperforms the benchmarks by a large margin. This result is even more remarkable when studying the results with a noise rate of 40%. For CI- FAR100, we show the results for 20% and 40% symmetric noise and imbalance ratio 10. We do not show results for larger imbalance ratios because it was not possible to build validation sets with 10 samples per class for the minority classes. Nevertheless, for the two CIFAR100 problems, our method shows substantially better results than previous SOTA methods. Our method can therefore be considered the new SOTA in this imbalanced noisy-label learning benchmark with Resnet32 model.
Open-set Noise
This type of noise refers to training images that belong to classes falling outside the C classes in D. We follow [31], which forms 3 benchmarks using CIFAR10 contaminated with images from CIFAR100 and ImageNet. We compare with Distill and other meta-learning methods [78,48,67,50,79] in Table 5, and our method shows significant improvements in all benchmarks. In particular, comparing to Distill, our method is 0.5% to 1% better. One interesting observation is that our method outperforms Distill in the open-set noise even though the selected validation set D (v) is largely contaminated with noisy samples (up to 40%) as shown in Fig. 2b. This is in contrast to our previous observation in symmetric and asymmetric noise settings where the more noisy samples in D (v) , the worse performance of the models trained with our method compared to Distill. Such difference might be attributed to the out-of-distribution characteristic of open-set noise. As open-set noisy-label datasets contain samples that do not belong to the set of known classes, such samples might help to regularise the learning on mislabelled data, reduc- ing the effect of confirmation bias, resulting in a better performance. Table 6 shows the results of our method and other SOTA approaches on real-world datasets. Except for HAR [4] that uses InceptionResnetV2, Table 6 (upper table) shows the performance on WebVision with Resnet50, while Table 6 (lower table) shows results on four different noise ratios evaluated on Red Mini-ImageNet. In general, our method outperforms many SOTA methods on WebVision and is competitive with the best method [47] on Red Mini-ImageNet. We note that our proposed method is more efficient in terms of memory footprint than most of Co-training based approaches [33,11,66] evaluated on Red Mini-ImageNet since we use only a single PreAct Resnet18 model with meta-learning instead of two separate PreAct Resnet18 models.
Real-world Datasets
Ablation Study and Discussion
We first study the optimisation in Eq. (7). In the lowerlever optimisation of Eq. (7), the function Info(.) not just select samples that maximise the training sample weight from Eq. (6), as that may lead to scenarios where most of selected samples are located in the same region of the feature space. Instead, we also maximise a diversity factor defined by maximising the maximum "information content" that each training sample can get from any sample in the clean validation set. In Table 7, we show an ablation study about the importance of this factor by redefining Info(.) in Eq. (7) with Eq. (6) (Weight in Eq. (6)). We also study the role of Clean(.) in Eq. (7) by optimising only the lower part of Eq. (7) (Info(.) Only). This ablation We also compare the validation set built with Eq. (7) with sets built with random sampling and most confident sampling based on the highest confidence score. Fig. 3 shows that most confident sampling shows inferior results compared to random sampling, but our method to build the validation set shows the best results. Table 7: Test accuracy (%) on CIFAR10 and CIFAR100 under asymmetric and imbalanced noisy-label problems. The 1 st row shows the results of the optimisation of the weight (col. Weight in Eq. (6)) instead of Eq. (7). The 2 nd row shows the results of optimising the lower part of Eq. (7) (col. Info(.) Only) without the upper part of Eq. (7) Clean(.). The last row (Whole Eq. (7)) shows our final model result. Traditional meta-learning approaches [13,52] always keep the clean validation set separate from the training set, while our method iteratively extracts D (v) from the training set. It can be argued that this non-separation of the training and validation sets can cause confirmation bias to happen during training. Hence, we tested our approach in a scenario where the candidate samples to form the validation set is always separate from the training set during training. However, results showed that such separate validation set causes a 2% drop in accuracy, on average. This can be explained by the smaller size of the training set and the restriction in potential choices for validation samples.
A final discussion point is the time needed to run our approach. The Distill model takes around 5 and 29 hours to train the Resnet29 and WideResnet28-10 models, respectively. When integrating our method with Distill, training takes around 5.5 hours on Resnet29 and 31 hours on WideResnet28-10. Hence, our algorithm adds an 10% traning time overhead. Experiments are conducted on a single NVIDIA V100 GPU.
Conclusion
We presented a novel meta-learning approach, called IN-OLML, that automatically and progressively selects a pseudo clean validation set from a noisily-labelled training set. This selection is based on our proposed validation set utility that takes into account sample informativeness, class distribution balance, and label correctness. Our proposed method is more effective and practical than prior meta-learning approaches since we do not require manually-labelled samples to include in the validation set. Compared with other meta-learning approaches that do not require a manually labelled validation set, e.g. FSR or FaMUS, our method is demonstrated to be more robust to high noise rate problems and to achieve state-of-the-art results on several synthetic and realistic label noise benchmarks.
A limitation of our approach is that the model can suffer from confirmation bias as it is based on a single model. As future work, we will tackle this problem by incorporating co-training in our meta-learning algorithm. Another limitation is the greedy and complex bi-level optimisation to form the validation set in Eq. (7), which can be improved in two ways: 1) the complexity can be reduced by replacing the bi-level optimisation by a single-level optimisation, and 2) the greedy strategy can be replaced by a relaxation method to solve the combinatorial optimisation problem. Additionally, optimising the clean validation set once per epoch is not ideal since the validation set can be outdated by the end of epoch. This issue will be addressed by updating the clean validation set more regularly. Initialise D (v) and D (t) from D (c) using Equation 7 13:
Reinitialize model f θ (.) 14: for t = 1 to T do 15: Meta-learn to train θ, ω and λ using Equation if T (u) ≡ t mod T (u) then 19: Update D (v) and D (t) from D (c) using Equation 7 20: return the trained model parameter θ
B Implementation Details
All CIFAR experiments use batches of size 100, which are trained on a single GPU. Similar to the Distill noise model [80], we use p = 5, k = 20 for CIFAR experiments, except the ones with the imbalance setting.
For Red Mini-ImageNet experiments, we trained the model on a single GPU with batches of size 100, with p = 5, k = 10.
For the WebVision experiment, we use p = 4, k = 8 with 4 NVIDIA V100 GPU and batches of size 16. All experiments use N = 200, K = 50, κ = 0.9.
C Additional Results of Symmetric Noise on CIFAR Datasets
We provide additional symmetric noise results of our proposed method and the Distill model [80] in Table 8. Note that our method is markedly better than Distill, particularly for the simpler model (RN29) with few samples per class (1 and 5) in the validation set. For the more complex model (WRN) and large validation set (10 samples per class), our method is still better than Distill, except for CIFAR100 at 0.8 symmetric noise rate. Table 8: Test accuracy (in %) comparison between our method ('INOLML') and the Distill noise ('DN') on symmetric noise using 1, 5 and 10 samples per class in the validation set on two backbone models: Resnet29 ('RN29') and Wideresnet28-10 ('WRN'). The results of the Distill model with WideResnet28-10 are collected from [80]. Recall that the Distill needs a clean set, while INOLML works with a pseudo-clean set.
Method
Val.
depends on the actual (hidden) proportion of clean samples in the pseudo clean set D (c) , while the efficiency depends on the size of D (c) . Hence, to reduce computational cost, the selection of the validation set in(7)uses a subset D (c) ⊂ {(x i , y i ) : (x i , y i ) ∈ D (c) ∧ arg max k∈{1,...,C} y i (k) = arg max k∈{1,...,C}ỹi (k)}.This subset contains N randomly-selected samples (x i , y i ) of each class in D (c) that have their observed labels y i consistent with the corresponding moving average robust label computed with the average prediction over the last E epochs, as inỹ i = κỹ i + (1−κ) /E E e=1 f θ (x i ), with κ ∈ [0, 1] being a hyperparameter. The details of the training process are in Algorithm 1 of Appendix A.
Figure 2 :
2Accuracy of the clean validation set D (v) as training progresses evaluated on different noise benchmarks.
Figure 3 :
3Accuracy (%) of our INOLML using different sample selection methods.
Training procedure of the proposed INOLML. 1: procedure TRAINING(D, η, T , T , T (u) , η, κ, N, M, f θ (.) with CE (.) from D 10: D (c) = PseudoCleanDetector (D)
samples in D (c) if t > T and η t < η 17: Update D (c) = PseudoCleanDetector (D) using Equation 3 if η t = 0 18:
Table 1 :
1Test accuracy (in %) of our INOLML and previous methods evaluated on various symmetric noise rates. Methods with superscript T represent meta-learning methods that need clean validation sets. The lower block contains meta-learning methods while the upper block shows methods with SOTA results.Method
CIFAR10
CIFAR100
0.2
0.4
0.8
0.2
0.4
0.8
GJS[78]
95.3 ± 0.2 93.6 ± 0.2 79.1 ± 0.3 78.1 ± 0.3 75.7 ± 0.3 44.5 ± 0.5
DivideMix[31] 95.7 ± 0.0
-
92.9 ± 0.0 76.9 ± 0.0
-
59.6 ± 0.0
CRUST[44]
91.1 ± 0.2 89.2 ± 0.2 58.3 ± 1.8
-
-
-
PENCIL[67]
-
-
-
73.9 ± 0.3 69.1 ± 0.6
-
ELR[38]
92.1 ± 0.4 91.4 ± 0.2 80.7 ± 0.6 74.7 ± 0.3 68.4 ± 0.4 30.2 ± 0.8
FaMUS [78]
-
95.3 ± 0.2
-
-
76.0 ± 0.2
-
Distill T [80]
96.2 ± 0.2 95.9 ± 0.2 93.7 ± 0.5 81.2 ± 0.7 80.2 ± 0.3 75.5 ± 0.2
MentorNet T [24] 92.
Table 2 :
2Test accuracy (in %) of our INOLML and pre-
vious methods on CIFAR10 with 0.4 asymmetric noise.
(table on left): comparison with Distill using a validation
set D (v) of sizes 1, 5 and 10 samples per class on Resnet29
and WideResnet28-10, and (table on right): comparison
with some leading methods. The superscript T indicates
the need for clean validation sets.
Method
D (v)
Resnet29 WRN28-10
Distill T
1 × C
76.8 ± 2.9 93.2 ± 0.2
INOLML
86.8 ± 0.9 93.6 ± 0.3
Distill T
5 × C
86.7 ± 0.5 94.5 ± 0.2
INOLML
89.3 ± 0.2 94.1 ± 0.1
Distill T
10 × C
88.6 ± 0.7 94.9 ± 0.1
INOLML
89.8 ± 0.3 94.2 ± 0.1
Method
Accuracy
GJS[16]
89.7 ± 0.4
F-Correction [48] 83.6 ± 0.3
PENCIL[67]
91.2 ± 0.0
DivideMix [33]
92.1 ± 0.0
MLNT [34]
92.3 ± 0.1
L2R T [50]
90.8 ± 0.3
FSR[79]
93.6 ± 0.3
INOLML
94.2 ± 0.1
Table 3 :
3Test accuracy (in %) of our INOLML and other
SOTA meta-learning approaches evaluated on the CIFAR
imbalanced learning (long-tailed) recognition task. The
reported results are from Zhang et al. [79] and Xu et
al. [66].
CIFAR10
CIFAR100
Imb. ratio
200
50
10
200
50
10
Softmax [79] 65.68 74.81 86.39 34.84 43.85 55.71
CB-Focal [79] 65.29 76.71 86.66 32.62 44.32 55.78
CB-Best [79] 68.89 79.27 87.49 36.23 45.32 57.99
L2R [50]
66.51 78.93 85.19 33.38 44.44 53.73
MWN [52]
68.91 80.06 87.84 37.91 46.74 58.46
GDW [6]
-
-
86.8
-
-
56.8
FaMUS [66]
-
83.32 87.90
-
49.93 59.03
FSR-DF [79] 66.15 79.78 88.15 36.74 44.43 55.60
FSR [79]
67.76 79.17 87.40 35.44 42.57 55.45
INOLML
74.91 84.43 90.81 41.52 51.35 62.07
Table 4 :
4Test accuracy (in %) of our INOLML and
other SOTA methods on CIFAR10 long-tailed recognition
mixed with symmetric noise. The reported results are col-
lected from [79] and [63].
Dataset
Cifar10
Cifar100
Noise ratio
0.2
0.4
0.2 0.4
Imb. ratio
10
50
200 10
50
200 10
10
CRUST[44]
65.7 41.5 34.3 59.5 32.4 28.8 -
-
LDAM[5]
82.4 -
-
71.4 -
-
48.1 36.7
LDAM-DRW[5] 83.7 -
-
74.9 -
-
50.4 39.4
BBN[81]
80.4 -
-
70.0 -
-
47.9 35.2
HAR-DRW[4] 82.4 -
-
77.4 -
-
46.2 37.4
ROLT-DRW[63] 85.5 -
-
82.0 -
-
52.4 46.3
FSR[79]
85.7 77.4 65.5 81.6 69.8 49.5 -
-
INOLML
90.1 80.1 66.6 89.1 78.1 61.6 59.8 56.1
Table 5 :
5Test accuracy (in %) of our INOLML and previ-
ous methods in open-set noise [3] using WideResnet28-10
with 10 samples per class for the validation set.
Method
ImageNet CIFAR100 BOTH
RoG [48] 83.4
87.1
84.4
L2R [50] 81.8
81.8
85.0
Distill [80] 94.0
92.3
93.0
INOLML 94.5 ± 0.1 93.6 ± 0.0 93.6 ± 0.1
Table 6 :
6SOTA prediction accuracy (in %) comparison on real-world datasets. (upper table): WebVision mini dataset (50 classes) using Resnet50 evaluated on Webvision and ImageNet test set; and (lower table): Red Mini-ImageNet. The results of other methods are reported from Zhang et al.[79], PropMix[11] and their own works.Method
WebVision ImageNet
top-1 top-5 top-1 top-5
HAR [4]
75.5 90.7 57.4 82.4
D2L [40]
62.7 84.0 57.8 81.4
Co-teaching [19] 63.6 85.2 61.5 84.7
Iterative-CV [8] 65.2 85.3 61.6 85.0
MentorNet [24] 63.0 81.4 63.8 85.8
CRUST [44]
72.4 89.6 67.4 87.8
FSR [79]
74.9 88.2 72.3 87.2
GJS [16]
78.0 90.6 74.4 91.2
MW-Net [52]
74.5 88.9 72.6 88.8
INOLML
81.7 93.8 78.1 92.9
Method
Noise ratio
0.2
0.4
0.6
0.8
Cross entropy [11] 47.36 42.70 37.30 29.76
Mix Up [11]
49.10 46.40 40.58 33.58
DivideMix [33]
50.96 46.72 43.14 34.50
MentorMix[23]
51.02 47.14 43.80 33.46
FaMUS[66]
51.42 48.06 45.10 35.50
PropMix[11]
61.24 56.22 52.84 43.42
MOIT [47]
63.14 60.78
-
45.88
INOLML
63.23 58.21 53.39 45.32
study is conducted on CIFAR10 and CIFAR100 under 0.4
asymmetric noise and 0.2 symmetric noise with imbal-
anced data. Overall, each component improves the per-
formance compared to just naively optimising the weight
in Eq. (6). Adding Info(.) and Clean(.) improves the
model significantly. Naively selecting samples based on
Eq. (6) facilitates the overfiting of the noisy-label sam-
ples, leading to confirmation bias. To mitigate this prob-
lem, Clean(.) limits the noise in the clean validation set,
while Info(.) prevents the gradient to go toward a single
wrong direction.
± 0.3 89.0 ± 0.3 83.5 ± 0.2 62.6 ± 0.4 58.8 ± 0.5 48.5 ± 0.5 INOLML-RN29 90.9 ± 0.2 90.9 ± 0.1 87.4 ± 0.2 66.6 ± 0.1 65.7 ± 0.1 59.0 ± 0.5 DN-RN29 10 91.0 ± 0.2 89.2 ± 0.1 87.0 ± 0.1 63.7 ± 0.2 60.5 ± 0.2 57.5 ± 0.5 INOLML-RN29 92.2 ± 0.1 91.0 ± 0.1 87.9 ± 0.2 67.1 ± 0.1 66.3 ± 0.1 59.2 ± 0.2 ± 0.2 95.9 ± 0.2 93.7 ± 0.5 81.2 ± 0.7 80.2 ± 0.3 75.5 ± 0.2 INOLML-WRN 96.9 ± 0.1 96.6 ± 0.1 95.0 ± 0.2 82.0 ± 0.2 81.3 ± 0.2 74.7 ± 0.1Set
size
Dataset
CIFAR10
CIFAR100
0.2
0.4
0.8
0.2
0.4
0.8
DN-RN29
1
87.0 ± 0.5 85.3 ± 0.5
FAIL
58.9 ± 0.5 55.8 ± 0.7
FAIL
INOLML-RN29
90.3 ± 0.2 89.1 ± 0.5 79.1 ± 0.3 65.9 ± 0.2 61.5 ± 0.2 55.1 ± 0.1
DN-RN29
5
90.7 DN-WRN
1
95.4 ± 0.6 94.5 ± 1.0 87.9 ± 5.1 77.4 ± 0.4 75.5 ± 1.1 62.1 ± 1.2
INOLML-WRN
96.0 ± 0.2 95.9 ± 0.2 94.3 ± 0.2 81.6 ± 0.2 79.5 ± 0.2 73.6 ± 0.3
DN-WRN
5
96.4 ± 0.0 95.5 ± 0.6 91.8 ± 3.0 80.4 ± 0.5 79.6 ± 0.3 73.6 ± 1.5
INOLML-WRN
96.4 ± 0.1 96.2 ± 0.1 94.6 ± 0.2 82.2 ± 0.2 81.5 ± 0.2 74.5 ± 0.2
DN-WRN
10
96.2
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| []
|
[
"Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications",
"Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications"
]
| [
"Z L Yuan \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n",
"M Lucamarini \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n",
"J F Dynes \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n",
"B Fröhlich \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n",
"M B Ward \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n",
"A J Shields \nCambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom\n"
]
| [
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom",
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom",
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom",
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom",
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom",
"Cambridge Research Laboratory\nToshiba Research Europe Limited\n208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom"
]
| []
| Since the introduction of the decoy-state technique, phase-randomised weak coherent light pulses have been the key to increase the practicality of quantum-based communications. Their ultra-fast generation was accomplished via compact gain-switched (GS) lasers, leading to high key rates in quantum key distribution (QKD). Recently, the question arose of whether the same laser could be employed to achieve high-speed measurement-device-independent-QKD, a scheme that promises long-haul quantum communications immune to all detector attacks. For that, a challenging highvisibility interference between independent picosecond optical pulses is required. Here, we answer the above question in the affirmative by demonstrating high-visibility interference from two independent GS lasers triggered at 1GHz. The result is obtained through a careful characterization of the laser frequency chirp and time jitter. By relating these quantities to the interference visibility, we obtain a parameter-free verification of the experimental data and a numerical simulation of the achievable key rates. These findings are beneficial to other applications making use of GS lasers, including random number generation and standard QKD. * [email protected] [1] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 00145 (2002). | 10.1103/physrevapplied.2.064006 | [
"https://arxiv.org/pdf/1501.01900v1.pdf"
]
| 111,733,343 | 1501.01900 | 8df3c4a0be505b2eeac6fa52d09f2e4b8111083b |
Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications
8 Jan 2015
Z L Yuan
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
M Lucamarini
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
J F Dynes
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
B Fröhlich
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
M B Ward
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
A J Shields
Cambridge Research Laboratory
Toshiba Research Europe Limited
208 Cambridge Science Park, Milton RoadCB4 0GZCambridgeUnited Kingdom
Interference of short optical pulses from independent gain-switched laser diodes for quantum secure communications
8 Jan 2015(Dated: January 9, 2015)
Since the introduction of the decoy-state technique, phase-randomised weak coherent light pulses have been the key to increase the practicality of quantum-based communications. Their ultra-fast generation was accomplished via compact gain-switched (GS) lasers, leading to high key rates in quantum key distribution (QKD). Recently, the question arose of whether the same laser could be employed to achieve high-speed measurement-device-independent-QKD, a scheme that promises long-haul quantum communications immune to all detector attacks. For that, a challenging highvisibility interference between independent picosecond optical pulses is required. Here, we answer the above question in the affirmative by demonstrating high-visibility interference from two independent GS lasers triggered at 1GHz. The result is obtained through a careful characterization of the laser frequency chirp and time jitter. By relating these quantities to the interference visibility, we obtain a parameter-free verification of the experimental data and a numerical simulation of the achievable key rates. These findings are beneficial to other applications making use of GS lasers, including random number generation and standard QKD. * [email protected] [1] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, Quantum cryptography, Rev. Mod. Phys. 74, 00145 (2002).
Interference lies at the heart of quantum information technologies. Novel protocols and schemes, such as quantum cryptography [1], quantum teleportation [2], quantum repeaters [3], or linear optics quantum computing [4] rely upon high visibility interference of light pulses. To achieve high visibility, the interfering pulses need to be indistinguishable in all possible degrees of freedom [5][6][7][8][9].
Weak coherent states of light have been long used to approximate single photon sources in quantum key distribution (QKD). This approximation guarantees high key rates if the decoy-state technique [10,11] is adopted. However, in order to apply it, the electromagnetic phase of coherent states is required to be random. Thankfully, semiconductor gain-switched (GS) laser diodes naturally generate optical pulses with random phases [12,13]. With a sufficient off period between subsequent events, each laser pulse is triggered by quantum-mechanical spontaneous emission and thus has random electromagnetic phase [12]. At the same time, GS short pulses (around 30 ps, see Fig. 1(a)) are perfectly suited to high bit rate [14,15] and noise-tolerant QKD [16]. This is remarkable given that time-jitter in GS lasers is about 10 ps, i.e., comparable to the pulse width. Furthermore, other potential sources of impairment like the pulses spectral distinguishability or a time-varying polarization, play only a minor role in standard QKD, where each generated pulse only interferes with itself to deliver a bit of the final key.
The situation is dramatically changed in measurementdevice-independent-QKD (MDI-QKD), a recent quantum protocol promising immunity against all detector attacks [17][18][19][20][21]. Similarly to conventional QKD, decoy states and phase randomization are also required in MDI-QKD. However, in this case, the successful distillation of the final key requires high visibility two pulse interfer-ence [17]. This poses stringent requirements on the system, as the interfering pulses have to be indistinguishable and perfectly overlapped to guarantee high visibility. Time jitter and frequency profile of the pulses play a very important role and it is unclear whether GS lasers represent a viable solution.
Until now it has not been possible to use GS pulses shorter than 2 ns and trigger rates higher than 1 MHz in an MDI-QKD experiment [20]. This is still orders of magnitude away from high bit-rate QKD [15,22], working at 1 GHz with pulse widths of tens of ps. The tolerance to time and frequency fluctuations could be improved by using the steady-state emission of GS laser diodes [12]. However, this would limit the prospect for high bit rate applications. Other MDI-QKD demonstrations [18,19,21] have improved the spectral stability of the pulses approximating the required source with continuous-wave (CW) lasers pulse-carved by an intensity modulator. Light pulses generated this way exhibit a negligible time-jitter, but also a constant, or slowly variable, phase, therefore violating the phase randomness requirement. Phase randomisation via separate modulation is possible [21,23], but at the expense of additional complexity of the setup [24].
Here, we investigate the relation between GS laser diodes and interference visibility and implement a solution to mitigate the detrimental effect of pulse distinguishability. By introducing a novel theoretical model, we identify frequency chirp of GS pulses as the main cause of poor interference visibility. Frequency chirp is common in fast-driven semiconductor laser diodes [25]. The rapid change in carrier density in the active region dynamically alters the refractive index thus chirping the laser frequency [26] and making the pulses far from transform limited [27] (see, e.g., Fig. 1(b)). The concomitant arXiv:1501.01900v1 [quant-ph] 8 Jan 2015 time jitter then prevents two chirped pulses from preserving a constant phase relation, which is a prerequisite for high-visibility interference. We verify this analysis by experimentally interfering short optical pulses emitted by two independent semiconductor GS laser diodes driven at 1 GHz and comparing the results with the theoretical prediction. When frequency chirp is taken into account, the theory provides a parameter-free fit of the experimental data, thus confirming the soundness of our analysis. This fact was then exploited to calibrate our system and achieve two-pulse interference visibility as high as 0.46, close to the theoretical limit 0.50 achievable with weak coherent states [6]. Combined with the intrinsic phase randomness of the pulses and the high trigger rate of the laser, this result demonstrates the usefulness of GS laser diodes in achieving high speed decoy-state MDI-QKD. Furthermore, it has implications for other types of high-speed quantum information applications, as discussed later on.
We start our analysis by describing the schematics of the experimental setup, depicted in Fig. 2, based on which we develop our theoretical model. The setup consists of a Hong-Ou-Mandel interferometer [5], with two attenuated GS distributed feedback laser diodes injecting light into a beam splitter through a pair of optical circulators and a tunable filter. After the beam splitter, light is detected by two single photon detectors, thus emulating a real MDI-QKD setup. Ideally, a tunable bandpass filter should appear in each arm of the setup to limit the bandwidth and hence the frequency chirp of each laser. For experimental convenience, we use a single tunable bandpass filter and two optical circulators to filter emissions of both lasers simultaneously [28]. Each laser is attenuated by more than 70 dB up to the single photon level before interference. Including the built-in isolation (30 dB) in each laser diode and the circulator extinction ratio of 50 dB, the total isolation between the light sources is greater than 150 dB. Considering each laser emitting an optical power of ∼200 µW at 1 GHz, this level of isolation ensures that the optical cross talk between the lasers is less than 10 −8 photons/pulse. We therefore conclude the laser diodes are optically independent.
Gain-switching in the laser diodes is achieved electrically by a superposition of a DC bias and a voltage square wave clocked at 1 GHz. Temporal alignment of the pulses, shown in Fig. 1(a), is achieved by tuning the delay of Laser 2, which is electronically adjustable in steps of 1 ps. Laser 1 is kept at room temperature, while laser 2 is cooled to −9 • C to tune its central frequency (193.47 THz) to approximately match Laser 1, as shown in Fig. 1(b). The second order correlation functions at 0-delay are measured for lasers 1 and 2 and amount to 99.3 ± 1.0% and 99.6 ± 1.3%, respectively, suggesting the Poisson statistics as expected for coherent state emission [29]. Optical pulses from both lasers exhibit temporal and spectral full widths at half maximum (FWHM's) of 30 ps and 70 GHz, respectively. Excluding the influence from the time jitter, which was measured to be 9.3 ps (FWHM), the laser pulses are far from Fouriertransform limited. Gaussian transform limited pulses of such duration correspond to a spectral broadening of about 15 GHz [27]. The excessive spectral broadening is attributed to laser frequency chirp, which requires a proper theoretical description to understand the results of the interference experiment in the presence of time jitter.
We focus on the beam splitter and evaluate the coincidence counts registered by the two detectors. The two optical pulses generated by the independent GS laser diodes enter the beam splitter through inputs a and b. The electric field at k = {a, b} and time t is:
E k (t) = I(t − t k ) exp{2πi[ν(t − t k ) + β(t − t k ) 2 + ϕ k ]},(1)where I(t) = exp{−t 2 /2τ 2 p }/(τ p √ 2π)
is the temporal profile of the laser pulse, assumed to be Gaussian, and τ p its temporal width; ν is the central frequency of the wavepackets; t k is the temporal distance of wavepacket k from the beam splitter at time t; β is a parameter accounting for frequency chirp, which is about 0.01 ps −2 in semiconductor lasers [25]; ϕ k is the (random) electromagnetic phase of the pulses. We also define for later convenience the time delay between the two pulses ∆t = t b −t a and the phase difference ∆ϕ = ϕ b − ϕ a . The time delay can be due to either systematic temporal misalignment or emission time uncertainty.
At the beam splitter, the pulses are aligned to same polarisation and interfere. The output intensities can be calculated from Eq. (1) and the beam splitter relations [30]. After integrating over the finite response time of the detectors, much longer than the pulse width, and assuming a 50/50 beam splitter, we obtain for the intensity at the output ports of the beam splitter, c and d,
I c,d = 1 ± cos(∆ϕ) exp[− (∆t) 2 8τ 2 p (1 + 16β 2 τ 4 p )],(2)
where the +(-) sign is associated with the c (d) mode. Figure 1(c) shows example traces of I c and I d recorded at the beam splitter output ports using a pair of fast photodiodes as detectors, after having set optical attenuation to 0 dB and spectral filtering to pass all frequency components (see setup in Fig. 2). Photodiodes record complementary outputs, as a result of energy conservation. Peak intensities fluctuate because of the random phase difference ∆ϕ in Eq. (2). Nearly complete constructive and destructive interference is observable because of the occasional perfect temporal alignment (∆t ≈ 0) of the two wavepackets. Since we do not limit the frequency chirp here, the observation of nearly complete interference suggests that the two lasers have a similar chirped profile. However, as we shall see, this is not sufficient to guarantee a high visibility in two-pulse interference. Most often ∆t = 0 because of the emission time jitter. In this case, frequency chirp will prevent complete interference and hence deteriorate the interference visibility. The differential phase between two pulses is no longer constant, but evolves as ∆ϕ(t) = ∆ϕ 0 + 2πβ∆t · t. Crudely speaking, whenever the differential phase evolves by more than 2π, half of the optical wave interferes constructively and the other half destructively, resulting an overall interference visibility approaching zero.
The average two-pulse interference visibility can be obtained as V (2) = 1 − P cd , where P cd ∝ I c · I d is the (normalized) coincidence rate seen by the two detectors under the assumption of attenuated intensities [6]. After averaging over the random phase difference ∆ϕ and under the experimentally fulfilled condition of pulses attenuated at the single photon level, we obtain the visibility as:
V (2) = 1 2 exp[− (∆t) 2 4τ 2 p (1 + 16β 2 τ 4 p )].(3)
It is worth remarking that the theoretical limit for the interference visibility is 50% in this case, not 100%, because attenuated coherent states, not single photons, are interfering at the beam splitter. In Fig. 3, we plot the interference visibility as a function of temporal misalignment and frequency chirp. We use τ p = 12.7 ps, corresponding to an FWHM of 30 ps for a Gaussian wavepacket. When both lasers are perfectly aligned and jitter free (∆t = 0), a visibility of V (2) = 0.5 is obtained irrespective of the amount of frequency chirp, as expected for perfectly indistinguishable, phase-randomized weak laser pulses. Similarly, in the absence of frequency chirp (β = 0), the two-pulse interference also exhibits high visibility as long as the temporal misalignment is insignificant as compared with the pulse duration. However, the visibility deteriorates rapidly when both temporal misalignment and frequency chirp are present. With a realistic temporal misalignment (∆t = 10 ps) and frequency chirp (βσ t = 70 GHz), the interference visibility drops to ∼0.10, a value too low for any practical applications.
As an example, low visibility reduces the secure key rate of MDI-QKD. This is because visibility directly affects the phase error rate in the protocol, thereby increasing the privacy amplification cost. Using realistic parameters for channel transmission of 0.2 dB/km and measurement efficiency of 30%, a maximum secure key rate (R max ) of the order of 10 kbps can be attained with a GHz-clocked MDI-QKD system over 100 km fiber [31,32]. Incidentally, this secure key rate is more than two orders of magnitude greater than what has previously been reported in the literature [18][19][20][21]. However, it will decrease rapidly with deterioration of the visibility. In Fig. 3, we plot contour lines illustrating the achievable secure key rates at corresponding interference visibilities. With a slight drop of the visibility from 0.50 to 0.45, the secure key rate is reduced to less than the half of R max . It will reduce to around 10% of R max if the visibility is less than 0.40. When the visibility is lower than 0.37, the generation of a secure key is no longer possible. Hence, high visibility interference is vital to maintain efficient secure key generation in MDI-QKD.
Having described our theoretical model for two-pulse interference visibility in relation to laser frequency chirp and time jitter, and its effect on the MDI-QKD key rate, we can now proceed and measure the real visibility obtained from an MDI-QKD-like setup like the one in Fig. 2, set in single photon counting mode. Specifically, both lasers are equally attenuated to < 0.05 photons per pulse and superconducting nanowire single-photon detectors with ∼5% quantum efficiency are employed [33] By setting the filter bandwidth to 2 THz, we record a coincidence histogram as shown in Fig. 4(a). The suppression at the zero delay corresponds to a visibility of V (2) = 0.25. The visibility is vastly improved to 0.46 by narrowing the filter bandwidth to 13.8 GHz, as shown in Fig. 4(b). We plot the interference visibility as a function of the filter bandwidth in Fig. 4. Three different regions can be distinguished in the data. In the first region with the filter bandwidth greater than 70 GHz, the visibility improves slowly when the filter narrows. In this region, the filter rejects only the spontaneous background and side-mode emissions, which are typically three orders of magnitude weaker than the lasing mode. The visibility improves from V (2) = 0.25 at 2 THz to 0.28 at 72.5 GHz. Then the visibility improves rapidly when the filter starts to limit the laser bandwidth until reaching a peak visibility of V (2) = 0.46 at 13.8 GHz. Note that this bandwidth value is readily obtainable by using appropriately chosen off-the-shelf telecom filters [34]. In the third region, the visibility starts to deteriorate for filter bandwidths less than 13.8 GHz.
We calculate the visibility as a function of the laser bandwidth using the measured time jitter values and the measured bandwidth-dependent pulse durations. The resulting theoretical curves fit the experimental data without any free parameters. As shown in Fig. 4, the model has reproduced the visibility improvement in the intermediate bandwidth region whereas for small bandwidth it shows a considerably higher visibility than actually measured. The discrepancy in the narrow bandwidth region is attributed to imperfection in the measurement setup. We use a single bandpass filter which has a finite backreflection ratio. The back-reflected light does not affect the lasers, protected by attenuators and optical isolators, but it can enter the 50/50 beam splitter and reach the detectors, thus causing accidental coincidences that spoil the interference visibility. To unveil the truly achievable visibility, we have interfered laser pulses emitted by a single laser diode. An asymmetric Mach-Zehnder interferometer is aligned to interfere optical pulses of adjacent clocks [12]. In this arrangement, the bandpass filter is placed before the interferometer and the back-reflection problem is thus avoided. The results are also plotted in Fig. 4. This time, a visibility of 0.48 is recorded at 11.5 GHz, which agrees well with the predicted visibility of 0.488. The small discrepancy is due to the imperfect splitting ratio in the 50/50 beam-splitter, which has been measured to be close to 53/47. Figure 5 shows the interference visibility as a function of systematic temporal misalignment (∆t) between two lasers. Here, the bandpass filter is fixed to give a band-width of 13.8 GHz. On top of laser timing jitter, the systematic misalignment further deteriorates the interference visibility. In the extreme case of large misalignment (|∆t| > 45 ps), the optical pulses have little overlap and the corresponding visibility approaches zero. Around the optimal delay ∆t = 0, the visibility varies slowly with the temporal misalignment. At a misalignment of 10 ps, the visibility remains as high as 0.41, a value that is still sufficient for positive key distillation in MDI-QKD. This temporal tolerance is readily achievable through remote optical synchronisation [16].
Our results are not limited to MDI-QKD and are useful to other quantum information applications. For instance, a fast random number generator could be envisaged if numbers are assigned to the complementary outcomes shown in Fig. 1(c), improving the flexibility of existing solutions based on first-order interference [12,35]. Optical interference could also be used to quantify sidechannel information in QKD implementations with multiple light sources. Currently information leakage is estimated trough a series of ad hoc measurements based on a few known degrees of freedom [36]. However information leakage from unknown degrees of freedom, sometimes referred to as side channels, cannot be ignored. This security risk could be unveiled by a decrease in the visibility of a multi-source interference experiment. Finally, our demonstration that Fourier-transform limited pulses are not necessary for high-visibility interference may allow weak coherent pulses to be tailored to interfere with quantum light sources, providing a plethora of opportunities, for example, a hybrid quantum relay [37] that bridges weak-pulse QKD and entangled photon pairs [38].
To summarise, we have demonstrated high-speed phase-randomised coherent state sources that can exhibit high visibility in two-pulse interference. The solution is based on semiconductor gain-switched laser diodes and characterized in their temporal and spectral properties. This result is highly beneficial to the recent application of MDI-QKD and to others exploiting similar principles. The achieved visibility of 0.46 (Fig. 4), limited by back-reflection in the filter, can already guarantee more than 50% of the maximum key rate in MDI-QKD. Despite high speed and narrow pulse width, the interference visibility obtained from frequency-filtered gain-switched laser diodes is comparable to, or better than, those achieved with optical pulses carved from continuous-wave lasers [18,19,21]. This suggests that this cheap and effective solution will play a major role in future quantumbased applications.
FIG. 1 .
1(a) Temporal and (b) spectral profiles of the two GS lasers used in this work's experimental apparatus. (c) Interference output traces recorded by an oscilloscope and two fast photodiodes (D0 and D1).
FIG. 2 .
2Schematics of the experimental setup. The shade illustrate the spectral filtering using a single filter with the help of two optical circulators. The same diagram is used for modelling the effect of time jitter and frequency chirp on the two-pulse interference visibility.
FIG. 3 .
3Calculated second order interference visibility V(2) as a function of frequency chirp (βσt) and temporal misalignment (∆t) of interfering pulses. We assume a temporal width of 30 ps (FWHM) for laser pulses. White lines indicate the achievable secure bit rates of MDI-QKD, compared to the rate Rmax achievable with perfect interference.FIG. 4. The second-order interference visibility V (2) as a function of filter bandwidth. Measurements using a single laser and theoretical calculation are also shown. Insets: Coincidence traces for the Hong-Ou-Mandel interference measurements are shown for two different bandwidths of (a) 2 THz and (b) 13.8 GHz.
FIG. 5 .
5Measured (symbols) and simulated (solid line) interference visibility V (2) as a function of temporal misalignment ∆t.
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[
"Families of Q-balls in a deformed O(4) linear sigma model",
"Families of Q-balls in a deformed O(4) linear sigma model",
"Families of Q-balls in a deformed O(4) linear sigma model",
"Families of Q-balls in a deformed O(4) linear sigma model"
]
| [
"A Alonso-Izquierdo \nUniversity of Salamanca\nPlaza de la Merced 137008SalamancaSpain\n",
"C Garzón Sánchez ",
"\nDepartamento de Matematica Aplicada\nUniversity of Salamanca\nCasas del Parque 237008SalamancaSpain (\n",
"A Alonso-Izquierdo \nUniversity of Salamanca\nPlaza de la Merced 137008SalamancaSpain\n",
"C Garzón Sánchez ",
"\nDepartamento de Matematica Aplicada\nUniversity of Salamanca\nCasas del Parque 237008SalamancaSpain (\n"
]
| [
"University of Salamanca\nPlaza de la Merced 137008SalamancaSpain",
"Departamento de Matematica Aplicada\nUniversity of Salamanca\nCasas del Parque 237008SalamancaSpain (",
"University of Salamanca\nPlaza de la Merced 137008SalamancaSpain",
"Departamento de Matematica Aplicada\nUniversity of Salamanca\nCasas del Parque 237008SalamancaSpain ("
]
| []
| In this paper the existence of analytical solutions describing Q-balls in a family of deformed O(4) sigma models in (1+1) dimensions has been investigated. These models involve two complex scalar fields whose coupling breaks the O(4) symmetry group to U (1) × U (1). It has been shown that there are two types of single Q-balls rotating around each of the components of the internal space and a one-parameter family of composite Q-balls. These composite solutions consist of two single Q-balls (separated by a distance determined by the family parameter) spinning around each complex field with the same internal rotation frequency. | 10.1016/j.physd.2023.133764 | [
"https://export.arxiv.org/pdf/2301.10739v1.pdf"
]
| 256,231,239 | 2301.10739 | 518740c1d5fc87716d3435e2cd1eadcef26821ac |
Families of Q-balls in a deformed O(4) linear sigma model
January 26, 2023
A Alonso-Izquierdo
University of Salamanca
Plaza de la Merced 137008SalamancaSpain
C Garzón Sánchez
Departamento de Matematica Aplicada
University of Salamanca
Casas del Parque 237008SalamancaSpain (
Families of Q-balls in a deformed O(4) linear sigma model
January 26, 2023
In this paper the existence of analytical solutions describing Q-balls in a family of deformed O(4) sigma models in (1+1) dimensions has been investigated. These models involve two complex scalar fields whose coupling breaks the O(4) symmetry group to U (1) × U (1). It has been shown that there are two types of single Q-balls rotating around each of the components of the internal space and a one-parameter family of composite Q-balls. These composite solutions consist of two single Q-balls (separated by a distance determined by the family parameter) spinning around each complex field with the same internal rotation frequency.
Introduction
Q-balls are time-dependent non-topological solutions defined in nonlinear field theories characterized by the presence of a global U (1) symmetry [1]. This symmetry gives place to a conserved Noether charge, which is associated to an angular motion with angular velocity ω in the internal space. The time dependence of these solutions allows them to avoid the severe restrictions of Derrick's theorem [2] whereas the internal rotation stabilizes the Q-balls, which in other case would decay to the vacuum solution [3][4][5]. Other possibilities for stabilizing non-topological defects in (1+1)-dimensions by introducing a potential barrier or a non-trivial target space have been described in [6][7][8]. In 1976 Friedberg, Lee and Sirlin studied the presence of this type of solutions in a theoretical model involving a complex scalar field coupled with a real scalar field in the seminal paper [9]. The authors describe the Q-balls arising in this model and thoroughly analyze the stability of these solutions versus small fluctuations that maintain the conserved Noether charge constant. After this pioneering work, the existence of Q-balls and its properties have been studied in different contexts, for example, in complex scalar field theories [10][11][12][13], in Abelian gauge theories [14][15][16][17], in non-Abelian theories [18][19][20], in models which include fermionic interactions [15,[22][23][24][25], in models with presence of gravity [26,27], etc. This interest is explained by the fact that Q-balls are thought to play a relevant role in some important physical phenomena. For example, in 1998 Kusenko and Shaposhnikov [28] showed that Q-balls can be produced in the early universe in supersymmetric extensions of the standard model in such a way that they can contribute to dark matter by means of the Affleck-Dine mechanism. The relevance of this fact is that it is conjectured that this mechanism could explain baryogenesis during the primordial expansion, after cosmic inflation. It has also been proposed that Q-balls present in models with gravity mediated supersymmetry breaking are long-lived, allowing in principle, for these Q-balls to be the source of both the baryons and the lightest supersymmetric particle dark matter particle [29].
In general, models involving Q-balls are so complicated that it is not possible to obtain analytical expressions for these non-topological solitons. For this reason an interesting research direction in the study of Q-balls is that of identifying models where these solutions can be analytically calculated. This knowledge opens the possibility of analyzing their properties further and in more detail. This scenario has been explored in theories with one complex scalar field in (1+1)-dimensions in recent works [30][31][32][33]. Here, the authors focus mainly on attaining analytical solutions with different features. For example, models involving compact Q-balls are constructed in [31]. Another relevant topic deals with the study of excited states of Q-balls. In [34,35] the authors investigate this class of configurations by performing stationary perturbations on spherical Q-balls and describe the implications to Cosmology derived from the magnetic fields generated by these excited solutions. Radial excitations are examined in [36] in the case of models with one complex scalar field. Additionally, vibrational modes of these solutions with a nearcritical charge or in theories with flat potential are analyzed in [37]. On the other hand, the interaction between these non-topological solitons leads to a very rich variety of behaviors. For example, processes such as charge transfer and fission has been identified in one, two and three space dimensions [38]. The scattering between Q-balls in (1+1)-dimensions has also been investigated in [39]. Curiously, it was found that attractive or repulsive forces arise depending upon the relative phase of the colliding Q-balls.
In this paper we shall investigate the existence of analytical non-topological solitons in a family of twocomponent complex scalar field theories in (1+1) dimensions with a global U (1) × U (1) symmetry. The coupling between the two fields breaks down a O(4) symmetry in such a way that the model parameter can be understood as a measure of the deformation of the model with respect to the rotationally invariant theory. The effect of introducing two complex fields have been previously explored by Loginov and Gauzshtein [40][41][42] although in a different framework. For example, in [40,41] the authors deal with a (2+1)-dimensional gauge model describing two complex scalar fields that interact through a common Abelian gauge field. In this framework composite solutions arise consisting of a vortex and a Q-ball. A similar model immersed in a (1+1)-dimensional space-time is addressed in [42]. It is shown that the model has again composite solutions consisting now of two Q-balls with opposite electric charges. Note that the Friedberg-Lee-Sirlin model [9] involves the coupling between a real and a complex field. It has been shown the existence of hairy Q-balls in the limiting case of the vanishing potential in the previously mentioned model [43]. Composite Q-balls involving different geometries have been studied in [44].
Obviously, the study of soliton solutions in these coupled theories is much more complex than the case where only one field is present. Despite this fact, Q-ball solutions of the models introduced in this paper are analytically identified. As previously mentioned the model involves the presence of two complex fields and exhibits a U (1) × U (1) symmetry, which gives place to the existence of two conserved Noether charges. In this scenario a general Q-ball is described by the profiles of these two fields, which in principle could rotate with different internal frequencies. In particular, it will be shown that for certain ranges of the internal rotation frequencies there are two types of single Q-balls rotating around one of the complex fields while the other field vanishes. These single energy lump solutions are stable. In addition, there exists a one-parameter family of composite Q-balls when the two previously mentioned internal rotation frequencies are synchronized. These solutions consist of two single Q-balls and are formed by two energy lumps separated by a distance determined by the family parameter. This scheme suggests that, for configurations where the internal rotation frequencies are different, forces between the different constituents of the solutions arise making the non-topological solutions depend non-trivially on time.
The study of the stability of these composite solutions is very complicated and some of the arguments introduced in [9] must be altered when applied to these models. For example, the existence of two negative eigenvalues in the spectrum of the second order small fluctuation operator is not a sufficient condition for proving the instability of the Q-balls. This argument is now valid if three of these eigenstates are considered. Numerical analysis seems to indicate that these composite solutions are long-lived but unstable states. The analysis of these instability channels could bring insight into the forces between the two single Q-balls when they approach each other. The organization of this paper is as follows: the family of deformed O(4) linear sigma models addressed in this work and its properties is introduced in Section 2. The previously mentioned single Q-balls and the family of composite Q balls are analytically identified and described in Section 3. Sum rules between the energies and the conserved charges of these solutions are also discussed. Section 4 is dedicated to investigate the linear stability of these non-topological solitons. Finally, the conclusions of this work are summarized in Section 5.
The model
We shall deal with a two-component complex scalar field theory immersed in a Minkowski spacetime characterized by the action functional
S = d 2 x 1 2 ∂ µ φ ∂ µ φ + 1 2 ∂ µ ψ ∂ µ ψ − U (|φ|, |ψ|) ,(1)
where φ = φ 1 +iφ 2 and ψ = ψ 1 +iψ 2 are dimensionless complex scalar fields, that is, φ, ψ ∈ Maps(R 1,1 , C) and φ stands for complex conjugate of φ. The Minkowski metric g µν is chosen as g 00 = −g 11 = 1 and g 12 = g 21 = 0. In order to alleviate the notation we introduce the two-component complex scalar field Φ = (φ, ψ), which let us define
|Φ| 2 = |φ| 2 + |ψ| 2 = |φ 1 | 2 + |φ 2 | 2 + |ψ 1 | 2 + |ψ 2 | 2 .(2)
With this notation, the potential term U (|φ|, |ψ|) which will be investigated in this paper can be written as
U (|φ|, |ψ|; σ, a, b) = 1 2 |Φ| 6 − a 2 |Φ| 4 + b 2 |Φ| 2 + 2σ 2 |ψ| 2 |Φ| 2 + σ 2 (σ 2 − a 2 )|ψ| 2(3)
with a, b, σ ∈ R. The relation (3) is a sixth-degree algebraic expression in the modulus of the complex fields φ and ψ. It is a deformation of the O(4)-invariant |Φ| 6 -model, where the parameter σ measures the asymmetry with respect to the rotationally invariant situation. This can be checked by noting that
for σ = 0 U (|φ|, |ψ|; 0, a, b) = 1 2 (|Φ| 6 − a 2 |Φ| 4 + b 2 |Φ| 2 ) .(4)
On the other hand, a and b are the usual parameters that allow to change the profile of the |Φ| 6potential [1]. The potential (3) has a critical point at (φ, ψ) = (0, 0), where the potential vanishes, U (0, 0) = 0. The Hessian matrix of (3) evaluated at this point is given by
H[0, 0] = ∂ 2 U ∂|φ| 2 ∂ 2 U ∂|φ|∂|ψ| ∂ 2 U ∂|φ|∂|ψ| ∂ 2 U ∂|ψ| 2 (0,0) = b 2 0 0 b 2 + σ 2 (σ 2 − a 2 ) ,
which means that (φ, ψ) = (0, 0) is a minimum point only if the condition
b 2 > σ 2 (a 2 − σ 2 )(5)
holds. We are interested in searching for Q-ball type solutions, so we shall restrict our study to this regime, which guarantees the linear stability of the vacuum solution Φ = 0. As previously mentioned, the O(4)-symmetry associated to (4) is broken in our model for σ = 0, see (3). However, it is not completely broken and two U (1)-symmetries involving each of the complex fields are still preserved. The model is invariant with respect to the transformations φ → e iα φ and ψ → e iβ ψ, which leads to the conserved Noether charges
Q 1 = 1 2i φ ∂ t φ − φ ∂ t φ dx , Q 2 = 1 2i ψ ∂ t ψ − ψ ∂ t ψ dx .(6)
The field equations obtained from the Euler-Lagrange equations associated to the action funcional (1) can be written as
∂ 2 φ ∂t 2 − ∂ 2 φ ∂x 2 + φ |φ| ∂U (|φ|, |ψ|) ∂|φ| = 0 , ∂ 2 ψ ∂t 2 − ∂ 2 ψ ∂x 2 + ψ |ψ| ∂U (|φ|, |ψ|) ∂|ψ| = 0 .(7)
In this framework, Q-balls are time-dependent regular localized solutions of (7) which rotates with constant frequency in the internal space of each complex field. They can be studied by substituting the ansatz
φ(x, t) = f 1 (x) e iω 1 t , ψ(x, t) = f 2 (x) e iω 2 t(8)
into the field equations (7). This leads to the ordinary differential equations
∂ 2 f 1 ∂x 2 = ∂U (f 1 , f 2 ) ∂f 1 − ω 2 1 f 1 , ∂ 2 f 2 ∂x 2 = ∂U (f 1 , f 2 ) ∂f 2 − ω 2 2 f 2(9)
for the real functions f 1 (x) and f 2 (x). The quantities ω 1 and ω 2 in (8) are respectively the internal rotation frequencies for the φ and ψ fields. Without loss of generality we can consider that ω 1 and ω 2 are non-negative. The potential term U in (9) reads now
U (f 1 , f 2 ; σ, a, b) = 1 2 |F | 6 − a 2 |F | 4 + b 2 |F | 2 + 2 σ 2 f 2 2 |F | 2 + σ 2 (σ 2 − a 2 )f 2 2(10)
where F = (f 1 , f 2 ) and |F | 2 = f 2 1 + f 2 2 . The integral over the space coordinate of the energy density
E[f 1 , f 2 ] = 1 2 ∂f 1 dx 2 + 1 2 ∂f 2 dx 2 + 1 2 ω 2 1 f 2 1 + 1 2 ω 2 2 f 2 2 + U (f 1 , f 2 ; σ, a, b)(11)
provides us with the total energy, i.e.,
E[f 1 , f 2 ] = ∞ −∞ E[f 1 , f 2 ] dx.
Finally, the conserved Noether charges (6) become
Q 1 = ω 1 ∞ −∞ (f 1 (x)) 2 dx , Q 2 = ω 2 ∞ −∞ (f 2 (x)) 2 dx(12)
in this case. Q-balls are finite energy solutions, which implies that the following asymptotic conditions
lim x→±∞ f i = lim x→±∞ df i dx = 0 with i = 1, 2,(13)
must hold. It is also clear from (11) that the problem involves the effective potential
U (f 1 , f 2 ; σ, a, b, ω 1 , ω 2 ) = U (f 1 , f 2 ; σ, a, b) − 1 2 ω 2 1 f 2 1 − 1 2 ω 2 2 f 2 2(14)
in such a way that the equations (9) can be written in the more compact form
∂ 2 f 1 ∂x 2 = ∂U (f 1 , f 2 ) ∂f 1 , ∂ 2 f 2 ∂x 2 = ∂U (f 1 , f 2 ) ∂f 2 .(15)
The effective potential (14) depends on the internal rotation frequencies. In Figure 1 the potential U (f 1 , f 2 ) has been depicted for several values of ω 1 and ω 2 with fixed values of the rest of the parameters. Note that for ω 1 = ω 2 = 0 the original potential U (f 1 , f 2 ) has an absolute minimum located at the origin of the internal space but in other cases this point becomes only a local minimum for the effective potential U (f 1 , f 2 ). This is a necessary condition for the existence of Q-balls.
Solving the system (15) together with the conditions (13) is tantamount to finding solutions asymptotically beginning and ending at the origin for Newton equations in which x plays the role of time, the particle position is determined by (f 1 , f 2 ) and the potential energy of the particle is V (f 1 , f 2 ) = −U (f 1 , f 2 ). We shall exploit this mechanical analogy in the next Section by using the Hamilton-Jacobi theory in this context. Note that the equations (15) can be derived from the static effective functional
E[f 1 , f 2 ] = dx 1 2 ∂f 1 dx 2 + 1 2 ∂f 2 dx 2 + U (f 1 , f 2 ; σ, a, b) .(16)
Another consequence of the previous interpretation is that
I 1 = 1 2 df 1 dx 2 + 1 2 df 2 dx 2 + 1 2 ω 2 1 f 2 1 + 1 2 ω 2 2 f 2 2 − U (f 1 , f 2 ; σ, a, b)(17)
is a first integral of the system (15). The asymptotic conditions (13) impose that Q-balls are characterized by the relation I 1 = 0.
Families of Q-balls
In this Section we shall investigate the existence of Q-ball solutions in the model (3) introduced in Section 1. It is clear that if one of the complex fields vanishes then the reduced potential becomes the usual sixth-order polynomial studied in the literature, see [1]. Consequently, the presence of two types of solutions is expected: Q-balls whose ψ-component is zero and Q-balls which comply with the condition φ = 0. They will be denoted respectively as B 1 (x) and B 1 (x). The subscript in the previous notation is used to indicate that they are single Q-balls involving only one energy lump, as will be seen later. As it is well known only internal rotational frequencies ω 1 and ω 2 belonging to certain intervals can lead to these solutions. Assuming that the first type of Q-balls arises for ω 1 ∈ (ω − 1 , ω + 1 ) and the second one for ω 2 ∈ (ω − 2 , ω + 2 ), it will be proved that for our model the lowest value of these frequencies coincides,
ω − 1 = ω − 2 ,
whereas the highest allowed frequencies verify ω + 2 ≤ ω + 1 . However, the main novelty of this model is the existence of Q-balls where the two complex fields are non-null. It will be shown that when the two internal rotational frequencies ω 1 and ω 2 are synchronized, that is, ω 1 = ω 2 = ω, a one-parameter family of Q-balls arises. Indeed, they can be analytically identified for every value of ω. Every member of this family can be interpreted as a non-linear combination of one B 1 (x)-type Q-ball and one B 1 (x)-type Q-ball, which are separated by a distance determined by the family parameter γ 1 . We will denote these solutions as B 2 (x; γ 1 ), where the subscript 2 in this notation emphasizes the fact that they are composite Q-balls displaying two distinct energy lumps. It will be proved that all the members of the family have both the same total energy E and the same sum of the Noether charges Q = Q 1 + Q 2 .
Single Q-balls
As previously mentioned, there are two types of single Q-balls:
• B 1 (x)-type Q-balls: A |φ| 6 -model is embedded in our model when ψ = 0. In this case the effective potential (14) reads
U (f 1 , 0; σ, a, b, ω 1 , ω 2 ) = 1 2 f 6 1 − a 2 f 4 1 + b 2 f 2 1 − 1 2 ω 2 1 f 2 1 . (18)
The values of the rotational frequency ω 1 are restricted by the following conditions: (1) The effective potential (18) must have a minimum at f 1 = 0 and (2) the effective potential (18) must have at least one nontrivial real root. Taking into account that
∂ 2 U (f 1 , 0) ∂f 2 1 f 1 =0 = b 2 − ω 2 1
and that the roots of (18) are determined by
f 1 = ± 1 √ 2 a 2 ± a 4 − 4(b 2 − ω 2 1 ) 1 2(19)
the previous conditions restrict the values of ω 1 as follows:
b 2 − 1 4 a 4 ≤ ω 2 1 ≤ b 2 .(20)
For this type of solutions the second equation of the system (15) (or equivalently (9)) is automatically satisfied. Solving the first equation leads to the solution
B 1 (x; ω 1 ) = (f 1 (x), f 2 (x)) = 2(b 2 − ω 2 1 ) a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh(2 b 2 − ω 2 1 x) , 0 (21)
where x = x − x 0 and x 0 can be interpreted as the Q-ball center. The Noether charges for this solution are given by
Q 1 [B 1 (x)] = ω 1 arctanh 2 b 2 − ω 2 1 a 2 , Q 2 [B 1 (x)] = 0(22)
whereas the total energy is
E[B 1 (x)] = 1 4 a 2 b 2 − ω 2 1 − Q 1 [B 1 (x)] 8ω 1 a 4 − 4b 2 − 4 ω 2 1 .(23)
• B 1 (x)-type Q-balls: On the other hand, a |ψ| 6 -model is found when φ = 0. The effective potential (14) in this case is given by
U (0, f 2 ; σ, a, b, ω 1 , ω 2 ) = 1 2 f 6 2 − (a 2 − 2 σ 2 )f 4 2 + [b 2 + σ 2 (σ 2 − a 2 )]f 2 2 − 1 2 ω 2 2 f 2 2 .(24)
The expression (24) follows the same functional form that (18). This can be checked by redefining the parameters
a 2 1 = a 2 − 2σ 2 , b 2 1 = b 2 − σ 2 (a 2 − σ 2 )(25)
and rewriting the expression of the restricted effective potential (24) in terms of these new parameters,
U (0, f 2 ) = 1 2 f 6 2 − a 2 1 f 4 2 + b 2 1 f 2 2 − 1 2 ω 2 2 f 2 2 .
We can take advantage of this fact to directly obtain the expressions that characterize the B 1 (x) solutions. Now, for example, the Hessian operator along the ψ-direction evaluated at the origin of the internal plane is
∂ 2 U (0, f 2 ) ∂f 2 2 = b 2 1 − ω 2 2 = b 2 − σ 2 (a 2 − σ 2 ) − ω 2 2
whereas the zeroes of the potential (24) along the |ψ|-axis are
f 2 = ± 1 √ 2 a 2 1 ± a 4 1 − 4(b 2 1 − ω 2 2 ) 1 2 = ± 1 √ 2 a 2 − 2σ 2 ± a 4 − 4(b 2 − ω 2 2 ) 1 2 .(26)
To obtain real roots from (26) we impose the additional condition
a 2 > 2 σ 2 .(27)
The constraints on the internal rotation frequencies of these Q balls now read
b 2 − 1 4 a 4 ≤ ω 2 2 ≤ b 2 − σ 2 (a 2 − σ 2 ) ,(28)
and for these frequencies the solutions can be written as
B 1 (x; ω 2 ) = (f 1 (x), f 2 (x)) = 0, 2(b 2 1 − ω 2 2 ) a 2 1 + a 4 1 − 4(b 2 1 − ω 2 2 ) cosh(2 b 2 1 − ω 2 2 x) .(29)
In this case, the Noether charges are
Q 1 [ B 1 (x)] = 0 , Q 2 [ B 1 (x)] = ω 2 arctanh 2 b 2 − σ 2 (a 2 − σ 2 ) − ω 2 2 a 2 − 2σ 2(30)
with a total energy Curiously, the lowest value of the frequency range defining the two types of Q balls coincides, see (20) and (28), that is,
E[ B 1 (x)] = 1 4 (a 2 − 2σ 2 ) b 2 − σ 2 (a 2 − σ 2 ) − ω 2 2 − Q 2 [ B 1 (x)] 8ω 2 a 4 − 4b 2 − 4ω 2 2 ) .(31)ω − 2 = ω − 1 .
On the other hand, the highest internal rotation frequency for the B 1 (x)-type Q balls is smaller than the value found for the B 1 (x) solutions, that is, ω + 2 ≤ ω + 1 . In Figure 2 Figure 2(c) it can be checked that these solutions consist of only one energy lump, that is, they are single Q-balls. In particular, the energy density of the first type of these non-topological solitons is more concentrated and compact than the energy density of the second type when the same internal rotational frequency ω is considered. Note that, in general, E[B 1 (0; ω)] < E[ B 1 (0; ω)] and the size of the Q-balls can be estimated respectively as Figure 3 the Noether charges and the total energies for the two types of Q-balls are plotted as a function of the internal rotational frequency ω. It can be checked that
∆B 1 (x; ω) = 1 √ b 2 − ω 2 , ∆ B 1 (x; ω) = 1 b 2 − σ 2 (a 2 − σ 2 ) − ω 2 such that ∆B 1 (x; ω) ≤ ∆ B 1 (x; ω). InQ 2 [ B 1 (x; ω)] ≤ Q 1 [B 1 (x; ω)] and E[ B 1 (x; ω)] ≤ E[B 1 (x; ω)].
If the Noether charges of these two solutions are compared when the asymmetry parameter σ is small while rotating with the same frequency ω, the relation
∆Q = Q 1 [B 1 (x; ω)] − Q 2 [ B 1 (x; ω)] = σ 2 ω √ b 2 − ω 2 + O(σ 4 )(32)
is found. From (32) it can be noted that the difference ∆Q between the Noether charges for the two types of Q-balls depends on σ 2 , which means that the two charges
Q 1 [B 1 (x; ω)] and Q 2 [ B 1 (x; ω)]
are approximately equal for a large range of small values of σ.
Composite Q-Balls
In this section we shall investigate the existence of Q-balls with two non-null complex scalar fields. Using the mechanical analogy these solutions could be interpreted as a particle which asymptotically leaves the origin of the internal (f 1 , f 2 )-plane, travels in a bounded region of this plane and asymptotically returns to the origin. A profitable method employed in the identification of solutions in some types of deformations of O(N ) invariant models consists of exploring the separability of the model when elliptic variables are used in the internal space. Bearing this in mind, we introduce the elliptic coordinates in the form
ξ * ± f 1 = 1 Ω u v , ξ * ± f 2 = ± 1 Ω (u 2 − Ω 2 )(Ω 2 − v 2 )(33)
where u ∈ [Ω, +∞) and v ∈ [−Ω, Ω]. The first condition for our model to be separable in elliptic variables is to set the eccentricity parameter Ω in (33) to the deformation parameter σ arising in the potential (10), i.e., Ω = σ .
This implies that the effective potential (14) can be expressed in the new coordinates as
ξ * σ U (f 1 , f 2 ) = 1 u 2 − v 2 f (u) + g(v) + ω 2 2 − ω 2 1 2σ 2 u 2 v 2 (u 2 − v 2 )(34)
where
f (u) = 1 2 u 2 (u 4 − a 2 u 2 + b 2 − ω 2 2 )(u 2 − σ 2 ) ,(35)g(v) = 1 2 v 2 (v 4 − a 2 v 2 + b 2 − ω 2 2 )(σ 2 − v 2 ) .(36)
From (34) it can be checked that separability is attained if and only if the internal rotation frequencies around each of the complex fields are equal, that is, ω 1 = ω 2 = ω. In this Section, we shall investigate this class of solutions and we shall show that they involve a very rich structure. Obviously, the possible values of ω are restricted to the intersection of the allowed values for ω 1 and ω 2 . The effective Lagrangian (16) can be written in elliptic variables as
E[ξ * f 1 , ξ * f 2 ] = 1 2 u 2 − v 2 u 2 − σ 2 du dx 2 + 1 2 u 2 − v 2 σ 2 − v 2 dv dx 2 + f (u) + g(v) u 2 − v 2 ,
which allows us to define the generalized momenta
p u = ∂L ∂ du dx = u 2 − v 2 u 2 − σ 2 du dx , p v = ∂L ∂ dv dx = u 2 − v 2 σ 2 − v 2 dv dx .
Now, the Hamiltonian associated to our problem reads
H = 1 u 2 − v 2 (h u + h v ) where h u = 1 2 (u 2 − σ 2 )p 2 u − f (u) and h v = 1 2 (σ 2 − v 2 )p 2 v − g(v) .
The Hamilton-Jacobi equation
∂J ∂x + H ∂J ∂u , ∂J ∂v , u, v = 0(37)
is now completely separable. In order to abbreviate the formulas we shall use 'prime'-notation to stand for derivatives with respect to the space coordinate x. For example, u = du dx and v = dv dx . If the separation ansatz for Hamilton's principle function J = J x (x) + J u (u) + J v (v) is substituted into (37) the relation
1 2 (u 2 − σ 2 )(J u ) 2 − Eu 2 − f (u) = F = − 1 2 (σ 2 − v 2 )(J v ) 2 − Ev 2 + g(v)
holds. The solutions of the PDE (37) can be expressed in terms of the solutions of the ODEs
J x = −E , J u = sign(u ) 2(F + Eu 2 + f (u)) u 2 − σ 2 , J v = sign(v ) 2(−F − Ev 2 + g(v)) σ 2 − v 2 which leads to J x = −Ex , J u = sign(u ) dx 2(F + Eu 2 + f (u)) u 2 − σ 2 , J v = sign(v ) dx 2(−F − Ev 2 + g(v)) σ 2 − v 2 .
Once Hamilton's principle function J has been obtained the solutions can be determined from the relations
∂J ∂F = γ 1 , ∂J ∂E = γ 2(38)
where γ 1 , γ 2 ∈ R identify every solution of the problem. The first equation in (38) provides the trajectories or orbits of the solutions in the (f 1 , f 2 )-plane parametrized by the value of γ 1 whereas the second equation in (38) specifies the dependence of the solutions with respect to the space x. Taking into account that the asymptotic conditions (13) implies that E = F = 0 the relations (38) lead to the quadratures
sign(u ) du (u 2 − σ 2 )f (u) − sign(v ) dv (σ 2 − v 2 )g(v) = √ 2 γ 1 ,(39)sign(u ) u 2 du (u 2 − σ 2 )f (u) − sign(v ) v 2 dv (σ 2 − v 2 )g(v) = √ 2 (x + γ 2 ) .(40)
For our model the expressions (39) and (40) become
sign(u ) du u(u 2 − σ 2 ) √ u 4 − a 2 u 2 + b 2 − ω 2 − sign(v ) dv v(σ 2 − v 2 ) √ v 4 − a 2 v 2 + b 2 − ω 2 = γ 1 , sign(u ) u du (u 2 − σ 2 ) √ u 4 − a 2 u 2 + b 2 − ω 2 − sign(v ) v dv (σ 2 − v 2 ) √ v 4 − a 2 v 2 + b 2 − ω 2 = x + γ 2 ,
where we recall that ω = ω 1 = ω 2 is the synchronized frequency around the complex coordinate axes at which the solutions we are looking for are spinning, that is, φ(x, t) = f 1 (x)e iωt and ψ(x, t) = f 2 (x)e iωt . Note that the parameter γ 2 only represents a translation of the solution along the x-axis. In order to alleviate the notation in the calculations of the previous quadratures we shall denote the roots of the polynomial
p(z) = z 2 − a 2 z + b 2 − ω 2(41)
as
r ± = 1 2 a 2 ± a 4 − 4(b 2 − ω 2 ) .(42)
Clearly, the inequation 0 ≤ v 2 ≤ σ 2 ≤ u 2 ≤ r − ≤ r + holds for the solutions which we are looking for. If we define
P (z) = (z − r − )(z − r + ) , R(z) = r + − z r − − z(43)
the expression of the Q-ball orbits in the (f 1 , f 2 )-space is given by
R(u 2 ) − R(0) R(u 2 ) + R(0) P (σ 2 ) P (0) R(u 2 ) + R(σ 2 ) R(σ 2 ) − R(u 2 ) sign(u ) R(v 2 ) − R(0) R(v 2 ) + R(0) P (σ 2 ) P (0) R(v 2 ) + R(σ 2 ) R(v 2 ) − R(σ 2 ) sign(v )
= e 2σ 2 P (σ 2 )γ 1 (44) while the spatial dependence of these solutions can be obtained from the relation
R(u 2 ) + R(σ 2 ) R(σ 2 ) − R(u 2 ) sign(u ) R(v 2 ) + R(σ 2 ) R(v 2 ) − R(σ 2 ) sign(v ) = e 2P (σ 2 ) x(45)
where x = x + γ 2 . It can be checked that relations (44) and (45) are invariant under the transformations γ 1 → −γ 1 and x → −x. This means that solutions with γ 1 < 0 can be constructed from solutions with γ 1 > 0 by simply flipping the space axis x. Plugging (45) into (44) the following more simple relation
R(u 2 ) − R(0) R(u 2 ) + R(0) sign(u ) R(v 2 ) − R(0) R(v 2 ) + R(0) sign(v ) = e 2P (0)(σ 2 γ 1 −x)(46)
is deduced. The set of equations (45) and (46) can be used to obtain an analytical expression of the families of Q-balls. Although it is a lengthy expression, it is worthwhile (from our point of view) to write it down in a sequential way because it can be used to plot the profiles of Q-balls as well as its energy densities, conserved Noether charges, etc. The equations (45) and (46) can be solved for the functions R(u 2 ) and R(v 2 ), which in turn provide us with the profile of the elliptic variables for the Q-balls in the form
u(x; σ, a, b, ω) = r + − r − (S 1 − S 2 ) 2 1 − (S 1 − S 2 ) 2 , v(x; σ, a, b, ω) = sign(v) r + − r − (S 1 + S 2 ) 2 1 − (S 1 + S 2 ) 2(47)
where
S 1 = E 2 [R 2 (σ 2 ) − R 2 (0)] 2(R(0) + E 1 E 2 R(σ 2 )) ,(48)S 2 = 4R(0)R(σ 2 )[R(σ 2 ) + E 1 E 2 R(0)][R(0) + E 1 E 2 R(σ 2 )] + E 2 2 [R 2 (σ 2 ) − R 2 (0)] 2 2(R(0) + E 1 E 2 R(σ 2 ))(49)
and The profile of the original variables f 1 and f 2 can be obtained from (47) by using the change of variables (33). We shall refer to these solutions as B 2 (x; ω, γ 1 ). In Figure 4 the profiles of the functions f 1 and f 2 associated with these Q-balls (determined by the expressions (47), (48) and (49)) have been plotted for several values of the orbit parameter γ 1 assuming the representative model parameters σ = 0.25, a = 1.75 and b = 2.0 chosen in this paper. In this case the internal rotational frequencies ω take values on the interval ω ∈ [1.28657, 1.95256]. The profiles of the B 2 (x; ω, γ 1 )-solutions have also been depicted for several values of these frequencies ω. Indeed, Figure 4 has been arranged in tabular form where rows are associated with a particular value of the frequency ω and columns correspond to different values of the orbit parameter γ 1 . Three different types of values of this parameter are considered in order to show the distinct behaviors of this family of Q-balls: in the first column γ 1 is set to zero while in the third one a large value of γ 1 is fixed (whose particular value depends on the rotational frequency ω). In the second column an intermediate value between the previous ones is considered. This categorization of the values of γ 1 is related to the gradual loss of the symmetry/antisymmetry in the components f 1 and f 2 . For γ 1 = 0 the function f 1 is an even configuration while f 2 is odd. As the orbit parameter γ 1 increases this symmetry breaks down, see the middle column in Figure 4. However, the most remarkable feature of the B 2 (x; ω, γ 1 )-family can be observed when the orbit parameter γ 1 is large enough, see the third column in Figure 4. In this case the profiles of the components f 1 and f 2 decouple each other. They are significantly different from zero for regions where the other component almost vanishes. The resulting configurations for each field correspond to the profiles of a single B 1 (x; ω)-type Q-ball and a single B 1 (x; ω)-type Q-ball. This suggests that the B 2 (x; ω, γ 1 ) solutions are non-linear combinations of the two types of single Q-balls where the orbit parameter γ 1 measures the distance between them. This justifies the subscript 2 included in the notation for these solutions. In particular, when |γ 1 | → ∞ the two single Q-balls follow the expressions (21) and (29) and they are infinitely separated. For γ 1 = 0 they are completely overlapped.
E 1 = tanh[P (σ 2 )x] , E 2 = tanh[P (0)(σ 2 γ 1 − x)] .
The previous remark is also clearly observed in Figure 5, where the orbits associated to the composite B 2 (x; ω, γ 1 )-solutions are displayed in the internal space (f 1 , f 2 ). For the sake of comparison we have employed in these graphics the same values of γ 1 and ω introduced in Figure 4. Using the previously mentioned mechanical analogy, the B 2 (x; ω, γ 1 )-orbit shown in the first plot in Figure 5 (for ω = 1.3 and γ 1 = 0) can be interpreted as the trajectory of a particle asymptotically leaving the maximum located at the origin of the plane (f 1 , f 2 ) in a certain direction. The particle travels up to a point where the potential barrier forces the particle to move backwards approaching to the f 1 -axis and crossing the focus point F = (σ, 0). At this point the particle is again expelled far away from the origin. A second bounce takes place which makes the particle approach again to the origin, which now is asymptotically reached. This behavior is replicated in the rest of the cases. As previously mentioned the orbits with γ 1 = 0 are symmetric with respect to reflections f 2 → −f 2 . This symmetry is lost for non-null values of γ 1 . Indeed, for |γ 1 | → ∞ the orbit becomes the union of two segments, one along the axis f 1 and the other along the axis f 2 , which correspond to the single Q-ball orbits. This behavior corroborates the previous claim that the composite B 2 (x; ω, γ 1 ) solutions consist of two single B 1 (x; ω) and B 1 (x; ω) lumps.
Finally, Figure 6 illustrates the energy density of the solutions plotted in Figure 4. As expected, when the orbit parameter γ 1 is large enough two different energy lumps are discerned for any value of the rotational frequency ω, each one associated to the previously mentioned single Q-balls. Note that almost for all the values of the internal rotational frequencies ω these two energy lumps arise even for γ 1 = 0 where the symmetric configuration takes place. Only for very high values of ω the two lumps merge for small values of the orbit parameter γ 1 . If γ 1 is large and negative the solution (47) describes a single B 1 (x; ω) lump placed to the left of a single B 1 (x; ω) solution, which are well separated. As γ 1 increases the two single Q-balls approach each other while losing its identity and eventually giving place to two identical lumps when γ = 0. In this situation the two lumps remains separated by a certain distance. However, as the value of γ 1 asymptotically increases the lump on the left gradually becomes a B 1 (x; ω) lump while the one on the right becomes a B 1 (x; ω)-type Q-ball.
There are another two interesting properties of the family of composite Q-balls introduced in this section. It can be proved that:
1. The B 2 (x; ω, γ 1 )-family is energy degenerate and complies with the energy sum rule that is, the energy of the composite Q-balls is equal to the sum of the energies of the two types of single Q-balls.
E[B 2 (x; ω, γ 1 )] = E[B 1 (x; ω)] + E[ B 1 (x; ω)](50)
2. The sum of the two Q i -charges is equal for every member of the family of composite Q-balls, which means that Q 1 [B 2 (x; ω, γ 1 )] + Q 2 [B 2 (x; ω, γ 1 )] is independent of the family parameter γ 1 . Besides, the relation
Q 1 [B 2 (x; ω, γ 1 )] + Q 2 [B 2 (x; ω, γ 1 )] = Q 1 [B 1 (x; ω)] + Q 2 [ B 1 (x; ω)](51)
holds.
In order to prove the previous statements, a Bogomolnyi arrangement of the functional (16) (the analogue mechanical energy functional) written in elliptic variables will be introduced in this framework. By definition, a superpotential expressed in certain generalized coordinates {u i } verifies
1 2 g ij ∂W ∂u i ∂W ∂u j = U (u i )
being U (u i ) the effective potential term (14). In our case, the generalized coordinates are the elliptic coordinates (u 1 = u, u 2 = v), the induced metric g ij is given by g 11 = u 2 −v 2 u 2 −σ 2 , g 22 = u 2 −v 2 σ 2 −v 2 and g 12 = g 21 = 0 and the effective potential term U (u, v) can be written as U (u, v) = 1
u 2 −v 2 [f (u) + g(v)]
where f (u) and g(v) were defined respectively in (35) and (36). By taking advantage of the separability of the model the superpotential can be figured out by means of quadratures. It follows the separated form
W (α,β) (u, v) = (−1) α W u (u) + (−1) β W v (v)(52)
with α, β = 0, 1 and
W z (z) = 2z 2 − a 2 8 z 4 − a 2 z 2 + b 2 − ω 2 − a 4 − 4(b 2 − ω 2 ) 16 log a 2 − 2z 2 − 2 z 4 − a 2 z 2 + b 2 − ω 2 (53)
for z = u, v. The expression (52) defines four different superpotentials depending on the values of α and β. Thus, the analogue mechanical energy E can be written as
E[u i (x)] = 1 2 N −1 k=0 x k+1 x k dx g ij du i dx + g mi ∂W (α k ,β k ) ∂u m du j dx + g nj ∂W (α k ,β k ) ∂u n + T (54) where T = N −1 k=0 W (α k ,β k ) [u i (x k+1 )] − W (α k ,β k ) [u i (x k )] .(55)
The partition −∞ = x 0 < x 1 < · · · < x N −1 < x N = ∞ has been introduced in (54) to let us choose different superpotentials for different intervals. Q-balls saturate (54) in a piecewise way and verify the first order differential equations
du i dx + g mi ∂W (α k ,β k ) ∂u m = 0 for x ∈ [x k , x k+1 ] and k = 1, . . . , N .
For the Q-balls studied in this section the previous equations become
du dx = (−1) α √ u 2 − σ 2 u 2 − v 2 2f (u) , dv dx = (−1) β √ σ 2 − v 2 u 2 − v 2 2g(v) ,(56)
which are completely equivalent to the relations (39) and (40) used previously. The first term in (54) vanishes and the total analogue mechanical energy of these solutions is simply given by E[u i (x)] = T . Therefore, E is completely determined by the projections of the orbits on each of the elliptic axes. For later use, note that the composite Q-balls asymptotically leave the origin (u, v) = (σ, 0) and travel away up to a turning point P 1 where du dx = 0, that is, f (u) = 0. Thus, the maximum value of the u-coordinate for these solutions is
u M = √ r − = 1 √ 2 a 2 − a 4 − 4(b 2 − ω 2 ) .
This value is independent of the family parameter γ 1 . Subsequently, the orbit crosses through the focus (u, v) = (σ, σ) of the elliptic coordinates, reaches another turning point P 2 with the same maximum value u M of the u-coordinate and finally asymptotically goes back to the origin (u, v) = (σ, 0). In sum, the orbit of the composite Q-balls transverses four times the interval u ∈ [σ, u M ] and twice the interval v = [0, σ]. On the other hand, the B 1 (x; ω)-orbit lies on the u-axis taking the values between σ and u M twice. The B 1 (x; ω)-orbit transverses the segment v ∈ [0, σ] with u = σ, crosses through the focus F = (σ, σ), follows the segment u ∈ [σ, u M ] with v = σ fixed and subsequently goes back to the origin in the reverse direction. Before computing the total energy of the composite Q-balls, we shall prove the second of the statements, which says that the sum of the Noether charges Q 1 + Q 2 is the same for all the members of the B 2 (x; ω, γ 1 )-family. The definition (12) of the Q i -charges allows us to write in the (u, v)-plane
Q 1 + Q 2 = ω ∞ −∞ dx u(x) 2 + v(x) 2 − σ 2 = ω ∞ −∞ dx u 2 (u 2 − σ 2 ) u 2 − v 2 + v 2 (σ 2 − v 2 ) u 2 − v 2 = = ω N k=0 x k+1 x k u √ u 4 − a 2 u 2 + b 2 − ω 2 du dx dx + ω N k=0 x k+1 x k v √ v 4 − a 2 v 2 + b 2 − ω 2 dv dx dx = = N k=0 W u [u(x k+1 )] − W u [u(x k )] + N k=0 W v [v(x k+1 )] − W v [v(x k )] where W z (z) = ω 2 log a 2 − 2z 2 − 2 z 4 − a 2 z 2 + b 2 − ω 2 .
The first order differential equations (56) have been used to find the exact differential of the previous integrands that defines the function W z (z). Therefore,
Q 1 [B 2 (x; ω, γ 1 )] + Q 2 [B 2 (x; ω, γ 1 )] = 4W u (u M ) − 2W u (σ) − 2W v (0) = = 2W u (u M ) − 2W v (0) + 2W u (u M ) − 2W u (σ) = Q 1 [B 1 (x; ω)] + Q 2 [ B 1 (x; ω)] .(57)
which justifies the sum rule between the Q i -charges. Finally, the total energy of the composite Q-balls can be obtained as
E[B 2 (x; ω, γ 1 )] = E[B 2 (x; ω, γ 1 )] + ω(Q 1 [B 2 (x; ω, γ 1 )] + Q 2 [B 2 (x; ω, γ 1 )]) = = 4W u (u M ) − 2W u (σ) − 2W v (0) + ω(Q 1 [B 2 (x; ω, γ 1 )] + Q 2 [B 2 (x; ω, γ 1 )]) = = 2W u (u M ) − 2W v (0) + ω Q 1 [B 1 (x; ω)] + 2W u (u M ) − 2W u (σ) + ω Q 2 [ B 1 (x; ω)] = = E[B 1 (x; ω)] + E[ B 1 (x; ω)]
where we have used (55) and that W z (z) is an increasing function in the interval z ∈ (0, u M ). Note that dWz(z) dz = z √ z 4 − a 2 z 2 + b 2 − ω 2 > 0 in the previously mentioned interval.
Stability analysis of the Q-balls in the model
In this Section the stability of the Q-balls identified in Section 3 will be studied following the nowstandard approach on this topic initially developed in [9]. In this seminal paper Friedberg, Lee and Sirlin analyze the classical stability of the Q-balls with respect to field fluctuations that maintain the value of the Noether charge Q constant. The model addressed in this paper involves two complex scalar fields, so the solution at every point can be perturbed in two different channels. Two different subsections will be included below to discuss separately the stability of the single and composite Q-balls. As previously mentioned we will thoroughly follow the procedure introduced in [9]. Details will only be provided to show some differences with respect to the standard approach.
Stability analysis of the single Q-balls
We shall begin analyzing the stability of the single B 1 (x; ω) type Q-balls. The standard approach consists in analyzing the behavior of the second variation of the energy functional E[f 1 , f 2 ] when small fluctuations δf 1 and δf 2 are respectively applied on each component of the solution B 1 (x; ω). In this context, these perturbations must preserve the conserved Noether charges Q 1 and Q 2 defined in (12). This relates the δf 1 -fluctuation to the variation δω 1 of the internal rotational frequency ω 1 in the form
δω 1 ∞ −∞ f 2 1 (x)dx = −2ω 1 ∞ −∞ f 1 δf 1 dx .
Recall that the second component of the solution B 1 (x; ω) vanishes and, consequently, Q 2 [B 1 (x; ω)] = 0. In order to preserve the value of Q 2 the fluctuations in the second component are assumed to be static. If we substitute the fluctuation δF = (δf 1 , δf 2 ) t into the energy functional E[f 1 , f 2 ] the term which determines the behavior of the functional at second order is given by
δE (2) | Q = ∞ −∞ dx 1 2 (δF ) t H[B 1 (x)] δF + 2ω 3 1 Q 1 ∞ −∞ f 1 δf 1 dx 2
where the second order small fluctuation operator H[B 1 (x)] reads
H[B 1 (x)] = − d 2 dx 2 + ∂ 2 U ∂f 2 1 B 1 (x) − ω 2 1 ∂ 2 U ∂f 1 ∂f 2 B 1 (x) ∂ 2 U ∂f 1 ∂f 2 B 1 (x) − d 2 dx 2 + ∂ 2 U ∂f 2 2 B 1 (x) .(58)
For our model we have
∂ 2 U ∂f 2 1 = b 2 − 6a 2 f 2 1 + 15f 4 1 − 2a 2 f 2 2 + 18f 2 1 f 2 2 + 3f 4 2 + 2σ 2 f 2 2 ,(59)∂ 2 U ∂f 1 ∂f 2 = −4a 2 f 1 f 2 + 12f 3 1 f 2 + 12f 1 f 3 2 + 4σ 2 f 1 f 2 ,(60)∂ 2 U ∂f 2 2 = b 2 − 2a 2 f 2 1 + 3f 4 1 − 6a 2 f 2 2 + 18f 2 1 f 2 2 + 15f 4 2 − a 2 σ 2 + 2σ 2 f 2 1 + 12σ 2 f 2 2 + σ 4 . (61)
It can be checked that the expression (60) evaluated on the solution (21)
H 11 ξ (i) 1 (z) = λ i 4(b 2 − ω 2 1 ) ξ (i) 1 (z)(62)
where
H 11 = 1 4(b 2 −ω 2 1 ) H 11 [B 1 (x, ω 1 )]
is given by
H 11 = − ∂ 2 ∂z 2 + 1 4 − 3a 2 a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z + 15(b 2 − ω 2 1 ) (a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z) 2(63)
with z = 2 b 2 − ω 2 1 x. It can be checked that
ξ 0 (z) = sinh z (a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z) 3 2(64)
is a zero mode of H 11 for any value of the internal rotation frequency ω 1 . The function (64) has a node. This implies that there must be one and only one negative eigenvalue λ − . In Figure 7 and ω + 1 . The continuous spectrum in this case emerges on the threshold value b 2 − ω 2 1 . At this point, the analysis of the stability of the B 1 (x; ω) solutions versus δf 1 -fluctuations is completely analogous to the study carried out by Friedberg, Lee and Sirlin in [9]. Theorem 3 (stated there) can be directly applied to this class of perturbations. The necessary and sufficient conditions for Q-balls to be
H 22 ξ (i) 2 (z) = µ i 4(b 2 − ω 2 1 ) ξ (i) 2 (z) where H 22 = 1 4(b 2 −ω 2 1 ) H 22 [B 1 (x; ω 1 )] follows the expression H 22 = − ∂ 2 ∂z 2 + b 2 − a 2 σ 2 + σ 4 4(b 2 − ω 2 1 ) − a 2 − σ 2 a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z + 3(b 2 − ω 2 1 ) (a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z) 2(65)
Here, z = 2 b 2 − ω 2 1 x and µ i are the corresponding eigenvalues of the original operator H 22 [B 1 (x; ω 1 )]. They have been numerically calculated for the particular model parameters considered in this paper. In Figure 7(right) these eigenvalues µ i have been plotted as a function of the rotational frequency ω 1 . For the smallest values of ω 1 there exist two discrete eigenvalues µ 0 and µ 1 whereas for large enough values of the frequency only one of them remains. The continuous spectrum in this case emerges on the threshold value b 2 − σ 2 (a 2 − σ 2 ). As we can see in Figure 7(right) there are no negative eigenvalues. In sum, the B 1 (x, ω 1 ) solutions for the model parameters σ = 0.25, a = 1.75, b = 2.0 are stable with respect to orthogonal perturbations. Given the form (65) it is difficult to introduce a general analysis of the stability for any value of the parameters. Despite this fact, it will be proved below that the B 1 (x; ω 1 ) balls are always stable when the asymmetry of the model is small enough. In order to do this, note that the operator
H 22 = − ∂ 2 ∂z 2 + b 2 − a 2 σ 2 + σ 4 4(b 2 − ω 2 1 ) − a 2 a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z + 3(b 2 − ω 2 1 ) (a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z) 2(66)
has a ground state
ξ 0 (x) = 1 (a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z) 1 2 with eigenvalue µ 0 = ω 2 1 − σ 2 (a 2 − σ 2 ) 4(b 2 − ω 2 1 )(67)
which clearly is positive if σ is small enough. It can be verified that
H 22 = H 22 + σ 2 a 2 + a 4 − 4(b 2 − ω 2 1 ) cosh z
which means that the potential well of H 22 is weaker than that in H 22 . As a consequence, the eigenvalue of the ground state for the operator H 22 must be higher than µ 0 in (67). Therefore, the B 1 (x)-type Q-balls are also stable with respect to orthogonal fluctuations for small values of σ.
Using a completely analogous reasoning as before and the continuity of the eigenvalues with respect to the model parameters it can be proved that the B 1 (x) solutions (the second type of single Q-balls described in Section 3) are stable for small values of σ. Indeed, numerical analysis applied to the case σ = 0.25, a = 1.75 and b = 2.0 leads to similar results to those found in Figure 7.
Stability analysis of the composite Q-balls
The stability analysis is much more complicated for the composite Q-balls, where the two complex components of the solutions are non null. As before a deformation of the solution B 1 (x; ω, γ 1 ) + (δf 1 , δf 2 ) which maintains the Q i -charges constant is introduced into the energy functional E[f 1 , f 2 ]. In this case, the variations of the two internal frequencies δω 1 and δω 2 must comply with the constraints
δω 1 = − 2ω 2 1 Q 1 ∞ −∞ f 1 δf 1 dx , δω 2 = − 2ω 2 2 Q 2 ∞ −∞ f 2 δf 2 dx .
If we substitute these fluctuations into the energy functional E[f 1 , f 2 ], the term at second order is given by
δE (2) | Q = ∞ −∞ dx 1 2 (δF ) t H[B 2 (x)] δF + 2ω 3 1 Q 1 ∞ −∞ f 1 δf 1 dx 2 + 2ω 3 2 Q 2 ∞ −∞ f 2 δf 2 dx 2 = = ∞ −∞ dx 1 2 (δF ) t H[B 2 (x)] δF + 2ω 3 ∞ −∞ (δF ) t · F dx 2 − 2 ∞ −∞ dx F 1 δf 1 · ∞ −∞ dx F 2 δf 2 (68) where δF = (δf 1 , δf 2 ) t and F = ( f 1 √ Q 1 , f 2 √ Q 2 ) t . The small fluctuation operator H[B 2 (x)] in (68) reads H[B 2 (x)] = − d 2 dx 2 + ∂ 2 U ∂f 2 1 B 2 (x) − ω 2 1 ∂ 2 U ∂f 1 ∂f 2 B 2 (x) ∂ 2 U ∂f 1 ∂f 2 B 2 (x) − d 2 dx 2 + ∂ 2 U ∂f 2 2 B 2 (x) − ω 2 2 .(69)
It can be proved that now the existence of two negative eigenvalues in the spectrum of the operator (69) is not a sufficient condition leading to the instability of the composite Q-balls because of the last term in (68). The argument becomes valid again if three of these eigenstates are considered. Obviously, it is not possible to analytically solve the spectrum of the matrix operator (69). Despite this fact it can be proved that this operator has two zero modes. Indeed, these eigenfunctions correspond to the expressions ∂B 2 (x;ω,γ 1 ) ∂x and ∂B 2 (x;ω,γ 1 ) ∂γ 1
. However, as far as we know, there are no mathematical results relating the nodes of these zero modes to the number of negative eigenvalues for matrix operators of the form (69). Therefore, numerical analysis must be necessarily applied to obtain this information. In Figure 8 the spectrum of the operator H[B 2 (x)] as a function of the family parameter γ 1 has been depicted for the parameter values σ = 0.25, a = 1.75, b = 2.0 and ω = 1.70. It can be observed that there are three negative eigenvalues, which from our previous claim implies that the composite Q-balls are unstable in this case although long-living. Similar numerical results have been found for other values of the parameters. The analysis of these instability channels could bring insight into the forces between the two single Q-balls rotating in each components of the internal space.
Conclusions and further comments
In this paper the existence of analytical solutions describing Q-balls in a family of deformed O(4) sigma models in (1+1) dimensions has been investigated. These models involve two complex scalar fields whose coupling breaks the O(4) symmetry group to U (1) × U (1). This leads to the existence of two Noether conserved Q i -charges. The model parameter σ can be understood as a measure of the deformation of the model with respect to the rotationally invariant theory. It has been shown that there are two types of single Q-balls rotating around each of the components of the internal space and a one-parameter family of composite Q-balls. All of these non-topological solitons have been analytically identified. The composite solutions consist of two single Q-balls (separated by a distance determined by the family parameter) spinning around each complex field with the same internal rotation frequency. Indeed, these solutions are formed by two energy lumps. It has been checked that the single Q-balls are linearly stable with respect to small fluctuations which preserve the Q i -charges. The study of the stability for the composite solutions is more complicated. Following the arguments introduced in [9] it has been proved that in this context the existence of three negative eigenvalues in the spectrum of the second order small fluctuation operator is a sufficient condition for proving the instability of these composite Q-balls. Numerical analysis has been used to analyze the evolution of these solutions when they are not initially perturbed. The results indicate that these composite solutions are long-lived. However, the numerical study of the Hessian operator (69) shows the presence of three negative eigenvalues, which means that these solutions are unstable states. In this context it has been found that some perturbations make the two constituents of the Q-balls travel away or approach each other while the synchronization of the two internal rotation frequencies is lost. These results make evident the presence of forces between the different constituents of the solutions, which makes the non-topological solutions depend non-trivially on time.
The research introduced in the present work opens up some possibilities for future work. The results point out that the interaction between the single Q-balls described in Section 3 are highly non-trivial. For this reason it is very interesting to tackle the study of the scattering between these two types of solutions following, for example, the scheme introduced in [39]. This could allow us to understand the forces between these constituents and the dependence of these forces with respect to the difference between the internal rotation frequencies of these Q-balls. The collision between excited Q-balls is also a future goal in order to discern if structure similar to those found in [45,46] arise in this context.
Figure 1 :
1Graphics of the effective potential U (f 1 , f 2 ) for the parameter values σ = 0.25, a = 1.75, b = 2.0 and several values of the internal rotation frequencies: (a) ω 1 = ω 2 = 0, (b) ω 1 = ω 2 = 1.29, (c) ω 1 = 1.29, ω 2 = 1.75 and (d) ω 1 = 1.95, ω 2 = 1.3.
Figure 2 :
2Profiles of the components f i (x) for the B 1 (x) (left) and B 1 (x) (middle) solutions for the parameter values σ = 0.25, a = 1.75, b = 2.0 and ω 1 = ω 2 = 1.93. Energy densities (right) for the previous solutions.
(a) and (b) the profiles of the functions f i (x) are respectively depicted for the two types of Q-balls for the specific choice of the model parameters σ = 0.25, a = 1.75 and b = 2.0. We consider these values as representative of the model. The behavior of the solutions is completely similar for other values of the model parameters. For our choice the internal rotational frequencies are approximately restricted to the values 1.28657 ≤ ω 1 ≤ 2.0 and 1.28657 ≤ ω 2 ≤ 1.95256. In
Figure 3 :
3Graphics of the conserved Noether charges Q i (left) and the total energies E (right) of the two types of single Q-balls as a function of ω for the parameter values σ = 0.25, a = 1.75 and b = 2.0.
Figure 4 :
4Graphics of the profiles f 1 (blue curve) and f 2 (orange curve) of the composite B 2 (x; ω, γ 1 )balls for several values of the internal rotational frequency ω and the family parameter γ 1 . The model parameters have been set as σ = 0.25, a = 1.75 and b = 2.0.
Figure 5 :
5Graphics of the B 2 (x; ω, γ 1 )-orbits (purple curve) for several values of the internal rotational frequency ω and the family parameter γ 1 . The model parameters have been set as σ = 0.25, a = 1.75 and b = 2.0. A contour plot for the effective potential density U (f 1 , f 2 ) has been used in the previous graphics.
Figure 6 :
6Graphics of the energy density of the B 2 (x; ω, γ 1 ) solutions for several values of the internal rotational frequency ω and the family parameter γ 1 . The model parameters have been set as σ = 0.25, a = 1.75 and b = 2.0.
is equal to zero. This means that the fluctuation operator (58) is diagonal and the longitudinal and orthogonal perturbations δf 1 and δf 2 to the B 1 (x; ω) solution are decoupled and evolve independently. The behavior of the longitudinal fluctuations are characterized by the eigenfunctions ξ
(left) the eigenvalues λ i of the operator H 11 [B 1 (x, ω)] (numerically identified for the representative model parameters σ = 0.25, a = 1.75, b = 2.0) have been depicted as a function of the frequency ω 1 . Note the presence of only one negative eigenvalue λ − , which tends to zero as the rotational frequency ω 1 approaches to the values ω −1
Figure 7 :
7Spectrum of the operators H 11 (left) and H 22 (right) for the parameter values σ = 0.25, a = 1.75, b = 2.0 as a function of the internal rotation frequency ω 1 . stable are: (1) the operator H 11 [B 1 (x; ω)] must have only one negative eigenvalue and (2) the derivative of the Noether charge Q 1 with respect to the internal rotational frequency ω 1 must comply with the 1 < 0. It has been shown that these two conditions are verified in this case. This implies that the B 1 (x)-type Q-balls are stable with respect to longitudinal fluctuations. On the other hand, the orthogonal eigenfluctuations ξ (i) 2 are governed by the spectral problem
Figure 8 :
8Spectrum of the operator H[B 2 (x)] for the parameter values σ = 0.25, a = 1.75, b = 2.0 and ω = 1.70 as a function of the family parameter γ 1 .
AcknowledgmentsThis research was funded by the Spanish Ministerio de Ciencia e Innovación (MCIN) with funding from the European Union NextGenerationEU (PRTRC17.I1) and the Consejería de Educación, Junta de Castilla y León, through QCAYLE project, as well as MCIN project PID2020-113406GB-I00 MTM.
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| []
|
[
"Surface Passivation Suppresses Local Ion Motion in Halide Perovskites",
"Surface Passivation Suppresses Local Ion Motion in Halide Perovskites"
]
| [
"Justin Pothoof \nDepartment of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States\n",
"Robert J E Westbrook \nDepartment of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States\n",
"Rajiv Giridharagopal \nDepartment of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States\n",
"Madeleine D Breshears \nDepartment of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States\n",
"David S Ginger \nDepartment of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States\n"
]
| [
"Department of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States",
"Department of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States",
"Department of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States",
"Department of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States",
"Department of Chemistry\nUniversity of Washington\n98195-1700Seattle, WashingtonUnited States"
]
| []
| We use scanning probe microscopy to study ion migration in the formamidinium (FA)-containing halide perovskite semiconductor Cs0.22FA0.78Pb(I0.85Br0.15)3 in the presence and absence of chemical surface passivation. We measure the evolving contact potential difference (CPD) using scanning Kelvin probe microscopy (SKPM) following voltage poling. We find that ion migration leads to a ~100 mV shift in the CPD of control films after poling with 3V for only a few seconds. Moreover, we find that ion migration is heterogeneous, with domain interfaces leading to a larger shift in the CPD. Application of (3-aminopropyl)trimethoxysilane (APTMS) as a surface passivator further leads to 5-fold reduction in the CPD shift from ~100 meV to ~20 meV. We use hyperspectral microscopy to show that APTMS-treated perovskite films undergo less photoinduced halide migration than control films. We interpret these results as due to a reduction in halide vacancy concentration due to passivation with APTMS.TOC Graphic | null | [
"https://export.arxiv.org/pdf/2304.05546v1.pdf"
]
| 258,078,922 | 2304.05546 | c93066e5ae92b0f3cff0c00cdf34b07cdc5b7f4c |
Surface Passivation Suppresses Local Ion Motion in Halide Perovskites
Justin Pothoof
Department of Chemistry
University of Washington
98195-1700Seattle, WashingtonUnited States
Robert J E Westbrook
Department of Chemistry
University of Washington
98195-1700Seattle, WashingtonUnited States
Rajiv Giridharagopal
Department of Chemistry
University of Washington
98195-1700Seattle, WashingtonUnited States
Madeleine D Breshears
Department of Chemistry
University of Washington
98195-1700Seattle, WashingtonUnited States
David S Ginger
Department of Chemistry
University of Washington
98195-1700Seattle, WashingtonUnited States
Surface Passivation Suppresses Local Ion Motion in Halide Perovskites
We use scanning probe microscopy to study ion migration in the formamidinium (FA)-containing halide perovskite semiconductor Cs0.22FA0.78Pb(I0.85Br0.15)3 in the presence and absence of chemical surface passivation. We measure the evolving contact potential difference (CPD) using scanning Kelvin probe microscopy (SKPM) following voltage poling. We find that ion migration leads to a ~100 mV shift in the CPD of control films after poling with 3V for only a few seconds. Moreover, we find that ion migration is heterogeneous, with domain interfaces leading to a larger shift in the CPD. Application of (3-aminopropyl)trimethoxysilane (APTMS) as a surface passivator further leads to 5-fold reduction in the CPD shift from ~100 meV to ~20 meV. We use hyperspectral microscopy to show that APTMS-treated perovskite films undergo less photoinduced halide migration than control films. We interpret these results as due to a reduction in halide vacancy concentration due to passivation with APTMS.TOC Graphic
Halide perovskite semiconductors are important materials for a range of optoelectronic applications such as photovoltaics and light-emitting devices, 1,2 and power conversion efficiencies for single-junction perovskite solar cells have increased to over 25.7%. 3 One challenge facing some applications is that perovskites are prone to undergoing ion migration, in which ions move through the crystal lattice. 4,5 Ion migration can lead to undesired interactions between the perovskite active layer and transport layers or electrodes, reduce operational stability, and lead to segregation of the perovskite into separate phases. [6][7][8][9][10][11] Due to a low energetic barrier, vacancymediated halide migration has been proposed to be the dominant ion migration process in the perovskite lattice. 5,12 Many studies have been performed using chemical passivation strategies as a method of reducing surface halide vacancies, which has resulted in increased photoluminescence quantum yields and ultimately device performance. [13][14][15][16][17][18][19][20] However, there are far fewer studies that examine how surface passivation affects ion migration by means other than hysteresis reduction, [21][22][23][24] especially at the local level.
Previous work from our group has shown that (3-aminopropyl)trimethoxysilane (APTMS) significantly reduces nonradiative recombination in halide perovskite semiconductors. 25,26 Since halide migration often involves halide vacancies, 5,12 the same sites that are often targeted by chemical surface passivation, 27,28 we hypothesize that surface passivation with molecules such as APTMS should also suppress halide migration.
Here, we examine this hypothesis, with a particular emphasis on investigating how APTMS surface passivation can affect ion migration locally, via scanning Kelvin probe microscopy (SKPM). We combine SKPM measurements on locally-poled perovskite samples with studies of photoluminescence using hyperspectral optical microscopy. The use of SKPM allows us to probe ion motion and effects of APTMS surface passivation below the optical diffraction limit of conventional photoluminescence measurements. [29][30][31] We focus our study on wide-gap (~1.7 eV), mixed-halide perovskites because such formulations are particularly relevant for perovskite-on-Si tandem photovoltaics and because ion migration often causes halide phase segregation in these compositions. We find that the contact potential difference (CPD) of the perovskite samples evolves with applied electric fields. We quantify the average shift in CPD for perovskite control films to be near ~100 mV at poling extremes of 3 V with a poling dwell time of only a few seconds, which is reduced to ~20 mV after surface passivation with APTMS. We attribute this reduction in CPD shift to the passivation of surface halide defects. Using photoluminescence hyperspectral imaging, we also observe a reduction in photoinduced halide segregation in the perovskite films after surface passivation with APTMS.
For this study, we prepared mixed-halide perovskite semiconductor films of the composition Cs0.22FA0.78Pb(I0.85Br0.15)3 on ITO substrates by using a one-step spin-coating technique as adapted from the literature 32 (see SI for full details). We refer to these as-grown films as "control" or "unpassivated" perovskites. To prepare passivated perovskite films, we exposed the films to APTMS for five minutes at room temperature in a low-vacuum chamber as previously described. 25,26 We verified the perovskite composition and structure using XRD, and the bandgap using UV-Vis absorption spectroscopy ( Figure S1). Figure 1 shows that APTMS passivation lengthens the photoluminescence (PL) lifetimes and increases the PL quantum yield of APTMStreated samples, as is consistent with a reduction in surface trap states. In order to probe ion migration in these films at the local level, we combine local electric-field poling with scanning Kelvin probe microscopy (SKPM) to measure time-dependent evolution of the surface potential following application of poling fields of both positive and negative bias. Figure 2, shows the general experimental approach, which is similar to the method Yun et al. and Richheimer et al. have used to study ion migration in unpassivated MA-based perovskites. . 33,34 First, we perform a single pass with the cantilever to measure the topography across a single line. Next, we lift the cantilever 10 nm above the sample, apply a potential to the tip, and perform a second pass across the same line. During this pass the poling bias causes mobile charges to move towards or away from the surface, depending upon the polarity of the poling bias. Finally, we remove the poling bias, engage the Kelvin probe at the same lift height, and we measure the contact potential difference (CPD) between the tip and sample after poling. We repeat this process for every line in the image. After measuring the sample at a range of difference tip voltages, we generate a stack of CPD images in the same region. This process ensures that there is no charge injected into the film that could complicate the measurement or cause electrochemical interactions, 35 and probes the samples in the dark.
Figure 2.
Schematic of poling-based SKPM measurement and pathways of data analysis. We measure the topography, followed by a poling step, and finally initiate a Kelvin probe loop to measure the CPD at each line in the image. The data is processed to look at the CPD shift as a function of the applied bias to the tip, the distribution in CPD to analyze heterogeneity, and differences that occur at domain interfaces relative to the domain centers.
We use this poling-based SKPM measurement to probe a Cs0.22FA0.78Pb(I0.85Br0.15)3 control film on ITO under biases ranging from -3 V to +3 V with steps of 1 V. Figures 3a-d shows the topography and the evolution of the CPD at applied biases of 0, +3, and -3 V. We observe a shift to more positive CPD values when applying a positive bias to the tip, which we attribute to the build up of negative charges at the surface that is consistent with accumulation of negative ionic surface charge at the film surface resulting in a vacuum level offset. We observe the opposite effect when a negative bias is applied to the tipa large shift to negative CPD values is seen as positive charges accumulate at the surface, resulting in a vacuum-level offset of the opposite sign, which we illustrat in Figure S2. Figure 3d, measured at a tip bias of -3 V, shows a significant degree of heterogeneity in the CPD as the applied tip bias becomes more negative and net positive charge accumulates at the perovskite surface (presumably due to driving negative ions away).
Figures 3e-h shows the topography and CPD evolution of an APTMS-passivated perovskite film on ITO which reveals two key differences between the passivated and unpassivated samples. First, the surface-passivated samples show lower overall shifts in their CPD following local poling. Second, the passivated perovskite films have much more homogenous CPD distributions, both before, and after poling. Figure 3i shows the distribution of the CPD shift, which is determined by the difference between the CPD after poling and the CPD measured at 0 V. Figures j-k show the average CPD shift and full-width half max (FWHM) of the CPD shift relative to the poling bias. We see that the average CPD shift measured at the bias extremes decreases from around 100 mV to only ~20 mV. Importantly, this observation that APTMS-based passivation reduces the CPD shift induced during poling is consistent with the hypothesized suppression of ionic conductivity due to the reduction of halide vacancies. Accordingly, the FWHM of the measured CPD should reflect the extent of heterogeneity in local ion migration in a given film. Figure 3k shows that upon applying more intense negative biases, we observe a broadening of the FWHM in the unpassivated film, while the APTMS-passivated film exhibits a consistent, narrow FWHM of only ~10 mV. We propose that this difference is due to the greater heterogeneity in the control film, likely due to differences in ion mobility between the perovskite domains and the domain interfaces. In Figure S7, we show a full picture of the average CPD shift and heterogeneity as a function of poling bias for both perovskite formulations. Overall, these results show that APTMS surface passivation reduces ion migration in these wide-gap perovskite formulations. We recognize based on previous literature that the topographic features observed are not necessarily grain boundaries, as a single crystallite can contain multiple domains, 36,37 but for the purposes of convenience here we refer to the crystallites as "domains" and the spaces between them as "interfaces". In order to separate the differences between the perovskite domains and interfaces, we compare the mean CPDs at the domains and interfaces using several images collected at various biases of the Cs0.22FA0.78Pb(I0.85Br0.15)3 film. To achieve this separation, we align each CPD image based on its associated topography image and masks were manually selected to distinguish individual topographic domains. Figures 4a-b show the topography of the unpassivated perovskite film, and the mask used to separate the domains and interfaces. Using this methodology, we aggregate the CPD as a function of distance to the nearest GB pixel for control and APTMS-passivated films. Figure 4c shows the average CPD as a function of distance to the nearest domain interface for control and APTMS-passivated films measured with a -3 V bias ( Figure S6 shows this calculation for all poling biases). We observe a large difference in the average CPD measured at or near domain interfaces compared to regions further awayat domain centers. In films that are surface passivated with APTMS, we see that the CPD becomes both more homogenous relative to its distance from the nearest interface, and uniform across all distances. Figure 4d shows the difference between the average CPD measured at domain centers and interfaces relative to the poling bias for unpassivated and APTMS-passivated films. We observe a negative linear trend in the CPD difference with poling biases increasing from -3 to +3 V for the unpassivated film. In contrast, we see the CPD difference remains relatively unchanged with varying biases for the passivated film. Figure S8 visualizes the difference between the domain centers and interfaces, in which we see a contrast inversion in the CPD for the unpassivated film. The larger CPD shifts at the visible interfaces are consistent with a range of literature reports suggesting increased ion motion near surface interfaces and domain interfaces. 33,35,38,39 Importantly, these new SKPM results also show that APTMS-based passivation preferentially treats domain interface-related defects, leading to significantly more homogeneous films in terms of their response to bias-induced ion motion. Finally, we use hyperspectral photoluminescence microscopy under laser light bias to further explore ion migration in the control and APTMS-treated films. For this measurement, we kept the perovskite samples in a dry-nitrogen environment and excited them with a 532 nm laser at 600 mW/cm 2 . Although moderately higher than 1 Sun (100 mW/cm 2 ), we selected this illumination intensity to accelerate the effects of ion migration within a reasonable time frame. We provide further details of the measurement parameters in the Supporting Information. Figures 5a-b show the hyperspectral photoluminescence maps of the unpassivated perovskite sample before and after light-soaking under laser light bias. Figure 5c shows the normalized photoluminescence spectra measured during the light-soaking process. Based on the timedependent evolution of the photoluminescence, we observe a shift in the PL spectra and spatial distribution of features, which we interpret as initial phase segregation with contributions from both iodide-rich (peak emission at 780 nm) and mixed (peak emission at 740 nm) phases. This phase segregation can be visualized in the overall spectra as a shoulder peak. These regions have been attributed to an inhomogeneous elemental distribution that forms during the crystallization process. [40][41][42] We see that the iodide-rich regions grow significantly in size after 30 minutes of light soaking. Figures 5d-e show the hyperspectral mapping and overall photoluminescence spectra for APTMS-passivated films undergoing light-soaking. Similar to the control film, we see initial halide segregation. In contrast to the control, we see that the growth of the iodide-rich regions is hindered by the APTMS surface passivation. In both films, we observe a slight red-shifting of the main mixed-phase emission peak, which may be attributed to demixing of the A-site cations as observed by Knight,et al. 43 Figure S9 shows the cumulative photoluminescence mapping for unpassivated and APTMS-passivated films, in which we see a consistently higher photoluminescence intensity for the surface passivated films.
We apply a wavelength threshold of 765 nm to separate the mixed and iodide-rich phases by binning the pixels based on their emission wavelength ( Figure S10a), with the aim of further understanding how the mixed and iodide-rich phases evolve over the course of light-soaking. Figure S10b shows the shift in emission wavelength relative to light-soaking duration for the mixed and iodide-rich phases in control and passivation films. We can see the conversion from mixed-phase perovskite to iodide-rich perovskite during light-soaking, where regions emitting at wavelengths shorter than 765 nm, prior to prolonged light exposure, shift to longer wavelengths over time. This effect is more pronounced in the control film as compared to the APTMSpassivated film.
In Figure S10c, we determined the extent of phase segregation by calculating the fraction of iodiderich pixels relative to mixed-phase pixels as a function of light-soaking time compared with the initial amount of phase-segregation prior to light exposure. While both the control and APTMSpassivated films show consistent red-shifting with light-soaking, the extent of phase-segregation increases to 7X its initial state after prolonged light-soaking in the control film as compared to an increase of 2X in the APTMS-passivated film. Overall, we see that APTMS passivation on these wide-gap perovskites treats defects like halide vacancies, which results increased photoluminescence quantum yields and reduce halide segregation. These photoluminescence observations are in agreement with and complement the results obtained with SKPM. (c,f) Histograms of the wavelength emission counts for each of the images in (a-d). Ensemble spectra for control (blue) and APTMS passivated (red) films calculated from hyperspectral maps. Samples were excited with a 532 nm laser with a fluence of ~600 mW/cm 2 .
Studying the evolution of the surface potential with SKPM following poling reveals insight into how APTMS passivation affects ion motion in halide perovskites. Notably, we observe a significant reduction in the concentration of charges that drift from poling after surface passivation with APTMS. We examined differences in the CPD of domain centers and domain interfaces as a function of poling bias and polarity, and we observed that domain interfaces exhibit a higher amount of ion migration compared to domain centers. This difference is suppressed following passivation with APTMS. To further study the role of APTMS on ion migration, we used hyperspectral photoluminescence microscopy to explore time-dependent halide segregation during illumination of these materials. We found that halide segregation, as measured by the PL red-shift, is significantly more pronounced in unpassivated films. Taken together, these data indicate that defect passivation reduces ion migration, both in terms of domain-to-domain variations in ion migration rate as well as the overall ion migration magnitude, and that these effects are correlated with lower amounts of photoinduced halide segregation. These results highlight the importance of developing new methods to measure ion migration and provide a simple method for screening new passivating agents via AFM for beneficial ion migration properties.
Figure 1 .
1(a) Time-resolved PL measurements for Cs0.22FA0.78Pb(I0.85Br0.15)3 control and APTMSpassivated films deposited on glass substrates. Stretched exponential fits for the decay curves are shown in black. We calculated average lifetimes of 120.78 and 1015.60 ns for control and APTMSpassivated films, respectively. (b) Steady-state PL spectra for perovskite control and APTMSpassivated films deposited on glass substrates.
Figure 3 .
3(a) Topography and (b-d) CPD of Cs0.22FA0.78Pb(I0.85Br0.15)3 control film measured with applied tip biases of 0, +3, and -3 V. (e) Topography and (f-h) CPD of APTMS passivated film measured with applied tip biases of 0, +3, and -3 V. All CPD images are shown on the same color scale to show the differences between the two samples. (i) Probability density distribution of the shift in CPD relative to the baseline CPD for control and APTMS passivated films measured at -3 V. (j) Shift in average CPD bias relative to the baseline CPD and (k) FWHM as a function of applied tip for control and APTMS passivated films. Error bars are shown as standard error of the mean for three different measurements performed on three different films each from different batches.
Figure 4 .
4(a) Topography of a perovskite control film. (b) Binary mask created to separate perovskite domains (red) from domain interfaces (blue). (c) Average CPD as a function of distance from the nearest domain interface for control and APTMS-passivated films. Error bars indicate the standard deviation of the CPD probed at varying distances from nearest domain interface. (d) Difference in the CPD measured at domain centers (DC) and interfaces (DI) for control and APTMS-passivated films.
Figure 5 .
5Hyperspectral microscopy emission images collected at light-soaking times of (a, d) 0 s and (b, e) 1865 s for (a, b) Cs0.22FA0.78Pb(I0.85Br0.15)3 control and (d,e) APTMS passivated films.
AcknowledgementsThis letter is based on work supported primarily by the U.S. Department of Energy (DOE-SC0013957). Part of this work was conducted with instruments supported by the University of Washington Student Technology Fee at the Molecular Analysis Facility, a National Nanotechnology Coordinated Infrastructure site at the University of Washington, which is a user facility supported in part by the National Science Foundation(Grant ECC-1542102
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]
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"Filippo Dell'oro ",
"Lorenzo Liverani ",
"Vittorino Pata "
]
| []
| []
| The exponential decay rate of the semigroup S(t) = e tA generated by the abstract damped wave equationü + 2f (A)u + Au = 0 is here addressed, where A is a strictly positive operator. The continuous function f , defined on the spectrum of A, is subject to the constraintswhich are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate S(t) ≤ Ce σ * t being σ * < 0 the supremum of the real part of the spectrum of A. This estimate always holds except in the resonant cases, where the negative exponential e σ * t turns out to be penalized by a factor (1 + t). The decay rate is the best possible allowed by the theory.2010 Mathematics Subject Classification. 35B35, 35P05, 47D06. Key words and phrases. Abstract damped wave equations, semigroups, exponential stability, decay rate, resonance. | null | [
"https://export.arxiv.org/pdf/2304.05816v1.pdf"
]
| 258,079,390 | 2304.05816 | 9ebc32517b0f87000b31a5899e2478412aeb824b |
ABSTRACT DAMPED WAVE EQUATIONS: THE OPTIMAL DECAY RATE
Filippo Dell'oro
Lorenzo Liverani
Vittorino Pata
ABSTRACT DAMPED WAVE EQUATIONS: THE OPTIMAL DECAY RATE
The exponential decay rate of the semigroup S(t) = e tA generated by the abstract damped wave equationü + 2f (A)u + Au = 0 is here addressed, where A is a strictly positive operator. The continuous function f , defined on the spectrum of A, is subject to the constraintswhich are known to be necessary and sufficient for exponential stability to occur. We prove that the operator norm of the semigroup fulfills the estimate S(t) ≤ Ce σ * t being σ * < 0 the supremum of the real part of the spectrum of A. This estimate always holds except in the resonant cases, where the negative exponential e σ * t turns out to be penalized by a factor (1 + t). The decay rate is the best possible allowed by the theory.2010 Mathematics Subject Classification. 35B35, 35P05, 47D06. Key words and phrases. Abstract damped wave equations, semigroups, exponential stability, decay rate, resonance.
The linear operator A is known to be the infinitesimal generator of a C 0 -semigroup S(t) = e tA : H → H of linear contractions (see, e.g., [8]). Hence, for any given u 0 = (u 0 , v 0 ) ∈ H, the unique mild solution u(t) to the equation above, subject to the initial condition u(0) = u 0 , is given by u(t) = (u(t),u(t)) = S(t)u 0 . We denote (twice) the energy of such a solution by
E(t) = u(t) 2 H = A 1 2 u(t) 2 + u(t) 2 .
1.2. Exponential decay of the semigroup. As far as the sole existence of S(t) is concerned, we could have taken any continuous function f : σ(A) → [0, ∞). However, as shown for instance in [8], the exponential stability of the semigroup occurs if and only if (1.1) is in place. Recall that S(t) is said to be exponentially stable if there exist ω > 0 and C ≥ 1 such that
(1.3) S(t) L(H) ≤ Ce −ωt ,
where the norm is taken in the Banach space L(H) of bounded linear operators on H. Once the exponential stability is known, one would like to find the best possible ω > 0 (i.e., the largest) for which (1.3) holds. This number is the (exponential) decay rate, defined as 1 ω * = sup ω > 0 : S(t) L(H) ≤ Ce −ωt , for some C = C(ω) ≥ 1. However, for general (exponentially stable) semigroups, computing ω * might not be an easy task. Another relevant quantity, usually much easier to detect, is the spectral bound of A
σ * = sup λ∈σ(A)
Re λ, 1 The number −ω * is usually called the growth bound of the semigroup. σ(A) being the spectrum of (the complexification of) A, which is related to ω * through the (possibly strict) inequality ω * ≤ −σ * (see [9,19]). Definition 1.1. The semigroup S(t) satisfies the spectrum determined growth (SDG) condition if ω * = −σ * .
In view of finding the decay rate, it is then of paramount importance to establish whether or not the SDG condition is satisfied. This is true, for instance, for eventually norm continuous (such as analytic or differentiable) semigroups [9]. Still, even if S(t) fulfills the SDG condition, this does not mean at all that (1.3) holds for ω = ω * , as the following simple example shows. Example 1.2. Consider the damped pendulum equation u + 2au + u = 0, a particular instance of (1.2) for H = R 2 , A = 1 and f ≡ a > 0. The related semigroup, being norm continuous, always fulfills the SDG condition. In particular, the decay rate is attained whenever a = 1. But, when a = 1, the norm of the solution corresponding to the initial datum u 0 = (u 0 , v 0 ) reads
S(t)u 0 H = u 2 0 + v 2 0 + 2(u 2 0 − v 2 0 )t + 2(u 0 + v 0 ) 2 t 2 e − t 2 .
The decay rate, equal to 1 2 in this case, is certainly not attained, and the reason is that a resonance phenomenon is encountered.
This motivates the next definition. Definition 1.3. The semigroup S(t) satisfies the strong spectrum determined growth (SSDG) condition if the decay rate ω * = −σ * is attained, that is, if (1.3) holds with ω = ω * = −σ * .
1.3. Earlier contributions. The equation (1.2) of this paper belongs to a general class of abstract differential models introduced in [2], and "exhibiting the empirically observed damping rates in elastic systems". For this reason, A is usually called the elastic operator, while f (A) is the damping operator. The prototypical case occurs when f (s) controls and is controlled by s θ for some θ ∈ [0, 1], namely, when f (A) is comparable with the powers A θ . Here, the semigroup is know to be analytic for θ ∈ [ 1 2 , 1] and differentiable (actually, of Gevrey type) for θ ∈ (0, 1 2 ). See, among others, the classical papers [3,4,5,13,14,15]. Further works, dealing with a damping operator not necessarily comparable with A θ , include [1,12,16,17,18]. Nevertheless, the literature concerning the analysis of the decay rate ω * of S(t), with particular reference to whether or not it is actually attained, is exceptionally bare. The full solution to the problem has been found only for the weakly damped wave equation, sometimes referred to as the telegrapher's equation, corresponding to the choice f (s) = a > 0 (see [6,10]). The much more general (and difficult) situation of a nonconstant f has been tackled in [11]. There, within a certain number of assumptions, which prevent f (s) to grow at infinity faster than s θ with θ < 1 2 , the authors obtain sharp exponential decay estimates for trajectories originating from sufficiently regular initial data, but they cannot generally exhibit the decay rate of the semigroup. On the other hand, the necessary and sufficient condition (1.1) for the exponential stability of S(t) clearly allows f (s) to have up to a linear growth at infinity. And indeed, the special case f (s) = s corresponds to the widely studied strongly damped wave equation, also known by the name of Kelvin-Voigt equation, which appears in several areas of Mathematical Physics.
1.4. Our result. The aim of this paper is to provide a complete answer. Within the sole condition (1.1), we show that the semigroup S(t) fulfills (1.3) with ω = ω * = −σ * , except in some particular cases, called resonant, where the term e −ω * t is penalized by a factor (1 + t). This result is optimal. The key idea of the proof is to decompose σ(A) into the disjoint union of suitably chosen subsets σ ı , and then perform energy estimates with ad hoc multipliers on each of the (orthogonal) subspaces E A (σ ı )H, making use of the functional calculus of A.
The paper is organized as follows. In the next Section 2, we introduce two objects that will play a crucial role in our analysis. In Section 3, we recall the properties of the spectrum of the operator A. The main result is stated and proved in Section 4, and an application is then discussed in Section 5. The final Sections 6-9 are devoted to the proofs of four lemmas encountered in the proof of the main theorem.
A word of warning. Although we work with a real Hilbert space H, having in mind the concrete examples of Mathematical Physics, all the results of this paper hold verbatim if H is a complex Hilbert space. The proofs are exactly the same, up to replacing in the calculations any occurrence of a scalar product with its real part.
The Function φ and the Number m *
A crucial object for our analysis is the continuous function
φ : σ(A) → (0, ∞) defined as φ(s) = f (s) if f (s) ≤ √ s , f (s) − f 2 (s) − s if f (s) > √ s , along with the number m * = inf s∈σ(A) φ(s) > 0.
The fact that m * > 0 merely follows by observing that
f (s) = φ(s) if f (s) ≤ √ s , s 2φ(s) + φ(s) 2 if f (s) > √ s .
Hence, m * = 0 would violate (1.1). Then, in the region f (s) > √ s , we have that
(2.1) f (s) s ≤ 1 2m * + m * 2s ,
this fact being equivalent to φ(s) ≥ m * . Finally, for every m > 0 and every sequence s n → ∞, we have the implication
(2.2) f (s n ) s n → 1 2m ⇒ φ(s n ) → m.
Indeed, the first convergence implies in particular that
s n f 2 (s n ) → 0 and f (s n ) > √ s n ,
the latter for all n large. Accordingly,
φ(s n ) = f (s n ) 1 − 1 − s n f 2 (s n ) ∼ s n 2f (s n ) → m.
The Spectrum of the Infinitesimal Generator
For every fixed s ∈ σ(A), we introduce the pair of complex numbers
λ ± s = −f (s) ± i s − f 2 (s) if f (s) ≤ √ s , −f (s) ± f 2 (s) − s if f (s) > √ s ,
which are nothing but the solutions to the second order equation
λ 2 + 2f (s)λ + s = 0. We define Σ = s∈σ(A) λ ± s .
We also consider the (possibly empty) set
Λ = λ < 0 : ∃ s n ∈ σ(A) such that s n → ∞ and lim n→∞ f (s n ) s n = − 1 2λ .
Then, as shown in [8], the spectrum of (the complexification of) the operator A reads
σ(A) = Σ ∪ Λ,
where the union is not necessarily disjoint. In light of (2.2), it is apparent that Λ belongs to the closure of Σ. This yields the equality
(3.1) σ * = sup λ∈σ(A) Re λ = sup λ∈Σ Re λ = − inf s∈σ(A) φ(s) = −m * .
Remark 3.1. Concerning the structure of the spectrum of A, we have that
σ(A) = σ p (A) ∪ σ c (A), where σ p (A) = s∈σp(A) λ ± s .
With standard notation, σ p and σ c are the point spectrum and the continuous spectrum, respectively. This characterization, although not explicitly stated in [8], can be inferred from the calculations of that paper, by arguing as in [7,Sec. 5].
The Theorem
4.1. Statement of the result. We begin with the rigorous definition of resonance.
Definition 4.1. The semigroup S(t) is said to be resonant if there exists s * ∈ σ(A) such that m * = φ(s * ) and f (s * ) = √ s * .
In other words, the resonance phenomenon appears whenever the real part of the spectrum of A attains its supremum −m * and the equality
−m * = λ + s * = λ − s *
holds for some s * ∈ σ(A). In fact, if such a point s * exists, it is easily seen to be unique.
Theorem 4.2.
There exists a constant C ≥ 1 such that
I. S(t) L(H) ≤ Ce −m * t , if S(t) not resonant; and II. S(t) L(H) ≤ C(1 + t)e −m * t , if S(t) resonant.
On account of (3.1), the theorem produces a straightforward consequence. It is sometimes possible to provide a precise estimate of the constant C.
Proposition 4.5. Assume that the semigroup is subdamped, that is,
sup s∈σ(A) f (s) √ s = < 1.
Then point I of Theorem 4.2 holds with
C = 1 + 1 − .
Also in this case, the result is optimal, as the following example shows.
S(t)u 0 H = 1 − a cos 2 √ 1 − a 2 t 1 − a e − a 2 t .
Choosing
t = t k = (2k + 1)π 2 √ 1 − a 2 ,
for any k ∈ N, we get exactly
S(t k ) L(H) ≥ S(t k )u 0 H = 1 + a 1 − a e − a 2 t k .
4.2. Some preparatory lemmas. For any given K ≥ 2 and ε ∈ (0, 1), we decompose the spectrum of A into the disjoint union
σ(A) = σ 0 ∪ σ 1 ∪ σ 2 ∪ σ 3 ,
where the four regions σ ı (some of which possibly empty) are defined as
σ 0 = s ∈ σ(A) : f (s) √ s > K , σ 1 = s ∈ σ(A) : f (s) √ s ≤ 1 − ε , σ 2 = s ∈ σ(A) : 1 + ε ≤ f (s) √ s ≤ K , σ 3 = s ∈ σ(A) : 1 − ε < f (s) √ s < 1 + ε .
The proofs of Theorem 4.2 and the subsequent Proposition 4.5 are based on the following four lemmas, which hold for all regular initial data, that is, for all (u 0 , v 0 ) in the domain of A. Given the trajectory
(u(t),u(t)) = S(t)(u 0 , v 0 ) ∈ dom(A),
we split the corresponding energy at time t ≥ 0 into the sum
E(t) = 3 ı=0 E ı (t), where E ı (t) = A 1 2 E A (σ ı )u 2 + E A (σ ı )u 2 . Note that E ı ≡ 0 if σ ı = ∅.
The constants K and ε in the statements of the lemmas are understood to be the ones appearing in the definitions of σ ı . Lemma 4.7. For every K ≥ 2 large enough we have the inequality
E 0 (t) ≤ 3E 0 (0)e −2m * t .
Lemma 4.8. For every ε ∈ (0, 1) we have the inequality
E 1 (t) ≤ 2 − ε ε E 1 (0)e −2m * t .
Lemma 4.9. For every ε ∈ (0, 1) and every K ≥ 2, we have the inequality
E 2 (t) ≤ 9K 2 ε E 2 (0)e −2m * t .
Lemma 4.10. For every ε ∈ (0, 1 16 ) such that σ 3 = ∅, we have the inequality
E 3 (t) ≤ 8 ε E 3 (0)e −2m 3 (1−4 √ ε )t ,
where m 3 = inf s∈σ 3 φ(s).
The rather technical proofs of the four lemmas, heavily based on the functional calculus of A, will be carried out in the final Sections 6, 7, 8 and 9 of the paper.
Proof of Theorem 4.2.
We consider the two cases separately.
I. Let us choose K ≥ 2 sufficiently large in order for Lemma 4.7 to hold. We claim that, up to fixing ε ∈ (0, 1 16 ) small enough,
E 3 (t) ≤ 8 ε E 3 (0)e −2m * t .
This is trivially true if σ 3 = ∅ for some ε. Let us then assume that σ 3 = ∅ for all ε ∈ (0, 1 16 ). We preliminarily observe that, for all ε small,
m 3 > m * .
If not, there would exist a sequence s n ∈ σ(A) such that
f (s n ) √ s n → 1 and φ(s n ) → m * = m 3 .
If s n is unbounded, then (up to a subsequence) s n → ∞. Therefore,
m * ∼ φ(s n ) ∼ f (s n ) ∼ √ s n → ∞,
a contradiction. This tells that s n must be bounded. But then (up to a subsequence) s n → s * ∈ σ(A), as σ(A) is closed. By continuity, we deduce that f (s * ) = √ s * and φ(s * ) = m * , against the assumption that S(t) is not resonant. At this point, the claim follows by fixing ε in Lemma 4.10 small enough that m 3 (1 − 4 √ ε ) ≥ m * (just note that m 3 is a decreasing function of ε). Once the claim is proven, setting M = 9K 2 ε , and taking advantage of Lemmas 4.7-4.9, we conclude that
E(t) = 3 ı=0 E ı (t) ≤ M E(0)e −2m * t .
Due to the continuity of the semigroup, the latter inequality remains valid by density for all initial data (u 0 , v 0 ) belonging to the phase space H. This finishes the proof.
II. In this case, we have the equality m 3 = m * for every ε ∈ (0, 1 16 ). Select then any K ≥ 2 sufficiently large in order for Lemma 4.7 to hold. From Lemmas 4.7-4.10 we learn that the inequality
E(t) ≤ 9K 2 ε E(0)e −2m * (1−4 √ ε )t
holds for every ε ∈ (0, 1 16 ). Fixing an arbitrary ε * ∈ (0, 1 16 ), we choose for every t ≥ 0 ε = ε(t) = ε * (1 + t) 2 . Then, calling
M = 9K 2 ε * e 8m * √ ε * ,
we finally obtain E(t) ≤ M (1 + t) 2 E(0)e −2m * t . As before, the sought inequality is valid by density for all initial data.
Proof of Proposition 4.5.
If the semigroup is subdamped, then σ(A) = σ 1 with ε = 1 − . Accordingly, the energy E reduces to E 1 , and point I of Theorem 4.2 is nothing but Lemma 4.8.
An Application: Wave Equations with Fractional Damping
For a > 0 and θ ∈ [0, 1], we consider equation (1.2) with f (s) = as θ , namely,
u + 2aA θu + Au = 0.
The function f clearly complies with (1.1), so that Theorem 4.2 and Proposition 4.5 apply. We denote by s 0 > 0 the minimum of the spectrum of A.
as θ − √ a 2 s 2θ − s if s < a 2 1−2θ , if θ < 1 2 , and φ(s) = as θ if s ≤ a 2 1−2θ , as θ − √ a 2 s 2θ − s if s > a 2 1−2θ , if θ > 1 2 , whereas φ(s) = a √ s if a ≤ 1, a − √ a 2 − 1 √ s if a > 1, if θ = 1 2 .
Note that φ(s) ≤ √ s . Besides, for θ = 1 2 , equality occurs only for s = a In general, the picture strongly depends on the exponent θ. To this end, we first observe that
f (s) √ s = as θ− 1 2
is decreasing for θ < 1 2 , constant and equal to a for θ = 1 2 , increasing and diverging to infinity for θ > 1 2 . In particular, this tells that, if θ > 1 2 and A is an unbounded operator, the semigroup S(t) is never subdamped.
θ = 0 θ ∈ (0, 1 2 ) θ ∈ ( 1 2 , 1) θ = 1C = 1 + a 1 − a .
Note that C is now independent of s 0 .
• Case θ ∈ ( 1 2 , 1). This is the most intriguing situation, since: φ is increasing for s < a • Case θ = 1. We have the equality φ(s) = as for s < 1 a 2 , where φ, thought again as defined on (0, ∞), reaches its maximum value, equal to 1 a . After that, φ is decreasing, and converges to 1 2a as s → ∞. Then, if A is unbounded,
m * = min as 0 , 1 2a ,
whereas if A is bounded, m * = min as 0 , φ(s M ) , where s M is the maximum of σ(A). Quite interestingly, when θ = 1 resonance cannot occur, except in the trivial situation where σ(A) = { 1 a 2 }. 6. Proof of Lemma 4.7
We choose K sufficiently large in order to have the set inclusion
(6.1) σ 0 ⊂ (K, ∞).
Indeed, recalling the second constraint in (1.1), if s ≤ K with K large then
f (s) √ s ≤ sup s∈σ(A) f (s) s √ K ≤ K.
Defining the function
g(s) = f (s) − m * 2 ,
we rewrite equation (1.2) in the equivalent form u + 2g(A)u + m * u + Au = 0.
Note that, due to (2.1),
(6.2) s ≥ 2m * g(s), ∀s ∈ σ 0 .
Next, setting w = E A (σ 0 )u, we introduce the functional
F 0 = E 0 + 2m 2 * w 2 + 4m * w,ẇ .
In what follows, as well as in the proofs of the remaining three lemmas, all the calculations are rigorous, since we are working with regular solutions. In particular, we use the fact that the selfadjoint projection E A (σ 0 ) commutes with all the functions of A. Multiplying the equation by the test function 2ẇ + 4m * w, we are led to the differential identity d dt
F 0 + 2m * F 0 + 2G 0 = 0, where G 0 = m * A 1 2 w 2 − (2m * g) 1 2 (A)w 2 + 2 g 1 2 (A)(m * w +ẇ) 2 − m * m * w +ẇ 2 .
In light of (6.2), we see at once that
A 1 2 w 2 − (2m * g) 1 2 (A)w 2 ≥ 0.
Besides, from the very definition of g and σ 0 ,
g(s) ≥ K √ s − m * 2 , ∀s ∈ σ 0 .
Thus, we infer from (6.1) that g(s) ≥ m * provided that K is large enough. Accordingly,
g 1 2 (A)(m * w +ẇ) 2 − m * m * w +ẇ 2 ≥ 0. In summary, we have proved that G 0 ≥ 0. Therefore, d dt F 0 + 2m * F 0 ≤ 0,
and by the Gronwall Lemma we conclude that
F 0 (t) ≤ F 0 (0)e −2m * t .
The proof is finished once we show the double inequality
1 2 E 0 ≤ F 0 ≤ 3 2 E 0 .
Indeed, from the Cauchy-Schwarz and the Young inequalities, and exploiting (6.1),
2m 2 * w 2 + 4m * | w,ẇ | ≤ 10m 2 * w 2 + 1 2 ẇ 2 ≤ 10m 2 * K A 1 2 w 2 + 1 2 ẇ 2 ≤ 1 2 E 0 ,
as soon as K is sufficiently large.
Proof of Lemma 4.8
Setting w = E A (σ 1 )u, we introduce the functional
F 1 = E 1 + 2 f (A)w,ẇ .
Multiplying (1.2) by the test function 2ẇ + 2f (A)w, we get the differential identity
d dt F 1 + 2m * F 1 + 2G 1 = 0,
having set (recall that in this region s − f 2 (s) > 0)
G 1 = f 1 2 (A)(A − f 2 (A)) 1 2 w 2 − m * (A − f 2 (A)) 1 2 w 2 + f 1 2 (A)(f (A)w +ẇ) 2 − m * f (A)w +ẇ 2 . Since m * ≤ φ(s) = f (s)
, ∀s ∈ σ 1 , it is apparent that G 1 ≥ 0. Therefore, we arrive at
(7.1) d dt F 1 + 2m * F 1 ≤ 0.
Finally, we claim that
(7.2) εE 1 ≤ F 1 ≤ (2 − ε)E 1 .
Indeed, recalling that f (s) ≤ (1 − ε) √ s for every s ∈ σ 1 , from the Cauchy-Schwarz and the Young inequalities, we get
2| f (A)w,ẇ | ≤ (1 − ε) A 1 2 w 2 + (1 − ε) ẇ 2 .
Collecting (7.1)-(7.2), and exploiting the Gronwall Lemma, the conclusion follows.
Proof of Lemma 4.9
In this proof, as well as in the following one, the symbol ·, · will also be used to denote the duality product. Setting w = E A (σ 2 )u, we introduce the functional
F 2 = E 2 + 2 (f 2 (A) − A)w, w + 2 f (A)w,ẇ .
Multiplying (1.2) by the test function 2ẇ + 2f (A)w we get the differential identity d dt
F 2 + 2m * F 2 + 2G 2 = 0,
having set
G 2 = (Af (A) + m * A − 2m * f 2 (A))w, w + (f (A) − m * )ẇ,ẇ + 2 (A − m * f (A))w,ẇ .
We claim that G 2 ≥ 0. Indeed, since
m * ≤ φ(s) = f (s) − f 2 (s) − s < f (s), ∀s ∈ σ 2 ,
exploiting the Cauchy-Schwarz and the Young inequalities,
2 (A − m * f (A))w,ẇ = 2 (f (A) − m * ) − 1 2 (A − m * f (A))w, (f (A) − m * ) 1 2ẇ ≥ − (f (A) − m * ) −1 (A − m * f (A)) 2 w, w − (f (A) − m * )ẇ,ẇ .
Therefore, since
sf (s) + m * s − 2m * f 2 (s) − (s − m * f (s)) 2 f (s) − m * = f 2 (s) − s f (s) − m * (s + m 2 * − 2m * f (s)), we get G 2 ≥ (f (A) − m * ) −1 (f 2 (A) − A)(A + m 2 * − 2m * f (A))w, w .
The claim follows by noting that, for every s ∈ σ 2 ,
s + m 2 * − 2m * f (s) = f (s) − f 2 (s) − s − m * f (s) + f 2 (s) − s − m * ≥ 0.
Accordingly, we end up with the differential inequality
(8.1) d dt F 2 + 2m * F 2 ≤ 0.
Next, we show that
(8.2) ε 3 E 2 ≤ F 2 ≤ 3K 2 E 2 .
To this end, we further apply the Cauchy-Schwarz and the Young inequalities, to get
2| f (A)w,ẇ | ≤ 1 τ f (A)w 2 + τ ẇ 2 ,
for every τ > 0. Since f 2 (s) ≤ K 2 s for every s ∈ σ 2 , by setting τ = 1 we obtain
F 2 ≤ E 2 + 2(K 2 − 1) A 1 2 w 2 + K 2 A 1 2 w 2 + ẇ 2 ≤ 3K 2 E 2 .
As far as the other inequality is concerned, we have
F 2 ≥ (1 − τ )E 2 + ((2 − τ −1 )f 2 (A) + (τ − 2)A)w, w .
Recalling that f 2 (s) ≥ (1 + ε) 2 s for every s ∈ σ 2 , and selecting τ = 1 − ε 3 , we end up with
F 2 ≥ ε 3 E 2 + (ε) A 1 2 w 2 ,
where, for every ε ∈ (0, 1),
(ε) = (3 − 2ε)(1 + ε) 2 3 − ε − 1 − ε 3 = 2ε(6 − ε − 3ε 2 ) 3(3 − ε) > 0.
At this point, the conclusion is drawn from (8.1)-(8.2) and the Gronwall Lemma.
Proof of Lemma 4.10
Along the proof, besides the constraint ε ∈ (0, 1 16 ), we will use the fact that m 3 ≤ φ(s) ≤ f (s) < (1 + ε) √ s , ∀s ∈ σ 3 .
Setting w = E A (σ 3 )u, we introduce the functional
F 3 = E 3 + 2(1 + 4 √ ε ) f (A)w,ẇ + 8 √ ε (1 + 4 √ ε ) f (A)w 2 .
Multiplying (1.2) by the test function 2ẇ + 2(1 + 4 √ ε )f (A)w, we get the differential identity d dt
F 3 + 2m 3 (1 − 4 √ ε )F 3 + 2G 3 = 0.
Here,
G 3 = p(A)w, w + (1 − 4 √ ε ) (f (A) − m 3 )ẇ,ẇ + 2(1 − 16ε) f (A)(f (A) − m 3 )w,ẇ ,
having set
p(s) = (1 + 4 √ ε )sf (s) − m 3 (1 − 4 √ ε )s − 8m 3 √ ε (1 − 16ε)f 2 (s) ≥ (1 + 4 √ ε )sf (s) − m 3 (1 − 4 √ ε )s − 8m 3 √ ε (1 − 16ε)(1 + ε) 2 s, = (1 + 4 √ ε )s(f (s) − m 3 ) + 8m 3 √ ε 1 − (1 − 16ε)(1 + ε) 2 s ≥ (1 + 4 √ ε )s(f (s) − m 3 ).
Accordingly, By an application of the Cauchy-Schwarz and the Young inequalities, we find
G 3 ≥ (1 + 4 √ ε ) A(f (A) −m2(1 − 16ε) f (A)(f (A) − m 3 )w,ẇ = 2 (1 + 4 √ ε )(1 − 4 √ ε ) 1 2 f (A)(f (A) − m 3 ) 1 2 w, (1 − 4 √ ε ) 1 2 (f (A) − m 3 ) 1 2ẇ ≥ −(1 − 16ε)(1 + 4 √ ε ) f 2 (A)(f (A) − m 3 )w, w − (1 − 4 √ ε ) (f (A) − m 3 )ẇ,ẇ ,
implying in turn
G 3 ≥ (1 + 4 √ ε ) (f (A) − m 3 )(A − (1 − 16ε)f 2 (A))w, w .
Observing that
s − (1 − 16ε)f 2 (s) ≥ 1 − (1 − 16ε)(1 + ε) 2 s ≥ 0,
we conclude that G 3 ≥ 0, and we arrive at
d dt F 3 + 2m 3 (1 − 4 √ ε )F 3 ≤ 0.
After an application of the Gronwall Lemma, the proof is finished once we show that
εE 3 ≤ F 3 ≤ 8E 3 .
Indeed, the second inequality is a straightforward consequence of the Cauchy-Schwarz and the Young inequalities, and is left to the reader. Concerning the first one, we have
2(1 + 4 √ ε ) f (A)w,ẇ ≥ −(1 − ε) ẇ 2 − (1 + 4 √ ε ) 2 1 − ε f (A)w 2 .
Therefore,
F 3 ≥ εE 3 + (1 − ε) A 1 2 w 2 + 8 √ ε (1 + 4 √ ε ) − (1 + 4 √ ε ) 2 1 − ε f (A)w 2 ≥ εE 3 + 1 − ε (1 + ε) 2 + 8 √ ε (1 + 4 √ ε ) − (1 + 4 √ ε ) 2 1 − ε f (A)w 2 ,
and the quantity in the square brackets is positive for all ε ∈ (0, 1 16 ).
.
The equation. Let H be a (separable) real Hilbert space, endowed with the scalar product and norm ·, · , and · , respectively, and let A : dom(A) ⊂ H → H be a strictly positive selfadjoint linear operator. We define the product Hilbert space 2 + v 2 . Denoting by σ(A) ⊂ (0, ∞) the spectrum of A, let f : σ(A) → (0, ∞)
For t > 0 ,f
0we consider the abstract damped wave equation in the unknown u = u(t) (1.2)ü + 2f (A)u + Au = 0, the dot standing for derivative with respect to time, subject to the initial conditions u(0) = u 0 andu(0) = v 0 , where the vector (u 0 , v 0 ) ∈ H is arbitrarily assigned. Here, f (A) is the selfadjoint linear operator on H constructed via the functional calculus of A, namely, (s) dE A (s), being E A the spectral measure of A (see, e.g., [20]). Calling u = (u, v), and introducing the linear operator A(u, v) = (v, −Au − f (A)v), with domain dom(A) = u ∈ H : Au ∈ H , we can rewrite (1.2) as the first order ODE in Ḣ u = Au.
Corollary 4. 3 .
3If S(t) is not resonant, then it fulfills the SSDG condition. Instead, if S(t) is resonant, it fulfills the SDG condition, but not the SSDG one.
Remark 4. 4 .
4In view of Example 1.2, the result is optimal.
Example 4. 6 .
6Consider the subdamped pendulum equationu + 2au + u = 0, a ∈ (0, 1),which meets the hypotheses of the proposition above with = a. Taking the unit vector u 0 = ( , the norm of the solution at time t reads
Remark 5 . 1 .
51Given a bounded domain Ω ⊂ R N with smooth boundary ∂Ω, a concrete instance of this model is obtained by taking H = L 2 (Ω) and the Laplace-Dirichlet operator A = −∆ with domain dom(−∆) = H 2 (Ω) ∩ H 1 0 (Ω). In particular, for θ = 0 we have the weakly damped wave equation, whereas for θ = 1 we have the strongly damped wave equation. Another physically relevant model is the beam (for N = 1) or plate (for N = 2) equation with fractional damping, subject to the hinged boundary conditions. In this case, the operator A is the Bilaplacian ∆ 2 with domain dom(∆ 2 ) = u ∈ H 2 (Ω) ∩ H 1 0 (Ω) : ∆u ∈ H 2 (Ω) ∩ H 1 0 (Ω) . For this choice of f , the function φ reads φ(s) = as θ if s
Figure 1 .•
1Profile of φ(s) versus √ s (dashed) for different values of θ. • Case θ ∈ [0, 1 2 ). The function φ is increasing (actually, strictly increasing if θ > 0). Therefore, Theorem 4.2 holds with m * = φ(s 0 ). The semigroup S(t) is resonant if and only if s 0 Case θ = 1 2 . Again, m * = φ(s 0 ). The semigroup is resonant if and only if a = 1, and is subdamped if and only if a < 1, with
.φ is decreasing for s ∈ (a2 1−2θ , s m ), where s m > a 2 1−2θ is the minimum of φ, here thought as defined on (0, ∞) and not only on σ(A), on the interval (a 2 1−2θ , ∞). φ is increasing and diverging to infinity for s > s m .The value m * is now less explicit, since it might not be equal to φ(s 0 ). 2θ , s b ) = ∅, where s b is the bigger of the two (distinct) solutions to the equation 2θ . In this case, s 0 is an eigenvalue of A, being an isolated point of σ(A), and in turn −m * is an eigenvalue of A.
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| []
|
[
"MUTATIONS OF NUMERICALLY EXCEPTIONAL COLLECTIONS ON SURFACES",
"MUTATIONS OF NUMERICALLY EXCEPTIONAL COLLECTIONS ON SURFACES"
]
| [
"Johannes Krah "
]
| []
| []
| A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin-Lunts and the latter a result of Kuleshov-Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work [Kra23] that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles. | null | [
"https://export.arxiv.org/pdf/2211.07724v2.pdf"
]
| 253,523,252 | 2211.07724 | 73adebf8c75c1317c2708cb91fd215482146e906 |
MUTATIONS OF NUMERICALLY EXCEPTIONAL COLLECTIONS ON SURFACES
Johannes Krah
MUTATIONS OF NUMERICALLY EXCEPTIONAL COLLECTIONS ON SURFACES
arXiv:2211.07724v2 [math.AG] 3 Apr 2023
A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin-Lunts and the latter a result of Kuleshov-Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work [Kra23] that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.
Introduction
Any smooth projective rational surface over an algebraically closed field admits a full exceptional collection by Orlov's projective bundle and blow-up formulae [Orl92], however a classification of exceptional collections on a given surface is widely open. To construct new exceptional collections from old ones, a key tool are so-called mutations of exceptional pairs, see Section 2.3; these give rise to an action of the braid group in n strands on the set of exceptional collections of length n on such a surface. Bondal and Polishchuk conjectured in more generality:
Conjecture 1.1 ([BP93, Conj. 2.2]). Let T be a triangulated category which admits a full exceptional collection T = E 1 , . . . , E n . Then any other full exceptional collection of T can be constructed from E 1 , . . . , E n by a sequence of mutations and shifts.
Recently, this conjecture was proven to be false [CHS23] and a counterexample is given by a Fukaya category of a certain smooth two-dimensional real manifold. To our knowledge, the conjecture still remains open for triangulated categories T = D b (Coh(X)), where X is a smooth projective variety.
This paper is concerned with the question of classifying (numerically) exceptional collections on a given algebraic surface. Exceptional collections on rational surfaces have been previously studied in [HP11] and [Per18] via considering their associated toric surfaces. A classification of surfaces admitting a numerically exceptional collection of maximal length was carried out in [Via17]. Conjecture 1.1 was first verified in the cases T = D b (X), where X is either P 2 or P 1 × P 1 . The case where X is a del Pezzo surfaces is treated in [KO94]. In [Kul97], similar results for surfaces with basepoint-free anticanonical class were obtained. A full discussion of exceptional collections on the Hirzebruch surface Σ 2 was worked out in [IOU21] and Conjecture 1.1 was settled for D b (Σ 2 ).
In the first part of the paper, we consider the images of exceptional collections in K num 0 (X) instead of the objects in the derived category itself. These so-called numerically exceptional collections on a surface X with χ(O X ) = 1 have been previously investigated by Perling and Vial [Per18;Via17]. Their lattice-theoretic arguments have been reworked by Kuznetsov in the abstract setting of surface-like pseudolattices, introduced in [Kuz17]. Independently, a similar notion of a surface-type Serre lattice was developed in [dTVdB16]. In Section 2 we unify both formalisms in order to prove in Section 3 part (i) of the following Theorem 1.2 (Theorem 3.1, Corollary 4.21). Let X be a smooth projective surface over a field k with χ(O X ) = 1 and let e • and f • be exceptional bases of K num 0 (X). (i) There exists a Z-linear automorphism φ : K num 0 (X) → K num 0 (X) preserving the Euler pairing and the rank of elements such that φ(e • ) can be transformed to f • by a sequence of mutations and sign changes.
(ii) If in addition rk K num 0 (X) ≤ 12, then e • and f • are related by a sequence of mutations and sign changes.
By definition, an exceptional basis of K num 0 (X) is the class of a numerically exceptional collection of maximal length in K num 0 (X), see Definition 2.2. Thus, we can reformulate Theorem 1.2 (i) as: Given two numerically exceptional collections (E 1 , . . . , E n ) and (F 1 , . . . , F n ) of maximal length on a surface X with χ(O X ) = 1 we can find a sequence of mutations and shifts σ such that χ(σ(E i ), σ(E j )) = χ(F i , F j ) and rk σ(E i ) = rk F i holds for all 1 ≤ i, j ≤ n.
Allowing automorphisms of K num 0 (X) preserving χ in addition to mutations and shifts was classically considered in the case of X = P 2 , where full exceptional collections can be interpreted as solutions of the Markov equation, see, e.g., [GK04,§ 7]. For lattices of higher rank this action was considered for instance in [Gor94].
To prove Theorem 1.2 (i) we can restrict to the case of X being either P 1 × P 1 or a blow-up of P 2 in a finite number of points by using Vial's classification recalled in Theorem 2.12. Moreover, the group Aut(K num 0 (X)) of isometries φ as in Theorem 1.2 (i) fits into a short exact sequence
1 → (Pic(X)/ ∼ num ) → Aut(K num 0 (X)) → O(Pic(X)/ ∼ num ) KX → 1,
where O(Pic(X)/ ∼ num ) KX = {f ∈ O(Pic(X)/ ∼ num ) | f (K X ) = K X } is the stabilizer of the canonical class in the orthogonal group of O(Pic(X)/ ∼ num ); see Lemma 2.10.
In Section 4 we address the question how to lift Theorem 1.2 (i) to D b (X) and prove Theorem 1.2 (ii). The following two conditions are sufficient to deduce from Theorem 1.2 (i) that mutations and shifts act transitively on the set of full exceptional collections on X:
(a) The action of an isometry φ : K num 0 (X) → K num 0 (X) as in Theorem 1.2 (i) can be realized as a sequence of mutations and shifts. (b) Two full exceptional collections sharing the same class in K num 0 (X) can be transformed into each other by a sequence of mutations and shifts. If X is a del Pezzo surface, the arguments of [KO94] prove (b), see Lemma 4.11, and for the Hirzebruch surface Σ 2 the condition (b) is verified in [IOU21,§ 6].
The main theorem of Elagin-Lunts in [EL16] states that any numerically exceptional collection consisting of line bundles on a del Pezzo surface is a full exceptional collection obtained from Orlov's blow-up formula applied to a minimal model. We extend this result to the blow-up X of 9 points in very general position in P 2 C . Theorem 1.3 (Corollary 4.10, Theorem 4.17). Let X be the blow-up of P 2 C in 9 points in very general position. Then (i) any numerically exceptional collection of maximal length consisting of line bundles is a full exceptional collection, and (ii) any two such collections are related by mutations and shifts.
The position of the 9 points is discussed in Remark 4.4. Further, our results in [Kra23] show that the statements of Theorem 1.3 do not hold for blow-ups of more than 10 points.
The proof of Theorem 1.3 (ii) is closely linked to the proof of Theorem 1.2 (ii). The key ingredient is the identification of the aforementioned group O(Pic(X)) KX with the Weyl group W X of a root system embedded in Pic(X), see Lemma 4.6. Although this lattice-theoretic equality holds for the blow-up of up to 9 points regardless of their position, our argument relies on a result of Nagata [Nag60] which uses the actual geometry of X. The equality O(Pic(X)) KX = W X then enables us to verify condition (a) for the blow-up in up to 9 points in very general position and thus we obtain Theorem 1.3 (ii) and Theorem 1.2 (ii).
In addition, our techniques provide a new proof of the fact that any two full exceptional collections on a del Pezzo surface are related by mutations and shifts; see Corollary 4.20. This result was proven in the first place by Kuleshov-Orlov in [KO94,Thm. 7.7].
Finally Section 5 discusses the lattice-theoretic behavior of the blow-up X of P 2 in 10 points. In this case the Weyl group W X ⊆ O(Pic(X)) KX has index two and Pic(X) admits an additional involution ι which fixes the canonical class K X ; see Lemma 5.1. While the action of W X on exceptional collections of line bundles can be modeled by Cremona transformations of P 2 , the action of ι gives rise to an extraordinary numerically exceptional collection of line bundles. In [Kra23] we show that the numerically exceptional collection obtained from ι is an exceptional collection of maximal length which is not full, provided the points are in general position. As a consequence, D b (X) contains a phantom subcategory and the braid group action on exceptional collections of maximal length is not transitive. If one could verify condition (b) for exceptional collections of maximal length on X, the results of [Kra23] would imply that the numerical bound in Theorem 1.2 (ii) is optimal, see Remark 5.4.
Conventions.
In this paper the term surface always refers to a smooth projective variety of dimension 2 over a field. The results in Section 3 are independent of the chosen base field, in Section 4 we exclusively work over the complex numbers.
Acknowledgements. This work is part of the author's dissertation, supervised by Charles Vial whom we wish to thank for numerous helpful discussions and explanations. Especially his encouragement to improve the results in this paper led to the developments in [Kra23]. Further, the author thanks Pieter Belmans for reading an earlier draft of this paper.
Numerically Exceptional Collections and Pseudolattices
We recall the necessary terminology of surface-like pseudolattices as it is presented in [Kuz17]. Independently, the notion of a surface-type Serre lattice was introduced in [dTVdB16]. After comparing both notions, we discuss the blow-up operation for pseudolattices in detail. Numerical blow-ups are explicitly mentioned in [dTVdB16] but were already used in [HP11; Via17; Kuz17] in a slightly different manner.
2.1. Exceptional collections. Let X be a smooth projective variety over a field k and let D b (X) := D b (Coh(X)) be the bounded derived category of coherent sheaves on X. An object
E ∈ D b (X) is exceptional if Hom(E i , E i ) = k and Hom(E i , E i [l]) = 0 for all l = 0. A full excep- tional collection in D b (X)
is a sequence of exceptional objects (E 1 , . . . , E n ) such that E 1 , . . . , E n generate D b (X) as a triangulated category and Hom(E i , E j [l]) = 0 for all l ∈ Z whenever i > j. When considering only their images in the Grothendieck group K 0 (X) := K 0 (D b (X)) homomorphism spaces have to be exchanged with alternating sums over their dimensions. For this, let
χ(E, F ) := j∈Z (−1) j dim k Hom(E, F [j])
be the Euler pairing. It gives rise to a bilinear form on K 0 (X) and an object E ∈ D b (X) is called numerically exceptional if χ(E, E) = 1.
Definition 2.1. A numerically exceptional collection in D b (X) is a sequence of numerically exceptional objects (E 1 , . . . , E n ) such that χ(E i , E j ) = 0 whenever i > j. The sequence is said to be of maximal length if [E 1 ], . . . , [E n ] ∈ K 0 (X) generate K num 0 (X) as a Z-module or equivalently if n = rk K num 0 (X).
Here K num 0 (X) := K 0 (X)/ ker χ denotes the numerical Grothendieck group. Note that the left and right kernels of χ coincide thanks to Serre duality. Clearly χ defines a non-degenerate bilinear form on K num 0 (X). Therefore studying numerically exceptional collections can be reduced to studying non-degenerate Z-valued bilinear forms, which will be formalized in the notion of a pseudolattice.
2.2.
Surface-like pseudolattices. We begin with recalling the notion of a pseudolattice in the sense of Kuznetsov.
Definition 2.2 ([Kuz17, Def. 2.1]). A pseudolattice is a finitely generated free abelian group G together with a non-degenerate bilinear form χ :
G ⊗ Z G → Z. An isometry φ : (G, χ G ) → (H, χ H ) between pseudolattices is a Z-linear isomorphism which satisfies χ G (v, w) = χ H (φ(v), φ(w)) for all v, w ∈ G.
• The pseudolattice (G, χ) is unimodular if χ induces an isomorphism G → Hom Z (G, Z).
• Let e • = (e 1 , . . . , e n ) be a basis of G, then (χ(e i , e j )) i,j is called the Gram matrix with respect to e • . • An element e ∈ G is called exceptional if χ(e, e) = 1.
• An ordered basis e • is called exceptional basis if the corresponding Gram matrix is upper unitriangular, i.e. χ(e i , e j ) = 0 whenever i > j and χ(e i , e i ) = 1 for all i.
• A Serre operator is an isometry S : G → G satisfying χ(v, w) = χ(w, S(v)) for all v, w ∈ G.
Note that the lattice G is unimodular if and only if the Gram matrix has determinant ±1. The Serre operator is unique, provided it exists, and if G is unimodular, it is given by M −1 M T , where M is the Gram matrix of χ with respect to a chosen basis. In case we need to pass to rational coefficients, we use the notation G Q := G ⊗ Z Q for a pseudolattice G (or more generally for any abelian group).
Definition 2.3 ([Kuz17, Def. 3.1]). A pseudolattice (G, χ) is surface-like if there exists a primitive element p ∈ G satisfying (i) χ(p, p) = 0, (ii) χ(p, v) = χ(v, p) for all v ∈ G, (iii) χ is symmetric on p ⊥ := {v ∈ G | χ(p, v) = 0}. Such an element p is called a point-like element.
The terminology is justified by the following geometric example. Remark 2.5. More generally, any 0-cycle of degree 1 in CH 0 (S) provides a point-like element by realizing it as a Chern character of a complex of skyscraper sheaves. However all the surfaces we consider are rational, for that reason we will only consider point-like elements as in Example 2.4.
From now on any surface-like pseudolattices K num 0 (S), where S is a surface over k, is implicitly assumed to be endowed with the Euler pairing and a point-like element given by the class of a skyscraper sheaf of a k-valued point. Recall that for a smooth projective surface S the Chern character induces an isomorphism
(2.6) ch : K num 0 (S) Q ∼ − → Q ⊕ (Pic(X)/ ∼ num ) Q ⊕ Q,χ(E, F ) = ef χ(O X ) + 1 2 f c 1 (E) 2 + e c 1 (F ) 2 − 2 c 1 (E) c 1 (F ) (2.7) − 1 2 K X (e c 1 (F ) − f c 1 (E)) − (f c 2 (E) + e c 2 (F )).
Given a surface-like pseudolattice G with point-like element p, we define the rank function with respect to p to be r(−) := χ(p, −) = χ(−, p). Then p ⊥ = ⊥ p = ker(r) and we obtain the analogue of the decomposition in (2.6).
Lemma 2.8 ([Kuz17, Lem. 3.10, Lem. 3.11]). If G is a surface-like pseudolattice and p a point-like element, there is a complex Z p − → G r − → Z with injective p and, if G is unimodular, surjective r. The middle cohomology of the above complex NS(G) := p ⊥ /p is a finitely generated free abelian group of rank rk(G) − 2.
On NS(G) the pairing −χ induces a well-defined non-degenerate symmetric bilinear form q, called the intersection form, which also will be denoted by the usual product − · −. Lemma 2.9 ([Kuz17, Lem. 3.12]). Let G be a surface-like pseudolattice with point-like element p and let λ :
2 G → p ⊥ be the alternating map sending v ∧ w → r(v)w − r(w)v.
Then there is a unique element K G ∈ NS(G) Q , called canonical class, satisfying
−q(K G , λ(v, w)) = χ(v, w) − χ(w, v) for all v, w ∈ G. If G is unimodular, K G is integral, i.e. K G ∈ NS(G).
The pair (NS(G), q) is called the Néron-Severi lattice and NS(G) the Néron-Severi group. One can check that for a surface S and pseudolattice G = K num 0 (S) as in Example 2.4 all these definitions agree with the usual ones up to sign. For example, via Riemann-Roch (2.7) one computes χ(O x , F) = − rk F for any coherent sheaf F on S and x ∈ S a k-valued point.
The following Lemma 2.10, which could not be found in the literature, will be important in the proof of Theorem 1.3. For that reason, we provide a proof here.
Lemma 2.10 (Self-isometries arise from orthogonal transformations). Let G be a surface-like pseudolattice of rk G ≥ 3 and let Aut(G) be the group of self-isometries φ : G → G with φ(p) = p. The map Ψ : Aut(G) → O(NS(G)) obtained by sending φ ∈ Aut(G) to the induced orthogonal transformation of NS(G) defines a group homomorphism. If G is unimodular, the image of Ψ equals the stabilizer of the canonical class O(NS(G)) KG
= {f ∈ O(NS(G)) | f (K G ) = K G }. Moreover if G = K num 0 (X)
for some surface X with χ(O X ) = 1 as in Example 2.4, the kernel of Ψ can be identified with the subgroup of automorphisms given by twists with line bundles. In other words we obtain a short exact sequence
1 → (Pic(X)/ ∼ num ) → Aut(K num 0 (X)) → O(Pic(X)/ ∼ num ) KX → 1.
Proof. Since r(−) = χ(p, −), any φ : G → G which preserves the point-like element p preserves the rank of elements. Hence it induces an orthogonal transformation of NS(G) which fixes the canonical class K G . If G is unimodular, we can choose a rank 1 vector v 0 ∈ G and a basis v 0 , v 1 , . . . , v n−1 such that p = v n−1 and p ⊥ = v 1 , . . . , v n−1 . By adding suitable multiples of p to the v i we can arrange
χ(v i , v 0 ) = 0 for all 1 ≤ i ≤ n − 2. Now χ(v 0 , v i ) = −q(K G , v i ) for 1 ≤ i ≤ n − 2.
Thus anyφ ∈ O(NS(G)) KG can be lifted to an isometry φ of G preserving p and fixing v 0 by choosing φ(v i ) to be the unique lift ofφ(v i ) ∈ p ⊥ /p which satisfies χ(φ(v i ), v 0 ) = 0. The construction of the lift depends on the choice of v 0 and for any other , q) has signature (1, rk G − 3), the canonical class K G is integral and K G is characteristic, i.e. q(D, D) ≡ q(K G , D)(mod 2) for all D ∈ NS(G). A surface-like pseudolattice G is minimal if it has no exceptional elements of rank zero. Equivalently NS(G) does not contain any (−1)-class.
choice v ′ 0 with χ(v 0 , v 0 ) = χ(v ′ 0 , v ′ 0 ) and r(v ′ 0 ) = 1 there exists exactly one lift φ which maps v 0 to φ(v 0 ) = v ′ 0 . Assume G = Kch(E) = rk E, c 1 (E), 1 2 (c 1 (E) 2 − 2 c 2 (E)) ,
It turns out that such geometric pseudolattices can be classified if we restrict to defect zero pseudolattices. Here the defect of G is the integer . Let G be a unimodular geometric pseudolattice of rank n ≥ 3 and zero defect such that G represents 1 by a rank 1 vector, i.e. there exists v ∈ G of rank 1 such that χ(v, v) = 1. Then the following holds:
δ(G) := K 2 G + rk(G) − 12. If G is• n = 3 and K G = −3H for some H ∈ NS(G) if and only if G is isometric to K num 0 (P 2 ); • n = 4, NS(G) is even and K G = −2H for some H ∈ NS(G) if and only if G is isometric to K num 0 (P 1 × P 1 ); • n ≥ 4, NS(G) is odd and K G is primitive if and only if G is isometric to K num 0 (X n−3 ).
Here X n−3 is the blow-up of P 2 in n − 3 points. Furthermore, G has an exceptional basis if and only if one of the three possibilities listed above is satisfied.
Remark 2.13. In fact, by [Via17, Thm. 3.1] the condition δ(G) = 0 is a necessary condition for admitting an exceptional basis if the pseudolattice results from taking the numerical Grothendieck group of a smooth projective surface with χ(O S ) = 1.
Let G be a surface-like pseudolattice with Serre operator S. Let v ∈ G, then
χ(p, (S − 1)(v)) = χ(v, p) − χ(p, v) = 0.
Furthermore, [Kuz17,Lem. 3.14] shows that S − 1 maps p ⊥ to Zp. Thus, we obtain a decreasing filtration
F 3 G = 0 ⊆ F 2 G = Zp ⊆ F 1 G = p ⊥ ⊆ F 0 G = G such that S − 1 maps F i G to F i+1 G.
If G is unimodular, the rank map induces an isomorphism r : G/p ⊥ → Z, thus the above filtration defines a so-called codimension filtration.
Definition 2.14 ([dTVdB16, Def. 5.1.1]). Let G be a pseudolattice with Serre operator S and let
V := G Q . A codimension filtration on V is a filtration 0 = F 3 V ⊆ F 2 V ⊆ F 1 V ⊆ F 0 V = V such that (S − 1)(F i V ) ⊆ F i+1 V , dim F 0 V /F 1 V = dim F 2 = 1 and χ(F 1 V, F 2 V ) = 0.
Conversely, any codimension filtration gives rise to a point-like element by choosing a generator of F 2 G = F 2 V ∩ G. This yields a 1:1-correspondence
{codimension filtrations F • on G} ↔ {point-like elements p}/{±1}.
We will refer to both of them, a point-like element and a codimension filtration, as a surface-like structure on the pseudolattice G. In Example 2.4 the codimension filtration coincides with the topological codimension filtration, as discussed in [Kuz17, Ex. 3.5].
2.3. Mutations. Given e ∈ G we define the left mutation L e and its right mutation R e as
L e (v) := v − χ(e, v)e R e (v) := v − χ(v, e)e
for all v ∈ G. Given an exceptional basis e • = (e 1 , · · · , e n ) of G we define L i,i+1 (e • ) := (e 1 , · · · , e i−1 , L ei (e i+1 ), e i , e i+2 , · · · , e n ),
R i,i+1 (e • ) := (e 1 , · · · , e i−1 , e i+1 , R ei+1 (e i ), e i+2 , · · · , e n ).
The sequences are again exceptional bases and the above operations are mutually inverse. By construction, these mutations match the known mutations of exceptional collections if G = K num 0 (S) as in Example 2.4. Indeed, if S is a surface and E ∈ D b (S) an exceptional object, the left mutation L E and right mutation R E are defined as
L E (F ) := Cone E ⊗ RHom(E, F ) ev − → F and R E (F ) := Cone F ev ∨ − − → E ⊗ RHom(F, E) ∨ [−1]
for any object F ∈ D b (S). Note that by construction the diagram
D b (S) K num 0 (S) D b (S) K num 0 (S) ME M [E] commutes, where M E = L E or M E = R E . Moreover, if D b (S) = E 1 , . . . , E n = E • is a full exceptional collection, the sequences L i,i+1 (E • ) := (E 1 , · · · , E i−1 , L Ei (E i+1 ), E i , E i+2 , · · · , E n ), R i,i+1 (E • ) := (E 1 , · · · , E i−1 , E i+1 , R Ei+1 (E i ), E i+2 , · · · , E n ).
are again full exceptional collections. Already on the level of D b (S) the operations L i,i+1 and R i,i+1 give rise to an action of the braid group B n , see, e.g., [BP93, Prop. 2.1]. Together with Z n acting by shifts, this yields an action of the semidirect product Z n ⋊ B n on the set of full exceptional collections, where the homomorphism B n → Aut(Z n ) is the composition of the canonical map B n → S n and the action of S n on Z n by permutations. If two exceptional bases lie in the same orbit of the Z n ⋊ B n -action, we say the exceptional collections are related by mutations up to shifts. On the level of K num 0 (S) shifts result in sign changes. More generally, if G is a pseudolattice of rank n with exceptional basis, then {±1} n ⋊ B n acts on the set of exceptional bases, where {±1} n acts by changing signs of basis elements. Moreover, this action commutes with the action of isometries φ : G → G. If two exceptional bases lie in the same orbit of {±1} n ⋊ B n , we say the exceptional bases are related by mutations up to signs. In this paper, we will only consider pseudolattices with surface-like structure. If we write that two exceptional bases e • , f • are related by mutations up to signs and isometry we always mean that there exists an isometry φ : G → G which preserves the point-like element φ(p) = p and φ(e • ) and f • are related by mutations up to signs.
Let G be a surface-like pseudolattice with exceptional basis. We will frequently mutate to norm-minimal bases, where the norm of an exceptional basis e • = (e 1 , . . . , e n ) is the number i r(e i ) 2 . We say an exceptional basis is norm-minimal if there is no exceptional basis related by mutations and sign changes with smaller norm. Recall that due to the work of Perling, normminimal exceptional bases can be understood via Perling's algorithm:
Theorem 2.15 ([Kuz17, Thm. 5.8], cf. [Per18,
Cor. 9.12, Cor. 10.7]). Let G be a geometric surfacelike pseudolattice. Any exceptional basis in G can be transformed by mutations and sign changes into a norm-minimal exceptional basis consisting of 3 or 4 elements of rank 1 and all other elements of rank 0.
2.4. Blow-up and blow-down. We recall the classical blow-up and blow-down construction for surface-like pseudolattices and give a detailed discussion of [dTVdB16, § 5] as we make use of these observations in Section 3. Let G be a unimodular surface-like pseudolattice with point-like element p. We denote the induced codimension filtration by F • G. Let e • = (e 1 , . . . , e n ) be a basis of G and let M be the Gram matrix of the pairing χ with respect to this basis. Choosing an element z ∈ F 2 G = Zp, we construct the numerical blow-up of G at z as follows: We extend the lattice G by adding a formal element f , i.e. we consider the free abelian group Bl z G := Zf ⊕ G. The pairing χ new on Bl z G is defined via
χ new | G⊗G := χ, χ new (g, f ) := 0 for all g ∈ G, χ new (f, f ) := 1,
and χ new (f, g) := χ(z, g) for all g ∈ G.
In abuse of notation we write χ also for the pairing on Bl z G. As outlined below, this definition matches the geometric situation of a blow-up. The Gram matrix with respect to the basis (f, e 1 , . . . , e n ) is of the form
1 χ(z, e 1 ) · · · χ(z, e n ) 0 . . . 0 M .
Note that Bl z G is again unimodular and surface-like with point-like element p ∈ G ⊆ Zf ⊕ G. The latter follows from writing z = np for some n ∈ Z which shows
χ(p, f ) = 0 = χ(z, p) = χ(f, p). The orthogonal complement of p in Bl z G is F 1 Bl z G = F 1 G ⊕ Zf and χ is symmetric on F 1 G ⊕ Zf as it is symmetric on both summands and χ(F 1 G, f ) = 0 = nχ(p, F 1 G) = χ(f, F 1 G). In particular, F 2 Bl z G = Zp = F 2 G.
Therefore the point-like element p does not change under blow-up; this allows us to blow up the same element multiple times. Note that the image of f in
NS(Bl z G) = NS(G) ⊥ ⊕ Zf
defines an element of self-intersection −1. It is the analogue of a (−1)-curve and can be blown down, but in contrast to the geometric setting, we cannot detect whether a divisor of self-intersection −1 is an actual curve or not. Again we compare the construction to the geometric one (cf. Example 2.4). Let S be a smooth projective surface and letS be the blow-up at a point p ∈ S with exceptional divisor E:
S S E {p}. π ψ j i For F ∈ D b (S) a Riemann-Roch computation shows χS(j * O E (−1), π * F) = − rk(F) = χ S (O p , F), see [Per18, Ex. 4.1].
Finally, Orlov's blow-up formula yields a semi-orthogonal decomposition
D b (S) = j * (ψ * D b ({p}) ⊗ O E (−1)), Lπ * D b (S) = j * O E (−1), Lπ * D b (S)
which coincides with the numerical blow-up construction. The inverse operation on a unimodular surface-like pseudolattice G is the blow-down or contraction. Let f ∈ G be a rank zero vector such that q
(f, f ) = −χ(f, f ) = −1. Then the contraction of f is the lattice G f := ⊥ f = {v ∈ G | χ(v, f ) = 0} ⊆ G with pairing χ|⊥ f ⊗ ⊥ f .
The pseudolattice G f is again surface-like with point-like element p and unimodular; see [Kuz17, Lem. 5.1]. If G is geometric, so is G f . In the following we prove a slightly modified version of [dTVdB16, Lem. 5.1], which will be a key tool towards establishing Theorem 3.1.
Proposition 2.16. Let G be a unimodular surface-like pseudolattice and f ∈ G a rank zero vector of self-intersection −1. Denote by S the Serre operator of G, then z :
= (S − 1)(f ) ∈ F 2 G f defines an element such that Bl z G f = G. Proof. Since f ∈ F 1 G = p ⊥ , we know that (S − 1)(f ) ∈ F 2 G = Zp is a multiple of p and lies in G f . Let H := Bl z (G f ) = Zg ⊕ G f be the blow-up of G f at z.
Then the pairing on H extends the pairing of G f with the property that g is an element of rank zero and of self-intersection −1. Consider the morphism G → H sending f → g and v → v for all v ∈ G f . We verify that this is an isometry: Obviously χ(g, g) = 1 = χ(f, f ) and
χ(v, f ) = 0 = χ(v, g) for v ∈ G f . Let v ∈ G f , then χ(g, v) = χ(z, v) = χ(v, z) = χ(v, S(f )) − χ(v, f ) = χ(f, v),
where we have used that z is a multiple of p and χ(v, f ) = 0 for all v ∈ G f . Clearly (Bl z G) f = G, thus blow-up and blow-down are mutually inverse.
Remark 2.17. Comparing the blow-down construction described above to the construction in [Kuz17, § 5], one observes that the contraction of an exceptional element of rank zero can also be defined as the right orthogonal f ⊥ .
We end this section by recalling formulae for the defect of the contraction.
Lemma 2.18 ([Kuz17, Lem. 5.7]). Let G be a surface-like pseudolattice and e ∈ G an exceptional element of rank zero. Then the defect of G equals
δ(G) = δ(G e ) + (1 − q(K G , e) 2 ). If G is geometric, then δ(G) ≤ δ(G e ) with equality if and only if q(K G , e) = ±1.
In the same manner a formula for the degree of the blow-up was obtained in [dTVdB16, Lem. 5.2.1]. The degree of a unimodular surface-like pseudolattice G is deg(G) = K 2 G and is related to the defect by the formula
(2.19) deg(G) = 12 + δ(G) − rk(G).
Lemma 2.20 ([dTVdB16, Lem. 5.2.1]). Let G be a unimodular surface-like pseudolattice and let σ ∈ G be an element such that its imageσ generates
Bl z G/F 2 Bl z G ∼ = Z. Then deg Bl z G = deg G − χ(σ, z) 2 .
Above Lemma 2.20 requires a justification in our context, as it is possibly not clear that the canonical class in the sense of [dTVdB16] coincides with the one in Lemma 2.9.
Proof of Lemma 2.20. The image ω of (S − 1)(σ) in NS(G) is the canonical class of G in the sense of [dTVdB16, Def. 3.5.1] and in [dTVdB16, Lem. 5.2.1] the statement is shown for deg G := q(ω, ω). Therefore it is enough to show:
Claim. Let G be a unimodular surface-like pseudolattice, σ ∈ G/F 2 G a generator and ω = (S − 1)(σ) ∈ NS(G). Then ω satisfies the defining equation in Lemma 2.9 up to sign, i.e.
±q(ω, λ(v, w)) = χ(v, w) − χ(w, v)
for all v, w ∈ G and λ as in Lemma 2.9.
Proof of the Claim. Since G is unimodular, the rank map induces an isomorphism r :
G/F 2 G → Z. Let σ ∈ G be a vector such thatσ generates G/F 2 G. Up to possibly replacing σ by −σ we can write any v ∈ G as r(v)σ + τ (σ) with τ (σ) ∈ F 2 G. Let ω = (S − 1)(σ) be the canonical class defined by σ and let d(v) := q(τ (v), ω) for all v ∈ G. By [dTVdB16, Prop. 3.6.2] the equality (2.21) χ(v, w) − χ(w, v) = det d(v) d(w) r(v) r(w) = r(w)q(τ (v), ω) − r(v)q(τ (w), ω)
holds for all v, w ∈ G. Let λ be the alternating form as in Lemma 2.9. Then
−q(ω, λ(v, w)) = −q(r(v)(r(w)σ + τ (w)) − r(w)(r(v)σ + τ (v)), ω) = −q(r(v)r(w)σ + r(v)τ (w) − r(w)r(v)σ − r(w)τ (v), ω) = q(r(w)τ (v) − r(v)τ (w), ω)
combined with (2.21) proves the claim.
Proof of Theorem 1.2 (i)
Throughout this section let X k be the blow-up of P 2 in k distinct points and let G k := K num 0 (X k ) be the pseudolattice obtained from X k . Using Vial's classification, see Theorem 2.12 and Remark 2.13, we can rephrase Theorem 1.2 (i) as follows:
Theorem 3.1. Let e • and f • be two exceptional bases of G k or of K num 0 (P 1 × P 1 ). Then there exists an isometry φ : G → G preserving the surface-like structure, i.e. φ(p) = p, such that φ(e • ) and f • are related by mutations up to signs.
In preparation for the proof of Theorem 3.1 we compute an explicit form of the pseudolattices G k . The surface P 2 admits a full exceptional sequence consisting of line bundles, namely the Beilinson sequence D b (P 2 ) = O P 2 , O P 2 (1), O P 2 (2) . This yields an exceptional basis of the numerical Grothendieck group G 0 := K num
0 (P 2 ) = Z[O P 2 ] ⊕ Z[O P 2 (1)] ⊕ Z[O P 2 (2)] with Gram matrix[0 → O P 2 (−2) = 2 O P 2 (−1) ⊕2 → O P 2 (−1) ⊕2 → O P 2 → 0] ∼ = O x .
Thus, we obtain after twisting by O P 2 (2) Hence, p = ±(1, −2, 1).
[O x ] = [O P 2 (2)] − 2[O P 2 (1)] + [O P 2 ] = e 3 − 2e 2 + e 1 ∈ K num 0 (P 2 ).
By the blow-up formula we compute Gram matrices M k of the pseudolattices G k , namely
M k :=
1 0 · · · 0 1 1 1 0 1 · · · 0 1 1 1 . . . . . . . . . . . . . . . 0 0 · · · 1 1 1 1 0 0 · · · 0 1 3 6 0 0 · · · 0 0 1 3 0 0 · · · 0 0 0 1 Proof. By [Kuz17,Cor. 4.25] norm-minimal exceptional bases of K num 0 (P 2 ) correspond to the Beilinson sequence O P 2 , O P 2 (1), O P 2 (2) . Therefore any two exceptional bases are related by mutations up to sign and isometry.
= id
Proposition 3.5. Any two exceptional bases in K num 0 (P 1 × P 1 ) are related by mutations up to sign and isometry.
Proof. If G admits a norm-minimal basis consisting of objects of nonzero rank, G is isometric to K num 0 (P 1 × P 1 ) and the norm-minimal basis corresponds to one of the full exceptional collections c + 1, 1) for some c ∈ Z; see [Kuz17,Cor. 4.27]. The corresponding Gram matrix is
D b (P 1 × P 1 ) = O P 1 ×P 1 , O P 1 ×P 1 (1, 0), O P 1 ×P 1 (c, 1), O P 1 ×P 1 (D c := 1 2 2c + 2 2c + 4 0 1 2c 2c + 2 0 0 1 2 0 0 0 1 , see [Kuz17, Ex. 3.7
]. Now we mutate the third and fourth basis vector and compute the corresponding Gram matrices:
L 3,4 (b 1 , . . . , b 4 ) = (b 1 , b 2 , −2b 3 + b 4 , b 3 ), 1 2 −(2(c − 1) + 2) 2(c − 1) + 4 0 1 −2(c − 1) 2(c − 1) + 2 0 0 1 −2 0 0 0 1 , R 3,4 (b 1 , . . . , b 4 ) = (b 1 , b 2 , b 4 , b 3 − 2b 4 ), 1 2 2(c + 1) + 2 −(2(c + 1) + 4) 0 1 2(c + 1) −(2(c + 1) + 2) 0 0 1 −2 0 0 0 1 .
Multiplying −2b 3 + b 4 by −1 in the first case and b 3 − 2b 4 in the second case, we observe that all bases corresponding with Gram matrices D c are related by mutations up to sign and isometry.
For later use in the proof of Theorem 3.1, we also treat the surfaces X 1 and X 2 by hand.
Proposition 3.6. Any two exceptional bases in K num 0 (X 1 ) are related by mutations up to sign and isometry.
Proof. Since K num 0 (X 1 ) is not isometric to K num 0 (P 1 × P 1 ), Perling's algorithm Theorem 2.15 shows that a norm-minimal basis has the form e 1 , . . . , e 4 , where e 1 is of rank zero and e 2 , e 3 and e 4 are of rank one. The contraction G e1 is isomorphic to K num 0 (P 2 ) with norm-minimal exceptional basis e 2 , e 3 , e 4 . Since blow-up and blow-down are mutually inverse, e 1 results from blowing up a point z = np ∈ Zp. Observe that δ(G) = δ(G e1 ) = 0, so by (2.19) the degree has to decrease by 1 from G to G e1 . Thus by Lemma 2.20 n = ±1 and χ(σ, z) = ±1. Possibly after changing the sign of e 1 , the Gram matrix with respect to e 1 , . . . , e 4 is M 1 .
The surface X 2 can be obtained from blowing up P 2 in 2 points or from blowing up P 1 × P 1 in 1 point. So a priori, there could potentially be two different types of norm-minimal exceptional bases. We compute that this is not the case.
Proposition 3.7. Let X 2 be the blow-up of P 2 in 2 points and let G 2 := K num 0 (X 2 ). Then any two exceptional bases are related by mutations up to sign and isometry. In particular, any normminimal exceptional basis is of norm 3.
Proof. We show that any exceptional basis can be mutated to an exceptional basis with Gram matrix M 2 . Let e • be an exceptional basis. Again with Perling's algorithm we mutate e • to a norm-minimal basis a 1 , . . . , a l , b 1 , . . . , b m with a i of rank zero and b i of rank one. Now m ∈ {3, 4}, since the (iterated) contraction of the rank zero elements yields a minimal geometric surface-like pseudolattice, which admits an exceptional basis; that implies it is isometric to K num 0 (P 2 ) or to K num 0 (P 1 × P 1 ). Assume for contradiction m = 4. Then the contraction G a1 has Gram matrix Moreover, as in the proof of Proposition 3.6, a 1 is obtained by blowing up K num 0 (P 1 × P 1 ) in ±p. After possibly changing the sign of a 1 , we can assume a 1 results from blowing up −p. Now we want to find a sequence of mutations, which reduces the norm of (a 1 , b 1 , . . . , b 4 ). We compute: a 1 + b 1 , a 1 , b 2 , b 3 , b 4 ) R2,3 −−→ (−a 1 + b 1 , b 2 , a 1 − b 2 , b 3
(a 1 , b 1 , b 2 , b 3 , b 4 ) L1,2 −−→ (−, b 4 ) R3,4 −−→ (−a 1 + b 1 , b 2 , b 3 , a 1 − b 2 − b 3 , b 4 ) L1,2 −−→ (a 1 − b 1 + b 2 , −a 1 + b 1 , b 3 , a 1 − b 2 − b 3 , b 4 ) L2,3 −−→ (a 1 − b 1 + b 2 , a 1 − b 1 + b 3 , −a 1 + b 1 , a 1 − b 2 − b 3 , b 4 ) R4,5 −−→ (a 1 − b 1 + b 2 , a 1 − b 1 + b 3 , −a 1 + b 1 , b 4 , a 1 − b 2 − b 3 + 3b 4 ).
(3.8)
Since the rank map is additive one easily computes that the last basis is of rank (0, 0, 1, 1, 1). But this contradicts the assumption that (a 1 , b 1 , . . . , b 4 ) was norm-minimal. Thus m = 3 and the exceptional basis a 1 , a 2 , b 1 , b 2 , b 3 results from blowing up K num 0 (P 2 ) in 2 points n 1 p and n 2 p. After possibly changing signs, we can assume n 1 , n 2 ≤ 0. The fact that G 2 and (G 2 ) a1,a2 = G 0 have defect zero implies that also (G 2 ) a1 has defect zero, since contraction only increases the defect by Lemma 2.18. Therefore the degree has to increase by 1 in each contraction and we have n 1 = n 2 = −1 by Lemma 2.20. Hence, the Gram matrix with respect to to a 1 , a 2 , b 1 , b 2 , b 3 is M 2 .
Remark 3.9. One can further compute the Gram matrix with respect to (a
1 − b 1 + b 2 , a 1 − b 1 + b 3 , −a 1 + b 1 , b 4 , a 1 − b 2 − b 3 + 3b 4 ) as 1 0 −1 −1 −1 0 1 −1 −1 −1 0 0 1 3 6 0 0 0 1 3 0 0 0 0 1
, but this will not be used subsequently.
Proof of Theorem 3.1. We show the following statement: Given any exceptional basis a • of G k we can find another exceptional basis related by mutations and sign changes to a • such that the Gram matrix is of the form M k and the first k basis elements have rank zero and the last 3 have rank one. The unimodularity then ensures that the involved isometry preserves the surface-like structure given by p, since the isometry respects the rank function. As we have treated the cases k ≤ 2 by hand, we can assume k > 2. Given any exceptional basis of G k we can mutate the basis to a norm-minimal basis (a 1 , . . . , a l , b 1 , . . . , b m ) where the elements a i are of rank zero and the b i are of rank one and m is equal to 3 or 4; see Theorem 2.15. Contracting the rank zero objects a i , we obtain a minimal unimodular geometric surface-like pseudolattice (G k ) a1,...,a l admitting an exceptional basis. Thus the defect of (G k ) a1,...,a l is zero by [Kuz17, Cor. 5.6]. Contraction of geometric pseudolattices only increases the defect, cf. Lemma 2.18, thus all intermediate pseudolattices (G k ) a1,...,ai are unimodular geometric surfacelike pseudolattices with defect zero and admit an exceptional basis. This implies that they are isometric to blow-ups of P 2 as long as k−i ≥ 2 by Theorem 2.12. Choosing i such that k−i = 2, the pseudolattice (G k ) a1,...,ai is isometric to the blow-up of 2 points in (G k ) a1,...,ai . By Proposition 3.7 any two exceptional bases of (G k ) a1,...,ai are related by mutations up to sign and we can mutate the exceptional basis to a basis of norm 3. Now mutations in the contraction lift to mutations of G k , which leave the contracted vectors invariant. Hence, m = 3 and l = k. In particular, (G k ) a1,...,a k is isometric to G 0 and we may assume that b 1 , b 2 , b 3 have Gram matrix M 0 .
As seen in Section 2.4, blowing up and contracting are mutually inverse operations. Thus the basis a 1 , . . . , a k , b 1 , b 2 , b 3 is a basis obtained from blowing up G 0 in k points. The point-like element of G 0 is unique up to sign, as discussed in [Kuz17,Ex. 3.5], hence we can assume p = b 3 − 2b 2 + b 1 . In each intermediate step (G k ) a1,...,ai+1 is obtained from (G k ) a1,...,ai by blowing up a point n i+1 p with n i+1 ∈ Z. As each (G k ) a1,...,aj has defect zero we deduce n j = ±1 for all j. Indeed, by (2.19) the degree has to decrease by −1 in each step and Lemma 2.20 yields deg((G k ) a1,...,ai ) = deg((G k ) a1,...,ai+1 ) − χ(σ, n i+1 p) 2 = deg((G k ) a1,...,ai+1 ) − n 2 i+1 . Up to possibly changing signs, we can arrange χ(a i , b j ) = 1 for all i, j. Thus the Gram matrix has the desired form.
Blow-up of 9 Points
Full exceptional collections on del Pezzo surfaces were studied in [KO94] and in [EL16]. In [EXZ21] and [IOU21] similar results for weak del Pezzo surfaces, i.e. surfaces with nef and big anticanonical divisor, were obtained. In this section, we expand the class of examples by considering the blow-up of P 2 in 9 points in very general position. In this situation, we can assume that there is a unique cubic curve in P 2 passing through each of the 9 points with multiplicity 1. Then the divisor class of the strict transform of this cubic coincides with the anticanonical divisor −K X = 3H − 9 i=1 E i of the blow-up X. Here H is the pullback of a hyperplane class in P 2 and E i is the exceptional divisor corresponding to the blow-up of the point p i . Therefore, −K X is nef but not big as (−K X ) 2 = 0, so X is not a weak del Pezzo surface. Additionally, −K X is not basepoint-free and for that reason the techniques developed in [Kul97] cannot be applied. In this section we exclusively work over the field of complex numbers.
Toric systems and numerically exceptional collections. We recall the necessary terminology of toric systems as introduced by Hille and Perling in [HP11, § § 2-3].
Definition 4.1. Let X be a smooth projective surface. A sequence of divisors A 1 , . . . , A n on X is a toric system if n ≥ 3 and one has A i · A i+1 = 1 = A 1 · A n for all 1 ≤ i ≤ n − 1, A i · A j = 0 for |i − j| > 1 except {i, j} = {1, n}, and A 1 + · · · + A n ∼ lin −K X .
If χ(O X ) = 1 and n = rk K 0 (X), we have a 1:1-correspondence between toric systems on X and numerically exceptional collections consisting of line bundles of length n up to twists with line bundles: toric systems (A 1 , . . . , A n ) / ∼ lin ↔ numerically exceptional collections of line bundles (O X (D 1 ), . . . , O X (D n )) /Pic(X)
A toric system (A 1 , . . . , A n ) and a choice of a divisor D 1 defines a numerically exceptional collection (O X (D 1 ), . . . , O X (D n )) given by
D i+1 := D 1 + A 1 + · · · + A i .
Conversely, any numerically exceptional collection of line bundles (O X (D 1 ), . . . , O X (D n )) gives rise to a toric system via
A i := D i+1 − D i for 1 ≤ i ≤ n − 1, D 1 − K X − D n for i = n.
A toric system is called exceptional if the corresponding collection of line bundles is exceptional. Moreover, (A 1 , . . . , A n ) is an (exceptional) toric system if and only if (A 2 , . . . , A n , A 1 ) is an (exceptional) toric system.
Equivalently, each divisor A i + · · · + A j (1 ≤ i ≤ j ≤ n − 1) is left-orthogonal (a divisor D is called left-orthogonal if h i (−D) = 0 for all i).
Orlov's blow-up formula for full exceptional collections can be transferred to toric systems via so-called augmentations; see [HP11, § 5] and [EL16, § 2.6]: If X ′ is a surface with toric system A ′ 1 , . . . , A ′ n and p : X → X ′ the blow-up of X in a closed point p ∈ X ′ with exceptional divisor E ⊆ X, denote by A i := p * A ′ i the pullback of the divisors. We obtain a toric system on X, namely E, A 1 − E, A 2 , . . . , A n−1 , A n − E.
This toric system and all its cyclic shifts are called augmentations. Conversely, a blow-down operation for toric systems can be defined. A 1 , . . . , A n be a toric system on a surface X such that there exists an index 1 ≤ m ≤ n with A m a (−1)-curve in X. Let p : X → X ′ be the blow-down of A m . Then A 1 , . . . , A n is an augmentation of a toric system A ′ 1 , . . . , A ′ n−1 on X ′ . An essential observation for the proof of Theorem 1.3 is that [EL16,Lem. 3.4] generalizes to the blow-up of P 2 in 9 points: Lemma 4.3. Let X be the blow-up of P 2 in 9 points in very general position. Then any divisor D with D 2 = −1 and χ(D) = 1 is linearly equivalent to a (−1)-curve.
Proposition 4.2 ([EL16, Prop. 3.3]). Let
Proof. First of all, Riemann-Roch yields χ(D) = 1 + 1 2 (D(D − K X )). As
D 2 = −1 we have −K X D = 1. Now χ(D) = h 0 (D) − h 1 (D) + h 2 (D) and h 2 (D) = h 0 (K X − D) by Serre duality. The intersection −K X (K X − D) = K X D = −1 implies that K X − D is not effective, since −K X is nef.
Therefore h 2 (D) = h 0 (K X − D) = 0 and in order for χ(D) = 1 to be fulfilled, D must have at least one nontrivial global section, i.e. D must be effective. We write D = i k i C i , where the C i are pairwise distinct integral curves in X and the k i are positive integers. From the equation 1 = −K X D = i k i (−K X )C i and the nefness of −K X we derive that among the curves C i there is one C 0 occurring with coefficient 1 and satisfying −K X C 0 = 1. All other C i lie in K ⊥ X . Note that by [Fer05,Prop. 2.3] any integral curve with negative self-intersection is a (−1)-curve. Therefore no curve in K ⊥ X can have negative self-intersection, as for (−1)-curves the intersection with the canonical class is nonzero. Hence, in order to achieve −1 = D 2 we must have C 2 0 = −1, C 2 i = 0 and C 0 C i = 0 for all i = 0. Let A := D − C 0 . Then A ∈ K ⊥ X and A 2 = 0. But this implies A = nK X , since any isotropic vector in K ⊥ X is a multiple of K X . Now C 0 K X = −1 together with C 0 A = 0 implies n = 0 and hence C 0 = D. [Nag60,Prop. 12] and assume that the points are in a position described in [Nag60,Prop. 9] to ensure that the surface carries no integral curve C with C 2 ≤ −2.
Remark 4.5. We will see in Section 5 that the conclusion of Lemma 4.3 does not hold for blow-ups of 10 or more points.
Roots in the Picard lattice.
Recall that any unimodular lattice Λ contains a root system with roots given by the elements α ∈ Λ such that α 2 = ±1 or α 2 = ±2. For such a root α ∈ Λ the reflection along α ⊥ is given by
s α (x) := x − 2 (x · α) α 2 α.
Any such reflection is an orthogonal transformation of Λ.
Let X be the blow-up of P 2 in n ≥ 3 points. Let H be the pullback of a hyperplane class and let E 1 , . . . , E n be the exceptional divisors. Then H, E 1 , . . . , E n is an orthogonal basis of the Picard lattice Pic(X) such that H 2 = 1 and E 2 i = −1. The elements α 1 := E 1 −E 2 , . . . , α n−1 := E n−1 −E n and α 0 := H −E 1 −E 2 −E 3 are roots in Pic(X) and we denote by W X the reflection group generated by s αi , 0 ≤ i ≤ n − 1. All roots α i lie in K ⊥ X , thus
W X ⊆ O(Pic(X)) KX ⊆ O(Pic(X)),
where O(Pic(X)) KX is the stabilizer of the canonical class K X = −3H + i E i . Lemma 4.6. Let X be the blow-up of P 2 in n points, where 3 ≤ n ≤ 9. Then the reflection group W X = s α0 , . . . , s αn−1 equals the stabilizer O(Pic(X)) KX .
Proof. First note that the equality W X = O(Pic(X)) KX does not depend on the position of points. Thus, we can assume that the points lie in very general position and −K X is class of an irreducible reduced curve in X. Then Lemma 4.3 (or [EL16,Lem. 3.4] if n ≤ 8) implies that any orthogonal transformation in O(Pic(X)) KX maps (−1)-curves to (−1)-curves. Hence, by [Har85, Thm. 0.1], which is essentially a reformulation of results in [Nag60], any such transformation is an element of W X and thus the lemma holds.
4.3.
A weak del Pezzo surface admitting a numerically exceptional collection of maximal length which is not exceptional. We cannot expect that the conclusion of Lemma 4.3 holds true for rational surface of higher Picard rank, as we show in Section 5. But already if we blow up less than 9 points in special position, the conclusion of Lemma 4.3 does not hold. As a consequence, in general a maximal numerically exceptional collection does not need to be exceptional. In Proposition 4.7 we construct such an example by blowing up 8 points in a special position. Similar examples were already obtained for Hirzebruch surfaces Σ d with even d in [EL16, Rmk. 2.18].
Proposition 4.7. Let π : X → P 2 be the blow-up of 8 points p 1 , . . . , p 8 such that p 1 , p 2 , p 3 lie on a line L and p 4 , . . . , p 8 on a smooth irreducible conic curve C such that p 1 , p 2 , p 3 / ∈ C and p 4 , . . . , p 8 / ∈ L. Then X is a weak del Pezzo surface, i.e. −K X is nef and big, but admits a maximal numerically exceptional collection consisting of line bundles which is not exceptional. Moreover, X admits an effective divisor D satisfying D 2 = −1, χ(D) = 1 and H 1 (X, O X (−D)) = 0.
Proof. Denote by E i the exceptional divisor corresponding to the blow-up of p i and let H be the pullback of the hyperplane class in P 2 . Then the anticanonical divisor satisfies −K X = 3H − 8 i=1 E i and thus is equal to the sum of the strict transformL of L and the strict transformC of C. Hence, the intersection of −K X with any other curve is non-negative and one checks that −K XL = 0 is zero and −K XC = 1. Therefore −K X is nef and hence (−K X ) 2 = 1 > 0 implies that −K X is big. Consider the divisor
D := 4H − 2E 1 − 2E 2 − 2E 3 − E 4 − · · · − E 8 ,
which satisfies D 2 = −1 and −K X D = 1. Arguing as in the proof of Lemma 4.3 one observes that D is effective. But D is not irreducible, which can be seen as follows: Assume for contradiction that D is irreducible. Since D is not one of the exceptional divisors E i , D must be the strict transform of a curve B in Finally, we complete D into a toric system in order to obtain a maximal numerically exceptional collection consisting of line bundles. The set of orthogonal transformations of Pic(X) fixing the canonical class K X coincides with the orthogonal group of K ⊥ X . A computation shows that K ⊥ X identifies with the E 8 -lattice and therefore the orthogonal group is the Weyl group of E 8 . It is known that the Weyl group acts transitively on the set of roots; see, e.g., [Hum78, § 10.4 Lem. C]. Further, we can write the exceptional divisor as E 1 = (K X + E 1 ) − K X and compute (K X + E 1 ) 2 = −2. Hence, there is an orthogonal transformation T fixing K X and sending the root
K X + E 1 to the root T (K X + E 1 ) = H − E 1 − E 2 − E 3 . Thus, E 1 is mapped to D = H − E 1 − E 2 − E 3 − K X under T .
Therefore, the image of the toric system associated to
D b (X) = O X , O X (E 1 ), . . . , O X (E 8 ), O X (H), O X (2H)
under T is a toric system which corresponds to a maximal numerically exceptional collection consisting of line bundles, which is not exceptional since H 1 (X, O X (D)) = 0.
4.4. Towards Theorem 1.3. The proof of Theorem 1.3 is separated in two steps. Recall that [EL16,Thm. 3.1] states that, on a del Pezzo surface, any toric system is obtained from a sequence of augmentations from an exceptional toric system on P 2 or a Hirzebruch surface. In the first step, we generalize this result to the blow-up X of 9 points in very general position. In the second step, we prove the transitivity of the braid group action, as stated in Theorem 1.3, by realizing each orthogonal transformation of Pic(X) fixing K X as a sequence of mutations.
The following Lemma 4.8 ensures that we can reduce X to a del Pezzo surface by contracting any (−1)-curve. As we were unable to find a suitable statement in the literature, we include a proof.
Lemma 4.8. Let X be the blow-up of P 2 in 9 points in very general position and let E ⊆ X be a (−1)-curve. Then the surface Y obtained from blowing down E is a del Pezzo surface.
Proof. Recall that the blow-up of less than 8 points in P 2 is a del Pezzo surface if and only if not 3 of the points lie on a line and not 6 lie on a conic; see, e.g., [Man86,Thm. 24.4]. Therefore the points are in special position if and only if the surface admits a (−2)-curve, namely the strict transform of a conic through 6 blown up points or the line through 3 blown up points. We further observe that the equivalence also holds true if the points are chosen infinitely near: If a point p is blown up on an exceptional divisor E, then the class of the strict transform of E is E − E p , where E p is the exceptional divisor corresponding to the blow-up of p. We compute (E − E p ) 2 = −2 in that case.
Let Y be the blow-down of the (−1)-curve and π : X → Y the blow-up map with center p ∈ Y and exceptional divisor E. Then for any curve C in Y , the strict transform in X has divisor class p * C − mE, where m is the multiplicity of C at p. Thus the self-intersection of the strict transform of C is C 2 − m. Hence, if X has no (−2)-curves, then Y has no (−2)-curves. Now Y is obtained from P 2 by a sequence of blow-ups of (possibly infinitely near) 8 points. As Y has no (−2)-curves, Y must be a del Pezzo surface.
Theorem 4.9. Let X be the blow-up of P 2 in 9 in very general position. Any toric system on X is a standard augmentation, i.e. it is obtained by a sequence of augmentations from a full exceptional toric system on P 2 or from a full exceptional toric system on a (non necessarily minimal) Hirzebruch surface.
Proof. Let A 1 , . . . , A 12 be a toric system on X. By Lemma 4.3, Lemma 4.8, and [EL16, Thm. 3.1] we only need to show that there is a divisor A i with A 2 i = −1. In this situation the argument of Elagin-Lunts still applies: By [HP11, Prop. 2.7] there exists a smooth toric surface Y with torus invariant irreducible divisors D 1 , . . . , D 12 such that D 2 i = A 2 i for any i. Since Y is not minimal, Y contains (−1)-curve which must be torus invariant as otherwise the self-intersection would be non-negative. We conclude that one of the D i squares to −1, hence there exists A i with A 2 i = −1.
Corollary 4.10. On the blow-up of P 2 in 9 very general points any numerically exceptional collection of maximal length consisting of line bundles is a full exceptional collection.
Proof. By [EL16, Prop. 2.21] a standard augmentation corresponds to a full exceptional collection.
In order to conclude the proof of Theorem 1.3 we are left to show that any two full exceptional collections resulting from two different sequences of augmentations are related by mutations and shifts. On a del Pezzo surface, an exceptional object is completely determined by its class in the Grothendieck group:
Lemma 4.11 (Exceptional objects on del Pezzo surfaces, [Gor88;KO94]). Let X be a del Pezzo surface and let E ∈ D b (X) be an exceptional object. Then E is isomorphic to some F [k], where F is an exceptional sheaf on X and k ∈ Z. Moreover F is either locally free or a torsion sheaf of the form O C (d), where C is a (−1)-curve. In particular, two exceptional objects with the same image in K 0 (X) only differ by an even number of shifts.
Pointer to references. That every exceptional object is a sheaf up to shift can be found in [KO94,Prop. 2.10] and [KO94,Prop. 2.9] states that an exceptional sheaf is locally free or a torsion sheaf of the form O C (d) where C is a (−1)-curve. In the latter case, such torsion sheaf is clearly uniquely determined by their Chern character and hence by their class in K 0 (X). The case of locally free sheaves is treated in [Gor88,Cor. 2.5].
For later use in the proof of Theorem 4.17 we compute in the following Lemma 4.12 a relation by mutations and shifts of two concrete exceptional collections on the blow-up of P 2 in 3 points. The statement of Lemma 4.12 can also be deduced from [KO94,Thm. 7.7]. We give an independent proof by computing an explicit sequence of mutations relating both collections.
Lemma 4.12. Let X be the blow-up of 3 points in P 2 which do not lie on a line. Then the full exceptional collections
D b (X) = O E1 (−1), O E2 (−1), O E3 (−1), O X , O X (H), O X (2H) and D b (X) = O H−E2−E3 (−1), O H−E1−E3 (−1), O H−E1−E2 (−1), O X , O X (2H − E 1 − E 2 − E 3 ), O X (4H − 2E 1 − 2E 2 − 2E 3 )
are related by mutations and shifts.
Proof. Since X is a del Pezzo surface it is enough to verify the claim in K 0 (X) by using Lemma 4.11. In K 0 (X) this becomes a lattice-theoretic computation: Let
a i := [O Ei (−1)] and b 1 := [O X ], b 2 := [O X (H)], b 3 := [O X (2H)].
Then the Gram matrix corresponding to the basis (a 1 , a 2 , a 3 , b 1 , b 2
, b 3 ) is (4.13) 1 0 0 −1 −1 −1 0 1 0 −1 −1 −1 0 0 1 −1 −1 −1 0 0 0 1 3 6 0 0 0 0 1 3 0 0 0 0 0 1 .
Similarly to (3.8) we have the following sequence of mutations
(a 1 , a 2 , a 3 , b 1 , b 2 , b 3 ) L5,6 −−→ R3,4 −−→ R2,3 −−→ L4,5 −−→ L3,4 −−→ R2,3 −−→ R1,2 −−→ L2,3 −−→ R3,4 −−→ R4,5 −−→ L2,3 −−→ L3,4 −−→ R5,6 −−→ (a 2 + a 3 + 2b 1 − 3b 2 + b 3 , −a 1 − a 3 − 2b 1 + 3b 2 − b 3 , −a 1 − a 2 − 2b 1 + 3b 2 − b 3 ,
a 1 + a 2 + a 3 + 3b 1 − 3b 2 + b 3 , b 2 , a 1 + a 2 + a 3 + 2b 1 − 3b 2 ).
After changing the sign of the first and last basis elements we obtain the exceptional basis
(−a 2 − a 3 − 2b 1 + 3b 2 − b 3 , −a 1 − a 3 − 2b 1 + 3b 2 − b 3 , −a 1 − a 2 − 2b 1 + 3b 2 − b 3 , (4.14) a 1 + a 2 + a 3 + 3b 1 − 3b 2 + b 3 , b 2 , −a 1 − a 2 − a 3 − 2b 1 + 3b 2 ),
which has Gram matrix (4.13). Recall that the Chern character on a surface is given by Thus (4.14) corresponds to the full exceptional collection (4.15)
ch = rk, c 1 , 1 2 (c 2 1 −2 c 2 ) .O H−E2−E3 , O H−E1−E3 , O H−E1−E2 , O X (−H + E 1 + E 2 + E 3 ), O X (H), O X (3H − E 1 − E 2 − E 3 ) .
We observe that H − E i − E j is the class of the strict transform of the line through the points p i and p j and K X can be rewritten as
K X = −3H + E 1 + E 2 + E 3 = −3(2H − E 1 − E 2 − E 3 ) + (H − E 2 − E 3 ) + (H − E 1 − E 3 ) + (H − E 1 − E 2 ), where 2H − E 1 − E 2 − E 3
can be identified with the pullback of a hyperplane class on P 2 considered as the blow-down of (
H − E 2 − E 3 ), (H − E 1 − E 3 ) and (H − E 1 − E 2 ).
Hence
O H−Ei−Ej (K X ) = O H−Ei−Ej (H − E i − E j ) = O H−Ei−Ej (−1),
where we have used the projection formula in the first equality. Recall that any twist with an integer multiple of the canonical line bundle can be realized as a sequence of mutations. Twisting (4.15) by K X yields
O H−E2−E3 (−1), O H−E1−E3 (−1), O H−E1−E2 (−1), O X (−4H + 2E 1 + 2E 2 + 2E 3 ), O X (−2H + E 1 + E 2 + E 3 ), O X .
Finally, by applying the sequence R 5,6 • R 4,5 • R 5,6 • R 4,5 of mutations, we obtain the desired full exceptional collection
D b (X) = O H−E2−E3 (−1), O H−E1−E3 (−1), O H−E1−E2 (−1), O X , O X (2H − E 1 − E 2 − E 3 ), O X (4H − 2E 1 − 2E 2 − 2E 3 ) .O X (D 1 ) ⊗ RHom(O X (D 1 ), O X (D 2 )) ev − → O X (D 2 ) → L OX (D1) (O X (D 2 )).
On the other hand
RHom(O X (D 1 ), O X (D 2 )) = H • (X, O X (E)) = C[0].
Therefore the ideal sheaf sequence
0 → O X (−E) → O X → O E → 0
yields an exact triangle
O X (D 1 ) ⊗ RHom(O X (D 1 ), O X (D 2 )) → O X (D 2 ) → O E (D 2 ).
As E is isomorphic to a projective line, we conclude that (4.18) is equivalent to
(4.19) D b (X) = O E (d), O X (D 1 ), O X (D 3 ), . . . , O X (D n )
for some d ∈ Z. Let p : X → X ′ be the blow-down of E; then K X = p * K X ′ + E. Using the projection formula to compute
O E (d)⊗ O X (K X ) = O E (d)⊗ O X (E)
, we can assume that d = −1 by twisting (4.19) with (d − 1)K X . This means that (4.18) is equivalent to a full exceptional collection obtained by the blow-up formula from a del Pezzo surface X ′ . Now Lemma 4.11 together with Theorem 3.1 implies that we can assume that the exceptional collection on X ′ comes from iterated blow-ups of a copy of P 2 . After repeating the computations as above for every (−1)-curve in the toric system from O X (D 1 ), O X (D 3 ), . . . , O X (D n ) we can assume that (4.19) is equivalent to a collection
O E ′ 1 (−1), O E ′ 2 (−d 2 ), . . . , O E ′ 9 (−d 9 ), O X (nH ′ ), O X ((n + 1)H ′ ), O X ((n + 2)H ′ ) , where E ′ 1 , .
. . , E ′ 9 are pairwise disjoint (−1)-curves with E ′ i H ′ = 0 and H ′2 = 1. Twisting the partial sequence
O E ′ 2 (−d 2 ), . . . , O E ′ 9 (−d 9 ), O X (nH ′ ), O X ((n + 1)H ′ ), O X ((n + 2)H ′ ) with K X ′ can be realized as a sequence of mutations, because (−⊗O X (K X ′ )[2]) is the Serre functor of O E ′ 2 (−d 2 ), . . . , O E ′ 9 (−d 9 ), O X (nH ′ ), O X ((n + 1)H ′ ), O X ((n + 2)H ′ ) ∼ = D b (X ′ )
. Thus we can assume d 2 = 1 and repeating this procedure, we can assume that d i = 1 for all i. We have an equivalence
O X (nH ′ ), O X ((n + 1)H ′ ), O X ((n + 2)H ′ ) ∼ = D b (P 2 ), where H ′ is identified with a hyperplane class. On P 2 we compute that O P 2 , O P 2 (H), O P 2 (2H) is equivalent to O P 2 (H), O P 2 (2H), O P 2 (−K P 2 ) = O P 2 (3H) ,
thus in our situation we can assume that n = 0. Therefore E ′ 1 , . . . , E ′ 9 , H ′ can be obtained from E 1 , . . . , E 9 , H by applying an orthogonal transformation of Pic(X) fixing the canonical class −3H +
i E i = −3H ′ + i E ′ i . It remains to show that the two sequences D b (X) = O X , O X (E 1 ), . . . , O X (E 9 ), O X (H), O X (2H) and D b (X) = O X , O X (E ′ 1 ), . . . , O X (E ′ 9 ), O X (H ′ )
, O X (2H ′ ) are equivalent. As explained in Section 4.2, by [Har85, Thm. 0.1] and Lemma 4.3 the group O(Pic(X)) KX coincides with the Weyl group generated by the reflections induced by the simple roots E 1 − E 2 , . . . , E 8 − E 9 , and H − E 1 − E 2 − E 3 . The reflection along the hyperplane orthogonal to the a (−2)-class v is given by
s v (x) = x + (x · v)v. Thus if v = E i − E i+1 ,(E i ), O X (E i+1 ) . Assume v = H − E 1 − E 2 − E 3
; then s v fixes E 4 , . . . , E 9 . Computing the corresponding mutation (on the blow-up of 3 points for simplicity) one observes that the full exceptional collection
D b (X) = O X , O X (E 1 ), O X (E 2 ), O X (E 3 ), O X (H), O X (2H) is changed to D b (X) = O X , O X (H − E 2 − E 3 ), O X (H − E 1 − E 3 ), O X (H − E 1 − E 2 ), O X (2H − E 1 − E 2 − E 3 ), O X (4H − 2E 1 − 2E 2 − 2E 3 ) .
This is the full exceptional collection obtained by the blow-up formula after blowing down the strict transforms of the lines through 2 of the points. By Lemma 4.12 this simple reflection can also be realized as a sequence of mutations and shifts. In general an element of the Weyl group is a composition of simple reflections s v1 • · · · • s vn . Recall that for reflections s v • s w • s v = s sv (w) holds. This gives s sv(w) • s v = s v • s w . Applying this to our composition of simple reflections we can rewrite s v1 • · · · • s vn = s sv 1 (v2) • · · · • s sv 1 (vn) • s v1 .
We conclude now by induction: After realizing s v1 by mutations and shifts, s sv 1 (v2) • · · · • s sv 1 (vn) is a sequence of n − 1 simple reflections with respect to the new basis of simple roots obtained after applying s v1 . Hence it can be realized as a sequence of mutations and shifts.
As a corollary we obtain a new proof of a result of Kuleshov-Orlov.
Corollary 4.20 (cf. [KO94,Thm. 7.7]). Let X be a del Pezzo surface, then any two full exceptional collections on X are related by mutations and shifts.
Proof. Recall that X is either P 1 × P 1 or a blow-up of less than 8 points in P 2 in general position. Given the latter case, suppose E • and F • are two full exceptional collections on X. By Theorem 1.2 we can assume that E • and F • consist of rank 1 objects. Now by Lemma 4.11 exceptional rank 1 objects on X are line bundles and we argue as in the proof of Theorem 4.17.
Assume X = P 1 × P 1 , then Pic(X) = ZH 1 ⊕ ZH 2 with H 1 H 2 = 1 and H 2 1 = H 2 2 = 0 and K X = −2H 1 − 2H 2 . One computes that the orthogonal transformations of Pic(X) fixing K X are exactly the permutations of H 1 and H 2 . Let E • be a full exceptional collection on X. As before we can assume that E • is a sequence consisting of line bundles. By Theorem 1.2 and Lemma 2.10 E • has the form
O X (a, b), O X (a + 1, b), O X (a, b + 1), O X (a + 1, b + 1) or O X (a, b), O X (a, b + 1), O X (a + 1, b), O X (a + 1, b + 1) .
Both are equivalent as the mutation L 2,3 permutes the middle factors. One computes that the right mutation R OX (a+1,b+1) (O X (a, b + 1)) is equal to O X (a + 2, b + 1) up to possible shifts and similarly R OX (a+1,b+1) (O X (a + 1, b)) identifies with O X (a + 1, b + 2). We deduce that E • is equivalent to O X (a, (b + 1)), O X (a + 1, (b + 1)), O X (a, (b + 1) + 1), O X (a + 1, (b + 1) + 1) , hence we realized the twist by O X (0, 1) as a sequence of mutations. Analogously one obtains that the twist by O X (1, 0) is a sequence of mutations and therefore E • is equivalent to O X (0, 0), O X (1, 0), O X (0, 1), O X (1, 1) .
Corollary 4.21. Let X be a smooth projective surface over a field k with χ(O X ) = 1 and rk K num 0 (X) ≤ 12. Then any two exceptional bases e • and f • of K num 0 (X) are related by a sequence of mutations and sign changes.
Proof. By Vial's classification, see Theorem 2.12, we can assume that X is a del Pezzo surface or the blow-up of P 2 in 9 points. In these cases K num 0 (X) is independent from the base field and the position of points, thus we can assume that the base field is C and the blown up points are in very general position. Moreover, by Perling's algorithm, see Theorem 2.15, we can assume that e • and f • only consist of rank 1 objects. Recall that for a numerically exceptional object E ∈ D b (X) the Riemann-Roch formula implies c 2 (E) = 0, thus we may assume that e • and f • arise from two numerically exceptional collections of maximal length consisting of line bundles. Hence, the corollary follows from Theorem 4.17 and Corollary 4.20.
Blow-up of 10 Points
Although the situation of 9 blown up points is similar to the case of del Pezzo surfaces, the situation changes if we blow up 10 points. In fact, the conclusions of Lemma 4.3 and Lemma 4.6 do not hold for the blow-up of 10 points.
Lemma 5.1. Let X be the blow-up of P 2 in 10 points. Then the stabilizer of the canonical class is
O(Pic(X)) KX = W X × ι ,
where W X is the reflection group generated by the simple reflections corresponding to the roots H − E 1 − E 2 − E 3 , E 1 − E 2 , . . . , E 9 − E 10 and ι is the involution of Pic(X) fixing K X and given by multiplication of −1 on K ⊥ X .
Proof. Denote the roots by α 0 := H − E 1 − E 2 − E 3 , α 1 := E 1 − E 2 , . . . , α 9 := E 9 − E 10 . Since K 2 X = −1, Pic(X) splits as an orthogonal direct sum Pic(X) = K ⊥ X ⊕ ZK X . One can compute that a basis of K ⊥ X is given by the roots α 0 , . . . , α 9 . This shows that K ⊥ X is an even unimodular lattice of signature (1, 9). It is known that II 9,1 (−1) is the unique even unimodular lattice of signature (1, 9). Its orthogonal group was computed by Vinberg [Vin75]. Vinberg's result was rewritten by Conway-Sloane which use the description of II 9,1 as the set x = (x 0 , . . . , x 9 ) ∈ Z 10 ∪ (Z + 1/2) 10 | x 0 + · · · + x 8 − x 9 ∈ 2Z ⊆ Q 10 with bilinear form (x, y) := 8 i=0 x i y i − x 9 y 9 . Now [CS99, § 27 Thm. 1] states that O(II 9,1 ) = W II9,1 × {± id II9,1 }, where W II9,1 is the Weyl group of the root system in II 9,1 with simple roots We observe that sending α 0 → β 0 , α i → β i−2 for 3 ≤ i ≤ 9, and α i → β i+7 for i = 1, 2 yields a suitable isomorphism of lattices K ⊥ X ∼ − → II 9,1 (−1) such that the α i are send to the simple roots β i . Clearly O(Pic(X)) KX = O(K ⊥ X ), thus the lemma follows.
A further computation shows that is a numerically exceptional collection of maximal length on X. We show in [Kra23] that the collection is exceptional but not full. The divisors D i are not effective but satisfy D 2 i = −1 and χ(D) = 1. This shows that the conclusion of Lemma 4.3 does not hold for blow-ups of 10 or more points.
Proposition 5.2. Let X be the blow-up of P 2 in 10 points in general position. Then Z 13 ⋊ B 13 does not act transitively on the set of exceptional collections of length 13.
Proof. Mutations and shifts do not change the generated subcategory of an exceptional collection. Thus the existence of a full and of a non-full exceptional collection of the same length shows that the action cannot be transitive.
Proposition 5.3. Let X be a smooth projective surface over a field k with χ(O X ) = 1 and rk K num 0 (X) = 13 such that K num 0 (X) admits an exceptional basis. Then the action of {±1} 13 ⋊ B 13 has at most 2 orbits.
Proof. Without loss of generality, we assume that X is the blow-up of P 2 in 10 points in general position. Applying Theorem 3.1, we know that each orbit contains an exceptional basis of the form and ϕ ∈ O(Pic(X)) KX . By Lemma 5.1, either ϕ ∈ W X or ϕ can be written as ι • w for some w ∈ W X . Thus, it is enough to show that for each ϕ ∈ W X the collections Remark 5.4. An characterization similar to Lemma 4.11 of exceptional objects in D b (X), where X is the blow-up of P 2 C in 10 points in general position, would be of particular interest. More precisely, if one could verify condition (b) from Section 1 for exceptional collections of maximal length on X, one could conclude that there are 2 orbits of the {±1} 13 ⋊ B 13 -action on exceptional bases of K num 0 (X). One orbit would consist of the images of full exceptional collections and the the other orbit of the images of exceptional collections of length 13 which are not full.
Example 2 . 4 (
24Pseudolattices from surfaces). Let S be a smooth projective surface over a field k which admits a k-valued point i : {x} ֒→ S. For example if S is rational, the existence of a k-valued point is guaranteed by the Lang-Nishimura Theorem. Let G := K num 0 (S) be the numerical Grothendieck group together with its Euler pairing. Then the class of the skyscraper sheaf i * k(x) = O x is a point-like element in G. An exceptional basis of G is the same as the image of a numerically exceptional collection of maximal length on S in K num 0 (S).
see, e.g., [EL16, Lem. 2.1]. For objects E, F ∈ D b (X) with e := rk E and f := rk F , Riemann-Roch yields
num 0 (
0X) for some surface X with χ(O X ) = 1 and choose the initial v 0 to be the class −[O X ]. For any numerically exceptional object E of rank 1, Riemann-Roch (2.7) implies that c 2 (E) = 0. Since
the condition c 2 (E) = 0 implies [E] = −[O X (c 1 (E))] in K num 0 (X). Now twisting with O X (c 1 (E)) defines an isometry of G which maps v 0 = −[O X ] to [E]. Let v i = [F ] for an object F ∈ D b (X). Then ch(F ) = (0, c 1 (F ), d), d ∈ Q. Multiplicativity of the Chern character gives ch(F (c 1 (E))) = (0, c 1 (F ), d) · (1, c 1 (E), d ′ ) = (0, c 1 (F ), c 1 (F ) c 1 (E) + d), for some d ′ ∈ Q. We observe that the first Chern class of F is invariant under twisting with a line bundle and also twisting with a line bundle does not change the point-like element defined by a skyscraper sheaf. Thus, twisting with O X (c 1 (E)) is the unique automorphism of G which maps v 0 to v ′ 0 = [E] and induces the identity on NS(G).
obtained as in Example 2.4 from a surface S which has an exceptional structure sheaf O S , one can compute δ(G) = 0; see [Kuz17, Lem. 5.4]. In general the defect can be interpreted as a suitable numerical replacement of the condition χ(O S ) = 1.
Theorem 2 .
212 ([Via17, Thm. 3.1], [Kuz17, Thm. 5.11])
Lemma 3. 2 .
2Let e • = (e 1 , e 2 , e 3 ) be an exceptional basis of G 0 with Gram matrix M 0 , then a point-like element is given by p := e 3 − 2e 2 + e 1 . Proof. For a closed point i : {x} ֒→ P 2 the skyscraper-sheaf i * k(x) = O x admits a Koszul resolution
Remark 3. 3 .
3The point-like element can also be computed directly from the pseudolattice, using the explicit description of [dTVdB16, Lem. 3.3.2]. Namely if V := G Q , then F 2 V = Im(S − 1) 2 and F 2 V = Qp. Thus p spans the line Im(S − 1) 2 over Q and is primitive. In the case of P 2 one computes S = M
χ(−p, e i ) = 1 for i = 1, 2, 3 .
3Denote by b 1 , . . . , b k , e 1 , e 2 , e 3 the exceptional basis corresponding to this Gram matrix. The elements b i are all orthogonal to p, so of rank zero and the corresponding images in NS(G) have self-intersection −1. Here, we have numerically blown up the point −p in order to obtain only positive signs in the Gram matrix. We first verify Theorem 3.1 in the minimal cases: Proposition 3.4 ([Kuz17, Cor. 4.25]). Any two exceptional bases in K num 0 (P 2 ) are related by mutations up to sign and isometry.
to b 1 , . . . , b 4 , see[Kuz17, Ex. 3.7]. By Proposition 3.5 we can assume that c = 0.
Remark 4 . 4 (
44On the position of the blown up points). The position of the 9 blown up points is important for only two facts: On the one hand we need to choose the position general enough so that there exists a unique cubic passing through the 9 points with multiplicity 1 and on the other hand in the proof of Lemma 4.3 we use the result of [Fer05] which depends on the position of the points. For the latter the assumptions are made more precise in [Fer05, Def. 2.1]. Alternatively one can replace [Fer05, Prop. 2.3] by
P 2 .
2Now by Bézout's theorem B · L = deg(B) deg(L) = deg(B) or B = L. The latter cannot occur sinceL has class H − E 1 − E 2 − E 3 = D. By the explicit form of D, we must have deg(B) = 4 and the multiplicity of B at p i must be 2 for i = 1, 2, 3. Hence, B · L ≥ 6, which contradicts to B · L = deg(B). Thus, D cannot be irreducible. Further we compute h 1 (−D) = 1: Riemann-Roch yields χ(−D) = 0 and since D is effective, −D admits no global sections. This gives h 1 (−D) = h 2 (−D) = h 0 (K X + D) = h 0 (H − E 1 − E 2 − E 3 ) = 1. Thus the conclusion of Lemma 4.3 does not hold for D.
a i ) = ch(O Ei (E i )) = 0, E i , − 1 2 , ch(b 1 ) = ch(O X ) = (1,0, 0), ch(b 2 ) = ch(O X (H)) = 1, H, 1 2 , ch(b 3 ) = ch(O X (2H)) = (1, 2H, 2).
Remark 4 . 16 (
416Geometric interpretation of Lemma 4.12). The surface X in Lemma 4.12 admits two different blow-up realizations. First one blows up 3 points p 1 , p 2 , p 3 in P 2 and then one contract the (−1)-curves H − E i − E j which are the strict transforms of the lines through the points p i , p j . The full exceptional collections compared in Lemma 4.12 are the collections resulting from Orlov's blow-up formula applied to these different realizations of X. Moreover, this construction defines a birational map P 2 P 2 , which is known as standard quadratic Cremona transformation.
Theorem 4 . 17 .
417On the blow-up X of P 2 in 9 points in very general position, any two full exceptional collections consisting of line bundles are related by mutations and shifts.Proof. For the sake of simplicity we call two full exceptional collections equivalent if they can be transformed into each other by a sequence of mutations and shifts. Let
( 4 .
418) D b (X) = O X (D 1 ), . . . , O X (D n ) be a full exceptional collection consisting of line bundles, then O X (D 2 ), . . . , O X(D n ), O X (D 1 − K X )is an equivalent collection. In particular, any twist with an integer multiple of the canonical class can be realized as a sequence of mutations and shifts. By Theorem 4.9 the toric system associated to (4.18) contains a (−1)-curve. After passing to an equivalent collection, we can assume that E := D 2 − D 1 is a (−1)-curve. The left mutation of the pair O X (D 1 ), O X (D 2 ) is defined by the exact triangle
D
i := ι(E i ) = −6H + 2 10 j=1 E j − E i and F := ι(H) = −19H + 6 10 i=1 E i . Thus O X , O X (D 1 ), . . . , O X (D 10 ), O X (F ), O X (2F )
(
[O X (D 1 )], . . . , [O X (D 13 )]), such that D 2 − D 1 = ϕ(A 1 ), D 3 − D 2 = ϕ(A 2 ), . . . , D 13 − D 12 = ϕ(A 12 ),where (A 1 , . . . , A 13) is the toric system associated to the collectionD b (X) = O X , O X (E 1 ), . . . , O X (E 10 ), O X (H), O X (2H)
(
[O X (D 1 )], . . . , [O X (D 13 )]) and ([O X ], [O X (E 1 )], . . . , [O X (E 10 )], [O X (H)], [O X (2H)]) lie in the same orbit. As ϕ ∈ W X sends (−1)-curves to (−1)-curves, D 2 − D 1 , . . . , D 11 − D 1 is a set of disjoint (−1)-curves. Thus, we can argue as in Theorem 4.17 to reduce to showing that ([O X ], [O X (E 1 )], . . . , [O X (E 10 )], [O X (H)], [O X (2H)]) and ([O X ], [O X (ϕ(E 1 ))], . . . , [O X (ϕ(E 10 ))], [O X (ϕ(H))], [O X (ϕ(2H))]) lie in the same orbit. But as ϕ can be factored in a sequence of simple reflections, this follows with the same argument as in Theorem 4.17.
then s v fixes E 1 , . . . E i−1 , E i+2 , . . . , H and permutes E i and E i+1 . This can be identified with a mutation of the exceptional pair O X
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. Mathematik Fakultät Für, address: [email protected], Germany EmailUniversität BielefeldFakultät für Mathematik, Universität Bielefeld, D-33501 Bielefeld, Germany Email address: [email protected]
| []
|
[
"Direct detection of spin polarization in photoinduced charge transfer through a chiral bridge †",
"Direct detection of spin polarization in photoinduced charge transfer through a chiral bridge †"
]
| [
"Alberto Privitera ",
"Emilio Macaluso ",
"Alessandro Chiesa ",
"Alessio Gabbani ",
"Davide Faccio ",
"Demetra Giuri ",
"Matteo Briganti ",
"Niccolò Giaconi ",
"Fabio Santanni ",
"Nabila Jarmouni ",
"Lorenzo Poggini ",
"Matteo Mannini ",
"Mario Chiesa ",
"Claudia Tomasini ",
"Francesco Pineider ",
"Enrico Salvadori ",
"Stefano Carretta ",
"Roberta Sessoli "
]
| []
| []
| It is well assessed that the charge transport through a chiral potential barrier can result in spin-polarized charges. The possibility of driving this process through visible photons holds tremendous potential for several aspects of quantum information science, e.g., the optical control and readout of qubits. In this context, the direct observation of this phenomenon via spin-sensitive spectroscopies is of utmost importance to establish future guidelines to control photo-driven spin selectivity in chiral structures.Here, we provide direct proof that time-resolved electron paramagnetic resonance (EPR) can be used to detect long-lived spin polarization generated by photoinduced charge transfer through a chiral bridge. We propose a system comprising CdSe quantum dots (QDs), as a donor, and C 60 , as an acceptor, covalently linked through a saturated oligopeptide helical bridge (c) with a rigid structure of $10Å.Time-resolved EPR spectroscopy shows that the charge transfer in our system results in a C 60 radical anion, whose spin polarization maximum is observed at longer times with respect to that of the photogenerated C 60 triplet state. Notably, the theoretical modelling of the EPR spectra reveals that the observed features may be compatible with chirality-induced spin selectivity, but the electronic features of the QD do not allow the unambiguous identification of the CISS effect. Nevertheless, we identify which parameters need optimization for unambiguous detection and quantification of the phenomenon.This work lays the basis for the optical generation and direct manipulation of spin polarization induced by chirality. h CNR-ICCOM, Via Madonna del Piano 10, Sesto Fiorentino, I-50019, Italy † Electronic supplementary information (ESI) available: Synthesis of organic ligands 1, 2, and 3. CdSe Quantum Dots (QDs) fabrication and morphological characterization. Ligand exchange methods. Optical measurements. X-ray photoelectron spectroscopy (XPS) analysis. DFT calculations. Time-resolved Electron Paramagnetic Resonance (trEPR) measurements and best-t simulations.Theoretical modeling. | 10.1039/d2sc03712b | [
"https://arxiv.org/pdf/2205.05353v1.pdf"
]
| 248,693,566 | 2205.05353 | 3960e21b8d12f4327d72abe96a374dfc899f6d70 |
Direct detection of spin polarization in photoinduced charge transfer through a chiral bridge †
Alberto Privitera
Emilio Macaluso
Alessandro Chiesa
Alessio Gabbani
Davide Faccio
Demetra Giuri
Matteo Briganti
Niccolò Giaconi
Fabio Santanni
Nabila Jarmouni
Lorenzo Poggini
Matteo Mannini
Mario Chiesa
Claudia Tomasini
Francesco Pineider
Enrico Salvadori
Stefano Carretta
Roberta Sessoli
Direct detection of spin polarization in photoinduced charge transfer through a chiral bridge †
10.1039/d2sc03712b
It is well assessed that the charge transport through a chiral potential barrier can result in spin-polarized charges. The possibility of driving this process through visible photons holds tremendous potential for several aspects of quantum information science, e.g., the optical control and readout of qubits. In this context, the direct observation of this phenomenon via spin-sensitive spectroscopies is of utmost importance to establish future guidelines to control photo-driven spin selectivity in chiral structures.Here, we provide direct proof that time-resolved electron paramagnetic resonance (EPR) can be used to detect long-lived spin polarization generated by photoinduced charge transfer through a chiral bridge. We propose a system comprising CdSe quantum dots (QDs), as a donor, and C 60 , as an acceptor, covalently linked through a saturated oligopeptide helical bridge (c) with a rigid structure of $10Å.Time-resolved EPR spectroscopy shows that the charge transfer in our system results in a C 60 radical anion, whose spin polarization maximum is observed at longer times with respect to that of the photogenerated C 60 triplet state. Notably, the theoretical modelling of the EPR spectra reveals that the observed features may be compatible with chirality-induced spin selectivity, but the electronic features of the QD do not allow the unambiguous identification of the CISS effect. Nevertheless, we identify which parameters need optimization for unambiguous detection and quantification of the phenomenon.This work lays the basis for the optical generation and direct manipulation of spin polarization induced by chirality. h CNR-ICCOM, Via Madonna del Piano 10, Sesto Fiorentino, I-50019, Italy † Electronic supplementary information (ESI) available: Synthesis of organic ligands 1, 2, and 3. CdSe Quantum Dots (QDs) fabrication and morphological characterization. Ligand exchange methods. Optical measurements. X-ray photoelectron spectroscopy (XPS) analysis. DFT calculations. Time-resolved Electron Paramagnetic Resonance (trEPR) measurements and best-t simulations.Theoretical modeling.
Introduction
The second quantum revolution is unfolding now, exploiting the enormous advancements in our ability to detect and coherently manipulate single quantum objects. 1 In this context, the possibility of controlling the electron spin of molecular qubits 2-4or reading out the information encoded in their spin statesthrough the use of visible photons represents an attractive approach toward the realization of smaller, faster, and more energy-efficient Quantum Information (QI) and spintronic technologies. [5][6][7] An emerging possibility to address this challenge is the use of chirality. [8][9][10] Chiral structures have recently received signicant attention thanks to their spin ltering behaviour in the phenomenon described as Chirality-Induced Spin Selectivity (CISS). 11,12 This mechanism has been adopted to interpret a wide range of experimental results in which chiral systems impart signicant spin selectivity in electron transport through chiral molecules, oligomers, and polymers. [8][9][10][11][12][13][14][15] Notably, the spin selectivity of the CISS effect is exceptionally high, even at room temperature. 16,17 In contrast to alternative methods previously used to achieve spin-to-light (or vice versa) interconversion, the CISS effectwhich operates at the molecular scalehas the potential to reach the sensitivity for the readout of individual spins. 18 The latter appears even more interesting considering that the molecular exibility achievable through chemical tunability allows controlling key features like the qubit-qubit interactions, crucial for implementing quantum gates. 19 Despite the recent experimental and theoretical efforts to rationalize the origin and potentialities of the CISS effect, the phenomenon is not fully understood yet. Most scientic results come from experiments performed on hybrid materials comprising chiral molecules supported on metallic substrates. 8,11,12 Conversely, the implementation of the CISS effect in molecular charge transfer (CT) through the use of light has tentatively been tested only recently via indirect methods, e.g. probing competitive nonradiative and radiative relaxation processes as a function of an external magnetic eld, light polarization, and molecular or helical handedness. 5 The challenge of directly detecting at the molecular level the non-Boltzmann spin populations that arise from photoinduced CISS originates from the lack of suitable donor-chiral bridgeacceptor (D-c-A) systems. The latter must simultaneously show good photoinduced CT efficiencies and an efficient spin-ltering effect through the chiral bridge. 20 In this regard, semiconductor quantum dots (QD) have demonstrated to be robust platforms for investigating the CISS effect. [21][22][23] In addition, systems comprising QD as a donor and an organic molecule as an acceptor have recently been demonstrated to transfer long-lived spin polarization from the photoexcited QD to the organic molecule. 24 The resulting non-Boltzmann spin populations can be investigated in detail through the use of timeresolved electron paramagnetic resonance (trEPR) spectroscopy. 24,25 The development of a sound theoretical framework to rationalize the trEPR spectra of these systems is mandatory for investigating spin selectivity in the CT process through an effective chiral potential. A deeper understanding of spin selectivity would dramatically advance our capabilities to control and harness CISS at a fundamental scale. 10 Here, we directly probe the spin-polarized CT process through a chiral bridge via time-resolved EPR (trEPR) spectroscopy in a model system comprising CdSe QD-chiral bridgefullerene (hereaer QD-c-C 60 ). The studied chiral system is favourable for several reasons: (i) QDs are an effective reservoir of electrons that can be donated via CT towards C 60 , [25][26][27] (ii) C 60 is a good electron acceptor, 26,28 and (iii) the used chiral bridge is a two-unit polypeptide belonging to the most investigated class of chiral linkers in CISS-based spintronic devices. 5,10,12,29 The QD-c-C 60 system is synthesized via a ligand exchange approach and characterized through optical spectroscopies and X-ray photoemission spectroscopy (XPS), the latter allowing a quantitative analysis that conrms the chemisorption of the C 60 chiral derivative. trEPR spectroscopy shows that the photoinduced CT from the QD to the fullerene generates a spin-polarized state assigned to the fullerene radical anion. Detailed modelling of the EPR features indicates that they are compatible with the presence of CISS, albeit the unambiguous detection of CISS is hampered by the equilibration of spin population on the photoexcited QD.
Though not yet conclusive, our investigation represents the rst attempt to probe the spin selectivity of the CISS effect by directly detecting the spin populations of the CT state generated following the photoinduced electron transfer through chiral bridges. This provides a fundamental step to formulate clear and handy guidelines for designing future materials based on CISS and ultimately championing the burgeoning eld of QI science.
Results and discussion
Chiral system engineering and synthesis
The QD-c-C 60 system investigated in this paper is shown in Fig. 1a. We engineered our system to optimize the CT process, which is the main focus of our analysis. CdSe QDs represent a sound model system thanks to their tailorable light absorption (through their size) and excellent electron-donating properties. 30 Prior studies, based on different experimental methods, e.g. photoelectron spectroscopy in air (PESA) and theoretical calculations, have shown that the approximate valence and conduction band energies of CdSe QDs with a diameter of 5 nm are −3.1 and −5.4 eV, respectively. 31 However, it should be noted that different techniques give different band energies, which are also strongly affected by the environment and the ligand nature. Thus, we represent these states as a distribution. The C 60 molecule is one of the best performing electron acceptors, 32 whose frontier energy levels, also distributed in energy, are approximately found at −4.0 and −6.2 eV. 33 The diagram in Fig. 1b also reports approximate energetics and a possible path for the CT in our system. Specically, aer the photoexcitation of the QD, an electron transfer to the C 60 can occur due to the favourable energy levels alignment. If C 60 is photoexcited, a hole transfer will occur, resulting in the same nal CT state (hole in the QD and electron in the C 60 ).
The selected linker, reported in Fig. 1a, is composed of two units comprising L-Alanine and the pseudo amino acid (4S,5R)-4-methyl-5-carboxyl-oxazolidin-2-one (D-Oxd). The graing thiolate group is on the N termination of the peptide chain, while NH 2 (CH 2 ) 2 -fulleropyrrolidine is bound to the C-termination of the polypeptide to give SH-(CH 2 ) 2 -(CO)-(L-Ala-D-Oxd) 2 -NH-(CH 2 ) 2 -fulleropyrrolidine, 1. This choice aims to nd an acceptable compromise between CT efficiency and effective chiral potential for the CT process. CISS can still be observed in a system where CT occurs in all directions, as suggested by our previous work, 18 though it is independent of the chosen enantiomer. Thus, we focused our analysis only on one of the two possible ligand enantiomers. Since the X-ray structure of the C 60 functionalized foldamer 1 is not accessible, to clarify the geometrical conformation of the chiral bridge aer C 60 graing, we performed density functional theory (DFT) geometry optimization of the isolated structure of 1 (see Computational Details in ESI †). In Fig. 1c, the nal optimized structure is reported. Minor deviations from the available X-ray structure of the S-protected oligopeptide lacking the fullerene unit [Tri-S-(CH 2 ) 2 -(CO)-(L-Ala-D-Oxd) 2 -OBn] 34 have been computed (see Fig. S1 †). The small root mean square deviation (RMSD) of 0.516 A between the optimized structure of the oligopeptide moiety in 1 and the crystal structure of the free oligopeptide suggests that the geometry of the pristine chiral bridge is preserved even aer the bonding to the fulleropyrrolidine. This can be rationalized considering that the chiral bridge is relatively rigid despite its shortness and the absence of intermolecular interactions in the crystal.
For this reason, we can conclude that the intramolecular hydrogen bonds between the two peptide units of the linker are enough to stabilize the structure of 1 and that the structure of 1 is reasonably preserved also upon graing on the QD. In addition, the insertion process of individual 1 ligands into a dense layer of capping molecules should prevent the folding of C 60 on the QD. [35][36][37] This is further corroborated by the observation of a trEPR signal (vide infra), suggesting slow charge recombination and, therefore, a signicant distance between the QD and C 60 . 24 Aer thoughtfully selecting the chiral linker, we turn to the synthesis and experimental characterization of the proposed QD-c-C 60 system. CdSe QDs were prepared via the hot injection method reported by Dai et al. with minor modications. 38 The average diameter of the QDs is $5 nm, as shown by TEM analysis. Further details on the synthesis and morphological characterization are reported in ESI. †
The fullerene-functionalized polypeptide 1 was prepared through liquid phase synthesis. The D-Oxd-OBn group was prepared from threonine and coupled with Boc-L-Ala. 39 Fullerene-C 60 was derivatized according to the Prato reaction to afford N-2-aminoethyl-fulleropyrrolidine. 40 Then, Boc-(L-Ala-D-Oxd) 2 -OBn was derivatized by replacing the OBn protecting group with the previously obtained N-2-aminoethyl-fulleropyrrolidine and the N-Boc protecting group with 3-(tritylthio)propanoic acid by standard coupling reactions. Finally, the S-protecting trityl group was removed, 41 and compound 1 was obtained. For further details, see ESI. †
The ligand-exchange reaction of the as-synthesized CdSe QDs capped with trioctylphosphine oxide (TOPO) was favoured by the strong affinity of thiols to the CdSe surface, according to HSAB theory for an X-type ligand. 42 We performed ligand exchange by adding an excess of 1 (10 times higher than the estimated quantity for maximum coverage of the QD surface) to a solution of QDs in chloroform and keeping the mixture overnight under mechanical stirring. Aer the exchange reaction, the newly-formed QDs (QD-c-C 60 system) precipitate, as the presence of C 60 in the ligand drastically reduces their colloidal stability. The change in solubility is the rst evidence of the successful exchange of the native ligands with 1. The QDc-C 60 system was subsequently re-dispersed in a solution of 1,2,4-trichlorobenzene and puried (details in ESI †). For comparative studies, two other molecular systems were synthesized: (i) SH-(CH 2 ) 2 -(CO)-(L-Ala-D-Oxd) 2 -L-Val-OMe (2) to obtain a similar functionalization of the QDs without the C 60 acceptor; (ii) fulleropyrrolidine-(CH 2 ) 2 -NH-COOtBu lacking the thiol graing group (3). For the molecular structures, see Schemes S1 and S2 † in the ESI. † To conrm the success of the ligand-exchange reaction, we started by carrying out optical measurements. In particular, we used UV-vis absorption, steady-state PL, and transient PL to investigate the photophysics of our model system as an initial platform on which to build the study of the spin dynamics mediated by light. In Fig. 2a, we compare the UV-vis and PL spectra of the pristine CdSe QDs and the CdSe QD-c-C 60 system. The UV-vis absorption spectrum of the CdSe QD-c-C 60 system can be rationalized as the sum of two main contributions. The rst originates from the CdSe QDs and shows a clear excitonic peak at around 600 nm, in analogy with the pristine Fig. 1 Engineering of the D-c-A system: system representation, energy levels, and DFT calculations of QD-c-C 60 . (a) Schematic representation of the CdSe QD-c-C 60 system. The QD acts as an electron donor and C 60 as an electron acceptor. The chiral bridge is characterized by a two-units peptidic chain, which offers an effective chiral potential through which the CT process occurs. (b) Energy level diagram of CdSe QD and C 60 . The dotted arrows represent the CT process after light absorption by the CdSe QD. Conversely, a hole transfer process can occur if the C 60 absorbs the light. Both processes generate a hole localized on the CdSe QDs and a radical anion localized on the C 60 . (c) DFT calculations of the ligand 1. The calculations highlight the presence of a rigid structure with a distance from the S atom to the C 60 of about 10Å.
CdSe QDs. The second is a broad absorption tail extending up to 700 nm typical of C 60 derivatives 43 and strongly resembles the UV-vis absorption spectrum of the free 1 ligand shown in Fig. S16. † The PL spectra provide a rst ngerprint of the success of the ligand exchange reaction. Specically, the inset of Fig. 2a shows that the PL intensity is strongly quenched in the CdSe QD-c-C 60 system compared to the pristine CdSe QDs. This quenching is mainly ascribed to the thiol-mediated hole trapping process becoming dominant over radiative recombination. 44 A comparable PL quenching was obtained for CdSe QDs functionalized with 2, where the C 60 molecule is absent, thus conrming the predominant role of thiols in the PL quenching mechanism (see Fig. S17 †).
We further probed the role of 1 ligand on the QD exciton decay by performing time-resolved photoluminescence (trPL) decay measurements on the QDs before and aer the ligand exchange. The luminescence decay curves recorded at the emission maximum (Fig. 2b) show a multiexponential decay that we modelled using triexponential decay kinetics, as shown in eqn (1):
PLðtÞ ¼ a 1 exp À t s 1 þ a 2 exp À t s 2 þ a 3 exp À t s 3(1)
The tting results are reported in Table S1 † in the ESI. † The explanation for the origin of the multiexponential PL decay has been thoroughly discussed in the literature. 45 The main reason is the presence of QD surface defects that give rise to trap states within the bandgap and affect the emission dynamics. [46][47][48] As a result, the photogenerated hole-electron pair exciton can follow different decay paths. 46 The most direct one consists of a rapid relaxation of the hole and the electron to the bottom of the valence and conduction bands, respectively, followed by the radiative relaxation to the ground state. This process contributes to the fastest lifetime decay. However, the hole (or the electron) can be localized in shallow trap states. These trapped charges can either repopulate the valence (or conduction) band or thermalize into deeper trap states. The former case contributes to the longer PL lifetimes, while the latter contributes to nonradiative mechanisms. The combination of all these processes and differences between the individual QDs give rise to multiexponential emission dynamics that occur over a nanosecond time scale, in agreement with literature values for similar systems. 45 In addition, the possibility of charge transfer from the CdSe QD to the C 60 fullerene in the QD-c-C 60 system can also occur, as reported in the next section. To compare the PL lifetimes of the QDs before and aer the ligand exchange, we calculated their average PL lifetime s using eqn (2): 45
s ¼ X i a i s i 2 , X i a i s i(2)
The results are s ¼ 29.0 AE 0.5 ns to 2.6 AE 0.5 ns for the pristine CdSe QDs and the CdSe QD-c-C 60 system, respectively. As expected from steady-state PL, the signicant decrease in the PL lifetimes aer ligand exchange conrms the fast nonradiative decay process induced by the thiol capping molecules, thereby conrming the success of the exchange interaction. 44 We gathered further information on the surface of the functionalized QDs with XPS experiments. We analysed the C 1s and S 2p/Se 3p regions of the pristine CdSe QD and the QD-c-C 60 system since they are the most relevant regions to investigate the chemical functionalization of the CdSe QDs. The analysis of Se 3d and Cd 3d regions is reported in ESI (Fig. S19-S23 †), together with the C 1s and S 2p XPS of the bulk phase of the individual building blocks (1, 2, and C 60 ). The spectra of the C 1s region are reported in Fig. 3c and S19. † In the spectrum of pristine CdSe QDs, the main component at 284.3 eV is attributed to aliphatic carbon atoms of the TOPO ligand, plus a minor component at 285.8 eV attributable to adventitious carbon. 49 Conversely, the C 1s region acquired on the QD-c-C 60 system features the components of both 1 and TOPO ligands (reported in Fig. S19 †), thereby demonstrating the coexistence of both ligands on the surface of CdSe QDs aer the exchange reaction. However, the components belonging to TOPO decrease aer the ligand exchange. Further conrmation of the presence of both ligands on the QDs surface is given by the P 2p XPS signal (132.4 eV) detectable on samples before and aer the exchange reaction (see Fig. S21 †). 50 Crucial insight regarding the assembly of molecules on the surface of the CdSe QDs can be deduced by analysing the S 2p/ Se 3p region ( Fig. 2d and S20 †). In the pristine CdSe QDs sample, the Se 3p signal (159.7 eV) and its relative spin-orbit contribution (+5.7 eV) are clearly visible. 51 In the QD-c-C 60 system, we observe a change in the lineshape of the Se 3p signal due to the overlap with additional components at 161.8 eV (highlighted in orange) attributable to sulfur atoms bound to the surface of the QDs. 52 Furthermore, the spectra do not feature signals at 163.5 eV and ca. 167 eV, which are characteristic of S-H and S-O n groups, thus excluding the presence of both physisorbed and oxidized species of 1.
Spin-polarized photoinduced charge transfer
With the synthesis of the CdSe QD-c-C 60 system conrmed, we turned to trEPR spectroscopy to investigate the photoinduced CT and its spin dynamics. 53,54 trEPR is sensitive to the presence of spin-polarized states, which show signals in enhanced absorption (A) and/or emission (E), 55 and, under favourable circumstances, can reach a time resolution as low as tens of nanoseconds. In trEPR spectroscopy, the detected signals result from non-Boltzmann population of the spin sublevels following the CT process. 20 The generation and time evolution of spin polarization is very informative about the spin dynamics of paramagnetic states. 56,57 We performed trEPR measurements at 40 K on our model system CdSe QD-c-C 60 (7.8 mM in 1,2,4-trichlorobenzene, red line, Fig. 3a). For comparison, we also investigated a solution containing both CdSe QD functionalized with 2 (CdSe QD-c) (7.8 mM) and 1 mM of 3 (black line, Fig. 3a). The concentration of QDs was chosen to have an optical density below 1 at the excitation wavelength used in the EPR experiment (450 nm) in the 0.3 cm EPR quartz tube (UV-vis spectrum shown in Fig. S18 †). In Fig. 3a, we show the trEPR spectra taken at 1 ms aer excitation at 450 nm. In both spectra, we observe a broad signal between 335-358 mT that we assign to the C 60 triplet state formed via intersystem crossing (ISC) promoted by spinorbit coupling (SOC). 25,58 This signal results from photogenerated singlet states in C 60 that do not undergo electron transfer. The C 60 triplet spectra in the two samples show some differences, specically the shoulders in the QD-c-C 60 sample are more pronounced. This small difference is most likely due to a different environment of the C 60 molecule: linked to the CdSe QDs in the target system or dispersed in the frozen solution in the control experiment. The different environment may affect the SOC-promoted ISC in the two C 60 triplets.
More interestingly, only in the spectrum of QD-c-C 60 we observe an additional intense and spectrally narrow feature in enhanced absorption centred at $346 mT (g-value z 2.00), which we attribute to the C 60 radical anion. 25,59 We propose that this absorptive signal results from the photoinduced electron transfer from the QDs to C 60 . Specically, aer the absorption of 450 nm light by the CdSe QDs, an exciton is generated, which undergoes a CT process thanks to the favourable energy alignment (see Fig. 1b). 25 As a result, a hole localized on the CdSe QD and a radical anion localized on the C 60 are formed. In agreement with literature reports, 24 the signal of the counterpart hole on the CdSe QDs is not visible in our spectra due to the fast spin relaxation induced by the large spin-orbit coupling of heavy Cd atoms. The lack of the trEPR signature of the hole was similarly reported by Olshansky et al. 24 As for the radical anion, the enhanced absorption polarization reminds of that recently observed in similar systems. 24,60 However, a signicant difference between our results and previous literature is the time evolution of spin polarization in the rst ms aer the laser pulse, reported in Fig. 3b. In our case, the spin polarization of the C 60 signal rises even more slowly than the polarization of the C 60 ISC triplet and shows a maximum at $1 ms, as further discussed in the theoretical modelling. In addition, most trEPR investigations of the CT process in similar QD/organic molecule systems involve D and A species that are nearby or connected by conjugated bonds, 24,25 while, in our case, the CdSe QD and the C 60 are covalently attached through a $10Å long saturated bridge which in principle hinders the electron transfer process. Notably, our trEPR observation appears even more interesting (1) Experimental trEPR spectra of QD-c-C 60 (red line) and QD-c + C 60 (black line) taken at 1 ms after 450 nm laser pulse (7 ns, 2 mJ) acquired at 40 K. Both spectra show a broad signal between $335-358 mT, which is assigned to the C 60 triplet, but only the spectrum of QD-c-C 60 displays a narrow signal in enhanced absorption centred at $346 mT, which is attributed to the photogenerated C 60 radical anion. Arrows legend: A ¼ enhanced absorption, E ¼ emission. The dashed line represents the zero. (b) Normalized trEPR transients of QD-c-C 60 excited at 450 nm (40 K). The transients are integrated on a magnetic field window of 0.5 mT and are centred at 343 and 346 mT for the C 60 triplet and the photoinduced C 60 radical anion, respectively. The transients show a slower generation of the photoinduced C 60 radical anion compared with the formation of the C 60 triplet. (c) Temperature-dependent trEPR spectra of QD-c-C 60 taken at 1 ms after 450 nm laser pulse (7 ns, 2 mJ). The spectra show a mostly enhanced absorptive signal associated with the C 60 radical anion. The dashed line represents the zero. The transients are available in Fig. S24. † (d) 2D experimental trEPR contour plot of QD-c-C 60 of the charge transfer signal acquired after 450 nm laser pulse (7 ns, 2 mJ) at T ¼ 10 K. Colour legend: red ¼ enhanced absorption, blue ¼ emission, green ¼ baseline.
considering the presence of a chiral bridge which has been suggested to favor the efficiency of the CT process. 5,12 To achieve better insight into the CT process, we investigated the signal of the spin-polarized C 60 anion by performing experiments at different temperatures (10-80 K), as reported in Fig. 3c. The absorptive signal of the C 60 radical anion is observed at all temperatures. Notably, the data show a reduction in the absorptive feature at low elds as the temperature increases. Polarization patterns of similar systems were tentatively rationalized in literature by considering two main contributions: (i) a main absorptive feature that originates from the triplet excited state of the photoexcited QD from which the CT process starts, and (ii) a minor absorption/emission (AE) contribution showing up at lower temperatures (<10 K) which originates from the spin-correlated radical pair (SCRP) mechanism. 24 However, the origin of spin polarization in our and similar systems is still little understood, and it is of paramount importance to theoretically understand the time evolution of the spin polarization of photoinduced CT states. This study appears even more fundamental in view of a rationalization of the role of the chiral linker in the spin-selectivity of the photoinduced CT process.
Simulation of trEPR spectra
In order to gain a deeper understanding of the CT process, we simulated trEPR spectra as a function of both time and static magnetic eld. [61][62][63][64][65] Theoretical modelling focuses solely on the magnetic eld region relevant for the charge transfer signal since the signal associated with the C 60 triplet (see Fig. 3b) is due to a spin-polarized triplet state originating from SOCpromoted ISC independent of CISS. In our simulation, the initial state of the radical pair is described by the density matrix r(0), written in the four-level basis composed by singlet and triplet spin states for the hole-electron pair. The time evolution of r(0) is computed using the stochastic Liouville equation (see ESI † and ref. 61-65 and 81), considering both coherent and incoherent contributions.
vr vt ¼ Ài½H; r À L½r(3)
Coherent evolution is determined by the Hamiltonian H, which in the high-eld approximation can be written in the rotating frame as follows:
H ¼ m B B 0 $(g D S D + g A S A ) + S D $D(U)$S A − ħu 0 (g D S D z + g A S A z ) + m B B 1 (g D S D x + g A S A x )(4)
Where g D and g A are the isotropic g-factors of the donor QD and the C 60 acceptor radical species, B 0 ¼ (0, 0, B 0 ) is the static magnetic eld, D is the spin-spin interaction tensor (including in principle both isotropic and dipole-dipole contributions), u 0 /2p ¼ 9.69 GHz is the microwave frequency and B 1 ¼ 0.02 mT is the microwave eld strength. The spin-spin coupling is much weaker than the Zeeman energy in the examined system, which fully justies the high-eld approximation. Here we have modeled for simplicity the hole on the QD as an isotropic spin 1/2. This framework can be further extended to a more complex spin structure of the hole; however, this would require an extensive characterization of the QD, which is beyond the scope of the present work. Incoherent evolution is accounted for by the superoperator L, which includes the effects of charge recombination from the singlet spin state (with rate k CR ), spin relaxation, and dephasing. Since the two g-factors are very different (see below), the eigenstates of the system are practically factorized. In addition, we expect a much shorter relaxation time for the electron spin on the QD. Hence, it is reasonable to assume two different relaxation times (T D 1 and T A 1 ) for the two electrons of the radical pair. Finally, the dephasing time T 2 (assumed to be the same for all transitions) induces a Lorentzian broadening of the EPR peaks.
By using the r(U, t) obtained for each time and magnetic eld value, we compute the spherical average to obtain the trEPR spectrum:
EPRðtÞ ¼ ð Tr n g DŜ D y þ g AŜ A y b rðU; tÞ o dU (5)
The result of eqn (5) is then convoluted with the exponential response function of the spectrometer, determined by its Qfactor of 6800, giving rise to a response time t R ¼ 2Q/u 0 ¼ 225 ns. 66 Simulated spectra are plotted as a function of both time and magnetic eld in Fig. 4. For a better comparison, in Fig. 3d, we report the 2D experimental trEPR acquired at 10 K as a contour plot similar to the computed ones.
This procedure allows us to investigate the initial state r(0), the Hamiltonian parameters, and relaxation/recombination rates. In Fig. 4, we consider three initial states:
A pure singlet state (S), as shown in Fig. 4, panel (a); A pure triplet state, with the three states equally populated (T), Fig. 4, panel (b);
A mixture of singlet and triplet (ST case), with all four levels equally populated, Fig. 4, panel (c).
Note that these three cases all share spherical symmetry. In Fig. 4, panels (a, b, c) show that only the ST case is in qualitative agreement with the measured time dependence, showing a negligible short-time signal rising slowly until reaching its maximum at z1.2 ms. We also explored more asymmetric states, where the singlet and triplet sublevels are not equally populated (see ESI † for further details). In this case, additional averaging over the spherical distribution of the chiral ligands with respect to the QD initial state is required. The calculations further corroborate the results of the symmetrical model and suggest that when the S population approaches z1/4 (and in turn T + + T 0 + T − z 3/4), the simulations do not substantially differ from the ST "symmetric" case, independently of the three triplet sublevels populations (see Fig. S26 †).
To better rationalise this result, it is worth revising the relevant spin relaxation times involved in the ET process. Equilibration between the S and T states in the QD occurs in a few picoseconds, well before the ET occurs, as reported for similar systems in the literature up to 5 K. 24 This suggests that the precursor state in the QD consists of thermally equilibrated S and T exciton states. 24 As a result, to have S population z1/4, S and T energy levels must be close in energy. Our QDs have an average diameter of 5 nm which should result in a ST energy difference of z1 meV, 59,60 thereby corroborating our assumption. 67,68 Table 1 reports the Hamiltonian parameters manually optimized to obtain our best simulation reported in Fig. 5 in the spherical symmetry assumption (ST state) at different times and temperatures. For simplicity, we neglected the isotropic exchange coupling and considered only an axial dipolar coupling D ¼ −26 MHz, which we determined from the separation between different peaks (Fig. 5, panels a, c). In turn, by using the point-dipole approximation for calculating D and a distance of 10 A between QD and C 60 and neglecting any distribution of conformations for simplicity, we obtain the gvalue of the QD g D z 1. The extracted g-value is consistent with what we expected for these nanoparticles 21 and justies the relatively weak dipolar coupling we determined from peak separation. At this distance, one could also expect an isotropic exchange contribution of the same order of magnitude. However, we prefer to limit the number of parameters and only assume a dipolar contribution. This simplied model leads to a higher g-value of the C 60 radical anion (g A ¼ 2.0037) than literature reports. 69,70 As a matter of clarity, it is worth stressing that in Fig. 4c, the absence of a signal at short times is due to the equally populated initial CT state. With time, the population of the four levels evolves because of incoherent processes, namely relaxation and recombination. A thorough study of the time dependence of the spectra allows us to t the relaxation parameters ( Table 1). The rate of charge recombination is estimated considering the time at which the maximum signal occurs (Fig. 4, panels c, f) and its value of k CR ¼ 1 ms −1 is aligned to what we expected for this system. 24 Spin relaxation is described by the two different characteristic times T D 1 ¼ 200 ns for the QD and T A 1 ¼ 10 ms for C 60 . As for the former, we are able to provide Table 1. Fig. 5 Simulated time-resolved EPR spectra as a function of magnetic field, temperature, and delay from the laser pulse. Experimental (blue line) and simulated (red line) spectra obtained at (a) t ¼ 1 ms, T ¼ 10 K, (b) t ¼ 1 ms, T ¼ 40 K, (c) t ¼ 2 ms, T ¼ 10 K, and (d) t ¼ 2 ms, T ¼ 40 K. All simulations were performed using the parameters in Table 1 and with the ST initial state.
an estimate thanks to its effect on the relative height of the two peaks, which is more evident at low temperature (Fig. 5, panels a, c). As for the latter, we can only infer a lower bound of about 2 ms, consistently to what expected for the C 60 anion, since the simulation does not signicantly change for larger values of T A 1 . An overall dephasing time T 2 ¼ 33 ns was obtained by tting the Lorentzian peak width. From the analysis of Fig. 5, we note that the main experimental spectral features, their time evolution and temperature dependence are in good agreement with our theoretical simulations, with only minor differences, e.g. an emissive calculated spin polarisation at 1 ms for 40 K.
We now consider the possibility of having a CT process that is spin ltered by the CISS effect. To this aim, we simulate the effect of CISS on the trEPR spectra for all three previously considered r(0). We model CISS as a "lter" which ideally keeps only the component of the transferred electron spin parallel (or anti-parallel, depending on the enantiomer) to the chiral bridge axis (see ESI †). 18,71,72 Hence, all the initial states are modied by CISS and produce different features in the resulting trEPR spectrum at short times on a randomly oriented solution. Notably, we considered only one of the two possible spin orientations because the isotropic QD generates identical EPR spectra for the two enantiomers. As shown in Fig. 4d and e, in the presence of an anisotropic dipole-dipole coupling, CISS lter strongly affects trEPR spectra for both the singlet and triplet cases, giving rise to opposite AE vs. EA features at short times. Conversely, CISS does not signicantly affect the spectrum with ST initial state (Fig. 4f). Note that the presence of an anisotropic spin-spin interaction is crucial to detect the occurrence of CISS in an isotropic solution. 18,73 Our current combined experimental and theoretical results do not allow distinguishing between a standard CT vs. a CISSmediated CT yet, due to the initial CT state composed of a mixture of singlet and triplet states. However, our results provide an essential step towards this ambitious goal since they allow drawing new guidelines for developing model systems for CISS detection. Specically, our calculations demonstrate that a "pure" precursor (or a precursor with different weights of S and T) would allow unravelling the occurrence of CISS even in a randomly oriented sample. This could occur in systems characterized by a larger singlet-triplet splitting than the CdSe QD employed here. For example, a larger splitting may be achieved by introducing a shell to increase the electron connement, as done in literature for similar QD-organic molecule dyads. 24 However, introducing a shell might further reduce the CT efficiency through the long and non-conjugated chiral bridge.
Conclusions
In conclusion, we have engineered and developed a model system comprising a CdSe QD as a donor and a C 60 derivative as an acceptor linked by a rigid and saturated bi-peptidic chiral bridge (c). This chiral system has shown spin polarization as a result of the photoinduced CT between the CdSe QD and the C 60 through c. The CdSe QD-c-C 60 system was fabricated through the ligand exchange approach and characterized through the combination of optical spectroscopies and XPS analysis. We then used time-resolved EPR to demonstrate that the photoinduced CT process generates an organic radical localized on the C 60 , which shows a peculiar spin-polarization evolution in the rst few ms aer the laser pulse. We modelled the trEPR signal of our system in two different cases: (i) a standard spin-polarized photoinduced CT, and (ii) a CISS-mediated photoinduced CT. Our calculations demonstrated that the observed EPR features might be compatible with the photoinduced CISS effect.
Though not conclusive yet, our combined experimental and theoretical work represents a rst promising attempt toward the direct spectroscopic observation of photodriven CISS effect. Although the search for the perfect model system simultaneously showing good CT efficiencies, efficient spin-ltering, and a well-dened precursor state is still in its infancy, our results suggest that QD-c-organic molecule dyads are very promising. In addition, our work builds up a sound theoretical framework that will allow a better understanding of spinpolarization arising from photoinduced CT processes in chiral hybrid systems comprising QD and organic molecules.
Ultimately, the possibility of observing a CISS-mediated charge transfer at the molecular level would provide a new tool for molecule-based quantum information processing. Indeed, the QD-c-C 60 could form an important building block of a quantum computing architecture, in which polarization resulting from CISS could be harnessed to initialize/readout qubits or implement quantum gates. Thanks to the remarkable efficiency displayed by CISS at high temperatures, this could pave the way toward room-temperature operation of a molecular quantum processor.
Experimental section
Synthesis
Details on the synthesis and characterisation of organic ligands 1, 2 and 3 are presented in ESI. †
DFT calculations
All DFT calculations were performed with ORCA 4.2.1 quantum chemistry package. 74 For the geometry optimizations, PBE0 functional 75 and D3 empirical dispersion correction 76,77 were used, while def2-TZVP basis set 78 was employed for all the atoms. The thresholds on the maximum force gradient and the energy change were set to 3 Â 10 −4 Hartree/Bohr and 5 Â 10 −6 Hartrees, respectively. The root mean square deviation was computed on the heavy atoms C, N, O, and S of the polypeptide chain within the formula RMSD ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 N X N i¼1 d i 2 s
where d i is the distance between a pair of equivalent atoms in the two structures and N is the total number of equivalent atoms.
CdSe quantum dots (QDs) fabrication
Details on the fabrication of pristine CdSe QDs, their morphological characterisation, and ligand exchange process are presented as ESI. †
Optical spectroscopy
UV-Vis spectra were recorded at room temperature by using a Cary 5000 spectrophotometer (Agilent) in the range 400-800. The PL spectra were measured at room temperature using an excitation wavelength (l ex ) of 400 nm on a FluoroMax P (Horiba). The measurements were performed using quartz cuvettes with a path length of 1 cm. PL spectra were normalized for the optical density of the same solution at the maximum of the excitonic peak.
Time-resolved photoluminescence (TRPL) measurements were carried out using the time-correlated single-photon counting (TCSPC) technique. The experimental apparatus is based on the FluoroMax P spectrouorometer detection unit (grating monochromator and photomultiplier tube), powered by the FluoroHub Single Photon Counting unit. The excitation source was a blue pulsed Horiba NanoLED, generating picosecond pulses in the UV (375 nm). The instrument response function (IRF) for the whole apparatus was determined by means of scattered light detection using a reference sample of LUDOX® colloidal silica. LED radiation was focused by means of a spherical lens on a sample holder, and sample emission was collected with a 90 geometry to minimize scattering interferences. The measurements were carried out on dilute solutions of TOPO-capped and 1-capped QDs solutions in 1,2,4 trichlorobenzene. Solutions were prepared to keep optical absorbance at 400 nm below 0.2 in 1 cm path quartz cuvettes.
X-ray photoelectron spectroscopy (XPS)
X-ray photoelectron spectroscopic (XPS) analyses were carried out in an UHV chamber with a base pressure lower than 10 −9 mbar. The chamber was equipped with non-monochromatized Mg Ka radiation (hn ¼ 1253.6 eV) and a hemispherical electron/ ion energy analyser (VSW mounting a 16-channel detector). The operating parameters of the X-ray source were 12 kV and 12 mA, and photoelectrons were collected normal to the sample surface, maintaining the analyser angle between analyser axis and X-ray source xed at 54.5 . All the samples were drop cast on In foil and on a slab of Au on mica and XPS spectra acquired in a xed analyser transmission mode with pass energy of 44.0 eV. The spectra were analysed by using the CasaXPS soware. Linear or Shirley functions were used to subtract the background. The deconvolution of the XPS spectra was performed by applying a combination of Gaussian and Lorentzian functions (70 : 30). The binding energy scale was calibrated using the Au 4f 7/2 peak or the In 3d 5/2 peak respectively at 84 eV, 79 and 443.9 eV. 80 Time-resolved electron paramagnetic resonance (trEPR)
All trEPR spectra were recorded on a Bruker Elexsys E580 Xband spectrometer equipped with a dielectric ring resonator (ER 4118X-MD5). The sample temperature was maintained using a helium gas-ow cryostat Oxford Instruments CF9350 and controlled with an Oxford Instruments ITC503. Laser excitation at different wavelengths was provided by a Litron AURORA II opto-parametric oscillator (OPO) tuneable laser (model number: A23-39-21, 21 Hz repetition rate, E/pulse z 2 mJ, l ¼ 410-700 nm, pulse duration ¼ 7 ns). The laser beam was coupled into the resonator through an optical window. No effects of laser beam polarisation are detected, which suggests the laser beam is non-polarised at the sample position. trEPR experiments were performed by direct detection with the transient recorder without lock-in amplication. The instrument response time was about 200 ns. The spectra were acquired with 2 mW microwave power and averaging 100 transient signals at each eld position. The magnetic eld was measured with a Bruker ER035M NMR Gaussmeter.
The trEPR measurements were performed on the model system CdSe QD-c-C 60 and the control sample, as a comparison, consisting of CdSe QD-c to which 1 mM of PCBM was added. The concentration of CdSe QD was 7.8 mM in 1,2,4-trichlorobenzene. The solutions were poured inside EPR quartz tubes that were sealed with Teon under N 2 atmosphere.
Aer data acquisition, baseline correction in both time and eld dimensions was performed. First, we subtracted the pretrigger offset for each eld point; second, we ltered out the laser induced background signal by subtracting the offresonance signal intensity from the spectra at each time point. For the narrow sweep spectra, the C 60 anion signal is superimposed to the C 60 triplet signal. We applied rst-order background subtraction for each time point to remove the driing baseline. The transient EPR spectrum at different time delays aer the laser pulse was extracted from the corrected dataset. The reported trEPR spectra were averaged over a time window of 0.2 ms.
Theoretical modelling
Details on theoretical modelling are presented in ESI. †
Data availability
All experimental and computed data are available from the authors upon request.
Author contributions
The manuscript was written through the contributions of all authors.
Conflicts of interest
The authors declare no competing nancial interests. agreement no. 862893. The authors acknowledge MatchLab Interdepartmental Research Unit (Università degli Studi di Firenze) for the XPS facilities. We acknowledge also nancial support from "Fondazione Cariparma". We acknowledge Michael R. Wasielewski and Robert Bittl for their critical manuscript reading.
Notes and references
Fig. 2
2Optical and XPS characterization of the QD-c-C 60 system. (a) UV-vis absorption and normalized photoluminescence spectra excited at l ex ¼ 400 nm (inset) of pristine QDs (black line) and CdSe QD-c-C 60 (red line) in 1,2,4-trichlorobenzene solution. Notably, the PL of the CdSe QD-c-C 60 system is drastically quenched due to the thiol binding and probably of the CT process. (b) Photoluminescence decay curves recorded at l ¼ 610 nm with l ex ¼ 370 nm. From the fitting, an average decay time s of 29.0 AE 0.5 ns and 2.6 AE 0.5 ns were obtained, respectively, for the pristine and chiral systems. (c) C 1s and (d) S 2p/Se 3p photoemission lines for ligand 1, the CdSe QDs, and the QD-c-C 60 system, as well as the single chemically shifted components from fit deconvolution.
Fig. 3
3Time-resolved EPR spectra of the QD-c-C 60 (model) and QD-c + C 60 (test) systems.
Fig. 4
4Simulated time-resolved EPR spectra as a function of both time and field. Panels (a, b, c): spectra calculated for the S, T, and ST initial states. Panels (d, e, f): corresponding spectra calculated for the S, T, and ST initial states, filtered by the CISS effect. Simulations are performed by considering a temperature of T ¼ 10 K and using the parameters found in
Table 1 Parameters
1obtained from the modelling of time-resolved
EPR data. D and A indexes refer to the CdSe QD donor and C 60
acceptor, respectively
g D
g A
D
T D
1
T A
1
T 2
k CR
1
2.0037
−26 MHz
200 ns
10 ms
33.4 ns
1 ms −1
© 2022 The Author(s). Published by the Royal Society of Chemistry
© 2022 The Author(s). Published by the Royal Society of Chemistry Chem. Sci., 2022, 13, 12208-12218 | 12215
© 2022 The Author(s). Published by the Royal Society of Chemistry Chem. Sci., 2022, 13, 12208-12218 | 12217
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[
"An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field",
"An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field"
]
| [
"P G Ivanov \nRudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3PUOxfordUK\n\nCulham Centre for Fusion Energy\nAtomic Energy Authority\nOX14 3DBAbingdonUnited Kingdom, UK\n",
"T Adkins \nRudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3PUOxfordUK\n\nCulham Centre for Fusion Energy\nAtomic Energy Authority\nOX14 3DBAbingdonUnited Kingdom, UK\n\nMerton College\nOX1 4JDOxfordUK\n"
]
| [
"Rudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3PUOxfordUK",
"Culham Centre for Fusion Energy\nAtomic Energy Authority\nOX14 3DBAbingdonUnited Kingdom, UK",
"Rudolf Peierls Centre for Theoretical Physics\nUniversity of Oxford\nOX1 3PUOxfordUK",
"Culham Centre for Fusion Energy\nAtomic Energy Authority\nOX14 3DBAbingdonUnited Kingdom, UK",
"Merton College\nOX1 4JDOxfordUK"
]
| []
| Starting from the equations of collisionless linear gyrokinetics for magnetised plasmas with an imposed inhomogeneous magnetic field, we present the first known analytical, closed-form solution for the resulting velocity-space integrals in the presence of resonances due to both parallel streaming and constant magnetic drifts. These integrals are written in terms of the well-known plasma dispersion function(Faddeeva & Terent'ev 1954;Fried & Conte 1961), rendering the subsequent expressions simpler to treat analytically and more efficient to compute numerically. We demonstrate that our results converge to the well-known ones in the straight-magnetic-field and two-dimensional limits, and show good agreement with the numerical solver by Gürcan (2014). By way of example, we calculate the exact dispersion relation for a simple electrostatic, ion-temperature-gradient-driven instability, and compare it with approximate kinetic and fluid models. † | 10.1017/s0022377823000077 | [
"https://export.arxiv.org/pdf/2212.02654v2.pdf"
]
| 254,275,023 | 2212.02654 | fd474d06582eedead947c862e2039cb79f44cacf |
An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field
P G Ivanov
Rudolf Peierls Centre for Theoretical Physics
University of Oxford
OX1 3PUOxfordUK
Culham Centre for Fusion Energy
Atomic Energy Authority
OX14 3DBAbingdonUnited Kingdom, UK
T Adkins
Rudolf Peierls Centre for Theoretical Physics
University of Oxford
OX1 3PUOxfordUK
Culham Centre for Fusion Energy
Atomic Energy Authority
OX14 3DBAbingdonUnited Kingdom, UK
Merton College
OX1 4JDOxfordUK
An analytical form of the dispersion function for local linear gyrokinetics in a curved magnetic field
(compiled on 18 January 2023)Under consideration for publication in J. Plasma Phys. 1
Starting from the equations of collisionless linear gyrokinetics for magnetised plasmas with an imposed inhomogeneous magnetic field, we present the first known analytical, closed-form solution for the resulting velocity-space integrals in the presence of resonances due to both parallel streaming and constant magnetic drifts. These integrals are written in terms of the well-known plasma dispersion function(Faddeeva & Terent'ev 1954;Fried & Conte 1961), rendering the subsequent expressions simpler to treat analytically and more efficient to compute numerically. We demonstrate that our results converge to the well-known ones in the straight-magnetic-field and two-dimensional limits, and show good agreement with the numerical solver by Gürcan (2014). By way of example, we calculate the exact dispersion relation for a simple electrostatic, ion-temperature-gradient-driven instability, and compare it with approximate kinetic and fluid models. †
Introduction
The investigation of the linear-stability properties of magnetically confined plasmas is crucial for the design of magnetic-confinement-fusion devices. The heat and particle losses in these devices are dominated by turbulent fluctuations, which are themselves excited by linear instabilities driven by the gradients of the plasma equilibrium (Rudakov & Sagdeev 1961;Pogutse 1968;Coppi et al. 1967;Guzdar et al. 1983;Hugill 1983;Liewer 1985;Waltz 1988;Wootton et al. 1990;Cowley et al. 1991;Kotschenreuther et al. 1995;Xanthopoulos et al. 2007;Ongena et al. 2016). In most cases, the strong toroidal magnetic field constrains the plasma fluctuations to have typical temporal scales that are slow compared to the frequency of the Larmor motion of the particles, and to be anisotropic in space: length scales along the magnetic field are comparable to the size of the device, while ones perpendicular to it are comparable to the Larmor radii of the particles. Therefore, the plasma dynamics can often be treated using the gyrokinetic formalism (Frieman & Chen 1982;Sugama et al. 1996;Howes et al. 2006;Abel et al. 2013;Catto 2019).
When solving the linear gyrokinetic equation, one inevitably encounters resonant velocity-space integrals that need to be evaluated, analytically or numerically, in order to obtain the dispersion relation for the linear modes present within the system. The most basic of these resonances results from the parallel (to the magnetic field) streaming of particles, first discussed by Landau (1946). However, in the presence of an inhomogeneous equilibrium magnetic field, one is presented with a qualitatively different type of resonance due to the magnetic drifts of the particles. Evaluating these resonant integrals analytically, in the presence of both parallel streaming and magnetic drifts that are constant along the magnetic field, and without further approximations (such as those used in, e.g., Terry et al. 1982;Kim et al. 1994) has remained an open research question, despite some progress being made numerically (Gürcan 2014;Gültekin & Gürcan 2018Parisi et al. 2020). On the other hand, it is well-known that there are instabilities that exist only in the presence of curved magnetic fields, e.g., the toroidal ion-temperature-gradient (ITG) instability (Pogutse 1968;Guzdar et al. 1983;Waltz 1988;Kotschenreuther et al. 1995). Often, such instabilities are the dominant ones in toroidal plasmas. Thus, the exact inclusion of the magnetic-drift resonance in the analytical theory of linear gyrokinetics is expected to lead to qualitative changes in the behaviour of the resulting dispersion relation and to allow for a more complete treatment of the linear-stability properties of strongly magnetised plasmas.
In this work, we present closed forms for the aforementioned resonant integrals. These are written in terms of the plasma dispersion function (Faddeeva & Terent'ev 1954;Fried & Conte 1961). They allow us to find a closed expression for the drift-kinetic dispersion relation, or an absolutely convergent series for the gyrokinetic one via Taylor expansions. The inclusion of magnetic drifts in the linear gyrokinetic problem introduces two distinct changes: (i) quantitatively, in that it significantly modifies the growth rates and frequencies of linear solutions; and (ii) qualitatively, by introducing a multivalued dispersion function. The latter has important consequences for the form of the dispersion relation and its solution, some of which have already been described in the literature (Kuroda et al. 1998;Sugama 1999).
The rest of the paper is organised as follows. We begin by summarising how the gyrokinetic dispersion relation, and the resonant velocity-space integrals of which it is comprised, emerge from the Fourier-Laplace transform of the linear gyrokinetic equations in §2. Then, in §3, we discuss already-known solutions for these integrals and the asymptotic limits in which they apply. The main result of this work is presented in §4, where we derive the exact solution to one particular resonant integral -the 'generalised plasma dispersion function' -to which all others will be related. In §5.1, we show both analytically and numerically that the generalised plasma dispersion function asymptotes to the known solutions in the cases of zero magnetic curvature and of two-dimensional perturbations, while §5.2 demonstrates that our expressions are in agreement with the numerical solver published by Gürcan (2014). Section §6 discusses the analytic continuation of these functions and the subsequent solution to the inverse-Laplacetransform problem by which we obtain the solution to the linear gyrokinetic system. In §7, we show how the results obtained in §4 can be generalised to the gyrokinetic case via absolutely convergent Taylor expansions. In §8, we give an example calculation for the electrostatic ITG instability and compare it with known kinetic and fluid limits. Finally, our results are summarised and possible extensions discussed in §9.
Collisionless gyrokinetic linear theory
In this section, we demonstrate how the resonant kinetic integrals that are the main focus of this paper emerge naturally from considerations of linear, collisionless local gyrokinetic theory with constant geometric coefficients [see the discussion following (2.10)]. Readers already familiar with gyrokinetic theory may wish to skip ahead to §2.3, working backwards where further clarification is required.
Gyrokinetics
As is often the case in the study of magnetically confined plasmas, we shall assume that the fluctuations within our plasma obey the standard gyrokinetic ordering (see, e.g., Abel et al. 2013or Catto 2019; that is, for fluctuations with a characteristic frequency ω and wavenumbers k and k ⊥ parallel and perpendicular to the equilibrium magnetic field direction b 0 = B 0 /B 0 , we have
ω Ω s ∼ ν ss Ω s ∼ k k ⊥ ∼ q s φ T 0s ∼ δB B 0 ∼ δB ⊥ B 0 ∼ ρ s L 1,(2.1)
where Ω s = q s B 0 /m s c is the cyclotron frequency of species s with charge q s , equilibrium density and temperature n 0s and T 0s , respectively, mass m s and thermal speed v ths = 2T 0s /m s , ν ss is the typical collision frequency, ρ s = v ths /|Ω s | is the thermal Larmor radius, δB and δB ⊥ are the fluctuations of the magnetic field parallel and perpendicular to the equilibrium direction, respectively, and L is a typical equilibrium length scale. It is assumed that all equilibrium quantities evolve on the (long) transport timescale τ −1 E ∼ (ρ s /L) 3 Ω s , and so will be considered static throughout the remainder of this paper.
Under the ordering (2.1), the perturbed distribution function δf s consists of the Boltzmann and gyrokinetic parts:
δf s (r, v, t) = − q s φ(r, t) T 0s f 0s (x, v) + h s (R s , v ⊥ , v , t), (2.2) where R s = r − b 0 × v ⊥
/Ω s is the guiding-centre position, and h s evolves according to the gyrokinetic equation
∂ ∂t h s − q s χ Rs T 0s f 0s + v b 0 + v ds · ∇h s + v χ · ∇ ⊥ (h s + f 0s ) = ∂h s ∂t c . (2.3)
In the above, and throughout this paper, ... Rs denotes the standard gyroaverage at constant R s . Here, χ = φ − v · A/c is the gyrokinetic potential (φ and A are the scalar and vector potential, respectively, under the Coulomb gauge ∇ · A = 0) that gives rise to the drift velocity
v χ = c B 0 b 0 × ∂ χ Rs ∂R s , (2.4)
which includes the E ×B drift, the parallel streaming along perturbed field lines, and the ∇B drift associated with the perturbed magnetic field. This gives rise to nonlinearities (with which we will not be concerned in this paper), as well as the familiar gyrokinetic drive associated with the equilibrium distribution f 0s , viz.,
v χ · ∇ ⊥ f 0s = − c B 0 b 0 × ∂ χ Rs ∂R s · ∇x 1 L ns + η s L ns v 2 v 2 ths − 3 2 f 0s , (2.5) where L −1 ns = − 1 n 0s ∂n 0s ∂x , L −1 Ts = − 1 T 0s ∂T 0s ∂x , η s = L ns L Ts , (2.6)
are the characteristic length scales associated with the radial equilibrium gradients of both density and temperature, respectively, η s is their ratio, and x is the direction of the equilibrium gradients. The magnetic drifts associated with the equilibrium field are
v ds = b 0 Ω s × v 2 b 0 · ∇b 0 + 1 2 v 2 ⊥ ∇ log B 0 . (2.7)
The last term on the right-hand side of (2.3) is the (linearised) collision operator, which we henceforth neglect given that we are interested in studying collisionless dynamics. The electromagnetic fields appearing in the gyrokinetic equation (2.3) are determined by the quasineutrality condition
0 = s q s δn s = s q s − q s φ T 0s n 0s + d 3 v h s r , (2.8)
where ... r denotes the gyroaverage at constant r, and by the parallel and perpendicular parts of Ampère's law, which are, respectively,
∇ 2 ⊥ A = − 4π c s q s d 3 v v h s r , (2.9) ∇ 2 ⊥ δB = − 4π c b 0 · ∇ ⊥ × s q s d 3 v v ⊥ h s r .
(2.10)
Together, (2.3) and (2.8)-(2.10) form a closed system of equations that, in principle, allows us to determine h s and thus the evolution of the fluctuations in our plasma. In this work, we solve the linear part of this system in the 'local' limit : we assume that the gradients of all equilibrium quantities are constant -including the geometric coefficients b 0 · ∇b 0 and ∇ log B 0 that appear in the magnetic driftsand choose orthonormal coordinates (x, y, z), in whichẑ = b 0 is the direction of the magnetic field,x is, as above, the direction of the equilibrium gradients (cf. the radial direction in toroidal geometry), andŷ ≡ b 0 ×x is the binormal direction (cf. the poloidal direction in toroidal geometry). One can think of this geometry as that of a Z-pinch (see Ricci et al. 2006;Ivanov et al. 2020;Adkins et al. 2022) due to the assumption of constant magnetic curvature and lack of magnetic shear, which we have implicitly assumed. Under these assumptions, the system of equations (2.3), (2.8)-(2.10) is homogeneous in space, allowing us to impose periodic boundary conditions in all three spatial dimensions.
In the next section, we consider the time evolution of a single Fourier mode and obtain the resulting gyrokinetic dispersion relation.
Linear gyrokinetic problem
Neglecting the nonlinear term and introducing the spatial Fourier decomposition:
h s (R s , v ⊥ , v , t) = k h sk (v ⊥ , v , t)e ik·Rs , χ(r, t) = k χ k (t)e ik·r ,(2.11)
with k = k ⊥ + k b 0 , the Fourier modes h sk and χ k can be shown to satisfy
∂ ∂t h sk − q s χ k Rs T 0s f 0s + ik v h sk + iω Ds h sk − iω T * s q s χ k Rs T 0s f 0s = 0, (2.12)
where we have defined the drift frequencies associated with the equilibrium gradients of species s [cf. (2.5)]:
ω T * s = ω * s 1 + η s v 2 v 2 ths − 3 2 , ω * s = − k y cT 0s q s B 0 L ns ,(2.13)
and with the equilibrium magnetic field curvature and gradient, respectively [cf. (2.7)]:
ω Ds = 2v 2 v 2 ths ω κs + v 2 ⊥ v 2 ths ω ∇Bs ,(2.14)
where
ω κs = v 2 ths 2Ω s k ⊥ · [b 0 × (b 0 · ∇)b 0 ] , ω ∇Bs = v 2 ths 2Ω s k ⊥ · (b 0 × ∇ log B 0 ) . (2.15)
Starting from the perpendicular force balance of the gyrokinetic equilibrium [see equation (128) in Abel et al. 2013], it is straightforward to show that the difference between these two drifts is given by
ω κs − ω ∇Bs = v 2 ths 2Ω s k ⊥ · (b 0 × ∇x) ∂ ∂x B0 s β s 2 , (2.16)
where β s = 8πn 0s T 0s /B 2 0 is the plasma beta of species s. Lastly, the gyroaveraged Fouriertransformed gyrokinetic potential is
χ k Rs = J 0 (b s ) φ k − v A k c + 2J 1 (b s ) b s T 0s q s v 2 ⊥ v 2 ths δB k B 0 , (2.17)
while the field equations (2.8)-(2.10) can be written as (see, e.g., Howes et al. 2006)
s q 2 s n 0s T 0s φ k = s q s d 3 v J 0 (b s )h s , (2.18) k 2 ⊥ A k = 4π c s q s d 3 v v J 0 (b s )h s , (2.19) δB k B 0 = − 1 2 s β s n 0s d 3 v v 2 ⊥ v 2 ths 2J 1 (b s ) b sh s ,(2.20)
where b s = k ⊥ v ⊥ /Ω s , and J 0 , J 1 are the Bessel functions of the first kind (Abramowitz & Stegun 1972) that capture finite-Larmor-radius effects. It will prove convenient to combine φ k , A k , and δB k into a single vector χ k given by
χ k = q r φ k T 0r , k |k | A k ρ r B 0 , δB k B 0 T .
(2.21)
Here, and in what follows, we normalise the electromagnetic fields using an arbitrary reference mass m r , density n 0r , thermal velocity v thr , temperature T 0r , and gyroradius ρ r . Following Landau (1946), we consider an initial-value problem and introduce the Laplace transformationŝ
h sk (v ⊥ , v , p) = ∞ 0 dt e −pt h sk (v ⊥ , v , t),χ k (p) = ∞ 0 dt e −pt χ k (t).
(2.22)
Assuming there exist positive real m and M such that
h sk (v ⊥ , v , t) , |χ k (t)| M e mt ,(2.23)
for all t > 0, and picking any real σ with σ > m, the integrals in (2.22) converge and the transformed distributionsĥ sk and fieldsχ k are analytic for all complex values of p with Re(p) σ. The inverse transformations are given by
h sk (v ⊥ , v , t) = 1 2πi
Cσ dp e ptĥ sk (v ⊥ , v , p), χ k (t) = 1 2πi Cσ dp e ptχ k (p), (2.24)
where the contour of integration C σ is along a straight line parallel to the imaginary axis and intersecting the real axis at Re(p) = σ, as in figure 1 (this is the so-called Bromwich contour). Figure 1: The complex p plane, with Re(p) and Im(p) shown on the horizontal and vertical axes, respectively. The contour of integration for the inverse Laplace transform Cσ is is a vertical straight line at Re(p) = σ, to the right of which (i.e, in the shaded grey region) the functionŝ h sk andχ k are guaranteed to be analytic. Singularities, such as poles (indicated by crosses) or branch cuts (indicated by the zigzag line), could exist at Re(p) < σ.
Re(p)
Im(p) σ × × × × C σ
Performing the Laplace transform as in (2.22), (2.12) straightforwardly becomeŝ
h sk = p + iω T * s p + ik v + iω Ds q s χ k Rs T 0s f 0s + g sk p + ik v + iω Ds , (2.25)
where g sk is the initial condition:
g sk (v ⊥ , v ) = h sk (v ⊥ , v , t = 0) − q s χ k (t = 0) Rs T 0s f 0s . (2.26)
Then, normalising the characteristic frequencies to the parallel-streaming rate 1 ζ s = ip |k |v ths , ζ * s = ω * s |k |v ths , ζ κs = ω κs |k |v ths , ζ ∇Bs = ω ∇Bs |k |v ths , (2.27) and defining the dimensionless velocity variables (2.29) in which L is the linear coefficient matrix and G is the vector of the initial conditions of 1 Note that normalising to |k |v ths rather than k v ths means that the condition for analyticity Re(p) σ > 0 implies Im(ζs) > 0, regardless of the sign of k .
u = k |k | v v ths , µ = v 2 ⊥ v 2 ths ,(2.Lχ k + G = 0,
the fields. The components of L are given by
L φφ = − s q 2 s n 0s T 0r q 2 r n 0r T 0s 1 + ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 I (s) a,b a=b=1 , (2.30) L φA = 2 s q 2 s n 0s v ths T 0r q 2 r n 0r v thr T 0s ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 J (s) a,b a=b=1 , (2.31) L φB = s q s n 0s q r n 0r ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ b K (s) a,b a=b=1 , (2.32) L Aφ = − s q 2 s n 0s v ths T 0r q 2 r n 0r v thr T 0s ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 J (s) a,b a=b=1 , (2.33) L AA = − B 2 0 (k ⊥ ρ r ) 2 8πn 0r T 0r − 2 s q 2 s n 0s m r q 2 r n 0r m s ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ a I (s) a,b a=b=1 , (2.34) L AB = s q s n 0s v ths q r n 0r v thr ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ b L (s) a,b a=b=1 , (2.35) L Bφ = − s β s 2 q s T 0r q r T 0s ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ b K (s) a,b a=b=1 , (2.36) L BA = s β s q s T 0r v ths q r T 0s v thr ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ b L (s) a,b a=b=1 , (2.37) L BB = −1 + s β s 2 ζ s − ζ * s + η s ζ * s ∂ a + ∂ b + 3 2 ∂ 2 b M (s) a,b a=b=1 , (2.38)
where we have defined the following integrals
I (s) a,b = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) J 2 0 (b s ), (2.39) J (s) a,b = 1 √ π ∞ −∞ du ∞ 0 dµ ue −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) J 2 0 (b s ), (2.40) K (s) a,b = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) 2J 0 (b s )J 1 (b s ) b s , (2.41) L (s) a,b = 1 √ π ∞ −∞ du ∞ 0 dµ ue −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) 2J 0 (b s )J 1 (b s ) b s , (2.42) M (s) a,b = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) 2J 1 (b s ) b s 2 .
(2.43)
Here, and throughout the remainder of this paper, the parameters a and b are assumed to be both real and positive, ensuring integral convergence. Finally, the components of
G = (G φ , G A , G B ) T are given by G φ = s q s n 0s q r n 0r 1 n 0s d 3 v g sk p + ik v + iω Ds J 0 (b s ), (2.44) G A = s q s n 0s v ths q r n 0r v thr 1 n 0s d 3 v v v ths g sk p + ik v + iω Ds J 0 (b s ), (2.45) G B = − s β s 2 1 n 0s d 3 v v 2 ⊥ v 2 ths g sk p + ik v + iω Ds 2J 1 (b s ) b s . (2.46)
The eigenvalue problem (2.29) can be inverted in order to solve for the fields in the usual way, viz.,χ
k (p) = (adj L)G det L , (2.47)
where adj L and det L are the adjugate matrix and determinant of the linear matrix L, respectively. The time-dependent fields are then determined by the inverse Laplace transform of (2.47). As discussed above, the integrals in (2.24) are, before analytic continuation, defined for Re(p) σ > 0. For these values of p, Im(ζ s ) > 0, and so the integrals in (2.39)-(2.43) converge and are analytic functions of p. Note that the equation
D(p) ≡ det L(p) = 0 (2.48)
is commonly known as the 'dispersion relation', while we shall refer to D itself as the 'dispersion function'.
Drift-kinetic limit
To evaluate the integrals (2.39)-(2.43), we specialise to the drift-kinetic limit, in which the perpendicular wavenumbers of the perturbations are assumed small in comparison to the species' gyroradii, viz., b s ∼ k ⊥ ρ s 1.
(2.49)
In this limit, the Bessel functions can be expanded as
J 0 (b s ) = 1 + O(b 2 s ), 2J 1 (b s )/b s = 1 + O(b 2 s ), (2.50)
meaning that, to leading order in b s , the contributions of the Bessel functions to the integrals (2.39)-(2.43) are equal to one, and we may write
I (s) a,b = I (s) a,b = K (s) a,b = M (s) a,b , J (s) a,b = J (s) a,b = L (s) a,b ,(2.51)
where I
I (s) a,b (ζ s , ζ κs , ζ Bs ) = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) , (2.52) J (s) a,b (ζ s , ζ κs , ζ Bs ) = 1 √ π ∞ −∞ du ∞ 0 dµ ue −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs )
.
(2.53) Furthermore, we consider the particular case in which the difference between the curvature and ∇B drifts, given by the right-hand side of (2.16), is zero and so their associated drift frequencies can be taken to be equal, viz.,
ω κs = ω ∇Bs ≡ ω ds ⇒ ζ κs = ζ ∇Bs ≡ ζ ds . (2.54)
Note that neither approximation should be interpreted as a consequence of some asymptotic ordering of the parameters describing our gyrokinetic system of equations. Instead, they should be viewed as formal approximations that allow us to obtain a solvable case of a more general one. Their relaxation is discussed in §7.
With these simplifications, we have reduced our problem to the evaluation of
I a,b (ζ, ζ d ) = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ + ζ d (2u 2 + µ) , (2.55) J a,b (ζ, ζ d ) = 1 √ π ∞ −∞ du ∞ 0 dµ ue −au 2 −bµ u − ζ + ζ d (2u 2 + µ) , (2.56)
where we have used (2.54) and have dropped the species index for the sake of compactness of notation.
Previous solutions
Before tackling the task of analytically integrating (2.55) and (2.56), we shall briefly discuss some special cases in which these expressions are already known within the literature. A reader already familiar with these solutions may wish to skip ahead to §4, working backwards if further clarification is required.
The plasma dispersion function and Landau's solution
In the absence of magnetic drifts (i.e., when ζ d = 0), (2.55) and (2.56) can straightforwardly be written in terms of the well-studied plasma dispersion function (Faddeeva & Terent'ev 1954;Fried & Conte 1961):
Z(ζ) ≡ 1 √ π ∞ −∞ du e −u 2 u − ζ , (3.1)
where the integral is defined for Im(ζ) > 0 with the integration contour along the real u axis, as in figure 2(a). In particular, we have that
I a,b | ζ d =0 = 1 b Z a (ζ), J a,b | ζ d =0 = 1 b 1 √ a + ζZ a (ζ) , (3.2)
where we have, for the sake of brevity, introduced the shorthand notation
Z a (ζ) ≡ Z( √ aζ). (3.3)
The integral in (3.1) can be analytically continued to Im(ζ) 0 by deforming the contour of integration in such a way as to always keep the pole above it, as shown in figure 2(b), (c). This is known as the Landau prescription, and the resultant contour is the well-known Landau contour C L (Landau 1946). The plasma dispersion function (3.1) is ubiquitous in calculations of linear waves and instabilities in systems with a spatially uniform magnetic field; notable examples include the electron-temperature-gradient (see, e.g., Liu 1971;Lee et al. 1987) and iontemperature-gradient (see, e.g., Rudakov & Sagdeev 1961;Coppi et al. 1966;Sauter et al. 1990;Brunner & Vaclavik 1998;Smolyakov et al. 2002) instabilities, the latter of which we shall consider in §8. It is also worth noting that the Bessel functions can easily be incorporated into the integrals if ζ d = 0 because the resonant denominators are independent of µ. The resulting expressions involve modified Bessel functions and are well-known in the literature (see, e.g., Howes et al. 2006).
Re(u) Im(u) C L × ζ Re(u) Im(u) C L × ζ Re(u) Im(u) C L × ζ (a) Im(ζ) > 0 (b) Im(ζ) = 0 (c) Im(ζ) < 0
Figure 2: The Landau prescription for the contour of integration CL that gives the analytic continuation of (3.1). As the Laplace transform demands Re(p) σ > 0, the pole u = ζ is located in the upper-half plane [where Im(ζ) > 0, see footnote 1], above the contour of integration, as in panel (a). Therefore, the appropriate analytic continuation for Re(p) 0 [i.e., Im(ζ) 0] demands that the contour must be deformed so as to always remain below the pole, as in panels (b), (c). Cauchy's integral theorem ensures that we are free to deform the contour without changing the value of the integral, so long as it does not cross the pole.
Two-dimensional limit
In the two-dimensional limit, k → 0 with ζ ∼ ζ d → ∞, it can be shown (via, e.g., a partial-fractions expansion of the integrand) that (2.55) can be expressed exactly in terms of products of the plasma dispersion function (Biglari et al. 1989), viz.,
I 1,1 = − 1 2ζ d Z( √ Ω) 2 + O ζ −2 d , J 1,1 = O ζ −2 d , Ω = ζ 2ζ d , (3.4)
with the integral for J a,b vanishing to leading order because the integrand in (2.56) is manifestly odd in x in this limit. The analytic continuation for (3.4) is significantly more subtle than in the case of the plasma dispersion function (3.1), owing to the presence of the branch point at ζ = 0; we shall delay discussion of these subtleties until §6. The solution (3.4) has been used extensively in the investigation of two-dimensional ITG instabilities (see, e.g., Similon et al. 1984;Biglari et al. 1989;Kuroda et al. 1998;Sugama 1999;Ricci et al. 2006;Helander et al. 2011;Mishchenko et al. 2018;Zocco et al. 2018).
Numerical methods
Owing to their analytical complexity, previous literature has also been devoted to the numerical evaluation of (2.55) and (2.56) (see Beer & Hammett 1996;Gürcan 2014;Gültekin & Gürcan 2018Parisi et al. 2020, and references contained therein). In many cases, this involves expressing these integrals in terms of one-dimensional integrals. For example, writing
1 u − ζ + ζ d (2u 2 + µ) = i sgn[Im(ζ)]∞ 0 dλ e −iλ[u−ζ+ζ d (2u 2 +µ)] ,(3.5)
allows the integration over u and µ in (2.55) and (2.56) to be done analytically, leaving an integral over λ that can be evaluated numerically (cf. Beer & Hammett 1996;Parisi et al. 2020). While this method is quite general -in that it also allows the direct inclusion of the Bessel functions in (2.39)-(2.43) -the numerical evaluation of the resultant expressions can often be slow, numerical errors may be difficult to quantify, and subtleties like multivaluedness and branch cuts easy to overlook. This motivates the goal of the present study, viz., to find expressions for these integrals in terms of known functions that can be better understood analytically and more readily computed numerically.
The generalised plasma dispersion function
In this section, we detail the method by which (2.55) and (2.56) can be expressed in terms of the plasma dispersion function (3.1), making the resultant expressions simpler to treat both analytically and numerically. When solving the integrals, we will assume that p remains within the region of analyticity Re(p) σ > 0, with σ defined after (2.23). The analytic continuation will be performed only after obtaining expressions for (2.55) and (2.56) in terms of known functions. In the main text, we present the integration of (2.55); all other required expressions follow directly from this single integral, and have been relegated to appendices A and B due to their complexity. The remainder of this section proceeds as follows. §4.1 discusses the multivalued nature of the integrand of (2.55) before evaluating the integral over u in terms of plasma dispersion function (3.1), allowing us, in §4.2, to obtain a closed form expression for (2.55) upon evaluating the remaining integral over µ. In §4.3, we discuss how the ∂ a and ∂ b derivatives of (2.55) and (2.56) can be obtained, with detailed calculations relegated to appendix B. Then, in §4.4, we discuss some important properties of (2.55) and (2.56).
Multivaluedness
To begin, it shall be useful to consider the integral over u separately, and so we write (2.55) as follows:
I a,b = ∞ 0 dµ e −bµĨ a ,Ĩ a = 1 √ π ∞ −∞ du e −au 2 u − ζ + ζ d (2u 2 + µ)
.
(4.1)
Now, for each value of µ, the denominator ofĨ a has two zeros at
u = −1 ± 1 + 8ζ d (ζ − ζ d µ) 4ζ d (4.2)
that produce poles on opposite sides of the integration contour along the real u axis. Unsurprisingly, given the presence of square roots in (4.2),Ĩ a is a multivalued function.
In particular, we shall find thatĨ a , and thus I a,b , has two branches, just like the square root. To define these two branches, we need to choose a branch cut, which will allow us to 'label' the two zeros in (4.2). Note that this choice cannot (and does not) affect the time evolution of the potentials that results from the inverse Laplace transform of (2.21). It turns out to be analytically convenient to consider the 'principal' branch cut for the square-root function, for which √ z is discontinuous across Re(z) < 0. We can then define the two branches of the square root, + √ z and − √ z, where the principal branch satisfies
+ √ z > 0 for all positive real z, and sgn[Im( + √ z)] = sgn[Im(z)]
. At this point, it is nontrivial to define the second branch of I a,b . The choice of a branch for the square root does not determine the branch of the integral (4.1) but only the labels of the zeros in (4.2) -observe that (4.1) makes no reference to any multivalued functions. Indeed, the function I a,b is defined as the integral in (4.1) only for Im(ζ) > 0; the multivaluedness becomes relevant after one considers the analytic continuation to Im(ζ) < 0. To make this explicit, until we perform said continuation, we will make use of the labelsĨ + a and I + a,b to indicate that our expressions only apply to this one branch. Choosing to work with + √ , the zeros (4.2) can be written as
u = ∓u ± , u ± ≡ + 1 + 8ζ d (ζ − ζ d µ) ± 1 4ζ d . (4.3)
Using a partial-fraction expansion of the integrand, it follows that
I + a = 1 2ζ d (u + + u − ) 1 √ π ∞ −∞ du e −au 2 u − u − − 1 √ π ∞ −∞ du e −au 2 u + u + . (4.4)
Now, given that Im(ζ) > 0, the sign of the imaginary part of
+ 1 + 8ζ d (ζ − ζ d µ) is determined by the sign of ζ d , viz., sgn Im + 1 + 8ζ d (ζ − ζ d µ) = sgn(ζ d ), (4.5)
and so (4.3) implies that Im(u ± ) > 0. Therefore, the first integral in the brackets in (4.4) is manifestly the plasma dispersion function, as the imaginary part of the pole at u = u − has the correct sign for the definition (3.1), i.e., Im(u − ) > 0. The second integral has a pole at u = −u + with the opposite sign of its imaginary part, i.e., Im(−u + ) < 0, meaning that it can also be turned into a plasma dispersion function under a straightforward change of variables u → −u (this effectively flips the pole from being below the real u axis to being above it). Thus, it follows that (4.4) can be written as
I + a = 1 2ζ d Z a (u + ) + Z a (u − ) u + + u − , (4.6)
where we have used the shorthand notation (3.3) for Z a .
Explicit evaluation of I
+ a,b
Using (4.6), our expression for I + a,b thus becomes:
I + a,b = 1 2ζ d ∞ 0 dµ e −bµ Z a (u + ) + Z a (u − ) u + + u − . (4.7)
Using (4.3), together with the property Z (u) = −2[1 + uZ(u)], it can be deduced that
µ = ζ ζ d + 1 4ζ 2 d − u 2 + + u 2 − (4.8)
and
e au 2 ± u + + u − = 1 √ a ∂ ∂µ e au 2 ± Z a (u ± ) . (4.9)
It is then a matter of straightforward algebra to show that (4.7) can be rewritten as
I + a,b = 1 2 √ aζ d ∞ 0 dµ e −(b−a)µ ∂ ∂µ e −aµ Z a (u + )Z a (u − ) .
(4.10)
Observe that (2.30)-(2.38) only reference I a,b , and its derivatives with respect to a and b, evaluated at a = b = 1. We thus set a = b in (4.10) and, noting that Z a (u ± ) → 0 as µ → ∞ since Im(u ± ) > 0, we find
I + a,a = − 1 2 √ aζ d Z a (ζ + + )Z a (ζ + − ), (4.11)
where we have introduced
ζ + ± = u ± | µ=0 = + √ 1 + 8ζ d ζ ± 1 4ζ d . (4.12)
From (4.11) and (4.12), it is clear that I a,a is a multivalued function with a branch point at ζ = −1/8ζ d . Its second branch can be obtained by considering the − √ branch of the square root in (4.12). This means that both branches can be summarised by defining
ζ λ ± ≡ λ √ 1 + 8ζ d ζ ± 1 4ζ d = λ + √ 1 + 8ζ d ζ ± 1 4ζ d , (4.13)
where λ = ± labels the branch. Therefore, I a,a can be written as
I λ a,a = − 1 2 √ aζ d Z a (ζ λ + )Z a (ζ λ − ).
(4.14)
Equation (4.14) is the key result of this paper. We shall henceforth refer to it as the generalised plasma dispersion function, in that it is the generalisation of the usual plasma dispersion function (3.1) to include the resonances associated with the magnetic drifts arising in a non-uniform magnetic field. In §5.1, we show that, in the appropriate limits, the generalised plasma dispersion function reduces to the already-known solutions discussed in §3. It is worth stressing that (4.14) is an exact result: no approximations have been made in deriving it from (2.55). Furthermore, the fact that (4.14) is composed of a product of plasma dispersion functions, for the evaluation of which there are numerous efficient algorithms, means that it is very fast to evaluate numerically. In §5.2, we compare our expression for I a,b with the numerical solver by Gürcan (2014).
It can be shown, via a similar procedure to the one used to obtain (4.10) (see appendix A), that the related integral J + a,b (2.56) can be expressed exactly in terms of I + a,b and plasma dispersion functions as
1 − 2b a J + a,b = 1 2aζ d Z a (ζ + + ) − Z a (ζ + − ) + b 2aζ d I + a,b ,(4.15)
and so
J λ a,a = − 1 2aζ d Z a (ζ λ + ) − Z a (ζ λ − ) − 1 2ζ d I λ a,a .
(4.16)
It is crucial to realise that the λ = + branch of the functions I λ a,a and J λ a,a is the 'more important' one, in the sense that it is the branch that is equal to the integrals (2.55) and (2.56) for Im(ζ) > 0. Thus, it is also the branch that is used in the inverse Laplace transform over C σ , as in (2.24). Therefore, we shall refer to the λ = +1 branch as the 'principal' branch of I λ a,a and J λ a,a .
Derivatives of the generalised plasma dispersion function
In addition to (2.55) and (2.56), the matrix elements (2.30)-(2.38) require the partial derivatives of these expressions with respect to a and b. There are two factors that conspire to simplify the necessary calculations. First, we only need I a,b , J a,b , and their derivatives at a = b = 1. Secondly, the derivatives ∂ a and ∂ b often appear in the combination ∂ a + ∂ b . Notice that, by the chain rule,
(∂ a + ∂ b )f a,b a=b=1 = ∂ a f a,a a=1
(4.17)
for any (appropriately smooth) function f . Using this, we can rewrite (2.30)-(2.38) in a way that involves only I a,a , J a,a , ∂ a I a,b | a=b , ∂ b I a,b | a=b , ∂ 2 b I a,b | a=b , and ∂ b J a,b | a=b . For example,
L φB = s q s n 0s q r n 0r ζ s − ζ * s + η s ζ * s ∂ a + 3 2 ∂ b I (s) a,b a=b a=1 ,(4.18)
where we have also taken advantage of (2.51). Due to their unwieldy length, the calculations of the required derivatives of I a,b and J a,b are relegated to appendix B.
Branches of the dispersion function
Our choice of the principal branch cut for the square root gives the branches of the dispersion function D (see §2.2) several nice properties stemming from the relationship
+ √ z * = + √ z * for any complex z.
In appendix C, we show that I a,a and J a,a satisfy
I λ a,a (−ζ * , −ζ d ) = −I λ a,a (ζ, ζ d ) * , (4.19) J λ a,a (−ζ * , −ζ d ) = J λ a,a (ζ, ζ d ) * ,(4.20)
and
I λ a,a (ζ * , ζ d ) = I −λ a,a (ζ, ζ d ) * , (4.21) J λ a,a (ζ * , ζ d ) = J −λ a,a (ζ, ζ d ) * . (4.22)
Relations (4.19)-(4.22) are also valid for the a and b derivatives of I a,b and J a,b . Of course, the functions I a,b and J a,b are double-valued for each of the species s, and so the dispersion function D has 2 N branches for a system with N species. Letting λ ≡ (λ 1 , λ 2 , ..., λ N ) be the vector of choices of the branch for each species, we can prove that D satisfies (see appendix C)
D λ (p * , −k) = D λ (p, k) * , (4.23) D λ (−p * , k) = D −λ (p, k) * . (4.24)
These imply two different pairings of roots of the dispersion relation (2.48); see figures 11 and 12 in appendix C for a visual illustration of (4.23) and (4.24). Note that when using the superscript λ, we are referring to a particular branch, while without it, D refers to all branches simultaneously. Relation (4.23) implies that solutions to the dispersion relation, i.e., D = 0, come in pairs (p, k y ) ↔ (p * , −k y ), which is the condition for the fields φ, A , and δB to remain real for all t. Therefore, such a pairing is bound to exist for all roots of D = 0, i.e., when all branches are considered. The choice of the principal branch of the square root makes this pairing also valid within each individual branch of D, hence justifying our adoption of it in §4.1. In §6, we shall see that there is a better choice of branch for the purposes of performing the inverse Laplace transform.
Additionally, (4.24) says that if p is a solution to D = 0 for a given poloidal wavenumber k y , then so is −p * for the same k y but for a different branch. At first glance, this might seem to imply that solutions to (2.48) always come in pairs, one stable and one unstable. While this is true if all branches of D are considered, the time evolution given by the inverse Laplace transform (2.24) does not necessarily pick up contributions from all solutions to D = 0; one cannot mix-and-match roots from different branches at will. In §6, we shall see that the roots of (2.48) picked up by (2.24) depend on the choice of branch cut. However, only the principal branch, given by λ = (+, +, ..., +), contributes linearly unstable solutions.
Comparison with known results
Asymptotic expansions of I a,b
Let us now show that (4.14) asymptotes to the known limits discussed in §3, as it should. First, in the limit of ζ d → 0, i.e., the limit of vanishing magnetic curvature, we
ζ + + = 1 2ζ d + ζ − 2ζ d ζ 2 + O ζ 2 d , (5.1) ζ + − = ζ − 2ζ d ζ 2 + 8ζ 2 d ζ 3 + O ζ 3 d ,(5.2)
and the asymptotic form Z(ζ) ∼ −ζ −1 for finite Im(ζ) but |Re(ζ)| → ∞, to find
Z a (ζ + + ) ∼ −2ζ d √ a , Z a (ζ + − ) ∼ Z a (ζ). (5.3)
Therefore, in the limit ζ d → 0, the principal branch satisfies
I + a,a ∼ 1 a Z a (ζ),(5.4)
in agreement with (3.2). This is visualised in figure 3. One can perform an analogous calculation with J + a,b to obtain the second expression in (3.2). Note that the second branch satisfies
I − a,a ∼ − 1 a Z a (−ζ) (5.5)
in the limit ζ d → 0, which is not related to the correct expression for I a,a at zero magnetic curvature. The 'connection' between the two branches, viz., the branch cut, is 'sent to infinity' as ζ d → 0 (see figure 3), and so the second branch I − a,a is 'lost' in the limit of zero magnetic curvature. In this way, the dispersion function loses all but one of its branches and becomes single-valued.
On the other hand, the 2D limit can be found by taking the limit ζ ∼ ζ d → ∞, which is equivalent to dropping the u term from the denominators in (2.55) and (2.56). In this case,
ζ + ± = + ζ 2ζ d ± 1 4ζ d + O(ζ −2 d ),(5.6)
and so one obtains (3.4). Figure 4: Plots of the asymptotic convergence of I a,b and J a,b to their known limits. Panels (a) and (b) demonstrate the convergence of I + 1,1 and J + 1,1 , given by (4.14) and (4.16), to their smalland large-ζ d limits, given by (3.2) and (3.4), respectively. We define the relative difference of two functions f and g as |f − g|/ min{|f |, |g|}. Note that this is ill-defined if one of the functions is identically zero, so for the J1,1 comparison in panel (b), we plot simply |J1,1| because we expect to recover J1,1 = 0 in the 2D limit (3.4). The solid and dotted lines in (a) and (b) show the average and maximum relative difference, respectively, as computed over a grid of 32 × 32 points for ζ, equally spaced in Re(ζ) ∈ [1, 1], Im(ζ) ∈ [0, 1], for each value of ζ d . Panel (c) demonstrates the convergence of the real (dashed) and imaginary (dash-dot) parts of I + 1,1 to the small-and large-ζ d limits for a fixed ζ = 1 + i, which are given by Z(ζ) and −Z 2 ( √ Ω)/ζ d , with Ω = ζ/2ζ d , respectively. Figure 4 compares the exact expressions (4.14) and (4.16) with their known asymptotic limits in the case of vanishing magnetic drifts (3.2) and 2D perturbations (3.4), respectively. It is evident that, while these known asymptotic limits are obtained in the cases of small and large ζ d , they are not a good approximation of I a,b and J a,b for ζ d ∼ 1, as one would expect.
Z(ζ) −Z( √ Ω) 2 /2ζ d (a) (b) (c)
Numerical comparison with Gürcan (2014)
Gürcan (2014) consider a very similar problem to the one on which this paper has focused but from a numerical perspective. In particular, they discuss the numerical integration of the function
I nm (ζ α , ζ β , b) = 2 √ π ∞ 0 dx ⊥ ∞ −∞ dx x n ⊥ x m J 2 0 ( √ 2bx ⊥ )e −x 2 −x 2 ⊥ x 2 + x 2 ⊥ /2 + ζ α − ζ β x ,(5.7)
defined for Im(ζ α ) > 0, real ζ β and b, and n > 1. With a few algebraic manipulations, it can be shown that, for odd n, Relative difference The above expression is actually correct only for ζ d < 0, otherwise (5.7) computes the second (λ = −) branch of I a,b , J a,b , and their derivatives. In the case ζ d < 0, the requirement Im(ζ α ) > 0 implies that Im(ζ) > 0. Figure 5 There is good agreement for all tested values of ζ and ζ d , with less than 1% relative difference in most cases. It is important to stress that as our solution uses only standard functions, e.g., the plasma dispersion function Z and √ , for which there exist very efficient numerical algorithms. We found that even a naïve, unoptimised Python implementation took anywhere between 20 and 80 times less time to compute I a,b , J a,b , and their derivatives than the direct numerical integration of (5.7) implemented in Fortran at https://github.com/gurcani/zpdgen.
I nm − ζ 2ζ d , − 1 2ζ d , 0 = − 1 ζ d (−∂ a ) m 2 (−∂ b ) n−1 2 I + a,b (ζ, ζ d )| a=b=1 ,I 1,1 J 1,1 ∂ a I a,b | a=b=1 ∂ b I a,b | a=b=1 ∂ 2 b I a,b | a=b=1 ∂ b J a,b | a=b=1
Analytic continuation for the inverse Laplace transform
Together, the expressions (4.14) and (4.16) for I a,a and J a,a , respectively, along with the derivatives (B 12)-(B 15), allow us to calculate L, and hence the Laplace-transformed fields (2.47). Recall that in order to determine the evolution of the system as a function of time, we need to compute the inverse Laplace transform χ k (t) = 1 2πi Cσ dp e ptχ k (p), (6.1)
where the contour of integration C σ is once again as in figure 1, and we remind the reader thatχ k (p) is given byχ where the vector of initial conditions G is given by (2.44)-(2.46). The results of §4.2 show that the entries of L have branch points at ζ s = −1/8ζ ds , or equivalently, at p = p s , where
k (p) = (adj L)G det L ,(6.p s ≡ ik 2 v 2 ths 8ω ds ,(6.3)
but are otherwise free of poles since, apart from the square roots and the associated branch cuts, they are composed of entire functions. Recall that we have defined the branches of the dispersion function D using the principal branch of the square root in (4.3). Therefore, the relevant branch D λ that enters the inverse Laplace transform is the principal branch given by λ = (+, ..., +). This has branch cuts that connect the branch points ζ s = −1/8ζ ds to ζ s → −sgn(ζ ds )∞, or, equivalently, p = p s to p → isgn(ζ ds )∞. While this choice of the principal branch and branch cuts was convenient for obtaining the closed forms of I a,b , J a,b , and D, and their properties, it is not necessarily the best one for performing the inverse Laplace transform (6.1). Instead, we would like to rotate the branch cuts by sgn(ζ ds )π/2 around p s , so that they are parallel to the real p axis, as shown in figure 6. Let us call the branch D of the dispersion function obtained this way the 'dispersion' branch. Crucially, the rotation of the branch cuts does not disturb the values of the dispersion function at Re(p) > 0. Therefore, D(p) = D (+,...+) (p) for Re(p) > 0. This ensures that the 'unphysical' unstable zeros of the other branches of the dispersion function, which are a consequence of (4.24), do not contribute to the solution (see also discussion in §4.4); the only unstable solutions that are picked up by the inverse Laplace transform are those of the principal branch. With this choice for the branch cuts of the dispersion function, we are ready to perform the inverse Laplace transform (6.1). This is done in the usual way, viz., by pushing the integration contour C σ towards Re(p) → −∞, with the proviso that it must be deformed so as not to cross any singularities, e.g., poles or branch cuts. Pushing the contour to the vertical line at Re(p) = ρ, we find the new integration contour C ρ (see figure 7). Since there are no singularities between C σ and C ρ , Cauchy's integral theorem ensures that the integrals over these two contours are equal. Taking the limit of ρ → −∞, it is evident that the contributions arising from the vertical segments of C ρ are exponentially Figure 7: Same as in figure 1, except that the contour associated with the inverse Laplace transformation (6.1) has now been shifted to Re(p) = ρ, deforming it such that it does not cross any of the poles or the branch cut. We denote this new contour Cρ. The original contour is shown by the vertical dashed line. The integrals along Cσ and Cρ are equal by Cauchy's integral theorem.
Re(p) Im(p) σ × × × × C σ C ρ ρ
small 2 , while those arising from the integration along the horizontal segments leading towards and away from the poles cancel, leaving the contributions from the poles. The integration around the branch cuts is more subtle and will be discussed shortly.
There are several singularities present in (6.2), and hence in the integrand in (6.1). The first is the so-called 'ballistic response' associated with the initial conditions contained within G, arising from simple poles located along Re(p) = 0, viz., lim ρ→−∞ 1 2πi Cρ dp e pt g sk p + ik v + iω Ds = g sk e −i(k v +ω Ds )t , (6.4)
where we have assumed that g sk is a smooth function. Plugging this into (2.44)-(2.46), we find that the contribution to χ k (t) due to the ballistic response can be written as
χ k0 (t) = s d 3 v L −1 (−ik v − iω Ds )g sk e −i(k v +ω Ds )t q s n 0s q r n 0r 1 n 0s J 0 (b s ), q s n 0s v ths q r n 0r v thr 1 n 0s v v ths J 0 (b s ), − β s 2 1 n 0s v 2 ⊥ v 2 ths 2J 1 (b s ) b s T . (6.5)
There is a wealth of interesting physics that can arise from the ballistic response, see, e.g., Ewart et al. (2022) and references therein, in the context of the Vlasov-Poisson system. However, this is not the focus of the present work and so will not be discussed further. Another source of non-analyticity are the solutions to the dispersion relation D = 0, should any of these exist. The contributions to (6.1) arising from the zeros p = p j of D can be written as j Res[χ k (p), p j ]e pj t .
(6.6) It is evident that unlike the ballistic response, whose time dependence is an oscillating exponential, the terms (6.6) can, in general, be exponentially decaying (i.e., stable) for Re(p j ) < 0 or growing (i.e., unstable) for Re(p j ) > 0. Finally, singularities may arise from the functions (2.39)-(2.43) that are contained within both adj L and det L. As discussed above, these functions are free of poles, but are multivalued. Deforming the integration contour C ρ around their branch cuts (see figure 7) gives a nontrivial contribution to (6.1). Letting B s (t) be the contribution from the integral around the branch cut due to a given species s, we can finally write the full solution for χ k (t) as
χ k (t) = χ k0 (t) + j Res[χ k (p), p j ]e pj t + s B s (t).
(6.7)
In appendix D, we show that, in the long-time limit t → ∞, the branch-cut contribution B s for each species is dominated by that arising from the branch point itself, and exhibits an algebraic decay ∝ t −3/2 . The same algebraic decay was found by Kim et al. (1994); Kuroda et al. (1998) in their treatment of the toroidal ITG mode. Such a 'continuum mode' (Kuroda et al. 1998;Sugama 1999) is a direct consequence of the multivaluedness of (2.55) and (2.56), in that such multivaluedness gives rise to a branch point and to the resulting discontinuity. This behaviour is qualitatively different from that of a plasma in a straight magnetic field, whose dispersion function is single-valued, meaning that there are no branch cuts and hence no continuum modes. Note that nonexponentially decaying solutions to similar initial-value problems can also be found in other contexts; see, e.g., Taylor (1965); Sedlàček (1995). Equation (6.7) is our final expression for the time evolution of χ k (t). Depending on whether there are any unstable solutions, we find that either: (i) there are solutions to D(p) = 0 for Re(p) > 0. In that case, the long-time solution is dominated by the solution with largest Re(p); or (ii) there are no solutions to D(p) = 0 for Re(p) > 0. In that case, the long-time solution is dominated by the ballistic response (6.4) and by waves with frequencies ω = ip s = −k 2 v 2 ths /8ω ds that exhibit a nonexponential decay ∝ t −3/2 .
From drift kinetics to gyrokinetics
The analytical forms of the integrals derived in §4 are not without their limitations: in their derivation, we assumed both the drift-kinetic limit and the case of equal magnetic drifts (see §2.3). We will now devote some space to a brief discussion of how one can relax these assumptions.
Bessel functions
The drift-kinetic assumption is perhaps the more egregious approximation, especially given that the presence of finite-Larmor-radius effects, or otherwise, can have a nontrivial impact on the plasma dynamics (see, e.g., Smolyakov et al. 2002;Parisi et al. 2020;Parisi et al. 2022, and references therein). Thankfully, however, it can be relaxed if one is willing to pay the price of complicated analytical expressions. Noting that 2J 0 (b s )J 1 (b s ) = −∂J 2 0 (b s )/∂b s , it is clear that the Bessel functions J 0 and J 1 always appear quadratically in (2.39)-(2.43), for which there are known, rapidly converging Taylor series (Neumann 1871;Watson 1966):
J 2 n (b s ) = ∞ m=0 (−1) m (2n + 2m)! m!(2n + m)![(n + m)!] 2 b s 2 2n+2m .
(7.1)
Using this expansion in (2.39)-(2.43), one can, in principle, compute each of the resulting integrals analytically, and thus obtain an absolutely convergent series for the resulting gyrokinetic dispersion relation. This is done by noticing that their argument b s only appears quadratically as b 2 s = µk 2 ⊥ ρ 2 s , and thus the additional factors of µ can be handled by partial differentiation with respect to b before setting a = b in (2.39)-(2.43). For example, (2.39) would give
I (s) a,b (ζ s , ζ ds , ζ ds ) = ∞ m=0 (2m)! (m!) 4 k ⊥ ρ s 2 2m ∂ m b I a,b (ζ s , ζ ds ), (7.2)
with the other required integrals, viz., J (s) a,b and ∂ a I (s) a,b , satisfying similar expressions. We remind the reader that I (s) a,b refers to the FLR-containing integral (2.39), while I a,b is the integral (2.55) on which we have focused throughout most of this paper. Doing this calculation by hand seems rather daunting given the complicated expressions even for the low-order derivatives ∂ 2 b I a,b and ∂ b J a,b [see (B 14) and (B 15), respectively]. In practice, however, only a few terms would be needed due to the rapid convergence of the Taylor series (7.1). Those wishing to compute these terms to an arbitrary order may want to do so by using symbolic libraries (e.g., those in Wolfram Mathematica) in order to calculate the derivatives analytically, which can then be imported into an associated numerical solver. An alternative approach would be to implement a recursive scheme to calculate numerically the m th -order derivatives from the (m − 1) th ones.
General magnetic drifts
Our second approximation was to neglect the difference between the curvature and ∇B drifts, taking their associated drift frequencies to be equal, i.e., ζ κs = ζ ∇Bs , as in (2.54). While this approximation is relatively well-satisfied in the context of magneticconfinement fusion, there are certainly other systems in which it is not, e.g., space and astrophysical plasmas. By a simple change of variables to µ = ζ Bs µ/ζ κs in (2.52), we find
I (s) a,b (ζ s , ζ κs , ζ Bs ) = 1 √ π ∞ −∞ du ∞ 0 dµ e −au 2 −bµ u − ζ s + (2u 2 ζ κs + µζ Bs ) = ζ κs √ πζ Bs ∞ −∞ du ∞ 0 dµ e −au 2 −(bζκs/ζ Bs )µ u − ζ s + ζ κs (2u 2 + µ ) . (7.3)
Therefore, the integral (2.52) that enters (2.30)-(2.38) at a = b can be found in terms of the known integrals (2.55) via I (s) a,a (ζ s , ζ κs , ζ Bs ) = ζ κs ζ Bs I a,b (ζ s , ζ κs ), (7.4)
where now b = aζ κs /ζ Bs . Finally, to find I a,b for a = b, one can Taylor expand
I a,b = ∞ m=0 (b − a) m m! ∂ m b I a,b | a=b . (7.5)
Fortunately, just as in §7.1, one can find closed, albeit complicated, analytical expressions for ∂ m b I a,b | a=b for any m. The expansion should converge for arbitrary positive a and b since (7.5) is equivalent to expanding e −(b−a)µ in (4.10) using its absolutely convergent Taylor series.
Electrostatic ITG: a detailed example
To illustrate the results of §4, we provide an explicit calculation of the dispersion relation in the simple case of an electrostatic, ion-scale, temperature-gradient-driven instability, and compare the solution with well-known kinetic and fluid limits.
In particular, we consider a two-species plasma of ions and electrons of comparable temperatures, T 0i ∼ T 0e . Since we want to consider electrostatic physics, we assume
β s ∼ (k ⊥ d s ) −2 1, (8.1)
where d s = m s c 2 /4πn 0s q 2 s is the skin depth. Therefore, to lowest order, A k and δB k do not contribute to (2.17), and (2.29) simplifies to
L φφ q rφk T 0r + G φ = 0. (8.2)
This implies that the dispersion relation is given simply by L φφ = 0. Furthermore, we consider the frequencies of the perturbations to be comparable to the parallel streaming and drift frequencies of the ions, as well as the magnetic-drift frequency, viz.,
p ∼ k v thi ∼ ω di ∼ ω * i ∼ η i ω * i . (8.3)
The relevant equilibrium length scales in our problem are thus the ion-density and iontemperature gradients L −1 ni and L −1 Ti , respectively [see (2.6)], and the gradient of the magnetic field L −1 B ≡ −∂ ln B 0 /∂x. In the small-mass-ratio limit, m e /m i 1, (8.3) implies
k v the ∼ m i m e k v thi p,(8.4)
i.e., the electrons stream quickly along the fields lines. Thus, ζ e ∼ ζ * e ζ i ∼ ζ * i , and the electron contributions to L φφ can be ignored. Choosing q r = q i = Ze, T 0r = T 0i , and n 0r = n 0i , the expression (2.30) simplifies to
− L φφ = 1 + τ + ζ i − ζ * i + η i ζ * i ∂ a + 3 2 I (i) a,a a=1 ,(8.5)
where τ ≡ T 0i /ZT 0e is the temperature ratio. To avoid carrying around an extra minus sign, we shall define D ≡ −L φφ , the object whose zeros we shall be interested in. Using (4.14), we obtain the principal branch of the ITG dispersion relation
D = 1 + τ − ζ − ζ * 2ζ d Z + Z − + ηζ * 2ζ d (ζ + Z − + ζ − Z + ) + ζ ζ d + 1 4ζ 2 d − 1 Z + Z − = 0, (8.6)
where we have dropped the i subscripts, ζ ± are given by (4.12), and we are using the shorthand notation Z ± ≡ Z(ζ ± ). Note that the principal branch (i.e., λ = +) is implicitly used everywhere, but we have dropped the associated superscripts to reduce the notational clutter.
We can use (5.2) and (5.6) to verify that (8.6) converges to the correct limits in the case of: vanishingly small magnetic gradients (i.e., ζ d → 0) and of 2D perturbations (i.e., ζ ∼ ζ * ∼ ζ d → ∞) (8.8) where Ω = ζ/2ζ d = ip/2ω d and Ω * = ζ * /2ζ d = ω * /2ω d . Note that (8.8) agrees with the expressions obtained by Biglari et al. (1989); Zocco et al. (2018) in a similar limit to (8.3).
D slab = 1 + τ + (ζ − ζ * )Z(ζ) + ηζ * ζ + ζ 2 Z(ζ) − 1 2 Z(D 2D = 1 + τ − (Ω − Ω * )Z( √ Ω) 2 + ηΩ * 2 √ ΩZ( √ Ω) + (2Ω − 1) Z( √ Ω) 2 = 0,
In figure 8, we compare the solutions to (8.6) and (8.7) for the case of zero density gradient, viz., ω * = 0, but nonzero temperature gradient, so ηω * ∝ L −1 Ti = 0. The growth rates agree well only at simultaneously large perpendicular and small parallel wavelengths; this is to be expected given that the slab dispersion relation (8.7) does not capture the effect of magnetic drifts, which are most important at large parallel wavelengths. There is poorer agreement between the frequencies of the two dispersion relations.
We can also compare the solutions to (8.6) with those obtained from a simple threefield fluid model of the ITG instability in a slab with magnetic curvature. The model consists of the following equations: (8.11) where ϕ ≡ Zeφ/T 0i , u , and δT i /T 0i are the perturbed electrostatic potential, ion parallel flow, and ion temperature, respectively. These equations can be derived by substituting a perturbed Maxwellian for h i in the ion gyrokinetic equation and taking the three relevant velocity moments (cf. Newton et al. 2010 or the cold-ion fluid model in but with additional τ ∼ 1 terms). Figure 9 shows a comparison between the kinetic and fluid growth rates at fixed value of τ and varying k L B . We see that the fluid approximation is decent for small k L B , but fails for larger ones because of its lack of kinetic effects. Making the ions cold, i.e., lowering τ , improves the accuracy of the fluid approximation, as in figure 10.
τ ∂ϕ ∂t + ∂u ∂z − ρ i v thi L B ∂ ∂y (1 + τ )ϕ + δT i T 0i = 0, (8.9) ∂u ∂t + v 2 thi 2 ∂ ∂z (1 + τ )ϕ + δT i T 0i − 2ρ i v thi L B ∂u ∂y = 0, (8.10) ∂ ∂t δT i T 0i + 2 3 ∂u ∂z + ρ i v thi 2L Ti ∂ϕ ∂y − 2 3 ρ i v thi L B ∂ ∂y (1 + τ )ϕ + 7 2 δT i T 0i = 0,
Summary and discussion
We have considered the problem of local linear gyrokinetics in a curved magnetic field, expressing the associated dispersion relation in terms of velocity-space integrals featuring resonances arising both from parallel streaming and from magnetic drifts ( §2). Previously, exact solutions for these integrals were known either in the absence of magnetic drifts -leading to the well-known plasma dispersion function Z(ζ) -or in the two-dimensional limit ( §3). In the case of drift kinetics (i.e., no finite-Larmorradius effects) and equal magnetic drifts, we showed that these resonances can in fact be handled simultaneously without any additional approximations or expansions, and that the integrals can be expressed exactly in terms of a generalised plasma dispersion function consisting of products of Z functions, and its derivatives ( §4). Since there exist known algorithms for the computation of the Z function, the resulting expressions are efficient to evaluate numerically, and can easily be handled analytically through known asymptotic expansions. Solutions to the exact dispersion relation for the electrostatic ITG instability, derived using this method, were then compared with approximate solutions in the previously known limits, showing poor agreement for the majority of parameters and wavenumbers considered ( §8). This demonstrates that, in order to properly capture the growth rate and frequency of kinetic instabilities in the presence of a curved magnetic field, one must simultaneously resolve the resonances associated with parallel streaming and magnetic drifts, for which this paper provides the first known exact analytical solution.
In §7, we discussed how the assumptions of no finite-Larmor-radius effects and equal magnetic drifts can be relaxed using absolutely convergent Taylor-series expansions, and thus solve the more general linear gyrokinetic system. This results in expressions that naturally capture the multivaluedness of the underlying dispersion relation and handle the integration of resonant denominators exactly.
An immediate practical application of this work would be to use the derived analytical expressions to implement an efficient and accurate solver for drift-kinetic/gyrokinetic instabilities in the local limit considered in this paper. Such a solver could be used to benchmark both reduced models and gyrokinetic solvers. It could also be exploited to explore the equilibrium parameter space in search of new instabilities or to investigate the properties of subdominant ones, i.e., those whose growth rate is smaller than the largest growth rate in the system; this is typically difficult to do in most gyrokinetic solvers. Such subdominant instabilities have been proposed as one of the possible explanations for the lack of saturation observed in certain electromagnetic gyrokinetic simulations. With this in mind, we consider the implementation of such a gyrokinetic dispersion-relation solver to be a natural extension of this work that will produce a useful practical tool in the study of gyrokinetic instabilities and turbulence.
Funding
This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014and 2019, and from the UKRI Energy Programme (EP/T012250/1). The views and opinions expressed herein do not necessarily reflect those of the European Commission. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC) [EP/R034737/1]. TA was supported by a UK EPSRC studentship.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Calculation of J a,b
In this appendix, we derive the expression (4.15) for J a,b . The calculation proceeds in a similar way to that of I a,b in §4.2. Starting from (2.56), we consider the integral over u separately, and so write
J a,b = ∞ 0 dµ e −bµJ a ,J a = 1 √ π ∞ −∞ du ue −au 2 u − ζ + ζ d (2u 2 + µ)
.
(A 1)
Defining u ± as in (4.3) and making the same choice for the branch cut and square-root branch, a partial-fractions expansion of the integrand yields
J + a = 1 2ζ d (u + + u − ) u − √ π ∞ −∞ du e −au 2 u − u − + u + √ π ∞ −∞ du e −au 2 u + u + . (A 2)
As previously, the sign of the imaginary part of u ± is always positive. Therefore, the first integral in the brackets in (A 2) is manifestly the plasma dispersion function, while the second can be turned into a plasma dispersion function under the change of variables u → −u. Thus, it follows that (A 2) can be written as
J + a = − 1 2ζ d u + Z a (u + ) − u − Z a (u − ) u + + u − . (A 3)
Using the property (confirmed by direct calculation)
u ± Z a (u ± ) = − 1 √ a + u + + u − a ∂Z a (u ± ) ∂µ , (A 4)
we can writeJ + a asJ
+ a = − 1 2aζ d ∂ ∂µ [Z a (u + ) − Z a (u − )] . (A 5)
Alternatively, using
u ± = 1 2 (u + + u − ) ± 1 4ζ d , (A 6)
we can also writeJ
+ a = − 1 4ζ d [Z a (u + ) − Z a (u − )] − 1 4ζ dĨ + a . (A 7)
Therefore, by substitution into the first expression in (A 1), we find
J + a,b = − 1 4ζ d ∞ 0 dµ e −bµ [Z a (u + ) − Z a (u − )] − 1 4ζ d ∞ 0 dµ e −bµĨ + a = − 1 4bζ d [Z a (ζ + ) − Z a (ζ − )] + a 2b J + a,b − 1 4ζ d I + a,b , (A 8)
where, in going from the first line to the second, we have integrated by parts in the first integral and have used (A 5). Equation (A 8) can be straightforwardly rearranged to yield (4.15).
Appendix B. Calculation of derivatives of I a,b
Even though we are unable to evaluate (4.7) exactly in the case where a and b are distinct, we are still able to find its derivatives with respect to a and b at a = b = 1, a task to which this appendix is devoted.
B.1. Derivatives for general a and b
To avoid clutter, we shall suppress the λ = + indices until appendix B.2. For this subsection, assume that all expressions I a,b , J a,b , Q a,b , and ζ ± come with a λ = +.
Using (4.3), we can show that
∂Z a (u ± ) ∂a = − u ± (u + + u − ) a ∂Z a (u ± ) ∂µ , (B 1)
and so the derivative of (4.7) with respect to a becomes
∂ a I a,b = − 1 2aζ d ∞ 0 dµ e −bµ u + ∂Z a (u + ) ∂µ + u − ∂Z a (u − ) ∂µ = − 1 2aζ d ∞ 0 dµ e −bµ 1 2ζ d ∂ ∂µ [Z a (u + ) − Z a (u − )] + u − ∂Z a (u + ) ∂µ + u + ∂Z a (u − ) ∂µ = 1 2aζ d [ζ − Z a (ζ + ) + ζ + Z a (ζ − )] + 1 2ζ d J a,b − 1 2a I a,b − b a Q a,b , (B 2)
where we have defined the integral
Q a,b = 1 2ζ d ∞ 0 dµ e −bµ [u − Z a (u + ) + u + Z a (u − )] . (B 3)
In going from the first line of (B 2) to the second, we made use of the fact, obvious from the definition (4.3), that
u ± = u ∓ ± 1 2ζ d , (B 4)
while going from the second to the third, we have recognised the first expression in the curly brackets as (A 5) and integrated by parts the second. Similarly, taking a derivative of (4.7) with respect to b, and making use of (4.8) and (4.12), we have that
∂ b I a,b = −(ζ 2 + + ζ 2 − )I a,b + 1 2ζ d ∞ 0 dµ e −bµ u 2 + + u 2 − u + + u − [Z a (u + ) + Z a (u − )] . (B 5) Since u 2 + + u 2 − [Z a (u + ) + Z a (u − )] = (u + + u − ) [u − Z a (u + ) + u + Z a (u − )] + (u + − u − ) [u + Z a (u + ) − u − Z a (u − )] , (B 6) (B 5) becomes ∂ b I a,b = −(ζ 2 + + ζ 2 − )I a,b − 1 2ζ d J a,b + Q a,b , (B 7)
where we have made use of (A 5) again. It is clear from (B 2) and (B 7) that we need to find Q a,b in order to obtain expressions for ∂ a I a,b and ∂ b I a,b . Though it is possible to do so via direct manipulation of the integrand of (B 3), we prefer an alternative approach.
Using
u u − ζ + ζ d (2u 2 + µ) = 1 + ζ − ζ d (2u 2 + µ) u − ζ + ζ d (2u 2 + µ) (B 8) in (2.56) gives J a,b = 1 √ ab + ζI a,b + ζ d (2∂ a + ∂ b )I a,b . (B 9)
Substituting (B 2) and (B 7) into (B 9), and rearranging, we obtain the following expression for Q a,b in terms of I a,b and J a,b :
1 − 2b a Q a,b = − 1 √ abζ d − 1 aζ d [ζ − Z a (ζ + ) + ζ + Z a (ζ − )] + 1 2ζ d J a,b + 1 a + 1 4ζ 2 d I a,b . (B 10)
In a similar way, taking a ∂ b derivative of (4.15), we find
1 − 2b a ∂ b J a,b = 2 a J a,b + 1 4ζ d I a,b + b 2aζ d ∂ b I a,b . (B 11) B.2. Derivatives at a = b
Finally, using (4.15), (B 2), (B 7), and (B 10), setting a = b, and simultaneously expressing both branches using (4.13), we obtain
∂ a I λ a,b a=b = − 1 a 3/2 ζ d + 1 2a − 1 4ζ 2 d I λ a,a − 1 2aζ 2 d Z a (ζ λ + ) − Z a (ζ λ − ) − 1 2aζ d ζ λ − Z a (ζ λ + ) + ζ λ + Z a (ζ λ − ) , (B 12) ∂ b I λ a,b a=b = 1 a 3/2 ζ d − 1 a + ζ ζ d I λ a,a + 1 aζ d ζ λ + Z a (ζ λ + ) + ζ λ − Z a (ζ λ − ) , (B 13) ∂ 2 b I λ a,b a=b = − 1 a 5/2 ζ d − 2 a Q λ a,a − 1 a + ζ 2ζ d + 1 2ζ 2 d ∂ b I λ a,b a=b − 1 ζ d ∂ b J λ a,b a=b , (B 14) ∂ b J λ a,b a=b = − 1 2a 3/2 ζ 2 d + 1 2ζ d 2 a + ζ ζ d I λ a,a − 1 2aζ 2 d ζ λ + Z a (ζ λ + ) + ζ λ − Z a (ζ λ − ) + 1 a 2 ζ d Z a (ζ λ + ) − Z a (ζ λ − ) , (B 15ζ λ ± (−ζ * , −ζ d ) = −ζ λ ± (ζ, ζ d ) * (C 1) ζ λ ± (ζ * , ζ d ) = ζ λ ± (ζ, ζ d ) * , (C 2)
and that (4.13) implies
ζ −λ ± = −ζ λ ∓ . (C 3)
Additionally, it is straightforward to show that the Z function satisfies Z(ζ * ) = −Z(−ζ) * .
(C 4)
Then, using (4.14) and (C 1)-(C 4), we have
I λ a,a (−ζ * , −ζ d ) = − 1 2 √ a(−ζ d ) Z a (−ζ λ * + )Z a (−ζ λ * − ) = 1 2 √ aζ d [−Z a (ζ λ + )] * [−Z a (ζ λ − )] * = −I λ a,a (ζ, ζ d ) * ,(C 5)
and I λ a,a (ζ * , ζ d ) = −
1 2 √ aζ d Z a (ζ λ * + )Z a (ζ λ * − ) = − 1 2 √ aζ d [−Z a (−ζ λ + )] * [−Z a (−ζ λ − )] * = − 1 2 √ aζ d Z a (ζ −λ − ) * Z a (ζ −λ + ) * = I −λ a,a (ζ, ζ d ) * . (C 6)
Similarly, using (4.16), we find J λ a,a (−ζ * , −ζ d ) = J λ a,a (ζ, ζ d ) * (C 7)
J λ a,a (ζ * , ζ d ) = J −λ a,a (ζ, ζ d ) * .
(C 8)
The derivatives of I a,b , given by (B 12)-(B 14), and ∂ b J a,b | a=b , given by (B 15), can also be shown to have the properties (C 5)-(C 6) and (C 7)-(C 8), respectively. Recall that the frequencies, which enter the dispersion matrix elements (2.30)-(2.38), are functions of ζ s ∝ p, ζ * s ∝ k y , and ζ ds ∝ k y [see (2.13), (2.15), (2.27), and (2.54)]. It is then evident that p → p * maps ζ s → −ζ * s , p → −p * maps ζ s → ζ * s , and the inversion k → −k results in ζ * s → −ζ * s and ζ ds → −ζ ds (recall that the sign of the parallel wavenumber k does not enter the normalised frequencies, as we noted in footnote 1). Combining this with (C 5)-(C 8), it is then straightforward to show that the dispersion matrix L and its elements (2.30)-(2.38) satisfy where the vector λ = (λ 1 , λ 2 , ..., λ N ) labels the branches the double-valued functions that constitute L, for each of the N particle species. Therefore, for the dispersion function
Appendix D. Integral around the branch cut
This appendix is devoted to calculating the asymptotic contribution to (6.1) in the limit t → ∞ arising from the integral around one of the branch cuts ofχ k (p). Similar calculations already exist in the literature (e.g., Kim et al. 1994;Kuroda et al. 1998); we are including one here for completeness.
Recall that there is one branch cut for each particle species, associated with the branch point p s (6.3). We choose the branch cut to be parallel to the real p axis and denote the Figure 12: The same as figure 11 but with the branch cuts rotated to point towards Re(p) → −∞. As previously, crossing the electron branch cut flips the sign of λe and so corresponds to jumping horizontally between the panels, while crossing the ion branch cut corresponds to jumping vertically between the panels. For practical purposes, we are only interested in the 'dispersion' branch D (see discussion in §6) shown in (a) as it is that one that enters the inverse Laplace transform.
contour around this branch cut C br , as in figure 13. C br consists of a semi-circular arc C ε of radius ε around the branch point, where we choose ε ∼ t −2 , and two horizontal, semi-infinite segments C ± along Im(p) = Im(p s ) ± ε, viz., C br dp e ptχ k (p) = C−+Cε+C+ dp e ptχ k (p). (D 1)
Let us calculate each of the contributions to (D 1) in turn. For C ε , we change variables to p = p s + εe iθ for θ ∈ [−π/2, π/2]. It straightforwardly follows that, since ε ∼ t −2 , Cε dp e ptχ k (p) |χ k (p s )| [1 + O(ε)] ε π/2 −π/2 dθ e εt cos θ = O(t −2 ). (D 2)
Turning our attention to C ± , we set p = p s + ξ ± iε, respectively, and find C± dp e ptχ k (p) = ∓ where we have split the integration interval using some positive real δ 1. The first Using 0 −δ dξ e ξt ξ = t −3/2 0 −tδ dη e η √ η ∼ t −3/2 i √ π 2 as t → ∞, (D 10)
we finally arrive at C br dp e ptχ k (p) ∼ t −3/2 e ipst √ π ∂χ k (p s ) ∂ + √ p − p s as t → ∞, (D 11) which is the required result.
Figure 3 :
3A plot of the principal branch I + 1,1 (ζ, ζ d ) in the complex plane for decreasing values of ζ d (from left to right). The black cross denotes the branch point ζ = −1/8ζ d . Panel (d) shows I + 1,1 (ζ, 0) = Z(ζ). As ζ d → 0 + , the branch point, alongside the entire branch cut, is pushed towards Re(ζ) → −∞. If ζ d were negative, the branch cut would instead join the branch point with Re(ζ) → +∞, to which the branch cut would be pushed in the limit of ζ d → 0 − . employ the expansions
2
J + a,b (ζ, ζ d )| a=b=1 , for m odd.
Figure 5 :
5Mean (solid) and maximum (dotted) relative difference (defined as in figure 4) between expressions (4.14), (4.16), (B 12)-(B 15) and their equivalents derived from (5.8), computed via the code published at https://github.com/gurcani/zpdgen. For each ζ d , we evaluated the respective functions at an equally spaced grid of 32 × 32 points in the region Re(ζ) ∈ [−10, 10], Im(ζ) ∈ [0, 10].
shows a comparison between the values obtained via the results of this work [represented by (4.14), (4.16), (B 12)-(B 15)] and the Gürcan (2014) result (5.7) in the region Re(ζ) ∈ [−10, 10], Im(ζ) ∈ [0, 10], ζ d ∈ [−10, −0.001].
Figure 6 :
6This diagram shows the 'principal' (in blue) and 'dispersion' (in black) branch cuts for a plasma with one negatively and one positively charged species, labelled as s1 and s2, respectively.
Figure 8 :Figure 9 :
89A comparison between the growth rate and frequency of the most unstable solution to the kinetic dispersion relation with magnetic effects (8.6) and the slab dispersion relation (8.7), represented by the solid and dotted lines, respectively. Here, ρs = ρi/ √ 2τ is the ion sound radius, and we have set τ = 0.1 and τ LB/2LT i = 2. A comparison between the growth rate and frequency of the most unstable solution to the kinetic dispersion relation (8.6) and that obtained from the fluid equations (8.9)-(8.11), represented by the solid and dotted lines, respectively. The parameters used are the same as infigure 8.
Figure 10 :
10A comparison between the growth rate and frequency of the most unstable solution to the kinetic dispersion relation (8.6) and that of the fluid equations (8.9)-(8.11), represented by the solid and dotted lines. Here we have set k LB = 1 and τ LB/2LT i = 2.
λ (−p * , k) = L −λ (p, k) * (C 10)
Figure 11 :
11A plot in the complex plane of the dispersion function D(p) = L φφ for an electrostatic, two-species plasma composed of ion and electrons, for the following parameters: mi/me = 2, qi = −qe = e, T0i = T0e, kyρi = 1, k LB = 1, and LT i = LB. The panels show the four branches of D, labelled by λ = (λi, λe) as shown (see §4.4). Here we are using the principal branch cut for the square root. The colour brightness shows the magnitude |D|, while its hue shows the phase arg D. The relation (C 12), D λ (−p * , k) = D −λ (p, k) * , is evident in the pairs (a),(d) and (b),(c): flipping the sign of λ corresponds to mirroring the real part of p and taking the complex conjugate of D (note the change in colour). Furthermore, crossing the electron branch cut flips the sign of λe and so corresponds to jumping horizontally between the panels; crossing the ion branch cut corresponds to jumping vertically between them. D = det L, we have, from (C 9),D λ (p * , −k) = D λ (p, k) * , (C 11)and, from (C 10),D λ (−p * , k) = D −λ (p, k) * .(C 12)Figures 11 and 12show an example of the four branches of the dispersion function in the case of a two-species plasma. In particular, the property (C 12) is illustrated clearly infigure 11.
e ξt e (ps±iε)tχ k (p + p s ± iε) e ξt e (ps±iε)tχ k (p + p s ± iε), (D 3)
)Q λ
a,a =
1
a 3/2 ζ d
−
1
a
+
1
4ζ 2
d
I λ
a,a +
1
aζ d
ζ λ
− Z a (ζ λ
+ ) + ζ λ
+ Z a (ζ λ
− ) −
1
2ζ d
J λ
a,a .
(B 16)
(4.13) satisfy
They are exponentially small at any t > 0 because the integrand of the inverse Laplace transformation (2.24) contains a factor e ρt .
P. G.Ivanov and T. Adkins
The principal branch is the appropriate one for √ p − ps only after performing the rotation of the branch cuts to align them in the horizontal direction in the p complex plane, see §6.
Appendix C. Properties of the branches of the dispersion functionThe main convenience of choosing the branch cut along the negative real line in §4.1 is the relationship + √ z * = + √ z * for any z ∈ C. It is then easy to see that the expressionsFigure 13: The contour of integration C br around the branch cut -chosen to be parallel to the real p axis -with the latter indicated by the zigzag line. C± are the horizontal, semi-infinite segments along Im(p) = Im(ps) ± ε that connect the vertical contour at Re(p) → −∞ (seefigure 7) to the semi-circular arc Cε around the branch point.integral in the square brackets of (D 3) is bounded by an exponential, viz.,and so it is exponentially small in the limit of t → ∞. For the second integral, we know that |ξ| δ 1, and so it is natural to Taylor-expand the integrand. Note that the functionχ k contains both parts that are discontinuous across the branch cut (related to species s), as well as some that are continuous (related to species other than s). The discontinuity is due to the square-root terms in I a,b and J a,b , manifest in the expression for ζ ± (4.13). These square roots appear only as arguments of analytic functions. Therefore, the discontinuity ofχ k across the branch cut can be made explicit by writinĝwhereχ k is an analytic function of both of its arguments. 3 Noting thatfor p = p s + ξ ± iε, respectively, we findwhere we have ignored terms O(δ) or O(ε) in the square brackets and ∂χ k (p s )/∂ + √ p − p s denotes the partial derivative ofχ k with respect to its second parameter in (D 5) evaluated at p = p s , i.e., at ξ = 0. Using (D 8), we then find C− + C+ dp e ptχ k (p) ∼ 2e pst 0 −δ dξ e ξt + ξ ∂χ k (p s ) ∂ + √ p − p s as t → ∞.(D 9)
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"The Space Coronagraph Optical Bench (SCoOB): 1. Design and Assembly of a Vacuum-compatible Coronagraph Testbed for Spaceborne High-Contrast Imaging Technology",
"The Space Coronagraph Optical Bench (SCoOB): 1. Design and Assembly of a Vacuum-compatible Coronagraph Testbed for Spaceborne High-Contrast Imaging Technology"
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"Jaren N Ashcraft \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n",
"Heejoo Choi \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n\nLarge Binocular Telescope Observatory\nUniversity Of Arizona\n933 N. Cherry Ave. Tucson85721AZ\n",
"Ewan S Douglas \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
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"Kyle Van Gorkom \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"Daewook Kim \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n\nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n\nLarge Binocular Telescope Observatory\nUniversity Of Arizona\n933 N. Cherry Ave. Tucson85721AZ\n",
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"Alex Carter \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n",
"Olivier Durney \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"Sebastiaan Haffert \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"Lori Harrison \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"Maggie Kautz \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n",
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"Jared R Males \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"Kian Milani \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n",
"Oscar M Montoya \nDepartment of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA\n",
"George A Smith \nJames C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ\n"
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"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"Large Binocular Telescope Observatory\nUniversity Of Arizona\n933 N. Cherry Ave. Tucson85721AZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"Large Binocular Telescope Observatory\nUniversity Of Arizona\n933 N. Cherry Ave. Tucson85721AZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"James C. Wyant College of Optical Sciences\nUniversity of Arizona\nMeinel Building 1630 E. University Blvd85721TucsonAZ",
"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
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"Department of Astronomy and Steward Observatory\nUniversity of Arizona\n933 N. Cherry Ave85719TucsonAZUSA",
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]
| []
| The development of spaceborne coronagraphic technology is of paramount importance to the detection of habitable exoplanets in visible light. In space, coronagraphs are able to bypass the limitations imposed by the atmosphere to reach deeper contrasts and detect faint companions close to their host star. To effectively test this technology in a flight-like environment, a high-contrast imaging testbed must be designed for operation in a thermal vacuum (TVAC) chamber. A TVAC-compatible high-contrast imaging testbed is undergoing development at the University of Arizona inspired by a previous mission concept: The Coronagraphic Debris and Exoplanet Exploring Payload (CDEEP). The testbed currently operates at visible wavelengths and features a Boston Micromachines Kilo-C DM for wavefront control. Both a vector vortex coronagraph and a knife-edge Lyot coronagraph operating mode are under test. The optics will be mounted to a 1 x 2 meter pneumatically isolated optical bench designed to operate at 10 −8 torr and achieve raw contrasts of 10 −8 or better. The validation of our optical surface quality, alignment procedure, and first light results are presented. We also report on the status of the testbed's integration in the vaccum chamber. | 10.1117/12.2628855 | [
"https://export.arxiv.org/pdf/2208.01156v1.pdf"
]
| 250,599,322 | 2208.01156 | 692f9c8e8bd018913875ab53153d1129cf4abe77 |
The Space Coronagraph Optical Bench (SCoOB): 1. Design and Assembly of a Vacuum-compatible Coronagraph Testbed for Spaceborne High-Contrast Imaging Technology
Jaren N Ashcraft
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Heejoo Choi
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Large Binocular Telescope Observatory
University Of Arizona
933 N. Cherry Ave. Tucson85721AZ
Ewan S Douglas
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Kevin Derby
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Kyle Van Gorkom
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Daewook Kim
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Large Binocular Telescope Observatory
University Of Arizona
933 N. Cherry Ave. Tucson85721AZ
Ramya Anche
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Alex Carter
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Olivier Durney
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Sebastiaan Haffert
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Lori Harrison
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Maggie Kautz
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Jennifer Lumbres
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Jared R Males
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
Kian Milani
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
Oscar M Montoya
Department of Astronomy and Steward Observatory
University of Arizona
933 N. Cherry Ave85719TucsonAZUSA
George A Smith
James C. Wyant College of Optical Sciences
University of Arizona
Meinel Building 1630 E. University Blvd85721TucsonAZ
The Space Coronagraph Optical Bench (SCoOB): 1. Design and Assembly of a Vacuum-compatible Coronagraph Testbed for Spaceborne High-Contrast Imaging Technology
space telescopessmall satellitesdebris disksTVACwavefront controlcoronagraphydeformable mirrors
The development of spaceborne coronagraphic technology is of paramount importance to the detection of habitable exoplanets in visible light. In space, coronagraphs are able to bypass the limitations imposed by the atmosphere to reach deeper contrasts and detect faint companions close to their host star. To effectively test this technology in a flight-like environment, a high-contrast imaging testbed must be designed for operation in a thermal vacuum (TVAC) chamber. A TVAC-compatible high-contrast imaging testbed is undergoing development at the University of Arizona inspired by a previous mission concept: The Coronagraphic Debris and Exoplanet Exploring Payload (CDEEP). The testbed currently operates at visible wavelengths and features a Boston Micromachines Kilo-C DM for wavefront control. Both a vector vortex coronagraph and a knife-edge Lyot coronagraph operating mode are under test. The optics will be mounted to a 1 x 2 meter pneumatically isolated optical bench designed to operate at 10 −8 torr and achieve raw contrasts of 10 −8 or better. The validation of our optical surface quality, alignment procedure, and first light results are presented. We also report on the status of the testbed's integration in the vaccum chamber.
INTRODUCTION
The Astro 2020 Decadal Survey has recommended the development of a future large infrared/optical/ultraviolet telescope optimized for observing habitable exoplanets as the highest priority for space frontier missions. 1 Such a telescope would need to be equipped with high-contrast imaging technology capable of achieving deep contrasts of 10 −10 and be capable of doing spectroscopy on the detected signal. Spaceborne high-contrast imaging instruments are necessary to achieve this goal because they can entirely bypass the limitations on wavefront stability and transmission imposed by the atmosphere. The atmosphere is a fundamentally turbulent environment, so a separate adaptive optics (AO) system is necessary just to correct for the wavefront error that results from fluctuations in the optical path. These AO systems typically require one or more deformable mirrors just to accommodate the atmospheric aberrations. 2 The atmosphere also attenuates heavily near the ultraviolet, eliminating an entire region of spectral observation. High-contrast imaging in space began with the Hubble Space Telescope (HST) and its repertoire of coronagraphic instrumentation. [3][4][5] The HST continues to be an integral tool for astrophysical discovery, producing the data for the largest number of refereed papers in astronomy 6 for a single observatory. Recently the James Webb Space Telescope (JWST) has launched, adding another promising space observatory equipped with high-contrast imaging instrumentation 7,8 to the list of available science instruments. To aid in the pursuit of developing a future space observatory capable of 10 −10 contrast, centers for testing technology and wavefront control methods are a necessity. Platforms for testing new technologies will invariably aid the pursuit of directly imaging faint astrophysical targets at high contrasts (e.g. exoplanets, debris disks). Several high-contrast imaging testbeds exist among research institutions and universities around the world. 9 However, few are designed to operate at vacuum, most notably the testbeds at NASA's Jet Propulsion Laboratory (Decadal Survey Testbed, 10 HCIT 11 ) are presently demonstrating wavefront sensing and control in vacuum.
A new testbed for testing very high-contrast wavefront sensing and coronagraph technology in vacuum, the Space Coronagraph Optical Bench (SCoOB) is being developed by the University of Arizona's Space Astrophysics Laboratory, in close collaboration with the Center for Astronomical Adaptive Optics * , the Large Optics Fabrication and Test Lab † , and the Extreme Wavefront Control lab ‡ . To meet the challenge of imaging rocky exoplanets in reflected light, SCoOB will provide support to ongoing experiments in vacuum to prototype new high-contrast imaging systems. In this proceedings we report on the updates to the design of the testbed previously intended to prototype the Coronagraphic Debris and Exoplanet Exploring Pioneer (CDEEP) payload.
REVIEW OF DESIGN AND ASSEMBLY
The Space Coronagraph Optical Bench is a vacuum-compatible high-contrast imaging testbed optimized for maximizing contrast on observatories with un-obscured pupils in small volumes. The testbench is meant to be a remotely-accessible high-contrast imaging laboratory that is uniquely suited to experiments in vacuum. Two coronagraph modes are currently undergoing modeling and development, a Vector-Vortex Coronagraph (VVC) and a Knife-edge Lyot Coronagraph (KLC). SCoOB's wavefront sensing and control algorithms are operated by the open-source CACAO platform. 12 This control software is used in other high-contrast imaging instruments (MagAO-X, 2 SCExAO 12 ). Consequently, algorithms developed for these testbeds are easily migratable to SCoOB, and vice-versa.
The baseline VVC mode was chosen for its achromaticity, high throughput, and ability to achieve small inner working angles (IWA). 13 This coronagraph was also the device chosen for the HabEx mission concept, a candidate next-generation observatory for the Astro 2020 Decadal Survey. 1 The KLC mode was a simple addition requiring relatively few specialized components to operate, so it was also used during the initial development phase of the coronagraph as a test for our wavefront sensing and control algorithms.
Changes to CDEEP Coronagraph Design
The SCoOB testbed concept was previously a laboratory realization of a small space telescope coronagraph designed to be compact and entirely vacuum-compatible. Changes were made to the nominal design of the space coronagraph to realize the instrument at a lower cost based on the existing parent parabolas that our vendor (Nu-Tek) had in stock. 14 The changes from the OAP's used in the CDEEP design to the in-stock parabolas from Nu-tek are shown below in Tables 1 and 2 OAP 0 34 27 7 OAP 1 72 40 32 OAP 2 73 55 18 OAP3 134 100 34 Table 2. Change in off-axis distance from the nominal CDEEP design
The final set of optics are more on-axis, and the design is unfolded so that room could be made for commercially-available off-the-shelf mounting hardware. The result is a testbed that is less prone to misalignment and polarization induced wavefront errors, increasing the potential performance beyond what was baselined for the original CDEEP concept. 14 The optical path (shown in figure 1) begins with a source simulator. Most experiments were conducted using a point-source microscope to illuminate a small (15um diameter) pinhole at this location, but recently this was switched to a single-mode fiber tip with a sufficiently small mode field diameter (< 5um) to act as a point source. The light from the source is collimated by OAP0 which propagates to the first pupil position at the first flat mirror. We plan to replace this flat with a piezo-electric fast-steering mirror for fine steering control and telescope pointing tests. A mirror tilted the same angle in the opposite direction follows to not induce polarization aberrations in the testbed and maintain a small footprint. Two OAPs follow this optic (OAP1, OAP2) which relay the image of the pupil onto a Boston Micromachines Kilo-C 1.5um stroke DM. After the DM, the light passes through a circular polarizer and then on to OAP3. OAP3 has the largest focal length of all optics in this system. This is to deliver a beam of appropriate focal ratio (F/48) to the focal plane mask. After the focal plane mask a fold mirror is inserted to redirect the beam to a collimating lens (O4) which passes the beam onto the next circular polarizer and then Lyot Stop. A lens group follows the Lyot stop to form an image on the AWO science camera.
The flat after the focal plane mask was selected in order to ensure that the testbed was sufficiently compact. Originally this optic was an off-axis parabola, but due to the space constraints on the testbed and the focal ratio constraint on the beam incident on the focal plane mask, the exit pupil was 1mm in radius. The corresponding Fresnel diffraction effects added amplitude variation across the pupil, which we believed could limit our contrast. The OAP was replaced with a flat and a collimating lens of a longer focal length to increase the exit pupil size. A diverging beam incident on a flat surface typically induces some spherical aberration. However, given the slow focal ratio of the beam the aberration induced is entirely negligible. We also don't expect this change to contribute substantive polarization aberrations that would corrupt our contrast because it is after the focal plane mask.
Mount Designs and Tolerance Analysis Results
High-contrast imaging instruments require a well-characterized optical design in order to achieve diffractionlimited optical imaging. A diffraction model informed tolerance analysis was conducted in order to asses the sensitivity of the coronagraph to low-order aberrations. Using HCIPy 15 we propagated wavefront error expressed with Zernike polynomials through a VVC and examined the average contrast in the dark hole region for various applied amplitudes of individual Zernike modes. A separate tolerance analysis in Zemax OpticStudio was done to examine the Zernike-decomposed wavefront error induced by perturbations of the optics in position and angle. Using these two analyses in tandem allowed us to examine the degradation in coronagraph contrast in response to mechanical aberration. The results of the tolerance analysis are shown in in Figure 6 and 7 of Maier et al, 14 but a table of the final tolerances for the optics used in SCoOB are shown in Table 3 below. Table 3. Sensitivities of each optic in 6 degrees of freedom such that the RSS of the contributions from individual Zernike modes do not degrade contrast beyond 10 −8 .
OAP0 OAP1 OAP2 OAP3 x y z x y z x y z x y z Decenter [mm] 0.1 0.1 0.3 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 Tilt [deg] 0.1 0.1 0.3 0.02 0.02 0.1 0.05 0.3 0.5 0.2 0.3 0.5
To isolate the optics from any residual vibration, the mounts are seated on 1" passivated posts from Newport. The mounts for the OAPs are vacuum-compatible variants of the THORLABS K6XS-SM1 6-axis kinematic mounts. A quarter-turn of each adjustment knob was defined as the "sensitivity" of the degree of freedom. The K6XS-SM1 is sufficiently sensitive to achieve the tolerances outlined in table 3.
To best match the CDEEP space coronagraph design, the clear aperture of the off-axis parabolas were smaller than are suitable to most commercially available optical mounts (1", 1/2", etc.). To adapt the mirrors to commercial off-the-shelf optomechanics, interface cups were designed to mount the mirrors in. The cups were designed such that the inner diameter of the mounting cylinder matched the outer diameter of the OAP. The OAP was bonded to the cup with 2216 epoxy adhesive and allowed to cure with a cylindrical shim in place to keep the optic centered during the curing. After the curing was completed the shim was removed and the OAP in the cup was mounted to the THORLABS K6XS mounts. The optics are atypical in aspect ratio with the length of the OAP being larger than the clear aperture. This was purely a consequence of using existing parabola parents from Nu-Tek, but we expect less mount-induced aberration because of it. Trefoil is a common aberration induced by 3-point optical mounts which result from the deformation of the surface by the mounting hardware. Because of the skewed aspect ratio and the interface cups, we expect our optics to be more resilient to surface deformation induced by the optomechanics.
OAP & Flat Measurement
Upon assembly of the optics into their associated mounting hardware, a preliminary characterization run was conducted to verify the quality of the OAPs. A simple and effective test of the surface quality is done interferometrically. We used a 4D PhaseCam 6000 Interferometer in a double-pass configuration. The OAP under test is placed in the path of the beam exiting the interferometer. After passing the OAP the beam is incident onto a reference optic which retroreflects the beam back through the system. We used the PhaseCam with a focusing optic to create a simulated source that the OAP could collimate onto a reference flat. The flat mirror only has two degrees of freedom (tip, tilt) because it needs to be normal to the incoming beam to retroreflect. This reduces the number of degrees of freedom that needed to be aligned, which simplifies the metrology process. The remaining complication was to position the OAP in the correct orientation, the procedure for which is outlined below. The theory of measuring the optics relies on aligning each degree of freedom individually, and iteratively repeating the process until a satisfactory interferogram is achieved. The coordinate system definitions for the following procedure are such that the z-axis is the initial propagation axis, the y-axis is in the direction transverse to the optical bench, and the x-axis is orthogonal to both. Before alignment of the OAP to the interferometer, we define the optical axis by placing two parallel irises on an optical rail in front of the interferometer. Translation stages under the interferometer were used to finely tune its position such that light exiting the interferometer passed through both irises. Upon sucessful definition of an optical axis, the position of the interferometer was fixed and the OAP could be aligned to it using the following procedure:
1. Clocking / Z-Tilt: Have the OAP vendor mark the direction of the parent axis of the OAP under test.
Then, clock the OAP until the marker is aligned to the horizontal on the 6DOF mount. This degree of freedom tends to not be as sensitive to misalignment (see Table 3), so this method should be sufficient.
2. X-Decenter : Place a reference target (e.g. a card) below the image produced by the interferometer. Then place the OAP on a rail in front of this target, confining the interferometer and OAP to the Y-Z plane.
3. Y-Decenter : Translate the OAP vertically such that the beam is approximately centered on the optical axis defined by the interferometer and two irises.
4. Z-Decenter : Translate the OAP along the rail a distance equal to its focal distance with respect to the target placed in step 2. This step is made easier using a rail with regular rulings, and can be refined with the adjustment knobs on the 6DOF mount later.
Y-Tilt:
Tilt the OAP about its Y-axis by the off-axis angle of the OAP. This can be accomplished coarsely with a protractor as a visual aid, and can be refined interferometrically with the tilt knob on the 6DOF mount.
6. X-Tilt: Adjust the X-tilt knob until the beam after the OAP is approximately level with the X-Z plane. This procedure will reveal what aberrations persist through the systems, so the remaining steps will largely depend on the interferogram observed. Dealing with residual coma and astigmatism is nontrivial because the aberrations will typically coupled to multiple degrees of freedom. They are indicative of primarily X-and Y-tilt errors, so adjusting these on the OAP mount and then accommodating for the adjustment with the tip/tilt knobs on the reference flat can slowly eliminate the aberration. If the aberration amplitude is too large, simply repeat the procedure above and then begin refining nulled interferogram until a satisfactory result is achieved. The results of our interferometric testing using this procedure are shown in Figure 3. Only the optics upstream of the focal plane mask were tested because they determine the shape of the PSF delivered to our coronagraphic mask, which sets our system's performance. OAP3 is slightly over-filled by the 4D beam due to the long focal length, so the OPD map is slightly smaller than the rest. There is some residual alignment-induced aberration that was challenging to null in OAP3 due to its sensitivity to tilt, but the measured wavefront error was below our specifications so we proceeded with assembly.
Alignment plan, Assembly, and performance validation tests
The philosophy for aligning the testbed optics is similar to the interferometric testing of the OAPs. Each degree of freedom is adjusted at a coarse and fine level, and then revisited until the interferogram is sufficiently nulled. We take the plane of the testbed to be the x-z plane, and the axis transverse to the testbed to be the y axis. All DOF are considered local to the optical element (e.g. z is a tilt about the axis through the center of an OAP).
1. Place OAP's with 6DOF mounts on a 4" post, this defines the nominal height of all optics in the testbed.
Place a 1" iris on a post and then in a post holder. Adjust the y-decenter of the iris to be coaxial with the
OAPs. This will be our rough y-decenter reference for the testbed.
3. Use digital calipers to set the clear diameter of the iris to be equal to the beam size at the deformable mirror (9.2mm) 4. Position a point-source microscope in a mount with adjustable height (translation stage, or stacked posts + spacers) such that the beam is coaxial to the iris.
5. Place the pinhole assembly on the testbed breadboard to define the first point of the optical axis.
6. Place the source with a focusing optic behind the pinhole and adjust the position until there is sufficient transmission through the pinhole. We used a point-source microscope for our initial source, which has an internal camera that is conjugate to the source focus, significantly reducing the pinhole alignment difficulty. After sufficient throughput is achieved the pinhole must remain fixed.
7. Place OAP0 one focal distance away from the pinhole plane using a ruler, and adjust the position until the beam after the mirror is approximately collimated when viewed on a card. Then fix the position of OAP0 to the testbed using a clamping fork.
8. Place a shear plate in the ensuing quasi-collimated space normal to the incoming beam and observe the interferogram. If the fringes are not parallel to the marked line, then there is residual focus in the beam.
9. Rotate the shear plate by 90 degrees by removing it from the post and placing it back in. If the fringes are tilted at a different angle than the previous configuration, there is astigmatism in the beam. Use the tip/tilt/z knobs to translate out the residual focus error shown in the shear plate. The astigmatism is addressed by altering the tilt angle about the X and Y axis to null the observed fringes, rotating the shear plate, and repeating the process. 10. Place a reference flat in the collimated space to retroreflect the beam back into the point-source microscope to view the PSF in detail. Examining the shape of the PSF is indicative of the remaining aberrations present in the system. This will serve as a guide for what DOF require fine-tuning before proceeding with the next optic.
This procedure is sufficient for collimating a beam from the source simulator and ensuring that minimal additional aberration is introduced to the ensuing beam. To align an OAP to a collimated space, begin from step 7 in the procedure above and instead use the following procedure:
1. Place OAP1 focal distance away from the intermediate pupil plane using a ruler, and adjust the position until the beam after the mirror forms a focus at approximately the correct distance when measured with a ruler. Then fix the position of OAP0 to the testbed using a clamping fork.
2. Place a card at the focal plane to assess the presence of any large aberrations that are easily nulled by a rotation about the y-or x-axes.
3. Once the point-spread function is smaller than is viewable by eye, place a detector at the focal plane. Ultimately the fine alignment of the OAP's are somewhat exploratative because of how dependent the alignment is on the initial setup. We found success in approaching the alignment of the testbed by iteratively exploring one degree of freedom at a time to minimize the coupling of wavefront error from orthogonal degrees of freedom. The procedures listed above are repeated for OAP2 and OAP3. Upon successfully aligning the OAPs up to OAP3 we evaluated the as-aligned wavefront error at the coronagraphic focal plane by placing a ball bearing in OAP3's image plane. The resultant interferogram was viewed in double-pass to observe if our alignment efforts were successful. At the test wavelength we achieved roughly λ/40 RMS wavefront error in single-pass, which was lower than our wavefront error specification outlined in Maier et al. 14 Given satisfactory optical alignment, we could proceed with the alignment of our spatial filters and begin coronagraphic imaging. Aligning to the knife-edge mask is fairly straightforward given that it should just block the core of the instrument PSF. Inserting the knife on a translation stage until a large amount of light is visibly reflected back is sufficient, and can be refined simply by observing the knife at the science camera. However, aligning to the VVC can be challenging because of its transparency. Luckily, the modern standard is that there is always a compact polarized screen somewhere. Aligning to the VVC is made simple by viewing it through a polarizer crossed with respect to the polarization of a screen (e.g. of a Mobile Phone). The result is a clear view of the optical vortex. Turning up the source power and/or opening the iris also allows you to view the light scattered off of the surface of the VVC from the PSF. Getting the center of the vortex with the spot of scattered light coarsely aligned will get the system mostly aligned. The fine allignment to the singularity can be achieved viewing the beam in a pupil plane downstream. Figure 5. Diagram illustrating the fine alignment to a VVC using a polarizer and mobile phone screen. The transparency of the mask and small singularity can be very challenging to align to. In the presence of polarized illumination the vortex pattern is easily seen. Simultaneously viewing the laser light reflecting off the mask and the vortex pattern considerably eases the alignment to these masks, and only uses materials that would be readily available in any laboratory.
FIRST LIGHT RESULTS
Upon successful installment of the hardware in the testbed we examined the point-spread function and it's influence on the two coronagraph modes. Before observation, whatever residual low-order aberrations were cleaned up with the Eye Doctor 16 algorithm developed for the Large Binocular Telescope. The eye doctor employs a grid search of low-order zernike aberrations applied to the DM to determine which coefficient maximizes the strehl ratio of the PSF. The summed shape of these aberration modes determines the "flat" position of the deformable mirror to further enhance our image quality. Below are figures of the testbed's point-spread function, and coronagraphic image plane. The coronagraphic mask and Lyot stop only serve to contribute part of the contrast. In order to reach deeper contrasts we employ wavefront sensing and control techniques suitable for high-contrast imaging. A more comprehensive review of the implementation and results of these efforts are outlined in Van Gorkom et al, 17 but a brief introduction to current efforts is written here.
Electric field conjugation (EFC) 18 is a method of focal plane wavefront sensing that has seen widespread implementation among high-contrast imaging instruments used for astrophysics. 19,20 The method relies on an accurate Fresnel/Angular Spectrum forward model to propagate the effects of phase errors on the optical surfaces to the image plane. This model-assisted approach is capable of reaching very deep contrast levels. In the small aberration regime, the relationship of the electric field in the pupil plane to that of the image plane is approximately linear. Using this relation solutions to the electric field of the image plane can be probed for a given control region. Upon succesful sensing of the electric field in this region, the opposite phase can be applied to the image field to create a high-contrast region (near 10 −8 has been demonstrated in SCoOB). In practice, this method is limited by the dynamic evolution of the electric field as a function of time. Vibrations, ambient turbulence, and temperature differentials are all sufficient to aberrate the electric field such that the contrast is lost. However, the EFC algorithm is computationally time-consuming and cannot be easily recalculated. Therefore, an efficient method of maintaining the dark hole is necessary for proper operation. Methods of circumventing this limitation are being explored by other investigators using an algorithm called Implicit Electric Field Conjugation. 21 To see the results of this method, please consult our other manuscript published in these proceedings. 17
The Self-Coherent Camera (SCC) 22, 23 is a method of focal-plane wavefront sensing undergoing experimentation on SCoOB that is uniquely suited to the dark hole maintenance problem. By placing a pinhole in the opaque part of the Lyot stop, the residual starlight that would have ordinarily been blocked is allowed to pass. Because stars are partially coherent with themselves, 24 this create interference fringes with the starlight, but not the planet light. Transforming the signal into the fourier domain reveals that this effect creates side-bands in the Optical Transfer Function (OTF) where the wavefront phase can be extracted. This technique requires no forward model in order to do wavefront sensing, so it can be computed faster than EFC. Given a sufficient starting solution and a small pinhole, the SCC is capable of maintaining deep contrasts in the dark hole. 23
VACUUM CHAMBER
SCoOB will be placed in an Rydberg Vacuum Sciences 104430 Thermal Vacuum (TVAC) Cycling chamber that has been acquired by the UA Space Astrophysics Laboratory. The chamber is in an underground room to minimize the coupling of external vibrations from the building. The TVAC chamber employs a dry screw pump and turbomolecular pump to achieve high vacuum. Upon achieving base pressure, temperature is regulated by a combination of nichrome heaters and gaseous nitrogen from boiled LN2. Thermal and vacuum cycling is conducted using a recipe-based system to simplify experimentation. We anticipate beginning preliminary in-chamber tests before the end of the year.
CONCLUSION AND FUTURE WORK
The Space Coronagraph Optical Bench is a new high-contrast imaging instrument undergoing active development at the UA Space Astrophysics Laboratory. We present the modified design of the high contrast imaging instruments, the quality of the optics, and the materials and process by which the testbed is assembled. In the immediate future the SCoOB team will focus on refining the contrast achievable out of vacuum and eliminating errant noise sources that could limit our testbed performance. Recently the SCC Lyot stop has been installed, and we will be conducting tests of dark hole stability under SCC control in the near future. To learn about our wavefront sensing and control efforts in detail, see the accompanying proceedings by Van Gorkom et al. 17 A candidate future realization of SCoOB includes two deformable mirrors (Fig 9) to correct for amplitude errors from pupil segmentation. 25 A list of the hardware used in this testbed is included in the appendix after the references. Figure 9. Raytrace of a candidate future layout for SCoOB including a second DM to correct for amplitude errors that arise from pupil discontinuities.
ACKNOWLEDGEMENTS
Many thanks to the many teams building and simulating existing testbed who have shared their expertise and know-how. In particular, the authors would like to thank Iva Laginja, Dimitri Mawet, Chris Mendillo, Mamadou N'Diaye, Marshall Perrin, A.J. Riggs, Garreth Ruane, and Remí Soummer for particularly helpful conversations and tours. Thanks to the HiCAT team for sharing their design for the BMC Kilo-DM's mounting hardware, 26 and a very helpful set of proceedings that guided the design phase of the testbed when it was called CDEEP. 27,28 Portions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).
APPENDIX: HARDWARE USED IN THE SCOOB TESTBED
Figure 1 .
1Raytrace model of SCoOB in Zemax OpticStudio. The optical path begins on the bottom left where a source simulator generates an isotropic point-source before OAP0. Light then propagates through the coronagraph into the science camera.
Figure 2 .
2Solid model of the interface cup holding an off-axis parabola. These cups make optics of any size under the mounting diameter adaptable to commercially available optomechanics at relatively low cost.
7 .
7Place the reference flat in the path of the ensuing quasi-collimated beam. Adjust the flat in tip and tilt until the beam is visible on the detector and no straight-line fringes are observed. 8. If there are a large amount of remaining focus fringes, carefully adjust the OAP's position on the rail in small increments. If there is a small amount of focus fringes remaining, adjust the tip/tilt/z knobs in equal amounts until the focus fringe is nulled.
Figure 3 .
3Figure withthe OPD of the testbed optics shown in units of waves tested at 632.8nm. On the top row from left to right is OAP0, the first, and second flat mirrors. The bottom row is OAP1, OAP2, and OAP3.
4 .
4Use the tip and tilt knobs to minimize the aberration in the PSF. The dominant modes of aberration from tip and tilt are astigmatism and coma. 5. Upon successful nulling a retroreflecting sphere can be placed in the image plane to view the PSF in the point-source microscope. Viewing the aberrations in double-pass enhances the small amplitude aberrations that are a result of decenter in x, y, and z.
Figure 4 .
4The as-aligned wavefront error map measured by the 4D interferometer in double pass. Units are in waves at the test wavelength of 632.8nm. The pupil of the system is smaller than that of the 4D interferometer so the interferogram occupied a region within the resultant interferogram file. On this scale, the spacing between actuators of the deformable mirror is visible.
Figure 6 .
6SCoOB Assembled in our laboratory's clean tent.
Figure 7 .
7(Left) First light measurement of the testbed point-spread function normalized to the peak of the airy disk. (Middle) The knife-edge Lyot coronagraph mode image plane plotted on the same vertical scale as the non-coronagraphic PSF. (Right) The vector vortex coronagraph mode image plane illustrating the on-axis starlight rejection being comparable to the background. The fringes observed in this image result from the self-coherent camera pinhole in the Lyot stop.
Figure 8 .
8RVS 104430 TVAC chamber at the basement of Steward Observatory. (Left) The exterior of the cylindrical testbed chamber. (Right) The interior of the vacuum chamber showing the support table. A 4-inch Newport high-vacuum optical breadboard will be installed in the chamber to support the SCoOB breadboard.
.Table 1. Change in radius of curvature from the nominal CDEEP design * https://www.as.arizona.edu/CAAO/ † https://www.loft.optics.arizona.edu/ ‡ https://xwcl.science/ CDEEP Design SCoOB Design DifferenceCDEEP Design SCoOB Design Difference
OAP 0
-142
-293.6
151.6
OAP 1
-200
-254
54
OAP 2
-202
-346
144
OAP3
-900
-914.4
14.4
• C to 150 • C Thermal Stability -Room Temperature ± 2 • C Thermal Stability -150 • C ± 10 • C Ramp Rate range ± 1 • C/hour to ± 2 • C/min Design vacuum 1e-8 torrSpecification
Value
Diameter
1.2 m
Length
2.2 m
Interior Finish
#4 grained finish
Roughing Pump
Varodry VD65 dry screw pump
High-vacuum Pump
Mag-Lev Turbomolecular pump
Temperature Range
-150
Table 4 .
4Table of TVAC chamber operational specifications, some of which are currently being evaluated.
Pathways to discovery in astronomy and astrophysics for the 2020's. The National Academies Press"Pathways to discovery in astronomy and astrophysics for the 2020's," The National Academies Press (2021).
MagAO-X: project status and first laboratory results. J R Males, L M Close, K Miller, L Schatz, D Doelman, J Lumbres, F Snik, A Rodack, J Knight, K Van Gorkom, J D Long, A Hedglen, M Kautz, N Jovanovic, K Morzinski, O Guyon, E Douglas, K B Follette, J Lozi, C Bohlman, O Durney, V Gasho, P Hinz, M Ireland, M Jean, C Keller, M Kenworthy, B Mazin, J Noenickx, D Alfred, K Perez, A Sanchez, C Sauve, A Weinberger, Conrad , A Close, L M Schreiber, L Schmidt, D , Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10703. 1070309Adaptive Optics Systems VIMales, J. R., Close, L. M., Miller, K., Schatz, L., Doelman, D., Lumbres, J., Snik, F., Rodack, A., Knight, J., Van Gorkom, K., Long, J. D., Hedglen, A., Kautz, M., Jovanovic, N., Morzinski, K., Guyon, O., Douglas, E., Follette, K. B., Lozi, J., Bohlman, C., Durney, O., Gasho, V., Hinz, P., Ireland, M., Jean, M., Keller, C., Kenworthy, M., Mazin, B., Noenickx, J., Alfred, D., Perez, K., Sanchez, A., Sauve, C., Weinberger, A., and Conrad, A., "MagAO-X: project status and first laboratory results," in [Adaptive Optics Systems VI], Close, L. M., Schreiber, L., and Schmidt, D., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 10703, 1070309 (July 2018).
NICMOS performance on the Hubble Space Telescope. R I Thompson, G Schneider, International Society for Optics and Photonics. Bely, P. Y. and Breckinridge, J. B.3356SPIESpace Telescopes and Instruments VThompson, R. I. and Schneider, G., "NICMOS performance on the Hubble Space Telescope," in [Space Telescopes and Instruments V ], Bely, P. Y. and Breckinridge, J. B., eds., 3356, 215 -221, International Society for Optics and Photonics, SPIE (1998).
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The space telescope imaging spectrograph design. B Woodgate, R Kimble, C Bowers, S Kraemer, M Kaiser, A Danks, J Grady, J Loiacono, M Brumfield, L Feinberg, T Gull, S Heap, S Maran, D Lindler, D Hood, W Meyer, C Vanhouten, V Argabright, S Franka, R Bybee, D Dorn, M Bottema, R Woodruff, D Michika, J Sullivan, J Hetlinger, C Ludtke, R Stocker, A Delamere, D Rose, I Becker, H Garner, J Timothy, M Blouke, C Joseph, G Hartig, R Green, E Jenkins, J Linsky, J Hutchings, H Moos, A Boggess, F Roesler, D Weistrop, Publications of the Astronomical Society of the Pacific. 110752Woodgate, B., Kimble, R., Bowers, C., Kraemer, S., Kaiser, M., Danks, A., Grady, J., Loiacono, J., Brumfield, M., Feinberg, L., Gull, T., Heap, S., Maran, S., Lindler, D., Hood, D., Meyer, W., VanHouten, C., Argabright, V., Franka, S., Bybee, R., Dorn, D., Bottema, M., Woodruff, R., Michika, D., Sullivan, J., Hetlinger, J., Ludtke, C., Stocker, R., Delamere, A., Rose, D., Becker, I., Garner, H., Timothy, J., Blouke, M., Joseph, C., Hartig, G., Green, R., Jenkins, E., Linsky, J., Hutchings, J., Moos, H., Boggess, A., Roesler, F., and Weistrop, D., "The space telescope imaging spectrograph design," Publications of the Astronomical Society of the Pacific 110(752), 1183-1204 (1998).
A Bibliometric Analysis of Observatory Publications for the Period. D Crabtree, 453.02American Astronomical Society Meeting Abstracts #233. 233Crabtree, D., "A Bibliometric Analysis of Observatory Publications for the Period 2012-2016," in [American Astronomical Society Meeting Abstracts #233], American Astronomical Society Meeting Abstracts 233, 453.02 (Jan. 2019).
The JWST/NIRCam coronagraph flight occulters. J E Krist, K Balasubramanian, R E Muller, S B Shaklan, D M Kelly, D W Wilson, C A Beichman, E Serabyn, Y Mao, P M Echternach, J T Trauger, K M Liewer, Jacobus M Oschmann, J Clampin, M C , Macewen , Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series. H. A.7731Space Telescopes and Instrumentation 2010: Optical, Infrared, and Millimeter WaveKrist, J. E., Balasubramanian, K., Muller, R. E., Shaklan, S. B., Kelly, D. M., Wilson, D. W., Beichman, C. A., Serabyn, E., Mao, Y., Echternach, P. M., Trauger, J. T., and Liewer, K. M., "The JWST/NIRCam coronagraph flight occulters," in [Space Telescopes and Instrumentation 2010: Optical, Infrared, and Mil- limeter Wave ], Oschmann, Jacobus M., J., Clampin, M. C., and MacEwen, H. A., eds., Society of Photo- Optical Instrumentation Engineers (SPIE) Conference Series 7731, 77313J (July 2010).
The mid-infrared instrument for the james webb space telescope, v: Predicted performance of the MIRI coronagraphs. A Boccaletti, P.-O Lagage, P Baudoz, C Beichman, P Bouchet, C Cavarroc, D Dubreuil, A Glasse, A M Glauser, D C Hines, C.-P Lajoie, J Lebreton, M D Perrin, L Pueyo, J M Reess, G H Rieke, S Ronayette, D Rouan, R Soummer, G S Wright, Publications of the Astronomical Society of the Pacific. 127Boccaletti, A., Lagage, P.-O., Baudoz, P., Beichman, C., Bouchet, P., Cavarroc, C., Dubreuil, D., Glasse, A., Glauser, A. M., Hines, D. C., Lajoie, C.-P., Lebreton, J., Perrin, M. D., Pueyo, L., Reess, J. M., Rieke, G. H., Ronayette, S., Rouan, D., Soummer, R., and Wright, G. S., "The mid-infrared instrument for the james webb space telescope, v: Predicted performance of the MIRI coronagraphs," Publications of the Astronomical Society of the Pacific 127, 633-645 (jul 2015).
Caotic, community of adaptive optics and high contrast testbeds. Caotic, community of adaptive optics and high contrast testbeds." https://sites.google.com/view/ highcontrastlabs/home?authuser=0.
The Decadal Survey Testbed: Demonstrating Technology for Imaging Earth-like Exoplanets with Future Space Telescopes. G Ruane, K Patterson, B J Seo, C Mejia Prada, B Crill, N Siegler, AGU Fall Meeting AbstractsRuane, G., Patterson, K., Seo, B. J., Mejia Prada, C., Crill, B., and Siegler, N., "The Decadal Survey Testbed: Demonstrating Technology for Imaging Earth-like Exoplanets with Future Space Telescopes," in [AGU Fall Meeting Abstracts ], 2019, P51H-3446 (Dec. 2019).
Testbed demonstration of high-contrast coronagraph imaging in search for Earth-like exoplanets. B.-J Seo, K Patterson, K Balasubramanian, B Crill, T Chui, D Echeverri, B Kern, D Marx, D Moody, C Mejia Prada, G Ruane, F Shi, J Shaw, N Siegler, H Tang, J Trauger, D Wilson, R Zimmer, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series], Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 11117, 111171V. Seo, B.-J., Patterson, K., Balasubramanian, K., Crill, B., Chui, T., Echeverri, D., Kern, B., Marx, D., Moody, D., Mejia Prada, C., Ruane, G., Shi, F., Shaw, J., Siegler, N., Tang, H., Trauger, J., Wilson, D., and Zimmer, R., "Testbed demonstration of high-contrast coronagraph imaging in search for Earth-like exoplanets," in [Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series], Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 11117, 111171V (Sept. 2019).
The compute and control for adaptive optics (CACAO) real-time control software package. O Guyon, A Sevin, D Gratadour, J Bernard, H Ltaief, D Sukkari, S Cetre, N Skaf, J Lozi, F Martinache, C Clergeon, B Norris, A Wong, J Males, International Society for Optics and Photonics, SPIE. Close, L. M., Schreiber, L., and Schmidt, D.10703Adaptive Optics Systems VIGuyon, O., Sevin, A., Gratadour, D., Bernard, J., Ltaief, H., Sukkari, D., Cetre, S., Skaf, N., Lozi, J., Mar- tinache, F., Clergeon, C., Norris, B., Wong, A., and Males, J., "The compute and control for adaptive optics (CACAO) real-time control software package," in [Adaptive Optics Systems VI ], Close, L. M., Schreiber, L., and Schmidt, D., eds., 10703, 469 -480, International Society for Optics and Photonics, SPIE (2018).
THE VECTOR VORTEX CORONAGRAPH: LABORATORY RESULTS AND FIRST LIGHT AT PALOMAR OBSERVATORY. D Mawet, E Serabyn, K Liewer, R Burruss, J Hickey, D Shemo, The Astrophysical Journal. 709Mawet, D., Serabyn, E., Liewer, K., Burruss, R., Hickey, J., and Shemo, D., "THE VECTOR VORTEX CORONAGRAPH: LABORATORY RESULTS AND FIRST LIGHT AT PALOMAR OBSERVATORY," The Astrophysical Journal 709, 53-57 (dec 2009).
Design of the vacuum high contrast imaging testbed for CDEEP, the Coronagraphic Debris and Exoplanet Exploring Pioneer. E R Maier, E S Douglas, D W Kim, K Su, J N Ashcraft, J B Breckinridge, H Choi, E Choquet, T E Connors, O Durney, K L Gonzales, C E Guthery, C A Haughwout, J C Heath, J Hyatt, J Lumbres, J R Males, E C Matthews, K Milani, O M Montoya, M N'diaye, J Noenickx, L Pogorelyuk, G Ruane, G Schneider, G A Smith, C C Stark, M Lystrup, M D Perrin, N Batalha, N Siegler, E C Tong, International Society for Optics and Photonics. 11443SPIESpace Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter WaveMaier, E. R., Douglas, E. S., Kim, D. W., Su, K., Ashcraft, J. N., Breckinridge, J. B., Choi, H., Choquet, E., Connors, T. E., Durney, O., Gonzales, K. L., Guthery, C. E., Haughwout, C. A., Heath, J. C., Hyatt, J., Lumbres, J., Males, J. R., Matthews, E. C., Milani, K., Montoya, O. M., N'Diaye, M., Noenickx, J., Pogorelyuk, L., Ruane, G., Schneider, G., Smith, G. A., and Stark, C. C., "Design of the vacuum high contrast imaging testbed for CDEEP, the Coronagraphic Debris and Exoplanet Exploring Pioneer," in [Space Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter Wave ], Lystrup, M., Perrin, M. D., Batalha, N., Siegler, N., and Tong, E. C., eds., 11443, 324 -340, International Society for Optics and Photonics, SPIE (2020).
High Contrast Imaging for Python (HCIPy): an open-source adaptive optics and coronagraph simulator. E H Por, S Y Haffert, V M Radhakrishnan, D S Doelman, M Van Kooten, S P Bos, Proc. SPIE 10703. SPIE 10703Adaptive Optics Systems VIPor, E. H., Haffert, S. Y., Radhakrishnan, V. M., Doelman, D. S., Van Kooten, M., and Bos, S. P., "High Contrast Imaging for Python (HCIPy): an open-source adaptive optics and coronagraph simulator," in [Adaptive Optics Systems VI ], Proc. SPIE 10703 (2018).
Large binocular telescope interferometer adaptive optics: on-sky performance and lessons learned. V P Bailey, P M Hinz, A T Puglisi, S Esposito, V Vaitheeswaran, A J Skemer, D Defrère, A Vaz, J M Leisenring, E Marchetti, L M Close, Véran , J.-P., eds., SPIEin [SPIE ProceedingsBailey, V. P., Hinz, P. M., Puglisi, A. T., Esposito, S., Vaitheeswaran, V., Skemer, A. J., Defrère, D., Vaz, A., and Leisenring, J. M., "Large binocular telescope interferometer adaptive optics: on-sky performance and lessons learned," in [SPIE Proceedings], Marchetti, E., Close, L. M., and Véran, J.-P., eds., SPIE (jul 2014).
The space coronagraph optical bench (SCoOB): 2. wavefront sensing and control in a vacuum-compatible coronagraph testbed for spaceborne high-contrast imaging technology. K Van Gorkom, E S Douglas, J N Ashcraft, S Y Haffert, D Kim, H Choi, R N Anche, J R Males, K Milani, K Derby, L Harrison, O Durney, Proc. SPIE 12180. SPIE 12180Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter WaveVan Gorkom, K. Douglas, E. S., Ashcraft, J. N., Haffert, S. Y., Kim, D., Choi, H., Anche, R. N., Males, J. R., Milani, K., Derby, K., Harrison, L., and Durney, O., "The space coronagraph optical bench (SCoOB): 2. wavefront sensing and control in a vacuum-compatible coronagraph testbed for spaceborne high-contrast imaging technology," in [Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave], Proc. SPIE 12180 (2022).
The electric field conjugation -a unified formalism for wavefront correction algorithms. A Give'on, OSA Optics & Photonics Technical Digest. 3Optica Publishing GroupOSA Optics & Photonics Technical DigestGive'on, A., "The electric field conjugation -a unified formalism for wavefront correction algorithms," in [Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest ], Frontiers in Optics 2009/Laser Science XXV/Fall 2009 OSA Optics & Photonics Technical Digest , AOWA3, Optica Publishing Group (2009).
The nancy grace roman space telescope coronagraph instrument (CGI) technology demonstration. N J Kasdin, V Bailey, B Mennesson, R Zellem, M Ygouf, J Rhodes, T Luchik, F Zhao, A J E Riggs, B.-J Seo, J Krist, B D Kern, H Tang, B Nemati, T D Groff, N T Zimmerman, B A Macintosh, M Turnbull, J Debes, E S Douglas, R E Lupu, M Lystrup, N Batalha, E C Tong, N Siegler, Perrin , M. D., eds., SPIESpace Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter WaveKasdin, N. J., Bailey, V., Mennesson, B., Zellem, R., Ygouf, M., Rhodes, J., Luchik, T., Zhao, F., Riggs, A. J. E., Seo, B.-J., Krist, J., Kern, B. D., Tang, H., Nemati, B., Groff, T. D., Zimmerman, N. T., Macintosh, B. A., Turnbull, M., Debes, J., Douglas, E. S., and Lupu, R. E., "The nancy grace roman space telescope coronagraph instrument (CGI) technology demonstration," in [Space Telescopes and Instrumentation 2020: Optical, Infrared, and Millimeter Wave], Lystrup, M., Batalha, N., Tong, E. C., Siegler, N., and Perrin, M. D., eds., SPIE (dec 2020).
. K Ahn, O Guyon, J Lozi, S Vievard, V Deo, N Skaf, R Belikov, S P Bos, M Bottom, T Currie, R Frazin, K V Gorkom, T D Groff, S Y Haffert, N Jovanovic, H Kawahara, T Kotani, J R Males, F Martinache, B A Mazin, K Miller, B Norris, A Rodack, A Wong, Scexao, a testbed for developing high-contrast imaging technologies for eltsAhn, K., Guyon, O., Lozi, J., Vievard, S., Deo, V., Skaf, N., Belikov, R., Bos, S. P., Bottom, M., Currie, T., Frazin, R., Gorkom, K. V., Groff, T. D., Haffert, S. Y., Jovanovic, N., Kawahara, H., Kotani, T., Males, J. R., Martinache, F., Mazin, B. A., Miller, K., Norris, B., Rodack, A., and Wong, A., "Scexao, a testbed for developing high-contrast imaging technologies for elts," (2021).
Advanced wavefront sensing and control demonstration with magao-x. S Y Haffert, J Males, K Van Gorkom, L Close, J Long, A Hedglen, O Guyon, L Schatz, M Kautz, A Rodack, J Knight, H Sun, K Forgarty, Adaptive Optics Systems VIII ], 12185, SPIE (2022Haffert, S. Y., Males, J., Van Gorkom, K., Close, L., Long, J., Hedglen, A., Guyon, O., Schatz, L., Kautz, M., Rodack, A., Knight, J., Sun, H., and Forgarty, K., "Advanced wavefront sensing and control demonstration with magao-x," in [Adaptive Optics Systems VIII ], 12185, SPIE (2022).
The self-coherent camera as a focal plane fine phasing sensor. P Janin-Potiron, P Martinez, P Baudoz, M Carbillet, Astronomy & Astrophysics. 592Janin-Potiron, P., Martinez, P., Baudoz, P., and Carbillet, M., "The self-coherent camera as a focal plane fine phasing sensor," Astronomy & Astrophysics 592, A110 (aug 2016).
Tolerance analysis of a self-coherent camera for wavefront sensing and dark hole maintenance. K Derby, S Y Haffert, J N Ashcraft, K Milani, H Choi, Y Kim, L Close, C Mendillo, C Chakrabarti, G Allan, L Pogorelyuk, K Cahoy, M N'daiye, D Kim, J R Males, E S Douglas, Proc. SPIE 12180, SPIE. SPIE 12180, SPIESpace Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter WaveDerby, K., Haffert, S. Y., Ashcraft, J. N., Milani, K., Choi, H., Kim, Y., Close, L., Mendillo, C., Chakrabarti, C., Allan, G., Pogorelyuk, L., Cahoy, K., N'Daiye, M., Kim, D., Males, J. R., and Douglas, E. S., "Tolerance analysis of a self-coherent camera for wavefront sensing and dark hole maintenance," in [Space Telescopes and Instrumentation 2022: Optical, Infrared, and Millimeter Wave], Proc. SPIE 12180, SPIE (Jul 2022).
. J W Goodman, Introduction to Fourier opticsGoodman, J. W., [Introduction to Fourier optics ] (1995).
Fundamental limits to high-contrast wavefront control. J Mazoyer, L Pueyo, S Shaklan, International Society for Optics and Photonics. 10400Techniques and Instrumentation for Detection of Exoplanets VIIIMazoyer, J. and Pueyo, L., "Fundamental limits to high-contrast wavefront control," in [Techniques and Instrumentation for Detection of Exoplanets VIII], Shaklan, S., ed., 10400, 297 -314, International Society for Optics and Photonics, SPIE (2017).
High-contrast imager for complex aperture telescopes (HiCAT): 5. first results with segmented-aperture coronagraph and wavefront control. E Choquet, A E Riggs, R Soummer, G R Brady, K Brooks, T Comeau, T Dillon, S Egron, R Gontrum, J Hagopian, I Laginja, L Leboulleux, M D Perrin, P Petrone, L Pueyo, J Mazoyer, M N'diaye, R Shiri, A Sivaramakrishnan, K S Laurent, A.-M Valenzuela, N T Zimmerman, H A Macewen, M Lystrup, G G Fazio, N Batalha, Optical, Infrared, and Millimeter Wave. Tong, E. C., and Siegler, N., eds., SPIESpace Telescopes andChoquet, E., Riggs, A. E., Soummer, R., Brady, G. R., Brooks, K., Comeau, T., Dillon, T., Egron, S., Gontrum, R., Hagopian, J., Laginja, I., Leboulleux, L., Perrin, M. D., Petrone, P., Pueyo, L., Mazoyer, J., N'Diaye, M., Shiri, R., Sivaramakrishnan, A., Laurent, K. S., Valenzuela, A.-M., and Zimmerman, N. T., "High-contrast imager for complex aperture telescopes (HiCAT): 5. first results with segmented-aperture coronagraph and wavefront control," in [Space Telescopes and Instrumentation 2018: Optical, Infrared, and Millimeter Wave], MacEwen, H. A., Lystrup, M., Fazio, G. G., Batalha, N., Tong, E. C., and Siegler, N., eds., SPIE (aug 2018).
High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design. M N'diaye, E Choquet, L Pueyo, E Elliot, M D Perrin, J K Wallace, T Groff, A Carlotti, D Mawet, M Sheckells, S Shaklan, B Macintosh, N J Kasdin, R Soummer, Shaklan, S., ed., SPIESPIE ProceedingsN'Diaye, M., Choquet, E., Pueyo, L., Elliot, E., Perrin, M. D., Wallace, J. K., Groff, T., Carlotti, A., Mawet, D., Sheckells, M., Shaklan, S., Macintosh, B., Kasdin, N. J., and Soummer, R., "High-contrast imager for complex aperture telescopes (HiCAT): 1. testbed design," in [SPIE Proceedings], Shaklan, S., ed., SPIE (sep 2013).
High-contrast imager for complex aperture telescopes (HICAT): II. design overview and first light results. M N'diaye, E Choquet, S Egron, L Pueyo, L Leboulleux, O Levecq, M D Perrin, E Elliot, J K Wallace, E Hugot, M Marcos, M Ferrari, C A Long, R Anderson, A Difelice, R Soummer, J M Oschmann, M Clampin, G G Fazio, Macewen , H. A., eds., SPIEin [SPIE ProceedingsN'Diaye, M., Choquet, E., Egron, S., Pueyo, L., Leboulleux, L., Levecq, O., Perrin, M. D., Elliot, E., Wallace, J. K., Hugot, E., Marcos, M., Ferrari, M., Long, C. A., Anderson, R., DiFelice, A., and Soummer, R., "High-contrast imager for complex aperture telescopes (HICAT): II. design overview and first light results," in [SPIE Proceedings ], Oschmann, J. M., Clampin, M., Fazio, G. G., and MacEwen, H. A., eds., SPIE (aug 2014).
| []
|
[
"What do the highest-energy cosmic-ray data suggest about possible new physics around 50 TeV?",
"What do the highest-energy cosmic-ray data suggest about possible new physics around 50 TeV?"
]
| [
"Vasiliki Pavlidou \nDepartment of Physics\nInstitute for Theoretical and Computational Physics\nUniversity of Crete\n\n\nFoundation for Research and Technology -Hellas\nIESL\n71110HeraklionGreece\n",
"Theodore Tomaras \nDepartment of Physics\nInstitute for Theoretical and Computational Physics\nUniversity of Crete\n\n"
]
| [
"Department of Physics\nInstitute for Theoretical and Computational Physics\nUniversity of Crete\n",
"Foundation for Research and Technology -Hellas\nIESL\n71110HeraklionGreece",
"Department of Physics\nInstitute for Theoretical and Computational Physics\nUniversity of Crete\n"
]
| []
| The latest observations of extensive air showers (EAS) induced by ultra-high-energy cosmic rays (UHECR) appear to indicate, prima facie, a transition to heavy primaries at the highest energies. However, this interpretation, which is based on extrapolations of the Standard Model (SM) to ultra-LHC energies, is strained from both astrophysical and particle phenomenology perspectives. We consider the alternative that after some energy threshold, the first collision of the primary in the atmosphere results in a state, the decay of which leads to a considerably increased shower particle multiplicity, so that light-primary EAS appear heavy-like. We show that a minimal implementation of such a model yields predictions for the average EAS depth and shower-to-shower fluctuations that are consistent with each other, and an excellent fit to Auger data. If such an effect indeed takes place, we predict that: (a) the center-of-momentum (CM) energy threshold for the effect is of order 50 TeV; (b) the probability with which the effect occurs is high, and it will be detected easily by next-generation accelerators; (c) the increase in multiplicity compared to the SM prediction grows with CM energy roughly as ∼ ECM; (d) the cosmic-ray composition at the highest energies is light. Remarkably, if the latter is confirmed electromagnetically this would necessitate the existence of new physics by these energies. | 10.1103/physrevd.99.123016 | [
"https://export.arxiv.org/pdf/1802.04806v2.pdf"
]
| 119,392,125 | 1802.04806 | 54362f1291b6335b7bc20691156a2d8b8747c663 |
What do the highest-energy cosmic-ray data suggest about possible new physics around 50 TeV?
28 May 2018
Vasiliki Pavlidou
Department of Physics
Institute for Theoretical and Computational Physics
University of Crete
Foundation for Research and Technology -Hellas
IESL
71110HeraklionGreece
Theodore Tomaras
Department of Physics
Institute for Theoretical and Computational Physics
University of Crete
What do the highest-energy cosmic-ray data suggest about possible new physics around 50 TeV?
28 May 2018(Dated: December 2, 2021)arXiv:1802.04806v2 [astro-ph.HE]
The latest observations of extensive air showers (EAS) induced by ultra-high-energy cosmic rays (UHECR) appear to indicate, prima facie, a transition to heavy primaries at the highest energies. However, this interpretation, which is based on extrapolations of the Standard Model (SM) to ultra-LHC energies, is strained from both astrophysical and particle phenomenology perspectives. We consider the alternative that after some energy threshold, the first collision of the primary in the atmosphere results in a state, the decay of which leads to a considerably increased shower particle multiplicity, so that light-primary EAS appear heavy-like. We show that a minimal implementation of such a model yields predictions for the average EAS depth and shower-to-shower fluctuations that are consistent with each other, and an excellent fit to Auger data. If such an effect indeed takes place, we predict that: (a) the center-of-momentum (CM) energy threshold for the effect is of order 50 TeV; (b) the probability with which the effect occurs is high, and it will be detected easily by next-generation accelerators; (c) the increase in multiplicity compared to the SM prediction grows with CM energy roughly as ∼ ECM; (d) the cosmic-ray composition at the highest energies is light. Remarkably, if the latter is confirmed electromagnetically this would necessitate the existence of new physics by these energies.
Introduction.
Ultra-high-energy cosmic rays (UHECR) are the highest-energy particles in the Universe. They are extremely rare (one particle per km 2 per year at energies above 10 18 eV). Even so, thanks to the operation of cosmic-ray observatories spanning thousands of km 2 , there has been, in the past fifteen years, an explosion of unprecedented-quality data [1][2][3][4]. Results from HiRes [5], the Pierre Auger Observatory [6], and Telescope Array [7], now allow the use of UHECR as probes of high-energy physics. The largest cumulative exposure at the highest energies (> 6.7 × 10 4 km 2 sr yr, [8]) has been achieved by the Auger Observatory, and it is the interpretation of the latest Auger data above 10 17.5 eV [9] that we focus on.
This plethora of high-quality data has exposed new puzzles in cosmic-ray physics. The most pressing one involves the composition of UHECR and its evolution with energy. All composition-sensitive observables appear to indicate, prima facie, that, at the highest energies, heavier nuclei start to dominate over protons [3,10,11]; however the results from these observables are not fully consistent with each other [9].
The distribution, in a given primary energy range, of the atmospheric slant depth X max (expressed as column density) where the energy deposition rate of EAS particles in the atmosphere reaches its maximum value is both composition-sensitive [12,13], and directly observable by fluorescence detectors. For this reason, its first two moments (average shower depth, X max , and standard deviation, σ Xmax ) are the most widely used compositionsensitive observables. Auger data on both X max and σ Xmax show a qualitative trend towards heavy-like EAS above ∼ 2×10 18 eV (see Fig. 2), however the two datasets are not straightforward to reconcile in detail, with the Auger Collaboration reporting strained fits to the ob-served X max distribution in more energy bins than what expected from random fluctuations alone: there is no primary composition that can fully reproduce the observed distributions [9]. Additional composition-sensitive quantities obtained from the surface water-Cherenkov detectors, when interpreted using SM EAS simulations, yield a mass composition heavier than the one derived from X max , with the discrepancy traced to an observed excess of muons compared to SM expectations [9]. This is not surprising, as the interpretation of composition-sensitive observables relies on simulations of EAS development, which in turn draw on extrapolations of SM results to ultra-LHC energies.
The alternative, therefore, to the UHECR composition getting heavier, is that there is some new physical effect, yet-unseen in accelerators, that takes place in the first collision of UHECR primaries in the atmosphere above some energy threshold E th and affects the shower development. That this scenario is an open possibility is widely recognized by the Auger Collaboration (e.g., [9,11,14]) and other authors (e.g., [15][16][17]). Here, we quantify phenomenological constraints encoded in Auger data for any new phenomenon that could be affecting EAS development.
Specifically, assuming that, at energies > 2 × 10 18 eV: (a) a single population of extragalactic cosmic rays dominates; (b) the composition of extragalactic cosmic rays remains light; (c) the -abnormal for protons and light nuclei -growth of X max with energy reflects the phenomenology of this new physical effect, we show that Auger data on X max and σ Xmax can be readily reproduced.
What kind of new physics? The primary require-ment for a candidate new physical effect is to make light-primary EAS appear "heavy-like", which in practice translates to (a) having a smaller X max and (b) having smaller σ Xmax than the SM prediction for protons. The phenomenology we consider is that the first collision of the primary in the atmosphere results, with high probability, in a state the decay of which leads to a considerably increased particle multiplicity early in the shower. A large number of particles injected early in the shower development will lead to showers that reach their maximum at smaller values of X, as well as smaller σ Xmax (as shower-to-shower fluctuations will average out).
Several candidate particles and new physics mechanisms that might lead to such a behavior are reviewed in [18,19]. They are based either on the possible existence of yet undiscovered particles (mini black holes, strangelets) or on special phases of QCD, such as the disoriented chiral condensate (DCC). The mini black hole paradigm has been analyzed in detail in [20], while a recent proposal based on chiral symmetry restoration in QCD can be found in [15].
The quantitative impact of such a scenario on composition-sensitive observables is model-dependent; a rough phenomenological estimate is however straightforward to make.
Growth of X max with energy. For a single shower, X max = X 1 +X D , with X 1 being the depth of the first interaction and X D being the additional column density required for the shower to reach its maximum development. For energies below E th , SM predictions hold. X 1 = m/σ p−air where m is the average atomic mass of air (≃ 14.5 proton masses, e.g. [21]) and σ p−air is the protonair cross section 1 . We parameterize σ p−air ≃ σ 0 + β log ǫ for ǫ ≤ 1, where ǫ = E/E th . Any new phenomenon will likely affect σ p−air , so that σ p−air ≃ σ 0 + β ′ log ǫ for ǫ ≥ 1, assuming that σ p−air is continuous as the slope changes 2 from its SM value β to β ′ . Thus, for ǫ ≥ 1,
X 1 ≃ (m/σ 0 ) − (mβ ′ /σ 2 0 ) log ǫ.
The change in X D is entirely due to an increase in partcile multiplicity at the first collision, since the products will have, on average, energies below E th . We parameterize the change in multiplicity by n(ǫ) ≡ N (ǫ)/N SM (ǫ) > 1 (for ǫ ≥ 1), where N (ǫ) and N SM (ǫ) are the actual and SM-predicted (by shower simulations) number of first collision products. We can then empirically model the shower as n(ǫ) "component-showers" (CS) of energy, on average, ǫ/n(ǫ), developing independently. Since for ǫ ≤ 1 the SM prediction [9] is
X D ≃ X D (1) + (65g/cm 2 ) log ǫ, for ǫ ≥ 1 we obtain X D ≃ X D (1) + (65g/cm 2 ) log ǫ/n(ǫ) (where we have assumed n(1) = 1).
The Auger Collaboration [9] fits, for E 2 × 10 18 eV, X max /g cm −2 ∼ (26 ± 2) log ǫ. In the simplest case where the new state is produced almost in every EAS for ǫ ≥ 1, assuming that the composition at these energies remains constant, and the difference with the SM prediction is purely due to new physics, we can obtain n(ǫ) by demanding that,
65 log[ǫ/n] − mβ ′ σ 2 0 log ǫ = 26 log ǫ .(1)
This yields
n(ǫ) ≃ ǫ 0.52−0.08δ ,(2)where δ = β ′ /β − 1.
Change of σ Xmax with energy. The X max spread between showers is the joint effect of fluctuations in X 1 and in shower development,
σ 2 Xmax = σ 2 X1 + σ 2 XD , with σ X1 = X 1 (Poisson statistics).
To estimate σ XD , we take the average (1/n) i X D,i of individual CS maxima to be a reasonable estimator of the overall X D . Then X D is the "sample mean" of n "draws" from the underlying distribution of X D,i , and the distribution of these "sample means" has a spread that is given by the "error in the mean" formula, σ XD = σ XD,i / √ n. Here σ XD,i is the spread of X D,i , and it can be assumed to follow the SM predictions, since the individual energies of the decay products initiating the CS are < E th . The SM predicts that σ XD,i is approximately constant (the mild decline with energy predicted by SM shower simulations for σ Xmax in the case of protons can be reproduced by the logarithmic rise of σ p−air with energy). Therefore
σ 2 Xmax (ǫ) = σ 2 X1 (1)−10.7 g cm 2 σ X1 (1)(1+δ) log ǫ+ σ 2 XD (1) n(ǫ) .
(3) A proof-of-principle minimal model. As a proof of principle for this concept, we show how a simple twocomponent astrophysical scenario (heavy Galactic cosmic rays cutting off; light extragalactic cosmic rays dominating at high energies) with EAS obeying Eqs. (2) and (3) above E th reproduces well Auger data on X max , σ Xmax , and yields reasonable flux spectra for the two populations.
For a mixture of Galactic and extragalactic cosmic rays with a fraction of Galactic over total particles f (ǫ), the probability density function of X max will be p(X max ) = f p G (X max ) + (1 − f )p EG (X max ), so that X max will be given by
X max = f X max G + (1 − f ) X max EG ,(4)
and σ 2 Xmax by
σ 2 Xmax = f σ 2 Xmax,G + (1 − f )σ 2 Xmax,EG +f (1 − f ) ( X max G − X max EG ) 2(5)
with subscripts G and EG referring to the Galactic and extragalactic populations respectively.
There is little freedom in this model. Assuming that extragalactic cosmic rays have completely dominated for E > 2 × 10 18 eV, the evolution of X max EG can be directly read off of the Auger data in this energy range, X max EG /g cm −2 = 728 + 26 log(ǫ/ǫ 17.5 ) , where ǫ 17.5 = 10 17.5 eV/E th . The continuity assumption for n(ǫ), and, consequently for X max EG (ǫ) then fully determines the behavior of X max EG at Auger energies, if the value of E th is known.
A similarly strong statement can be made for f . The shape of the extragalactic population flux spectrum is affected by intergalactic losses (which in turn depend on the composition of extragalactic cosmic rays, the distribution and cosmic evolution of extragalactic cosmic-ray sources, and the cosmic density of diffuse photon backgrounds) and the pileup of particles down-cascading from higher energies [24][25][26][27][28]. These are non-trivial to calculate theoretically, because of the uncertainties involved in the inputs, but also because any systematic uncertainties in the energy reconstruction of cosmic-ray events shift the energy location where specific absorption features appear. In contrast, the Galactic cosmic-ray flux is reasonably expected to be a declining power law (from Fermi acceleration) with an exponential cutoff (induced by Galactic accelerators reaching the maximum energy they can achieve), F G (ǫ) = F G,0 (ǫ/ǫ 17.5 ) −γG exp [−ǫ/ǫ G ]. The values of F G,0 and γ G are well-constrained by KASCADE-Grande data at lower energies 3 , with F G,0 ≃ 2 × 10 −15 km −2 yr −1 sr −1 eV −1 and γ G ≃ 3 (see Fig. 1). The value of ǫ G = E G /E th can then be constrained by the requirement that the flux residuals F total,Auger (ǫ) − F G (ǫ) in the lower-energy part of the Auger range, before any intergalactic propagation losses set in, are consistent with a power law (again assuming Fermi acceleration for extragalactic sources). For values outside the range 6.5 × 10 17 eV < E G < 8.5 × 10 17 eV the low-energy Auger residuals (see Fig. 1, upper panel, green open circles) start to exhibit curvature in a log-log plot. We adopt E G = 7.5 × 10 17 eV, in the middle of this range (purple line, Fig. 1, upper panel). This then fixes f (ǫ) to F G (ǫ)/F total,Auger (ǫ) (Fig. 1, lower panel).
The Galactic component is heavy. The exact composition is subject to various systematic uncertainties [22,23], so for simplicity, we take the SM predictions for carbon nuclei ( X max G,0 ≃ 670g/cm 2 and σ Xmax G,0 ≃ 38g/cm 2 at 10 17.5 eV, from a naive extrapolation of data presented in [9,29]) to be representative, on average, of the behavior of EAS initiated by Galactic cosmic rays 4 . We have however verified that more complex mixes also give good fits with other model inputs within their respective allowed ranges. Since σ Xmax evolves very little for heavier nuclei in the energy range relevant for the Galactic population, we take it to be constant for simplicity. Because f (ǫ) is highly suppressed by the energy new physics sets in, these choices affect neither our fit to Auger data at the high end of their energy range, nor our conclusions on possible new physics phenomenology.
For both a pure proton population and any reasonable light mix, σ Xmax EG,0 will be 68 ± a few g/cm 2 at 10 17.5 eV [9]. We take σ Xmax EG,0 = 68 g/cm 2 .
A nominally free parameter in our model is the threshold energy, E th , where new physics sets in. However its value is very well bounded. By the requirement that X max EG does not, at any energy, exceed (within systematic uncertainties) the SM predictions for protons, E th 10 17.5 eV (see Fig. 2, upper panel). This corresponds to E CM,th 25 TeV, in agreement with the nondetection by the LHC of any effects deviating from SM predictions. By the assumption that new physics has already set in by the break observed by Auger in X max , E th 10 18.3 eV. Good fits to the Auger dataset can be obtained throughout this narrow range, given the uncertainties in the Auger data and the allowed range in other model inputs. In what follows, we will use E th ≃ 10 18 eV (E CM,th ≃ 45 TeV). For heavier primary nuclei, the pernucleon threshold for mass number A is reached at a higher primary energy, AE th . For this reason, the new physics never becomes relevant for Galactic cosmic rays, as extragalactic cosmic rays have completely dominated before AE th is reached, for any reasonable A (hence the "agnostic" dotted lines for the Galactic population at high energies in Fig. 2).
This leaves a single free parameter in our model, δ, which affects X 1 . X max shows no sensitivity to δ, because it is dominated by X D . In contrast σ Xmax is more sensitive to δ; however, at the high energies where its effect becomes important, Auger σ Xmax data have large statistical uncertainties. In Fig. 2, we show two cases: δ = 0 (σ p−air is not affected by new physics, orange line), and δ = 2.9 (cyan line). Note that even the latter case is consistent with SM predictions within uncertainties [21].
Results and Discussion. The resulting X max (E) and σ Xmax (E) curves are shown in Fig. 2. In the same energy range, the two datasets resemble broken logarithmic growth with two different slopes; the Auger Collaboration fits them as such [9]. Each such relation involves four free parameters, so fitting the two datasets in this way would require eight free parameters. We have incorporated in our model the slope and normalization of the second branch of X max , so a purely empirical model would need another six free parameters to fit both datasets well. Without using any of this freedom, we have produced model curves for two very different values of δ that perform better than Astrophysical scenarios (extragalactic accelerator composition getting heavier) [11,28,[30][31][32]; and all other inputs in our model are driven by astrophysics and/or the requirement of consistency with the In addition, astrophysical scenarios with a transition to heavier composition at the highest energies generally do not attempt to reproduce the entire Auger energy range (e.g., [11,28,30,31]), but focus instead above ∼ 5 × 10 18 eV, leaving room for a possible third component between Galactic cosmic rays and the highest-energy cosmic rays, an issue explicitly addressed by [11] (see however [32,33] for models that treat the entire Auger energy range).
Astrophysical explanations of the shallow growth of X max at the highest energies have to invoke two "cosmic coincidences": (a) the Galactic/extragalactic accelerator coincidence at 10 18.5 eV: the energy where the Galactic accelerators cut off is close to the energy where the composition of extragalactic accelerators starts getting heavier; (b) the extragalactic accelerator / cosmic photon background coincidence at 10 19.5 eV: the maximum energy achievable by extragalactic accelerators is close to the energy threshold for photopion/photodissociation energy losses (the Greisen -Zatsepin -Kuzmin, GZK, cutoff [34,35]). Neither issue appears in our scenario, where extragalactic accelerators remain efficient and their output light throughout the Auger energy range. In our scenario, the energy scale of 2×10 18 eV where the slopes of X max and σ Xmax are seen to change in the data does not represent the energy where new physics sets in; rather, this break is astrophysical, and signifies extragalactic cosmic rays dominating over the Galactic population. The new effect has already appeared at a lower energy. Our empirical model does not treat the muon excess; we note however that both production of mini black holes and the restoration of chiral symmetry paradigms might in principle alleviate the muon deficit problem. The simple implementation of the new effect we have presented here is only meant as a proof of principle. Ultimately, the impact of specific models on EAS phenomenology, including their ability to alleviate the muon excess, can be best studied using EAS simulations as, e.g, in [15,20].
The phenomenology we have considered here leads to four specific predictions with important implications for future astroparticle and particle physics experiments.
1. The increase in multiplicity relative to the SM, n(E), grows with lab-frame primary energy as ∼ E 0.52−0.08δ (and with CM energy as E 1.04−0.16δ CM ). Curiously, the multiplicity of the decay of mini black holes depends on the black hole mass M BH ∝ E CM as M (n+2)/(n+1) BH (where n is the number of extra dimensions), in general agreement with the empirical relation; however the estimated cross-section for mini black hole production is generally too small to affect the majority of EAS.
2. The energy threshold E th for the new effect lies between 10 17.5 − 10 18.3 eV ( CM energy 25 -60 TeV), within reach of any next-generation accelerators.
3. The compositon of the extragalactic cosmic ray population is light and stable with energy. This could, in principle, be independently tested electromagnetically, for example by propagation studies in the Galactic magnetic field, provided that an accurate tomographic mapping for the latter becomes available. Should such a confirmation be made, it would necessitate the existence of new physics around 50 TeV. Another central factor in such efforts is good statistics at the highest energies. Next-generation cosmic-ray experiments will thus play a key role in our ability to use UHECR as probes of new physics.
FIG. 1 .FIG. 2 .
12Upper panel: cosmic ray spectrum between 10 16 and 10 20 eV. Filled circles: Auger 2017 ICRC spectrum [9] (error bars are statistcical). Brown triangles: KASCADE-Grande 2015 all-particle spectrum [22], QGSJET II -04 reconstruction (error bars are systematic). Purple line: Galactic population model spectrum (this work). Open green circles: Auger total flux minus Galactic model. The vertical black dotted line indicates the lowest energy for which there are spectrum measurements from Auger. Lower panel: fraction of cosmic rays of Galactic origin as a function of energy, derived from the Galactic flux model over the total observed flux. Upper panel: Xmax as a function of energy. Filled circles: Auger 2017 ICRC data (error bars are systematic). Red/blue dashed lines: SM (Sibyll) predictions for protons/iron, from [8]. The hatched boxes indicate the systematic uncertainty of SM predictions (result of using EPOS/QGSJet instead of Sibyll). Thick lines: our model (purple: Galactic; green: extragalactic; orange: total). Lower panel: σX max as a function of energy. Filled circles: Auger ICRC 2017 data (error bars are statistical). Orange line: δ = 0.Cyan line: δ = 2.9. Other lines as above. For clarity, the extragalactic model is only shown for δ = 0. SM predictions at low energies.
We use the Sibyll 2.1 extrapolation σ p−air ≃ 520 mb + 60 mb log(E/10 17.5 eV)[21]; our results are not sensitive to this choice.2 More generally, σ p−air might also exhibit a discontinuity at ǫ = 1.For simplicity, we do not make use of this extra freedom.
We adopt purely empirically, the 2015 ICRC QGSJetII-04based energy reconstruction of KASCADE-Grande events[22], which results in a near-perfect continuity with Auger measurements at overlapping energies, seeFig. 1.4 The composition of Galactic cosmic rays evolves strongly between the knee (≃ 10 15.5 eV) and their final cutoff at E G . Our simple assumption cannot capture this behavior and thus we do not expect to fit the data below 10 17.5 eV.
ACKNOWLEDGMENTSWe thank P. Sphicas, N. Kylafis, and K. Tassis for useful discussions, and M. Unger for helpful feedback on an early version of this work. TT wishes to thank CERN-TH for their hospitality during the late stages of this work.
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| []
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[
"Random matrix models with log-singular level confinement: method of fictitious fermions *",
"Random matrix models with log-singular level confinement: method of fictitious fermions *"
]
| [
"E Kanzieper \nDepartment of Physics\nThe Jack and Pearl Resnick Institute of Advanced Technology\nBar-Ilan University\n52900Ramat-GanIsrael\n",
"V Freilikher \nDepartment of Physics\nThe Jack and Pearl Resnick Institute of Advanced Technology\nBar-Ilan University\n52900Ramat-GanIsrael\n"
]
| [
"Department of Physics\nThe Jack and Pearl Resnick Institute of Advanced Technology\nBar-Ilan University\n52900Ramat-GanIsrael",
"Department of Physics\nThe Jack and Pearl Resnick Institute of Advanced Technology\nBar-Ilan University\n52900Ramat-GanIsrael"
]
| []
| Joint distribution function of N eigenvalues of U (N ) invariant randommatrix ensemble can be interpreted as a probability density to find N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic, log-singular level confinement. An effective one-particle Schrödinger equation for wave-functions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson's density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk, and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.cond-mat/9704149 | 10.1080/13642819808205006 | [
"https://arxiv.org/pdf/cond-mat/9704149v1.pdf"
]
| 2,082,936 | cond-mat/9704149 | f17320846951ba9aa1089c6e779e8071d6d92eb4 |
Random matrix models with log-singular level confinement: method of fictitious fermions *
9704149v1 17 Apr 1997 April 17, 1997
E Kanzieper
Department of Physics
The Jack and Pearl Resnick Institute of Advanced Technology
Bar-Ilan University
52900Ramat-GanIsrael
V Freilikher
Department of Physics
The Jack and Pearl Resnick Institute of Advanced Technology
Bar-Ilan University
52900Ramat-GanIsrael
Random matrix models with log-singular level confinement: method of fictitious fermions *
9704149v1 17 Apr 1997 April 17, 1997arXiv:cond-mat/ * Presented at the MINERVA Workshop on Mesoscopics, Fractals and Neural Networks, Eilat, Israel, March 1997 1
Joint distribution function of N eigenvalues of U (N ) invariant randommatrix ensemble can be interpreted as a probability density to find N fictitious non-interacting fermions to be confined in a one-dimensional space. Within this picture a general formalism is developed to study the eigenvalue correlations in non-Gaussian ensembles of large random matrices possessing non-monotonic, log-singular level confinement. An effective one-particle Schrödinger equation for wave-functions of fictitious fermions is derived. It is shown that eigenvalue correlations are completely determined by the Dyson's density of states and by the parameter of the logarithmic singularity. Closed analytical expressions for the two-point kernel in the origin, bulk, and soft-edge scaling limits are deduced in a unified way, and novel universal correlations are predicted near the end point of the single spectrum support.cond-mat/9704149
Introduction and basic relations
Random matrices are the field-theoretical models which study the universal features of physical phenomena resulting from the symmetry constraints only. This is the reason why quite different physical problems get a unified mathematical description within the framework of the random-matrix theory [1]. In particular, the applicability of so-called invariant matrix model given by the joint distribution function
ρ N ({λ}) d {λ} = 1 Z N N k=1 dλ k |λ k | αβ exp {−βv (λ k )} N i>j=1 |λ i − λ j | β(1)
of N eigenvalues {λ} of a large N ×N random matrix H ranges from the problem of two-dimensional gravity [2], through the spectral properties of the Dirac operator in quantum chromodynamics [3], to the mesoscopic electron transport in normal and superconducting quantum dots [4,5]. Here the eigenvalues {λ} belong to entire real axis, −∞ < {λ} < +∞, and the partition function Z N is determined from the normalization condition ρ N ({λ}) d {λ} = 1. Parameter β in Eq. (1) accounts for the symmetry of the problem, α is a free parameter associated with a logarithmic singularity, while v (λ) is a non-singular part of the confinement potential
V (λ) = v (λ) − α log |λ| .(2)
It is implied that dimension N of the matrix H is large enough, N ≫ 1. In this thermodynamic limit the matrix model Eq. (1) becomes exactly solvable. Different physics, that is behind the model introduced, deals with different regions of spectrum that can be explored in the corresponding scaling limits. Up to now, the most study received the random-matrix ensemble Eq. (1) with U (N ) symmetry (β = 2), where three types of universal correlations have been established in the origin [6,7], bulk [8,9,10,11], and soft-edge [12,13] scaling limits. Corresponding eigenvalue correlations are described by the universal Bessel, sine, and G−multicritical kernels, respectively. Various scaling limits of the model Eq. (1) have been investigated by using different methods, so that a unified treatment of the problem of spectral correlations in U (N ) invariant ensembles is still absent. The purpose of this paper is to present a unified approach allowing us to explore the spectral properties of the U (N ) invariant matrix model Eq. (1) with effective log-singular level confinement in an arbitrary spectrum range.
The following representation [1] of the joint distribution function ρ N ({λ}) is well-known in the random-matrix theory:
ρ N ({λ}) = |Ψ 0 (λ 1 , ..., λ N )| 2 ,(3)Ψ 0 (λ 1 , ..., λ N ) = 1 √ N ! det ϕ j−1 (λ i ) i,j=1...N .(4)
As far as Ψ 0 takes the form of the Slater determinant, ρ N ({λ}) can be thought of as a probability density to find N non-interacting fictitious fermions in the quantum states ϕ 0 , ..., ϕ N −1 at the "spatial" points λ 1 , ..., λ N . The "wavefunctions" of such fermions are uniquely determined by the set of polynomials P n (λ) orthogonal on the entire real axis with respect to the measure
dµ (λ) = exp {−2V (λ)} dλ, ϕ n (λ) = P n (λ) exp {−V (λ)}(5)
so that the orthogonality relation
+∞ −∞ dλϕ n (λ) ϕ m (λ) = δ nm(6)
holds. It follows from Eq. (3) that the joint distribution function ρ N ({λ}) can be represented as
ρ N ({λ}) = 1 N ! det K N (λ i , λ j ) i,j=1...N ,(7)
where K N (λ, λ ′ ) (referred to as the "two-point kernel")
K N (λ, λ ′ ) = N −1 k=0 ϕ k (λ) ϕ k (λ ′ ) .(8)
is completely determined by the wave-functions ϕ n . Due to an additional constraint on the wave-functions of three successive quantum states that results from the recurrence equation Eq. (10) below, only the highly excited states, ϕ N −1 and ϕ N , contribute to the two-point kernel in accordance with the Christoffel-Darboux theorem [14]:
K N (λ, λ ′ ) = c N ϕ N (λ ′ ) ϕ N −1 (λ) − ϕ N (λ) ϕ N −1 (λ ′ ) λ ′ − λ .(9)
This formula simplifies significantly the mathematical calculations in the thermodynamic limit N ≫ 1. Effective Schrödinger equation for ϕ N , that is the cornerstone of our unified approach, will be derived in the next Section.
Effective Schrödinger equation
In the particular case of the Gaussian unitary ensemble (GUE) the wave-functions ϕ n (λ) are well-known. They are eigenfunctions of a fermion confined by a parabolic potential [1]. For general non-Gaussian ensemble Eq. (1) the calculation of such effective wave-functions can be done by an extension of the Shohat's method [15,16] that previously has been used by the authors [13] to treat the problem of eigenvalue correlations in random-matrix ensembles with non-singular, strong level confinement. This method allows us to map a threeterm recurrence equation
λP n−1 (λ) = c n P n (λ) + c n−1 P n−2 (λ)(10)
for polynomials P n (λ) orthogonal on the entire real axis with respect to the measure dµ
(λ) = exp {−2V (λ)} dλ, +∞ −∞ dµ (λ) P n (λ) P m (λ) = δ nm ,(11)
onto a second-order differential equation for corresponding fictitious wave-functions ϕ n . Coefficients c n appearing in Eq. (10) are uniquely determined by the measure dµ.
In order to derive an effective Schrödinger equation, we note the following identity dP n (λ)
dλ = A n (λ) P n−1 (λ) − B n (λ) P n (λ) ,(12)
with functions A n (λ) and B n (λ) to be determined from the following consideration. Since dP n (λ) /dλ is a polynomial of the degree n − 1, it can be represented [14] through the Fourier expansion in the terms of the kernel Q n (t, λ) = n−1 k=0 P k (λ) P k (t) as:
dP n (λ) dλ = +∞ −∞ dµ (t) dP n (t) dt Q n (t, λ) .(13)
Integrating by parts in the last equation we get that
dP n (λ) dλ = 2 +∞ −∞ dµ (t) Q n (t, λ) dV dt − dV dλ P n (t) .(14)
Now, making use of the Christoffel-Darboux theorem, we conclude that unknown functions A n (λ) and B n (λ) in Eq. (12) are
A n (λ) = 2c n +∞ −∞ dµ (t) t − λ dV dt − dV dλ P 2 n (t) ,(15)B n (λ) = 2c n +∞ −∞ dµ (t) t − λ dV dt − dV dλ P n (t) P n−1 (t) .(16)
We also notice the identity that directly follows from Eqs. (15), (16), (10) and from oddness of dV /dλ:
B n (λ) + B n−1 (λ) − λ c n A n−1 (λ) = −2 dV dλ .(17)
Differentiating Eq. (12), making use of the recurrence equation Eq. (10), and bearing in mind relation Eq. (5) between P n (λ) and ϕ n (λ), one can obtain an exact differential equation for the wave-functions of fictitious fermions, that is valid for arbitrary n:
d 2 ϕ n (λ) dλ 2 − F n (λ) dϕ n (λ) dλ + G n (λ) ϕ n (λ) = 0,(18)
where
F n (λ) = 1 A n dA n dλ ,(19)G n (λ) = dB n dλ + c n c n−1 A n A n−1 − B n B n + 2 dV dλ + 1 A n dA n dλ (20) + d 2 V dλ 2 − dV dλ 2 − 1 A n dA n dλ dV dλ .
Previously, equation of this type was known in the context of the randommatrix theory only for GUE, where V (λ) = λ 2 /2. For such a confinement potential both functions A n and B n can easily be computed from Eqs. (15) and (16), and are given by A n (λ) = 2c n and B n (λ) = 0. Taking into account that for GUE c n = n/2 we end up with F n (λ) = 0 and G n (λ) = 2n + 1 − λ 2 . This allows us to interpret ϕ n (λ) as a wave-function of the fermion confined by a parabolic potential:
d 2 ϕ GUE n (λ) dλ 2 + 2n + 1 − λ 2 ϕ GUE n (λ) = 0.(21)
In principle, the effective Schrödinger equation Eq. (18) applies to general non-Gaussian random-matrix ensembles as well, although the explicit calculation of F n (λ) and G n (λ) in this situation may be a rather complicated task. However, significant simplifications arise in the thermodynamic limit n = N ≫ 1.
To proceed with derivation of the asymptotic Schrödinger equation, we have to specify the form of confinement potential V introduced by Eq. (2). Choosing the regular part v (λ) to be an even function, we set
V (α) (λ) = p k=1 d k 2k λ 2k − α log |λ| ,(22)
with d p > 0. The signs of the rest d k 's can be arbitrary, allowing for nonmonotonic level confining, but they should lead to an eigenvalue density supported on a single connected interval (−D N , +D N ). Confinement potential V (α) (λ) determines its own set of orthogonal polynomials P
N (λ) = |λ| α P (α) N (λ) exp {−v (λ)}.
[Here upper index α reflects the presence of the log-singular component in V (α) (λ), and the restriction α > − 1 2 takes place due to normalization Eq. (11)]. In accordance with Eqs. (15) and (22)
it is convenient to represent A (α) N in the form A (α) N (λ) = A (N ) reg (λ) + αA (N ) sing (λ) ,(23)
where
A (N ) reg (λ) = 2c N +∞ −∞ dµ (t) t − λ P (α) N (t) 2 dv dt − dv dλ ,(24)A (N ) sing (λ) = 2c N +∞ −∞ dµ (t) t P (α) N (t) 2 .(25)
Analogously, Eq. (16) leads to the similar representation
B (α) N (λ) = B (N ) reg (λ) + αB (N ) sing (λ) ,(26)
with
B (N ) reg (λ) = 2c N +∞ −∞ dµ (t) t − λ P (α) N (t) P (α) N −1 (t) dv dt − dv dλ ,(27)B (N ) sing (λ) = 2c N λ +∞ −∞ dµ (t) t P (α) N (t) P (α) N −1 (t) .(28)
In the above formulas A reg can be done along the lines presented in Ref. [13]. Then, we immediately obtain that A (α) N (λ) is expressed in terms of Dyson's density ν D (λ) as follows:
A (α) N (λ) = πν D (λ) 1 − (λ/D N ) 2 ,(30)ν D (λ) = 2 π 2 P DN 0 ξdξ ξ 2 − λ 2 dv dξ 1 − (λ/D N ) 2 1 − (ξ/D N ) 2 ,(31)
where D N = 2c N should be identified with the soft edge of the spectrum. [It is easy to see that a log-singular part of the confinement potential does not contribute to the Dyson's density, so that in the thermodynamic limit there are no changes in D N due to logarithmic singularity of confinement potential]. Expression for B
(N ) reg can be obtained by the use of the large−N version of the identity Eq. (17), that yields
B (N ) reg (λ) = λ D N A (α) N (λ) − dv dλ ,(32)
whence B (α)
N (λ) = λ D N πν D (λ) 1 − (λ/D N ) 2 − dv dλ + α 1 − (−1) N λ .(33)
Now, having asymptotic representations for A
d 2 ϕ (α) N dλ 2 − d dλ log πν D (λ) 1 − (λ/D N ) 2 dϕ (α) N dλ (34) + π 2 ν 2 D (λ) + (−1) N α − α 2 λ 2 ϕ (α) N (λ) = 0
Also, due to Eq. (12), one can verify that the wave-functions of two successive quantum states are connected by the relationship
dϕ (α) N dλ = πν D (λ) 1 − (λ/D N ) 2 ϕ (α) N −1 (λ) − λ D N ϕ (α) N (λ) + (−1) N α λ ϕ (α) N (λ) . (35)
Equations (34) and (35) provide a general basis for the study of eigenvalue correlations in non-Gaussian random-matrix ensembles in an arbitrary spectral range. In particular case of GUE, the Dyson's density of states is the celebrated semicircle, ν GUE
D (λ) = π −1 D 2 N − λ 2 with D N = √ 2N .
The square-root law for ν GUE D (λ) immediately removes the first derivative dϕ (α) N /dλ in Eq. (34), providing us the possibility to interpret the fictitious fermions as those confined by a quadratic potential (α = 0). As far as the semicircle is a distinctive feature of density of states in GUE only, one will always obtain a first derivative in the effective Schrödinger equation for the non-Gaussian unitary ensembles of random matrices. Therefore, fictitious non-interacting fermions associated with non-Gaussian ensembles of random matrices live in a non-Hermitian quantum mechanics.
An interesting property of these equations is that they do not contain the regular part of confinement potential explicitly, but only involve the Dyson's density ν D (analytically continued on the entire real axis) and the spectrum end point D N . In contrast, the logarithmic singularity (that does not affect the Dyson's density) introduces additional singular terms into Eqs. (34) and (35), changing significantly the behavior of the wave-function ϕ (α) N near the origin λ = 0. The influence of the singularity decreases rather rapidly outward from the origin.
Structure of the effective Schrödinger equation leads us to the following fundamental statements: (i) Eigenvalue correlations are stable with respect to nonsingular deformations of the confinement potential. (ii) In the random-matrix ensembles with well-behaved confinement potential the knowledge of Dyson's density (that is rather crude one-point characteristics coinciding with the real density of states only in the spectrum bulk) is sufficient to determine the genuine density of states, as well as the n−point correlation function, everywhere. The latter conclusion is rather unexpected since it considerably reduces the knowledge required for computing n−point correlators.
Local eigenvalue correlations
Effective Schrödinger equation obtained in the preceding Section allows us to examine in a unified way the eigenvalue correlations in non-Gaussian ensembles with U (N ) symmetry in different scaling limits. As we show below, it inevitably leads to the universal Bessel correlations in the origin scaling limit [6,7], to the universal sine correlations in the bulk scaling limit [8,9,10,11], and to the universal G−correlations in the soft-edge scaling limit [13]. Corresponding twopoint kernels are given by Eqs. (40), (42) and (52), respectively.
Origin scaling limit and the universal Bessel law
Origin scaling limit deals with the region of spectrum close to λ = 0 where confinement potential displays the logarithmic singularity. In the vicinity of the origin the Dyson's density can be taken as being approximately a constant, ν D (0) = 1/∆ N (0), where ∆ N (0) is the mean level spacing at the origin in the absence of the logarithmic deformation of potential v. In the framework of this approximation, Eq. (34) reads
d 2 ϕ (α) N dλ 2 + π 2 ∆ 2 N (0) + (−1) N α − α 2 λ 2 ϕ (α) N (λ) = 0.(36)
Solution to this equation that remains finite at λ = 0 can be expressed by means of Bessel functions:
ϕ (α) 2N (λ) = a √ λJ α− 1 2 πλ ∆ (0) ,(37)ϕ (α) 2N +1 (λ) = b √ λJ α+ 1 2 πλ ∆ (0) ,(38)
where a and b are constants to be determined later, and ∆ (0) = ∆ 2N (0) ≈ ∆ 2N +1 (0). In accordance with Eq. (9), the two-point kernel can be written down as
K (α) 2N (λ, λ ′ ) = c √ λλ ′ λ ′ − λ J α+ 1 2 πλ ∆ (0) J α− 1 2 πλ ′ ∆ (0) (39) −J α+ 1 2 πλ ′ ∆ (0) J α− 1 2 πλ ∆ (0) ,
where the unknown factor c can be found from the requirement K (α=0) 2N
(λ, λ) = 1/∆ (0). This immediately yields us c = −π/∆ (0). Defining now the scaled variable s = λ s /∆ (0), we obtain that in the origin scaling limit the two-point
kernel K orig (s, s ′ ) = lim N →∞ K (α) N (λ s , λ s ′ ) dλ s /ds takes the universal Bessel law K orig (s, s ′ ) = π 2 √ ss ′ J α+ 1 2 (πs) J α− 1 2 (πs ′ ) − J α− 1 2 (πs) J α+ 1 2 (πs ′ ) s − s ′ .(40)
Formula (40) is valid for arbitrary α > − 1 2 . Note, that a recent proof of universality of the Bessel kernel given in Ref. [7] was based on the Christoffel theorem [14], that imposed an artificial restriction on parameter α to be only positive integer.
Bulk scaling limit and the universal sine law
Bulk scaling limit has been explored in a number of works [8,9,10,11]. It is associated with a spectrum range where the confinement potential is well behaved (that is far from the logarithmic singularity λ = 0), and where the density of states can be taken as being approximately a constant on the scale of a few levels. In accordance with this definition one has
K bulk (s, s ′ ) = lim s,s ′ →∞ K orig (s, s ′ ) ,(41)
where s and s ′ should remain far enough from the end point D N of the spectrum support.
Taking this limit in Eq. (40), we arrive at the universal sine law
K bulk (s, s ′ ) = sin [π (s − s ′ )] π (s − s ′ ) .(42)
3.3 Soft-edge scaling limit and the universal G−multicritical law
Soft-edge scaling limit is relevant to the tail of eigenvalue support where crossover occurs from a non-zero density of states to a vanishing one. It is known [17] that by tuning coefficients d k which enter the regular part v of confinement potential [see Eq. (22)], one can obtain a bulk (Dyson's) density of states which possesses a singularity of the type 1)]. Such an m−th multicriticality can be achieved by many means, and the corresponding plethora of multicritical potentials V (m) is given by the equation
ν D (λ) = 1 − λ 2 D 2 N m+1/2 R N λ D N(43)dV (m) (λ) dλ = P +DN −DN dt λ − t 1 − t 2 D 2 N m+1/2 R N t D N .(44)
So-called minimal multicritical potentials which correspond to R N = const can be found in Refs. [17,18]. Below we demonstrate that as long as multicriticality of order m is reached, the eigenvalue correlations in the vicinity of the soft edge become universal, and are independent of the particular potential chosen. The order m of the multicriticality is the only parameter which governs spectral correlations in the soft-edge scaling limit.
Let us move the spectrum origin to its endpoint D N , making the replacement
λ s = D N 1 + s · 1 2 2 πD N R N (1) 1/ν * ,(45)
that defines the m−th soft-edge scaling limit provided s ≪ (D N R N (1)) 1/ν * ∝ N 1/ν * , with
ν * = m + 3 2 .(46)
It is straightforward to show from Eqs. (34) and (35) that in the vicinity of the end point D N the function ϕ N (s) = ϕ (α) N (λ − D N ) obeys the universal differential equation
ϕ ′′ N (s) − ν * − 3 2 s ϕ ′ N (s) − s 2(ν * −1) ϕ N (s) = 0,(47)
and that the following relation takes place:
ϕ N −1 (s) = ϕ N (s) + (−1) ν * − 3 2 2 πD N R N (1) 1 2ν * s 3 2 −ν * ϕ ′ N (s) .(48)
It follows from Eq. (52) that the density of states in the same scaling limit
ν (m) soft (s) = d ds G (s|ν * ) 2 s 3 2 −ν * − [G (s|ν * )] 2 s ν * − 1 2(54)
is also universal. The large− |s| behavior of ν (m) soft can be deduced from the known asymptotic expansions of the Bessel functions:
ν (m) soft (s) = |s| ν * −1 π + (−1) ν * − 1 2 4π|s| cos 2|s| ν * ν * , s → −∞, exp − 2s ν * ν * 4πs cos 2 ( π 4ν * ) sin( π 4ν * )+(−1) ν * − 3 2 , s → +∞.(55)
Note that the leading order behavior as s → −∞ is consistent with the |s| ν * −1 singularity of the bulk density of states, Eq. (43).
Concluding remarks
We have presented a general formalism for a treatment of the problem of eigenvalue correlations in spectra of U (N ) invariant ensembles of large random matrices with log-singular level confinement. An important ingredient of our analysis is an effective one-particle Schrödinger equation [see Eqs. (18) and (34)] for fictitious non-interacting fermions naturally appearing in the determinantal representation of the joint distribution function of N eigenvalues of large N × N Hermitian random matrix. The structure of the asymptotic equation Eq. (34) allowed us to conclude that: (i) Eigenvalue correlations are stable with respect to non-singular deformations of confinement potential. (ii) In the random-matrix ensembles with well-behaved confinement potential the knowledge of Dyson's density (that is rather crude one-point characteristics coinciding with the real density of states only in the spectrum bulk) is sufficient to determine the genuine density of states, as well as the n−point correlation function, everywhere.
We have also demonstrated that effective Schrödinger equation contains all the information about eigenvalue correlations in arbitrary spectrum range: the universal Bessel kernel Eq. (40) was found to describe eigenvalue correlations in the origin scaling limit; the universal sine kernel Eq. (42) was revealed in the bulk scaling limit; finally, we have shown that the soft-edge scaling limit is described by the novel universal G−multicritical kernel Eq. (52).
needed to construct an asymptotic second-order differential equation for the function ϕ
caused by its log-singular part. First, it is easy to see that A (N ) sing (λ) ≡ 0 due to evenness of the measure dµ. Second, the exact expression for B
N
given by Eqs. (30) and (33), and taking into account Eqs. (18), (19)and(20), it is straightforward to obtain the following remarkable effective asymptotic Schrödinger equation for the wave-functions ϕ
N
(λ) exp {−v (λ)} of highly excited states (N ≫ 1) of fictitious fermions:
with the multicritical index m = 0, 2, 4, etc., and R N being a well-behaved function with R N (±1) = 0. [Odd indices m are inconsistent with our choice that the leading coefficient d p , entering the regular component v (λ) of confinement potential, be positive in order to keep a convergence of integral for partition function Z N in Eq. (
AcknowledgmentOne of the authors (E. K.) acknowledges the support of the Levy Eshkol Fellowship from the Ministry of Science of Israel.Solution to Eq. (47) which decreases at s → +∞ (that is at far tails of the density of states) can be represented through the functionwhere a is an unknown constant. Making use of Eq. (48), we obtain that in the vicinity of the soft edge the two-point kernel iswhere b is an unknown constant again. It can be found by fitting[12]These G−multicritical correlations are universal in the sense that they do not depend on the details of confinement potential, but only involve such an "integral" characteristic of level confinement as the index m of the multicriticality. In particular case of m = 0, that is inherent in random-matrix ensembles with monotonic confinement potential, the function G coincides with the Airy function, G s| 3 2 = Ai (s), and the previously supposed universal Airy correlations[12]K
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| []
|
[
"Deeply Virtual Compton Scattering at Future Electron-Ion Colliders",
"Deeply Virtual Compton Scattering at Future Electron-Ion Colliders"
]
| [
"Gang Xie \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Wei Kou \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Qiang Fu \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n",
"Zhenyu Ye \nDepartment of Physics\nUniversity of Illinois\n60607ChicagoILUSA\n",
"Xurong Chen \nInstitute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina\n\nSchool of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina\n"
]
| [
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina",
"Department of Physics\nUniversity of Illinois\n60607ChicagoILUSA",
"Institute of Modern Physics\nChinese Academy of Sciences\n730000LanzhouChina",
"School of Nuclear Science and Technology\nUniversity of Chinese Academy of Sciences\n100049BeijingChina"
]
| []
| The study of hadronic structure has been carried out for many years. Generalized parton distribution functions (GPDs) give broad information on the internal structure of hadrons. Combining GPDs and high-energy scattering experiments, we expect yielding three-dimensional physical quantities from experiments. Deeply Virtual Compton Scattering (DVCS) process is a powerful tool to study GPDs. It is one of the important experiments of Electron Ion Collider (EIC) and Electron ion collider at China (EicC) in the future. In the initial stage, the proposed EicC will have 3 ∼ 5 GeV polarized electrons on 12 ∼ 25 GeV polarized protons, with luminosity up to 1 ∼ 2 × 10 33 cm −2 s −1 . EIC will be constructed in coming years, which will cover the variable c.m. energies from 30 to 50 GeV, with the luminosity about 10 33 ∼ 10 34 cm −2 s −1 . In this work we present a detailed simulation of DVCS to study the feasibility of experiments at EicC and EIC. Referring the method used by HERMES Collaboration, and comparing the model calculations with pseudo data of asymmetries attributed to the DVCS, we obtained a model-dependent constraint on the total angular momentum of up and down quarks in the proton.I. INTRODUCTIONIn high energy nuclear physics, the internal structure and dynamics of the proton is still not fully understood. Although decades have passed since the discovery that the proton internal structure consisted of quarks [1-4] and gluons (partons) [5-8], we still know a little about how the partons contribute to the global properties of the proton such as its mass and spin. The measurement of the fraction of the proton spin carried by quarks by the European Muon Collaboration (EMC) in 1987 indicated that only small percentages of the proton's spin comes from quarks[9]. The data of nucleon's polarized structure function g 1 (x B ) in EMC has deviated significantly from the Ellis-Jaffe sum rule [10]. These results created the so-called "spin crisis", or more appropriately, the "spin puzzle". The discrepancy has since inspired many intensive experimental and theoretical studies of spin dependent nucleon structure[11][12][13][14][15][16][17]. It was proposed that the missing fraction of the proton spin comes from the polarized gluon contribution. Recent measurements of the polarized gluon density showed that gluons indeed contribute, but could not fill the gap in the spin puzzle[16]. The orbital angular momenta of the quarks and gluons play an important role in the proton spin. According to the generator of Lorentz transformation we can define the angular momentum operator in QCD [18],where M 0jk is the angular momentum density, which can be expressed by the energy-momentum tensor TT µν has the Belinfante-Improved form and is symmetric, gauge-invariant, and conserved. It can be divided into gauge-invariant quark and gluon contributions,and ⃗ J has a gauge-invariant form,In pure gauge theory, ⃗ J g is a conserved angular momentum charge by itself, generating spin quantum numbers for glueballs. We can see that ⃗ J q and ⃗ J g are interactiondependent. To study the orbital angular momentum of the partons, one needs to study beyond one-dimentional parton distributions.One-dimensional parton distribution functions (PDFs) provide significant informations about the structure of the proton. Although the PDFs have provided us with much knowledge on the proton, one-dimensional distributions can not give us a complete picture. Therefore, theorists developed a new density function about 30 years ago, which called GPDs. GPDs provide information including both transverse spacial and longitudinal momentum distributions. Besides the momentum fraction, GPDs depend on another independent variable, the negative value of momentum transfer square t = − (p − p ′ ) 2 between the initial and final states of a proton. Thus, the GPDs give extensive informations about three-dimensional dynamics of nucleon, which includes the composition of spin and pressure distribution[19][20][21][22][23][24]. Similar to the one dimensional PDFs, GPDs include non-polarized and polarized functions. | null | [
"https://export.arxiv.org/pdf/2306.02357v1.pdf"
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| 259,075,595 | 2306.02357 | 4f0059f92549502b890ddf10f28f6a4a590f82dc |
Deeply Virtual Compton Scattering at Future Electron-Ion Colliders
Gang Xie
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
School of Nuclear Science and Technology
University of Chinese Academy of Sciences
100049BeijingChina
Wei Kou
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
School of Nuclear Science and Technology
University of Chinese Academy of Sciences
100049BeijingChina
Qiang Fu
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
School of Nuclear Science and Technology
University of Chinese Academy of Sciences
100049BeijingChina
Zhenyu Ye
Department of Physics
University of Illinois
60607ChicagoILUSA
Xurong Chen
Institute of Modern Physics
Chinese Academy of Sciences
730000LanzhouChina
School of Nuclear Science and Technology
University of Chinese Academy of Sciences
100049BeijingChina
Deeply Virtual Compton Scattering at Future Electron-Ion Colliders
(Dated: June 6, 2023)
The study of hadronic structure has been carried out for many years. Generalized parton distribution functions (GPDs) give broad information on the internal structure of hadrons. Combining GPDs and high-energy scattering experiments, we expect yielding three-dimensional physical quantities from experiments. Deeply Virtual Compton Scattering (DVCS) process is a powerful tool to study GPDs. It is one of the important experiments of Electron Ion Collider (EIC) and Electron ion collider at China (EicC) in the future. In the initial stage, the proposed EicC will have 3 ∼ 5 GeV polarized electrons on 12 ∼ 25 GeV polarized protons, with luminosity up to 1 ∼ 2 × 10 33 cm −2 s −1 . EIC will be constructed in coming years, which will cover the variable c.m. energies from 30 to 50 GeV, with the luminosity about 10 33 ∼ 10 34 cm −2 s −1 . In this work we present a detailed simulation of DVCS to study the feasibility of experiments at EicC and EIC. Referring the method used by HERMES Collaboration, and comparing the model calculations with pseudo data of asymmetries attributed to the DVCS, we obtained a model-dependent constraint on the total angular momentum of up and down quarks in the proton.I. INTRODUCTIONIn high energy nuclear physics, the internal structure and dynamics of the proton is still not fully understood. Although decades have passed since the discovery that the proton internal structure consisted of quarks [1-4] and gluons (partons) [5-8], we still know a little about how the partons contribute to the global properties of the proton such as its mass and spin. The measurement of the fraction of the proton spin carried by quarks by the European Muon Collaboration (EMC) in 1987 indicated that only small percentages of the proton's spin comes from quarks[9]. The data of nucleon's polarized structure function g 1 (x B ) in EMC has deviated significantly from the Ellis-Jaffe sum rule [10]. These results created the so-called "spin crisis", or more appropriately, the "spin puzzle". The discrepancy has since inspired many intensive experimental and theoretical studies of spin dependent nucleon structure[11][12][13][14][15][16][17]. It was proposed that the missing fraction of the proton spin comes from the polarized gluon contribution. Recent measurements of the polarized gluon density showed that gluons indeed contribute, but could not fill the gap in the spin puzzle[16]. The orbital angular momenta of the quarks and gluons play an important role in the proton spin. According to the generator of Lorentz transformation we can define the angular momentum operator in QCD [18],where M 0jk is the angular momentum density, which can be expressed by the energy-momentum tensor TT µν has the Belinfante-Improved form and is symmetric, gauge-invariant, and conserved. It can be divided into gauge-invariant quark and gluon contributions,and ⃗ J has a gauge-invariant form,In pure gauge theory, ⃗ J g is a conserved angular momentum charge by itself, generating spin quantum numbers for glueballs. We can see that ⃗ J q and ⃗ J g are interactiondependent. To study the orbital angular momentum of the partons, one needs to study beyond one-dimentional parton distributions.One-dimensional parton distribution functions (PDFs) provide significant informations about the structure of the proton. Although the PDFs have provided us with much knowledge on the proton, one-dimensional distributions can not give us a complete picture. Therefore, theorists developed a new density function about 30 years ago, which called GPDs. GPDs provide information including both transverse spacial and longitudinal momentum distributions. Besides the momentum fraction, GPDs depend on another independent variable, the negative value of momentum transfer square t = − (p − p ′ ) 2 between the initial and final states of a proton. Thus, the GPDs give extensive informations about three-dimensional dynamics of nucleon, which includes the composition of spin and pressure distribution[19][20][21][22][23][24]. Similar to the one dimensional PDFs, GPDs include non-polarized and polarized functions.
GPDs, also named as the off-forward PDFs, have attracted a lot of attention since spin decomposition rule was first proposed [18]. It was proposed to factorize the hard exclusive processes. The corresponding factorization structure functions including the structure of nucleon are the GPDs H q (x B , ξ, t), E q (x B , ξ, t), H q (x B , ξ, t) andẼ q (x B , ξ, t). These functions correspond to the Fourier transform of the non-diagonal operators [18,20,22,25]:
P + 2π dy − e jx B P + y − p ′ Ψ q (0)γ + Ψ q (y) p y + =⃗ y ⊥ =0 = H q (x B , ξ, t)N (p ′ ) γ + N (p) +E q (x B , ξ, t)N (p ′ ) iσ +v ∆v 2M N N (p), P + 2π dy − e x B P + y − p ′ Ψ q (0)γ + γ 5 Ψ q (y) p y + =⃗ y ⊥ =0 = H q (x B , ξ, t)N (p ′ ) γ + γ 5 N (p) + E q (x B , ξ, t)N (p ′ ) γ 5 ∆ + 2M N N (p),(5)
where y is the coordinate of the two correlated quarks, the P is the average nucleon four-momentum in lightfront frame: P = (p + p ′ ) /2 and ∆ = p ′ −p. The "+" superscript means the plus component of four-momentum in light-front frame. Each GPD function defined above is for a specified flavor of quark: H q , E q , H q , E q (q = u, d, s, . . .). H q and H q are spin non-flipped GPD functions and E q and E q are spin flipped ones. The ordinary parton distributions and nucleon form factors are both included in the off-forward parton distributions. In t → 0 and ξ → 0 limit, we get
H(x B , 0, 0) = f 1 (x B ), H (x B , 0, 0) = g 1 (x B ),(6)
where f 1 (x B ) is quark distribution and g 1 (x B ) is quark helicity distribution. According to Dirac and Pauli form factors F 1 , F 2 and axial-vector and pseudo-scalar form factor G A , G P , the sum rules are obtained,
dx B H (x B , ξ, t) = F 1 (t) , dx B E (x B , ξ, t) = F 2 (t) , dx BH (x B , ξ, t) = G A (t) , dx BẼ (x B , ξ, t) = G P (t) .(7)
The most interesting Ji's sum rules related to the nucleon spins are described through GPDs [22],
1 −1 dx B x B [H (x B , ξ, t) + E (x B , ξ, t)] = A(t) + B(t).
(8) Then the total spin of the proton can be expressed as:
J q,g = 1 2 [A q,g (0) + B q,g (0)] , J q + J g = 1 2 ,(9)
where A q,g (0) gives the momentum fractions carried by quarks and gluons in the nucleon (A q (0) + A g (0) = 1), and B-form factor is analogous to the Pauli form factor for the vector current. By extrapolating the sum rule to t = 0, one gets J q,g . The GPDs can be measured in deep-exclusive processes such as DVCS and deeply virtual meson production (DVMP) [18,22,[26][27][28][29][30]. Both of these processes are exclusive hard scattering processes in lepton-nucleon collisions. Theoretical research on these topics has been conducted for many years, and many theoretical models and predictions were created by researchers [18,21,22,[31][32][33][34][35][36][37][38][39]. During the past 20 years, the collaborations at HERA and Jefferson Lab (JLab) have spent a lot effort to get information of GPDs from electro-production of a real photon (DVCS processes) [31,, such as DESY with H1 [40,43], ZEUS [41] and HERMES [45,46], JLab Halls A [31,50,[52][53][54][55] and Halls B [48,51,[56][57][58][59], and COMPASS [60,61]. These experiments have important contributions to our exploration of the internal structure of the proton. Although there are many data from above experiments, the data don't have high precision and wide range of kinematic region. Accurate measurement of the DVCS process is a huge challenge, which requires high luminosity to compensate for very small cross section and good detector design to ensure the exclusive measurement of the final states. Both EicC and EIC are important experiments in the future that will have very high luminosity and excellent detectors for particle detection. In this work, we discuss the relation of GPDs and DVCS observables [22], and carry out a Monte-Carlo simulation of DVCS + Bethe-Heithler (BH) events and do a projection to get the statistical errors of asymmetry observables of DVCS experiments for the future EicC and EIC.
Since the contribution of GPDs to amplitude is not independent, the acquisition of GPDs from the exclusive reactions is indirect. We need to use the appropriate GPDs model. After years of development, there are many theoretical models of GPD, and two of those are based on double distributions (DDs) [20,62,63], one has been given by Vanderhaeghen, Guichon and Guidal, which called VGG model [26,27,64,65], another was presented by Goloskokov and Kroll called GK model [28,66,67]. By accessing the available experimental data, the researchers examined different GPD models, and show that the data from different experiments can match well with the VGG model calculation [25,48,54,57,58]. Based on these results, we perform theoretical calculations with VGG model. In VGG model, the observable A sin(ϕ−ϕs) cos ϕ U T is more sensitive to the quark total angular momentum in the nucleon than other parameters [31,68,69]. Thus we make a constraint on J u and J d by the pseudo data of Transverse Target-Spin Asymmetry (TTSA) A sin(ϕ−ϕs) cos ϕ U T . The organization of the paper is as follows. The relationship between GPDs and DVCS is illustrated in Sec. II. The phenomenological parametrization of GPDs is described in Sec. III. The invariant kinematic and final state kinematic distributions of the simulation are shown in Sec. IV. The projections of DVCS experiment are shown in Sec. V. Finally, some discussions and a concise summary is given in Sec. VI.
II. GENERALIZED PARTONS DISTRIBUTION AND DEEPLY VIRTUAL COMPTON SCATTERING
Deeply virtual Compton scattering on a necleon shown in Fig. 1 left panel is the simplest process to access GPDs, it's an important role in exploring the internal structure of necleon. In addition to the DVCS, there also exists another process shares the same final state with DVCS process, see Fig. 1 middle and right panels, called the BH process. processes. e, e ′ and p, p ′ are the initial and final states electron and proton respectively. And t is the four-momentum square transition between the initial and final state proton. The five-fold differential cross section for electroproduction of real photon ep → e ′ p ′ γ is defined as [32]:
dσ dx B dyd |∆ 2 | dϕdφ = α 3 x B y 16π 2 Q 2 √ 1 + ϵ 2 T e 3 2 .(10)
This cross section depends on the common Bjorken scaling variable x B , the squared momentum transfer ∆ = (P 2 − P 1 ) 2 , the lepton energy fraction y = P 1 · q 1 /P 1 · k, with q 1 = k−k ′ . The azimuthal angle between the lepton plane and the recoiled proton momentum is ϕ. There, φ is the angle between the polarization vector and the scattered hadron shown in Fig. 2, and ϵ = 2x B M/Q that incorporates nonvanishing target mass effects [32,70]. The reaction amplitude T is the linear superposition sum of the BH and DVCS amplitudes,
T 2 = |T BH | 2 + |T DV CS | 2 + T I ,(11)
where T I = T DV CS T * BH + T * DV CS T BH . The squared BH term |T BH | 2 , squared DVCS amplitude |T DV CS | 2 , and interference term T I are given by:
|T BH | 2 = e 6 x 2 B y 2 (1 + ϵ 2 ) 2 ∆ 2 P 1 (ϕ)P 2 (ϕ) c BH 0 + 2 n=1 c BH n cos(nϕ) + s BH 1 sin(ϕ) ,(12)|T DVCS | 2 = e 6 y 2 Q 2 c DVCS 0 + 2 n=1 c DVCS n cos(nϕ) + s DVCS n sin(nϕ) ,(13)T I = ±e 6 x B y 3 ∆ 2 P 1 (ϕ)P 2 (ϕ) c I 0 + 3 n=1
c I n cos(nϕ) + s I n sin(nϕ) .
The results for the Fourier coefficients can be found in [32,70]. The variables ξ and t (or ∆ 2 ) can be computed from the kinematic variables. Since we cannot directly obtain x B from experiment, the Compton form factors (CFF) are obtained by integrating the GPDs,
1 −1 F q (x B , ξ, t) x B − ξ + iϵ dx B = P 1 −1 F q (x B , ξ, t) x B − ξ dx B − iπF q (ξ, ξ, t),(15)
where F q are H q , H q , E q , or E q . These real and imaginary part of Eq. 15, which can be expressed in eight GPD-related quantities that can be extracted from DVCS observables [25]:
H Re (ξ, t) ≡ P 1 0 dx B [H (x B , ξ, t) − H (−x B , ξ, t)] C + , H Im (ξ, t) ≡ H(ξ, ξ, t) − H(−ξ, ξ, t), E Re (ξ, t) ≡ P 1 0 dx B [E (x B , ξ, t) − E (−x B , ξ, t)] C + , E Im (ξ, t) ≡ E(ξ, ξ, t) − E(−ξ, ξ, t), H Re (ξ, t) ≡ P 1 0 dx B H (x B , ξ, t) − H (−x B , ξ, t) C − , H Im (ξ, t) ≡ H(ξ, ξ, t) − H(−ξ, ξ, t), E Re (ξ, t) ≡ P 1 0 dx B E (x B , ξ, t) − E (−x B , ξ, t) C − , E Im (ξ, t) ≡ E(ξ, ξ, t) − E(−ξ, ξ, t).(16)
The case with subscript "Re" is accessed by observables sensitive to the real part of the DVCS amplitude, while the case with subscript "Im" is accessed by observables sensitive to its imaginary part, where the coefficient C ± defined as:
C ± = 1 x B − ξ ± 1 x B + ξ .(17)
As a result, the Compton form factors with four complex functions are written as:
H(ξ, t) ≡ H Re (ξ, t) − iπH Im (ξ, t), H(ξ, t) ≡ H Re (ξ, t) − iπ H Im (ξ, t), E(ξ, t) ≡ E Re (ξ, t) − iπE Im (ξ, t), E(ξ, t) ≡ E Re (ξ, t) − iπ E Im (ξ, t).(18)
For the measurement of CFFs, it is mandatory to consider the interference term from BH events. The production of BH events is a pure QED process, which can be measued precisely from the form factor F 1 and F 2 . In addition to the absolute cross section, another way to obtain the CFF is by measuring the asymmetries. The beam charge asymmetries are defined as:
A C = σ + (ϕ) − σ − (ϕ) σ + (ϕ) + σ − (ϕ) ,(19)
where σ + and σ − refer to cross sections with lepton beams of opposite charge. We can see that the asymmetries only depends on ϕ. The observables of interest in this paper are the correlated charge and transversely polarized target-spin asymmetries, defined as:
A U T,DV CS = (σ + + (ϕ)−σ + − (ϕ))+(σ − + (ϕ)−σ − − (ϕ)) σ + + (ϕ)+σ + − (ϕ)+σ − + (ϕ)+σ − − (ϕ) , A U T,I = (σ + + (ϕ)−σ + − (ϕ))−(σ − + (ϕ)−σ − − (ϕ)) σ + + (ϕ)+σ + − (ϕ)+σ − + (ϕ)+σ − − (ϕ) ,(20)
where A with subscripts denote the cross section asymmetries of ep → e ′ p ′ γ at certain beam (first subscript) and target (second subscript) polarization sign ("U" stands for unpolarized and "T" for transverse polarized). Note that there are two independent transverse polarization direction of proton: U T x is in the hadronic plane and U T y is perpendicular to it. There, the uperscript and subscript of σ refers to the charge of the lepton beam and beam (or target) spin projection. One can measure exclusive ep → e ′ p ′ γ cross section with different beam and target polarization since the spin asymmetries give the access to different CFFs through the interference term I, the BH and DVCS process. At leading-order and leading-twist, the relation linking observables and CFFs for ep → e ′ p ′ γ process have been derived as [32,71,72]:
A sin(ϕ−ϕs) UT, DVCS ∝ Im (HE * ) − ξ Im H E * ,(21)A sin(ϕ−ϕs) cos ϕ UT,I ∝ Im − t 4M 2 (F 2 H − F 1 E) + ξ 2 F 1 + t 4M 2 F 2 (H + E) −ξ 2 (F 1 + F 2 ) H + t 4M 2 E .(22)
These approximations illustrate that different experimental observables are sensitive to different CFFs. We can see that the above asymmetries have dependence on CFF E, which is important implication for our following study of the total angular momentum of different quarks within the proton.
III. PHENOMENOLOGICAL PARAMETRIZATION OF GPDS
Assuming a factorized t-dependence, the quark GPD H q is given by [26]:
H q (x, ξ, t) = H q (x, ξ) · F q 1 (t).(23)
The nucleon form factors in dipole form is given by:
F dipole 1 (t) = 1 − 1 + κ P t/4m 2 N 1 − t/4m 2 N 1 (1 − t/0.71) 2 . (24) For the function H q (for each flavor q), the t-independent part H q (x, ξ) ≡ H q (x, ξ, t = 0) is parametrized by a two- component form, H q (x, ξ) ≡ H q DD (x, ξ, t = 0) + θ(ξ − |x|)D q x ξ ,(25)
where D q x ξ is the D-term, set to 0 in our following calculation. And H q DD is the part of the GPD which is obtained as a one-dimensional section of a two-variable double distribution (DD) F q , imposing a particular dependence on the skewedness ξ,
H q DD (x, ξ) = 1 −1 dβ 1−|β| −1+|β| dαδ(x − β − αξ)F q (β, α).(26)
For the double distributions, entering Eq. 26, we use the following model,
F q (β, α) = h(β, α)q(β),(27)
where q(β) is the forward quark distribution (for the flavor q) and where h(β, α) denotes a profile function. In the following estimates, we parametrize the profile function through a one-parameter ansatz, following [26,62,63]:
h(β, α) = Γ(2b + 2) 2 2b+1 Γ 2 (b + 1) (1 − |β|) 2 − α 2 b (1 − |β|) 2b+1 .(28)
For β > 0, q(β) = q val (β) +q(β) is the ordinary PDF for the quark flavor q. In this work, we use IMParton as input [73]. The negative β range corresponds to the antiquark density: q(−β) = −q(β). The parameter b characterizes to what extent the GPD depends on the skewness ξ, and fixed to 1 in this work. The spin-flip quark GPDs E q in the factorized ansatz are given by:
E q (x, ξ, t) = E q (x, ξ) · F q 2 (t)/κ q .(29)
Here F q 2 (t) denotes the Pauli FF for quark flavor q, and is parameterized by:
F q 2 = κ q 1 − t/4m 2 p · (1 − t/m 2 D ) 2 ,(30)
where κ q is the anomalous magnetic moment of quarks of flavor q, κ u = 2κ p + κ n = 1.67, κ d = κ p + 2κ n = −2.03. Same as Eq. 25, the t-independent part of the quark GPDs, E q (x, ξ) is defined as:
E q (x, ξ) = E DD q (x, ξ) − θ(ξ − |x|)D q x ξ .(31)
The part of the GPD E that can be obtained from the double distribution has a form analogous to the spinnonflip case:
E DD q (x, ξ) = 1 −1 dβ 1−|β| −1+|β| dαδ(x − β − αξ)K q (β, α),(32)there, K q (β, α) is given by: K q (β, α) = h(β, α)e q (β),(33)
and e q (β) denotes the spin-flip can be written as:
e q (x) = A q · q val (x) + B q · δ(x),(34)
with:
A q = 2J q − M (2) q M (2) q val , B u = 2 1 2 κ u − 2J u − M (2) u M (2) u val , B d = κ d − 2J d − M (2) d M (2) d val .(35)
By defining the total fraction of the proton momentum carried by the quarks and antiquarks of flavor q as:
M q 2 = 1 0 dxx[q(x) +q(x)] = 1 0 dxx [q val (x) + 2q(x)] ,(36)
and the momentum fraction carried by the valence quarks as:
M q val 2 = 1 0 dxxq val (x).(37)
The parameterizations of H and E are introduced in [26,27,64,65]. While parameterization of H, we use polIMParton as input [74]. In this model, the total angular momentum carried by u-quarks and d-quarks, J u and J d , are free parameters in the parameterization of the spin-flip GPD E q (x, ξ, t). Therefore, this parameterization can be used to study the sensitivity of hard electroproduction observables to variations in J u and J d .
IV. DISTRIBUTIONS OF INVARIANT AND FINAL-STATE KINEMATICS
There is a package of Monte-Carlo (MC) simulations of DVCS and BH processes called MILOU [75]. We use this software to generate 5 million events of EicC and EIC. We use the PARTONS (PARtonic Tomography Of Nucleon Software) package as the observables input [76]. Thus, we can make some pseudo data for subsequent theoretical calculations. We focus on two future experiments (EIC and EicC), and assume the beam energy of incoming electron and incoming proton with E e = 3.5 GeV, E p = 20 GeV at EicC [77], E e = 5 GeV, E p = 100 GeV at EIC [78]. We propose to do the measurement of spin azimuthal asymmetries in deeply virtual Compton scattering on transverse polarized proton. Besides the scattered electron, real photon and the scattered proton will be measured after the incoming unpolarized electron. Transverse Target-Spin Asymmetry (A sin(ϕ−ϕs) cos ϕ U T ) will be extracted from the data. The EicC facility can offer the beam integrated luminosity up to 50 fb −1 , which corresponds to the effective running time within one year [77]. EicC also has a large kinematic acceptance capacity, which can complement the current vacant data. Compared to EicC, EIC offer the beam integrated luminosity up to 60 fb −1 in less running time [78,79]. Combining with the EIC and EicC experiments, high precision data of most kinematic regions will be availabled.
In order to efficiently generate the events in the kinematic region of interests, we apply the following kinematical ranges for the Monte-Carlo sampling: 10 −4 < x B < 1, 1 GeV 2 < Q 2 < 100 GeV 2 , and 10 −3 GeV 2 < −t < 3 GeV 2 . Fig. 3 and Fig 4 show the coverage of the momentum vs polar angles for final state electrons, real photons and scattered protons coming from DVCS and BH process at EicC and EIC. We see that the final proton having a large fraction of the momentum of the incoming proton and a small scattering angle. Especially, most protons locate at very small polar angles, and the momentum difference with beam is so small that we need very good momentum resolution for the forward detector. The final electron having a larger scattering angle than the final proton. According to the distribution of the final state particles, we can place the detectors appropriately to collect more valid examples. Fig. 5 and Fig. 6 show the cross-section weighted invariant kinematics distributions of ep → e ′ p ′ γ reaction at EicC and EIC. These color z-axis distribution were weighted by the cross section computed in VGG model built in the MILOU software and shown in Log z scale. We can see that, the range of Q 2 covers from 1.0 GeV 2 to 10.0 GeV 2 , x B lies between 0.003 and 0.05, and t goes from 0 down to -0.2 GeV 2 , most of the events are in this area. Comparing the results of EicC and EIC, we can see that EIC has more data in the smaller x B and smaller −t region than EicC.
V. PROJECTION OF DVCS EXPERIMENT
The statistical uncertainty of the measured experimental observable is directly related to the number of events collected during an experiment. To estimate the number of events of an experiment, we need to know the cross section of the reaction, the integrated luminosity of the experiment, and the events selection criteria of the reaction. EIC yields an integrated luminosity of 1.5 fb −1 per month [78]. We assume the integrated luminosity of the experiment of EicC to be 50 fb −1 , which takes three to four years. The integrated luminosity of EIC is assumed to be 60 fb −1 about three years. To make sure the collected events are valid for our study, we have applied the following conditions for the event selection: 0.01 < y < 0.85, t > −0.5 GeV 2 , W > 2.0 GeV, P e ′ > 0.5 GeV. Fig. 7 shows the kinematic regions of EIC and EicC, which is the simulated region in this work. EIC and EicC will provide data in small x region. Red area is indicating EIC and green area is indicating EicC. In the small Q 2 region, EicC can provide data, where x is close to x ∼ 0.005. Since EIC has higher center-of-mass energy, it can provide data for more smaller x-region in the range of x ∼ 0.0007. DVCS experiment poses strong challenges to us on the detection of recoiled proton with small t. In order to make sure that the recoiled proton can be detected by forward detector, we assumed some constraints on the detection of final state protons. This low-t acceptance eliminates many forward events, taking EicC as an example (Fig. 8). Based on the event selection criteria discussed above, the number of events in each bin is calculated with the following formula, (38) where N is the total events in each kinematical bins, σ avg is the average of the four cross section with different electron and proton beam polarization directions, "Lumi" is the beam luminosity, "Time" is the beam duration, and ϵ eff is the overall efficiency of detector, and the rest denotes the sizes of the kinematical bins. In this work, we conservatively assumed an acceptance of final state particles, which is 25 % at EIC and 20 % at EicC [77,78].
N = σ avg · Lumi · Time · ϵ eff · ∆x B · ∆t · ∆Q 2 ,
The counts of events in each bin is denoted as N ++ , N +− , N −+ , and N −− , corresponding to different electron and nucleon polarization directions. One can obtain the asymmetries quantities of the target spin asymmetry (A T S ):
A T S = N ++ + N −+ − N +− − N −− N ++ + N +− + N −+ + N −− 1 P T ,(39)
where P T stands for the polarization degree of nucleon (assumed as 70 %) [77,78]. Considering that the asymmetries quantities are in several percent level, we use the unpolarized events by MILOU to do the projection, and the total event number of all polarization conditions is denoted as N . Thus the absolute statistical uncertainty of the asymmetries quantities can be expressed approximately as: Fig. 9 and Fig. 10 show the statistical errors projection in a low Q 2 bin between 1 and 3 GeV 2 for EicC and EIC experiments. We focus on small x B and −t region, and divide the x B vs. −t plane into very small bins. We see in these plots that the statistical uncertainty goes up with x B increasing. For most of the data at EicC and EIC, the projected statistical uncertainty is smaller than 3 %. When x B increasing to around 0.12, the statistical uncertainty is around 5 %. These precise data will be of great help to theoretical research in the future. Now we can give the pseudo-data of the asymmetry of the crosssection in the area of interest at EicC and EIC. We divide x B , t, and Q 2 in different bins, show in Tab. I. This table corresponds to the Fig. 11 and Fig. 12. For the case where only x B , t or Q 2 changes, we applied a similar division approach. Here x B ranges from 0.01 to 0.17 in steps of 0.02 (t : −0.11 ∼ −0.09 GeV 2 , Q 2 : 1.13 ∼ 1.38 GeV 2 ), t ranges from -0.19 GeV 2 to -0.03 GeV 2 in steps of 0.02 (x B : 0.01 ∼ 0.03, Q 2 : 1.13 ∼ 1.38 GeV 2 ) and Q 2 ranges from 1.13 GeV 2 to 3.13 GeV 2 in steps of 0.25 (x B : 0.01 ∼ 0.03, t : −0.11 ∼ −0.09 GeV 2 ). As shown in Fig. 11, EicC provides large phase space coverage and good statistics, especially for small x B ,−t and Q 2 regions. The similar results at EIC [83] are shown in Fig. 12. Since we also divide the Q 2 into small bins, the statistical errors of pseudo-data in Fig. 11 and Fig. 12 are much larger than those shown in Fig. 9 and Fig. 10. We develop a code to calculate observables in the exclusive reaction ep → e ′ p ′ γ to LO precision in perturbative theory. This calculation follows the VGG model described in Sec. III. In order to compare the results from theoretical calculations with the TTSA amplitudes pseudo data in Fig. 13, the χ 2 exp is defined as: There we need to consider the systematic errors. Based on the previous experiments [31,, we make a conservative estimate for EicC and EIC. Thus, for EicC and EIC, we assume experimental systematic errors are 10 %. The constraints on J u and J d obtained for the extracted TTSA amplitudes from the pseudo data are shown in Fig. 13. We calculate the TTSA amplitudes for J u (J d ) ranging from 0 to 1 (-1 to 1) in steps of 0.2, and set the D-term = 0 (D q x ξ in Eq. 25). Fig. 14 shows the model-dependent constraint on u-quark total angular momentum J u vs d-quark total angular momentum J d in the same kinematic region as HERMES [68,69]. Here we only consider the influences from statistical errors. The result of EicC, which is shown in Fig. 14, can be expressed as
δA T S ≈ 1 P T 1 √ N .(40)χ 2 exp (J u , J d ) =J u + J d /2.9 = 0.41 ± 0.06,(42)
and the result of EIC is
J u + J d /3.0 = 0.39 ± 0.04.(43)
If we consider both statistical and systematic errors (A sin(ϕ−ϕs) cos ϕ U T = −0.142 ± 0.020 ± 0.014 at EicC, A sin(ϕ−ϕs) cos ϕ U T = −0.020 ± 0.002 ± 0.002 at EIC), the result (shown in Fig. 15) is
J u + J d /2.9 = 0.41 ± 0.08,(44)
for EicC, and
J u + J d /3.0 = 0.39 ± 0.06.(45)
for EIC. The uncertainty is propagated from the TTSA amplitudes uncertainty of the pseudo data, and experimental systematic errors dominate. According to the results of HERMES [68,69,84],
J u + J d /2.9 = 0.42 ± 0.21,(46)
we ignore the effects of parameter b and D-term. As the Fig. 15 shows, EicC and EIC have higher accuracy to obtain smaller uncertainty for constraint on u-quark and d-quark total angular momentum. Since EIC and EicC can provide a large amount of accurate data in the small x region, we performed some calculations in this region. Both statistical and systematic errors are considered in these results. At x = 0.01, the results of EicC and EIC are shown in Fig. 16, where EicC is
J u + J d /2.6 = 0.39 ± 0.05,(47)
and EIC is
J u + J d /2.7 = 0.38 ± 0.05.(48)
In the smallest x area that EicC can provide, we obtained the flowing results, where is the result of EicC shown in Fig. 17. The result of EIC in this kinematic region is
J u + J d /2.5 = 0.38 ± 0.05,(49)J u + J d /2.5 = 0.39 ± 0.05.(50)
As Fig. 7 shows, EIC also provides accurate data in the area of x ∼ 0.002. In this very small x region, we present the result of EIC,
J u + J d /2.4 = 0.35 ± 0.04,(51)
which is shown in Fig. 18. precise experiments, it is difficult for theoretical work to move forward. These precise experimental data will help us gain a deeper understanding of the nucleon structure in the future.
VI. DISCUSSIONS AND SUMMARY
The internal structure of the nucleon is mysterious, and we explore it by various methods. After the EMC experiment, the researchers conducted many detailed studies of nucleon spins. The proposed GPDs theory opens new paths for the study of the three-dimensional structure and spin of nucleon. By Ji's sum rule, we find that GPDs are directly related to the total angular momentum carried by the partons. DVCS experiments are a good choice to obtain GPDs, although not quite directly extracted. In contrast to the great progress in studying GPDs on the theoretical side, relatively little progress has been made on the experimental side. Because the experiment requires high statistical accuracy, this means that extremely good detectors and very high luminosity are required.
In this work, we simulated the DVCS process at EicC and EIC to study the internal structure of proton. The statistical errors of these two future experiments are predicted. According to the very small statistical errors, we find that the measurement accuracy of future DVCS experiment will be limited mostly by systematical errors. It seems that the accuracy of the EIC and EicC data will be greatly improved in the future when compared with the existing real data from different experiment groups. Advanced experiment equipment to reduce systematic errors and better detection of final state particles to reduce statistical errors. We believe that future EicC and EIC experiments will yield more accurate data than those predicted in this work. This has significant implications for future experimental studies of the internal structure of nucleon. With the excellent detectors and high accelerator luminosity, DVCS experiments at EicC and EIC will have a bright prospect.
Based on the EIC and EicC measurements of TTSA high-precision pseudo-data, we can have a good study of the nucleon helicity-flip GPD E. Through the VGG model, the GPD E is parameterized by the total angular momentum of the up and down quarks in the nucleon. With this model we combine DVCS experiments with nu-cleon spin studies. According to the HERMES and JLab experiments constraint on the total angular momentum of quarks in the proton and neutron, we constraint on the total angular momentum carried by up quarks and down quarks inside the proton in future EIC and EicC experiments. There are different GPD models based on experimental and theoretical research to study the mysterious nucleon structure. Current research relies on models too heavily, we look forward to more precise experimental data to verify these theoretical research in the future.
FIG. 1 .
1The Feynman diagram of DVCS (left) and BH (right)
FIG. 2 .
2The reference frame of scattering plane and kinematic variables of ep → e ′ p ′ γ reaction in the laboratory[25].
FIG. 3 .
3The cross-section weighted momentum and polar angles distributions of the final-state particles (scattered protons, scattered electrons and real photons) in the MC simulation at EicC.
FIG. 4 .
4The cross-section weighted momentum and polar angles distributions of the final-state particles (scattered protons, scattered electrons and real photons) in the MC simulation at EIC.
FIG. 5 .FIG. 6 .
56The cross-section weighted distributions of the invariant kinematics in the MC simulation at EicC. The cross-section weighted distributions of the invariant kinematics in the MC simulation at EIC.
FIG. 7 .FIG. 8 .
78Kinematic range in the x, Q 2 plane at EicC ( √ s = 16.7 GeV) and EIC ( √ s = 45 GeV)[80][81][82]. The hatched areas indicate the areas simulated in this work, which correspond to 0.01 ≤ y ≤ 0.85. The red dashed line and green dashed line indicate y = 0.6. The cross-section weighted momentum and polar angles distributions of the scattered protons with the geometric cut. The square breach at the right side shows the eliminated data with proton momentum larger than 99 % of beam momentum and scattering angle smaller than 2 mrad.
FIG. 9 .FIG. 10 .
910The statistical errors projection of the Transverse Target-Spin Asymmetry at low Q 2 at EicC. We calculate the statistical errors at each bin center. The right axis shows how large the statistical errors are. The statistical errors projection of the Transverse Target-Spin Asymmetry at low Q 2 at EIC. We calculate the statistical errors at each bin center. The right axis shows how large the statistical errors are.TABLE I. Binning scheme for xB, t, and Q 2 . xB t (GeV 2 ) Q 2 (GeV 2 )
FIG. 11 .FIG. 12 .
1112xB t (GeV 2 ) Q 2 (GeV 2 ) A sin(ϕ−ϕs) cos ϕ Asymmetries with polarized electron beam and proton beam in some typical bins at EicC. Asymmetries with polarized electron beam and proton beam in some typical bins at EIC.
FIG. 13 .
13Asymmetries with polarized electron beam and proton beam in small x region at EicC (Tab. II) and EIC (Tab. III).
FIG. 14 .
14The result of model-dependent constraint on u-quark total angular momentum Ju vs d-quark total angular momentum J d at EIC and EicC compared with HERMES[68,69]. Only statistical errors are considered.FIG.15. The result of model-dependent constraint on u-quark total angular momentum Ju vs d-quark total angular momentum J d at EIC and EicC compared with HERMES[68,69]. Both statistical and systematic errors are considered.
FIG. 16 .FIG. 17 .
1617The results of EicC and EIC are both within the error range of HERMES and both have small errors. Without The result of model-dependent constraint on u-quark total angular momentum Ju vs d-quark total angular momentum J d in the region of x ∼ 0.01 at EIC and EicC. Both statistical and systematic errors are considered. The result of model-dependent constraint on u-quark total angular momentum Ju vs d-quark total angular momentum J d in the region of x ∼ 0.006 at EIC and EicC. Both statistical and systematic errors are considered.
FIG. 18 .
18The result of model-dependent constraint on u-quark total angular momentum Ju vs d-quark total angular momentum J d in the region of x ∼ 0.002 at EIC. Both statistical and systematic errors are considered.
TABLE III .
IIIAsymmetries with polarized electron beam and
proton beam at EIC.
xB
t (GeV 2 ) Q 2 (GeV 2 ) A
sin(ϕ−ϕs) cos ϕ
U T
± stat
0.002
0.10
1.25
-0.225±0.005
0.006
0.10
1.25
-0.172±0.008
0.01
0.10
1.25
-0.121±0.007
0.1
0.12
2.50
-0.020±0.002
3
−
ACKNOWLEDGMENTSWe thank Prof. J. P. Chen and Dr. Korotkov for suggestions and discussions. This work is supported by the Strategic Priority Research Program of Chinese Academy of Sciences under the Grant NO. XDB34030301.
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| []
|
[
"SIGN CHANGING BUBBLE TOWER SOLUTIONS TO A SLIGHTLY SUBCRITICAL ELLIPTIC PROBLEM WITH NON-POWER NONLINEARITY",
"SIGN CHANGING BUBBLE TOWER SOLUTIONS TO A SLIGHTLY SUBCRITICAL ELLIPTIC PROBLEM WITH NON-POWER NONLINEARITY"
]
| [
"Shengbing Deng ",
"Fang Yu "
]
| []
| []
| We study the following elliptic problem involving slightly subcritical non-power nonlinearitywhere Ω is a bounded smooth domain in R n , n ≥ 3, 2 * = 2n n−2 is the critical Sobolev exponent, ε > 0 is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as ε goes to zero.2020 Mathematics Subject Classification. 35B33; 35B40; 35J15. | null | [
"https://export.arxiv.org/pdf/2306.02973v1.pdf"
]
| 259,075,611 | 2306.02973 | f2a245d3d614102468c739e5d3b4b84ccbb76074 |
SIGN CHANGING BUBBLE TOWER SOLUTIONS TO A SLIGHTLY SUBCRITICAL ELLIPTIC PROBLEM WITH NON-POWER NONLINEARITY
5 Jun 2023
Shengbing Deng
Fang Yu
SIGN CHANGING BUBBLE TOWER SOLUTIONS TO A SLIGHTLY SUBCRITICAL ELLIPTIC PROBLEM WITH NON-POWER NONLINEARITY
5 Jun 2023
We study the following elliptic problem involving slightly subcritical non-power nonlinearitywhere Ω is a bounded smooth domain in R n , n ≥ 3, 2 * = 2n n−2 is the critical Sobolev exponent, ε > 0 is a small parameter. By the finite dimensional Lyapunov-Schmidt reduction method, we construct a sign changing bubble tower solution with the shape of a tower of bubbles as ε goes to zero.2020 Mathematics Subject Classification. 35B33; 35B40; 35J15.
Introduction
In this paper, we consider the following elliptic problem involving slightly subcritical non-power nonlinearity −∆u = |u| 2 * −2 u [ln(e+|u|)] ε in Ω, u = 0 on ∂Ω,
(1.1)
where Ω is a bounded smooth domain in R n , n ≥ 2, 2 * = 2n n−2 is the critical Sobolev exponent for the embedding H 1 0 (Ω) ֒→ L 2 * (Ω), ε > 0 is a small parameter. The main feature of problem (1.1) is the non-power type nonlinearity, which is first proposed by Castro and Pardo [12], they proved the existence of a priori L ∞ bounds for positive solutions of Laplacian problem involving the nonlinearity f (u) = u n+2 n−2 ln(2+u) α with α > 2 n−2 . Then, Mavinga and Pardo [36] obtained a priori estimates for positive classical solutions to the following Hamiltonian elliptic system
−∆u = v p [ln(e+v)] α in Ω, −∆v = u q [ln(e+u)] β
in Ω,
u = v = 0 on ∂Ω,
where Ω is a bounded convex domain with smooth boundary in R n for n > 2, 1 < p, q < ∞ and α, β > 0, 1 p+1 + 1 q+1 = n−2 n . For more results of non-power nonlinearity, we refer to [13,22,39] for slightly subcritical problem and [26] for supercritical problem.
On the one hand, problem (1.1) is related to the following slightly subcritical elliptic problem −∆u = |u| 2 * −2−ε u in Ω, u = 0 on ∂Ω. (1.2) When ε = 0, Pohozaev [41] proved that the non existence of nontrivial solution if Ω is a starsharped domain. When Ω is a annulus, Kazdan and Warner [33] obtained the existence of a positive radial solution. Bahri and Coron [1] studied a positive solution provided that Ω has nontrivial topology. For the existence of sign changing solution, there are few results. When Ω = R n , Ding [27] showed the infinitely many sign changing solutions by Ljusternik-Schnirlman category theory. In specific case like torii, Hebey and Vaugon [32] investigated the existence and multiple sign changing solutions. The existence and multiplicity of sign changing solutions are also treated in some contractible domains with an involution symmetry by Clapp and Weth [19].
When ε is a positive parameter, problem (1.2) has a positive least energy solution u ε , that is, u ε is a solution for the variational problem inf u 2 =ˆΩ |∇u| 2 dx : u ∈ H 1 0 (Ω),ˆΩ |u| 2 * −ε dx = 1 .
The blow-up phenomenon for positive and sign changing solutions to (1.2) has been studied extensively. When ε goes to 0, Rey [42] and Han [31] studied that the solution to (1.2) blows up and concentrates at a critical point of Robin function. Moreover, Flucher and Wei [28] proved that the concentration point is the minimum point of the Robin function. Furthermore, if ξ * is a stable critical point of Robin function, then (1.2) has a positive solution which blows up at ξ * , this result is obtained in [37,42]. For the multiple concentration points, Rey [43] showed that the two blow up and concentration points (ξ * 1 , ξ * 2 ), which is a critical point of a function involving Robin function and Green's function. If the domain is convex, Grossi and Takahashi in [30] proved that (1.2) does not admit any positive solution blowing up more than two points. The positive solution to (1.2) concentrate simultaneously at different points ξ 1 , · · · , ξ k ∈ Ω, k ≥ 2, has been established in [2,37]. If any ξ i , i = 1, · · · , k, is a simple blow up point, Li [34] characterized the form of solution u ε near each blow up point ξ i as
u ε (x) ∼ µ i √ ε (µ 2 i ε + |x − ξ i | 2 )
n− 2 2 , with µ i > 0.
On the other hand, the existence of one sign changing solution to (1.2) is first proved in [5,11], and multiple sign changing solutions with their nodal properties are treated in [6,7] for ε ∈ (0, 4 n−2 ). Moreover, they proved that (1.2) has a least energy nodal solution with two nodal domains. Ben Ayed et al. [8] obtained that the low energy sign-changing solutions blow up at two points, and the energy converges to the value 2S n 2 , where S is the Sobolev constant for the embedding H 1 0 (Ω) into L 2n n−2 (Ω). Bartsch et al. [4] considered that (1.2) has k pairs of sign changing solutions ±u (i) ε , i = 1, · · · , k, which satisfies that u [3] proved a sign changing four-bubble solution with two positive and two negative blow-up points provided that Ω is convex and satisfies some symmetry conditions.
In contrast to the result of positive and sign changing solutions to (1.2), there are some papers of bubble tower. If Ω is a smooth bounded domain in R n symmetric with respect to x 1 , · · · , x n and contains the origin, Pistoia and Weth [40] constructed a sign changing bubble tower solution u ε concentrating at the center of symmetry of Ω. The same consequence in any bounded smooth domain is considered in [38], and they removed the assumption on non-degeneracy of critical point of Robin's function. If the domain has holes like Ω \ B(a, ε) ∪ B(b, ε) with center at point a, b and radius ε > 0, Ge et al. [29] constructed sign changing solutions blowing up both at a and b. For any other bubble tower results of elliptic problem, see [14,15,17,20,24] and references therein. In particular, we refer to the papers [21,23] for fractional and biharmonic operators involving almost critical Soblev exponent.
Recently, by Lyapunov-Schmidt reduction method, Clapp et al. [9,18] constructed solutions to problem (1.1). Before stating the results, let us introduce some definitions and notations. For ξ ∈ Ω and µ > 0, let
(1.3) U (x) = α n 1 (1 + |x| 2 ) n−2 2 , U µ,ξ (x) = α n µ n−2 2 (µ 2 + |x − ξ| 2 ) n−2 2 , with α n = (n(n − 2)) n−2 4 ,
which are the only solutions of the equation
(1.4) −∆u = u 2 * −1 , u > 0 in R n .
Let us denote by G(x, y) the Green's function of −∆ in Ω with Dirichlet boundary condition, and by H(x, y) its regular part, so that
H(x, y) = 1 (n − 2)|∂B| 1 |x − y| n−2 − G(x, y) , for every x, y ∈ Ω,
where |∂B| denotes the surface area of the unit sphere in R n . The Robin function is defined as ϕ(x) = H(x, x) for every x ∈ Ω. Let ξ * is a non-degenerate critical point of Robin function, Clapp et al. [18] constructed a single bubble solution of the form [35] established a solution concentrating at the origin point for a critical Hénon problem with non-power type. Ben Ayed et al. [9] obtained positive as well as sign changing solutions concentrating at several points, which involving Robin function and Green's function.
u ε = U µε,ξε + φ ε , with µ ε | ln ε| ε 1 n−2 → d > 0, ξ ε → ξ * , φ ε ∈ H 1 0 (Ω) such that´Ω |∇φ ε | 2 dx = O ε | ln ε| as ε → 0. Liu et al.
In present paper, motivated by several results [9,18,38,40], we construct a solution with the shape of a tower of sign changing bubbles to problem (1.1) by finite Lyapunov-Schmidt dimensional reduction procedure.
Our result can be stated as follows.
Theorem 1.1. Assume that n ≥ 3, for any integer k ≥ 1, there exists ε 0 > 0 such that for every ε ∈ (0, ε 0 ), there are some points ξ iε ∈ Ω and positive constants d iε for i = 1, · · · k, problem (1.1) has a solution u ε of the form,
u ε (x) = α n k i=1 (−1) i d iε ( ε | ln ε| 2 ) 2i−1 n−2 d iε ( ε | ln ε| 2 ) 2i−1 n−2 2 + |x − ξ iε | 2 n−2 2 + Θ ε (x), where Θ ε → 0 as ε → 0, ϕ(ξ iε ) → min z∈Ω ϕ(z) and d iε → d i > 0 for i = 1, · · · k.
Observe that in the above construction, the solutions behaves like a superposition of bubbles of different blow up orders centered at around the minimum point of the Robin function, thus, it is called bubble tower solutions. It was first studied by del Pino et al. [25] for a slightly supercritical Brezis-Nirenberg problem in a ball, and this type solutions has been constructed in many problems, see [16,21,23,29,38,40] and references therein.
The paper is organized as follows. In Section 2, we give the scheme of proof for Theorem 1.1. We show the finite dimensional reduction process in Section 3. Proposition 2.2 is proved in Section 4. Finally, there are some estimates in the Appendix. We will use C > 0 to denote various positive constants.
Scheme of the proof
In this section, let us give the sketch proof of Theorem 1.1. We first introduce some notations. The Sobolev space H 1 0 (Ω) is endowed with inner product ·, · defined by
u, v =ˆΩ ∇u∇vdx, for all u, v ∈ H 1 0 (Ω), and L p (Ω) is the Lebesgue space with the norm |u| q = ´Ω |u| q dx 1 q , for 1 < q < ∞. Let i * : L 2n n+2
(Ω) ֒→ H 1 0 (Ω) be the adjoint operator of the embedding i :
H 1 0 (Ω) ֒→ L 2n n−2 (Ω), that is, for v ∈ L 2n n+2 (Ω), u = i * (v) if and only if −∆u = v in Ω, u = 0 on ∂Ω.
Then, it holds
(2.1) i * (v) ≤ c|u| 2n n+2 ,
for some constant c > 0 depending only on Ω and n. Using these definitions and notations, problem (1.1) is equivalent to the following equation
u = i * [f ε (u)], u ∈ H 1 0 (Ω), where f ε (u) = |u| 2 * −2 u [ln(e+|u|)] ε .
In order to describe the shape of the solutions to problem (1.1), we give an integer number k, and define the positive parameters µ i as
(2.2) µ i = ε | ln ε| 2 2i−1 n−2 d i , with d i > 0, i = 1, · · · , k.
Let ξ be a point in Ω, ξ i ∈ Ω, i = 1, · · · , k, is given by
(2.3) ξ i = ξ + µ i σ i , for some points σ i ∈ R n ,
where σ k = 0. We will assume the following bounds on the parameters and points appearing in (2.2) and (2.3): given η > 0 small,
(2.4) dist(ξ, ∂Ω) > η, η < d i < 1 η , |σ i | ≤ 1 η , i = 1, · · · , k.
It is an immediate observation that
µ 1 = ε | ln ε| 2 1 n−2 d 1 and µ i+1 µ i = ε | ln ε| 2 2 n−2 d i+1 d i .
We denote by P U µ,ξ the projection onto
H 1 0 (Ω) of U µ,ξ , that is −∆P U µ,ξ = −∆U µ,ξ in Ω, P U µ,ξ = 0 on ∂Ω.
Let k ≥ 1, the approximate solutions are given by
(2.5) u(x) = V (x) + φ(x), V (x) = Vd ,σ,ξ (x) = k i=1 (−1) i P U µ i ,ξ i (x),
where (2.6)d = (d 1 , · · · , d k ) ∈ R k + ,σ = (σ 1 , · · · , σ k ) ∈ (R n + ) k . The term φ is small in some sense. Let us describe φ. As it is shown in [10], any solution of (2.7)
−∆ψ = f ′ 0 (U µ,ξ )ψ in R n , can be expressed as a linear combination of
(2.8) ψ 0 (y) = (n − 2)α n 2 |y| 2 − 1 (1 + |y| 2 ) n 2 , ψ h (y) = (n − 2)α n y h (1 + |y| 2 ) n 2 , for h = 1, · · · , n.
Moreover, we set
ψ 0 µ,ξ (x) = µ − n−2 2 ψ 0 ( x − ξ µ ) = n − 2 2 α n µ n−2 2 |x − ξ| 2 − µ 2 (µ 2 + |x − ξ| 2 ) n 2 , ψ h µ,ξ (x) = µ − n−2 2 ψ h ( x − ξ µ ) = (n − 2)α n µ n 2 x h − ξ h (µ 2 + |x − ξ| 2 ) n 2 , for h = 1, · · · , n, (2.9) then (2.10) ψ 0 µ,ξ (x) = µ ∂U µ,ξ ∂µ , ψ h µ,ξ (x) = µ ∂U µ,ξ ∂ξ h .
We denote that P ψ h µ,ξ is the projection of ψ h µ,ξ , h = 0, · · · , n, and define the subspace of H 1 0 (Ω)
E µ,ξ = span P ψ h µ,ξ : h = 0, 1, · · · , n, i = 1, · · · , k , E ⊥ µ,ξ = φ ∈ H 1 0 (Ω) : φ, P ψ h µ,ξ = 0 : h = 0, 1, · · · , n, i = 1, · · · , k . Let Π µ,ξ : H 1 0 (Ω) → E µ,ξ and Π ⊥ µ,ξ : H 1 0 (Ω) → E ⊥ µ,ξ ,
be the corresponding projections. To solve (1.1), it is equivalent to solve the couple of following equations
(2.11) Π ⊥ µ,ξ V + φ − i * [f ε (V + φ] = 0, and (2.12) Π µ,ξ V + φ − i * [f ε (V + φ)] = 0.
We solve equation (2.11) in the following result, whose proof can be found in Section 3.
Proposition 2.1. There exists ε 0 > 0 such that for any ξ ∈ Ω,d ∈ R k + ,σ ∈ (R n + ) k satisfying (2.4), for ε ∈ (0, ε 0 ), there is a unique function φ ∈ E ⊥ µ,ξ which solves (2.11). Moreover
φ = O ε | ln ε| 2 ln ε | ln ε| 2 ln ln ε | ln ε| 2 if 3 ≤ n ≤ 6, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 7.
(2.13) From Proposition 2.1, there is a unique φ ∈ E ⊥ µ,ξ such that (2.11) holds, it means that there are some constants c il s (i = 1, · · · , k, l = 0, · · · , n) such that (2.14)
V + φ − i * [f ε (V + φ)] = k i=1 n l=0 c il P ψ l µ i ,ξ i ,
which equals to solve equation (2.12), that is, the following result is valid, whose proof is postponed to Section 4.
V + φ − i * [f ε (V + φ)], P ψ h µ jε ,ξ jε = 0, for h = 0, · · · , n, (2.15) where j = 1, · · · , k. Then V + φ is a solution of problem (1.1). Part b. For ξ ∈ Ω,d = (d 1 , · · · , d k ) ∈ R k + ,σ = (σ 1 , · · · , σ k ) ∈ (R n + ) k , there holds V + φ − i * [f ε (V + φ)], P ψ h µ j ,ξ j = ε | ln ε| 2 G ε 0 (d,σ, ξ) − 2k 2 (n−2) 2 a 4 ε ln ε | ln ε| 2 for h = 0, ε | ln ε| 2 G ε h (d,σ, ξ) for h = 1, · · · , n, where j = 1, · · · , k, and G ε = (G ε 0 , G ε h ) is given by G ε 0 (d,σ, ξ) = α n a 1 d n−2 1 ϕ(ξ) + a 3 k−1 i=1 d i+1 d i n−2 2 g(σ i ) − a 4 k i=1 2 2i−1 | ln d i | + o(1), G ε h (d 1 , ξ) = αn 2 a 2 ∂ ξ h ϕ(ξ)d n−1 1 for h = 1, · · · , n, with G ε 0 : [0, +∞] × [0, +∞] × Ω → R × R × R n , G ε h : [0, +∞] × Ω → R × R n and a 1 = (2 * − 1)ˆR n U 2 * −2 (y)ψ 0 (y)dy, a 2 =ˆR n U 2 * −1 (y)dy, a 3 = n − 2 2 α p+1 n , a 4 =ˆR n 1 (1 + |y − σ i | 2 ) n+2 2 ln 1 (1 + |y − σ i | 2 ) n+2 2 ψ 0 (y) dy > 0, g(σ) =ˆR n y 2−n (1 + |y − σ| 2 ) n+2 2
dy.
From Propositions 2.1 and 2.2, we view that V + φ is the solution to problem (1.1) if there are d ε > 0, σ ε > 0 and ξ ε ∈ Ω such that c il (d ε , σ ε , ξ ε ) are zero when ε small enough.
The sequel of this section is devoted to the proof of the main result.
α n a 1 d n−2 1 ϕ(ξ) + a 3 k−1 i=1 d i+1 d i n−2 2 g(σ i ) − a 4 k i=1 2 2i−1 | ln d i | = o(1), αn 2 a 2 ∂ ξ h ϕ(ξ)d n−1 1 = 0,(2.16)
for h = 1, · · · , n. We note that G ε → G uniformly on compact set of [0, +∞] × [0, +∞] × Ω, and the vector functional
G(d,σ, ξ) = G 0 (d,σ, ξ), G h (d 1 , ξ) is the principal part of defined by G 0 (d,σ, ξ) = α n a 1 d n−2 1 ϕ(ξ) + a 3 k−1 i=1 d i+1 d i n−2 2 g(σ i ) − a 4 k i=1 2 2i − 1 | ln d i |, G h (d 1 , ξ) = α n 2 a 2 ∂ ξ h ϕ(ξ)d n−2 1
, for h = 1, · · · , n.
Let us set s 1 = d 1 , s i = d i d i−1 , i = 2, · · · , k, then in the new variabless = (s 1 , · · · , s k ), G h (d 1 , ξ) and G 0 (d,σ, ξ) can be rewrite as
G h (s 1 , ξ) = α n 2 a 2 ∂ ξ h ϕ(ξ)s n−2 1 , for h = 1, · · · , n, G 0 (s,σ, ξ) =α n a 1 s n−2 1 ϕ(ξ) + a 3 k i=2 s n−2 2 i g(σ i ) − a 4 k i=1 2 2i − 1 | ln s i |.
We denoteḠ = Ḡ 0 (s,σ, ξ),Ḡ h (s 1 , ξ) . Let ξ 0 ∈ Ω be a strict minimum point of Robin function ϕ, which is the zero point of functionḠ h for h = 1, · · · , n. Observe that σ i = 0 is a strict minimum point of g. On the other hand, when s i is close to 0, the functionḠ 0 tends to −∞, and G 0 > 0 as s i > 0 large enough, thus, by intermediate value theorem, there exists as 0 such that G(s 0 , 0, ξ 0 ) = 0. Moreover, (s 0 , 0, ξ 0 ) is an isolated zero ofḠ whose Brouwer degree is not zero. Therefore, if ε is small enough, (2.16) has a solution (s ε ,σ ε , ξ ε ) near (s 0 , 0, ξ 0 ). We conclude that the right hand side of (2.14) is zero, i.e.,
k i=1 n l=0 c il P ψ l µ i ,ξ i , P ψ h µ j ,ξ j = 0.
Moreover, by Lemma 5.3, we conclude that c il are zero. We finish the proof of this theorem.
The finite dimensional reduction
In this section, we prove Proposition 2.
1. Let L µ,ξ : E ⊥ µ,ξ → E ⊥ µ,ξ be the linear operator defined by (3.1) L µ,ξ (φ) = φ − Π ⊥ µ,ξ i * [f ′ ε (V )φ] ,
where V is defined in (2.5). In the following, we establish the invertibility of L µ,ξ on E ⊥ µ,ξ .
Lemma 3.1. There exist ε 0 > 0 and C > 0 such that for any ξ ∈ Ω,d ∈ R k + ,σ ∈ (R n + ) k satisfying (2.4), for ε ∈ (0, ε 0 ), it holds (3.2) L µ,ξ (φ) ≥ C φ , ∀φ ∈ E ⊥ µ,ξ .
Proof. We argue by contradiction. Assume there exist sequences ε m → 0, ξ ∈ Ω,σ m ∈ (R n + ) k and
d m = (d 1m , · · · , d km ) ∈ R k + with ξ m → ξ ∈ Ω, σ im → σ i and d im → d i > 0, i = 1, · · · , k, φ m , h m ∈ Λ ⊥ µm,ξm such that (3.3) L µm,ξm (φ m ) = h m , φ m = 1 and h m → 0. From (3.1), there exists ω m ∈ E µm,ξm such that (3.4) φ m − i * [f ′ ε (V m )φ m ] = h m + ω m , where V m = V (d m ,σ m , ξ m ) = k i=1 P U µ im ,ξ im .
Step 1. We prove that
(3.5) ω m → 0. Let ω m = k i=1 n l=0 c il m P ψ l µ i m ,ξ i m , we multiply (3.4) by P ψ h µ l m ,ξ l m
, and integrating in Ω, then
(3.6) k i=1 n l=0 c il m P ψ l µ i m ,ξ i m , P ψ h µ j m ,ξ j m =ˆΩ f ′ ε (V m )φ m P ψ h µ j m ,ξ j m dx. From Lemma 5.3, we obtain k i=1 n l=0 c il m P ψ l µ i m ,ξ i m , P ψ h µ j m ,ξ j m =c jh m c h (1 + o(1)) + O(1) n l=0,l =h c jl m + o ε | ln ε| 2 n n−2 k i=1,i =j n l=0 c il m . (3.7)
On the other hand, by (5.4), (5.6), (5.7), (5.9), (5.11), (5.13) and the orthogonality condition
φ m , P ψ h µ j m ,ξ j m = 0, we havê Ω f ′ ε (V m )φ m P ψ h µ j m ,ξ j m dx =ˆΩ f ′ ε (V m ) − f ′ 0 (V m ) φ m P ψ h µ j m ,ξ j m − ψ h µ j m ,ξ j m dx +ˆΩ f ′ ε (V m ) − f ′ 0 (V m ) φ m ψ h µ j m ,ξ j m dx +ˆΩ f ′ 0 (V m ) − k i=1 (−1) i f ′ 0 (P U µ i m ,ξ i m ) φ m P ψ h µ j m ,ξ j m dx + k i=1 (−1) iˆΩ f ′ 0 (P U µ i m ,ξ i m ) − f ′ 0 (U µ i m ,ξ i m ) φ m P ψ h µ j m ,ξ j m dx ≤ f ′ ε (V m ) − f ′ 0 (V m ) n 2 |φ m | 2n n−2 P ψ h µ j m ,ξ j m − ψ h µ j m ,ξ j m 2n n−2 + f ′ ε (V m ) − f ′ 0 (V m ) n 2 |φ m | 2n n−2 |ψ h µ j m ,ξ j m | 2n n−2 + f ′ 0 (V m ) − k i=1 (−1) i f ′ 0 (P U µ i m ,ξ i m ) n 2 |P ψ h µ j m ,ξ j m | 2n n−2 |φ m | 2n n−2 + f ′ 0 (P U µ i m ,ξ i m ) − f ′ 0 (U µ i m ,ξ i m ) n 2 |P ψ h µ j m ,ξ j m | 2n n−2 |φ m | 2n n−2 = O ε ln ln ε | ln ε| 2 . (3.8)
Consequently, from (3.6)-(3.8), we obtain (3.5).
Step 2. We prove that
(3.9) lim inf m→∞ˆΩ f ′ ε (V m )u 2 m dx = C > 0, where u m satisfies −∆u m = f ′ ε (V m )u m + f ′ ε (V m )(h m + ω m ) in Ω, u m = 0 on ∂Ω, (3.10) with (3.11) u m = φ m − h m − ω m , u m → 1.
We prove that
(3.12) lim inf m→∞ u m = C > 0. From (3.10), there holds (3.13) u m = i * f ′ ε (V m )u m + f ′ ε (V m )(h m + ω m )
. Moreover, by (2.1), (5.9), (5.11) and (5.13), we get
|u m | 2n n+2 ≤C |f ′ ε (V m )u m | 2n n+2 + |f ′ ε (V m )(h m + ω m )| 2n n+2 ≤C|f ′ ε (V m ) − f ′ 0 (V m )| n 2 |u m | 2n n−2 + C f ′ 0 (V m ) − k i=1 (−1) i f ′ 0 (P U µ i m ,ξ i m ) n 2 |u m | 2n n−2 + C|f ′ ε (V m ) − f ′ 0 (V m )| n 2 |h m + ω m | 2n n−2 + C f ′ 0 (V m ) − k i=1 (−1) i f ′ 0 (P U µ i m ,ξ i m ) n 2 |h m + ω m | 2n n−2 + C k i=1 (−1) i f ′ 0 (P U µ i m ,ξ i m ) − f ′ 0 (U µ i m ,ξ i m ) n 2 |h m + ω m | 2n n−2 ≤ C u m + o(1). (3.14)
It follows that |u m | 2n n+2 → 0 provided that u m → 0, this contradicts with (3.11). Therefore,
(3.12) holds.
We multiply (3.13) by u m , that is
(3.15) u m 2 =ˆΩ f ′ ε (V m )u 2 m dx +ˆΩ f ′ ε (V m )(h m + ω m )u m dx. By (3.3) and (3.5), one haŝ Ω f ′ ε (V m )(h m + ω m )u m dx ≤|f ′ ε (V m )| n 2 |h m + ω m )| 2n n−2 |u m | 2n n−2 ≤ h m + ω m u m = o(1). (3.16)
Therefore (3.9) follows by (3.11), (3.12), (3.15) and (3.16).
Step 3. Let us prove that a contradiction arises, by showing that
(3.17)ˆΩ f ′ ε (V m )u 2 m dx = o(1).
In order to deal with this conclusion, we decompose B(ξ, ρ) into the union of non-overlapping
annuli, that is B(ξ, ρ) = k i=1 A i , where (3.18) A i = B(ξ, √ µ i µ i−1 ) \ B(ξ, √ µ i µ i+1 ), i = 1, · · · , k,
with µ 0 = ρ 2 µ 1 and µ k+1 = 0. We set a smooth cut-off function χ i m as
χ i m (x) = 1 if µ i m µ i+1 m ≤ |x − ξ m | ≤ µ i m µ i−1 m , 0 if |x − ξ m | ≤ √ µ i m µ i+1 m 2 or |x − ξ m | ≥ 2 µ i m µ i−1 m ,(3.19)
and
(3.20) |∇χ i m (x)| ≤ 2 µ i m µ i−1 m and |∇ 2 χ i m (x)| ≤ 4 µ i m µ i−1 m , for any i = 1, · · · , k. We define (3.21)ũ i m (y) = (µ i m ) n−2 2 u m (µ i m y + ξ m )χ i m (µ i m y + ξ m )
. First, the following results will be showed in Step 4,
(3.22)ũ i m → 0 weakly in D 1,2 (R n ),ũ i m → 0 strongly in L q loc (R n )
for any q ∈ [2, 2 * ). Let us prove (3.17). There holdŝ
Ω f ′ ε (V m )u 2 m dx =ˆΩ \B(ξ,ρ) f ′ ε (V m )u 2 m dx + k i=1ˆA i f ′ ε (V m )u 2 m dx, whereˆΩ \B(ξ,ρ) f ′ ε (V m )u 2 m dx ≤ C k i=1 (µ n i ) 2ˆΩ \B(ξ,ρ) u 2 m dx = o(1). Since ( 1 1+|x| 2 ) 2 ∈ L n 2 (R n ) and (3.22) hold, we conclude that´( A i m −ξm)/µ i m ( 1 1+|y−σ in | 2 ) n−2 2 (p−1) (ũ i m ) 2 dy → 0.
On the other hand, we set x − ξ m = µ i m y and by a fact that, let h ∈ L 1 rad (R n ), performing the proper change of variable: for any i = l,
A i m −ξm µ i m h(|x|)dx = O ( µ l µ i ) n 2 if i ≤ l − 1 < l, O ( µ i µ l ) n 2 if i ≥ l − 1 > l.
(3.23)
By (3.23) and the choice of µ i in (2.2), we deduce that
(3.24) if h ∈ L 1 rad (R n ), i = l,ˆA l m −ξm µ i m h(|x|)dx = O ε | ln ε| 2 n n−2 . Then, there holdŝ A i m −ξm µ i m U (p−1) n 2 µ im ,ξ im dx = O ˆ√ µ i m µ i+1 m µ i m ≤|y|≤ √ µ i m µ i−1 m µ i m 1 (1 + |y − σ im | 2 ) n dy .
Consquently, from (5.13), we havê
A i m f ′ ε (V m )u 2 m dx =ˆA i m f ′ ε (V m ) − f ′ 0 (V m ) u 2 m dx +ˆA i m f ′ 0 (V m )u 2 m dx ≤C|f ′ ε (V m ) − f ′ 0 (V m )| n 2 |u m | 2 n n−2 + C k i=1ˆA i m U p−1 µ im ,ξ im u 2 m dx ≤Cε ln ln ε | ln ε| 2 + C(µ i m ) 2− n−2 2 (p−1)ˆA i m −ξm µ i m 1 1 + |y − σ im | 2 n−2 2 (p−1) (ũ i m ) 2 dy + C k j=1,j =i ˆA i m U (p−1) n 2 µ jm ,ξ jm dx 2 n |u m | 2 2n n−2 = o(1).
Step 4. We prove (3.22).
From the definition ofũ i m , i = 1, · · · , k, in (3.21), when x − ξ m = µ i m y, we get (3.25) ∇ũ i m (y) = (µ i m ) n 2 ∇u m (x) χ i m (x) + u m (x) ∇χ i m (x) , and (3.26) ∆ũ i m (y) = (µ i m ) n+2 2 ∆u m (x) χ i m (x) + 2∇u m (x)∇χ i m (x) + u m (x) ∆χ i m (x) .
Then, from (3.19), (3.20) and (3.25), it holds that ũ i m D 1,2 (R n ) ≤ C. It follows that, up to a subsequence, u i m →ũ i weakly in D 1,2 (R n ),ũ i m →ũ i strongly in L q loc (R n ) for any q ∈ [2, 2 * ). Next, we show thatũ i is the solution of the following problem
(3.27) −∆ũ i = f ′ 0 (U 1,σ i )ũ i in R n ,
and satisfies the orthogonality conditions
(3.28)ˆR n ∇ψ h 1,σ i ∇ũ i dx = 0, h = 0, 1, · · · , n.
It follows thatũ i = 0. This is a contradiction by the result of [10], which concludes the proof.
Step 5. We prove (3.27) and (3.28).
(1) Let us prove (3.27). By (3.25) and (3.26)
, if x − ξ m = µ i m y, y ∈ Ω i m = Ω−ξm µ i m , we have −∆ũ i m (y) = (µ i m ) 2 f ′ ε V m (x) ũ i m (y) + (µ i m ) n+2 2 f ′ ε V m (x) h m (x) + ω m (x) χ i m (x) +2(µ i m ) n+2 2 ∇u m (x)∇χ i m (x) + 2(µ i m ) n+2 2 u m (x)∆χ i m (x), u i m = 0 on ∂Ω i m . Therefore, if ̟ ∈ C ∞ 0 (R n ), one haŝ R n ∇ũ i m (y)∇̟(y)dy =ˆR n (µ i m ) 2 f ′ ε V m (µ i m y + ξ m ) ũ i m (y)̟(y)dy +ˆR n (µ i m ) n+2 2 × f ′ ε V m (µ i m y + ξ m ) h m (µ i m y + ξ m ) + ω m (µ i m y + ξ m ) χ i m (µ i m y + ξ m )̟(y)dy + 2(µ i m ) n+2 2ˆR n ∇u m (µ i m y + ξ m )∇χ i m (y) + u m (µ i m y + ξ m )∆χ i m (µ i m y + ξ m ) ̟(y)dy. (3.29)
From (3.19) and (3.20), we deduce that the second and the third term tends to 0. For the first
term, if √ µ i m µ i+1 m 2 ≤ |µ i m y| ≤ 2 µ i m µ i−1 m , there holds f ′ ε V m (µ i m y + ξ m ) =f ′ ε P U µ in ,ξ in (µ i m y + ξ m ) + k i=1,j =i P U µ jm ,ξ jm (µ i m y + ξ m ) =f ′ ε (µ i m ) − n−2 2 U (y − σ in ) + U µ jm ,ξ jm (µ i m y + ξ m ) + o(1) , (3.30) where U µ jm ,ξ jm (µ i m y + ξ m ) = O (µ j m ) − n−2 2 if i > j, O (µ j m ) n−2 2 (µ i m ) n−2 y − µ j m µ i m σ j m −(n−2) if i < j.
(3.31)
Moreover, by (3.30), (3.31) and Lebesgue's dominated convergence theorem, it holds that
R n (µ i m ) 2 f ′ ε V m (µ i m y + ξ m ) ũ i m (y)̟(y)dy →ˆR n f ′ 0 U (y − σ i ) ũ i (y)̟(y)dy.
Then, (3.27) follows by passing to the limit in (3.29).
(2) Let us prove (3.28). We set x − ξ m = µ i m y, then
R n ∇ψ h 1,σ i m (y)∇ũ i m (y)dy =ˆR n f ′ 0 U 1,σ i m (y) ψ h 1,σ i m (y)ũ i m (y)dy =µ i mˆ√ µ i m µ i+1 m 2 ≤|x−ξ|≤2 √ µ i m µ i−1 m f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x)χ i m (x)dx =µ i m ˆA i m f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x)dx + o(1) . (3.32)
Now, we show that
(3.33) µ i mˆA i m f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x)dx = o(1).
Therefore, (3.28) follows by (3.32) and (3.33), taking into account that σ i m → σ i . From (3.5) and (3.11), one has
(3.34) µ i mˆΩ ∇P ψ h µ i m ,ξ i m (x)∇u m (x)dx = o(1). On the other hand, Ω ∇P ψ h µ i m ,ξ i m (x)∇u m (x)dx =ˆΩ f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x)dx =ˆΩ \B(ξm,ρ) · · · dx + k l=1, l =iˆA l m · · · dx +ˆA i m · · · dx =ˆA i m · · · dx + o 1 µ i m , whereˆΩ \B(ξm,ρ) f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x) dx ≤C|ψ h µ i m ,ξ i m | 2n n−2 |u m | 2n n−2 ˆΩ \B(ξm,ρ) U 2n n−2 µ i m ,ξ i m dx 2 n = O(µ i m ).µ i m ,ξ i m dx = O ε | ln ε| 2 n n−2 , then A l m f ′ 0 U µ i m ,ξ i m (x) ψ h µ i m ,ξ i m (x)u m (x) dx ≤C|ψ h µ i m ,ξ i m | 2n n−2 |u m | 2n n−2 ˆA l m U 2n n−2 µ i m ,ξ i m dx 2 n = O ε | ln ε| 2 n n−2 .
We finish the proof of this lemma. Now, by means of the previous result, we show the following proof. Proof of Proposition 2.1: First of all, we point out that φ solves equation (2.11) if and only if φ is a fixed point of the map T µ,ξ :
E ⊥ µ,ξ → E ⊥ µ,ξ defined by T µ,ξ (φ) =L −1 µ,ξ Π ⊥ µ,ξ i * f ε (V + φ) − f ε (V ) − f ′ ε (V )φ + f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) φ + k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) φ + f ε (V ) − k i=1 (−1) i f 0 (P U µ i ,ξ i ) + k i=1 (−1) i f 0 (P U µ i ,ξ i ) − k i=1 (−1) i f 0 (U µ i ,ξ i ) .
From Lemma 3.1 and Sobolev inequality, we have
T µ,ξ (φ) ≤C f ε (V + φ) − f ε (V ) − f ′ ε (V )φ 2n n+2 + C f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) φ 2n n+2 + C f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) φ 2n n+2 + C f ε (V ) − k i=1 (−1) i f 0 (P U µ i ,ξ i ) 2n n+2 + C k i=1 (−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) 2n n+2 =H 1 + · · · + H 5 .
Estimate of H 1 : From the mean value theorem, we choose t = t(x) ∈ (0, 1), then
H 1 = f ε (V + φ) − f ε (V ) − f ′ ε (V )φ) 2n n+2 = f ε (V + tφ) − f ′ ε (V )φ) 2n n+2 . (3.36)
When n < 6, Lemma 5.5 follows that
H 1 ≤C |φ| p 2n n+2 + |U p−2 µ i ,ξ i φ 2 | 2n n+2 ≤ C |φ| p−2 2n n−2 + |U µ i ,ξ i | p−2 p−2 |φ| 2 2n n−2 = C |φ| p−2 + 1 φ 2 .
When n = 6, by Sobolev inequality, one has
H 1 ≤ C |φ| p 2n n+2 + |φ 2 | 2n n+2 = C |φ| p 2n n−2 + Ω |φ| 2n n−2 dx n+2 2n = 2C|φ| p p+1 ≤ 2C φ 2 .
When n > 6, there holds
H 1 ≤C |φ| p 2n n+2 + ε|U p−1 µ i ,ξ i φ| 2n n+2 = C |φ| p 2n n−2 + ˆΩ (U p−1 µ i ,ξ i |φ|) 2n n+2 dx n+2 2n ≤C |φ| p 2n n−2 + ε|U µ i ,ξ i | p−1 2n n−2 |φ| 2n n−2 = C |φ| p−1 2n n−2 + ε|U µ i ,ξ i | p−1 2n n−2 |φ| 2n n−2 ≤C( φ p−1 + ε) φ .
Sum up these estimates, we have
H 1 ≤ C(|φ| p−2 + 1) φ 2 if 3 ≤ n ≤ 5, C φ 2 if n = 6, C( φ p−1 + ε) φ if n ≥ 7.
(3.37)
Estimate of H 2 : From Hölder's inequality and (5.12), we get
H 2 = f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) φ 2n n+2 ≤ f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) n 2 |φ| 2n n−2 ≤ Cε ln ln ε | ln ε| 2 φ .
Estimate of H 3 : By Hölder's inequality, (5.9) and (5.11), there holds
H 3 = f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) φ 2n n+2 = f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) φ 2n n+2 + k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i ) φ 2n n+2 ≤ f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) n 2 |φ| 2n n−2 + k f ′ 0 (P U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i ) n 2 |φ| 2n n−2 ≤ O ε | ln ε| 2 φ if 3 ≤ n ≤ 5, ε | ln ε| 2 ln ε | ln ε| 2 φ if n = 6, O ε | ln ε| 2 −n+8 n−2 φ if n ≥ 7.
Estimate of H 4 and H 5 : From (5.12) and (5.8), one has
|H 4 | 2n n+2 + |H 5 | 2n n+2 = O(R ε ),
and R ε satisfies
R ε = O ε | ln ε| 2 ln ε | ln ε| 2 ln ln ε | ln ε| 2 if 3 ≤ n ≤ 6, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 7.
From H 1 -H 5 , there is a constant C * > 0 and µ 0 > 0 such that for each µ ∈ (0, µ 0 ), we obtain
T µ,ξ (φ) ≤ C * R ε for every φ ∈B = {φ ∈ E ⊥ µ,ξ : φ ≤ C * R ε }. Next, we prove that T µ,ξ is a contraction map. If φ 1 , φ 2 ∈B, then T µ,ξ (φ 2 ) − T µ,ξ (φ 1 ) ≤C f ε (V + φ 2 ) − f ε (V + φ 1 ) − f ′ ε (V )(φ 2 − φ 1 ) 2n n+2 + C f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) (φ 2 − φ 1 ) 2n n+2 + C k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) (φ 2 − φ 1 ) 2n n+2 = K 1 + K 2 + K 3 .
Estimate of K 1 : Similar to the computations of H 1 -H 3 . By mean value theorem, we choose ̺ = ̺(x) ∈ (0, 1) and φ ̺ = (1 − ̺)φ 1 + ̺φ 2 , then
K 1 = f ε (V + φ 2 ) − f ε (V + φ 1 ) − f ′ ε (V )(φ 2 − φ 1 ) 2n n+2 = f ′ ε (V + φ ̺ ) − f ′ ε (V ) (φ 2 − φ 1 ) 2n n+2 .
When n < 6, by Lemma 5.5 and Hölder's inequality, we get
K 1 ≤C |φ ̺ | p−1 (φ 2 − φ 1 ) 2n n+2 + k i=1 U µ i ,ξ i p−2 φ ̺ (φ 2 − φ 1 ) 2n n+2 ≤C |φ ̺ | p−1 2n n−2 |φ 2 − φ 1 | 2n n−2 + ˆΩ k i=1 U µ i ,ξ i p−2 φ ̺ (φ 2 − φ 1 ) 2n n+2 dx n+2 2n ≤C |φ ̺ | p−1 2n n−2 + k i=1 |U µ i ,ξ i | p−2 2n n−2 |φ ̺ | 2n n−2 |φ 2 − φ 1 | 2n n−2 .
When n = 6, we have
K 1 ≤ C|φ ̺ | p−1 2n n−2 |φ 2 − φ 1 | 2n n−2 .
When n > 6, there holds
K 1 ≤C |φ ̺ | p−1 (φ 2 − φ 1 ) 2n n+2 + ε k i=1 U µ i ,ξ i p−1 φ ̺ (φ 2 − φ 1 ) 2n n+2 ≤C |φ ̺ | p−1 2n n−2 |φ 2 − φ 1 | 2n n−2 + εˆΩ k i=1 U µ i ,ξ i p−1 φ ̺ (φ 2 − φ 1 ) 2n n+2 dx n+2 2n ≤C |φ ̺ | p−1 2n n−2 + ε |φ 2 − φ 1 | 2n n−2 .
Hence, by Sobolev inequality, we obtain
K 1 ≤ C |φ ̺ | p−1 + max{ φ ̺ , ε} φ 2 − φ 1 .
Estimate of K 2 : Similar to the proof of H 2 and H 3 , from (5.13) and (5.11), there holds
K 2 = f ′ ε (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) (φ 1 − φ 2 ) 2n n+2 ≤ f ′ ε (V ) − f ′ 0 (V ) n 2 |φ 2 − φ 1 | 2n n−2 + f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) n 2 |φ 2 − φ 1 | 2n n−2 ≤ C ε | ln ε| 2 −n+8 n−2 ln ln ε | ln ε| 2 φ 2 − φ 1 .
Estimate of K 3 : By (5.9), one has
K 3 = k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) (φ 2 − φ 1 ) 2n n+2 ≤ k f ′ 0 (P U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i ) n 2 |φ 2 − φ 1 | 2n n−2 ≤ O ε | ln ε| 2 φ 2 − φ 1 if 3 ≤ n ≤ 5, O ε | ln ε| 2 ln ε | ln ε| 2 1 2 φ 2 − φ 1 if n = 6, O ε | ln ε| 2 2 n−2 φ 2 − φ 1 if n ≥ 7. From K 1 -K 3 , if ε is sufficient small, there exists a constant L * ∈ (0, 1) such that T µ,ξ (φ 2 ) − T µ,ξ (φ 1 ) ≤ L * φ 2 − φ 1 .
It follows that T µ,ξ is a contraction mapping fromB toB, then, it has a unique fixed point φ ∈B. This concludes the proof.
Proof of Proposition 2.2
This section is devoted to prove Proposition 2.2. Proof of Part a. We consider the following perturbation problem
−∆(V + φ) = f ε (V + φ) + k i=0 n l=0 c il U p−1 µ εi ,ξ εi P ψ l µ εi ,ξ εi in Ω, k i=1´Ω
U p−1 µ εi ,ξ εi P ψ l µ εi ,ξ εi φdx = 0 for l = 0, 1, · · · , n.
(4.1) From (2.15), we have
(4.2)ˆΩ ∆(V + φ)P ψ h µ j ,ξ j dx +ˆΩ f ε (V + φ)P ψ h µ j ,ξ j dx. = 0,
Thus, by (4.1) and (4.2), we obtain
c ilˆΩ U p−1 µ εi ,ξ εi P ψ l µ εi ,ξ εi P ψ h µ j ,ξ j dx = 0,
which means that c il = 0 for i = 1, · · · , k and l = 0, 1, · · · , n. Then V + φ is a solution of problem (1.1).
Proof of Part b. There holds
V + φ − i * [f ε (V + φ)], P ψ h µ j ,ξ j = k i=1 P U µ i ,ξ i , P ψ h µ j ,ξ j −ˆΩ f ε (V + φ)P ψ h µ j ,ξ j dx = k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i )P ψ h µ j ,ξ j dx −ˆΩ f ε (V + φ)P ψ h µ j ,ξ j dx = k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i ) − f 0 (P U µ i ,ξ i ) ψ h µ j ,ξ j dx + k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i ) − f 0 (P U µ i ,ξ i ) (P ψ h µ j ,ξ j − ψ h µ j ,ξ j )dx +ˆΩ k i=1 (−1) i f 0 (P U µ i ,ξ i ) − f ε (V ) ψ h µ j ,ξ j dx +ˆΩ k i=1 (−1) i f 0 (P U µ i ,ξ i ) − f ε (V ) (P ψ h µ j ,ξ j − ψ h µ j ,ξ j )dx −ˆΩ f ε (V + φ) − f ε (V ) − f ′ ε (V )φ P ψ h µ j ,ξ j dx −ˆΩ[f ′ ε (V ) − f ′ 0 (V )]φP ψ h µ j ,ξ j dx −ˆΩ f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) φP ψ h µ j ,ξ j dx − k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )φ(P ψ h µ j ,ξ j − ψ h µ j ,ξ j )dx − k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )φψ h µ j ,ξ j dx =P 1 + · · · , P 9 .
Estimate of P 1 : It holds
P 1 = k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i ) − f 0 (P U µ i ,ξ i ) ψ h µ j ,ξ j dx = − k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i )ψ h µ j ,ξ j dx − k i=1ˆΩ (−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′
If h = 0, by (5.1) and (2.9), we have
−ˆΩ f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i )ψ 0 µ j ,ξ j dx = pα n µ n−2 2 iˆΩ U p−1 µ i ,ξ i H(x, ξ i )ψ 0 µ j ,ξ j dx = pα n µ n−2 i H(ξ i , ξ i ) + O(µ i ) ˆΩ −ξ i µ i 1 (1 + |y| 2 ) 2 ψ 0 µ i y + ξ i − ξ j µ j dy = pα n µ n−2 i H(ξ i , ξ i ) + O(µ i ) ´R n U p−1 (y)ψ 0 (y)dy if j = i, (n−2)α 2 n p 2 µ n−2 i H(ξ i , ξ i ) + O(µ i ) ´Ω −ξ i µ i 1 (1+|y| 2 ) 2 |µ i y+ξ i −ξ j | 2 −µ 2 j (µ 2 j +|µ i y+ξ i −ξ j | 2 ) n 2 dy if j > i, = α n a 1 H(ξ, ξ)µ n−2 i + O(µ n−1 i ) if j = i and h = 0, CH(ξ, ξ) + O(µ n−1 i ) if j > i and h = 0. If h = 1, · · · , n and j = i, we set ∂ ξ h i ϕ(ξ) = ∂ϕ(ξ) ∂ξ h i , by (2.10), one has −ˆΩ f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i )ψ h µ j ,ξ j dx =pα n µ n−2 2 iˆΩ U p−1 µ i ,ξ i H(x, ξ i )ψ h µ i ,ξ i dx =α n µ n 2 iˆΩ H(x, ξ i ) ∂U p µ i ,ξ i ∂ξ h i dx =α n µ n 2 i ∂ ∂ξ h iˆΩ U p µ i ,ξ i H(x, ξ i )dx −ˆΩ U p µ i ,ξ i ∂H(x, ξ i ) ∂ξ h i dx =α n µ n 2 i µ n−2 2 i ∂ ∂ξ h iˆΩ −ξ i µ i U p µ i ,ξ i H(µ i y + ξ i , ξ i )dy − µ n−2 2 iˆΩ −ξ i µ i U p µ i ,ξ i ∂H(µ i y + ξ i , ξ i ) ∂ξ h i dy =α n a 2 µ n−1 i ∂(H(ξ i , ξ i )) ∂ξ h i − ∂H(ξ i , ξ i ) ∂ξ h i + O(µ i ) =α n a 2 µ n−1 i 1 2 ∂ ξ h ϕ(ξ) + O(µ i ) .
If h = 1, · · · , n and j > i, by Lemma 5.1 and (2.9), one haŝ
Ω f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i )ψ h µ j ,ξ j dx =pα n µ n−2 2 iˆΩ U p−1 µ i ,ξ i H(x, ξ i )ψ h µ j ,ξ j dx =(n − 2)pα 2 n µ n−2 2 i µ n 2 jˆΩ µ 2 i (µ 2 i + |x − ξ i | 2 ) 2 H(x, ξ i ) (x − ξ j ) h (µ 2 j + |x − ξ j | 2 ) n 2 dx =O(µ n−2 2 i µ n 2 j ) = O ε | ln ε| 2 n+1 n−2 .
On the other hand, by (5.6), (5.7) and (5.10), we get
Ω f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′ ≤ f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) n 2 |ψ h µ j ,ξ j | n n−2 =o ε | ln ε| 2 n−1 n−2
for h = 0, · · · , n. Therefore, we get
P 1 = α n a 1 k i=1 µ n−2 i H(ξ i , ξ i ) + O(µ i ) + o(µ n−2 i ) if j = i and h = 0, k i=1 O H(ξ i , ξ i ) + O(µ i ) + o(µ n−2 i ) if j > i and h = 0, α n a 2 k i=1 µ n−1 i 1 2 ∂ ξ h ϕ(x) + O(µ i ) + o(µ n−1 i ) if j = i and h = 1, · · · , n, O( k i=1 µ n−2 2 i µ n 2 j ) + o(µ n−1 i ) if j > i and h = 1, · · · , n, = α n a 1 H(ξ, ξ) ε | ln ε| 2 d n−2 1 + o ε | ln ε| 2 if j = i and h = 0, CH(ξ, ξ) + o ε | ln ε| 2 if j > i and h = 0, 1 2 α n a 2 ∂ ξ h ϕ(ξ) ε | ln ε| 2 n−1 n−2 d n−2 1 + o ε | ln ε| 2 n−1 n−2 if j = i and h = 1, · · · , n, O ε | ln ε| 2 n−1 n−2 if j > i and h = 1, · · · , n.
Estimate of P 2 : By (5.8) and (5.4), we deduce
P 2 = k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i ) − f 0 (P U µ i ,ξ i ) (P ψ h µ j ,ξ j − ψ h µ j ,ξ j ) dx ≤ C k i=1 f 0 (U µ i ,ξ i ) − f 0 (P U µ i ,ξ i ) 2n n+2 P ψ h µ j ,ξ j − ψ h µ j ,ξ j 2n n−2 = o ε | ln ε| 2 if h = 0, o ε | ln ε| 2 n−1 n−2 if h = 1, · · · , n.
Estimate of P 3 : The main proof of P 3 shows in Lemma 5.7, the final result is
P 3 =ˆΩ k i=1 (−1) i f 0 (P U µ i ,ξ i ) − f ε (V ) ψ h µ j ,ξ j dx = α n a 1 ε | ln ε| 2 H(ξ, ξ)d n−2 1 + a 3 ε | ln ε| 2 k−1 i=1 d i+1 d i n−2 2 g(σ i ) − 2k 2 (n−2) 2 a 4 ε ln ε | ln ε| 2 −a 4 ε | ln ε| 2 k i=1 2 2i−1 | ln d i | + o ε | ln ε| 2 if h = 0, 1 2 α n a 2 ε | ln ε| 2 n−1 n−2 d n−1 1 ∂ ξ h ϕ(ξ) + o ε | ln ε| 2 n−1 n−2 if h = 1, · · · , n,
Estimate of P 4 : From (5.12) and (5.4), we get
P 4 =ˆΩ k i=1 (−1) i f 0 (P U µ i ,ξ i ) − f ε (V ) (P ψ h µ j ,ξ j − ψ h µ j ,ξ j ) dx ≤ C f ε (V ) − k i=1 (−1) i f 0 (P U µ i ,ξ i ) 2n n+2 P ψ h µ j ,ξ j − ψ h µ j ,ξ j 2n n−2 = o ε | ln ε| 2 if h = 0 and 3 ≤ n ≤ 4, o ε | ln ε| 2 n−1 n−2 if h = 0 and n ≥ 5, o ε | ln ε| 2
if h = 1, · · · , n and n ≥ 3.
Estimate of P 5 : From (5.4), (5.6), (5.7), (3.36) and (3.37), we have
P 5 =ˆΩ f ε (V + φ) − f ε (V ) − f ′ ε (V )φ P ψ h µ j ,ξ j dx = O f ε (V + φ) − f ε (V ) − f ′ ε (V )φ) 2n n+2 |P ψ h µ j ,ξ j | 2n n−2 ≤ O(1 + φ p−2 |) φ 2 if 3 ≤ n ≤ 5, O( φ 2 ) if n = 6, O(ε + φ p−1 |) φ 2 if n ≥ 7, = o ε | ln ε| 2 if n = 3, o ε | ln ε| 2 n−1 n−2 if n ≥ 4.
Estimate of P 6 : By (5.13) and (2.13), it holds
P 6 =ˆΩ [f ′ ε (V ) − f ′ 0 (V )]φP ψ h µ j ,ξ j dx = O |f ′ ε (V ) − f ′ 0 (V )| n 2 |φ| 2n n−2 |P ψ h µ j ,ξ j | 2n n−2 = o ε | ln ε| 2 if h = 0, o ε | ln ε| 2 n−1 n−2 if h = 1, · · · , n..
Estimate of P 7 : By (5.11), (2.13), (5.7), (5.6), (5.4), (5.9), there holds
P 7 =ˆΩ f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) φP ψ h µ j ,ξ j dx = O f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (U µ i ,ξ i ) P ψ h µ j ,ξ j 2n n+2 φ = O f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) n 2 |P ψ h µ j ,ξ j | 2n n+2 φ + O k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i ) n 2 |P ψ h µ j ,ξ j | 2n n+2 φ = o ε | ln ε| 2 if h = 0, o ε | ln ε| 2 n−1 n−2 if h = 1, · · · , n.
Estimate of P 8 : For h = 0, by (5.5), (2.13) and (5.4), it follows that
P 8 = k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )φ(P ψ 0 µ j ,ξ j − ψ 0 µ j ,ξ j ) dx = O k i=1 f ′ 0 (U µ i ,ξ i ) n 2 |φ| 2n n−2 P ψ 0 µ j ,ξ j − ψ 0 µ j ,ξ j 2n n−2 = O k i=1 µ n−2 2 i φ = o ε | ln ε| 2 if n = 3, o ε | ln ε| 2 n−1 n−2 if n ≥ 4,
and for h = 1, · · · , n, we obtain
P 8 = k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )φ(P ψ h µ j ,ξ j − ψ h µ j ,ξ j ) dx = O k i=1 f ′ 0 (U µ i ,ξ i ) n 2 |φ| 2n n−2 |P ψ h µ j ,ξ j − ψ h µ j ,ξ j | 2n n−2 = O k i=1 µ n 2 i φ = o ε | ln ε| 2 if 3 ≤ n ≤ 5, o ε | ln ε| 2 n−1 n−2 if n ≥ 6.
Estimate of P 9 : We use φ to multiply (2.7) and integral in the Ω, then
Ω f ′ 0 (U µ i ,ξ i )φψ h µ j ,ξ j dx = 0.
From P 1 -P 9 , we complete the proof.
Appendix
We collect some well known estimates.
Lemma 5.1. [42] Let ξ ∈ Ω, µ > 0 is small, there holds
(5.1) P U µ,ξ (x) = U µ,ξ (x) − α n µ n−2 2 H(x, ξ) + O(µ n+2 2 ), (5.2) P ψ 0 µ,ξ (x) = ψ 0 µ,ξ (x) − n − 2 2 α n µ n−2 2 H(x, ξ) + O(µ n+4 2 ), (5.3) P ψ h µ,ξ (x) = ψ h µ,ξ (x) − α n µ n 2 ∂ ξ h H(x, ξ) + O(µn+2
2 ), as µ → 0 uniformly with respect to ξ in compact subsets of Ω, where h = 1, · · · , n and α n is given in (1.3). Moreover,
|P ψ h µ j ,ξ j − ψ h µ j ,ξ j | 2n n−2 = O ε | ln ε| 2 1 2 if h = 0, O ε | ln ε| 2 n 2(n−2)
if h = 1, · · · , n, (5.4) for j = 1, · · · , k.
Lemma 5.2. [18] There holdŝ
Ω U q µ,ξ (x)dx = O ε | ln ε| 2 q 2 if 0 < q < n n−2 , O ε | ln ε| 2 n 2(n−2) ln ε | ln ε| 2 if q = n n−2 , O ε | ln ε| 2 n n−2 − q 2 if n n−2 < q ≤ 2n n−2 , (5.5)ˆΩ |ψ 0 µ,ξ (x)| q dx = O ε | ln ε| 2 q 2 if 0 < q < n n−2 , O ε | ln ε| 2 n 2(n−2) ln ε | ln ε| 2 if q = n n−2 , O ε | ln ε| 2 n n−2 − q 2 if n n−2 < q ≤ 2n n−2 , (5.6) andˆΩ |ψ h µ,ξ (x)| q dx = O ε | ln ε| 2 nq 2(n−2) if 0 < q < n n−1 , O ε | ln ε| 2 n 2 2(n−1)(n−2) ln ε | ln ε| 2 if q = n n−1 , O ε | ln ε| 2 n n−2 − q 2 if n n−1 < q ≤ 2n n−2 ,(5.
7)
for h = 1, · · · , n. Moreover,
f 0 (P U µ,ξ ) − f 0 (U µ,ξ ) 2n n+2 = O ε | ln ε| 2 if 3 ≤ n ≤ 5, O ε | ln ε| 2 ln ε | ln ε| 2 2 3 if n = 6, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 7,
(5.8) f ′ 0 (P U µ,ξ ) − f ′ 0 (U µ,ξ ) n 2 = O ε | ln ε| 2 if n = 3, O ε | ln ε| 2 ln ε | ln ε| 2 1 2 if n = 4, O ε | ln ε| 2 2 n−2 if n ≥ 5, (5.9) f 0 (P U µ,ξ ) − f 0 (U µ,ξ ) − f ′ 0 (U µ,ξ )(P U µ,ξ − U µ,ξ ) n 2 = O ε | ln ε| 2 5 2 if n = 3, O ε | ln ε| 2 3 2 ln ε | ln ε| 2 1 2 if n = 4, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 5.
(5.10)
Lemma 5.3. It holds
P ψ l µ i ,ξ i , P ψ h µ j ,ξ j = o ε | ln ε| 2 n n−2 if j > i, O(1) if l = h, c h (1 + o(1)) if i = j, l = h,
for some positive constants c 0 and c 1 = · · · , c n , where i, j = 1, · · · , k and h, l = 0, · · · , n,
Proof. We have P ψ l µ i ,ξ i , P ψ h µ j ,ξ j =ˆΩ f ′ 0 (U µ i ,ξ i )ψ l µ i ,ξ i ψ h µ j ,ξ j dx +ˆΩ f ′ 0 (U µ i ,ξ i )ψ l µ i ,ξ i (P ψ h µ j ,ξ j − ψ h µ j ,ξ j )dx.
From (5.5), (5.7) and (5.4), there holdŝ
Ω f ′ 0 (U µ i ,ξ i )ψ l µ i ,ξ i (P ψ h µ j ,ξ j − ψ h µ j ,ξ j )dx ≤ |f ′ 0 (U µ i ,ξ i )| n 2 |ψ l µ i ,ξ i | 2n n−2 |P ψ h µ j ,ξ j − ψ h µ j ,ξ j | 2n n−2 = o ε | ln ε| 2 .
On the other hand, if l, h = 1, · · · , n, the change of variables x − ξ i = µ i y shows that
Ω f ′ 0 (U µ i ,ξ i )ψ l µ i ,ξ i ψ h µ j ,ξ j dx = (n − 2) 2 α 2n n−2 n µ n+4 2 i µ n 2 jˆΩ (x − ξ i ) l (µ 2 i + |x − ξ i | 2 ) n+4 2 (x − ξ j ) h (µ 2 j + |x − ξ j | 2 ) n 2 dx = (n − 2) 2 α 2n n−2 n µ n−2 2 i µ n 2 jˆΩ −ξ i µ i y l (1 + |y| 2 ) n+4 2 (µ i y + ξ i − ξ j ) h (µ 2 j + |µ i y + ξ i − ξ j | 2 ) n 2 dy = O ( µ j µ i ) n 2 if j > i, O(1) if i = j, l = h, c h (1 + o(1)) if i = j, l = h.
If l = 1, · · · , n and h = 0, we havê
Ω f ′ 0 (U µ i ,ξ i )ψ l µ i ,ξ i ψ 0 µ j ,ξ j dx = (n − 2) 2 2 α 2n n−2 n µ n+4 2 i µ n−2 2 jˆΩ (x − ξ i ) l (µ 2 i + |x − ξ i | 2 ) n+4 2 |x − ξ j | 2 − µ 2 j (µ 2 j + |x − ξ j | 2 ) n 2 dx = (n − 2) 2 α 2n n−2 n µ n−2 2 i µ n−2 2 jˆΩ −ξ i µ i y l (1 + |y| 2 ) n+4 2 |µ i y + ξ i − ξ j | 2 − µ 2 j (µ 2 j + |µ i y + ξ i − ξ j | 2 ) n 2 dy = o ε | ln ε| 2 .
Finally, if l = 0 and h = 0, one haŝ
Ω f ′ 0 (U µ i ,ξ i )ψ 0 µ i ,ξ i ψ 0 µ j ,ξ j dx = (n − 2) 2 4 α 2n n−2 n µ n−2 2 i µ n−2 2 jˆΩ |x − ξ i | 2 − µ 2 i (µ 2 i + |x − ξ i | 2 ) n 2 |x − ξ j | 2 − µ 2 j (µ 2 j + |x − ξ j | 2 ) n 2 dx = (n − 2) 2 α 2n n−2 n µ − n−2 2 i µ n−2 2 jˆΩ −ξ i µ i (|y − σ i | 2 − 1) (1 + |y − σ i | 2 ) n 2 |µ i y + ξ i − ξ j | 2 − µ 2 j (µ 2 j + |µ i y + ξ i − ξ j | 2 ) n 2 dy = o ε | ln ε| 2 if j > i, c 0 (1 + o(1)) if i = j.
Therefore, this lemma follows from above estimates.
Lemma 5.4. [37] Let ξ ∈ Ω, there holds
P ψ 0 µ,ξ (x) = n − 2 2 a 2 µ n−2 2 G(x, ξ) + o ε | ln ε| 2 , x ∈ Ω, and P ψ h µ,ξ (x) = a 2 µ n 2 ∂G ∂ξ h (x, ξ) + o ε | ln ε| 2 n−1 n−2 if h = 1, · · · , n, x ∈ Ω,
as ε → 0 uniformly on compact sets of Ω \ {ξ}, where a 2 is given in Proposition 2.2.
Lemma 5.5. [18] Let θ > 0 and u = k i=1 u i , i = 1, · · · , n, if ε > 0 small enough, for u, u i , v ∈ R, p = 2 * − 1, it holds that (1) |f ε (u) − f 0 (u)| ≤ ε|u| p ln ln(e + |u|), (2) f ′ ε (u) ≤ C|u| p−1 , (3) |f ′ ε (u) − f ′ 0 (u)| ≤ ε|u| p−1 p ln ln(e + |u|) + 1 ln(e+|u|) ,(4)|f ′ ε (u + v) − f ′ ε (u)| ≤ C(|u| p−2 + |v| p−2 )|v| if n ≤ 6, C(|v| p−1 + ε|u| p−1 ) if n > 6,(5)f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) n 2 = O ε | ln ε| 2 if 3 ≤ n ≤ 5, O ε | ln ε| 2 ln ε | ln ε| 2 if n = 6, O ε | ln ε| 2 −n+8 n−2 if n ≥ 7, (5.11) f ε (V ) − k i=1 (−1) i f 0 (P U µ i ,ξ i ) 2n n+2 = O ε ln ln ε | ln ε| 2 if 3 ≤ n ≤ 6, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 7, (5.12)
(5.13) |f ′ ε (V ) − f ′ 0 (V )| n 2 = O ε ln ln ε | ln ε| 2 .
Proof. Let us estimate (5.11). One haŝ
Ω f ′ 0 (V ) − k i=1 (−1) i f ′ 0 (P U µ i ,ξ i ) n 2 dx =ˆΩ \B(ξ,ρ) V p−1 − k i=1 (−1) i (P U µ i ,ξ i ) p−1 n 2 dx + k l=1ˆA l V p−1 − k i=1 (−1) i (P U µ i ,ξ i ) p−1 n 2 dx.
We estimate the first term
Ω\B(ξ,ρ) V p−1 − k i=1 (−1) i (P U µ i ,ξ i ) p−1 n 2 dx ≤ k i=1ˆΩ \B(ξ,ρ) U (p−1) n 2 µ i ,ξ i dx ≤ C k i=1 µ n i = O ε | ln ε| 2 n n−2 .
For any l, by the mean value theorem, there exists t = t(x) ∈ [0, 1] such that
A l V p−1 − k i=1 (−1) i (P U µ i ,ξ i ) p−1 n 2 dx =ˆA l (−1) l P U µ l ,ξ l + k i =l (−1) i P U µ i ,ξ i p−1 − (−1) l (P U µ l ,ξ l ) p−1 − k i =l (−1) i (P U µ i ,ξ i ) p−1 n 2 dx ≤CˆA l (−1) l P U µ l ,ξ l + t k i =l (−1) i P U µ i ,ξ i p−2 k i =l (−1) i P U µ i ,ξ i n 2 dx + C k i =lˆA l |P U µ i ,ξ i | (p−1) n 2 dx ≤CˆA l (−1) l+i (P U µ l ,ξ l ) p−2 k i =l P U µ i ,ξ i n 2 dx + C k i =lˆA l |P U µ i ,ξ i | (p−1) n 2 dx ≤C k i =lˆA l U p−2 µ l ,ξ l U µ i ,ξ i n 2 dx + C k i =lˆA l |U µ i ,ξ i | (p−1) n 2 dx.
If i = l, by (3.24), let x − ξ = µ i y, then
A l |U µ i ,ξ i | (p−1) n 2 dx ≤CˆA l µ n−2 2 i (µ 2 i + |x − ξ i | 2 ) n−2 2 (p−1) n 2 dx =Cµ n− n−2 2 (p−1) n 2 iˆA l µ i 1 (1 + |y − σ i | 2 ) n dy = O ε | ln ε| 2 n n−2 .
If n > 6, by (3.24), one haŝ
A l U p−2 µ l ,ξ l U µ i ,ξ i n 2 dx ≤ CˆA l µ −n+6 2 l (µ 2 l + |x − ξ l | 2 ) −n+6 2 n 2 µ n−2 2 i (µ 2 i + |x − ξ i | 2 )σ l | n(−n+6) 2 1 (1+|y−σ i | 2 ) n−2 2 n 2 dy if l > i, O(µ n− n−2 2 n 2 i µ − −n+6 2 n 2 l )´A l µ i 1 (1+|y−σ i | 2 ) n−2 2 n 2 dy if l < i, = O(µ n− n−2 2 n 2 i ( µ l µ i ) −n+6 2 n 2 ( ε | ln ε| 2 ) n n−2 if l > i, O ( µ i µ l ) −n+6 2 n 2 ( ε | ln ε| 2 ) n n−2 if l < i, = O ε | ln ε| 2 −n+6 n−2 n 2 + n n−2 .
If n < 6, it holdŝ Similar to the proof of (5.11), we obtain
f 0 (V ) − k i=1 (−1) i f 0 (P U µ i ,ξ i ) 2n n+2 = O ε | ln ε| 2 if 3 ≤ n ≤ 5, O ε | ln ε| 2 ln ε | ln ε| 2 if n = 6, O ε | ln ε| 2 n+2 2(n−2)
if n ≥ 7.
(5.15)
On the other hand, by Lemma 5.5, there holdŝ
Ω f ε (V ) − f 0 (V ) 2n n+2 dx ≤εˆΩ |V p ln ln(e + V )| 2n n+2 dx ≤εˆΩ k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx ≤εˆΩ \B(ξ,ρ) k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx + ε k l=1ˆA l k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx. (5.16)
We now observe that
Ω\B(ξ,ρ) k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx ≤C k i=1ˆΩ \B(ξ,ρ) U p µ i ,ξ i ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx ≤C k i=1 µ n i ln ln e + k i=1 (−1) i µ n−2 2 i 2n n+2 dy ≤C ε | ln ε| 2 n n−2 ln ln ε | ln ε| 2 2n n+2 .
For the second integral in (5.16), from (3.18), and let x − ξ = µ l y, then
A l k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i 2n n+2 dx =ˆA l (−1) l U µ l ,ξ l + k i =l (−1) i U µ i ,ξ i p ln ln e + (−1) l U µ l ,ξ l + k i =l (−1) i U µ i ,ξ i 2n n+2 dx =ˆA l (−1) l U p µ l ,ξ l + O k i =l (−1) i U µ i ,ξ i ln ln e + (−1) l U µ l ,ξ l + k i =l (−1) i U µ i ,ξ i 2n n+2 dx =ˆA l (−1) l U p µ l ,ξ l ln ln e + (−1) l U µ l ,ξ l + k i =l (−1) i U µ i ,ξ i 2n n+2 dx + C k i =lˆA l U µ i ,ξ i ln ln e + (−1) l U µ l ,ξ l + k i =l (−1) i U µ i ,ξ i 2n n+2 dx. (5.17) =ˆΩ k i=1 (−1) i f 0 (U µ i ,ξ i ) − f 0 (V ) ψ h µ j ,ξ j dx +ˆΩ k i=1 (−1) i f 0 (P U µ i ,ξ i ) − k i=1 (−1) i f 0 (U µ i ,ξ i ) ψ h µ j ,ξ j dx +ˆΩ f 0 (V ) − f ε (V ) ψ h µ j ,ξ j dx = J 1 + J 2 + J 3 .
Estimate of J 1 : One has
J 1 =ˆΩ k i=1 (−1) i f 0 (U µ i ,ξ i ) − f 0 (V ) ψ h µ j ,ξ j dx = k i=1ˆΩ (−1) i f 0 (U µ i ,ξ i )ψ h µ j ,ξ j dx −ˆΩ f 0 (V )ψ h µ j ,ξ j dx = k i=1 (−1) iˆΩ f 0 (U µ i ,ξ i )P ψ h µ j ,ξ j dx + =a 2 2 µ n 2 j µ n−2 2 i ∂G ∂ξ h j (ξ i , ξ j ) + o ε | ln ε| 2 n−1 n−2 . (5.22)
If h = 1, · · · , n and i = j, from (5.3) and (2.10), we havê
Ω f 0 (U µ i ,ξ i )P ψ h µ i ,ξ i dx =ˆΩ U 2 * −1 µ i ,ξ i P ψ h µ i ,ξ i dx =ˆΩ U 2 * −1 µ i ,ξ i ψ h µ,ξ i (x)dx − α n µ n 2 iˆΩ U 2 * −1 µ i ,ξ i ∂ ξ h i H(x, ξ i )dx + o ε | ln ε| 2 n−1 n−2 =µ n− n+2 2 +1− n−2 2 iˆΩ −ξ i µ i U 2 * −1 (y)ψ h (y)dy − α n µ n 2 − n+2 2 +n i ∂ ξ h i H(ξ i , ξ i )ˆΩ −ξ i µ i U 2 * −1 (y)dy + o ε | ln ε| 2 n−1 n−2 =µ iˆΩ −ξ i µ i U 2 * −1 (y)ψ h (y)dy − α n µ n−1 i ∂ ξ h i H(ξ i , ξ i )ˆΩ −ξ i µ i U 2 * −1 (y)dy + o ε | ln ε| 2 n−1 n−2 = − α n a 2 µ n−1 i ∂H ∂ξ h i (ξ i , ξ i ) + o ε | ln ε| 2 n−1 n−2 . (5.23)
It remains to estimate the second term in (5.19), if h = 0, by (5.2), it holdŝ If h = 1, · · · , n, by (5.3), we obtain Similarly, for h = 1, · · · , n, we get =o ε | ln ε| 2 n−1 n−2 .
Ω f 0 (U µ i ,ξ i )(ψ 0 µ j ,ξ j − P ψ 0 µ j ,Ω f 0 (U µ i ,ξ i )(ψ h µ j ,ξ j − P ψ h µ j ,
For h = 1, · · · , n and j > i,
A l k i =l (−1) i P U µ i ,ξ i ψ h µ j ,ξ j dx ≤ k i =lˆA l |U µ i ,ξ i ψ h µ j ,ξ j |dx =(n − 2)α 2 n k i =l µ n 2 jˆA l µ l µ − n−2 2 i (1 + |y − σ i | 2 ) n−2 2 (y h − σ j µ j µ l )µ i ( µ j µ i ) 2 + |y − σ j µ j µ i | 2 n 2 µ n i µ n i dy =o ε | ln ε| 2 n−1 n−2 .
Therefore, we haveˆΩ
f 0 (V )ψ h µ j ,ξ j dx = a 3 µ l+1 µ l n−2 2 g(σ l ) + o ε | ln ε| 2 n−1 n−2 if l = 1, · · · , k − 1, h = 0,J 2 = k i=1ˆΩ (−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) ψ h µ j ,ξ j dx = k i=1ˆΩ (−1) i f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i )ψ h µ j ,ξ j dx + k i=1ˆΩ (−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) ψ h µ j ,ξ j dx =(2 * − 1) k i=1 µ n−2 2 iˆΩ U 2 * −2 µ i ,ξ i − α n H(x, ξ i ) + O(µ 2 i ) ψ h µ j ,ξ j dx −ˆΩ(−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′ 0 (U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) ψ h µ j ,ξ j dx.
Moreover, for l = 0, · · · , n, by Hölder inequality, (5.6), (5.7) and (5.10), one haŝ
Ω (−1) i f 0 (P U µ i ,ξ i ) − f 0 (U µ i ,ξ i ) − f ′ Moreover,ˆΩ V p ln ln(e + V ) ψ h µ j ,ξ j dx =ˆΩ k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i ψ h µ j ,ξ j dx − ˆΩ k i=1 (−1) i U µ i ,ξ i p ln ln e + k i=1 (−1) i U µ i ,ξ i − V p ln ln(e + V ) ψ h µ j ,ξ j dx. (5.33)
Let us set h(u) = u p ln ln(e + u), by the mean value theorem, one has
0 ≤ h(u) − h(v) ≤ Cu p−1 ln ln(e + u) + 1 (u − v) for 0 ≤ v ≤ u. ThenˆΩ k i=1 (−1) i U µ i ,ξ i p−1 ln ln e + k i=1 (−1) i U µ i ,ξ i + 1 k i=1 (−1) i U µ i ,ξ i − V ψ h µ j ,ξ j dx = k i=1ˆA i k i=1 (−1) i U µ i ,ξ i p−1 ln ln e + k i=1 (−1) i U µ i ,ξ i + 1 k i=1 (U µ i ,ξ i − P U µ i ,ξ i )ψ h µ j ,ξ j dx + o ε | ln ε| 2 .
Moreover, on each annulus A i , i = 1, · · · , k, if h = 0, by Lemma 5.1 , (2.9) and the Lemma 5.5, we change variable setting µ i y = x − ξ, then
A i (−1) i U µ i ,ξ i + (−1) j k j =i U µ j ,ξ j p−1 ln ln e + (−1) i U µ i ,ξ i + (−1) j k j =i U µ j ,ξ j + 1 × k i=1 (−1) i (U µ i ,ξ i − P U µ i ,ξ i )ψ 0 µ j ,ξ j dx = α n k i=1 µ n−2 2 iˆA i (−1) i U p−1 µ i ,ξ i ln ln(e + (−1) i U µ i ,ξ i ) + 1 (H(x, ξ i ) + O(µ 2 i ))ψ 0 µ j ,ξ j dx +o ε | ln ε| 2 = α n k i=1 µ n−2 2 iˆA i µ i µ −2 i (1 + |y − σ i | 2 ) 2 ln ln e + µ − n−2 2 i (1 + |y − σ i | 2 ) n−2 2 + 1 × H(ξ i + µ i y − µ i σ i , ξ i ) + O(µ 2 i ) |µ i y − µ j σ j | 2 − µ 2 j µ 2 j + |µ i y − µ j σ j | 2 n 2 dy + o ε | ln ε| 2 ≤ α n k i=1 µ n−2 2 i H(ξ i , ξ i ) + O(µ 2 i ) µ −(n−2) j (ln | ln µ i | + 1) ×´A i µ i 1 (1+|y−σ i | 2 ) 2 1 |y| n−2 dy + o ε | ln ε| 2 if j < i, α n k i=1 µ − n−2 2 i (ln | ln µ i | + 1) H(ξ i , ξ i ) + O(µ 2 i ) ×´A i µ i 1 (1+|y−σ i | 2 ) 2 1 |y− µ j µ i σ j | n−2 dy + o ε | ln ε| 2 if j > i,
Bartsch et al.
ln ln(e + µ −θ u) = ln ln(µ −θ ) + ln 1 + ln(e 1−θ| ln µ| +u) θ| ln µ| , (6) lim µ→0 | ln µ| ln 1 + ln(e 1−θ| ln µ| +u) θ| ln µ| = 1 θ ln u, where C is a positive constant. Lemma 5.6. There holds
2 l + |µ i y − µ l σ l | 2 ) |y − σ i | 2 )
H
(µ i y + ξ i , ξ j )dy + o(µ
ξ j )H(µ i y + ξ i , ξ j )dy + o(
0 (V )ψ h µ j ,ξ j dx =ˆΩ \B(ξ,ρ) f 0 (V )ψ h µ j ,ξ j dx + k l=1ˆA l f 0 (V )ψ h µ j ,ξ j dx.
3 and g(σ l ) are defined in Proposition 2.2. From (5.19)-(5.25) and (5.28), we obtain l = k, h = 1, · · · , n.
Estimate of J 2 : By Lemma 5.1, there holds
(U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) ψ h µ j ,ξ j dx.
(U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) ψ h µ j ,ξ j dx
(U µ i ,ξ i )(P U µ i ,ξ i − U µ i ,ξ i ) ψ h µ j ,ξ j dx
Acknowledgments. The authors were supported by National Natural Science Foundation of China 11971392.A similar estimate can be obtained for n = 6, we prove thatThus,(5.11)holds.Let us now estimate(5.12), one has.(5.14)For i > l, by Lemma 5.5, we havê A l U p µ l ,ξ l ln ln e + (−1) l U µ l ,ξ l + k i =lIn a same way, the estimate of second term in (5.17) isThus,Combining (5.14), (5.15) and (5.18), we obtain(5.12). Similar to the proof of (5.16), the estimate (5.13) holds.Lemma 5.7. There holdswhere a 1 -a 4 and g(σ i ) are given in Proposition 2.2 for i = 1, · · · , k.Proof. We haveFor the first term in(5.19), if h = 0 and i = j, from Lemma 5.4, we deducêLet h = 0 and i = j, from (5.2) and (2.10), there holdŝIf h = 1, · · · , n and i = j, from Lemma 5.4, we getIf h = 0, by (2.9), a direct computation shows thatIf h = 1, · · · , n, we obtainOn the other hand, we estimate the second term in(5.26),The second term in(5.27).The last term in(5.27). For h = 0 and j > i,If h = 0 and j = i, by (2.9), we getwhere a 1 is given in Proposition 2.2. If h = 1, · · · , n and j = i, by (2.10), one hasIn a same way, if h = 1, · · · , n and j = i, we haveIf h = 0 and j = i,As a consequence, there holdsEstimate of J 3 : By Taylor expansion with respect to ε, we haveFor the second term in (5.30), from Lemma 5.5 and the annulus given in(3.18), it holdŝand on each annulus A i , we change variable setting µ i y = x − ξ, for h = 0, by (2.9), thenSimilarly, for h = 1, · · · , n, we havêThus the second term in(5.30)becomeŝConsequently,By the same argument, for h = 1, · · · , n, we havêTherefore, the second term in(5.33)becomeŝFor the first term in (5.33), if j = i, from Lemma 5.5, (2.9), let x − ξ = µ i y, then, · · · , n. Since ψ(y) is a odd function, we deduce´R n Λ(y)dy = 0. Further, by Lemma 5.5, there holdsˆRHence, for µ i small enough, we conclude thatWhen h = 0, since U µ i ,ξ i is the unique positive solution of problem (1.4) and also, ψ 0On the other hand,ˆΩBy the same argument, if j = i, one haŝConsequently, if h = 1, · · · , n.Combining J 1 -J 3 , the proof of this lemma is completed.Finally, let us state that a 4 is a positive constant. From (2.8), polar coordinates, integrating by parts, changing variables (s = r 2 , dr = 1 2 s − 1 2 ds), and a fact that |∂B 1 (0)| = 2π
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People's Republic of China. Deng) School of Mathematics and Statistics Southwest University Chongqing. 400715Email address: [email protected]) School of Mathematics and Statistics Southwest University Chongqing 400715, People's Republic of China. Email address: [email protected]
People's Republic of China. Yu) School of Mathematics and Statistics Southwest University Chongqing. 400715Email address: [email protected]) School of Mathematics and Statistics Southwest University Chongqing 400715, People's Republic of China. Email address: [email protected]
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[
"On some conjectural series containing binomial coefficients and harmonic numbers",
"On some conjectural series containing binomial coefficients and harmonic numbers"
]
| [
"Chuanan Wei [email protected] \nSchool of Biomedical Information and Engineering\nHainan Medical University\n571199HaikouChina\n"
]
| [
"School of Biomedical Information and Engineering\nHainan Medical University\n571199HaikouChina"
]
| []
| Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper. | null | [
"https://export.arxiv.org/pdf/2306.02641v1.pdf"
]
| 259,075,622 | 2306.02641 | fb3183604257b18e9926ea989e4dceee9a384846 |
On some conjectural series containing binomial coefficients and harmonic numbers
5 Jun 2023
Chuanan Wei [email protected]
School of Biomedical Information and Engineering
Hainan Medical University
571199HaikouChina
On some conjectural series containing binomial coefficients and harmonic numbers
5 Jun 2023binomial coefficientsharmonic numbershypergeometric series AMS Subject Classifications: 33D1505A15
Binomial coefficients and harmonic numbers are important in many branches of number theory. With the help of the operator method and several summation and transformation formulas for hypergeometric series, we prove eight conjectural series of Z.-W. Sun containing binomial coefficients and harmonic numbers in this paper.
Introduction
For ℓ, n ∈ Z + , define the generalized ℓ-order harmonic numbers as When ℓ = 0, they reduce to classical harmonic numbers:
H n = n k=1 1 k .
For a nonnegative integer m, define the shifted-factorial to be (x) 0 = 1 and (x) m = x(x + 1) · · · (x + m − 1) when m ∈ Z + .
For a differentiable function f (x), define the derivative operator D x by
D x f (x) = d dx f (x).
Then it is ordinary to find that
D x (1 + x) r = (1 + x) r H r (x).
Several nice harmonic number identities from differentiation of the shifted-factorials can be viewed in the papers [8,10,11,17]. Define the digamma function ψ(x) as
ψ(x) = d dx log Γ(x) ,
where Γ(x) is the familiar gamma function. Furthermore, we can define the polygamma function ψ (n) (x) to be
ψ (n) (x) = d n+1 dx n+1 log Γ(x) = d n dx n ψ(x).
It is famous that the polygamma function satisfies the recurrence relation:
ψ (n) (x + 1) = ψ (n) (x) + (−1) n n! x n+1 . (1.1)
Two related special values of ψ ′ (x) (cf. [8]) read
ψ ′ 1 4 = π 2 + 8G, (1.2) ψ ′ 3 4 = π 2 − 8G, (1.3)
where G is the Catalan constant:
G = ∞ k=0 (−1) k (1 + 2k) 2 .
Sun [13,Corollary 1.4] provided the following supercongruence:
p−1 k=0 2k k 3k k (−216) k ≡ p 3 p−1 k=0 2k k 3k k 24 k (mod p 2 ), p−1 k=0 2k k 4k 2k (−192) k ≡ −2 p p−1 k=0 2k k 4k 2k 48 k (mod p 2 ), p−1 k=0 2k k 4k 2k (−4032) k ≡ −2 p p−1 k=0 2k k 4k 2k 63 k (mod p 2 ), p−1 k=0 2k k 4k 2k 576 k ≡ −2 p p−1 k=0 2k k 4k 2k 72 k (mod p 2 ),
where p > 3 is any prime and ( . p ) denotes the Legendre symbol. Some related supercongruences can be viewed in the papers [9,12,16].
Motivated by the works just mentioned, Sun [15,Equations (2.21), (2.25), (2.26), (2.23), (2.24)] proposed the following five conjectures containing binomial coefficients and harmonic numbers.
Theorem 1.1. ∞ k=0 2k k 3k k (−216) k (3H 3k − H k ) = log 8 9 ∞ k=0 2k k 3k k (−216) k , (1.4) ∞ k=0 2k k 4k 2k (−192) k (2H 4k − H 2k ) = 1 2 log 3 4 ∞ k=0 2k k 4k 2k (−192) k , (1.5) ∞ k=0 2k k 4k 2k (−4032) k (2H 4k − H 2k ) = 1 2 log 63 64 ∞ k=0 2k k 4k 2k (−4032) k , (1.6) ∞ k=0 2k k 4k 2k 72 k (2H 4k − H 2k ) = (log 3) ∞ k=0 2k k 4k 2k 72 k , (1.7) ∞ k=0 2k k 4k 2k 576 k (2H 4k − H 2k ) = 1 2 log 9 8 ∞ k=0 2k k 4k 2k 576 k . (1.8)
Motivated by the series from Mathematica:
∞ k=0 2k k 2 32 k = Γ(1/4) 2 2π √ π ,
Sun [15,Equation (2.19)] conjectured the following series containing binomial coefficients and harmonic numbers.
Theorem 1.2. ∞ k=0 2k k 2 32 k H (2) 2k − 1 4 H (2) k = Γ 1 4 2 π 2 − 8G 32π √ π . (1.9)
There exist a lot of interesting π-formulas in the literature. Two series for 1/π 2 due to Guillera [5,6] can be laid out as follows:
∞ k=0 (20k 2 + 8k + 1) 2k k 5 (−2 12 ) k = 8 π 2 , (1.10) ∞ k=0 (820k 2 + 180k + 13) 2k k 5 (−2 20 ) k = 128 π 2 .
(1.11)
For more conclusions on π-formulas, the reader is referred to the papers [1,4,7,14,18]. Encouraged by (1.10) and (1.11), Sun [15,Equations (4.19) and (4.24)] proposed the following two conjectures containing binomial coefficients and harmonic numbers.
Theorem 1.3. ∞ k=0 2k k 5 (−2 12 ) k (20k 2 + 8k + 1) 8H (2) 2k − 3H (2) k + 4 = 8 3 , (1.12) ∞ k=0 2k k 5 (−2 20 ) k (820k 2 + 180k + 13) 11H (2) 2k − 3H (2) k + 43 = 128 3 . (1.13)
The rest of the paper is organized as follows. According to the operator method and several summation and transformation formulas for hypergeometric series, we shall certify Theorems 1.1-1.3 in Sections 2-4, respectively.
Proof of Theorem 1.1
Above all, we shall give the following parametric generalizations of (1.4)-(1.8).
Theorem 2.1. Let x be a complex number. Then ∞ k=0 2k k 2 x k (2H 2k − H k ) = 1 2 log x x − 16 ∞ k=0 2k k 2 x k , (2.1) where |x| > 16, ∞ k=0 2k k 3k k x k (3H 3k − H k ) = log x x − 27 ∞ k=0 2k k 3k k x k , (2.2) where |x| > 27, ∞ k=0 2k k 4k 2k x k (2H 4k − H 2k ) = 1 2 log x x − 64 ∞ k=0 2k k 4k 2k x k , (2.3) where |x| > 64, ∞ k=0 3k k 6k 3k x k (6H 6k − 3H 3k − 2H 2k + H k ) = log x x − 432 ∞ k=0 3k k 6k 3k x k , (2.4)
where |x| > 432.
Proof. Following Bailey [2], define the hypergeometric by
r+1 F r a 1 , a 2 , . . . , a r+1 b 1 , b 2 , . . . , b r ; z = ∞ k=0 (a 1 ) k (a 2 ) k · · · (a r+1 ) k (1) k (b 1 ) k · · · (b r ) k z k .
Then Euler's transformation formula connecting two 2 F 1 series (cf. [2, P. 2]) may be stated as
2 F 1 a, b c ; x = (1 − x) c−a−b 2 F 1 c − a, c − b c ; x , (2.5)
where |x| < 1. It is routine to understand that the two series in (2.5) are both uniformly convergent for a ∈ C. Apply the operator D a on both sides of (2.5) to discover
∞ k=0 (a) k (b) k (1) k (c) k x k H k (a − 1) = −(1 − x) c−a−b {log(1 − x)} ∞ k=0 (c − a) k (c − b) k (1) k (c) k x k − (1 − x) c−a−b ∞ k=0 (c − a) k (c − b) k (1) k (c) k x k H k (c − a − 1). The c = a + b case of it becomes ∞ k=0 (a) k (b) k (1) k (a + b) k x k H k (a − 1) = −{log(1 − x)} ∞ k=0 (a) k (b) k (1) k (a + b) k x k − ∞ k=0 (a) k (b) k (1) k (a + b) k x k H k (b − 1). (2.6) Replacing x by 1/x, equation (2.6) can be manipulated as ∞ k=0 (a) k (b) k (1) k (a + b) k H k (a − 1) + H k (b − 1) x k = log x x − 1 ∞ k=0 (a) k (1 − a) k (1) k (a + b) k 1 x k . (2.7)
Choosing (a, b, x) → ( 1 2 , 1 2 , x 16 ) in (2.7) and using the following two relations:
( 1 2 ) 2 k (1) 2 k = 2k k 2 16 k , H k − 1 2 = 2H 2k − H k , we obtain (2.1). Fixing (a, b, x) → ( 1 3 , 2 3 , x 27 ) in (2.7)
and utilizing the following two relations:
( 1 3 ) k ( 2 3 ) k (1) 2 k = 2k k 3k k 27 k , H k − 1 3 + H k − 2 3 = 3H 3k − H k ,
we deduce (2.2). Setting (a, b, x) → ( 1 4 , 3 4 , x 64 ) in (2.7) and using the following two relations:
( 1 4 ) k ( 3 4 ) k (1) 2 k = 2k k 4k 2k 64 k , H k (− 1 4 ) + H k (− 3 4 ) = 4H 4k − 2H 2k ,
we arrive at (2.3). Taking (a, b, x) → ( 1 6 , 5 6 , x 432 ) in (2.7) and utilizing the following two relations:
( 1 6 ) k ( 5 6 ) k (1) 2 k = 3k k 6k 3k 432 k , H k (− 1 6 ) + H k (− 5 6 ) = 6H 6k − 3H 3k − 2H 2k + H k ,
we are led to (2.4
2 F 1 a, 1 − a b ; 1 2 = Γ( b 2 )Γ( 1+b 2 ) Γ( a+b 2 )Γ( 1−a+b 2 )
.
(3.1)
Now we begin to prove Theorem 1.2.
Proof of Theorem 1.2. We know that the series in (3.1) is uniformly convergent for a ∈ C. Employ D a on both sides of (3.1) to get
∞ k=0 1 2 k−1 (a) k (1 − a) k (1) k (b) k {H k (a − 1) − H k (−a)} = Γ( b 2 )Γ( 1+b 2 ) Γ( a+b 2 )Γ( 1−a+b 2 ) × ψ 1 − a + b 2 − ψ a + b 2 . (3.2)
Via the relation:
H k (a − 1) − H k (−a) = k i=1 1 a − 1 + i − k i=1 1 −a + i = k i=1 1 − 2a (a − 1 + i)(−a + i) ,
equation (3.2) can be reformulated as
∞ k=0 1 2 k−1 (a) k (1 − a) k (1) k (b) k k i=1 1 (a − 1 + i)(−a + i) = Γ( b 2 )Γ( 1+b 2 ) Γ( a+b 2 )Γ( 1−a+b 2 ) × ψ( 1−a+b 2 ) − ψ( a+b 2 ) 1 − 2a . (3.3)
By the L'Hôpital rule, we have
lim a→ 1 2 ψ( 1−a+b 2 ) − ψ( a+b 2 ) 1 − 2a = ψ ′ ( 1+2b 4 ) 2 .
Letting a → 1 2 in (3.3) and using the upper limit, there is
∞ k=0 1 2 k ( 1 2 ) 2 k (1) k (b) k H (2) 2k − 1 4 H (2) k = Γ( b 2 )Γ( 1+b 2 ) 16Γ( 1+2b 4 ) 2 ψ ′ 1 + 2b 4 . (3.4)
Choosing b = 1 in (3.4) and using (1.2), we find (1.9).
In this section, we shall also establish the following theorem similar to Theorem 1.2.
Theorem 3.1. ∞ k=0 2k k 2 32 k (1 + k) H (2) 2k − 1 4 H (2) k = Γ 3 4 2 π 2 + 8G − 16 4π √ π . (3.5) Proof. Fixing b = 2 in (3.4), there is ∞ k=0 2k k 2 32 k (1 + k) H (2) 2k − 1 4 H (2) k = Γ( 3 2 ) 16Γ( 5 4 ) 2 ψ ′ 5 4 . (3.6)
According to (1.1) and (1.2), it is easy to show that
ψ ′ 5 4 = ψ ′ 1 4 − 16 = π 2 + 8G − 16. (3.7)
So the combination of (3.6) and (3.7) produces (3.5).
Proof of Theorem 1.3
In order to prove Theorem 1.3, we need Dougall's 5 F 4 summation formula (cf. [2, P. 27]):
5 F 4 a, 1 + a 2 , b, c, d a 2 , 1 + a − b, 1 + a − c, 1 + a − d ; 1 = Γ(1 + a − b)Γ(1 + a − c)Γ(1 + a − d)Γ(1 + a − b − c − d) Γ(1 + a)Γ(1 + a − b − c)Γ(1 + a − b − d)Γ(1 + a − c − d) , (4.1) where R(1 + a − b − c − d) >∞ k=0 (c) k (d) k (e) k (1 + a − b − c) k (1 + a − b − d) k (1 + a − b − e) k (1 + a − c) k (1 + a − d) k (1 + a − e) k (1 + 2a − b − c − d − e) k × (−1) k (1 + a − b) 2k α k (a, b, c, d, e) = ∞ k=0 (a + 2k) (b) k (c) k (d) k (e) k (1 + a − b) k (1 + a − c) k (1 + a − d) k (1 + a − e) k , (4.2) where R(1 + 2a − b − c − d − e) > 0 and α k (a, b, c, d, e) = (1 + 2a − b − c − d + 2k)(a − e + k) 1 + 2a − b − c − d − e + k + (1 + a − b − c + k)(1 + a − b − d + k)(e + k) (1 + a − b + 2k)(1 + 2a − b − c − d − e + k)
.
Setting e = a in (4.2) and calculating the series on the right-hand side by (4.1), we discover
∞ k=0 (a) k (c) k (d) k (1 − b) k (1 + a − b − c) k (1 + a − b − d) k (1) k (1 + a − c) k (1 + a − d) k (2 + a − b − c − d) k × (−1) k (1 + a − b) 2k β k (a, b, c, d) = Γ(1 + a − b)Γ(1 + a − c)Γ(1 + a − d)Γ(2 + a − b − c − d) Γ(1 + a)Γ(1 + a − b − c)Γ(1 + a − b − d)Γ(1 + a − c − d) , where β k (a, b, c, d) = k(1 + 2a − b − c − d + 2k) a + (a + k)(1 + a − b − c + k)(1 + a − b − d + k) a(1 + a − b + 2k) . The (a, b, c) = ( 1 2 , 1 2 , 1 − d) case of it can be expressed as ∞ k=0 −1 4 k ( 1 2 ) k (d) 2 k (1 − d) 2 k (1) 3 k ( 1 2 + d) k ( 3 2 − d) k (d − d 2 + 2k + 5k 2 ) = 1 − 2d π tan(dπ). (4.3)
Notice that the series in (4.3) is uniformly convergent for d ∈ C. Apply D d on both sides of (4.3) to obtain
∞ k=0 −1 4 k ( 1 2 ) k (d) 2 k (1 − d) 2 k (1) 3 k ( 1 2 + d) k ( 3 2 − d) k (d − d 2 + 2k + 5k 2 ) × 2H k (d − 1) − 2H k (−d) + H k 1 2 − d − H k d − 1 2 + ∞ k=0 −1 4 k ( 1 2 ) k (d) 2 k (1 − d) 2 k (1) 3 k ( 1 2 + d) k ( 3 2 − d) k (1 − 2d) = (1 − 2d) sec 2 (dπ) − 2 π tan(dπ).
Dividing both sides of the last equation by (1 − 2d), we have
∞ k=0 −1 4 k ( 1 2 ) k (d) 2 k (1 − d) 2 k (1) 3 k ( 1 2 + d) k ( 3 2 − d) k (d − d 2 + 2k + 5k 2 ) × k i=1 2 (d − 1 + i)(−d + i) − k i=1 1 (d − 1 2 + i)( 1 2 − d + i) + ∞ k=0 −1 4 k ( 1 2 ) k (d) 2 k (1 − d) 2 k (1) 3 k ( 1 2 + d) k ( 3 2 − d) k = sec 2 (dπ) − 2 π tan(dπ) 1 − 2d . (4.4)
By the L'Hôpital rule, there holds
lim d→ 1 2 sec 2 (dπ) − 2 π tan(dπ) 1 − 2d = 2 3 . (4.5)
Letting a → 1 2 in (4.4) and using (4.5), we deduce (1.12). Recollect the following transformation formula for hypergeometric series (cf. [3, Theorem 32]):
∞ k=0 (−1) k (b) k (c) k (d) k (e) k (1 + a − b − c) k (1 + a − b − d) k (1 + a − b − e) k (1 + a − b) 2k (1 + a − c) 2k (1 + a − d) 2k (1 + a − e) 2k × (1 + a − c − d) k (1 + a − c − e) k (1 + a − d − e) k (1 + 2a − b − c − d − e) 2k λ k (a, b, c, d, e) = ∞ k=0 (a + 2k) (b) k (c) k (d) k (e) k (1 + a − b) k (1 + a − c) k (1 + a − d) k (1 + a − e) k ,(4.6)
where
R(1 + 2a − b − c − d − e) > 0 and λ k (a, b, c, d, e) = (1 + 2a − b − c − d + 3k)(a − e + 2k) 1 + 2a − b − c − d − e + 2k + (e + k)(1 + a − b − c + k) (1 + a − b + 2k)(1 + a − d + 2k) × (1 + a − b − d + k)(1 + a − c − d + k)(2 + 2a − b − d − e + 3k) (1 + 2a − b − c − d − e + 2k)(2 + 2a − b − c − d − e + 2k) + (c + k)(e + k)(1 + a − b − c + k)(1 + a − b − d + k) (1 + a − b + 2k)(1 + a − c + 2k)(1 + a − d + 2k)(1 + a − e + 2k) × (1 + a − b − e + k)(1 + a − c − d + k)(1 + a − d − e + k) (1 + 2a − b − c − d − e + 2k)(2 + 2a − b − c − d − e + 2k)
.
Taking e = a in (4.6) and evaluating the series on the right-hand side by (4.1), we get
∞ k=0 (−1) k (a) k (b) k (c) k (d) k (1 − b) k (1 − c) k (1 − d) k (1) 2k (1 + a − b) 2k (1 + a − c) 2k (1 + a − d) 2k × (1 + a − b − c) k (1 + a − b − d) k (1 + a − c − d) k (2 + a − b − c − d) 2k θ k (a, b, c, d) = Γ(1 + a − b)Γ(1 + a − c)Γ(1 + a − d)Γ(2 + a − b − c − d) Γ(1 + a)Γ(1 + a − b − c)Γ(1 + a − b − d)Γ(1 + a − c − d) ,
where θ k (a, b, c, d)
= 2k(1 + 2a − b − c − d + 3k) a + (a + k)(1 + a − b − c + k) a(1 + a − b + 2k) × (1 + a − b − d + k)(1 + a − c − d + k)(2 + a − b − d + 3k) (1 + a − d + 2k)(2 + a − b − c − d + 2k) + (a + k)(c + k)(1 − b + k)(1 − d + k) a(1 + 2k)(1 + a − b + 2k)(1 + a − c + 2k) × (1 + a − b − c + k)(1 + a − b − d + k)(1 + a − c − d + k) (1 + a − d + 2k)(2 + a − b − c − d + 2k) . The (a, b, c) = ( 1 2 , 1 2 , 1 − d) case of it reads ∞ k=0 (−1) k ( 1 2 ) 4 k (d) 3 k (1 − d) 3 k (1) 3 2k ( 1 2 + d) 2k ( 3 2 − d) 2k Ω k (d) = 1 − 2d π tan(dπ),(4.7)
where Ω k (d) = 2k(1 + 6k) + (d + k)(1 − d + k)(2 − d + 3k) 3 − 2d + 4k
+ (d + k)(1 − d + k) 3 (1 + 2d + 4k)(3 − 2d + 4k)
.
Realize that the series in (4.7) is uniformly convergent for d ∈ C. Employ D d on both sides of (4.7) to gain
∞ k=0 (−1) k ( 1 2 ) 4 k (d) 3 k (1 − d) 3 k (1) 3 2k ( 1 2 + d) 2k ( 3 2 − d) 2k Ω k (d) × 3H k (d − 1) − 3H k (−d) + H 2k 1 2 − d − H 2k d − 1 2 + ∞ k=0 (−1) k ( 1 2 ) 4 k (d) 3 k (1 − d) 3 k (1) 3 2k ( 1 2 + d) 2k ( 3 2 − d) 2k D d Ω k (d) = (1 − 2d) sec 2 (dπ) − 2 π tan(dπ).
Dividing both sides of the last equation by (1 − 2d), we have
∞ k=0 (−1) k ( 1 2 ) 4 k (d) 3 k (1 − d) 3 k (1) 3 2k ( 1 2 + d) 2k ( 3 2 − d) 2k Ω k (d) × k i=1 3 (d − 1 + i)(−d + i) − 2k i=1 1 (d − 1 2 + i)( 1 2 − d + i) + ∞ k=0 (−1) k ( 1 2 ) 4 k (d) 3 k (1 − d) 3 k (1) 3 2k ( 1 2 + d) 2k ( 3 2 − d) 2k D d Ω k (d) 1 − 2d = sec 2 (dπ) − 2 π tan(dπ) 1 − 2d . (4.8)
Letting a → 1 2 in (4.8) and utilizing (4.5), we catch hold of (1.13).
k) ℓ ,where x is a complex variable. The x = 0 case of them are the ℓ-order harmonic numbers
).Now we are ready to prove Theorem 1.1.
Proof of Theorem 1.1. The x = −216 case of (2.2) is (1.4). Selecting x = −192, −4032, 72
and 576 in (2.3), we catch hold of (1.5), (1.6), (1.7) and (1.8), respectively.
3 Proof of Theorem 1.2
For the goal of proving Theorem 1.2, we require Bailey's 2 F 1 summation formula (cf. [2, P.
17]):
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| []
|
[
"A Simple Yet High-Performing On-disk Learned Index: Can We Have Our Cake and Eat it Too?",
"A Simple Yet High-Performing On-disk Learned Index: Can We Have Our Cake and Eat it Too?"
]
| [
"Hai Lan ",
"Zhifeng Bao ",
"J Shane Culpepper ",
"Renata Borovica-Gajic \nThe University of Melbourne\n† PingCAP\n",
"Yu Dong ",
"Hai Lan ",
"Zhifeng Bao ",
"J Shane Culpepper ",
"Renata Borovica-Gajic \nThe University of Melbourne\n† PingCAP\n",
"Yu Dong ",
"\nRMIT University\n\n"
]
| [
"The University of Melbourne\n† PingCAP",
"The University of Melbourne\n† PingCAP",
"RMIT University\n"
]
| []
| While in-memory learned indexes have shown promising performance as compared to B+-tree, most widely used databases in real applications still rely on disk-based operations. Based on our experiments, we observe that directly applying the existing learned indexes on disk suffers from several drawbacks and cannot outperform a standard B+-tree in most cases. Therefore, in this work we make the first attempt to show how the idea of learned index can benefit the on-disk index by proposing AULID, a fully on-disk updatable learned index that can achieve state-of-the-art performance across multiple workload types. The AULID approach combines the benefits from both traditional indexing techniques and the learned indexes to reduce the I/O cost, the main overhead under disk setting. Specifically, three aspects are taken into consideration in reducing I/O costs: (1) reduce the overhead in updating the index structure;(2) induce shorter paths from root to leaf node; (3) achieve better locality to minimize the number of block reads required to complete a scan. Five principles are proposed to guide the design of AULID which shows remarkable performance gains and meanwhile is easy to implement. Our evaluation shows that AULID has comparable storage costs to a B+-tree and is much smaller than other learned indexes, and AULID is up to 2.11x, 8.63x, 1.72x, 5.51x, and 8.02x more efficient than FITing-tree, PGM, B+-tree, ALEX, and LIPP. | null | [
"https://export.arxiv.org/pdf/2306.02604v1.pdf"
]
| 259,075,757 | 2306.02604 | cdb196b7cde1722952a65fcc5aa7fc1d789a6db7 |
A Simple Yet High-Performing On-disk Learned Index: Can We Have Our Cake and Eat it Too?
Hai Lan
Zhifeng Bao
J Shane Culpepper
Renata Borovica-Gajic
The University of Melbourne
† PingCAP
Yu Dong
Hai Lan
Zhifeng Bao
J Shane Culpepper
Renata Borovica-Gajic
The University of Melbourne
† PingCAP
Yu Dong
RMIT University
A Simple Yet High-Performing On-disk Learned Index: Can We Have Our Cake and Eat it Too?
ACM Reference Format:
While in-memory learned indexes have shown promising performance as compared to B+-tree, most widely used databases in real applications still rely on disk-based operations. Based on our experiments, we observe that directly applying the existing learned indexes on disk suffers from several drawbacks and cannot outperform a standard B+-tree in most cases. Therefore, in this work we make the first attempt to show how the idea of learned index can benefit the on-disk index by proposing AULID, a fully on-disk updatable learned index that can achieve state-of-the-art performance across multiple workload types. The AULID approach combines the benefits from both traditional indexing techniques and the learned indexes to reduce the I/O cost, the main overhead under disk setting. Specifically, three aspects are taken into consideration in reducing I/O costs: (1) reduce the overhead in updating the index structure;(2) induce shorter paths from root to leaf node; (3) achieve better locality to minimize the number of block reads required to complete a scan. Five principles are proposed to guide the design of AULID which shows remarkable performance gains and meanwhile is easy to implement. Our evaluation shows that AULID has comparable storage costs to a B+-tree and is much smaller than other learned indexes, and AULID is up to 2.11x, 8.63x, 1.72x, 5.51x, and 8.02x more efficient than FITing-tree, PGM, B+-tree, ALEX, and LIPP.
INTRODUCTION
Nowadays, most widely used database systems still rely on on-disk indexing techniques for (at least) two reasons. First, the total index size may be larger than the main memory available -a consequence of growing data sizes in real applications [1]. Also, multiple indexes (not just one index) might be built to optimize workload-specific performance [40]; they are usually operationalized as a "secondary index", where the leaf nodes should be included when calculating the total storage requirements. Second, main memory is also a precious resource for efficient query processing to store intermediate results, e.g., a hash table in a hash join [9]. If most of the available memory is used to hold the indexes, query performance can be significantly degraded. On the other hand, most existing learned indexes are designed for main memory setting and try to reduce the search/insert overhead via different approaches: (1) use modelbased search instead of binary search [4,35] ( (1) vs. (log )); (2) have a smaller search range when employing a binary search [7,8];
(3) reduce the tree height by modeling the data distribution [7,35]; (4) support queries using cache-aware techniques [39]; (5) use a gapped array instead of a packed array [4,35].
When adapting the idea of learned indexes to a fully on-disk setting, most of these techniques are no longer useful since I/O costs are the dominant bottleneck. For example, when issuing a lookup query in a four-layer B+-tree, we find 93.6% of the total execution time is spent on I/O operations. Hence, reducing the number of block reads (and writes) is a critical performance factor.
An immediate question to ask then is how existing learned indexes perform on disk? To answer that, we implemented four stateof-the-art updatable learned indexes [4,7,8,35] on disk and compared them against a standard B+-tree across six workload types commonly encountered by a database. Figures 1(a)-(b) present the normalized throughput on COVID and FB, which are a representative of easy and hard dataset respectively, as per the dataset profiling in a recent experimental study on learned indexes [34]. We observe that although these learned indexes exhibit different strengths and weaknesses depending on the workload type and dataset distribution, none of them outperforms or achieves competitive performance to the B+-tree across all workload types on any dataset. This should come as no surprise to database designers, given that most research on learned indexes has focused on in-memory performance. The benefit of learned indexes in main memory and the shortage of current learned index on disk motivate us to develop a high-performing on-disk learned index. In the rest of this section, we will outline the challenges when building on-disk learned indexes, our design principles to mitigate them, followed by our solutions that align with these principles. Readers can refer to Figure 2 for an overview from challenges to our proposed design principles and solutions.
Challenges in Building a Fully On-disk Learned Index
When (re)implementing existing in-memory learned indexes on disk, several critical challenges discussed next frequently arise. For the reason of providing an intuitive illustration of these challenges, some experimental comparison and analysis are highlighted in between. Challenge 1: a learned index cannot guarantee to reduce I/O costs when searching data on disk. Figure 1(c) shows the average number of inner nodes, inner blocks, and total blocks per query for Lookup-Only and Scan-Only workloads on the FB dataset 1 . For LIPP, the total number of fetched nodes is reported, and the number of nodes in the scan is highlighted in the bracket of the fifth row. When combining the results for a Lookup-Only workload and a Scan-Only workload in Figure 1(b), we observe that the performance rank is directly correlated with the number of fetched blocks. In contrast to in-memory indexes, reducing the search overhead for each step does not help on-disk indexes. Instead, reading or writing a block from/to disk is the main overhead. In a Lookup-Only workload, among all the learned indexes, only LIPP fetches fewer blocks than a B+-tree and is more efficient than the B+-tree.
We also observe that, in contrast to the B+-tree, existing learned indexes have larger scan overheads, which, in practical terms, means fetching the next item becomes more expensive. For example, to support a scan, LIPP traverses many nodes, which incurs a higher I/O cost and leads to poorer performance.
Challenge 2: large insertion overheads. Current indexing techniques support an insertion using four steps: (1) find a slot to hold a new key-payload pair (Search); (2) do the insertion (Insert); (3) induce an index structure modifications operation (SMO) if necessary; (4) update various statistics (Stats), such as the total number of lookups and insertions, which determine when to induce SMOs. An SMO may create new nodes or re-construct an entire sub-tree during the insertion, which is necessary for the index to allocate empty slots, and for a learned index to benefit from future modelbased operations. Figure 1(d) shows the average latency breakdown per query for a Write-Only workload on the FB dataset. We observe that learned indexes have several shortcomings. ALEX and LIPP have a large overhead when updating statistics and performing an SMO. The FITing-tree, PGM, and ALEX incur a large overhead for the insertion. LIPP frequently induces an SMO to resolve conflicts between two keys, while ALEX re-writes large nodes (the leaf node) in every SMO. Both lead to large SMO overheads. The shift operations, which are used to obtain an empty slot to store a new key-payload pair in a FITing-tree, PGM, and ALEX, may span multiple blocks on disk -leading to more writes on disk.
AULID -Simple Is Better
In this paper, we show that the idea of learned index can benefit the on-disk index design by proposing AULID, an updatable learned index on disk. To address the above challenges, we propose five design principles for such an on-disk learned index in Section 3.1, directed toward supporting lower tree heights and lower SMO overheads. Then, we design AULID to meet these principles by PGM, a FITing-tree, and a B+-tree fetch the same number of inner nodes and blocks needed for a lookup and a scan. leveraging the idea of B+-tree and learned indexes in three ways (an overview from the challenges to our design principles and solutions can be found in Figure 2):
1 Leaf Node Layout. To reduce the insertion overheads, instead of using a learned model to search items for all layers, we use modelbased search only for inner nodes. This helps reduce the burden on leaf nodes in maintaining the benefits of model-based search. Specifically, we use a B+-tree styled layout for leaf nodes, which has a low overhead when updating the index. Since the majority of SMOs are on leaf nodes, a lightweight SMO mechanism for leaf nodes, as achieved with AULID, reduces the insertion overheads significantly. A B+-tree styled leaf node design also benefits scan operations in fetching the next item.
2 Inner Node Layout. After building a B+-tree styled leaf node, the path from the root node to a leaf node should be shorter than that in a B+-tree. Otherwise, the learned index cannot outperform a B+-tree for on-disk operations. The results in Figure 1(a)-(b) inspired us to adopt the Fastest Minimum Conflict Degree (FMCD) algorithm in LIPP [35] to reduce the tree depth. Although it is not the best indexing method for most workload types, it has the smallest number of fetched blocks for a lookup, making it suitable for on-disk indexes when attempting to reduce I/O costs. Moreover, a lookup is often the first step in other operations. For example, in a scan operation to find the position of a start key and in an insert operation to find a position to hold a new key-payload pair. Thus, a better performance of the lookup operation should boost the performance of other operations too. However, for certain workload types and datasets, e.g., OSM (a hard dataset) [34], directly applying the aforementioned inner node layout still reveals several shortcomings -a larger storage size and a lower throughput. To overcome these shortcomings, we introduce two new inner node types and design processing algorithms upon the new layout. For example, with our design, AULID is 1.18x more efficient on the Lookup-Only workload while only taking 0.84x storage on OSM (as compared to a B+-tree).
3 Structural Modification Operations. With a B+-tree styled leaf node, AULID already manages to achieve a lighter overhead in modifying the index structure due to a lower frequency in updating the inner nodes and a lighter overhead in updating the leaf node. However, the tree height in some region could grow and even become larger than that of a standard B+-tree. In turn, AULID will have a worse performance after lots of insertions. To avoid that from happening, we monitor the tree height of each branch and trigger a re-construct process to bound their tree height if needed.
, ,
To summarize, we make the following technical contributions:
• To the best of our knowledge, AULID is the first approach to employ the ideas of learned index to a fully on-disk setting to replace a traditional B+-tree 2 . • We propose five principles to guide the design of an on-disk updatable learned index (Section 3.1), and carefully design the indexing structure (Section 3.2 -3.3), the query processing algorithms (Section 4.1 -4.3), and an SMO mechanism, to achieve efficient reads and writes (Section 4.4). • We implement AULID in C++, conduct comprehensive experiments across a wide range of datasets and workloads and compare it against a B+-tree and on-disk implementations of existing in-memory learned indexes. Our evaluation shows that AULID has competitive storage costs to a B+tree and is much smaller than most other learned indexes. Performance-wise, AULID achieves up to 2.11x, 8.63x, 1.72x, 5.51x, and 8.02x larger throughput than FITing-tree, PGM, B+-tree, ALEX, and LIPP, respectively. We also conduct an in-depth evaluation on the benefits of the AULID design (Section 5).
RELATED WORK
Given that our work focuses on developing a fully on-disk learned index under the single-threaded setting, we start the literature review on learned indexes outside main memory, followed by an overview of in-memory ones and a discussion on the difference in concurrency support between in-memory case and on-disk case.
Learned Indexes Outside Main Memory. The authors of [1,3] studied how to use learned indexes on disk in a log-structured merge tree (LSM) [23]. A learned model is constructed for each Sorted Strings Table (SSTable), which is immutable after being created. Modification operations (insert, update, delete) are supported in an LSM framework. Models are rebuilt and dropped during periodic compaction processes. The LSM framework supports efficient writes at the cost of reads. In contrast, our work focuses on building an on-disk learned index with the hope of replacing the B+-tree. Two most recent studies [19,38] focus on the larger than main memory setting, where they pin part of the index in main memory and introduce different caching strategies. TreeLine [38] uses the partitioning algorithm proposed in PGM [7] to generate the leaf nodes and adopts a B+-tree to index them. The B+-tree in TreeLine is pinned in main memory, and a record-level caching strategy is used to cache frequently accessed items in main memory. FILM [19] builds a PGM index, stores it in main memory and uses one bit for each item to indicate the location of that item, i.e., in main memory or on disk. Moreover, FILM introduces a global chain and a local chain, to organize the segments at the last level and the items in the segment based on their access time, respectively. In this way, FILM can quickly locate cold items. However, FILM is designed for append-only insertions. Differently, our work focuses on storing the whole index on disk rather than only leaf nodes.
Lu et al. [18] propose APEX, a learned index for persistent memory (PM) [33]. APEX is a variant of ALEX, with several tailored designs: (1) different node size settings used in APEX -a larger inner node and a smaller data node to reduce SMO overhead; (2) a new probe-and-stash mechanism to resolve collisions without introducing unnecessary nodes' access; (3) concurrency control and recovery mechanisms are introduced to support simultaneous inserts and instant recovery. Different from APEX, on-disk operations are our focus where I/O costs are the main overhead.
Learned Indexes in Main Memory. Kraska et al. [14] was the first group to propose the idea of learned index, where a hierarchy of models, called RMI, was built to replace a B+ Tree for sorted 1-d data. The new approach can achieve 3x performance boost and 10x smaller index size. Given that RMI only supports lookup queries, subsequent studies [4,7,8,35] address this limitation using tailored index structures and new mechanisms for index structure modification. A FITing-tree [8] replaces the last layer of a B+tree with model-based search, and supports insertions by introducing buffers for each segment. PGM [7] uses a similar idea to the FITingtree, but it leverages model-based search for every layer based on an optimal partitioning algorithm [24]. PGM handles arbitrary insertions in an LSM tree [23] manner. Although the FITing-tree and PGM leverage model-based search, additional binary search operations are needed. ALEX [4] inserts a key-payload pair using model prediction, and hence manages to have accurate predictions in the inner nodes without binary search. At each data node (leaf node), ALEX uses a gapped array, where the key-payload pairs and empty slots are interleaved. The gapped array can reduce the frequency of shifts for new insertions. Without an error bound for the search range, ALEX uses an exponential search to find the target position. Model prediction in LIPP [35] tends to be accurate in every layer. LIPP has been shown to have better performance in practice than other learned indexes in most settings [34]. However, LIPP requires much more memory and is not efficient for a range query. Wu et al. [36] use Normalizing Flows [28] to transform a dataset so that it can be easily modeled with linear models, and extend the idea of LIPP by introducing a bucket node type. CARMI [39] is a cache-aware learned index which uses tailored partitioning algorithms and a prefetching mechanism for the in-memory setting. To verify the efficiency of learned indexes, several studies have conducted comprehensive experimental studies [21,34], detailed theoretical analysis [6], and performance analysis [20].
There are also several studies that propose read-only learned indexes [11,13,30] or use the model to boost B+-tree's performance [10,17] for main memory operations. Several other learned indexes have also been proposed as secondary indexes [12,37], tailored for multi-dimensional data [2,5,22], spatial data [16,[25][26][27], or string data [29,32].
Learned Index With Concurrency Support. Among these inmemory learned indexes, the XIndex [31], FINEdex [15], ALEX+ [34], and LIPP+ [34] support concurrent operations. All of them use optimistic locking, which associates a versioning lock at each node. With the node size being larger than a block at the leaf node, Tree-Line [38] proposes some locking strategies based on the node-level lock and block-level lock to support concurrent operations on the LSM data structure. Aligned with the latest study on the larger than main memory case [19], we focus on the single-threaded setting in this work. In a fully on-disk learned index with different sizes of node and block, the node-level lock and the block-level lock need to be introduced at the same time. Without a tailored mechanism, one cannot achieve a good scalability in a multi-core setting due to the block-level lock. Although TreeLine [38] has introduced these two locks, its inner nodes are pinned in main memory, which is easier than the above case. We believe our work is an important first step and our findings will help the community in the future, when designing a fully on-disk concurrent learned index.
AN OVERVIEW OF AULID
In this section, we first introduce the principles and highlights of the AULID design -addressing the challenges discussed in Section 1.2. Then, we present the AULID layout. Figure 2 illustrates two identified challenges and the associated design principles used to resolve them, as well as how these principles are reflected in our proposed solutions.
Design Principles
Based on the key properties arising from disk and learned indexes, we propose a number of principles to guide the design of AULID:
• P1. Reducing the Tree Height of the Index. Accessing each level in an index requires at least one disk access when an index is stored on disk. Reducing the tree height can reduce the number of disk access. • P2. Model-based Operations (Search and Insert). An index with a reduced height usually has larger nodes in certain levels of the index. Model-based operations help AULID quickly find search keys in a specific part of the node, without the need to access the entire node on disk. • P3. Lightweight Structure Modification Operations. Structure modification operations (SMOs) for the existing learned indexes incur a substantial amount of writes on disk. AULID should reduce the overhead of such SMO calls. • P4. Support Duplicate Index Keys. Duplicate (i.e. non-unique) index keys are common in real systems. Typically, they can be supported using a linked list in a main memory setting [35], but not on disk, since it leads to additional disk reads. .AULID must provide a lightweight method to fetch the next item efficiently.
Design Highlights
AULID uses a combination of existing and novel techniques to meet the above principles and achieve high performance on disk. AULID consists of inner nodes and leaf nodes, both of which are stored Figure 3: AULID Index Structure on disk. Leaf nodes, where most SMOs occur, are organized in a B+-tree manner. A low update cost at leaf nodes reduces the SMO overhead (P3). Moreover, AULID only uses the idea of learned index for inner nodes to index the maximum key of each leaf node, which leads to less frequent SMOs in updating the inner nodes (P3) and a low tree height in inner part (P1). Each leaf node is a packed array -it stores pointers to its siblings and its size is equal to the block size. Using the packed array and links to siblings, AULID can support efficient scan operations (P5). We optimize our inner nodes based on properties of the disk drive. Fast lookup time with the learned model means that AULID can efficiently locate target leaf nodes (P1, P2). To achieve robust performance on different datasets (i.e., different distributions), we also introduce two new node types for the inner nodes, a packed array and a two-layer B+-tree, with the purpose of reducing the number of SMOs for non-leaf nodes (P3). By proposing a tailored mechanism to handle duplicate keys inserted in inner nodes, AULID manages to store duplicate values with reduced on-disk costs (P4). To maintain the performance gains achieved from the learned model, AULID adjusts the index structure based on the tree height and bounds the tree height during insertions (P1).
Node Structure
The index structure of AULID is presented in Figure 3. Similar to existing indexes, AULID is composed of two components: the inner nodes which store the route information to leaf nodes, and the leaf nodes which store the key-payload pairs.
3.3.1 Metanode. Metanode in AULID stores (1) the physical address of the root node, (2) the linear model of the root node, and (3) the physical address of the last leaf node, as well as the minimum and maximum keys of that node. We store the metanode in main memory, which requires only 80 bytes, a negligible main memory overhead.
Inner
Nodes. AULID has two node types in the inner part, a mixed node type and a packed array node type. And there are three types of slot in the inner part: NODE, NULL, and DATA. The NODE slot stores the pointer to its child. The NULL slot is the empty slot and can be converted to NODE or DATA. The DATA slot stores the key-payload pair. Each mixed node has a model to predict the slots for a key search and can include three different slot types above. AULID stores the model in the parent node, combined with the physical address. If we store the model at the starting address of a mixed node, the large fanout for mixed nodes increases the chance that the predicted position and the model are located in different blocks. Thus, two blocks must be fetched from disk for each level in the tree. In contrast, when storing the model in the parent node, AULID only fetches one block per level.
The NODE slot in AULID can be further divided into three types: (1) a pointer to the packed array of fixed size. That is, the first slot in Node A in Figure 3; (2) a small B+-tree node that contains at most four child nodes -the fourth slot in Node A; (3) a pointer to another mixed node -the sixth slot in Node A. For the first case, we introduce four different packed array types, each with a fixed size. The ℎ packed array type can store 2 +2 items of DATA types, where ranges from 1 to 4. A DATA slot in the inner nodes stores the physical address of a leaf node and the largest key it contains, i.e., the key-block pair. The second case is proposed to improve the performance for scenarios where the number of keys to be inserted into the same slot is greater than 64, but smaller than 1020 (which will be explained later in Section 3.3.2). This indicates the conflict degree for the region. If we create a new mixed nodeto hold these keys, there can be key conflicts in the new node. This leads to a larger tree height (more than two layers). Conversely, a two-layer B+-tree can be used to hold the nodes and help AULID to bound the tree height for the region. Therefore, AULID is able to bound the number of fetched blocks. Also, the design of the packed array and the two-layer B+tree can support more newly inserted key-block pairs (the routing knowledge to the leaf nodes) to be stored with low overheads. Using the packed array and a two-layer B+-tree, AULID achieves a better empty slot ratio, and this translates to smaller storage costs.
The B+-tree node contains only four child nodes for the following reasons: (1) The conflict degrees in most of the test datasets we have used (except for one) 3 is less than 1000, which can be stored easily using a two-layer B+-tree. (2) A larger fanout requires more metadata (pivot keys and physical addresses) in a slot, and it increases the total storage cost significantly. A key-block pair occupies 16 bytes on disk in our implementation. Thus, a block with 4 KB can store 256 pairs. The first item records the item count for a two-layer B+-tree's leaf node. Four children can store at most 1020 items.
Leaf Nodes.
The leaf nodes have the same structure as a standard B+-tree. The DATA slot in the leaf node stores the keypayload pairs to be indexed. This layout design is based on the observation that most SMOs happen on leaf nodes as new keypayload pairs are added. Learned indexes need to read all of the items in a large leaf node and re-write them to disk to maintain the benefits of their unique structure, which incurs large I/O costs on disk. A lightweight SMO overhead for a leaf node design can help significantly reduce the number of SMO operations required (see the experiments in Section 5.2.2).
This simple yet elegant design in the leaf nodes has many other benefits. First, the link between siblings when using a packed storage layout requires no additional utility structures to perform efficiently when scan operations must locate the start of a query range. Second, the storage costs of the inner nodes can be significantly reduced by only storing the largest keys. In our experiments, 3 We have also tested all of the datasets proposed in a recent benchmark paper [34].
AULID has a similar storage size and bulkload time as a B+-tree on disk, which is better than other learned indexes. Third, reducing the items inserted into the inner nodes also decreases the SMO frequency and the number of items that must be processed. Last, AULID is able to efficiently support duplicate index keys when using a B+-tree styled leaf node.
AULID OPERATIONS
First, we present how AULID supports each type of operation, and then we discuss how structural modifications are supported.
Bulkload
AULID supports bulkload using two steps. In the first step, it creates leaf nodes to store the key-payload pairs using B+-tree styled leaf nodes. When building leaf nodes -with the exception on the last leaf node -AULID records the maximum key, and the physical address for each leaf node, i.e., the key-block pairs to be indexed in the inner nodes. For the final leaf node, AULID stores the minimum and maximum keys, as well as its physical address in a meta-node.
The second step builds the inner nodes for AULID over the keyblock pairs. We first use the Fastest Minimum Conflict Degree (FMCD) algorithm in LIPP to generate a linear model for a node. Given the collection of keys to be indexed and the number of slots that can be used, FMCD aims to generate a linear model under which the maximum number of keys inserted into the same slot is minimized, i.e., the smallest "conflict degree". Then, we insert the key using the resulting model. If only one key is inserted into a slot, this slot is labeled as DATA and used to store the key-block pair. Different from LIPP, AULID does not aggressively create a new node if more than one item is mapped to the same slot. Instead, we divide them into three cases depending on the size of the items that are mapped into the same slot: (1) If the size of the items mapped to one slot is smaller than 64, a packed array is created. (2) If the size is greater than 64 and less than 1020 (see explanation at the end of Section 3.3.2), a two-layer B+-tree is created with at most four child nodes. (3) Otherwise, a new mixed node is created to hold the keys.
Lookup & Scan
4.2.1 Lookup. Given a search key, we first check whether it belongs to the last leaf node by comparing it with the minimum key and the maximum key that are stored in the meta-node. The overhead of this operation is negligible as the meta-node resides in main memory. If the key belongs to the last leaf node, the leaf node is read from disk, and then a binary search is initiated. Otherwise, the inner nodes are searched to find the leaf node address where the search key should reside.
When traversing from root node to leaf node, five different cases of model prediction can occur (assume a mixed node is the root node):
• DATA Slot: A leaf node is fetched based on the physical address contained in it. If the key in the DATA slot is less than the search key, then we fetch the successor. • NODE Slot for a Packed Array: The packed array content is retrieved from disk, and a DATA slot is located to hold the search key. It is then processed in the same way as the DATA Slot case.
• NODE Slot for a B+-tree: Just as in a standard B+-tree, a child node is found which holds the search key (if it exists), and then it is fetched and processed in the same way as the NODE Slot for a Packed Array. • NODE Slot for another mixed Node: The model from the node is used to predict which node to access next, and the search process is repeated. • NULL Slot: Using the monotonic linear function from AULID and indexing the largest keys for each leaf node, to find the next DATA slot we must search forward. For example, given a search key, suppose the predicted position is the 5 ℎ slot in Node A of Figure 3, which is a NULL slot. AULID will scan forward to find the next DATA slot using Node C and Node F.
Scan.
Given a query range [ , ], we first call a lookup operation to locate the leaf node where should reside, and the position of in the node. Then, we scan forward until reaching the last key . Using the links to sibling leaf nodes and the packed array, the next item can be quickly accessed without any additional utility structures, such as bitmaps in ALEX to differentiate empty slot, or traversing many nodes in LIPP.
Optimization for Reading.
When storing only the largest key of each leaf node in the inner nodes, AULID could issue additional I/O requests for two reasons:
• Issue 1: When traversing the inner nodes, AULID may need to find the predecessor (the left sibling of a target node) and extra I/O is needed to fetch that target. • Issue 2: If the predicted location is a NULL slot, a scan operation is triggered to locate the next DATA slot. For example, if the predicted position is the 5 ℎ slot in Node A of Figure 3, AULID needs to access Node C and Node F. Due to the large fanout of the inner nodes, Nodes A, C, and F are stored in different blocks, which could incur additional I/O costs. Note that with a monotonic linear function in AULID, these two cases cannot happen in a single lookup at the same time.
To address the first issue, if a DATA slot is found, instead of fetching the leaf node directly, we first check whether the key it contains is smaller than the search key. If so, we scan forward to find the next DATA slot. In contrast to a NULL slot, the scan operation is initiated only on the currently fetched block. The DATA slot found before is used if no new DATA slots are found in the same block. Otherwise, scanning forwards reads at least one additional block.
To address the second issue, AULID fulfills the preceding NULL slots for one DATA slot until reaching the previous DATA node during the bulkloading process. This operation has negligible overhead for the bulkloading process while it only works with Read-Only workloads. For any workload involving a write operation, inserting a new key-block pair into the inner nodes will incur an update for empty slots and hence leads to increased latency during an insertion. The effectiveness of these two optimizations have also been verified in our evaluation at Section 5.4.
Insert & Duplicate Index Keys Support
4.3.1 Insert. The full insertion process is presented in Algorithm 1, and the key process is depicted in Figure 4. Given a new key-payload pair ( , ), a lookup operation is first called to locate the leaf node . If the item count in this leaf node is less than a predefined threshold , ( , ) is added into this node (Line 30). Note that, if this is the last leaf node, and ( , ) was the first or last item in the node, the minimum key or maximum key is updated in the meta-node. If the item count exceeds the threshold, a splitting process is triggered (Line 4). Unlike a B+-tree, which keeps the smaller half of the items in the original node, AULID keeps the larger half of the items in the original node. Otherwise, the address of the last key is updated in the original inner node, which requires extra writes. After a new leaf node is created, the links to the sibling nodes are also updated. After the leaf node is generated, it is indexed (the largest key and physical address) in the inner nodes (Lines 5-28). A lookup process is initiated to find the first non-mixed node slot to hold the new key. Just as in the search process, there are five different cases: 1) If we encounter a NULL slot, we insert the new key-block pair into it and the insertion is completed (Line 8). 2) If we encounter a NODE slot pointing to a mixed node, the model is fetched and the search process is repeated. 3) For a packed array, if full, a larger packed array type will be allocated. If the maximum supported packed array is already being used, it is converted into a two-layer B+-tree (Line 20-24). Otherwise, we insert the new key-payload pair into the empty slot and complete the process (Line 26). 4) If the B+-tree is not full, the process proceeds as in a standard B+-tree (Line 19).
5) Otherwise, it is converted into a mixed node (Lines 15-17).
After completing the insertion, the statistics of the mixed nodes (accessed_nodes) are updated in the access path to guide later SMOs (Line 27). AULID records the number of items in a third layer or a deeper layer. Finally, we check if we need to initiate an SMO operation by calling the Adjust function (Line 28).
Handling Duplicate
Keys. Duplicate keys are common in real databases. A linked list used in main memory is however not appropriate when on-disk, as it leads to additional I/O costs when fetching items from the list.
Using a B+-tree styled leaf node, AULID can efficiently store and search for duplicate keys in the leaf nodes. This process is the same as a standard B+-tree. If a duplicate key must be inserted into the inner nodes of AULID, one potential way is to directly insert the duplicate key into the inner nodes. AULID can handle key conflicts using the packed array/two-layer B+-tree proposed earlier. However, in this case, the maximum number of duplicate keys that can be supported is 163, 840 for a block size of 4 KB 4 . Using the link between two sibling leaf nodes, another way is that we can only store the first leaf node's address for the duplicate key. With the larger half of items stored in the original block after the leaf node is split, the address (block number) stored in the original slot in the inner nodes needs to be updated. The write overhead is the same as the last case while we can support an arbitrary number of duplicate keys.
Structural Modification of Inner Nodes
As shown in Figure 4, in AULID, inserting new key-block pairs into the packed array and the two-layer B+-tree will not increase the tree height, i.e., not incur an additional I/O request. However, when a two-layer B+-tree node is converted into a new mixed node, certain regions may have a larger tree height. In our test datasets with up to 800M key-payload pairs, a B+-tree has at most three layers in the inner nodes. Therefore, the index structure must be carefully modified to bound the height of the branches in AULID's inner nodes to at most three layers. Otherwise, a B+-tree on disk will be the best. Packed arrays and two-layer B+-tree nodes do not have an impact on the tree height. Here, we focus on when to re-construct mixed nodes in AULID and how to perform a re-construction.
When Should the Rebuilding Occur?
To bound the tree height of the inner nodes and avoid aggressive node updates, we introduce two new constraints to determine when a mixed node (Line 2 in Algorithm 2) should be reconstructed. Criterion 1: the percentage of the items in a subtree rooted at node n in the third layer or a deeper layer (l3_item) is larger than . This guarantees that no more than leaf nodes can have a longer path than a B+-tree on disk, with a high probability. Criterion 2: The number of current items rooted at node n is larger than times the initial size.
In corner cases where a region has a high degree of conflict, a mixed node can have more than items, even when it is initially being created. To avoid reconstructing this node frequently, we adjust it after observing a sufficient number of new items.
A smaller value for and leads to a more frequent node reconstruction. By default, we set = 0.05 and = 1.2 to balance the tree height, i.e., lookup performance and SMO overheads. We have verified the impact of different settings for and in Section 5.4.3.
How to Reconstruct a Node.
If a mixed node meets both of the above criteria, all key-payload pairs stored in the inner nodes rooted at that node are collected, and then the bulkload process is called again to build a new mixed node.
Other Operations
To support a delete operation, AULID first locates the items to be deleted at the leaf node, and then deletes it in the same manner as a standard B+-tree. If no SMO is required (merging the sibling nodes), the delete operation is finalized. In this case, even if we delete the last key-payload pair in the leaf node, AULID still does not update the inner nodes. If a merge is required, a delete operation in the inner nodes is required. If the key-block pair to be deleted is in a mixed node, this slot is marked as an empty slot. If it is contained in the packed array or a two-layer B+-tree node, it will be removed. AULID will convert the packed array or a two-layer B+-tree node into a DATA slot if there is only one key-block pair remaining.
There are two types of update operations, updating the payload and updating the key. In the former, an in-place update is used 5 . For the latter, a delete operation and an insert operation are initiated.
EXPERIMENTS
We have conducted extensive experiments to answer the following questions: Q1: How good is AULID as compared to other learned indexes and a B+-tree when disk-resident? Q2: How well does AULID scale to large datasets? Q3: Do the proposed index structure design and structural modification operation help improve the performance?
We start with the experimental setup as described in Section 5.1. Then, we present our answers to Q1-Q3. To answer Q1, we compare AULID against five competitors across six different workload types and four different datasets. We demonstrate that AULID is superior in terms of throughput and storage cost in Section 5.2. To answer Q2, in Section 5.3 we use another four datasets with 800M keys to study the performance of AULID on large datasets of varying hardness. Notably, most existing studies [4,19,35,38] use at most 200M keypayload pairs to study index performance. Finally, to answer Q3, in Section 5.4 we compare AULID to its variants, with and without the proposed data structures and structure modification operations, in order to reveal the performance benefits of our proposed design choices.
Experimental Setup
Baselines.
We implement a standard B+-tree and four stateof-the-art (SOTA) updatable in-memory learned indexes-PGM [7], FITing-tree [8], ALEX [4], and LIPP [35]-all modified to work on disk. To improve the FITing-Tree's performance and reduce the segment count, we replace the greedy partitioning algorithm in the FITing-Tree with a streaming algorithm [24] originally used in PGM. Additionally, to support arbitrary insertions, we support a Delta Insert Strategy [8] in the FITing-tree, which allocates a buffer for each segment. LIPP and ALEX use their default settings. For PGM and FITing-tree, we set the error bound as 64, where they achieve good performance in most test cases. PGM supports the insertion operation via the same mechanism as the studies [1,3], and hence PGM can also reflect their pros and cons.
Datasets.
The most recent experimental study [34] on memoryresident learned indexes introduced 11 real datasets in their evaluation. Based on its profiling results, these datasets can be roughly divided into four categories (see Figure 2 in [34]): C1: Globally easy and locally easy, C2: globally normal and locally normal, C3: globally normal and locally hard, and C4: globally hard and locally normal. We select one dataset from each of these categories for our experiments: COVID (C1), PLANET (C2), GENOME (C3), and OSM (C4). Each dataset has 200M keys of type uint64. The performance of AULID and LIPP correlates to the conflict degree (the maximum number of keys being inserted into the same slot) in one dataset due to the usage of the FMCD algorithm. Consequently, datasets with a greater conflict degree are more challenging for AULID and LIPP. A summary of the conflict degrees of the tested datasets is presented in Table 1. To test the scalability of AULID on large datasets, we use OSM from [21], which contains 800M uint64 keys. The generator proposed in [34] is used to generate another three datasets, each of which has different levels of hardness (details presented in Section 5.3). All of the generated datasets contain 800M uint64 keys. It is worth highlighting that, as compared to the latest work on the larger than the main memory setting [19,38] where at most 30M keys are used, the number of keys in each dataset used in our experiment is much larger.
For all datasets, we generate a uint64 payload for each key with key plus 1 as their value. The first four datasets require 2.98 GB of storage space on disk, and the last four datasets (used for the scalability testing) require 11.92 GB.
Workloads.
We compare AULID against all baselines across six different workload types typically encountered in a database. W1 -Lookup-Only workload, where each index is built on 200M key-payload pairs and the workload contains 20,000 randomly sampled search keys. W2 -Scan-Only workload, where the start key of the search range is set to the same key in the lookup-only workload, and the search range is set to 100. The queries are issued on indexes prebuilt on the full dataset. W3 -Write-Only workload, where the initial index is built with 10M key-payload pairs that are randomly selected from a dataset, and then another 10M key-payload pairs are inserted. W4 -Read-Heavy workload includes 90% lookup queries and 10% write operations. W5 -Balanced workload consists of 50% lookup queries and 50% write operations. W6 -Write-Heavy workload includes 90% write operations and 10% lookup queries.
We refer to W4-W6 as mixed workloads, with the only difference between them being the ratio between reads and writes. For mixed workloads, the initial index is built over 10M key-payload pairs randomly sampled from a dataset, and then lookup queries and write operations are issued (10M queries in total). The search keys in all mixed workloads are randomly sampled from the existing keys of an index.
Metrics & Environment.
The primary metric we measure is throughput. We also report the number of fetched blocks, the storage size of each index, and the tail latency. We conduct the experiments on a SATA HDD using a Red Hat Enterprise Server 7.9 on an Intel Xeon CPU E5-2690 v3 @ 2.60GHz with 256 GB memory and a 1TB HDD. The block size is 4 KB in all experiments.
Efficiency Comparisons on Disk
In this section, we compare AULID against four state-of-the-art learned indexes, and a B+-tree on disk. AULID outperforms all five indexes on every dataset and workload tested. Figure 5 shows the throughput and the average number of fetched blocks per query, for each index. Overall, AULID is the most efficient indexing method. Specifically, it achieves up to 1.68x, 2.10x, 1.62x, 1.76x, and 1.55x higher throughput than the FITing-tree, PGM, B+-tree, ALEX, and LIPP, respectively. LIPP is the second most efficient index across the majority of datasets. The performance of each index is directly correlated to the number of fetched blocks where more fetched blocks lead to a lower throughput. The FITing-tree, PGM, and ALEX cannot outperform the B+-tree on disk even on the COVID dataset, which is considered to be an easy dataset as per [34]. The improvements from each search step attained in the main memory setting -e.g., a smaller search range in FITing-tree and PGM, or model-based search in ALEX -do not provide any tangible benefits for on-disk operations if they cannot reduce the number of fetched blocks.
Lookup-Only Workload.
The performance of the FITing-tree, PGM, and B+-tree is similar across all datasets. ALEX however has the worst performance on OSM. The performance of AULID and LIPP vary across different datasets; specifically, the performance is related to the conflict degree of a dataset, where a higher number of conflicts usually leads to a greater tree height, and in turn more fetched blocks. Figure 6 summarizes the throughput and the average number of fetched blocks for the Scan-Only workload. In terms of throughput, AULID outperforms FITing-tree, PGM, B+-tree, ALEX, and LIPP by up to 2.11x, 2.44x, 1.65x, 3.04x, 7.94x, respectively. Just as in the Lookup-Only workload, the performance of the scans is determined by the number of fetched blocks.
Scan-Only Workload.
To support a scan query, all indexes first initiate the search process for a lookup query to locate the start key in the search range, and then scan forward until reaching the end key. Consequently, better performance in Lookup-Only workloads yields better performance in Scan-Only workloads. Using the packed array in leaf nodes and links between siblings, the B+-tree and AULID reap the benefits from efficient lookup queries, and are the two top performing algorithms. In contrast, LIPP does not gain any benefits from lookup queries. LIPP only has one node type, where key-payload pairs, pointers to child nodes, and empty slots are all interleaved. Thus, when fetching the next item, LIPP may have to traverse multiple nodes. Since LIPP has a large fanout, there is a greater chance that these nodes are in different blocks. Also, we observe that the performance of ALEX decreases more quickly than the FITing-tree and PGM. This is because with a gapped array in the leaf node, ALEX uses a bitmap to indicate whether a slot is occupied, and thus incurs additional I/O cost when fetching it. Figure 7 shows the throughput for the workloads that include write operations. AULID is still the best performer across all workloads and datasets. The superiority of AULID is attributed to three reasons: (1) a lower latency to locate where a new key-payload should be inserted, i.e., benefiting from the best lookup performance; (2) a lower SMO overhead on leaf nodes with the B+-tree styled leaf node design; and (3) a lower SMO overhead for the inner nodes, and fewer SMOs required. Based on our design of the packed array and the two-layer B+-tree nodes, most new key-block pairs can be stored without creating new mixed nodes. In the tested datasets, no dataset required AULID creation of new mixed nodes. Other learned indexes, however, require more SMOs, e.g., on the Write-Only workload, ALEX and LIPP require 45,897 and 4.5M SMOs on GENOME, respectively.
Write-Only and Mixed Workloads.
From the other learned indexes, PGM outperforms other approaches on the Write-Only workload, but it performs worse when the ratio of read queries increases. Better insertion support stemming from the LSM tree [23] allows PGM to be competitive for write operations. However, since multiple files are maintained as static PGM indexes, PGM may access more than one file for a lookup query, which increases the I/O cost. The performance gain from a faster lookup time can benefit the workloads containing more reads, e.g., the FITing-tree and LIPP on the Read-Heavy workload. However, as the number of writes increases, the SMO overhead and the cost of updating statistics (for ALEX and LIPP) [34] can outweigh the benefits gained from faster lookups. Figure 8 reports the bulkloading time and the on-disk index size after bulkloading. When calculating index sizes, instead of only reporting the inner node sizes, we report the total size of the index file on disk. This ensures that the entire on-disk size is reported for a fair comparison in practice. In terms of bulkloading time, AULID is similar to the B+-tree, and both of them are significantly smaller than the other indexes. AULID also achieves similar storage cost to the B+-tree. The FITing-tree and LIPP have different storage sizes across different datasets. In the case of FITing-tree, harder datasets will create more leaf nodes (segments), and allocate additional buffers on disk for later key-payload pair insertions. For LIPP, a dataset with a larger degree of conflict will result in more nodes being created on disk, and in turn occupy more space. ALEX and LIPP have larger bulkloading times than the other methods due to model training, and more on-disk writes. Figure 9 presents the storage occupancy of all indexes after finishing workloads comprising writes (W3-W6). Overall, AULID achieves similar storage overheads to the B+-tree across all workload and dataset combinations. Among the rest of the competitors, PGM has the smallest storage size. This is attributed to the LSM tree used in PGM to support arbitrary insertions, i.e. after an index has been merged, we can delete it from disk. For LIPP, a dataset with a higher degree of conflict usually has a larger storage cost due to the creation of additional nodes. The FITing-tree has a large space occupancy, regardless of the dataset or workload. For a hard dataset, re-segmenting a leaf node can generate many more leaf nodes compared to an easy one. This results in more buffers being created. For an easy dataset, a leaf node holds more items. Thus, each SMO operation writes more blocks on disk.
Bulkloads.
Index Size.
Tail Latency.
To study the robustness of each index, in Figure 10 we report the p99 latency and standard deviation on the Lookup-Only and Write-Only workloads. Overall, AULID has the smallest p99 latency in the Lookup-Only workload. AULID, PGM, and B+-tree have similar p99 latencies for the Write-Only workload -all of which are better than the FITing-tree, ALEX, and LIPP. However, all learned indexes have a larger standard deviation than the B+-tree across both workloads. Due to an unbalanced tree structure of LIPP and ALEX, accessing some regions may issue more I/O requests for the Lookup-Only workload. When indexing only the largest key of each leaf node in the inner nodes, in a lookup, AULID may access more blocks to fetch the next DATA slot or read an extra block to locate the target leaf node as discussed in Section 4.2.3. PGM will periodically merge items into a larger index. Heavy SMOs for certain queries result in a larger latency in the Write-Only workload, which in turn results in a larger variance.
Scalability Test
In this section, we study the performance of AULID on large scale datasets using different workload types. Setting. Since existing learned indexes perform worse than the B+tree overall, in this section, we compare the scalability of AULID against the B+-tree only. To test the performance of AULID on datasets of different hardness, we include OSM800 [21], and three other datasets of size 800M generated using the method from [34]. For each, we set the local hardness and global hardness to 4x of COVID, PLANET, and GENOME and name them as COVID800, PLANET800, and GENOME800, respectively.
For the Lookup-Only and Scan-Only workloads, we issue 800,000 queries over the index built on the entire dataset. Search keys are randomly sampled from the entire dataset. For the workloads that contain writes, we build an initial index containing 150M keys sampled from a dataset and issue a total of 50M operations, where the write ratio is the same as used for W3-W6 in Section 5.1.3. Figure 11 presents the throughput speedup of AULID compared to the B+-tree, across the four large datasets. AULID beats the B+-tree with up to 1.75x throughput gains on all tested workloads and datasets. AULID and B+-tree have the same leaf node layout. Due to a carefully designed inner node structure and an SMO mechanism to bound the tree height, AULID is more efficient when locating the target leaf node, and also benefits scans and writes.
Performance Speedup.
The superiority of AULID on large datasets also comes from the smaller SMO overhead for write operations. When indexing the largest key for each leaf node of the learned model, AULID reduces the number of SMOs needed to reap the benefits of model-based search. Moreover, a packed array and a two-layer B+-tree hold the new key-block pairs without increasing the tree height (See Figure 4), while incurring only small update overheads.
Bulkload Time & Storage Usage.
We report the bulkloading time and index size for AULID and B+-tree in Figure 12. To build an index for 800M key-payload pairs on disk, the B+-tree takes around 20 and AULID is competitive at 27 . Both are much more efficient than other learned indexes, even on small datasets.
An In-depth Study of AULID Design
In this section, we study how the design of inner nodes meets our proposed design principles and address the two challenges aforementioned in Section 1.1. Typically, basic operations include the lookup and insertion, which in turn define the studied workload types (W1, W3-W6) and are also the key step in W2. Specifically, we first study the impact of the AULID design on these two operations, and then investigate the impact of the adjustment strategy proposed in Section 4.4.
Impact of Different Designs on Lookup-Only Workloads.
To study the effectiveness of the proposed design, we compare AULID against LIPP-B+ -an approach which directly adopts LIPP as the inner nodes, and organizes the leaf nodes in the same vein as B+tree. We report the throughput of the Lookup-only Workload (W1) in Table 2. Across all datasets, AULID outperforms LIPP-B+ and fetches fewer blocks. The performance of AULID is attributed to our read optimization strategies (Section 4.2.3), packed array design, and two-layered B+tree nodes (Section 3.3.2). The first helps reduce the number of blocks being fetched, and the latter two help reduce the tree height. Benefits of Read Optimizations. As discussed in Section 4.2.3, in AULID, there are two cases that may incur additional I/O costs: Case 1 -located in the predecessor of the target leaf node and require an additional block being fetched; Case 2 -located in a NULL slot but need to scan forward until the next DATA slot is found. For the first case, AULID scans forward (ScanFward) to determine whether the current block has another DATA slot. For the second case, AULID Table 3). From Table 3, we observe that most additional fetched blocks are from Case 2.
With the ScanFward optimization, AULID significantly reduces the number of extra fetched blocks for COVID, PLANET, and GENOME. With the Fulfill optimization, AULID avoids fetching extra blocks in Case 1.
When enabling both operations, AULID can reduce the number of extra fetched blocks by at least 50%, particularly for COVID and GENOME. By default, we only enable the ScanFward optimization, which can reduce 0.08, 0.15, 0.18, and 0.03 blocks per query for COVID, PLANET, GENOME, and OSM, respectively. Thus, improvements in AULID on COVID and GENOME in Table 2 are produced by ScanFward optimization only. Benefits of Data Structure Design. The design of the packed array and the two-layer B+-tree node in AULID helps further reduce the number of fetched blocks for the PLANET and OSM datasets due to the lower tree height (as compared to LIPP-B+). Table 4 reports the average node heights after a bulkload on COVID and GENOME. AULID and LIPP-B+ both have the minimal average node height, where most DATA slots are located in the first level. However, for datasets with a larger conflict degree, AULID has a smaller average node height. This is because LIPP-B+ creates more nodes to eliminate the number of conflicts in each dataset with a larger conflict degree, which leads to a greater height. A lower tree height can be achieved with the packed data structures discussed above, and thus AULID requires less storage space than LIPP-B+ on the hard datasets as shown in Table 4. Table 5 reports the throughput for the Write-Only workload (W3). We observe that AULID and LIPP-B+ have similar performance for COVID, PLANET, and GENOME, but AULID outperforms LIPP-B+ by 1.19x on OSM.
To further understand the performance, we break down the insertion process into three steps: (1) the search step (Search) to find the leaf node that will hold the new key-payload pair, (2) the insertion step on a leaf node (Leaf), and (3) the update step in the inner nodes (Inner). From Figure 13, we can see that the main overhead is brought by the first two steps. The overhead from indexing the new key-block pairs in the inner nodes (Inner) is negligible. AULID and LIPP-B+ update the inner nodes when a leaf node is split. When compared against LIPP, both have fewer SMO operations. For example, on GENOME, AULID and LIPP-B+ only require 49,038 SMOs on the leaf nodes, where the leaf nodes must be split; in contrast, LIPP requires 4.6M SMOs and most of them are caused by creating LIPP nodes to eliminate conflicts. On OSM, AULID has a more efficient Search step, which contributes to the higher throughput, as compared to LIPP-B+. The design of the packed array and the two-layer B+-tree nodes in AULID consistently results in shorter paths to leaf nodes, and hence smaller search cost. Hot Region Insertions. Another potential problem with LIPP are insertions in a hot region which occur in the inner nodes. This produces a high number of conflicts and triggers additional SMOs. We use here an Append-Only workload to analyze this case. The throughput and latency breakdown for this case are shown in Table 6 and Figure 14, respectively. To support new key-payload pairs, AULID and LIPP-B+ index the last leaf node in the meta-node. Thus, they have similar performance on the Search and Leaf in Figure 6. AULID has a lower latency when updating the inner nodes (Inner) across all datasets. To study the overhead of the inner nodes, we further break down the process of the inner nodes into five steps: (1) search to find a slot to hold the new key-block pair (Search), (2) create a new node or convert the node type (Create), (3) insert the key-block pair into a DATA slot (Insert), (4) adjust the tree structure (Adjust), and (5) update the statistics for later adjustment operations (Update).
In Figure 15, we observe that, except for the Insert, AULID has a lower latency than LIPP-B+. The packed array and two-layer B+-tree node design of AULID yield a shorter path to target slots holding the new key-block pairs, which contributes to lower search and update times. Moreover, conflicts in the Append-Only workload require more operations when creating LIPP nodes and adjusting tree structures in LIPP-B+. For example, AULID only requires 1,163 and 37 SMOs in creating new nodes and adjusting the tree structure, while LIPP-B+ requires 38,837 and 1,625 SMOs, respectively. 5.4.3 Adjustment Study. Last, we study the effectiveness of index adjustments and provide an analysis on the parameters in AULID -and presented in Algorithm 2. We use the OSM dataset, and build an initial index using 50M keys. Then, another 50M queries are issued with different write ratios, just as in W3-W6. To study the impact of and , we set = 1.07 and = 0.0025 as the default. Using small default values makes the corresponding condition true (Line 2 in Algorithm 2), and hence we can study the impact of other parameters in isolation. Figure 16 illustrates the throughput when using different settings. We also report the throughput without adjusting the index (the dashed lines). In Figure 16, we observe: (1) The index adjustment in AULID significantly improves the throughput, especially for workloads with write operations. (2) Larger values of and usually result in worse performance, and the workloads with more writes are more sensitive to these two parameters. If index is not adjusted, certain regions in the inner nodes can result in longer paths to the leaf nodes. This incurs more reads to fetch leaf nodes and more writes to update statistics on disk. We find that the default values of (0.05) and (1.2) in AULID result in good performance across all tested workloads.
CONCLUSION
In this paper, we propose AULID, a novel simple yet efficient ondisk learned index. In contrast to in-memory indexes, I/O cost is the main bottleneck for disk-resident indexes. Toward that end, we propose five principles to build a learned index on disk, focusing on reducing the I/O cost. We carefully craft the index structure, propose the query processing algorithms, and introduce an index adjustment mechanism to meet the proposed principles. AULID outperforms all of the baselines across a wide range of settings.
Our evaluation shows that AULID has competitive storage cost to the B+-tree (the smallest of the alternatives), and achieves up to 2.11x, 8.63x, 1.72x, 5.51x, and 8.02x higher throughput than the FITing-tree, PGM, B+-tree, ALEX, and LIPP respectively.
This paper is published under the Creative Commons Attribution 4.0 International (CC-BY 4.0) license. Authors reserve their rights to disseminate the work on their personal and corporate Web sites with the appropriate attribution. , , © 2023 IW3C2 (International World Wide Web Conference Committee), published under Creative Commons CC-BY 4.0 License.
Figure 1 :
1Throughput Comparison and Analysis. Each index's throughput in (a)-(b) is normalized by the largest under the same workload (higher is better), (c) is an analysis on the fetched blocks per query, and (d) is a latency breakdown per query.
Figure 2 :
2An Overview of AULID: from Challenges to Design Principles and Solutions
• P5. Better Scan Performance. Existing learned indexes have their own limitations when supporting scans on disk (see Figure 1(a)-(b))
Algorithm 1 :
1Insert(I, , ) Input: I: the AULID index, : the key, : the payload, : maximum slot count in the leaf node 1 accessed_nodes = []; 2 leaf_node = GetLeafNode(I, ); 3 if leaf_node.size ≥ then 4 block_id, k = SplitNode(leaf_node); ⊲ max key and block id of left child; 5 =
Figure 4 :
4AULID 's Insertion Process that should contain and the position of in the node (Line 2)
pointer to in its parent to point to ;6 StatsUpdate(ancestors of P); ⊲ This step can be merged with Line 27 in Algorithm 1 to reduce the write overhead;
Figure 5 :
5Throughput on Lookup-Only Workload (W1).
Figure 6 :
6Throughput on Scan-Only Workload (W2).
Figure 7 :
7Throughput of Mixed Workloads (W3-W6).
Figure 8 :
8Comparison of Bulkload Time and Storage Usage.
Figure 9 :
9Storage Occupancy of Mixed Workloads (W3-W6).
Figure 10 :
10Tail Latency on Lookup-Only (W1) and Write-Only (W3) Workloads.
Figure 11 :
11Throughput Speedup on Large Datasets.
Figure 12 :
12Bulkload Time and Storage Usage Compared to B+-tree on Large Datasets.
Figure 13 :
13Latency Breakdown of Write-Only Workload (W3).
Figure 14 :
14Latency Breakdown of Append-Only Workload.
Figure 15 :
15Inner Nodes' Latency Breakdown of Append-Only Workload.
Figure 16 :
16Throughput under Different Settings. The dashed line indicates the throughput of the corresponding workload with the same color without any adjustments.
Table 1 :
1Conflict Degree of Each DatasetDataset
COVID PLANET GENOME OSM
Conflict Degree
27
22
585
4,106
Table 2 :
2Throughput Comparison of AULID and LIPP-B+ on Lookup-Only Workload (W1).Metric
Index
COVID PLANET GENOME
OSM
Throughput
LIPP-B+ 158,489
133,404
153,851
104,659
AULID
164,897
141,543
163,825
123,749
Blocks
LIPP-B+
2.15
2.72
2.25
3.30
AULID
2.07
2.50
2.07
2.96
Table 3 :
3Extra Fetched Blocks under Different Optimizations Fulfill) the empty slot with the next DATA slot during a bulkloading process. With the Fulfill optimization, AULID avoids accessing extra blocks in Case 2. Thus, all of the fetched extra blocks in the Fullfill optimization are stemming from Case 1 (the third column inDataset
w/o Opt. Fulfill ScanFward Fulfill & ScanFward
COVID
26,107
18,337
9,266
277
PLANET
59,711
52,619
30,090
21,027
GENOME
47,229
40,727
9,456
710
OSM
30,148
22,368
23,232
14,924
fulfills (
Table 4 :
4Impact of Packed Array and Two-Layer B+-tree on the Average DATA Slot Height and Storage.Metric
Index
COVID PLANET GENOME OSM OSM800
Avg. Height
LIPP-B+
1.00
1.60
1.01
2.29
2.28
AULID
1.00
1.36
1.00
1.83
1.93
Storage (GB)
LIPP-B+
4.27
4.47
4.28
5.11
19.51
AULID
4.27
4.28
4.27
4.29
15.77
Table 5 :
5Throughput Comparison of AULID and LIPP-B+ on Write-Only Workload (W3) Packed Array, Two-Layer B+-tree Nodes, and B+-tree Styled Leaf Nodes for Write-Only Workloads.Metric
Index
COVID PLANET GENOME
OSM
Throughput
LIPP-B+ 111,017
100,430
109,631
76,865.4
AULID
111,669
104,816
107,661
91,707.1
5.4.2
Table 6 :
6Throughput Comparison of AULID and LIPP-B+ on Append-Only WorkloadMetric
Index
COVID PLANET GENOME
OSM
Throughput
LIPP-B+ 144,432
153,060
152,633
153,214
AULID
182,732
188,015
187,886
184,051
For a Scan-Only workload, we set the start key to the same key that was used in the Lookup-Only workload, and then we scan forward 99 keys. This ensures that ALEX,
Two recent studies[1,3] on disk are built upon LSM tree[23] and suffer from poor read performance. Details are presented in Section 2.
A B+-tree node with 4 children can store at most 640 leaf node addresses and each leaf node can store 256 items.
Currently, AULID supports key-payload pairs of a fixed length.
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"Revisiting the Alpha Algorithm To Enable Real-Life Process Discovery Applications - Extended Report",
"Revisiting the Alpha Algorithm To Enable Real-Life Process Discovery Applications - Extended Report"
]
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"\nProcess and Data Science (PADS)\nRWTH Aachen University\nGermany\n"
]
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"Process and Data Science (PADS)\nRWTH Aachen University\nGermany"
]
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| Aaron Küsters [0009−0006−9195−5380] , Wil M.P. van der Aalst [0000−0002−0955−6940]Abstract. The Alpha algorithm was the first process discovery algorithm that was able to discover process models with concurrency based on incomplete event data while still providing formal guarantees. However, as was stated in the original paper, practical applicability is limited when dealing with exceptional behavior and processes that cannot be described as a structured workflow net without short loops. This paper presents the Alpha+++ algorithm that overcomes many of these limitations, making the algorithm competitive with more recent process mining approaches. The different steps provide insights into the practical challenges of learning process models with concurrency, choices, sequences, loops, and skipping from event data. The approach was implemented in ProM and tested on various publicly available, real-life event logs. | 10.48550/arxiv.2305.17767 | [
"https://export.arxiv.org/pdf/2305.17767v1.pdf"
]
| 258,960,318 | 2305.17767 | f290abea57c2683db9f95a0bc732134bd7b305a0 |
Revisiting the Alpha Algorithm To Enable Real-Life Process Discovery Applications - Extended Report
28 May 2023
Process and Data Science (PADS)
RWTH Aachen University
Germany
Revisiting the Alpha Algorithm To Enable Real-Life Process Discovery Applications - Extended Report
28 May 2023Process Discovery · Process Mining · Process Models · Petri Nets
Aaron Küsters [0009−0006−9195−5380] , Wil M.P. van der Aalst [0000−0002−0955−6940]Abstract. The Alpha algorithm was the first process discovery algorithm that was able to discover process models with concurrency based on incomplete event data while still providing formal guarantees. However, as was stated in the original paper, practical applicability is limited when dealing with exceptional behavior and processes that cannot be described as a structured workflow net without short loops. This paper presents the Alpha+++ algorithm that overcomes many of these limitations, making the algorithm competitive with more recent process mining approaches. The different steps provide insights into the practical challenges of learning process models with concurrency, choices, sequences, loops, and skipping from event data. The approach was implemented in ProM and tested on various publicly available, real-life event logs.
Introduction
The original Alpha algorithm was developed over twenty years ago [1,3]. The goal of the algorithm was to show the challenges related to discovering process models with concurrency from example traces. It was formally proven that, a process modeled as a structured workflow net without short loops, can be rediscovered from an event log that is directly-follows complete [3]. Despite this remarkable theoretical result, the Alpha algorithm has limited practical relevance for two main reasons:
-The original algorithm did not attempt to filter out infrequent behavior.
Since exceptional behavior is not separated from frequent behavior, it is generally impossible to uncover structure from real-life event logs. -The original algorithm assumed that the process can be modeled as a freechoice Petri net with unique visible activity labels. Most real-life processes can not be modeled as a structured workflow net without short loops and unique visible labels.
These limitations were already acknowledged in the papers proposing the algorithm, e.g., the focus of [3] was on showing the theoretical limits of process discovery based on directly-follows complete event logs. Many of the later process discovery approaches use these insights. Various extensions of the Alpha algorithm have been proposed, e.g., [18] extends the core algorithm to deal with long-term dependencies, and [19] extends the core algorithm to deal with invisible activities (e.g., skipping). Region-based process-discovery approaches provide formal guarantees. State-based regions were introduced by Ehrenfeucht and Rozenberg [9] in 1989 and generalized by Cortadella et al. [7]. In [2], it is shown how these state-based regions can be applied to process mining by first creating a log-based automaton using different abstractions. In [6,14], refinements are proposed to tailor state-based regions toward process discovery. Language-based regions work directly on traces without creating an automaton first; see, for example, the approaches presented in [5,20,21]. Variants of the Alpha algorithm and the region-based approaches have problems dealing with infrequent behavior and are rarely used in practice. The regionbased approaches are also infeasible for larger models and logs. Approaches such as the eST-Miner [12] and the different variants of the inductive miner [10,11] aim to provide formal guarantees but can also handle infrequent behavior. Variants of the inductive miner have also been implemented in various commercial systems (e.g., Celonis). The so-called split-miner uses a combination of approaches to balance recall and precision [4].
The goal of this paper is to go back to the original ideas used by the Alpha algorithm and make the algorithm work in practical settings. The result is the Alpha+++ algorithm, which, not only extends the core algorithm, but also removes problematic noisy activities, adds invisible activities, repairs loops, and post-processes the resulting Petri net. The approach uses a broad combination of novel ideas, making the Alpha algorithm competitive when compared with the state-of-the-art. The ideas incorporated in the Alpha+++ algorithm may also be used in combination with other approaches (e.g., identifying problematic activities and introducing artificially created invisible activities).
The remainder of this paper is organized as follows. Section 2 introduces event logs, directly-follows graphs, and the original Alpha algorithm. Section 3 describes the Alpha+++ algorithm. The algorithm has been implemented in ProM (cf. Section 4) and evaluated using various event logs (cf. Section 5). Section 6 concludes the paper.
Preliminaries
Event Logs
Process mining starts from event data. An event may have many different attributes. However, here we focus on discovering the control flow and assume that each event has a case attribute, an activity attribute, and a timestamp attribute. We only use the timestamps to order events related to the same case. Therefore, each case can be described as a sequence of activities, also called trace. An event log is a multiset of traces, as different cases can exhibit the same trace.
Definition 1 (Event Log). U act is the universe of activity names. A trace σ = ⟨a 1 , a 2 , . . . , a n ⟩ ∈ U act * is a sequence of activities. An event log L ∈ B(U act * ) is a multiset of traces.
For example, L 1 = [⟨a, b, c, d⟩ 400 , ⟨a, b, d⟩ 250 , ⟨d, a, b, c⟩ 4 , ⟨d, a, b⟩ 2 ] is an event log containing 656 cases with 4 different variants. Variant ⟨a, b, c, d⟩ is the most frequent one, i.e., L 1 (⟨a, b, c, d⟩) = 400.
We write actMult(L) = [ σ(i) | σ ∈ L ∧ 1 ≤ i ≤ |σ| ] for the multiset of activities in an event log L and act(L) = {a | a ∈ actMult(L)} for the set of activities.
Directly-Follows Graphs
A Directly-Follows Graph (DFG) is a graph showing how often one activity is followed by another. A DFG consists of the activities as nodes and has an arc from an activity a ∈ U act to an activity b ∈ U act if a is directly followed by b. Two special nodes, corresponding to a start and an end node, are added additionally. The construction of a DFG from an event log is straightforward.
Definition 3 (Constructing DFGs from Event Logs). Let L ∈ B(U act * ) be an event log. We can construct a DFG disc dfg (L) = (A, L = ⇒) based on the directly-follows relations of event log L, with the set of activities A = {a ∈ σ | σ ∈ L} and the multiset of arcs L = ⇒ = [(σ i , σ i+1 ) | σ ∈ L ′ ∧ 1 ≤ i < |σ|] , where L ′ = [⟨▶⟩ · σ · ⟨■⟩ | σ ∈ L] denotes the event log where artificial start and end activities have been added.
Given an event log L, we can construct a DFG disc dfg (L) = (A, L = ⇒), and in the context of L refer to the directly-follows relations in L represented by L = ⇒ directly.
Petri Nets
We would like to discover process models which can represent more complex control-flow structures, like choices, loops, and concurrency. Therefore, we use labeled Petri nets as a target format for process discovery. The reader is assumed to be familiar with the Petri net basics. Definition 4 (Labeled Petri Net). A labeled Petri net is a tuple N = (P, T, F, l) with a set of places P , a set of transitions T (where T ∩ P = ∅), a flow relation, F ⊆ (P × T ) ∪ (T × P ), and a labeling function l ∈ T ̸ → U act . We write l(t) = τ if t ∈ T \ dom(l) (i.e., t is a silent transition that cannot be observed).
A marking is represented by a multiset of places M ∈ B(P ). For a node x ∈ P ∪ T , we define the preset of x as •x = {y ∈ P ∪ T | (y, x) ∈ F } and the postset of x as x• = {y ∈ P ∪ T | (x, y) ∈ F }. We focus on so-called accepting Petri nets, i.e., Petri nets with a defined initial and final state.
Definition 5 (Accepting Petri Net). An accepting Petri net is a triplet AN = (N, M init , M final ) where N = (P, T, F, l) is a labeled Petri net, M init ∈ B(P ) is the initial marking, and M final ∈ B(P ) is the final marking. U AN is the set of accepting Petri nets.
The language defined by an accepting Petri net is then simply given by the set of traces corresponding to all firing sequences that start in the initial marking M init and end in the final marking M f inal . A firing sequence leading from M init to M f inal is converted into a trace, i.e., a sequence of activities. Note that transitions that fire are mapped onto the corresponding activities. If a transition t is silent (i.e., l(t) = τ ), no corresponding activity is created when firing t. Hence, the language of an accepting Petri net is a set of traces.
Alpha Algorithm
A process discovery algorithm aims to discover a model from event data such that the language of the model best characterizes the example behavior seen in the event log.
Definition 6 (Process Discovery Algorithm). A process discovery algorithm is a function disc ∈ B(U act * ) → U AN , i.e., based on a multiset of traces, an accepting Petri net is discovered.
The classical Alpha process discovery algorithm was introduced in [3]. To be able to better explain the extensions presented in this paper, we split the description into three main parts. From an input event log L, place candidates are constructed based on the directly-follows relations of the log. The resulting set of place candidates is pruned to remove dominated candidates. Finally, the discovered Petri net is constructed.
Candidate Building
Cnd = {(A, B) | ∅ ⊊ A, B ⊆ act(L) ∧ ∀ a∈A ∀ b∈B (a L = ⇒ b) ∧ ∀ a,a ′ ∈A (a L = ⧸ ⇒ a ′ ) ∧ ∀ b,b ′ ∈B (b L = ⧸ ⇒ b ′ )} Candidate Pruning Sel = {(A 1 , A 2 ) ∈ Cnd | ∀ (A ′ 1 ,A ′ 2 )∈Cnd ((A 1 ⊆ A ′ 1 ∧ A 2 ⊆ A ′ 2 ) ⇒ (A 1 , A 2 ) = (A ′ 1 , A ′ 2 ))} Petri Net Construction Let P N = ((P, T, F, l), M init , M f inal ), where: -P = {p (A,B) | (A, B) ∈ Sel } ∪ {i W , o W } -T = {t a | a ∈ act(L)} -F = {(t a , p (A,B) ) | (A, B) ∈ Sel ∧ a ∈ A} ∪ {(p (A,B) , t b ) | (A, B) ∈ Sel ∧ b ∈ B} ∪ {(i W , t s ) | ∃ σ ⟨s⟩ · σ ∈ L} ∪ {(t e , o W ) | ∃ σ σ · ⟨e⟩ ∈ L} -l = {(t a , a) | a ∈ act(L)} -M init = [i W ] -M f inal = [o W ]
Alpha+++
In this section, we introduce the Alpha+++ process discovery algorithm based on the classical Alpha algorithm. Through certain pre-processing steps on the event log and a corresponding DFG, as well as fitness-based place filtering, this algorithm is especially well suited for real-life event logs.
The input for this process discovery algorithm is an event log L. In particular, only ordered traces of activities with corresponding frequencies are required. For the main steps of the algorithm, a DFG based on the event log L is used exclusively. Traces of the event log are only used for replay to remove unfitting place candidates. For simplicity, we assume that the traces of L already include artificial start and end activities, in particular, we assume {▶, ■} ⊆ act(L).
We introduce the steps of the algorithm in the following order:
1. Determine Activities, where the set of activities used throughout the algorithm is determined. Problematic activities are removed from the event log and artificial activities are added, resulting in a repaired event logL. 2. Create an Advising DFG, where an advising DFG is constructed based on the DFG corresponding to the repaired logL, retaining only some of the original DFG edges. 3. Candidate Building, where a set of place candidates is built based on the directly-follows relation of the activities. 4. Candidate Pruning, where through efficient multistep filtering unfit or undesirable place candidates are discarded. 5. Petri Net Construction, where a Petri net is constructed based on the activities of the event log, the added artificial activities and the remaining place candidates. 6. Post-Processing Petri Net, where the repaired event log is replayed on the Petri net to remove problematic places.
Determine Activities
First, we determine the set of activities used in the later steps. Initially, starting with the set of activities occurring in the log, we first remove problematic activities that can cause issues with discovering place candidates later on. Next, we also add artificial activities to allow discovery of place candidates for certain loop and skip constructs.
Removing Problematic Activities Problematic activities can significantly alter the directly-follows relations of an event log, which are used in the later steps to identify place candidates. In the most extreme case, if a problematic activity randomly occurs between any other two activities in all traces, all the directly-follows information between two other activities would be lost. We select a subset A L ⊆ act(L) of activities to keep and remove the other problematic activities act(L) \ A L . There are many possible approaches to identifying problematic activities, such as calculating a problem-score per activity and considering all values above a certain threshold as problematic. For instance, for a simple problem-score, the fraction of directly-follows relation involving an activity which are parallel, i.e., also occur in the opposite direction, could be considered. This would, for example, allow to correctly identify the problem in the aforementioned extreme case.
Adding Artificial Activities Discovering Petri net constructs involving silent transitions is a non-trivial task for a DFG-based algorithm. Additionally, in later steps, we want to use traces from the log to assess the fitness of place candidates. Silent activities make calculating fitness scores significantly harder, as then token-based replay is no longer sufficient and computationally expensive alignments have to be computed. As a solution, we propose adding artificial activities to traces. They are not part of the activity set of the event log and In the first event log (L ⟲ ) the directly-follows relation between ▶ and a is the same as between c and a. This causes issues, as the corresponding place candidates all have very low fitness. The added artificial activity τ inserted between the looped sequence ⟨a, b, c⟩ solves this problem, as the problematic directly-follows relation between c and a is replaced.
are only used to find and evaluate place candidates. In the final discovered Petri net, these artificial activities are then translated as silent transitions. This allows discovering Petri nets with silent transitions, while still retaining the advantages of token-based replay fitness evaluation during the algorithm steps. We add artificial activities for two types of constructs: Loops and Skips.
Adding artificial activities for loops is necessary, as the directly-follows relation between an end activity and a start activity of a loop can cause the discovery of problematic places. For example, consider the event log L ⟲ = [⟨a, b, c, d⟩, ⟨a, b, c, a, b, c, d⟩]. Clearly, this event log can be nicely expressed by a Petri net containing a loop construct, which allows repeating the activities a, b, c. However, the directly-follows relation c L ⟲ = = ⇒ a prevents discovering this loop accurately, as shown in Figure 2.
We detect loop constructs based on the directly-follows relations of the input event log L. For a given threshold d ∈ R + , we can define the set of detected loops:
Definition 7 (Detected Loops). Let loops be the function that maps an event log to the set of detected loop start and end activities.
loops(L) = {(b, a) ∈ act(L) × act(L) | ∃ (x1,...,x k )∈act(L) * ,i∈{1,...,k} (x i = a ∧ ∀ i∈{1,...,k−1} (x i L = = ⇒ ≥d x i+1 ) ∧ x k L = = ⇒ ≥d b ∧ b L = = ⇒ ≥d a)}
The parameter d determines the minimal DFG edge weight to consider when looking for loops. For example, with a threshold d=1 and the event log L ⟲ , we can calculate loops(L ⟲ ) = {(c, a)}. As loop constructs can make a process model very imprecise, we do not want to falsely detect loop behavior from rather infrequent directly-follows relations. For convenience, we can also consider threshold values d relative to the mean directly-follows weight.
For each detected loop endpoint pair (b, a) ∈ loops(L), we want to add an artificial activity loop b,a ̸ ∈ act(L). We write A loop = {loop b,a | (b, a) ∈ loops(L)} to denote the set of added artificial loop activities. Additionally, we define a transformation function which transforms a trace σ ∈ L to a trace σ ′ ∈ (A L ∪ A loop ) * .
Definition 8 (Loop Repair Function). Let repair ⟲ be the function that transforms a trace σ into a repaired trace with added artificial loop activities.
repair ⟲ (σ, L) = ⟨b, loop b,a , a⟩ · repair ⟲ (σ ′ , L) if ∃ (b,a)∈loops(L) σ = ⟨b, a⟩ · σ ′ ⟨⟩ if σ = ⟨⟩ ⟨x⟩ · repair ⟲ (σ ′ , L) otherwise, with σ = ⟨x⟩ · σ ′
This function will be later used to transform the input event log L into a repaired event log, in which artificial activities have been added to relevant traces. Next, we describe how artificial activities can assist in correctly discovering activity Skips, as shown in Figure 3. For a directly-follows-weight threshold d ∈ R + , the detected skips for event log L are defined by the following function, which provides the set of activities that have been detected as being "skippable" after an activity a ∈ (A L ∪ A loop ).
skips(a, L) = {b ∈ act(L) | a L = ⇒ b ∧ a L = ⧸ ⇒ a ∧ b L = ⧸ ⇒ ≥d a ∧ b L = ⧸ ⇒ ≥d b ∧ a, b ̸ ∈ {▶, ■} ∧ ∅ ⊊ {x ∈ act(L) | b L = = ⇒ ≥d x} ⊆ {x ∈ act(L) | a L = = ⇒ ≥d x}}
If B ∈ skips(a, L) we assume that all activities b ∈ B are optional steps after a. To allow appropriate model discovery in the rest of the algorithm, the log is repaired using a new artificial activity skip a,B ̸ ∈ act(L). The set of all artificial skip activities is denoted by A skip . This artificial skip activity is inserted everywhere in a trace σ of L, where activity a is not directly followed by an activity b ∈ B in σ (i.e., b was skipped). For that, we define the following transformation function:
repair τ (σ, L) = ⟨⟩ if σ = ⟨⟩ ⟨x⟩ · repair τ (σ ′ , L) if σ = ⟨x⟩ · σ ′ ∧ skips(x, L) = ∅ ⟨a, skip a,B ⟩ · repair τ (⟨x⟩ · σ ′ , L) if σ = ⟨a, x⟩ · σ ′ ∧ x ̸ ∈ skips(a, L) ⟨a, x⟩ · repair τ (σ ′ , L) if σ = ⟨a, x⟩ · σ ′ ∧ x ∈ skips(a, L)
We can now construct a repaired event logL from the input event log L based on the previously identified set of detected loops loops(L) and skips skips(L). For that, we use their corresponding artificial activity set A loop and A skip as well as their corresponding trace transformation functions repair ⟲ and repair τ to transform the input event log L into a repaired event logL. Note that act(L) = (A L∪ A loop∪ A skip ).
L = [repair τ repair ⟲ (σ, L↾A L ), L↾A L | σ ∈ L↾A L ]
Create an Advising DFG
Next, we extract a pruned DFG from the repaired event logL, which ignores infrequent directly-follows relations. This DFG is used as guidance using the following algorithm steps. Note that this step does not modify the repaired event log: The output of this step is a pruned DFG containing the activities act(L) as nodes. Edges between activities a and b are retained if their weight corresponds to at least 1% of the sum of the weights of all incoming edges to b or 1% of the sum of all outgoing edges from a. The value of 1% was determined as a good cutoff through experimentation. In addition, edges with weights below an absolute threshold value n ∈ N 0 are also removed.
For the repaired event logL and a given DFG-weight threshold n ∈ N 0 , we define the advising DFG (abbreviated as aDFG) as follows:
minW (a, b) = 0.01 · min c∈act(L)L = ⇒(c, b), c∈act(L)L = ⇒(a, c) aDFG = act(L), (a, b) ∈ act(L) 2 L = ⇒(a, b) ≥ max {n, minW (a, b)}
Candidate Building
With the repaired event log and the aDFG, we can continue with building place candidates. Place candidates are composed of two sets of activities: The first set corresponds to the transitions that should add a token to this place in a Petri net. The second set corresponds to transitions that should remove a token from this place. The set of all place candidates is given by:
Cnd 0 = {(A 1 , A 2 ) | A 1 , A 2 ⊆ act(L) ∧ ∀ a1∈A1 ∀ a2∈A2 (a 1 aDFG ===⇒ a 2 ) ∧ ∀ a1∈A1 ∀ a2∈A1\A2 (a 1 aDFG = = ⧸ = ⇒ a 2 ) ∧ ∀ a1∈A2\A1 ∀ a2∈A2 (a 1 aDFG = = ⧸ = ⇒ a 2 ) ∧ ∃ a1∈A1\A2 ∃ a2∈A2\A1 (a 2 aDFG = = ⧸ = ⇒ a 1 )}
Candidate Pruning
The set of place candidates Cnd 0 includes many unfit places, which would produce process models with very low fitness. Furthermore, some place candidates might be dominated by others (e.g., the place candidate ({a}, {f }) is dominated by the candidate ({a, b}, {e, f })). Pruning the set of place candidates requires an efficient approach, as the number of place candidates can easily grow huge. We propose a three-step pruning approach. First, place candidates are filtered purely based on activity counts. If the difference in frequency of the input and output activity set is relatively large, the place candidate is rather unfit. This condition can be checked very efficiently. Next, the local fitness of the place candidate is calculated based on local trace replay. Local trace replay takes the order of the activities in the traces into account, and thus can detect even more unfit place candidates. Finally, to remove dominated place candidates, we retain only maximal place candidates.
Balance-based Pruning: For the balance-based pruning, we consider the number of activity occurrences in the logL using actMult(L). For a set of activities, A ⊆ act(L) we can then sum the frequencies together as count(L, A) = a∈A actMult(L)(a). Based on that, we define the balance of a candidate (A 1 , A 2 ):
balance(L, A 1 , A 2 ) = |count(L, A 1 ) − count(L, A 2 )| max{count(L, A 1 ), count(L, A 2 )}
The balance of a candidate is between 0 and 1. Higher values are an indication that the place candidate is unfit. Based on a balance threshold b ∈ [0, 1], candidates with a higher balance value than b can be filtered out:
Cnd 1 = {(A 1 , A 2 ) ∈ Cnd 0 | balance(L, A 1 , A 2 ) ≤ b}
Fitness-based Pruning: Let fit(σ, (A 1 , A 2 ), k) be defined as follows:
fit(σ, (A 1 , A 2 ), k) = 1 if σ = ⟨⟩, k = 0 0 if σ = ⟨⟩, k ̸ = 0 0 if σ = ⟨a⟩ · σ ′ , k = 0, a ̸ ∈ A 1 , a ∈ A 2 fit(σ ′ , (A 1 , A 2 ), k + 1) if σ = ⟨a⟩ · σ ′ , a ∈ A 1 , a ̸ ∈ A 2 fit(σ ′ , (A 1 , A 2 ), k − 1) if σ = ⟨a⟩ · σ ′ , k ≥ 1, a ̸ ∈ A 1 , a ∈ A 2 fit(σ ′ , (A 1 , A 2 ), k) if σ = ⟨a⟩ · σ ′ , (a ∈ A 1 ∩ A 2 ∨ a ̸ ∈ A 1 ∪ A 2 )
Note that fit(σ, (A 1 , A 2 ), 0) = 1 if the place candidate (A 1 , A 2 ) fits the trace; otherwise it takes the value 0. The traces relevant for a place candidate (A 1 , A 2 ) are defined by the following function:
rel (A 1 , A 2 ) = σ = ⟨a 1 , . . . , a n ⟩ ∈L | ∃ i∈{1,...,n} (a i ∈ A 1 ∨ a i ∈ A 2 )
We consider traces relevant for a place candidate, if they contain at least one activity that is in the set of outgoing or ingoing activities of that place candidate. For a single activity, we use the notation rel (a) := rel ({a}, ∅) to denote the traces containing that activity.
We write fit(σ,
(A 1 , A 2 )) := fit(σ, (A 1 , A 2 ), 0) and fit(L, (A 1 , A 2 )) := σ∈rel(A1,A2) fit(σ, (A 1 , A 2 ))
for ease of notation. For a given local candidate fitness threshold t ∈ [0, 1], the candidates remaining after the local fitness replay pruning are then given as:
mfit(A 1 , A 2 ) = min σ∈rel(a) fit(σ, (A 1 , A 2 )) |rel (a)| a ∈ A 1 ∪ A 2 Cnd 2 = (A 1 , A 2 ) ∈ Cnd 1 fit(L, (A 1 , A 2 )) |rel (A 1 , A 2 )| ≥ t ∧ mfit(A 1 , A 2 ) ≥ t
Maximal Candidate Selection: Finally, as the last candidate pruning step, all dominated place candidates are removed, just like in the original Alpha algorithm.
Sel = {(A 1 , A 2 ) ∈ Cnd 2 | ∀ (A ′ 1 ,A ′ 2 )∈Cnd2 ((A 1 ⊆ A ′ 1 ∧ A 2 ⊆ A ′ 2 ) ⇒ (A 1 , A 2 ) = (A ′ 1 , A ′ 2 ))}
Petri Net Construction
Based on the remaining place candidates, an accepting Petri net is constructed as the tuple ((P,
T, F, l), M init , M f inal ), where -P = {p (A1,A2) | (A 1 , A 2 ) ∈ Sel } -T = {t a | a ∈ act(L) \ {▶, ■}} -F = {(t a , p (A1,A2) ) | (A 1 , A 2 ) ∈ Sel ∧ a ∈ A 1 \ {▶, ■}} ∪ {(p (A1,A2) , t a ) | (A 1 , A 2 ) ∈ Sel ∧ a ∈ A 2 \ {▶, ■}} -l = {(t a , a) | a ∈ A L } ∪ {(t a , τ ) | a ∈ (A loop ∪ A skip } -M init = p (A1,A2) ∈ P | ▶ ∈ A 1 -M f inal = p (A1,A2) ∈ P | ■ ∈ A 2
are the components defined using the results of the previous steps.
Post-Processing Petri Net
Let replay(p, P N, σ) be the replay function, which takes the value 1 exactly when the place p of the Petri net P N can replay trace σ (i.e., there is no missing or remaining token in p at any time when replaying σ on P N ).
For a given local place replay fitness threshold r ∈ [0, 1], we can then define the result of the post-process replay as (
(P ′ , T, F ′ , l), M ′ init , M ′ f inal ),
where the set of updated places P ′ is given by:
P ′ = p (A1,A2) | (A 1 , A 2 ) ∈ Sel ∧ σ∈rel(A1,A2) replay(p, P N, σ) |rel(A 1 , A 2 )| ≥ r
The flow relation and initial and final markings are also updated correspondingly:
-F ′ = {(i, o) ∈ F | i ∈ P ′ ∧ o ∈ P ′ } -M ′ init = [ p ∈ M init | p ∈ P ′ ] -M ′ f inal = [ p ∈ M f inal | p ∈ P ′ ]
The final accepting Petri net discovered is (
(P ′ , T, F ′ , l), M ′ init , M ′ f inal ).
Implementation
We implemented the Alpha+++ algorithm as a ProM 1 plugin (Java) and also created a Python implementation 2 for large-scale evaluation on a variety of reallife event logs. The ProM plugin (AlphaRevisitExperiments 3 ) can be installed in ProM Nightly versions and can be used in standard mode to simply discover a Petri net or in interactive mode to experiment with different algorithm step options and view additional information (e.g., how many place candidates were pruned in which step). In both versions, the Alpha+++ preset can be selected out of the preset list on the top. The parameters used throughout the algorithm steps can then be changed. Additionally, the different algorithm steps can be swapped with alternatives or skipped, allowing for further experimentation. there are multiple possible algorithm implementations available, which can be selected from a dropdown-menu. Different presets that form a complete process discovery algorithm, e.g., the Alpha+++ algorithm presented here, can be applied at the top.
Evaluation
To evaluate the proposed Alpha+++ algorithm (α+++), we discovered Petri nets for five real-life event logs, shown in Table 1. For comparison, we also discovered models using the Inductive Miner Infrequent (IMf) and the standard Alpha algorithm (α). We subsequently calculated alignment-based fitness, precision and F1-scores using PM4Py 4 . For IMf, we evaluated four models per event log using noise thresholds of 0.1, 0.2, 0.3 and 0.4. For α, we used four variant filtering approaches upfront: Either only selecting the 10 most common variants or the n most common variants to cover at least 10%, 50% or 80% of traces. For α+++, we chose artificial activity thresholds of 2 and 4 (relative to the mean directly-follows weight) for the log repair steps. Here, a lower threshold value causes more artificial activities to be added. For each artificial activity threshold, we selected five combinations of the balance b, local candidate fitness t and local place replay fitness r thresholds. Note, that for t and r a value closer to 1 and for b a value closer to 0 is more restrictive. We did not apply problematic activities filtering.
The evaluation results are shown in Table 2. Overall, the fitness and F1-scores of α+++ are competitive compared to the IMf. 8 of the 20 models discovered with α are not easy sound (i.e., no final marking is reachable), and thus no alignment scores could be computed. The remaining 12 models exhibit rather low fitness for some logs but very high precision across the board, significantly boosting the corresponding F1-values. Although our approach does not guarantee easy soundness, all 50 Petri nets discovered with α+++ are easy sound and allow computation of alignments. There are notable differences across the different event logs: α+++ performs significantly worse compared to the IMf on the Sepsis log in terms of F1-score, caused by lower precision scores, as the models discovered with α+++ seem to be underfitting. On the two BPI Challenge 2020 logs, α+++ outperforms the IMf in most configurations, often also exhibiting better fitness and precision scores simultaneously.
The influence of the parameters of α+++ is mostly as expected: More restrictive b, t, r values improve the fitness of the models while decreasing the precision.
Manual inspection of the discovered models reveals that the models discovered with α+++ are mostly rather simple and often consist of several disconnected model fragments. Furthermore, multiple models exhibit redundant structures involving silent transitions (e.g., a place with one labeled transition as preset and one silent transition as postset). Such constructs could be removed by further post-processing of the Petri net. For more details and a comprehensive list of the discovered models, see section B and section C of the appendix.
Conclusion
In this paper, we revisited the Alpha algorithm to overcome its limitations, focusing on real-life event logs. For that, we presented the Alpha+++ algorithm which, like the Alpha algorithm, primarily uses directly-follows relations to discover Petri nets. Alpha+++ pre-processes event logs by adding artificial activities for potential loop or skip constructs. This allows discovering silent transitions while still assessing the fitness of places by easily computable tokenbased replay instead of expensive alignment computations. Subsequently, place candidates are generated based on a pruned DFG. A multistep candidate filtering approach efficiently removes place candidates with low fitness, configurable through parameters. We implemented the Alpha+++ algorithm both as a ProM plugin and in Python. The ProM plugin is available in ProM nightly builds and also features an interactive mode to allow experimenting with different algorithm steps and parameters. We evaluated the Alpha+++ on five real-life event logs and compared the results to the classical Alpha algorithm and the widely adapted Inductive Miner Infrequent. Overall, the results indicate that the Al-pha+++ algorithm is competitive in terms of fitness and precision. In general, the different step parameter configurations tested reliably determine the tradeoff between fitness and precision.
Further research should include further evaluation of the algorithm. For that, additional performance metrics like simplicity or generality could be included and also compared to other process discovery algorithms. It is particularly interesting to see if there are any patterns regarding algorithm parameters, event log properties, and model performance. Such observations could enable automatic parameter selection based on the log, and thus simplify Alpha+++ to a wellperforming one-in-all algorithm. Additionally, a more comprehensive qualitative analysis of the discovered models is needed. Further research could also explore if any theoretical guarantees, such as easy-soundness, are attainable, e.g., using more sophisticated post-processing of the discovered Petri net.
A Generality and Simplicity Evaluation
We additionally evaluated all the Petri nets discovered for the evaluation in section 5 in terms of simplicity and generalization. For that, we again utilized PM4Py using the corresponding evaluation functions. In those, simplicity is measured using the inverse arc degree of nodes in the Petri net, while generalization relies on how frequent model elements are revisited during replay 5 .
B Disconnected Transitions
In many of the models discovered with Alpha+++ there are one or more transitions that are disconnected from the rest of the model. As expected, this phenomenon is most frequently observable for more restricting algorithm parameters, which aggressively filter out low-fitness place candidates and places. In Table 4, we counted the number of disconnected labeled transitions for each of the models and calculated what percentage of the activities of the log are represented by disconnected transitions on average.
In Alpha+++ disconnected transitions are not handled separately and always included in the final Petri net. Note, that always including the transitions without restrictions is not the only option, and has non-negligible effects on the fitness and precision of the complete model. In particular, the decision whether an activity should be included as a disconnected transition, i.e., without restrictions, or not at all can be made separately for each transition based on frequency information of the corresponding activity.
C Discovered Process Models
In the next pages, we include all the process models used for evaluation (see section 5). The logs are enumerated in the following order: To stay concise, we abbreviate the algorithms as before and additionally introduce the following notation for the different configurations of the Alpha+++ algorithm:
α+++; 2.0 DF Threshold ; b 0.3 Balance Thresh.
; t 0.7
Local Candidate Fitness Thresh.
; r 0.6
Replay Fitness Thresh.
For example, the configuration α+++;2.0;b0.3;t0.7;r0.6 represents that the model was discovered with α+++ using a relative DF threshold of 2.0 (used for adding artificial activities), a balance threshold of 0.3 (used to filter place candidates), a local candidate fitness threshold of 0.7 (used to prune the candidates further) and finally a replay place fitness threshold of 0.6 (used to post-process the Petri net by removing unfit places). We (visually) post-processed the models discovered with α+++, to allow for easier interpretation and comparison of the models, as the models often exhibit multiple disconnect parts. In particular, we added artificial start and end transitions, marked by ▶ or ■, and connected them to an added place, which also connects all disconnected process parts. Note, that this does not alter the underlying semantic of the Petri net, if interpreted correctly.
C.1 RTFM
Definition 2 (
2Directly-Follows Graph). A Directly-Follows Graph(DFG) is a pair G = (A, G = ⇒), where A ⊆ U act is a set of activities and G = ⇒ ∈ B((A × A) ∪ ({▶} × A) ∪ (A × {■}) ∪ ({▶} × {■})) is a multiset of arcs. ▶ is the start node and ■ is the end node.Note that a DFG has arc weights. Hence, G = ⇒ is a multiset, where G = ⇒(a, b) denotes how often a is followed by b. We write a G = ⇒ b if and only if G = ⇒(a, b) > 0 holds. Similarly, we say that a G = = ⇒ ≥t b holds if and only if G = ⇒(a, b) ≥ t.
(b) Weighted DFG for event log L1 with annotated frequencies of the directly-follows relations and activities.
Fig. 1 :
1Example DFGs for event log L 1 .
Fig. 2 :
2Two event logs (traces shown on the left) and their DFGs.
Fig. 3 :
3Two event logs and their DFGs showcasing the motivation for repairing implicit skips. The directly-follows relation between a and d would suggest considering place candidates with poor fitness. The second log, where an artificial activity τ is inserted where b and c are skipped, mitigates this problem by replacing the directly-follows relation between a and d.
Fig. 4 :
4A screenshot of the interactive mode of the developed ProM plugin Alpha Revisit Experiments. On the right, the steps and different step parameters can be configured. The main section on the left shows the discovered Petri net.
Fig. 5 :
5A closeup of the configuration panel of the ProM plugin. For each step,
Fig. 95 :
95Model discovered using α+++;4.0;b0.1;t0.9;r0.9 on BPI Challenge 2020 (Domestic Declarations)
Table 1 :
1Overview of the event logs used for evaluation. We used a random sample of 3000 cases from the BPI Challenge 2019 log for computational reasons, as it allowed for alignment-based evaluation of the discovered models.Event Log
#Events #Activities #Traces #Variants Reference
RTFM
561,470
11
150,370
231
[8]
Sepsis
15,214
16
1,050
846
[13]
BPI Challenge 2019
(Sample of 3000 Cases)
18,972
34
3,000
470
[15]
BPI Challenge 2020
(Request for Payment)
36,796
19
6,886
89
[17]
BPI Challenge 2020
(Domestic Declaration)
56,437
17
10,500
99
[16]
Table 2 :
2Evaluation ResultsArtificial Activity Threshold of 2.0Inductive Miner Infrequent
Alpha Algorithm
Alpha+++ Algorithm
Noise Threshold
Variant Filtering
Artificial Activity Threshold of 4.0
0.1
0.2
0.3
0.4
Top10 10%
50%
80%
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
Table 3 :
3Simplicity and Generalization ResultsInductive Miner Infrequent
Alpha Miner
Alpha+++ Algorithm
Noise Threshold
Variant Filtering
Artificial Activity Threshold of 2.0
Artificial Activity Threshold of 4.0
0.2
0.3
0.4
0.5
Top10
10%
50%
80%
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
Table 4 :
4Disconnected labeled transitions in models discovered with Alpha+++. The average percentage refers to the average fraction of activities in the event log that are labels of disconnected transitions.Alpha+++ Algorithm
Average %
Artificial Activity Threshold of 2.0
Artificial Activity Threshold of 4.0
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
b=0.5
t=0.5
r=0.5
b=0.3
t=0.7
r=0.6
b=0.2
t=0.8
r=0.7
b=0.2
t=0.8
r=0.8
b=0.1
t=0.9
r=0.9
RTFM
1
1
2
2
2
1
1
2
2
2
14.55
Sepsis Cases
1
5
5
7
7
6
7
12
12
13
46.88
BPI Challenge 2019
(Sample of 3000 Cases)
8
10
14
14
22
7
10
14
14
24
40.29
BPI Challenge 2020
(Requests for Payment)
0
0
0
0
0
0
0
0
0
0
0
BPIC 2021
0
0
0
0
0
1
1
1
1
1
2.94
Fig. 6: Model discovered using IMf 0.1 on RTFM Receive Result Appeal from Prefecture Send for Credit Collection Insert Date Appeal to Prefecture Notify Result Appeal to OffenderFig. 8: Model discovered using IMf 0.3 on RTFMFig. 11: Model discovered using α 10%Cov on RTFM Fig. 12: Model discovered using α 50%Cov on RTFM Fig. 28: Model discovered using α Top10 on Sepsis CasesFig. 29: Model discovered using α 10%Cov on Sepsis CasesFig. 30: Model discovered using α 50%Cov on Sepsis CasesFig. 31: Model discovered using α 80%Cov on Sepsis CasesFig. 32: Model discovered using α+++;2.0;b0.5;t0.5;r0.5 on Sepsis CasesFig. 33: Model discovered using α+++;2.0;b0.3;t0.7;r0.6 on Sepsis CasesFig. 34: Model discovered using α+++;2.0;b0.2;t0.8;r0.7 on Sepsis CasesFig. 35: Model discovered using α+++;2.0;b0.2;t0.8;r0.8 on Sepsis CasesFig. 36: Model discovered using α+++;2.0;b0.1;t0.9;r0.9 on Sepsis Cases Fig. 37: Model discovered using α+++;4.0;b0.5;t0.5;r0.5 on Sepsis Cases Record Service Entry Sheet SRM: In Transfer to Execution Syst. SRM: In Transfer to Execution Syst.Fig. 53: Model discovered using α+++;2.0;b0.2;t0.8;r0.8 on BPI Challenge 2019 (Sample of 3000 Cases) SRM: In Transfer to Execution Syst. SRM: In Transfer to Execution Syst.Fig. 57: Model discovered using α+++;4.0;b0.2;t0.8;r0.7 on BPI Challenge 2019 (Sample of 3000 Cases)Request For Payment APPROVED by PRE_APPROVER Request For Payment APPROVED by SUPERVISOR Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by MISSING Request For Payment APPROVED by ADMINISTRATION Request Payment Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment REJECTED by ADMINISTRATION Request For Payment FOR_APPROVAL by SUPERVISOR Request For Payment REJECTED by EMPLOYEE Request For Payment SUBMITTED by EMPLOYEE Request For Payment SAVED by EMPLOYEE Request For Payment REJECTED by SUPERVISOR Payment Handled Request For Payment REJECTED by PRE_APPROVER Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment REJECTED by SUPERVISOR Request For Payment REJECTED by PRE_APPROVER Request For Payment APPROVED by BUDGET OWNER Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by ADMINISTRATION Payment Handled Request For Payment SUBMITTED by EMPLOYEE Request For Payment REJECTED by EMPLOYEE Request For Payment FINAL_APPROVED by SUPERVISOR Request Payment Request For Payment APPROVED by ADMINISTRATION Fig. 67: Model discovered using α 80%Cov on BPI Challenge 2020 (Request for Payment) Request For Payment FOR_APPROVAL by ADMINISTRATION Request For Payment APPROVED by BUDGET OWNER Payment Handled Request For Payment REJECTED by EMPLOYEE Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment APPROVED by SUPERVISOR Request For Payment REJECTED by MISSING Request For Payment SUBMITTED by EMPLOYEE Request For Payment REJECTED by SUPERVISOR Request For Payment REJECTED by BUDGET OWNER Request For Payment REJECTED by ADMINISTRATION Request For Payment APPROVED by ADMINISTRATION Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment REJECTED by EMPLOYEE Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by SUPERVISOR Request For Payment REJECTED by MISSING Request For Payment REJECTED by ADMINISTRATION Request For Payment REJECTED by SUPERVISOR Request For Payment REJECTED by MISSING Request For Payment APPROVED by SUPERVISOR Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment SUBMITTED by EMPLOYEE Request For Payment APPROVED by PRE_APPROVER Payment Handled Request For Payment APPROVED by BUDGET OWNER Request For Payment APPROVED by ADMINISTRATION Request For Payment REJECTED by EMPLOYEE Request For Payment FINAL_APPROVED by SUPERVISOR Request Payment Request For Payment SAVED by EMPLOYEE Request For Payment FOR_APPROVAL by ADMINISTRATION Request For Payment FOR_APPROVAL by SUPERVISOR Request For Payment REJECTED by PRE_APPROVER Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment REJECTED by ADMINISTRATION Request For Payment REJECTED by BUDGET OWNER Request For Payment FOR_APPROVAL by ADMINISTRATION Request Payment Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment APPROVED by SUPERVISOR Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment REJECTED by EMPLOYEE Request For Payment REJECTED by PRE_APPROVER Request For Payment SAVED by EMPLOYEE Request For Payment SUBMITTED by EMPLOYEE Request For Payment REJECTED by MISSING Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment APPROVED by PRE_APPROVER Request For Payment APPROVED by ADMINISTRATION Payment Handled Request For Payment REJECTED by ADMINISTRATION Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by SUPERVISOR Request For Payment FOR_APPROVAL by SUPERVISOR Request For Payment REJECTED by BUDGET OWNER Fig. 73: Model discovered using α+++;4.0;b0.5;t0.5;r0.5 on BPI Challenge 2020 (Request for Payment) Request For Payment APPROVED by SUPERVISOR Request For Payment APPROVED by ADMINISTRATION Request For Payment REJECTED by BUDGET OWNER Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by EMPLOYEE Request For Payment REJECTED by ADMINISTRATION Request For Payment REJECTED by SUPERVISOR Request For Payment FOR_APPROVAL by ADMINISTRATION Payment Handled Request For Payment APPROVED by PRE_APPROVER Request Payment Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment REJECTED by PRE_APPROVER Request For Payment SUBMITTED by EMPLOYEE Request For Payment SAVED by EMPLOYEE Request For Payment REJECTED by MISSING Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment FOR_APPROVAL by SUPERVISOR Fig. 74: Model discovered using α+++;4.0;b0.3;t0.7;r0.6 on BPI Challenge 2020 (Request for Payment) Request For Payment APPROVED by ADMINISTRATION Request Payment Request For Payment REJECTED by PRE_APPROVER Request For Payment SAVED by EMPLOYEE Payment Handled Request For Payment REJECTED by EMPLOYEE Request For Payment APPROVED by BUDGET OWNER Request For Payment FOR_APPROVAL by SUPERVISOR Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment FOR_APPROVAL by ADMINISTRATION Request For Payment REJECTED by SUPERVISOR Request For Payment APPROVED by PRE_APPROVER Request For Payment SUBMITTED by EMPLOYEE Request For Payment REJECTED by ADMINISTRATION Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment REJECTED by MISSING Request For Payment REJECTED by BUDGET OWNER Request For Payment APPROVED by SUPERVISOR Request For Payment REJECTED by SUPERVISOR Request For Payment SUBMITTED by EMPLOYEE Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment FOR_APPROVAL by SUPERVISOR Request For Payment REJECTED by EMPLOYEE Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment SAVED by EMPLOYEE Request For Payment APPROVED by SUPERVISOR Payment Handled Request For Payment REJECTED by MISSING Request For Payment FOR_APPROVAL by ADMINISTRATION Request For Payment REJECTED by ADMINISTRATION Request Payment Request For Payment APPROVED by PRE_APPROVER Request For Payment REJECTED by BUDGET OWNER Request For Payment APPROVED by ADMINISTRATION Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment APPROVED by BUDGET OWNER Request For Payment REJECTED by PRE_APPROVER Fig. 76: Model discovered using α+++;4.0;b0.2;t0.8;r0.8 on BPI Challenge 2020 (Request for Payment) Request For Payment REJECTED by BUDGET OWNER Request For Payment FINAL_APPROVED by DIRECTOR Request For Payment APPROVED by PRE_APPROVER Request For Payment REJECTED by ADMINISTRATION Request For Payment APPROVED by ADMINISTRATION Request For Payment FINAL_APPROVED by SUPERVISOR Request For Payment FOR_APPROVAL by SUPERVISOR Payment Handled Request For Payment FINAL_APPROVED by BUDGET OWNER Request For Payment SUBMITTED by EMPLOYEE Request For Payment SAVED by EMPLOYEE Request For Payment REJECTED by MISSING Request For Payment REJECTED by SUPERVISOR Request Payment Request For Payment FOR_APPROVAL by ADMINISTRATION Request For Payment REJECTED by PRE_APPROVER Request For Payment REJECTED by EMPLOYEE Request For Payment APPROVED by BUDGET OWNER Request For Payment APPROVED by SUPERVISOR Fig. 77: Model discovered using α+++;4.0;b0.1;t0.9;r0.9 on BPI Challenge 2020 (Request for Payment) C.5 BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by MISSING Declaration REJECTED by BUDGET OWNER Declaration REJECTED by ADMINISTRATION Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by ADMINISTRATION Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by PRE_APPROVER Declaration SAVED by EMPLOYEE Declaration FOR_APPROVAL by SUPERVISOR Declaration FINAL_APPROVED by SUPERVISOR Declaration APPROVED by BUDGET OWNER Payment Handled Request Payment Declaration REJECTED by SUPERVISOR Declaration REJECTED by EMPLOYEE Fig. 79: Model discovered using IMf 0.2 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by MISSING Declaration SAVED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration SUBMITTED by EMPLOYEE Declaration REJECTED by PRE_APPROVER Declaration APPROVED by BUDGET OWNER Declaration REJECTED by BUDGET OWNER Declaration FINAL_APPROVED by SUPERVISOR Declaration FOR_APPROVAL by PRE_APPROVER Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by SUPERVISOR Payment Handled Declaration REJECTED by ADMINISTRATION Declaration APPROVED by ADMINISTRATION Request Payment Fig. 80: Model discovered using IMf 0.3 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by BUDGET OWNER Declaration REJECTED by ADMINISTRATION Declaration APPROVED by ADMINISTRATION Request Payment Declaration REJECTED by EMPLOYEE Declaration FINAL_APPROVED by SUPERVISOR Declaration SAVED by EMPLOYEE Declaration REJECTED by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by BUDGET OWNER Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by MISSING Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by SUPERVISOR Fig. 81: Model discovered using IMf 0.4 on BPI Challenge 2020 (Domestic Declarations) Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by BUDGET OWNER Declaration REJECTED by ADMINISTRATION Declaration REJECTED by EMPLOYEE Declaration REJECTED by PRE_APPROVER Declaration REJECTED by SUPERVISOR Declaration APPROVED by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Declaration APPROVED by PRE_APPROVER Fig. 82: Model discovered using α Top10 on BPI Challenge 2020 (Domestic Declarations) Fig. 84: Model discovered using α 50%Cov on BPI Challenge 2020 (Domestic Declarations) Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by ADMINISTRATION Payment Handled Declaration APPROVED by BUDGET OWNER Declaration FINAL_APPROVED by SUPERVISOR Fig. 85: Model discovered using α 80%Cov on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by BUDGET OWNER Declaration REJECTED by ADMINISTRATION Declaration APPROVED by BUDGET OWNER Declaration APPROVED by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Declaration SUBMITTED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration FOR_APPROVAL by SUPERVISOR Declaration SAVED by EMPLOYEE Declaration REJECTED by PRE_APPROVER Declaration FOR_APPROVAL by ADMINISTRATION Declaration REJECTED by SUPERVISOR Declaration APPROVED by PRE_APPROVER Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by MISSING Fig. 86: Model discovered using α+++;2.0;b0.5;t0.5;r0.5 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by SUPERVISOR Declaration FOR_APPROVAL by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration APPROVED by PRE_APPROVER Declaration APPROVED by ADMINISTRATION Declaration APPROVED by BUDGET OWNER Payment Handled Declaration REJECTED by MISSING Declaration SAVED by EMPLOYEE Declaration FINAL_APPROVED by SUPERVISOR Request Payment Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by BUDGET OWNER Declaration REJECTED by PRE_APPROVER Declaration REJECTED by ADMINISTRATION Declaration FOR_APPROVAL by ADMINISTRATION Fig. 87: Model discovered using α+++;2.0;b0.3;t0.7;r0.6 on BPI Challenge 2020 (Domestic Declarations) Declaration FOR_APPROVAL by PRE_APPROVER Declaration APPROVED by PRE_APPROVER Request Payment Declaration FINAL_APPROVED by SUPERVISOR Payment Handled Declaration SAVED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration APPROVED by BUDGET OWNER Declaration REJECTED by ADMINISTRATION Declaration APPROVED by ADMINISTRATION Declaration REJECTED by BUDGET OWNER Declaration REJECTED by PRE_APPROVER Declaration REJECTED by SUPERVISOR Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by MISSING Declaration FOR_APPROVAL by ADMINISTRATION Declaration SUBMITTED by EMPLOYEE Fig. 88: Model discovered using α+++;2.0;b0.2;t0.8;r0.7 on BPI Challenge 2020 (Domestic Declarations) Declaration APPROVED by ADMINISTRATION Declaration APPROVED by BUDGET OWNER Declaration REJECTED by PRE_APPROVER Declaration REJECTED by SUPERVISOR Declaration FOR_APPROVAL by SUPERVISOR Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by MISSING Declaration REJECTED by ADMINISTRATION Declaration SAVED by EMPLOYEE Declaration FINAL_APPROVED by SUPERVISOR Declaration APPROVED by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Payment Handled Declaration REJECTED by EMPLOYEE Request Payment Declaration FOR_APPROVAL by ADMINISTRATION Declaration REJECTED by BUDGET OWNER Fig. 89: Model discovered using α+++;2.0;b0.2;t0.8;r0.8 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by EMPLOYEE Declaration SAVED by EMPLOYEE Declaration APPROVED by PRE_APPROVER Request Payment Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by PRE_APPROVER Declaration REJECTED by ADMINISTRATION Declaration APPROVED by BUDGET OWNER Declaration FOR_APPROVAL by SUPERVISOR Declaration FOR_APPROVAL by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by ADMINISTRATION Declaration REJECTED by SUPERVISOR Declaration REJECTED by MISSING Declaration REJECTED by BUDGET OWNER Payment Handled Fig. 90: Model discovered using α+++;2.0;b0.1;t0.9;r0.9 on BPI Challenge 2020 (Domestic Declarations) Declaration APPROVED by BUDGET OWNER Request Payment Declaration REJECTED by EMPLOYEE Declaration FOR_APPROVAL by PRE_APPROVER Declaration FOR_APPROVAL by ADMINISTRATION Declaration REJECTED by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Declaration REJECTED by PRE_APPROVER Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by BUDGET OWNER Payment Handled Declaration APPROVED by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by ADMINISTRATION Declaration REJECTED by MISSING Declaration REJECTED by SUPERVISOR Declaration SAVED by EMPLOYEE Fig. 91: Model discovered using α+++;4.0;b0.5;t0.5;r0.5 on BPI Challenge 2020 (Domestic Declarations) Declaration FOR_APPROVAL by PRE_APPROVER Declaration FOR_APPROVAL by ADMINISTRATION Declaration REJECTED by ADMINISTRATION Payment Handled Declaration REJECTED by PRE_APPROVER Declaration FINAL_APPROVED by SUPERVISOR Declaration APPROVED by BUDGET OWNER Declaration REJECTED by SUPERVISOR Declaration REJECTED by BUDGET OWNER Declaration APPROVED by ADMINISTRATION Declaration SAVED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration FOR_APPROVAL by SUPERVISOR Request Payment Declaration REJECTED by MISSING Declaration APPROVED by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Fig. 92: Model discovered using α+++;4.0;b0.3;t0.7;r0.6 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by SUPERVISOR Declaration FOR_APPROVAL by ADMINISTRATION Declaration REJECTED by BUDGET OWNER Declaration FINAL_APPROVED by SUPERVISOR Declaration FOR_APPROVAL by PRE_APPROVER Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by ADMINISTRATION Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by EMPLOYEE Declaration SAVED by EMPLOYEE Declaration REJECTED by ADMINISTRATION Declaration REJECTED by MISSING Declaration APPROVED by PRE_APPROVER Declaration REJECTED by PRE_APPROVER Declaration APPROVED by BUDGET OWNER Payment Handled Fig. 93: Model discovered using α+++;4.0;b0.2;t0.8;r0.7 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by MISSING Declaration APPROVED by ADMINISTRATION Declaration SAVED by EMPLOYEE Declaration FOR_APPROVAL by SUPERVISOR Declaration REJECTED by BUDGET OWNER Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by EMPLOYEE Declaration REJECTED by SUPERVISOR Declaration REJECTED by ADMINISTRATION Declaration SUBMITTED by EMPLOYEE Declaration APPROVED by PRE_APPROVER Payment Handled Request Payment Declaration REJECTED by PRE_APPROVER Declaration APPROVED by BUDGET OWNER Declaration FOR_APPROVAL by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Fig. 94: Model discovered using α+++;4.0;b0.2;t0.8;r0.8 on BPI Challenge 2020 (Domestic Declarations) Declaration REJECTED by ADMINISTRATION Declaration SUBMITTED by EMPLOYEE Declaration REJECTED by EMPLOYEE Declaration REJECTED by MISSING Declaration FOR_APPROVAL by SUPERVISOR Declaration FOR_APPROVAL by PRE_APPROVER Declaration REJECTED by SUPERVISOR Declaration APPROVED by ADMINISTRATION Declaration SAVED by EMPLOYEE Request Payment Payment Handled Declaration FOR_APPROVAL by ADMINISTRATION Declaration FINAL_APPROVED by SUPERVISOR Declaration APPROVED by BUDGET OWNER Declaration REJECTED by PRE_APPROVER Declaration APPROVED by PRE_APPROVER Declaration REJECTED by BUDGET OWNERSend Fine
■
Send for Credit Collection
Send Appeal to Prefecture
Create Fine
Payment
Notify Result Appeal to Offender
Insert Fine Notification
Receive Result Appeal from Prefecture
Appeal to Judge
Insert Date Appeal to Prefecture
Add penalty
•
Payment
Send Fine
Appeal to Judge
Send Appeal to Prefecture
Notify Result Appeal to Offender
Add penalty
Send for Credit Collection
Insert Fine Notification
■
Insert Date Appeal to Prefecture
Create Fine
Receive Result Appeal from Prefecture
•
Fig. 7: Model discovered using IMf 0.2 on RTFM
Create Fine
■
Send Appeal to Prefecture
Appeal to Judge
Payment
Insert Fine Notification
Add penalty
Send Fine
•
Appeal to Judge
Payment
■
Send Fine
Add penalty
Notify Result Appeal to Offender
Insert Fine Notification
Insert Date Appeal to Prefecture
Send for Credit Collection
Create Fine
Receive Result Appeal from Prefecture
•
Fig. 9: Model discovered using IMf 0.4 on RTFM
Payment
Insert Fine Notification
Send Fine
Create Fine
Add penalty
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
■
Send for Credit Collection
•
Fig. 10: Model discovered using α Top10 on RTFM
Insert Fine Notification
Send for Credit Collection
■
Add penalty
Create Fine
Send Fine
•
Send Fine
Send for Credit Collection
■
Add penalty
Insert Fine Notification
Create Fine
Payment
•
Add penalty
Create Fine
Payment
■
Insert Fine Notification
Send Fine
Send for Credit Collection
•
Fig. 13: Model discovered using α 80%Cov on RTFM
Add penalty
Receive Result Appeal from Prefecture
Notify Result Appeal to Offender
Create Fine
Send Fine
Payment
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
Insert Fine Notification
Appeal to Judge
Send for Credit Collection
Fig. 14: Model discovered using α+++;2.0;b0.5;t0.5;r0.5 on RTFM
Create Fine
Appeal to Judge
Add penalty
Send Appeal to Prefecture
Send Fine
Receive Result Appeal from Prefecture
Send for Credit Collection
Notify Result Appeal to Offender
Insert Date Appeal to Prefecture
Payment
Insert Fine Notification
Fig. 15: Model discovered using α+++;2.0;b0.3;t0.7;r0.6 on RTFM
Create Fine
Appeal to Judge
Payment
Insert Fine Notification
Send for Credit Collection
Receive Result Appeal from Prefecture
Send Fine
Insert Date Appeal to Prefecture
Notify Result Appeal to Offender
Send Appeal to Prefecture
Add penalty
Fig. 16: Model discovered using α+++;2.0;b0.2;t0.8;r0.7 on RTFM
Add penalty
Receive Result Appeal from Prefecture
Send for Credit Collection
Insert Fine Notification
Notify Result Appeal to Offender
Payment
Insert Date Appeal to Prefecture
Create Fine
Appeal to Judge
Send Appeal to Prefecture
Send Fine
Fig. 17: Model discovered using α+++;2.0;b0.2;t0.8;r0.8 on RTFM
Add penalty
Receive Result Appeal from Prefecture
Notify Result Appeal to Offender
Send for Credit Collection
Insert Fine Notification
Create Fine
Send Fine
Payment
Send Appeal to Prefecture
Appeal to Judge
Insert Date Appeal to Prefecture
Fig. 18: Model discovered using α+++;2.0;b0.1;t0.9;r0.9 on RTFM
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
Payment
Add penalty
Notify Result Appeal to Offender
Create Fine
Insert Fine Notification
Appeal to Judge
Send Fine
Send for Credit Collection
Receive Result Appeal from Prefecture
Fig. 19: Model discovered using α+++;4.0;b0.5;t0.5;r0.5 on RTFM
Insert Fine Notification
Appeal to Judge
Send Fine
Receive Result Appeal from Prefecture
Add penalty
Payment
Send Appeal to Prefecture
Insert Date Appeal to Prefecture
Create Fine
Send for Credit Collection
Notify Result Appeal to Offender
Fig. 20: Model discovered using α+++;4.0;b0.3;t0.7;r0.6 on RTFM
Appeal to Judge
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
Notify Result Appeal to Offender
Add penalty
Send for Credit Collection
Receive Result Appeal from Prefecture
Insert Fine Notification
Payment
Send Fine
Create Fine
Fig. 21: Model discovered using α+++;4.0;b0.2;t0.8;r0.7 on RTFM
Payment
Notify Result Appeal to Offender
Appeal to Judge
Send Fine
Add penalty
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
Create Fine
Send for Credit Collection
Receive Result Appeal from Prefecture
Insert Fine Notification
Fig. 22: Model discovered using α+++;4.0;b0.2;t0.8;r0.8 on RTFM
Appeal to Judge
Payment
Receive Result Appeal from Prefecture
Notify Result Appeal to Offender
Insert Fine Notification
Add penalty
Create Fine
Send Fine
Send for Credit Collection
Insert Date Appeal to Prefecture
Send Appeal to Prefecture
Fig. 23: Model discovered using α+++;4.0;b0.1;t0.9;r0.9 on RTFM
C.2 Sepsis Cases
ER Triage
ER Registration
ER Sepsis Triage
Return ER
IV Antibiotics
■
Admission NC
Admission IC
Release E
CRP
LacticAcid
IV Liquid
Release D
Leucocytes
Release A
Release C
•
Fig. 24: Model discovered using IMf 0.1 on Sepsis Cases
Release A
Release C
Return ER
LacticAcid
IV Liquid
ER Triage
Leucocytes
CRP
ER Sepsis Triage
IV Antibiotics
■
Admission NC
Admission IC
ER Registration
Release D
•
Fig. 25: Model discovered using IMf 0.2 on Sepsis Cases
LacticAcid
ER Triage
ER Sepsis Triage
■
Admission NC
ER Registration
Admission IC
Leucocytes
IV Antibiotics
CRP
IV Liquid
•
Fig. 26: Model discovered using IMf 0.3 on Sepsis Cases
ER Triage
ER Registration
IV Antibiotics
IV Liquid
Admission NC
ER Sepsis Triage
Admission IC
■
LacticAcid
Leucocytes
CRP
•
Fig. 27: Model discovered using IMf 0.4 on Sepsis Cases
Release A
■
ER Registration
ER Triage
IV Antibiotics
IV Liquid
LacticAcid
ER Sepsis Triage
Leucocytes
Admission NC
CRP
•
Leucocytes
LacticAcid
IV Antibiotics
■
IV Liquid
ER Registration
ER Sepsis Triage
ER Triage
CRP
•
Release A
ER Registration
Return ER
LacticAcid
CRP
Leucocytes
Admission NC
Release D
Release E
Admission IC
IV Liquid
Release C
ER Sepsis Triage
ER Triage
IV Antibiotics
Release B
•
■
Release A
Return ER
Release C
ER Sepsis Triage
Leucocytes
ER Triage
Release D
Release E
IV Liquid
ER Registration
LacticAcid
CRP
Admission NC
IV Antibiotics
Release B
Admission IC
•
■
Release A
Admission IC
CRP
IV Antibiotics
ER Registration
Return ER
Release C
Admission NC
Leucocytes
Release E
Release B
ER Sepsis Triage
ER Triage
LacticAcid
IV Liquid
Release D
Return ER
Release D
Admission NC
LacticAcid
ER Sepsis Triage
Admission IC
CRP
Release A
Release B
IV Liquid
Leucocytes
IV Antibiotics
Release E
Release C
ER Registration
ER Triage
ER Triage
ER Sepsis Triage
Release C
IV Liquid
LacticAcid
Admission IC
Return ER
Release B
Release E
Release D
Admission NC
IV Antibiotics
Release A
CRP
Leucocytes
ER Registration
Release D
LacticAcid
ER Registration
IV Antibiotics
Release B
ER Triage
CRP
Release A
IV Liquid
Release C
Return ER
Release E
ER Sepsis Triage
Admission NC
Admission IC
Leucocytes
Admission NC
Leucocytes
ER Sepsis Triage
Release B
Return ER
Release E
IV Liquid
CRP
IV Antibiotics
Release C
Release D
Admission IC
LacticAcid
ER Triage
ER Registration
Release A
Leucocytes
Release B
IV Antibiotics
Release D
ER Sepsis Triage
Return ER
Release A
IV Liquid
Release E
LacticAcid
Admission NC
CRP
ER Triage
Release C
Admission IC
ER Registration
ER Triage
Admission IC
Leucocytes
Release A
Release D
ER Sepsis Triage
Release E
Release B
Release C
ER Registration
LacticAcid
CRP
Return ER
IV Liquid
IV Antibiotics
Admission NC
Fig. 38: Model discovered using α+++;4.0;b0.3;t0.7;r0.6 on Sepsis Cases
Admission IC
Release E
Release D
ER Triage
Admission NC
LacticAcid
Release A
Return ER
CRP
IV Antibiotics
Release B
Release C
Leucocytes
ER Sepsis Triage
ER Registration
IV Liquid
Fig. 39: Model discovered using α+++;4.0;b0.2;t0.8;r0.7 on Sepsis Cases
ER Triage
LacticAcid
ER Sepsis Triage
IV Liquid
Release D
Release E
IV Antibiotics
Leucocytes
ER Registration
Return ER
Admission IC
Release B
Release A
Admission NC
Release C
CRP
Fig. 40: Model discovered using α+++;4.0;b0.2;t0.8;r0.8 on Sepsis Cases
Release D
Return ER
Release E
ER Triage
LacticAcid
Admission NC
IV Liquid
Admission IC
Release A
IV Antibiotics
Release C
ER Registration
CRP
Release B
Leucocytes
ER Sepsis Triage
Fig. 41: Model discovered using α+++;4.0;b0.1;t0.9;r0.9 on Sepsis Cases
C.3 BPI Challenge 2019 (Sample of 3000 Cases)
Receive Order Confirmation
SRM: Created
Cancel Subsequent Invoice
Vendor creates invoice
SRM: Complete
Record Invoice Receipt
Record Service Entry Sheet
Change Price
Change Delivery Indicator
Create Purchase Requisition Item
Release Purchase Requisition
Delete Purchase Order Item
SRM: Change was Transmitted
SRM: Transfer Failed (E.Sys.)
■
Create Purchase Order Item
SRM: Document Completed
SRM: Ordered
Vendor creates debit memo
Release Purchase Order
Remove Payment Block
Change Storage Location
Change Quantity
Clear Invoice
Block Purchase Order Item
SRM: In Transfer to Execution Syst.
Update Order Confirmation
Reactivate Purchase Order Item
Record Subsequent Invoice
Record Goods Receipt
SRM: Awaiting Approval
SRM: Deleted
Cancel Goods Receipt
Change Approval for Purchase Order
Cancel Invoice Receipt
•
Fig. 42: Model discovered using IMf 0.1 on BPI Challenge 2019 (Sample of 3000
Cases)
Create Purchase Order Item
SRM: Awaiting Approval
Record Goods Receipt
Delete Purchase Order Item
Cancel Goods Receipt
SRM: Complete
Change Price
Cancel Subsequent Invoice
Cancel Invoice Receipt
Block Purchase Order Item
Release Purchase Order
Change Approval for Purchase Order
SRM: Transfer Failed (E.Sys.)
■
Clear Invoice
Vendor creates invoice
Record Service Entry Sheet
Receive Order Confirmation
Change Quantity
SRM: Document Completed
SRM: In Transfer to Execution Syst.
SRM: Change was Transmitted
Change Delivery Indicator
Release Purchase Requisition
Create Purchase Requisition Item
Change Storage Location
SRM: Created
SRM: Ordered
Reactivate Purchase Order Item
Record Invoice Receipt
Update Order Confirmation
Remove Payment Block
Vendor creates debit memo
Record Subsequent Invoice
•
Fig. 43: Model discovered using IMf 0.2 on BPI Challenge 2019 (Sample of 3000
Cases)
Clear Invoice
Record Goods Receipt
Record Invoice Receipt
SRM: Complete
Change Quantity
Vendor creates debit memo
Cancel Subsequent Invoice
SRM: Transfer Failed (E.Sys.)
■
Create Purchase Order Item
SRM: Awaiting Approval
Delete Purchase Order Item
Cancel Goods Receipt
Release Purchase Order
Vendor creates invoice
Cancel Invoice Receipt
Record Subsequent Invoice
SRM: Created
SRM: In Transfer to Execution Syst.
Reactivate Purchase Order Item
Change Approval for Purchase Order
Change Delivery Indicator
Update Order Confirmation
Remove Payment Block
Create Purchase Requisition Item
SRM: Document Completed
SRM: Change was Transmitted
Block Purchase Order Item
Receive Order Confirmation
Release Purchase Requisition
Change Storage Location
SRM: Ordered
Change Price
Record Service Entry Sheet
•
Fig. 44: Model discovered using IMf 0.3 on BPI Challenge 2019 (Sample of 3000
Cases)
SRM: Change was Transmitted
Change Approval for Purchase Order
Vendor creates debit memo
Receive Order Confirmation
Change Storage Location
Record Goods Receipt
Change Price
Release Purchase Requisition
SRM: Complete
Remove Payment Block
Create Purchase Order Item
SRM: In Transfer to Execution Syst.
Cancel Invoice Receipt
Record Invoice Receipt
Block Purchase Order Item
Release Purchase Order
Delete Purchase Order Item
Record Service Entry Sheet
SRM: Created
Change Quantity
SRM: Transfer Failed (E.Sys.)
■
Create Purchase Requisition Item
SRM: Awaiting Approval
SRM: Ordered
Reactivate Purchase Order Item
Change Delivery Indicator
Update Order Confirmation
SRM: Document Completed
Vendor creates invoice
Cancel Goods Receipt
Record Subsequent Invoice
Clear Invoice
•
Fig. 45: Model discovered using IMf 0.4 on BPI Challenge 2019 (Sample of 3000
Cases)
Remove Payment Block
Receive Order Confirmation
Vendor creates invoice
Record Invoice Receipt
Create Purchase Requisition Item
Clear Invoice
■
Delete Purchase Order Item
Create Purchase Order Item
Record Goods Receipt
•
Fig. 46: Model discovered using α Top10 on BPI Challenge 2019 (Sample of
3000 Cases)
Vendor creates invoice
Record Goods Receipt
Record Invoice Receipt
Create Purchase Order Item
Clear Invoice
■
•
Fig. 47: Model discovered using α 10%Cov on BPI Challenge 2019 (Sample of
3000 Cases)
Remove Payment Block
Receive Order Confirmation
Vendor creates invoice
Create Purchase Order Item
Record Goods Receipt
Record Invoice Receipt
Clear Invoice
■
Create Purchase Requisition Item
•
Fig. 48: Model discovered using α 50%Cov on BPI Challenge 2019 (Sample of
3000 Cases)
Remove Payment Block
Vendor creates invoice
Receive Order Confirmation
Record Invoice Receipt
Change Price
Record Goods Receipt
Change Quantity
Clear Invoice
■
Delete Purchase Order Item
Create Purchase Order Item
Create Purchase Requisition Item
Record Service Entry Sheet
•
Fig. 49: Model discovered using α 80%Cov on BPI Challenge 2019 (Sample of
3000 Cases)
Cancel Subsequent Invoice
Receive Order Confirmation
Delete Purchase Order Item
SRM: Created
SRM: Complete
SRM: Deleted
Clear Invoice
Change Approval for Purchase Order
Release Purchase Order
SRM: Ordered
Record Subsequent Invoice
Release Purchase Requisition
Record Service Entry Sheet
Change Delivery Indicator
Block Purchase Order Item
SRM: Transfer Failed (E.Sys.)
SRM: In Transfer to Execution Syst.
Record Goods Receipt
Vendor creates debit memo
Create Purchase Order Item
Update Order Confirmation
Reactivate Purchase Order Item
Cancel Invoice Receipt
Create Purchase Requisition Item
SRM: Awaiting Approval
SRM: Change was Transmitted
Change Storage Location
Change Quantity
Vendor creates invoice
Change Price
Remove Payment Block
SRM: Document Completed
Cancel Goods Receipt
Record Invoice Receipt
Fig. 50: Model discovered using α+++;2.0;b0.5;t0.5;r0.5 on BPI Challenge
2019 (Sample of 3000 Cases)
Receive Order Confirmation
Vendor creates debit memo
Record Subsequent Invoice
Change Quantity
Create Purchase Requisition Item
Delete Purchase Order Item
Cancel Subsequent Invoice
Create Purchase Order Item
Update Order Confirmation
Vendor creates invoice
SRM: Created
SRM: Document Completed
Clear Invoice
Record Goods Receipt
SRM: Awaiting Approval
SRM: Transfer Failed (E.Sys.)
SRM: Change was Transmitted
Change Storage Location
SRM: Ordered
SRM: Complete
Cancel Goods Receipt
Record Invoice Receipt
Remove Payment Block
Block Purchase Order Item
Change Approval for Purchase Order
Change Delivery Indicator
Release Purchase Order
Release Purchase Requisition
SRM: Deleted
Cancel Invoice Receipt
Change Price
Reactivate Purchase Order Item
Fig. 51: Model discovered using α+++;2.0;b0.3;t0.7;r0.6 on BPI Challenge
2019 (Sample of 3000 Cases)
SRM: Deleted
Receive Order Confirmation
Change Delivery Indicator
SRM: Document Completed
Cancel Subsequent Invoice
Release Purchase Requisition
Record Invoice Receipt
Release Purchase Order
Remove Payment Block
SRM: In Transfer to Execution Syst.
SRM: Complete
Delete Purchase Order Item
SRM: Transfer Failed (E.Sys.)
Vendor creates debit memo
Change Price
Record Service Entry Sheet
Cancel Invoice Receipt
SRM: Ordered
Change Quantity
Clear Invoice
Record Goods Receipt
Create Purchase Requisition Item
Record Subsequent Invoice
Create Purchase Order Item
SRM: Created
Cancel Goods Receipt
Block Purchase Order Item
Update Order Confirmation
Vendor creates invoice
SRM: Change was Transmitted
Change Storage Location
Change Approval for Purchase Order
Reactivate Purchase Order Item
SRM: Awaiting Approval
Fig. 52: Model discovered using α+++;2.0;b0.2;t0.8;r0.7 on BPI Challenge
2019 (Sample of 3000 Cases)
SRM: Created
SRM: Ordered
Release Purchase Order
Update Order Confirmation
SRM: Deleted
Cancel Subsequent Invoice
Delete Purchase Order Item
Change Approval for Purchase Order
Clear Invoice
Receive Order Confirmation
Record Subsequent Invoice
Remove Payment Block
Release Purchase Requisition
SRM: Awaiting Approval
SRM: Complete
Change Quantity
Record Invoice Receipt
Reactivate Purchase Order Item
Block Purchase Order Item
Record Service Entry Sheet
SRM: Change was Transmitted
Cancel Goods Receipt
Change Price
Create Purchase Requisition Item
Vendor creates invoice
Vendor creates debit memo
Change Delivery Indicator
Change Storage Location
Record Goods Receipt
Create Purchase Order Item
SRM: Transfer Failed (E.Sys.)
SRM: Document Completed
Cancel Invoice Receipt
Remove Payment Block
SRM: Document Completed
Change Price
Cancel Invoice Receipt
Clear Invoice
SRM: Change was Transmitted
Record Subsequent Invoice
Vendor creates invoice
SRM: Transfer Failed (E.Sys.)
Cancel Subsequent Invoice
Change Approval for Purchase Order
Create Purchase Order Item
Cancel Goods Receipt
Change Delivery Indicator
Change Quantity
Delete Purchase Order Item
Reactivate Purchase Order Item
Change Storage Location
Block Purchase Order Item
SRM: Deleted
Record Goods Receipt
Vendor creates debit memo
SRM: Awaiting Approval
Release Purchase Order
SRM: Ordered
Record Service Entry Sheet
Update Order Confirmation
SRM: Created
Record Invoice Receipt
Create Purchase Requisition Item
Release Purchase Requisition
Receive Order Confirmation
SRM: Complete
Fig. 54: Model discovered using α+++;2.0;b0.1;t0.9;r0.9 on BPI Challenge
2019 (Sample of 3000 Cases)
SRM: Document Completed
Vendor creates invoice
SRM: Change was Transmitted
Block Purchase Order Item
Vendor creates debit memo
Cancel Goods Receipt
Cancel Invoice Receipt
Create Purchase Requisition Item
Record Service Entry Sheet
Change Delivery Indicator
Cancel Subsequent Invoice
Remove Payment Block
Change Storage Location
Change Price
Reactivate Purchase Order Item
SRM: Deleted
Change Approval for Purchase Order
Record Goods Receipt
SRM: Transfer Failed (E.Sys.)
Change Quantity
SRM: Complete
Receive Order Confirmation
Release Purchase Requisition
Record Subsequent Invoice
SRM: Awaiting Approval
Clear Invoice
Delete Purchase Order Item
SRM: Ordered
SRM: Created
Create Purchase Order Item
Release Purchase Order
Update Order Confirmation
Record Invoice Receipt
Fig. 55: Model discovered using α+++;4.0;b0.5;t0.5;r0.5 on BPI Challenge
2019 (Sample of 3000 Cases)
Change Price
Release Purchase Requisition
Block Purchase Order Item
Release Purchase Order
SRM: Document Completed
Change Approval for Purchase Order
Cancel Goods Receipt
Create Purchase Order Item
Record Subsequent Invoice
Clear Invoice
Remove Payment Block
Reactivate Purchase Order Item
Change Delivery Indicator
Record Goods Receipt
SRM: Complete
SRM: Transfer Failed (E.Sys.)
Update Order Confirmation
Record Service Entry Sheet
SRM: Change was Transmitted
SRM: Ordered
Create Purchase Requisition Item
Delete Purchase Order Item
Change Quantity
Record Invoice Receipt
SRM: Deleted
Receive Order Confirmation
Cancel Invoice Receipt
SRM: Created
SRM: In Transfer to Execution Syst.
Vendor creates invoice
Cancel Subsequent Invoice
SRM: Awaiting Approval
Vendor creates debit memo
Change Storage Location
Fig. 56: Model discovered using α+++;4.0;b0.3;t0.7;r0.6 on BPI Challenge
2019 (Sample of 3000 Cases)
Cancel Subsequent Invoice
Change Approval for Purchase Order
Change Price
SRM: Awaiting Approval
Release Purchase Requisition
SRM: Created
Cancel Goods Receipt
Vendor creates debit memo
Record Invoice Receipt
Remove Payment Block
SRM: Ordered
Change Quantity
SRM: Complete
Update Order Confirmation
SRM: Change was Transmitted
Create Purchase Requisition Item
Block Purchase Order Item
Delete Purchase Order Item
SRM: Transfer Failed (E.Sys.)
Release Purchase Order
Clear Invoice
Change Storage Location
SRM: In Transfer to Execution Syst.
Record Subsequent Invoice
Cancel Invoice Receipt
Vendor creates invoice
SRM: Document Completed
Receive Order Confirmation
Record Service Entry Sheet
Reactivate Purchase Order Item
Record Goods Receipt
SRM: Deleted
Create Purchase Order Item
Change Delivery Indicator
Record Subsequent Invoice
SRM: Deleted
Record Goods Receipt
Vendor creates invoice
Cancel Goods Receipt
Create Purchase Order Item
SRM: Document Completed
Change Delivery Indicator
Clear Invoice
Cancel Subsequent Invoice
SRM: Created
SRM: Complete
Record Service Entry Sheet
Delete Purchase Order Item
Change Quantity
SRM: Ordered
Reactivate Purchase Order Item
Release Purchase Requisition
Create Purchase Requisition Item
Change Storage Location
Update Order Confirmation
Remove Payment Block
Block Purchase Order Item
Change Price
Cancel Invoice Receipt
SRM: Transfer Failed (E.Sys.)
SRM: Change was Transmitted
SRM: Awaiting Approval
SRM: In Transfer to Execution Syst.
Record Invoice Receipt
Change Approval for Purchase Order
Receive Order Confirmation
Release Purchase Order
Vendor creates debit memo
Fig. 58: Model discovered using α+++;4.0;b0.2;t0.8;r0.8 on BPI Challenge
2019 (Sample of 3000 Cases)
Create Purchase Order Item
Cancel Subsequent Invoice
Vendor creates invoice
Delete Purchase Order Item
Clear Invoice
Receive Order Confirmation
Release Purchase Requisition
Change Quantity
SRM: Document Completed
Reactivate Purchase Order Item
Change Storage Location
Release Purchase Order
Change Price
Record Subsequent Invoice
Cancel Goods Receipt
Record Invoice Receipt
Update Order Confirmation
SRM: Complete
SRM: Ordered
SRM: Change was Transmitted
SRM: Transfer Failed (E.Sys.)
Record Goods Receipt
SRM: Awaiting Approval
Change Delivery Indicator
Record Service Entry Sheet
Create Purchase Requisition Item
Block Purchase Order Item
Vendor creates debit memo
Change Approval for Purchase Order
Cancel Invoice Receipt
SRM: In Transfer to Execution Syst.
SRM: Created
Remove Payment Block
SRM: Deleted
Fig. 59: Model discovered using α+++;4.0;b0.1;t0.9;r0.9 on BPI Challenge
2019 (Sample of 3000 Cases)
C.4 BPI Challenge 2020 (Request for Payment)
Request For Payment REJECTED by ADMINISTRATION
Payment Handled
Request For Payment FINAL_APPROVED by BUDGET OWNER
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment REJECTED by EMPLOYEE
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment FOR_APPROVAL by SUPERVISOR
■
Request For Payment APPROVED by BUDGET OWNER
Request For Payment APPROVED by PRE_APPROVER
Request For Payment SAVED by EMPLOYEE
Request For Payment APPROVED by ADMINISTRATION
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment APPROVED by SUPERVISOR
Request Payment
•
Fig. 60: Model discovered using IMf 0.1 on BPI Challenge 2020 (Request for
Payment)
Request For Payment REJECTED by PRE_APPROVER
Request For Payment APPROVED by PRE_APPROVER
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment APPROVED by SUPERVISOR
Request For Payment REJECTED by SUPERVISOR
Payment Handled
Request For Payment APPROVED by BUDGET OWNER
Request For Payment REJECTED by MISSING
Request For Payment APPROVED by ADMINISTRATION
Request For Payment FINAL_APPROVED by BUDGET OWNER
Request For Payment REJECTED by ADMINISTRATION
Request For Payment FOR_APPROVAL by SUPERVISOR
■
Request For Payment REJECTED by EMPLOYEE
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment SAVED by EMPLOYEE
Request Payment
•
Fig. 61: Model discovered using IMf 0.2 on BPI Challenge 2020 (Request for
Payment)
■
•
Fig. 62: Model discovered using IMf 0.3 on BPI Challenge 2020 (Request for
Payment)
Request For Payment APPROVED by ADMINISTRATION
Request For Payment FINAL_APPROVED by SUPERVISOR
Request Payment
Request For Payment FINAL_APPROVED by BUDGET OWNER
Request For Payment FOR_APPROVAL by SUPERVISOR
■
Request For Payment REJECTED by EMPLOYEE
Request For Payment REJECTED by MISSING
Request For Payment SAVED by EMPLOYEE
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment APPROVED by PRE_APPROVER
Payment Handled
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment REJECTED by ADMINISTRATION
Request For Payment APPROVED by SUPERVISOR
•
Fig. 63: Model discovered using IMf 0.4 on BPI Challenge 2020 (Request for
Payment)
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment APPROVED by PRE_APPROVER
Request For Payment APPROVED by ADMINISTRATION
Request For Payment REJECTED by SUPERVISOR
Request For Payment REJECTED by EMPLOYEE
Request For Payment APPROVED by BUDGET OWNER
Request For Payment REJECTED by ADMINISTRATION
Request Payment
Payment Handled
■
•
Fig. 64: Model discovered using α Top10 on BPI Challenge 2020 (Request for
Payment)
Request For Payment SUBMITTED by EMPLOYEE
Payment Handled
■
Request Payment
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment APPROVED by ADMINISTRATION
•
Fig. 65: Model discovered using α 10%Cov on BPI Challenge 2020 (Request for
Payment)
Request For Payment SUBMITTED by EMPLOYEE
Request Payment
Request For Payment APPROVED by BUDGET OWNER
Payment Handled
■
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment APPROVED by ADMINISTRATION
•
Fig. 66: Model discovered using α 50%Cov on BPI Challenge 2020 (Request for
Payment)
■
•
Request Payment
Request For Payment APPROVED by PRE_APPROVER
Request For Payment SAVED by EMPLOYEE
Request For Payment FINAL_APPROVED by BUDGET OWNER
Request For Payment REJECTED by PRE_APPROVER
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment FOR_APPROVAL by SUPERVISOR
Fig. 68: Model discovered using α+++;2.0;b0.5;t0.5;r0.5 on BPI Challenge
2020 (Request for Payment)
Request For Payment REJECTED by BUDGET OWNER
Request For Payment FOR_APPROVAL by ADMINISTRATION
Request For Payment FOR_APPROVAL by SUPERVISOR
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment APPROVED by SUPERVISOR
Request For Payment REJECTED by EMPLOYEE
Request For Payment SAVED by EMPLOYEE
Request For Payment REJECTED by ADMINISTRATION
Request For Payment APPROVED by PRE_APPROVER
Request For Payment REJECTED by MISSING
Request For Payment REJECTED by SUPERVISOR
Request For Payment SUBMITTED by EMPLOYEE
Request Payment
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment APPROVED by ADMINISTRATION
Request For Payment APPROVED by BUDGET OWNER
Payment Handled
Request For Payment REJECTED by PRE_APPROVER
Request For Payment FINAL_APPROVED by BUDGET OWNER
Fig. 69: Model discovered using α+++;2.0;b0.3;t0.7;r0.6 on BPI Challenge
2020 (Request for Payment)
Request For Payment APPROVED by ADMINISTRATION
Request For Payment FOR_APPROVAL by ADMINISTRATION
Request For Payment REJECTED by PRE_APPROVER
Request Payment
Payment Handled
Request For Payment SAVED by EMPLOYEE
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment REJECTED by BUDGET OWNER
Request For Payment APPROVED by PRE_APPROVER
Request For Payment APPROVED by SUPERVISOR
Request For Payment FOR_APPROVAL by SUPERVISOR
Fig. 70: Model discovered using α+++;2.0;b0.2;t0.8;r0.7 on BPI Challenge
2020 (Request for Payment)
Request For Payment SAVED by EMPLOYEE
Request For Payment FOR_APPROVAL by SUPERVISOR
Request For Payment REJECTED by ADMINISTRATION
Request For Payment REJECTED by PRE_APPROVER
Request For Payment REJECTED by SUPERVISOR
Request For Payment REJECTED by EMPLOYEE
Request For Payment FOR_APPROVAL by ADMINISTRATION
Request For Payment REJECTED by MISSING
Payment Handled
Request Payment
Request For Payment APPROVED by BUDGET OWNER
Request For Payment APPROVED by ADMINISTRATION
Request For Payment FINAL_APPROVED by SUPERVISOR
Request For Payment FINAL_APPROVED by BUDGET OWNER
Request For Payment FINAL_APPROVED by DIRECTOR
Request For Payment APPROVED by PRE_APPROVER
Request For Payment REJECTED by BUDGET OWNER
Request For Payment SUBMITTED by EMPLOYEE
Request For Payment APPROVED by SUPERVISOR
Fig. 71: Model discovered using α+++;2.0;b0.2;t0.8;r0.8 on BPI Challenge
2020 (Request for Payment)
Fig. 72: Model discovered using α+++;2.0;b0.1;t0.9;r0.9 on BPI Challenge
2020 (Request for Payment)
Fig. 75: Model discovered using α+++;4.0;b0.2;t0.8;r0.7 on BPI Challenge
2020 (Request for Payment)
Declaration REJECTED by MISSING
Declaration REJECTED by ADMINISTRATION
Declaration FINAL_APPROVED by SUPERVISOR
Declaration REJECTED by EMPLOYEE
Declaration SUBMITTED by EMPLOYEE
Payment Handled
■
Declaration SAVED by EMPLOYEE
Declaration FOR_APPROVAL by SUPERVISOR
Request Payment
Declaration APPROVED by BUDGET OWNER
Declaration FOR_APPROVAL by PRE_APPROVER
Declaration APPROVED by ADMINISTRATION
•
Fig. 78: Model discovered using IMf 0.1 on BPI Challenge 2020 (Domestic Dec-
larations)
■
•
■
•
Payment Handled
■
•
Request Payment
Payment Handled
■
•
Declaration FINAL_APPROVED by SUPERVISOR
Declaration SUBMITTED by EMPLOYEE
Request Payment
Payment Handled
■
Declaration APPROVED by ADMINISTRATION
•
Fig. 83: Model discovered using α 10%Cov on BPI Challenge 2020 (Domestic
Declarations)
Payment Handled
■
Declaration APPROVED by BUDGET OWNER
Declaration FINAL_APPROVED by SUPERVISOR
Request Payment
Declaration SUBMITTED by EMPLOYEE
Declaration APPROVED by ADMINISTRATION
•
Request Payment
■
•
Request Payment
Payment Handled
Request Payment
https://promtools.org/ 2 https://github.com/aarkue/alpha-revisit-python 3 https://svn.win.tue.nl/repos/prom/Packages/AlphaRevisitExperiments/
https://pm4py.fit.fraunhofer.de/ (Version 2.6.1)
See also https://pm4py.fit.fraunhofer.de/documentation#evaluation.
Discovering Workflow Performance Models from Timed Logs. W M P Van Der Aalst, B F Van Dongen, International Conference on Engineering and Deployment of Cooperative Information Systems. Y. Han, S. Tai, and D. WikarskiBerlinSpringer-Verlag2480W.M.P. van der Aalst and B.F. van Dongen. Discovering Workflow Performance Models from Timed Logs. In Y. Han, S. Tai, and D. Wikarski, editors, International Conference on Engineering and Deployment of Cooperative Information Systems (EDCIS 2002), volume 2480 of Lecture Notes in Computer Science, pages 45-63. Springer-Verlag, Berlin, 2002.
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Road Traffic Fine Management Process. M De Leoni, F Mannhardt, M. de Leoni and F.Mannhardt. Road Traffic Fine Management Pro- cess, 2015. https://data.4tu.nl/articles/dataset/Road_Traffic_Fine_ Management_Process/12683249.
Partial (Set) 2-Structures -Part 1 and Part 2. A Ehrenfeucht, G Rozenberg, Acta Informatica. 274A. Ehrenfeucht and G. Rozenberg. Partial (Set) 2-Structures -Part 1 and Part 2. Acta Informatica, 27(4):315-368, 1989.
Discovering Block-Structured Process Models from Event Logs Containing Infrequent Behaviour. S J J Leemans, D Fahland, W M P Van Der Aalst, Business Process Management Workshops, International Workshop on Business Process Intelligence (BPI 2013). N. Lohmann, M. Song, and P. WohedBerlinSpringer-Verlag171S.J.J. Leemans, D. Fahland, and W.M.P. van der Aalst. Discovering Block- Structured Process Models from Event Logs Containing Infrequent Behaviour. In N. Lohmann, M. Song, and P. Wohed, editors, Business Process Management Workshops, International Workshop on Business Process Intelligence (BPI 2013), volume 171 of Lecture Notes in Business Information Processing, pages 66-78. Springer-Verlag, Berlin, 2014.
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| Generalizing policies across different domains with dynamics mismatch poses a significant challenge in reinforcement learning. For example, a robot learns the policy in a simulator, but when it is deployed in the real world, the dynamics of the environment may be different. Given the source and target domain with dynamics mismatch, we consider the online dynamics adaptation problem, in which case the agent can access sufficient source domain data while online interactions with the target domain are limited. Existing research has attempted to solve the problem from the dynamics discrepancy perspective. In this work, we reveal the limitations of these methods and explore the problem from the value difference perspective via a novel insight on the value consistency across domains. Specifically, we present the Value-Guided Data Filtering (VGDF) algorithm, which selectively shares transitions from the source domain based on the proximity of paired value targets across the two domains. Empirical results on various environments with kinematic and morphology shifts demonstrate that our method achieves superior performance compared to prior approaches. | 10.48550/arxiv.2305.17625 | [
"https://export.arxiv.org/pdf/2305.17625v1.pdf"
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| 258,960,613 | 2305.17625 | bcf4b63c32f20aab48355276fa90782db3b6321a |
Cross-Domain Policy Adaptation via Value-Guided Data Filtering
Kang Xu
Fudan University
Shanghai Artificial Intelligence Laboratory
Chenjia Bai
Shanghai Artificial Intelligence Laboratory
† Xiaoteng Ma
Tsinghua University
Dong Wang
Shanghai Artificial Intelligence Laboratory
Bin Zhao
Shanghai Artificial Intelligence Laboratory
Northwestern Polytechnical University
Zhen Wang
Northwestern Polytechnical University
Xuelong Li
Shanghai Artificial Intelligence Laboratory
Northwestern Polytechnical University
Wei Li
Fudan University
Cross-Domain Policy Adaptation via Value-Guided Data Filtering
Generalizing policies across different domains with dynamics mismatch poses a significant challenge in reinforcement learning. For example, a robot learns the policy in a simulator, but when it is deployed in the real world, the dynamics of the environment may be different. Given the source and target domain with dynamics mismatch, we consider the online dynamics adaptation problem, in which case the agent can access sufficient source domain data while online interactions with the target domain are limited. Existing research has attempted to solve the problem from the dynamics discrepancy perspective. In this work, we reveal the limitations of these methods and explore the problem from the value difference perspective via a novel insight on the value consistency across domains. Specifically, we present the Value-Guided Data Filtering (VGDF) algorithm, which selectively shares transitions from the source domain based on the proximity of paired value targets across the two domains. Empirical results on various environments with kinematic and morphology shifts demonstrate that our method achieves superior performance compared to prior approaches.
Introduction
Reinforcement Learning (RL) has demonstrated the ability to train highly effective policies with complex behaviors through extensive interactions with the environment [59, 56, 2]. However, in many situations, extensive interactions are infeasible due to the data collection costs and the potential safety hazards associated with domains such as robotics [30] and medical treatments [51]. To address the issue, one approach is to interact with a surrogate environment, such as a simulator, and then transfer the learned policy to the original domain. However, an unbiased simulator may be unavailable due to the complex system dynamics or unexpected disturbances in the target scenario, leading to a dynamics mismatch. Such a mismatch is crucial for the sim-to-real problem in robotics [1, 35,48] and may cause performance degradation of the learned policy in the target domain. In this work, we focus on the dynamics adaptation problem, where we aim to train a well-performing policy for the target domain, given the source domain with the dynamics mismatch.
Recent research has tackled the adaptation over dynamics mismatch through various techniques, such as domain randomization [53,50,42], system identification [73], or simulator calibration [7], that require domain knowledge or privileged access to the physical system. Other methods have explored Figure 1: Semantic illustration of main settings for dynamics adaptation problem. Methods in the first three categories require different assumptions, such as a wide range of source domains, demonstrations from the target domain, or a manipulable simulator. We focus on a more general setting, online dynamics adaptation, only requiring limited online interactions with the target domain.
the adaptation problem in specific scenarios, such as those with expert demonstrations [38,29] or offline datasets [39,46], while the effectiveness of these methods heavily depends on the optimality of demonstrations or the quality of the datasets. In contrast to these works, we consider a more general setting called online dynamics adaptation, where the agent can access sufficient source domain data and a limited number of online interactions with the target domain. We compare the settings for the dynamics adaptation problem in Figure 1.
To address the online dynamics adaptation problem, prior works mainly focus on the single-step dynamics discrepancy and practically eliminating the gap via different ways [14,11]. However, we empirically demonstrate the limitation of the methods through a motivation example, suggesting their effectiveness heavily relies on strong assumptions about the transferability of paired domains. Theoretically, we formulate the performance bound of the learned policy with respect to the dynamics discrepancy term, which provides an explicit interpretation of the results. To address the problem, we focus on the value discrepancy between paired transitions across domains, motivated by the key idea: the transitions with significant dynamics discrepancy but consistent value targets can be seen as equivalent for policy adaptation. Based on the insight, we proposed a simple yet efficient algorithm called Value-Guided Data Filtering (VGDF) for online dynamics adaptation via selective data sharing. Specifically, we use a learned target domain dynamics model to obtain paired transitions based on the source domain state-action pair. The transitions are shared from the source to the target domain only if the value targets of the imagined target domain transition and that of the source domain transition are close. Compared to previous methods that utilize the single-step dynamics gap, our method measures value discrepancies to capture long-term differences between two domains for better adaptation.
Our contributions can be summarized as follows: 1) We reveal the limitations of prior dynamics-based methods and propose the value discrepancy perspective with theoretical analysis. 2) To provide a practical instantiation, we propose VGDF for online dynamics adaptation via selective data sharing.
3) We extend VGDF to a more practical setting with an offline source domain dataset and propose a variant algorithm motivated by novel theoretical results. 4) We empirically demonstrate the superior performance of our method given significant dynamics shifts, including kinematics and morphology mismatch, compared to previous methods.
Related Work
Domain adaptation in RL. Different from domain adaptation in supervised learning where different domains correspond to distinct data distributions [31], different domains in RL can differ in observation space [23], transition dynamics [53, 73,14], embodiment [75,40], or reward functions [13,77,54]. In this work, we focus on domain adaptation with dynamics discrepancies. Prior works utilizing meta RL [72,45,52], domain randomization [53,50,42], and system identification [76,73,12,70] all assume the access to the distribution of training environments and rely on the hypothesis that the source and target domains are drawn from the same distribution. Another line of work has proposed to handle domain adaptation given expert demonstrations from the target domain [38,29,24]. These approaches align the state visitation distributions of the trained policy in the source domain to the distribution of the expert demonstrations in the target domain through state-action correspondences [75] or imitation learning [25,18,68]. However, near-optimal demonstrations can be challenging to acquire in some tasks. More recent works have explored the dynamics adaptation given an offline dataset collected in the target domain [39,46], while the performance of the trained policy depends on the quality of the dataset [47]. Orthogonal to these settings, we concentrate on a general paradigm where a relatively small number of online interactions with the target domain are accessible.
Online dynamics adaptation . Given limited online interactions with the target domain, several works calibrate the dynamics of the source domain by adjusting the physical parameters of the simulator [7, 55, 12, 44], while they assume the access of a manipulable simulator. Action transformation methods correct the transitions collected in the source domain by learning dynamics models of the two domains [22,11,74]. However, the learned model can be inaccurate, which results in model exploitation and performance degradation [27,28]. Furthermore, the work that compensates the dynamics gap by modifying the reward function [14] is practical only if the policy that performs well in both domains exists. Instead, we do not assume the dynamics-agnostic policy exists and demonstrate the effectiveness of our method when such an assumption does not hold.
Knowledge transfer in RL. Knowledge transfer has been proposed to reuse the knowledge from other tasks to boost the training for the current task [66,34]. The transferred knowledge can be modules (e.g., policy) [49,8,4
Preliminaries and Problem Statement
We consider two infinite-horizon Markov Decision Processes (MDP) M src := (S, A, P src , r, γ, ρ 0 ) and M tar := (S, A, P tar , r, γ, ρ 0 ) for the source domain and the target domain, respectively. The two domains share the same state space S, action space A, reward function r : S × A → R with range [0, r max ], discount factor γ ∈ [0, 1), and the initial state distribution ρ 0 : S → [0, 1]. The two domains differ on the transition probabilities, i.e., P src (s ′ |s, a) and P tar (s ′ |s, a).
We define the probability that a policy π encounters state s at the time step t in MDP M as P π M,t (s). We denote the normalized probability that a policy π encounters state s in M as ν π M (s) :
= (1 − γ) ∞ t=0 γ t P π
M,t (s), and the normalized probability that a policy encounters state-action pair (s, a) in M is ρ π M (s, a) := (1 − γ) ∞ t=0 γ t P π M,t (s)π(a|s). The performance of a policy π in M as is formally defined as η M (π) := E s,a∼ρ π M [r(s, a)]. We focus on the online dynamics adaptation problem where limited online interactions with the target domain are accessible, which can be defined as follows:
Definition 3.1. (Online Dynamics Adaptation) Given source domain M src and target domain M tar with different dynamics, we assume sufficient data from the source domain (online or offline) and a relatively small number of online interactions with M tar (e.g., Γ := # source domain data # target domain data = 10), hoping to obtain a near-optimal policy π concerning the target domain M tar .
The prior work [14] also focuses on the online dynamics adaptation problem with online source domain interactions. The proposed algorithm DARC estimates the dynamics discrepancy via learned domain classifiers and further introduces a reward correction (i.e., ∆r(s, a, s ′ ) ≈ log (P tar (s ′ |s, a)/P src (s ′ |s, a))) to optimize policy together with the task reward r (i.e., r(s, a) + ∆r(s, a, s ′ )), discouraging the agent from dynamics-inconsistent behaviors in the source domain.
Guaranteeing Policy Performance from a Value Difference Perspective
In this section, we will first present an example demonstrating the limitation of the prior method considering the dynamics discrepancy. Following that, we provide a theoretical analysis of the Four triangles represent four actions; the darker color suggests a higher value estimation. Our method learns the optimal Q table whose greedy policy leads the agent to the goal of the target domain, while DARC fails due to pessimistic values of the crucial state-action pairs with dynamics mismatch.
dynamics-based method to provide an interpretation of the experiment results. Finally, we introduce a novel perspective on value discrepancies across domains for the online dynamics adaptation problem.
Motivation Example
We start with a 2D grid world task shown in Figure 2 (a), where the agent represented by the red dot needs to navigate to the green square representing the goal. We design source and target domains with different layouts and train a policy to reach the goal successfully in the target domain. We investigate the performance of DARC [14] that trains the policy with dynamics-guided reward correction and our proposed method (Section 5), using tabular Q-learning [69] as the backbone for all methods. Detailed environment settings are shown in Appendix D.
As the empirical state visitations and the learned Q tables show in Figure 2, DARC is stuck in the room and fails to obtain near-optimal Q-values, leading to poor performance. Specifically, we circle out four positions where specific actions will lead to the states with a dynamics mismatch concerning the two domains. Due to the introduced reward correction on the source domain transitions with dynamics mismatch, DARC learns overly pessimistic value estimations of particular state-action pairs, which hinders the agent from the optimal trajectory concerning the target domain. However, the values of the following inconsistent states, induced by the particular state-action pairs, are not significantly different concerning the target domain. The value difference quantifies the discrepancy of the long-term behaviors rather than single-step dynamics. Motivated by the value difference perspective, our proposed method (Section 5.1) demonstrates superior performance.
Theoretical Interpretations and Value Discrepancy Perspective
To provide rigorous interpretations for the results, we derive a performance guarantee for the dynamicsguided methods, which mainly build on the theories proposed in prior methods [27,14].
η Mtar (π) ≥ η Msrc (π) − 2γr max (1 − γ) 2 · E ρ π src [D TV (P src (·|s, a)∥P tar (·|s, a))]
(a) dynamics discrepancy .
(1)
The proof of Theorem 4.1 is given in Appendix B. We observe that the derived performance bound in (1) is controlled by the dynamics discrepancy term (a). Intuitively, the performance difference would be minor when the dynamics discrepancy between the two domains is negligible. DARC [14] applies the Pinsker's inequality [10] and derives the following form:
η Mtar (π) ≥ η Msrc (π) − γr max (1 − γ) 2 · 2E ρ π src [D KL (P src (·|s, a)∥P tar (·|s, a))]
= η Msrc (π) + γr max (1 − γ) 2 · 2E ρ π src ,Psrc [log (P tar (s ′ |s, a)/P src (s ′ |s, a))].
(2)
Based on the result in (2), DARC optimizes the policy by converting the second term in RHS to a reward correction (i.e., ∆r := log(P tar (s ′ |s, a)/P src (s ′ |s, a))), leading to the dynamics discrepancybased adaptation. However, given the transition from the source domain (i.e., P src (s ′ |s, a) ≈ 1), the reward correction will lead to significant penalty (i.e., log(P tar (s ′ |s, a)/P src (s ′ |s, a)) ≪ 0) if the likelihood estimation of the transition concerning the target domain is low (i.e., P tar (s ′ |s, a) ≈ 0). Consequently, the value estimation of the transition with dynamics mismatch tends to be overly pessimistic as shown in Figure 2 (c), which hinders learning an effective policy concerning the target domain.
Instead of myopically considering the single-step dynamics mismatch, we claim that the transitions with significant dynamics mismatch can be equivalent concerning the value estimations that evaluate the long-term behaviors. Due to the dynamics shift across domains, a state-action pair (i.e., (s, a)) would lead to two different next-states (i.e., s ′ src , s ′ tar ), the paired transitions are nearly equivalent for temporal different learning if the induced value estimations are close (i.e., |V (s ′ src ) − V (s ′ tar )| ≤ ϵ). Motivated by this, we derive a performance guarantee from the value difference perspective. Theorem 4.2. (Performance bound controlled by value difference.) Denote source domain and target domain as M src and M tar , respectively. We have the performance guarantee of any policy π over the two MDPs:
η Mtar (π) ≥ η Msrc (π) − γ 1 − γ · E ρ π Msrc E Psrc V π Mtar (s ′ ) − E Ptar V π Mtar (s ′ ) (a): value difference . (3)
The proof of Theorem 4.2 is given in Appendix B. The value difference term provides a novel perspective: the performance can be guaranteed if the transitions from the source domain lead to consistent value targets in the target domain. The result further highlights the value consistency perspective for the online dynamics adaptation problem.
Value-Guided Data Filtering
In this section, we propose Value-Guided Data Filtering (VGDF), a simple yet efficient algorithm for online domain adaptation via selective data sharing. Then we introduce the setting with offline source domain data and a variant of VGDF based on novel theoretical results. The pseudocodes are shown in Appendix A, and the illustration of VGDF is shown in Figure 3.
Dynamics Adaptation by Selective Data Sharing
Inspired by the performance bound proposed in Theorem 4.2, we can guarantee the policy performance by controlling the value difference term in (3). As discussed in Section 4.2, the paired transitions concerning two domains, induced by the same state-action pair, can be regarded as equivalent for temporal difference learning when the corresponding values are close. Thus, we propose to select source domain transitions with minor value discrepancies for dynamics adaptation.
To select rational transitions from the source domain, we need to compare the value differences of paired transitions based on the same source domain state-action pair (s src , a src ). Formally, given a state-action pair (s src , a src ) from the source domain, our objective is to estimate whether the value-difference between s ′ tar and s ′ src is sufficiently small, i.e., ∆(s src , a src ) :
= 1 V π Mtar (s ′ tar ) − V π Mtar (s ′ src ) ≤ ϵ ,(4)
where s ′ tar ∼ P tar (·|s src , a src ), s ′ src ∼ P src (·|s src , a src ), 1 denotes the indicator function and ϵ can be a predefined threshold.
To obtain ∆(s src , a src ), we need to perform policy evaluation over the states to obtain the value estimations given the paired next states (i.e., s ′ src , s ′ tar ), as formulated in Eq. (4). Monte Carlo (MC)
Source Domain
Target Domain Figure 3: Semantic illustration of VGDF. We tackle online dynamics adaptation by selectively sharing the source domain data, and the RL denotes any off-the-shelf off-policy RL algorithm. evaluation can provide unbiased values by rolling the policy starting from specific states [63]. However, since the environment is not manipulable, we cannot perform MC evaluation from arbitrary states. Thus, we propose to use an estimated value function for policy evaluation. In this work, we adopt the Fitted Q Evaluation (FQE) [43] that is widely used in off-policy RL algorithms [37,20,21]. Specifically, we utilize a learned Q function Q θ : S × A → R for evaluation.
! !"# Data Filtering RL ! !"# (#, %, &, # !"# $ ) ( % & ' (!"# $ % {*!"# & } % !!"# $ (#%#& ' ) , ℙ ⋅ Mean {,)*+ , } -, Var {,)*+ , } - Λ (# !"# $ ) (#, %) Rejection Sampling ! 67"
Furthermore, one problem is that the corresponding target domain next state s ′ tar induced by (s src , a src ) is unavailable in practice. To achieve this, we train a dynamics model with the collected data from the target domain. Following prior works [33,9], we employ an ensemble of Gaussian dynamics models {T ϕi (s ′ |s, a)} M i=1 , in an attempt to capture the epistemic uncertainty due to the insufficient target domain samples. Given the source domain state-action pair (s src , a src ), we generate an ensemble of fictitious states and obtain the corresponding values for each state-action pair, which we call fictitious value ensemble (FVE) Q π tar (s src , a src ):
Q π tar (s src , a src ) := Q θ (s ′ i , a ′ i )| s ′ i ∼T ϕ i (·|ssrc,asrc),a ′ i ∼π(·|s ′ i ) M i=1 .(5)
In practice, the choice of ϵ in Eq. (4) is also nontrivial due to task-specific scales of the values and the non-stationary value function during training. We replace the absolute value difference with the likelihood estimation to address the problem. Specifically, we construct a Gaussian distribution with the mean and variance of FVE denoted as N (Mean(Q π tar (s src , a src )), Var(Q π tar (s src , a src ))). Estimating the value of the source domain state as V π tar (s ′ src ) := Q θ (s ′ src , a ′ src )| a ′ src ∼π(·|s ′ src ) , we introduce Fictitious Value Proximity (FVP) representing the likelihood of the source domain state value in the distribution: Λ(s src , a src , s ′ src ) := P(V π tar (s ′ src ) | Mean(Q π tar (s src , a src )), Var(Q π tar (s src , a src ))). (6) Based on the likelihood estimation, we utilize the rejection sampling to select fixed percentage data (i.e., 25%) with the highest likelihood from a batch of source domain transitions at each training iteration. Specifically, we train the value function by optimizing the following objective:
θ ← arg min θ 1 2 E (s,a,r,s ′ )∼Dtar (Q θ − T Q θ ) 2 + 1 2 E (s,a,r,s ′ )∼Dsrc ω(s, a, s ′ ) (Q θ − T Q θ ) 2 , where ω(s, a, s ′ ) := 1 Λ(s, a, s ′ ) > Λ ξ% .(7)
Λ ξ% is the top ξ-quantile likelihood estimation of the minibatch sampled from source domain data, T represents the Bellman operator, and D src , D tar denote replay buffers of two domains.
Consider the case when the agent can perform online interactions with the source domain, the training data mostly comes from the source domain, while we aim to train a policy for the target domain. Hence, exploring the source domain is essential to collect transitions that might be high-value concerning the target domain. Thus, we introduce an exploration policy π E that maximizes the approximate upper confidence bound of the Q-value, i.e.,
π E ← arg max π E E s∼Dtar∪Dsrc Q UB (s, a)| a∼π E (·|s) , where Q UB (s, a) := max {Q θi (s, a)} 2 i=1
under the implementation with SAC [21] backbone. Importantly, the exploration policy π E is separate from the main policy π learned via vanilla SAC. π E and π are used for data collection in the source domain and target domain, respectively.
Adaptation with Offline Dataset of Source Domain
So far, we have discussed the setting where the agent can interact with the source domain to collect data actively. Nonetheless, simultaneous online access to the source and target domain might sometimes be impractical. In order to address the limitation, we aim to extend our method to the setting we refer to as Offline Source with Online Target, in which the agent can access a source domain offline dataset and a relatively small number of online interactions with the target domain.
To adapt VGDF to such a setting, we propose a novel theoretical result of the performance guarantee: Theorem 5.1. Under the setting with offline source domain dataset D whose empirical estimation of the data collection policy is π D (a|s) := D 1(s,a) D 1(s) , let M src and M tar denote the source and target domain, respectively. We have the performance guarantee of any policy π over the two MDPs:
η Mtar (π) ≥ η Msrc (π) − 4γr max (1 − γ) 2 E ρ π D Msrc ,Psrc [D T V (π D ||π)] (a): policy regularization − γ 1 − γ E ρ π D Msrc ζ(s, a) (b): value difference , (8) where ζ(s, a) := E Psrc,π Q π Mtar (s ′ , a ′ ) − E Ptar,π Q π Mtar (s ′ , a ′ ) .
The proof of Theorem 5.1 is given in Appendix B. This theorem highlights the importance of policy regularization and value difference for achieving desirable performance. It is worth noting that the policy regularization term can shed light on the impact of behavior cloning, which has been proven effective for offline RL [19]. Additionally, the value difference term has a similar structure to that of Theorem 3. Thus, we propose a variant called VGDF + BC that combines behavior cloning loss with the original selective data sharing scheme. The pseudocode is shown in Algorithm 2, Appendix A.
Experiments
In this section, we present empirical investigations of our approach. We examine the effectiveness of our method in scenarios with various dynamics shifts, including kinematic change and morphology change. Furthermore, we provide ablation studies and qualitative analysis of our method. Details of environment settings and the implementation are shown in Appendix D and Appendix E, respectively. Additional results are in Appendix F.
Adaptation Performance Evaluation
To systematically investigate the adaptation performance of the methods, we construct two types of dynamics shift scenarios, including kinematic shift and morphology shifts, for four environments (HalfCheetah, Ant, Walker, Hopper) from Gym Mujoco [67,6]. We use the original environment as the source domain across all experiments. To simulate kinematic shifts, we limit the rotation angle range of specific joints to simulate the broken joint scenario. As for morphology shifts, we modify the size of specific limbs while the number of limbs keeps unchanged to ensure the state/action space consistent across domains. Full details of the environment settings are deferred to Appendix D.
We compare our algorithm with four baselines: (i) DARC [14] trains the domain classifiers to compensate the agent with an extra reward for seeking dynamics-consistent behaviors; (ii) GARAT [11] trains the policy with an adversarial imitation reward in the grounded source domain via action transformation [22]; (iii) IW Clip (Importance Weighting Clip) performs importance-weighted bellman updates for source domain samples. The importance weights (i.e., P tar (s ′ |s, a)/P src (s ′ |s, a)) are approximated by the domain classifiers proposed in DARC, and we clip the weight to [10 −4 , 1] to stabilize training; (iv) Finetune uses the 10 5 target domain transitions to finetune the policy trained in the source domain with 10 6 samples. Furthermore, Zero-shot shows the performance of directly transferring the learned policy in the source domain to the target domain, and Oracle demonstrates the performance of the policy trained in the target domain from scratch with 10 6 transitions. We run all algorithms with the same five random seeds. The implementation details are given in Appendix E.1.
As the results in Figure 4 show, our method outperforms GARAT and IW Clip in all environments. DARC demonstrates competitive performance only in the first two environments, while it does not work in other environments. We believe that the assumption of DARC does not hold in the failure cases due to the significant dynamics mismatch. GARAT fails in almost all environments, which we believe is caused by the impractical action transformation from inaccurate dynamics models. The performance of Zero-shot suggests that the policies trained in the source domains barely work in the target domains due to dynamics mismatch. Finetune achieves promising results and outperforms our method in two of eight environments. We believe that the temporally-extended behaviors of the pre-trained policy benefit learning in the downstream tasks with the assistance of efficient exploration. Nonetheless, our method is the only one that outperforms or matches the asymptotic performance of Oracle in four out of eight environments.
Ablation Studies
To investigate the impact of design components in our method, we perform ablation analysis on the ratio of transitions Γ, data selection ratio ξ%, and the optimistic exploration. Data selection ratio ξ%. We employ different data ratios (10%, 25%, 50%, 75%) for the variants of our algorithm. Furthermore, we propose a baseline algorithm Mix that learns with all source domain samples without selection (ω(s, a, s ′ ) ≡ 1 in Eq. (7)). The results, shown in Figure 6, indicate that our algorithm performs robustly under various ratios within a specific range (e.g., ξ% ≤ 50%). Surprisingly, Mix performs exceptionally well in environments with kinematic mismatches but fails in scenarios with morphology shifts. We attribute this to the less significant dynamics shift induced by kinematic changes compared to morphology changes.
Optimistic data collection. To validate the effect of the optimistic exploration π E , we introduce a variant of our method without π E . The results are shown in Figure 7. Removing the optimistic Table 1 demonstrate that our method outperforms the other methods in four out of six environments, indicating that filtering the source domain data with the value consistency paradigm is effective in the offline-online setting.
Quantifying Dynamics Mismatch via Fictitious Value Proximity
Although the empirical results suggest that our method can adapt the policy in the face of various dynamics shifts, the degree of the dynamics mismatch can only be evaluated via the adaptation performance rather than be quantified directly. Here, we propose quantifying the dynamics shifts via the proposed Fictitious Value Proximity (FVP) (Section 5.1).
Kinematic Shift
Morphology Shift We show the approximated FVP in Ant environments with kinematic or morphology shifts in Figure 8. We observe a significant gap between the FVP values of the paired domains, which suggests the target domain with the morphology shifts is "closer" to the source domain than the target domain with the kinematic shifts with respect to the value difference. FVP measured by value differences quantifies the long-term effect on the expected return. Such a measurement can be regarded as a way to quantify the domain discrepancies.
Conclusion
This work addresses the online dynamics adaptation problem by proposing VGDF that selectively shares the source domain transitions from a value consistency paradigm. Starting from the motivation example, we reveal the limitation of the prior dynamics-based method. Then we introduce a novel value discrepancy perspective with theoretical analysis, motivated by the insight that paired transitions with consistent value targets can be regarded as equivalent for training. Practically, we propose VGDF and the variant for the offline source domain setting. Empirical studies demonstrate the effectiveness of our method under significant dynamics gaps, including kinematics shifts and morphology shifts.
Limitation and future directions. One limitation of our method is the reliance on the ensemble dynamics models. However, the recent work estimating the epistemic uncertainty with a single model [16] could be applicable. Furthermore, value-aware model learning [15] may improve our method by training dynamics models with accurate value predictions of the generated samples. Finally, exploring the effectiveness of value consistency for generalizing across reward functions can be another direction for future research. [8] Ching-An Cheng, Andrey Kolobov, and Alekh Agarwal. Policy improvement via imitation of multiple oracles. Advances in Neural Information Processing Systems, 33:5587-5598, 2020.
References
[9] Kurtland Chua, Roberto Calandra, Rowan McAllister, and Sergey Levine. Deep reinforcement learning in a handful of trials using probabilistic dynamics models. Advances in neural information processing systems, 31, 2018.
[10] Imre Csiszár and János Körner.
A Algorithm Description
The pseudocode of VGDF is presented in Algorithm 1. We utilize SAC [21] as our backbone algorithm. We employ a fixed entropy temperature coefficient in all experiments, demonstrating sufficient empirical performance. The training of the dynamics model ensemble follows prior works [9,27] with the MLE loss. The calculation of the Fictitious Value Proximity follows Eq. (6) proposed in Section 5.1. Furthermore, the pseudocode of VGDF + BC is presented in Algorithm 2. We introduce the value-normalized tradeoff between the behavior cloning loss and the policy gradient following the prior work [19]. # Optimize value function with data filtering 16:
θ i=1,2 ← − arg min θi 1 2B btar (Q θi − T Q θi ) 2 + 17: 1 ⌊2B · ξ%⌋ bsrc 1 Λ(s, a, s ′ ) > Λ ξ% (Q θi − T Q θi )2 18:
# Optimize policies 19: Obtain FVP quantile Λ ξ% of {Λ(s, a, s ′ )} B
π E ← arg max π E 1 2B btar∪ bsrc max {Q θ1 (s, a), Q θ2 (s, a)} | a∼π E (·|s) + λH[π E ]
12:
# Optimize value function with data filtering 13: η Mtar (π) ≥ η Msrc (π) − 2γr max (1 − γ) 2 · E ρ π src [D TV (P src (·|s, a)∥P tar (·|s, a))] .
θ i=1,2 ← − arg min θi 1 2B btar (Q θi − T Q θi ) 2 + 14: 1 ⌊2B · ξ%⌋ bsrc 1 Λ(s, a, s ′ ) > Λ ξ% (Q θi − T Q θi )
Proof. We have
η src (π) − η tar (π) = γ 1 − γ E ρ π src (s,a) s ′ P src (s ′ |s, a)V π tar (s ′ ) − s ′ P tar (s ′ |s, a)V π tar (s ′ )ds ′ (Lemma C.1) = γ 1 − γ E ρ π src (s,a) s ′ (P src (s ′ |s, a) − P tar (s ′ |s, a))V π tar (s ′ )ds ′ ≤ γ 1 − γ E ρ π src (s,a) s ′ |(P src (s ′ |s, a) − P tar (s ′ |s, a))V π tar (s ′ )| ds ′ ≤ γ 1 − γ · r max 1 − γ E ρ π src (s,a) s ′ |P src (s ′ |s, a) − P tar (s ′ |s, a)| ds ′ = 2γr max (1 − γ) 2 E ρ π src (s,
Theorem B.2. (Performance bound controlled by value difference.) Denote the source domain and target domain as M src and M tar , respectively. We have the performance guarantee of any policy π over the two MDPs:
η Mtar (π) ≥ η Msrc (π) − γ 1 − γ · E ρ π Msrc E Psrc V π Mtar (s ′ ) − E Ptar V π Mtar (s ′ ) .
Proof. We have
η src (π) − η tar (π) = γ 1 − γ E ρ π src (s,a) s ′ P src (s ′ |s, a)V π tar (s ′ ) − s ′ P tar (s ′ |s, a)V π tar (s ′ )ds ′ (Lemma C.1) = γ 1 − γ E ρ π src (s,a) s ′ (P src (s ′ |s, a) − P tar (s ′ |s, a))V π tar (s ′ )ds ′ ≤ γ 1 − γ E ρ π src (s,a) s ′ (P src (s ′ |s, a) − P tar (s ′ |s, a))V π tar (s ′ )ds ′ = γ 1 − γ · E ρ π Msrc E Psrc V π Mtar (s ′ ) − E Ptar V π Mtar (s ′ ) Theorem B.3.
Under the setting with offline source domain dataset D whose empirical estimation of the data collection policy is π D (a|s) := D 1(s,a) D 1(s) , let M src and M tar denote the source and target domain, respectively. We have the performance guarantee of any policy π over the two MDPs:
η Mtar (π) ≥ η Msrc (π) − 4γr max (1 − γ) 2 E ρ π D Msrc ,Psrc [D T V (π D ||π)] − γ 1 − γ E ρ π D Msrc ζ(s, a) ,(10)
where ζ(s, a) := E Psrc,π Q π Mtar (s ′ , a ′ ) − E Ptar,π Q π Mtar (s ′ , a ′ ) .
Proof. We have
η Mtar (π) − η Msrc (π) = η Msrc (π D ) − η Msrc (π) (a) − η Msrc (π D ) − η Mtar (π) (b)
.
According to Lemma C.2, we have
η Msrc (π D ) − η Msrc (π) ≥ − 1 1 − γ E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) E a ′ ∼π D (·|s ′ ) Q π Msrc (s ′ , a ′ ) − E a ′ ∼π(·|s ′ ) Q π Msrc (s ′ , a ′ ) = − 1 1 − γ E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) A (π D (a ′ |s ′ ) − π(a ′ |s ′ )) Q π Msrc (s ′ , a ′ ) ≥ − 1 1 − γ E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) A (π D (a ′ |s ′ ) − π(a ′ |s ′ )) r max 1 − γ ≥ − r max (1 − γ) 2 E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) A |π D (a ′ |s ′ ) − π(a ′ |s ′ )| = − 2r max (1 − γ) 2 E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) [D T V (π D (·|s ′ ) ∥ π(·|s ′ ))] , and − η Msrc (π D ) − η Mtar (π) = − γ 1 − γ E s,a∼ρ π D Msrc G π1,π2 M1,M2 (s, a) ≥ − 2γr max (1 − γ) 2 E s,a∼ρ π D Msrc s ′ ∼Psrc(·|s,a) [D T V (π D (·|s ′ ) ∥ π(·|s ′ ))] − γ 1 − γ E s,a∼ρ π D Msrc E s ′ ,a ′ ∼Psrc,π Q π Mtar (s ′ , a ′ ) − E s ′ ,a ′ ∼Ptar,π Q π Mtar (s ′ , a ′ ) . (Lemma C.3)
Combining the two inequalities above completes the proof.
C Proofs of Lemmas
This section provides proof of several lemmas used for our theoretical results. The first lemma is adopted from [41], and the proof is essentially the same as the original paper. Lemma C.2 and Lemma C.3 support the derivation of the performance difference bound in Theorem B.3. Lemma C.1. (Telescoping Lemma, Lemma 4.3 in [41].) Let M 1 := (S, A, P 1 , r, γ) and M 2 := (S, A, P 2 , r, γ) be two MDPs with different dynamics P 1 and P 2 . Given a policy π, let
G π M1,M2 (s, a) := E s ′ ∼P1 V π M2 (s ′ ) − E s ′ ∼P2 V π M2 (s ′ ) , we have η M1 (π) − η M2 (π) = γ (1 − γ) E s,a∼ρ π M 1 G π M1,M2 (s, a) .
Proof. Define W j as the expected return when executing π on M 1 for the first j steps, then switching to π and M 2 for the remainder. That is
W j := ∞ t=0
γ t E t<j:st,at∼P1,π t≥j:st,at∼P2,π2
[r(s t , a t )] = E t<j:st,at∼P1,π t≥j:st,at∼P2,π ∞ t=0 γ t r(s t , a t ) .
Then we have W 0 = E s,a∼ρ M 2 ,π [r(s t , a t )] = η M2 (π),
and W ∞ = E s,a∼ρ M 1 ,π [r(s t , a t )] = η M1 (π).
Thus we can obtain
η M1 (π) − η M2 (π) = ∞ j=0 (W j+1 − W j ).(11)
Convert W j and W j+1 as following:
W j = R j + E sj ,aj ∼P1,π E sj+1∼P2 γ j+1 V π M2 (s j+1 ) W j+1 = R j + E sj ,aj ∼P1,π E sj+1∼P1 γ j+1 V π M2 (s j )
Plug back to Eq.11 and we obtain
η M1 (π) − η M2 (π) = ∞ j=0 (W j+1 − W j ) = ∞ j=0 γ j+1 E s,a∼P π M 1 ,j E s ′ ∼P1 V π M2 (s ′ ) − E s ′ ∼P2 V π M2 (s ′ ) = γ (1 − γ) E s,a∼ρ π M 1 E s ′ ∼P1 V π M2 (s ′ ) − E s ′ ∼P2 V π M2 (s ′ ) = γ (1 − γ) E s,a∼ρ π M 1 G π M1,M2 (s, a) .
Lemma C.2. (Extension of Telescoping Lemma.) Let M 1 := (S, A, P 1 , r, γ) and M 2 := (S, A, P 2 , r, γ) be two MDPs with different dynamics P 1 and P 2 . Given two policies π 1 , π 2 , let
G π1,π2 M1,M2 (s, a) := E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) , we have η M1 (π 1 ) − η M2 (π 2 ) = 1 (1 − γ) E s,a∼ρ π 1 M 1 G π1,π2 M1,M2 (s, a) .
Proof. Define W j as the expected return when executing π 1 on M 1 for the first j steps, then switching to π 2 and M 2 for the remainder. That is γ t r(s t , a t ) .
W j := ∞ t=0 γ t E t<j
Then we have
W 0 = E s,a∼ρ M 2 ,π 2 [r(s t , a t )] = η M2 (π 2 ), and W ∞ = E s,a∼ρ M 1 ,π 1 [r(s t , a t )] = η M2 (π 1 ).
Thus we can obtain
η M1 (π 1 ) − η M2 (π 2 ) = ∞ j=0 (W j+1 − W j ).(12)
Convert W j and W j+1 as following:
W j = R j + E sj ,aj ∼P1,π1 E sj+1,aj+1∼P2,π2 γ j+1 Q π2 M2 (s j+1 , a j+1 ) W j+1 = R j + E sj ,aj ∼P1,π1 E sj+1,aj+1∼P1,π1 γ j+1 Q π2 M2 (s j+1 , a j+1 )
Plug back to Eq.12 and we obtain
η M1 (π 1 ) − η M2 (π 2 ) = ∞ j=0 (W j+1 − W j ) = ∞ j=0 γ j+1 E s,a∼P π 1 M 1 ,j E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) = γ (1 − γ) E s,a∼ρ π 1 M 1 E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) = γ (1 − γ) E s,a∼ρ π 1 M 1 G π1,π2 M1,M2 (s, a) . Lemma C.3. (Bound of G π1,π2 M1,M2 (s, a).) Let G π1,π2 M1,M2 (s, a) := E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) ,
we have
G π1,π2 M1,M2 (s, a) ≤ 2r max 1 − γ E s ′ ∼P1 [D T V (π 1 (·|s ′ ) ∥ π 2 (·|s ′ ))] + E s ′ ,a ′ ∼P1,π2 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) .
Proof. We have
G π1,π2 M1,M2 (s, a) :=E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) = E s ′ ,a ′ ∼P1,π1 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P1,π2 Q π2 M2 (s ′ , a ′ ) (a) + E s ′ ,a ′ ∼P1,π2 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) (b)
.
For (a), we have
(a) = E s ′ ∼P1 a ′ π 1 (a ′ |s ′ )Q π2 M2 (s ′ , a ′ ) − π 2 (a ′ |s ′ )Q π2 M2 (s ′ , a ′ ) ≤ E s ′ ∼P1 a ′ |π 1 (a ′ |s ′ ) − π 2 (a ′ |s ′ )| r max 1 − γ = r max 1 − γ E s ′ ∼P1 a ′ |π 1 (a ′ |s ′ ) − π 2 (a ′ |s ′ )| = 2r max 1 − γ E s ′ ∼P1 [D T V (π 1 (·|s ′ ) ∥ π 2 (·|s ′ ))] .
For (b), we have
(b) = E s ′ ,a ′ ∼P1,π2 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) ≤ E s ′ ,a ′ ∼P1,π2 Q π2 M2 (s ′ , a ′ ) − E s ′ ,a ′ ∼P2,π2 Q π2 M2 (s ′ , a ′ ) .
Adding these two bounds together yields the desired result.
D Detailed Environment Setting
D.1 Grid World
In the grid world environment, the agent obtains the X-Y coordination as the state and executes one of the four actions (Up, Down, Left, Right) at each time step. A non-zero reward 1.0 is provided only if the agent reaches the goal. Each episode terminates when the agent reaches the goal or the episode length of 256 is reached. The source domain and the target domain of the grid world are shown in Figure 9. For each algorithm, the agent interacts with the source and target domains for 5e 5 and 5e 4 steps, respectively.
D.2 Mujoco Environments
To investigate the performance of the algorithm thoroughly, we design eight environments based on four Mujoco [67] Detailed modifications of the environments with morphology shifts are shown below:
HalfCheetah -no thighs: We modify the size of both thighs. Detailed modifications of the xml file are:
1 < geom fromto = " 0 0 0 -0.0001 0 -0.0001 " name = " bthigh " size = " 0.046 " type = " capsule " / > 2 < body name = " bshin " pos = " -0.0001 0 -0.0001 " > 1 < geom fromto = " 0 0 0 0.0001 0 0.0001 " name = " fthigh " size = " 0.046 " type = " capsule " / > 2 < body name = " fshin " pos = " 0.0001 0 0.0001 " > Ant -short feet: We modify the size of feet on leg 1 and leg 2. Detailed modifications of the xml file are: 1 < geom fromto = " 0.0 0.0 0.0 0.1 0.1 0.0 " name = " left_ankle_geom " size = " 0.08 " type = " capsule " / > 1 < geom fromto = " 0.0 0.0 0.0 -0.1 0.1 0.0 " name = " right_ankle_geom " size = " 0.08 " type = " capsule " / > Walker -no right thigh: We modify the size of thigh on the right leg. Detailed modifications of the xml file are: 1 < body name = " thigh " pos = " 0 0 1.05 " > 2 < joint axis = " 0 -1 0 " name = " thigh_joint " pos = " 0 0 1.05 " range = " -150 0 " type = " hinge " / > 3 < geom friction = " 0.9 " fromto = " 0 0 1.05 0 0 1.045 " name = " thigh_geom " size = " 0.05 " type = " capsule " / > 4 < body name = " leg " pos = " 0 0 0.35 " > 5 < joint axis = " 0 -1 0 " name = " leg_joint " pos = " 0 0 1.045 " range = " -150 0 " type = " hinge " / > 6 < geom friction = " 0.9 " fromto = " 0 0 1.045 0 0 0.3 " name = " leg_geom " size = " 0.04 " type = " capsule " / > 7 < body name = " foot " pos = " 0.2 0 0 " > 8 < joint axis = " 0 -1 0 " name = " foot_joint " pos = " 0 0 0.3 " range = " -45 45 " type = " hinge " / > 9 < geom friction = " 0.9 " fromto = " -0.0 0 0.3 0.2 0 0.3 " name = " foot_geom " size = " 0.06 " type = " capsule " / > 10 </ body > 11 </ body > 12 </ body > Hopper -big head: We modify the size of the head. Detailed modifications of the xml file are: 1 < geom friction = " 0.9 " fromto = " 0 0 1.45 0 0 1.05 " name = " torso_geom " size = " 0.125 " type = " capsule " / > E Algorithms and Implementation Details E.1 Implementation Details
The details of our algorithm and baseline methods are specified as follows:
SAC: We first specify the implementation of the shared backbone algorithm SAC utilized in all algorithms. The policy and the value function are two-layer MLP with 256 hidden units using ReLU activation. The learning rate is 3e −4 . Discount γ is set as 0.99 in all environments. The temperature coefficient is fixed as 0.2. The batch size is 128. The smoothing coefficient of the target networks is 0.005. The training delay of the policy is set as 2. The replay buffer size is 1e 6 .
VGDF:
We use a five-layer MLP with 200 units as the dynamics model using Swish activation following prior works [9,27]. The ensemble size is 7. We set the data selection ratio ξ% as 25% in the experiments shown in Section 6.1. For each probabilistic dynamics model T ϕi (s t+1 , r t |s t , a t ) = N (µ ϕi (s t , a t ), Σ ϕi (s t , a t )), i = 1, . . . , M , we train the model by maximizing the objective:
J(ϕ i ) := E (st,at,rt,st+1)∼Dtar µ ϕi (s t , a t )− (s t+1 , r t ) ⊤ Σ −1 ϕi (s t , a t ) [µ ϕi (s t , a t ) − (s t+1 , r t )] + log detΣ ϕi (s t , a t ) .(13)
The exploration policy is a two-layer MLP with 256 hidden units. We warm-start the algorithm by utilizing samples from both domains without selection for the first 1e5 steps in the source domain.
DARC:
We follow the default configurations of the public implementation (https://github. com/google-research/google-research/tree/master/darc). The domain classifiers q ψ SAS (s t , a t , s t+1 ), q ψ SA (s t , a t ) are trained by maximizing the cross-entropy losses:
J(ψ SAS ) := E (st,at,st+1)∼Dtar [log q ψ SAS (tar|s t , a t , s t+1 )] + E (st,at,st+1)∼Dsrc [log(1 − q ψ SAS (tar|s t , a t , s t+1 ))] , J(ψ SA ) := E (st,at)∼Dtar [log q ψ SA (tar|s t , a t )] + E (st,at)∼Dsrc [log(1 − q ψ SA (tar|s t , a t ))] .
Following the original implementation, we use the standard Gaussian noise for the domain classifier training. During training, a reward correction ∆r(s t , a t ) is augmented to the original reward r(s t , a t ) of each source domain transition, i.e.r(s t , a t ) := r(s t , a t ) + ∆r(s t , a t ). The reward correction is calculated by: ∆r(s t , a t ) := log q ψ SAS (tar|s, a, s ′ ) q ψ SAS (src|s, a, s ′ ) q ψ SA (src|s, a) q ψ SA (tar|s, a) . We warm-start the algorithm by training with samples from both domains for the first 10 5 steps following the original implementation.
GARAT:
We use the author implementation with default configurations (Supplemental in https://proceedings.neurips.cc/paper/2020/hash/ 28f248e9279ac845995c4e9f8af35c2b-Abstract.html). We add the XML files of our customized environments to rl_gat/envs/assets/ folder. We limit the extra interactions with the grounded source environments as 10 5 for fair comparisons with other algorithms.
Importance Weighting Clip (IW Clip): We use the domain classifiers same as DARC to calculate the importance weight w(s, a, s ′ ). The importance weighting is calculated by: w(s, a, s ′ ) := P tar (s ′ |s, a) P src (s ′ |s, a) ≈ q ψ SAS (tar|s, a, s ′ ) q ψ SAS (src|s, a, s ′ ) q ψ SA (src|s, a) q ψ SA (tar|s, a) ,
where q ψ SAS and q ψ SA are the domain classifiers proposed in [14]. We use the importance weighing to reweight the value training with source domain samples. Specifically, θ ← arg min θ 1 2 E (s,a,r,s ′ )∼Dsrc w(s, a, s ′ )(Q θ − T Q θ ) 2 .
To stabilize training, we clip the importance weight between [1e −4 , 1], same as the prior work [46].
Finetune:
We first train a policy in the source domain with 10 6 steps. Then we transfer the policy to the target domain and further train the policy for 10 5 steps.
The detailed hyperparameters of all algorithms are listed in Table. 2, and we use the same hyperparameters across all environments.
E.2 Implementation Details of the Offline-Online Experiments
To evaluate the performance of our algorithm in the offline source online target setting, we use medium datasets from D4RL [17] for three environments (i.e., HalfCheetah, Hopper, Walker). We use the same source domain offline dataset for each environment's two different target domains. For the algorithms performing online learning using offline data (i.e., Symmetric sampling, H2O, VGDF + BC), we perform the online interactions with the target domain for 10 5 steps and use 10 6 source domain transitions, the training is repeated for 10 times per step in the target domain. The details of the methods are specified as follows:
Offline only: We directly transfer the policy learned through CQL [32] with the source domain offline dataset. For the CQL implementation, we follow the suggested configurations in a public CQL implementation (https://github.com/tinkoff-ai/CORL). We perform training for 10 6 steps with the offline dataset and report the zero-shot performance of the learned policy in the target domain.
Symmetric sampling [3]: We perform the value function training by combining CQL optimization (with offline transitions) and SAC optimization (with online transitions). For each training step, we sample 50% of the data from the target domain replay buffer and the remaining 50% from the source domain offline dataset. The CQL and SAC loss is computed with the corresponding transitions.
H2O [46]: We follow the original implementation that learns the classifiers to estimate the dynamics discrepancy across domains and perform the clipped importance weighting on the CQL loss on the source domain data. Same as Symmetric sampling, we repeat the training for 10 times per step in the target domain.
VGDF + BC: We adapt VGDF to the Offline-Online setting by simply integrating the behavior cloning loss following (10). The training is repeated for 10 times per step with the target domain the same as the baseline methods. For the trade-off between the policy gradient and behavior cloning, we use the value-normalized regularization following the TD3 + BC [19] work and set the constant α as 5. Furthermore, we remove the exploration policy proposed in Section 5.1 since the online access to the source domain is no longer available in the offline-online setting.
F Additional Experiment Results
F.1 Quantifying Dynamics Shifts via FVP
In this section, we investigate whether the estimation of the value differences can quantify the difference across domains. Specifically, in different target domains of the same source domain, we demonstrate the estimation of FVP in two target domains. As the results show in Figure 11, the FVP differs in environments with different dynamics shifts (Kinematic or morphology). We observe that the FVP values in two target domains gradually approach each other in three out of four environments (HalfCheetah, Walker, Hopper), while the values in Ant remain relatively stationary. Furthermore, the FVP values in target domains with kinematic shifts are lower than those with morphology shifts across all four environments, which could result from the mismatched state space due to the limited joint ranges of robots in the target domain. Given the differences across different environments, we believe the FVP estimation could be used to quantify the domain differences.
F.2 Sensitivity to Ensemble Size
We have introduced the dynamics model ensemble to capture the epistemic uncertainty induced by the limited samples from the target domain. However, training the ensemble of the dynamics model takes extra computation resources. Unlike prior works in model-based RL [27,58] that utilize the variant with importance weighting in almost all environments. The accuracy of the value proximity
Figure 2 :
2The illustrations and results of the motivation experiment. (a) Illustration of the source and target domains in the grid world environment. The red dot and green square represent the agent and goal, respectively. (b) Visualization of the state visitation in both domains. The darker color suggests higher visitation probabilities. Our method guides the agent to reach regions with high target domain values while the agent trained by DARC is stuck in the room. (c) Visualization of the learned Q tables.
Theorem 4.1. (Performance bound controlled by dynamics discrepancy.) Denote the source domain and target domain with different dynamics as M src and M tar , respectively. We have the performance difference of any policy π evaluated under M src and M tar be bounded as below,
Figure 4 :
4Adaptation performance in the target domain with kinematic mismatch (Top) or morphology mismatch (Bottom). Solid curves are average returns over five runs with different random seeds, and shaded areas indicate one standard deviation. We use data ratio Γ = 10, which indicates all algorithms perform 10 6 online interactions with the source domain except Oracle.
Figure 5 :
5Effect of transition ratio Γ.Data ratio Γ. We employ different ratios of transitions from the source domain versus those from the target domain (Γ = 5, 10, 20) for variants of our algorithm. The results shown inFigure 5demonstrate that the performance of our algorithm improves with more source domain transitions when the number of target domain transitions is the same. This finding indicates that VGDF can fully exploit the reusable source domain transitions to enhance the training efficiency concerning the target domain.
Figure 6 :
6Effect of data selection ratios ξ%.
Figure 7 :
7Effect of the optimistic exploration technique (i.e., π E ).
Figure 8 :
8Quantification analysis of the approximated FVP in Ant environments. We approximate the FVP in Eq. (5) by calculating the average likelihood of a batch of samples from the source domain by E[Λ(s, a, s ′ )] ≈ 1 B (s,a,s ′ )Λ (s, a, s ′ ).
Algorithm 1
1Value-Guided Data Filtering (VGDF) Input: Source domain M src , target domain M tar , and transition ratio Γ (= 10) (source vs. target). Initialization: Policy π, exploration policy π E , value functions {Q θi } i=1,2 , replay buffers{D src , D tar }, dynamics model ensemble {T ϕi } M i=1 , data selection ratio ξ, batch size B, entropy temperature coefficient λ. (s src , a src , r src , s ′ src ) using π E in M src4:D src ← D src ∪ (s src , a src , r src , s ′ src ) (s tar , a tar , r tar , s ′ tar ) using π in M tar 8:D tar ← D tar ∪ (s tar , a tar , r tar , s ′ tar ) ensemble {T ϕi } M i=1 with Dtar via Eq. (13) 11: Sample b src := {(s, a, r, s ′ )} B src from D src 12: Sample b tar := {(s, a, r, s ′ )} B tar from D tar 13: Obtain Fictitious Value Proximity (FVP) {Λ(s, a, s ′ )} B via Eq. (6) for transitions in b src 14:Obtain FVP quantile Λ ξ% of {Λ(s, a, s ′ )} B 15:
btar∪ bsrc min {Q θ1 (s, a), Q θ2 (s, a)} | a∼π(·|s) + λH[π] 21: end for B Proofs of the Performance GuaranteesThis section presents the proof of our main results. Specifically, we propose that the value discrepancy can be leveraged for the performance guarantee across different domains Lemma C.3. In Theorem B.1, we convert the performance bound induced by the value discrepancy into a novel form for the offline source domain setting.Theorem B.1. (Performance bound controlled by dynamics discrepancy.) Denote the source domain and target domain with different dynamics as M src and M tar , respectively. We have the Algorithm 2 Value-Guided Data Filtering + Behavior Cloning (VGDF + BC) Input: Source domain offline dataset D src , target domain M tar , max interaction steps with the target domain T max , and transition ratio Γ (:= |Dsrc| Tmax = 10) (source vs. target). Initialization: Policy π, value functions {Q θi } i=1,2 , target domain replay buffer D tar , dynamics model ensemble {T ϕi } M i=1 , data selection ratio ξ, batch size B, entropy temperature coefficient λ, train repeat K, behavior cloning constant α. (s tar , a tar , r tar , s ′ tar ) using π in M tar 4: D tar ← D tar ∪ (s tar , a tar , r tar , s ′ tar ) ensemble {T ϕi } M i=1 with D tar via Eq. src := {(s, a, r, s ′ )} B src from D src 9: Sample b tar := {(s, a, r, s ′ )} B tar from D tar 10: Obtain Fictitious Value Proximity (FVP) {Λ(s, a, s ′ )} B via Eq. (6) for transitions in b src 11:
min {Q θ1 (s, a), Q θ2 (s, a)} a∼π(·|s) 17: π ← arg max π β 2B btar∪ bsrc min {Q θ1 (s, a), Q θ2 (s, a)} a∼π(·|s) + λH[π]performance difference of any policy π evaluated under M src and M tar be bounded as below,
a) [D TV (P src (·|s, a)∥P tar (·|s, a))] .
Figure 9 :
9The source domain (Left) and the target domain (Right) of the grid world environments.
Figure 10 :
10Illustration of all environments, including all source domains (Top), all target domains with kinematic shifts (Middle), and all target domains with morphology shifts (Bottom). and Hopper-v2. For each benchmark, we propose two variants with kinematic shift or morphology shift. We run all experiments with the original environment as the source domain and the variation environment as the target domain. Detailed modifications of the environments are shown below, and the illustration of the environments is shown in Figure 10. For algorithms that access interactions with both domains, the agent interacts with the source and target domains for 10 6 and 10 5 steps, respectively. Detailed modifications of the environments with kinematic shifts are shown below: HalfCheetah -broken back thigh: We modify the rotation range of the joint on the thigh of the back leg from [−0.52, 1.05] to [−0.0052, 0.0105]. Ant -broken hips: We modify the rotation range of the joints on the hip of leg 1 and leg 2 from [−30, 30] to [−0.3, 0.3]. Walker -broken right foot: We modify the rotation range of the joint on the foot of the right leg from [−45, 45] to [−0.45, 0.45]. Hopper -broken joints: We modify the rotation range of the joint on the head from [−150, 0] to [−0.15, 0] and the joint on foot from [−45, 45] to [−18, 18].
Figure 11 :
11Quantification analysis of the approximated FVP in all environments with different dynamics shifts. The dots are averaged values, and the error bars indicate the standard error across five runs.
Theories on learning with dynamics mismatch. The performance guarantee of a policy trained with imaginary transitions from an inaccurate dynamics model has been analyzed in prior Dynastyle [61, 62, 64] model-based RL algorithms[41,27, 58]. The theoretical results inspire us to formulate performance guarantees in the context of dynamics adaptation.], representations [5], and experiences [26, 36, 71, 65]. Our method is
related to works transferring experiences. However, prior works focus on transferring between tasks
with different reward functions instead of dynamics. When the dynamics changes, the direct adoption
of commonly used temporal difference error [60] or advantage function [57] in previous works
[26, 36, 65] would be inappropriate due to the shifted transition probabilities across domains. In
contrast, we introduce novel measurements to evaluate the usefulness of the source domain transitions
to tackle the dynamics shift problem specifically.
Table 1 :
1Results in the offline source online target setting. We evaluate the algorithms via the performance of the learned policy in the target domain and report the mean and std of the results across five runs with different random seeds. Offline only that directly transfers the offline learned policy via CQL[32] to the target domain; Symmetric sampling [3] that samples 50% of the data from the target domain replay buffer and the remaining 50% from the source domain offline dataset for each training step; H2O[46] that penalizes the Q function learning on source domain transitions with the estimated dynamics gap via learned classifiers. All algorithms have limited interactions with the target domain to 10 5 steps. The experimental details are shown in Appendix E.2. The results shown inOffline only
Symmetric sampling
H2O
VGDF + BC
halfcheetah -broken back thigh 1128 ± 156
2439 ± 390
5761 ±148 4834 ± 250
halfcheetah -no thighs
361 ± 39
2211 ± 77
3023 ± 77
3910 ±160
hopper -broken hips
155 ± 19
2607 ± 181
2435 ± 325
2785 ±75
hopper -short feet
399 ± 5
2144 ± 509
868 ± 73
3060 ±60
walker -broken right thigh
1453 ± 412
709 ± 128
3743 ±50
3000 ± 388
walker -no right thigh
975 ± 131
872 ± 301
2600 ± 355
3293 ±306
exploration technique results in performance degradation in three out of four environments concerning
the sample efficiency, validating the effectiveness of the exploration policy.
6.3 Performance under Offline Source with Online Target
In this subsection, we extend our method to the setting with a source domain offline dataset and
limited online interactions with the target domain, investigating the performance of our method
without online access to the source domain. We use the D4RL medium datasets [17] of three
environments (i.e., HalfCheetah, Walker, Hopper) for evaluation. We compare the proposed VGDF +
BC (Section 5.2) with the following baselines:
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Table 2 :
2Hyperparameters. "-" denotes the hyperparameter is not used in the algorithm. "←" denotes the same choice as the algorithm in the first column.Hyperparameters
VGDF DARC GARAT IW Clip Finetune
Hidden layers (Policy)
2
←
←
←
←
Hidden units per layer (Policy)
256
←
←
←
←
Hidden layers (Value)
2
←
←
←
←
Hidden units per layer (Value)
256
←
←
←
←
Hidden layers (Classifier)
-
2
-
2
-
Hidden units per layer (Classifier)
-
256
-
256
-
Hidden layers (Dynamics model)
5
-
-
-
-
Hidden units per layer (Dynamics model)
200
-
-
-
-
Ensemble size
7
-
-
-
-
Learning rate
3e −4
←
←
←
←
Batch size
128
←
←
←
←
Fixed temperature coefficient
0.2
←
←
←
←
Target smoothing coefficient
0.005
←
←
←
←
Policy training delay
2
←
←
←
←
Buffer size
1e 6
←
←
←
←
Data selection ratio ξ%
25%
-
-
-
-
Warm-start steps
1e 5
1e 5
-
1e 5
-
Importance weight clipping range
-
-
-
[1e −4 , 1]
-
Interactions with grounded src environment
-
-
1e 5
-
-
The results validate that a smaller ensemble size is sufficient to achieve competitive asymptotic performance compared to the variant with a large ensemble size in most environments. the generated samples for training, we measure the value difference with the help of the generated samples. Therefore, we aim to investigate whether a smaller ensemble size is sufficient to achieve competitive asymptotic performance. Here we set the ensemble size as different values (M = 7 in the original implementation) and run experiments in four environments. As the results show inFigure 12, variants with a small ensemble size (e.g., M = 3 or M = 5) can achieve identical asymptotic performance compared to the variant with a large ensemble size (e.g., M = 7) in three out of four environments.F.3 What about Importance Weighting via FVP instead of Rejection Sampling?Steps inTargetThe results demonstrate that the original algorithm using rejection sampling outperforms the variant using importance weighting via FVP in almost all environments.In the case of data selection based on the estimated FVP (fictitious value proximity in Eq. (6), one may wonder about using importance weighting via the FVP rather than rejection sampling, which might be sample-inefficient due to the discarded partial data. Here we implement a variant of our algorithm that performs importance weighting with the estimated fictitious value proximity. Specifically, we train the value functions following:We compare the variant with the original algorithm using rejection sampling in all eight environments and demonstrate the results inFigure 13. The original algorithm using rejection sampling outperforms depends on the generated state and the value function. Thus, the estimation of FVP could be biased due to the inaccurate dynamics models and value functions in the early training stage, in which case naively utilizing the source domain samples weighted by the FVP can harm the policy performance concerning the target domain. In contrast, rejection sampling that only utilizes a small portion of source domain samples alleviates the negative effect of the source domain samples.F.4 What about Data Filtering via Value instead of FVP?Steps inTargetPrior works have examined sharing data across tasks with different reward functions rather than dynamics[71]. To investigate whether selectively sharing data with a high Q value can address the online dynamics adaptation problem, we propose a variant of our algorithm that shares partial data with a relatively high Q value from the source domain. Specifically, we train the value functions following:where Q ξ% is the top ξ-quantile Q value of a batch of source domain samples. We set ξ% as 25%, the same as our implementation. We compare the variant with the original algorithm in all eight environments and demonstrate the results inFigure 14. The results demonstrate that the original algorithm outperforms the variant using data filtering via value in four of eight environments. Due to the dynamics mismatch, a state-action pair from the source domain will lead to inconsistent states concerning two domains. Therefore, directly utilizing the transitions with high Q value without considering the consistency of the next state would provide a counterfactual value target for the state-action pair, which can result in an improper value estimation for learning.F.5 Comparison with Dynamics-guided Data FilteringTo investigate the effect of value consistency, we perform the ablation study by comparing VGDF to a variant that shares partial data based on dynamics discrepancies, i.e., Dynamics-guided Data Filtering (DGDF). Specifically, we estimate the dynamics discrepancy via the learned classifiers following the prior works[14,46]. Same as VGDF, we share the source domain transitions whose estimated dynamics difference is smaller than the quantile value. We set the selection ratios as 25%, the same as our implementation. The results demonstrate that the original algorithm outperforms the variant in three out of four environments, validating the superior effect of the value consistency compared to the dynamics discrepancy.
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| [
"https://github.com/tinkoff-ai/CORL)."
]
|
[
"Lazy states, discordant states and entangled states for 2-qubit systems",
"Lazy states, discordant states and entangled states for 2-qubit systems"
]
| [
"Jianwei Xu \nCollege of Science\nNorthwest A&F University\n712100YanglingShaanxiChina\n"
]
| [
"College of Science\nNorthwest A&F University\n712100YanglingShaanxiChina"
]
| []
| We investigate the lazy states, entangled states and discordant states for 2-qubit systems. We show that many lazy states are discordant, many lazy states are entangled, and many mixed entangled states are not lazy. With these investigations, we provide a laziness-discord-entanglement hierarchy diagram about 2-qubit quantum correlations. | 10.1142/s0217979215501210 | [
"https://arxiv.org/pdf/1401.4260v1.pdf"
]
| 118,476,374 | 1401.4260 | 6ad65353a2ee3c9a3fc7608f1ad7cb64cdfc3309 |
Lazy states, discordant states and entangled states for 2-qubit systems
17 Jan 2014
Jianwei Xu
College of Science
Northwest A&F University
712100YanglingShaanxiChina
Lazy states, discordant states and entangled states for 2-qubit systems
17 Jan 2014(Dated: January 20, 2014)numbers: 0365Ud0367Mn0365Aa
We investigate the lazy states, entangled states and discordant states for 2-qubit systems. We show that many lazy states are discordant, many lazy states are entangled, and many mixed entangled states are not lazy. With these investigations, we provide a laziness-discord-entanglement hierarchy diagram about 2-qubit quantum correlations.
I. INTRODUCTION
Quantum correlation is one of the most striking features of quantum theory. Entanglement is the most famous kind of quantum correlation, and leads to powerful applications [1]. Discord is another kind of quantum correlation, which captures more correlation than entanglement in the sense that a disentangled state may have no zero discord [2]. Due to the theoretical and applicational interests, discord has been extensively studied [2] and still in active research (for examples see [3][4][5][6]).
A bipartite state is called lazy, if the entropy rate of one subsystem is zero under any coupling to the other subsystem. Necessary and sufficient conditions have recently been established for a state to be lazy [7], and it was shown that almost all states are pretty lazy [8]. It is shown that a maximally entangled pure state is lazy [9]. This indicates that the correlation described by lazy states is not the same by entanglement. So we are interested to clarify the question that, whether there are many lazy states which are entangled, and whether there are many entangled states which are lazy. This paper answers this question for the 2-qubit case. This paper is organized as follows. In Section 2, we briefly review the definitions of entangled states, discordant states and lazy states. In Section 3, we establish a necessary and sufficient condition for 2-qubit lazy states. In Section 4, we show that there are many 2-qubit lazy states which are discordant states. In Section 5, we show that there are many disentangled states which are not lazy. In Section 6, we show that there are many 2-qubit mixed lazy states which are entangled. In section 7, we briefly summary this paper by providing a lazinessdiscord-entanglement hierarchy diagram to characterize the bipartite quantum correlations. * Electronic address: [email protected]
II. ENTANGLED STATES, DISCORDANT STATES, LAZY STATES
We briefly review the definitions about entangled states, discordant states and lazy states.
Finite-dimensional quantum systems A and B are described by the Hilbert spaces H A and H B respectively, the composite system AB is then described by the Hilbert
space H A ⊗ H B . Let n A = dim H A , n B = dim H B .
A state ρ AB is called a disentangled state (or separable state) if it can be written in the form
ρ AB = i p i ρ A i ⊗ ρ B i ,(1)
where
p i ≥ 0, i p i = 1, {ρ A i } i are density operators on H A , {ρ B
i } i are density operators on H B .If ρ AB is disentangled we then say E(ρ AB ) = 0.
A state ρ AB is called a zero-discord state with respect to A if it can be written in the form
ρ AB = nA i=1 p i |ψ A i ψ A i | ⊗ ρ B i ,(2)
where
p i ≥ 0, i p i = 1, {|ψ A i } i is an orthonormal basis for H A , {ρ B i } i are density operators on H B .If ρ AB is in the form Eq.(2) we then say D A (ρ AB ) = 0.
Evidently,
D A (ρ AB ) = 0 ⇒ E(ρ AB ) = 0.(3)
A state ρ AB is called a lazy state with respect to A if
[7] C A (ρ AB ) = [ρ AB , ρ A ⊗ I B ] = 0,(4)
where ρ A = tr B ρ AB , I B is the identity operator on H B . An important physical interpretation of lazy states is that the entropy rate of A is zero in the time evolution under any coupling to B,
C A (ρ AB (t)) = 0 ⇔ d dt tr A [ρ A (t) log 2 ρ A (t)] = 0. (5)
D A (ρ AB ) = 0 and C A (ρ AB ) = 0 has the inclusion relation below [9]
D A (ρ AB ) = 0 ⇒ C A (ρ AB ) = 0.(6)
Maximal pure entangled states are the examples of C A (ρ AB ) = 0 but D A (ρ AB ) = 0 [9]. The direct product states have the form
ρ AB = ρ A ⊗ ρ B ,(7)
they are obviously zero-discord states.
III. THE FORM OF 2-QUBIT LAZY STATES
Any 2-qubit state can be written in the form [10] ρ AB = 1
4 (I ⊗ I + 3 i=1 x i σ i ⊗ I + 3 j=1 y j I ⊗ σ j + 3 i,j=1 T ij σ i ⊗ σ j ),(8)
where I is the two-dimensional identity operator,
{σ i } 3 i=1 are Pauli operators, {x i } 3 i=1 , {y j } 3 j=1 , {T ij } 3 i,j=1
, are all real numbers satisfying some conditions (we will explore these conditions when we need them) to ensure the positivity of ρ AB , ρ A and ρ B . We often omit I for simplicity without any confusion.
{x i } 3 i=1 //{T ij } 3 i=1 for j = 1, 2, 3.(9)
Proof. For state in Eq. (8),
ρ A = 1 2 (I + 3 k=1 x k σ k ⊗ I),(10)[ρ AB , ρ A ] = 1 8 3 ijk=1 T ij x k [σ i ⊗ σ j , σ k ⊗ I] = 1 8 3 ijk=1 T ij x k [σ i , σ k ] ⊗ σ j = i 4 3 ijkl=1 T ij x k ε ikl σ l ⊗ σ j .(11)
In the last line, ε ikl is the permutation symbol.
Let [ρ AB , ρ A ] = 0, then
It is easy to check that C A (ρ AB ) = 0 defined in Eq.(4) is invariant under locally unitary transformations for arbitrary n A and n B . Under locally unitary transformations, any 2-qubit state in Eq. (8) can be written in the form [11] ρ AB = 1
4 (I ⊗ I + 3 i=1 x i σ i ⊗ I + 3 j=1 y j I ⊗ σ j + 3 i=1 λ i σ i ⊗ σ i ),(13)where 0 ≤ λ 1 ≤ λ 2 ≤ λ 3 being the singular values of {T ij } ij in Eq.(8). Note that {x i } 3 i=1 , {y j } 3 j=1
in Eq.(9) are not the same with in Eq. (8).
We now look for the conditions such that D A (ρ AB ) = 0. Suppose D A (ρ AB ) = 0, then according to Eq.(2), there exists real vector − → n = {n 1 , n 2 , n 3 } with n 2 1 + n 2 2 + n 2 3 = 1 such that
ρ AB = Π 0 ⊗ Iρ AB Π 0 ⊗ I + Π 1 ⊗ Iρ AB Π 1 ⊗ I, (14) with Π 0 = 1 2 (I + − → n · − → σ ),(15)Π 1 = 1 2 (I − − → n · − → σ ).(16)
It can be check that
Π 0 σ i Π 0 + Π 1 σ i Π 1 = n i − → n · − → σ .(17)
Then Eq. (14) becomes
ρ AB = 1 4 (I ⊗ I + 3 i=1 x i n i − → n · − → σ ⊗ I + 3 j=1 y j I ⊗ σ j + 3 i=1 λ i n i − → n · − → σ ⊗ σ i ) = 1 4 (I ⊗ I + 3 ij=1 x i n i n j σ j ⊗ I + 3 j=1 y j I ⊗ σ j + 3 ij=1 λ i n i n j σ j ⊗ σ i ).(18)
Comparing to Eq.(13), then for j = 1, 2, 3,
3 i=1 x i n i n j = x j ⇒ − → n // − → x ,(19)λ i n i n j = δ ij λ j = δ ij λ i ⇒ λ i = 0 or n i = ±1.(20)
(i).If λ 1 = λ 2 = λ 3 = 0, let − → n // − → x , then D A (ρ AB ) = 0.
(ii).If 0 = λ 1 = λ 2 < λ 3 = 0, then − → n = (0, 0, ±1), to satisfy − → n // − → x , we see that only when − → x = (0, 0, x 3 ) we have D A (ρ AB ) = 0.
(iii).If 0 = λ 1 < λ 2 < λ 3 = 0, then Eq.(20) can not be satisfied, so ρ AB is discordant.
(iv).If 0 < λ 1 < λ 2 < λ 3 = 0, then Eq.(20) can not be satisfied, so ρ AB is discordant.
Comparing with Proposition 1, we then get Proposition 2 below. We make a note that some constraints about {y j } 3 j=1 , λ 1 , λ 2 , λ 3 are required to guarantee the positivity of ρ AB , ρ A , ρ B in Proposition 2.These constraints are rather complex since there are so many parameters. To show there indeed exist many states described in Proposition 2, we choose some special states. For the state
ρ AB = 1 4 (I ⊗ I + 3 j=1 y j I ⊗ σ j + 3 i=1 λ i σ i ⊗ σ i ),(21)
where 0 ≤ λ 1 ≤ λ 2 , 0 < λ 2 < λ 3 , we have ρ A = I,and
ρ B = 1 2 (I + 3 j=1 y j σ j ). (22) ρ B is positive then 3 j=1 y 2 j ≤ 1.(23)
Let y 2 = y 3 = λ 1 = 0, then the four eigenvalues of ρ AB in Eq.(21) are
1 4 (1 ± y 2 1 + (λ 3 ± λ 2 ) 2 ).(24)
These eigenvalues are all nonnegtive then we need
0 < λ 2 < λ 3 ,(25)y 2 1 + (λ 3 + λ 2 ) 2 ≤ 1.(26)
There are many triples {y 1 , λ 3 , λ 2 } satisfy Eqs. (25,26), then the corresponding states in Eq.(21) are lazy but discordant states.
V. SOME DISENTANGLED BUT NOT LAZY 2-QUBIT STATES
To show there exist many 2-qubit states which are disentangled but not lazy, we consider the states of the form
ρ AB = p|ψ A 1 ψ A 1 | ⊗ ρ B 1 + (1 − p)|ψ A 2 ψ A 2 | ⊗ ρ B 2 , (27) where p ∈ (0, 1), {|ψ A i } 2 i=1 are normalized states in H A but not necessarily orthogonal,{ρ B i } 2 i=1
are density operators on H B . Note that p = 0 or p = 1 leads to direct product states, so we do not consider such cases.
Under locally unitary transformations, we let
|ψ A 1 ψ A 1 | = I + (0, 0, 1) · − → σ 2 ,(28)|ψ A 2 ψ A 2 | = I + (sin α, 0, cos α) · − → σ 2 ,(29)ρ B 1 = I + a(0, 0, 1) · − → σ 2 ,(30)ρ B 2 = I + b(sin β, 0, cos β) · − → σ 2 ,(31)
where α, β ∈ [0, π], a, b ∈ [0, 1]. Some special states can be apparently specified.
{T i1 } i = (b(1 − p) sin α sin β, 0, b(1 − p) cos α sin β),(32){T i2 } i = (0, 0, 0), (34) {T i3 } i = (b(1 − p) sin α cos β, 0, ap + b(1 − p) cos α cos β).(33)
From Proposition 1, ρ AB in Eq. (27) is lazy if and only if
− → x //{T i1 } i and − → x //{T i3 } i . Since x 1 = (1 − p) sin α = 0, then − → x //{T i1 } i and − → x //{T i3 } i lead to b sin β = 0. (36) a = b cos β.(37)
VI. SOME LAZY BUT ENTANGLED STATES
We know that a bipartite pure state is lazy only if under locally unitary transformations it can be written in the form [7]
|ψ AB = 1 √ s s i=1 |ψ A i |ψ B i , where {|ψ A i } i are orthonormal sets in H A , {|ψ B
i } i are orthonormal sets in H B , s ≤ min{n A , n B }. When s = min{n A , n B } it is maximally entangled state. In this section we look for more 2-qubit mixed states which are lazy but entangled.
From Proposition 1, we know the following 2-qubit Bell-diagonal states are lazy
ρ AB = 1 4 (I ⊗ I + 3 i=1 λ i σ i ⊗ σ i ),(38)
where {λ i } 3 i=1 are real numbers satisfying some constraints to ensure the positivity of ρ AB .
In this section, for convenience, we do not assume
{λ i } 3
i=1 are all nonnegative. We represent the states in Eq.(38) in the (λ 1 , λ 2 , λ 3 ) space.
The eigenvalues of ρ AB in Eq.(38) are
1 4 {1 − λ 1 + λ 2 + λ 3 , 1 + λ 1 − λ 2 + λ 3 , 1 + λ 1 + λ 2 − λ 3 , 1 − λ 1 − λ 2 − λ 3 }.(39)
Then the positivity of ρ AB requires that {λ i } 3 i=1 are in the tetrahedron (with its boundary) with the vertices (−1, −1, −1), (−1, 1, 1), (1, −1, 1), (1, 1, −1) in the (λ 1 , λ 2 , λ 3 ) space [12]. Disentangled states in Eq.(38) are in the octahedron (with its boundary) with the vertices (±1, 0, 0), (0, ±1, 0), (0, 0, ±1) [12]. From Proposition 2, we know the zero-discord states in Eq.(38) are only three line segments (λ 1 , 0, 0) with λ 1 ∈ [−1, 1], (0, λ 2 , 0) with λ 2 ∈ [−1, 1], (0, 0, λ 3 ) with λ 3 ∈ [−1, 1].
Then the states in the tetrahedron (with its boundary) but not in the octahedron (with its boundary) are lazy but entangled. Among these, only the states at the vertices of tetrahedron are (maximally entangled) pure states.
VII. SUMMARY: A HIERARCHY DIAGRAM
We explored some 2-qubit states, showed that many states are lazy but discordant, many states are lazy but entangled, and many states are disentangled but not lazy. With these investigations, we can surely give a hierarchy diagram (Figure 1) of 2-qubit states, including lazy states, disentangled states and zero-discord states.
Proposition 1 .
1The 2-qubit state ρ AB in Eq.(8) is lazy if and only if
Proposition 2 .
2A 2-qubit state in Eq.(13) is lazy but discordant if and only if − → x = 0 and 0 < λ 2 < λ 3 . Since any locally unitary transformation keeps − → x = 0 invariant in Eq.(8), then we rewrite Proposition 2 as Proposition 2 ′ below. Proposition 2 ′ . A 2-qubit state in Eq.(8) is lazy but discordant if and only if − → x = 0 and the matrix {T ij } ij have at least two positive singular values.
(v).α = 0, ρ AB in Eq.(27) are direct product states; (vi).α = π, ρ AB in Eq.(27) are zero-discord states; (vii).a = b = 0, ρ AB in Eq.(27) are direct product states. Now we consider the cases excluding (v), (vi), (vii) above. Taking Eqs.(28-31) into Eq.(27), and using the notations in Eq.(8), we get − → x = ((1 − p) sin α, 0, p + (1 − p) cos α),
FIG. 1
1: laziness-discord-entanglement diagram This hierarchy diagram enriches the entanglementdiscord hierarchy, then provides more understandings about the structures of quantum correlations. This work was supported by the National Natural Science Foundation of China (Grant No.11347213) and the Research Start-up Foundation for Talents of Northwest A&F University of China (Grant No.2013BSJJ041). The author thanks Zi-Qing Wang and Chang-Yong Liu for helpful discussions.
ik=1 T ij x k ε ikl = 0,(12)this evidently leads to Eq.(9).IV. LAZY BUT DIACORDANT 2-QUBIT STATES
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| []
|
[
"Dead Zones as Thermal Barriers to Rapid Planetary Migration in Protoplanetary Disks",
"Dead Zones as Thermal Barriers to Rapid Planetary Migration in Protoplanetary Disks"
]
| [
"Yasuhiro Hasegawa yh:[email protected] ",
"Ralph E Pudritz rep:[email protected] \nOrigins Institute\nMcMaster University\nCanada -2L8S 4M1HamiltonON\n",
"\nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonONCanada\n"
]
| [
"Origins Institute\nMcMaster University\nCanada -2L8S 4M1HamiltonON",
"Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonONCanada"
]
| []
| Planetary migration in standard models of gaseous protoplanetary disks is known to be very rapid (∼ 10 5 years) jeopardizing the existence of planetary systems. We present a new mechanism for significantly slowing rapid planetary migration, discovered by means of radiative transfer calculations of the thermal structure of protoplanetary disks irradiated by their central stars. Rapid dust settling in a disk's dead zone -a region with very little turbulence -leaves a dusty wall at its outer edge. We show that the back-heating of the dead zone by this irradiated wall produces a positive gradient of the disk temperature which acts as a thermal barrier to planetary migration which persists for the disk lifetime.Although we analyze in detail the migration of a Super-Earth in a low mass disk around an M star, our findings can apply to wide variety of young planetary systems. We compare our findings with other potentially important stopping mechanisms and show that there are large parameter spaces for which dead zones are likely to play the most important role for reproducing the observed massperiod relation in longer planetary periods. | 10.1088/2041-8205/710/2/l167 | [
"https://arxiv.org/pdf/1001.5022v1.pdf"
]
| 118,833,263 | 1001.5022 | 4da9c86814087b0f27bbf89078a56e97e5655c46 |
Dead Zones as Thermal Barriers to Rapid Planetary Migration in Protoplanetary Disks
27 Jan 2010
Yasuhiro Hasegawa yh:[email protected]
Ralph E Pudritz rep:[email protected]
Origins Institute
McMaster University
Canada -2L8S 4M1HamiltonON
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonONCanada
Dead Zones as Thermal Barriers to Rapid Planetary Migration in Protoplanetary Disks
27 Jan 2010Received ; acceptedSubject headings: accretion, accretion disks -radiative transfer -turbulence - planets and satellites: formation -protoplanetary disks
Planetary migration in standard models of gaseous protoplanetary disks is known to be very rapid (∼ 10 5 years) jeopardizing the existence of planetary systems. We present a new mechanism for significantly slowing rapid planetary migration, discovered by means of radiative transfer calculations of the thermal structure of protoplanetary disks irradiated by their central stars. Rapid dust settling in a disk's dead zone -a region with very little turbulence -leaves a dusty wall at its outer edge. We show that the back-heating of the dead zone by this irradiated wall produces a positive gradient of the disk temperature which acts as a thermal barrier to planetary migration which persists for the disk lifetime.Although we analyze in detail the migration of a Super-Earth in a low mass disk around an M star, our findings can apply to wide variety of young planetary systems. We compare our findings with other potentially important stopping mechanisms and show that there are large parameter spaces for which dead zones are likely to play the most important role for reproducing the observed massperiod relation in longer planetary periods.
Introduction
Extrasolar planets (ESPs) have an unexpected distribution of orbital radii around their host stars (Udry et al. 2009) -ranging from about 0.02 to 70 astronomical units (AU). 1 In particular, ESPs are observed to obey a mass-period (M-P) relation wherein lower mass planets end up in short period orbits around their host stars (Udry & Santos 2007). The predominance of very short period planets is generally thought to arise as a consequence of planetary migration. As an example, the tidal interaction of a planet with its surrounding gaseous disk excites density waves in the disk at so-called Lindblad resonances.
These waves exert a torque back on the planet which results in a net angular momentum transfer between them (Goldreich & Tremaine 1980). Planets may also exchange angular momentum with the gas inside of their horseshoe region (Ward 1991). For locally isothermal protoplanetary disks with smoothly declining distributions of disk column density and temperature with radius, the net torque generally leaves a planet spiraling inwards through the disk (Tanaka et al. 2002), i.e. the torque exerted by the outer wake is marginally stronger than that of the inner wake (Ward 1997). Many calculations and simulations show that the migration timescale of planets in such "standard" disk models is very short -roughly two orders of magnitude smaller than the disk lifetime (one to ten million years (Myr)) (Ward 1997;Nelson et al. 2000;Tanaka et al. 2002;D'Angelo et al. 2003). Why are there any planetary systems at all?
The key to understanding the M-P relation and the survival of planetary systems is in how the dynamics of planetary motion is coupled to the properties and structure of the protoplanetary disks. As an example, Schlaufman et al. (2009) focused on the surface density transition that can be produced at the location of the ice-line, where a local pressure maximum can act as an accumulation point for planetesimals (Kretke & Lin 2007). If it is assumed that type I migration is much slower than predicted in locally isothermal disk models, this feature could account for planets with orbital radii 0.1 -2 AU. Obviously, a physical explanation for slower migration is needed.
In this Letter, we present a new slowing mechanism of rapid type I migration -which may occur for planets with masses that are too low to open up a gap in their disks (massive planets can tidally form a gap and undergo type II migration). We show, by means of Monte Carlo radiative transfer simulations, that dead zones -the dense inner disk region wherein turbulence cannot be readily excited (Gammie 1996) -support a thermal barrier to migration. One of the most important consequences is that the thermal barrier could account for planets at larger orbital radii. In § 2, we outline our disk model and discuss tidal torques. In § 3, we analyze how the presence of a thermal barrier impacts the migration rates of low-mass planets. In § 4, we discuss potential issues for the M-P relation by comparing our stopping mechanism with others.
Disk model & tidal torques
Protoplanetary disks are known to be heated by radiation from the central star (Chiang & Goldreich 1997;D'Alessio et al. 1998;Hasegawa & Pudritz 2010a, hereafter, HP10). This radiation mainly determines the thermal structure of disks (Kenyon & Hartmann 1987). This is because viscous heating dominates stellar irradiation heating only within about 1 AU for the classical T Tauri star systems (CTTSs; D'Alessio et al. 1998) and only within about 0.1 AU for lower mass stars such as M stars (HP10).
Detailed modeling of the spectral energy distributions emitted by disks has shown that s = −1 for disks where the disk surface density Σ ∝ r s (D' Alessio et al. 1998). It is well established that the sign of a net torque exerted on planets depends on two central properties of disks, their surface density and temperature (e.g., Ward 1997). The thermal structure therefore plays a critical role in controlling the direction of planetary migration.
In disks without internal structure, the disk temperature T ∝ r t at the mid-plane steadily decreases (t < 0) and planetary migration is steadily inwards.
The point is that disks are not simple power-law structures. The strength of turbulence within them varies considerably, with very low levels occurring in dense regions called dead zones (Gammie 1996) that initially extend over roughly 10 AU in disks (Matsumura & Pudritz 2006) and then gradually shrink in size as disk material is accreted onto the central star (Matsumura et al. 2009, hereafter, MPT09). Turbulence in disks is most likely excited by the magnetorotational instability (MRI) (Balbus & Hawley 1991).
The MRI requires good coupling between ions and magnetic fields, which is largely absent in the dead zone -that inner, high density region in which the ionization due to the X-rays from the central star and cosmic rays is suppressed.
Dust is the dominant absorber of stellar radiation in disks although its total mass is 100 times smaller than that of gas (Dullemond et al. 2009). Many observations imply that it has a density distribution that is different from the gas distribution, which is derived assuming vertical hydrostatic equilibrium (Kenyon & Hartmann 1987). Dust settling, a consequence of its size distribution (Dullemond & Dominik 2004a), is ubiquitous in protoplanetary disks around any young star (HP10, references herein). The dust scale height depends upon the amplitude of disk turbulence which keeps it suspended in the gravitational field of the disk (which is determined by the central star) (Dubrulle et al. 1995).
We demonstrated in HP10 that dust settling in dead zones results in the appearance of a limited region in which the disk temperature can actually increase with radius -a radial temperature inversion. In this Letter, we adopt the disk model developed in HP10 (see Table. 1) and focus our computations on M dwarf systems, such as the recently discovered Super-Earth (∼ 5M ⊕ ; M ⊕ is the Earth's mass) with an orbital radius of 2 AU (Beaulieu et al. 2006). We refer the readers to HP10 for the detail. The low mass of disks around M stars allows much more comprehensive Monte Carlo simulations to be performed, but our analysis in principle applies to any protoplanetary disk.
We adopt the torque formula (Ward 1997;Menou & Goodman 2004;Jang-Condell & Sasselov 2005) in which only the Lindblad torque is considered. This is because the corotation torque is readily saturated (i.e. is canceled out) in dead zones in our radiatively heated disk model.
We compare the libration timescale (ie the timescale for gas to complete an orbit in the horse- (Shakura & Sunyaev 1973), and the disk scale height is h. Our numerical results give h p /r p ≃ 0.05, so that the critical value of turbulence, α crit , below which the corotation torque is saturated (and therefore negligible),
shoe region), τ lib ≈ 8πr p /3Ω p x s , with the viscous timescale, τ vis ≈ x 2 s /3ν, where the half- width of the horseshoe region is x s /r p = 1.68(M p r p /M * h p ) 1/2 (Paardekooper & Papaloizou 2009), the kinematic viscosity is ν = αh 2 Ω Kepis α crit = 0.01 M p 5M ⊕ 3/2 M * 0.1M ⊙ −3/2 h p /r p 0.05 −7/2 .
(1)
Since the dead zone has α = 10 −5 , we can safely neglect the corotation torque for M p 0.5M ⊕ in the dead zone of our disk model. In addition, we confirmed that, in the active region where the corotation torque is generally unsaturated, both Lindblad and corotation torques result in inward migration in our disk model (Paardekooper et al. 2009).
Thus, exclusion of the corotation torque in the active region does not affect our findings in our disk model. We note that corotation torque may be unsaturated in dead zones for sufficiently small planetary masses, but the exact limit will depends on knowing the disk scale height that is established by the host star. Furthermore, our torque formula takes into account the effects of vertical disk thickness by diluting the gravitational force of a planet by z (Jang-Condell & Sasselov 2005).
Results
We performed numerical simulations of the thermal structure of 2D disks by solving the radiative transfer equation with a Monte Carlo method (Dullemond & Dominik 2004b, HP10). We included the effects of vertical hydrostatic balance, dust settling, a dead zone, and the gravitational field of a planet embedded in the disk. The tidal torque is calculated as described in § 2, and incorporates our numerical data, in order to calculate the migration time. Figure 1 presents the thermal and density structure of a disk with a 5M ⊕ planet placed at 8 AU. The top and bottom panels show the dust and gas densities by color, respectively.
Since we define the disk temperature as the mass-averaged temperature of dust, both panels
show the identical temperature structure which is represented by contours. The thick line denotes the Hill radius r H = r p (M p /3M * ) 1/3 . In this Letter, we adopt, without loss of generality, a dead zone which is 6 AU in size. Dead zones enhance dust settling because of the low turbulent amplitude there. Since disks have inner dead, and outer active regions for turbulence, the transition of the density distribution of dust occurs at the outer edge of the dead zone. This leaves a marked step in the dust scale height behind -in effect a wall of dust. The additional stellar energy absorbed at the wall is distributed by radiative diffusion (Hasegawa & Pudritz 2010b) since the optical depth at this region is high. The resulting radial "thermal inversion" -ie a region of increasing temperature with increasing disk radius -is shown in Figure 2. An analytic fit to our data shows that this back-heated region in the dead zone has a positive temperature gradient described by a power-law T ∝ r t ′ with t ′ > 3/2.
We show in Figure 3 (top) that this radial thermal inversion causes the migration rate to be positive (the migration time becomes negative -bottom panel), -i.e. planets migrate outward in the region with the positive temperature gradient. The physical explanation of this behavior is that the increasing function of disk temperature changes the disk's pressure distribution which in turn causes the position of the outer Lindblad resonances to be further from the planet than the inner ones (Artymowicz 1993). This results in outer torques that are much weaker than the inner ones. 2 Figure 3 (bottom) also shows that the planets very slowly enter the region of torque reversal -as seen by the strong positive "spike" in the migration time. These regions correspond to radii at which dT /dr ≃ 0 (see Figure 2) at the inner and outer resonances, making the torque difference between them very small.
We emphasize that the positive temperature gradient arising from the wall-like dust structure is achieved for the case of a finite transition region, △r ≤ 10h, in the value of turbulent α (HP10). Although, for simplicity, we adopt a sharp spatial transition from the active to the dead zone in this Letter (△r = 0), our above results are valid for the case of △r ≃ h, since the positions of Lindblad resonances are typically offset from the planets by 2h/3 (Artymowicz 1993). In addition, it is interesting that the migration timescale of the M star system is similar to that of CTTS (∼ 10 4 − 10 5 years) for the other two cases (well mixed, dust settling). This is because the tidal torque is scaled by Σ(h/r) −2 . A more detailed discussion of them is presented in Hasegawa & Pudritz (2010b).
Discussion
A thermal vs a density barrier at outer dead zone radius
We find that the dusty wall produces a radial temperature inversion that is a thermal barrier for rapid type I planetary migration. Whereas the torque balance in the well coupled active zone forces planets to migrate inward, once they encounter the radial thermal 2 We will show in a forthcoming paper that the Lindblad torque becomes negative when s − t/2 < −7/4 (Hasegawa & Pudritz 2010b).
inversion region, the torque balance reverses, and they move out of the region. Thus, planets are trapped there if they originally migrate from the active region beyond the dead zones, or even if they formed close to the outer edge of the dead zone.
The astrophysical implications of this result are very important since we have shown that dusty protoplanetary disks with dead zones possess an innate mechanism for strongly slowing planetary migration within them, provided that the corotation torque is saturated.
While such a thermal barrier exists for any size of dead zone (HP10), its effectiveness is probably most important for lower mass disks, as we now show.
The density structure of disks evolves with time due to viscous evolution. MPT09
found that the difference of α between the active and dead regions produces a steep density gradient at the boundary and the location of their jump moves inward with time over the long (∼ 10 Myr) viscous timescale of the dead zone. This density gradient at the outer dead zone boundary also plays an important role in slowing down or stopping type I migration,
provided that planets migrate from larger disk radii. The inner torques become larger than the outer ones in the density gradient region, resulting in the reflection of migrating planets off the density gradient (MPT09).
The relative importance of these two dead zone mechanisms is controlled by the ratio of dust settling τ set ≈ Σ/ √ 2πρ d aΩ Kep and the viscous τ vis timescales, where ρ d is the bulk density of dust and a is the grain size of dust. We find that the critical condition τ set /τ vis ≥ 1 for dominance of the density vs thermal barriers presented by a dead zone is
Σ h r 2 ≥ 25 α 10 −2 −1 ρ d 1g cm −3 a 1mm ≡ C crit .(2)
This implies that a density barrier is dominant for sufficiently large values of Σ and h/r.
For disks around CTTSs with a typical dead zone size (≈ 10 AU), Σ(h/r) 2 ∼ 300 g cm −2 × (0.4) 2 ≈ 2C crit (Chiang & Goldreich 1997), which indicates that a density barrier is dominant. For disks around M stars with a typical dead zone size (≈ 5 AU), Σ(h/r) 2 ∼ 20 g cm −2 × (0.05) 2 = 2 × 10 −3 C crit (Scholz et al. 2007), which implies that a thermal barrier is dominant. Generally, we find that a thermal barrier becomes weaker in the late stages because of accretion which reduces the density at the outer edge of the dead zones.
Comparisons with other possible stopping mechanisms and potential effects on the M-P relation
It is well known that tidal interaction and angular momentum exchange with the central star (Lin et al. 1996) and the creation of a hole in the inner part of disks by the presence of a stellar magnetosphere (Shu et al. 1994;Lin et al. 1996) cannot reproduce the whole of the observed M-P relation. This is because such barriers become important only for planets approaching very close to the central star. Most ESPs, however, have observed orbital radii from about 0.02 AU to 10 AU (Udry & Santos 2007). Thus, while these barriers may play a role in piling up ESPs in the vicinity of the star, they have difficulty in predicting planets with larger orbital radii.
Stochastic migration provides another mechanism for controlling planetary migration.
It arises when disks undergo magnetohydrodynamic turbulence (Nelson & Papaloizou 2004;Laughlin et al. 2004), which is the outcome of the MRI (Balbus & Hawley 1991). Stochastic torques tend to reduce the timescale for planetary survival, however (Johnson et al. 2006).
In a few exceptional cases, planets can diffuse out to large disk radii where they can survive longer. Planets within the dead zones cannot perform random walks because turbulent torques are reduced by about two orders of magnitude there and consequently would undergo steady inward type I migration (Oishi et al. 2007). Thus, stochastic migration is unlikely to be the main barrier to rapid type I migration.
Is our thermal barrier sufficiently wide to stop planets scattered into the dead zone by stochastic effects? We consider a characteristic length scale for turbulent diffusion defined by △r turb = √ ντ c = h ατ c /Ω −1 , where τ c is the correlation time of turbulence. We adopt a value τ c = 0.5Ω −1 (Nelson & Papaloizou 2004) and find that △r turb ≈ 0.02 AU at r = 6 AU. This is shorter than the width of the thermal barrier (∼ 2 AU). Therefore, stochastic motions due to turbulence are unlikely to affect the migration stopping mechanism at the thermal barrier.
Corotation torques are also potentially important for slowing or stopping planets. In both barotropic and adiabatic disks, corotation torque associated with a radial vortensity gradient may work as a barrier around the region of inner edge of the dead zone (∼ 0.01-0.1 AU; Fromang et al. 2002;Masset et al. 2006, MPT09) because it becomes positive due to a positive surface density gradient, resulting in outward planetary migration (Masset et al. 2006). However, the location of the barrier is almost constant with time because stellar irradiation controls it, and consequently this barrier is important only for planets in the vicinity of the star. In adiabatic disks, corotation torque associated with a radial entropy gradient may also act as a barrier because it becomes large and positive around the region with a large (in magnitude), negative entropy gradient (Paardekooper & Mellema 2006;Baruteau & Masset 2008;Paardekooper & Mellema 2008;Paardekooper & Papaloizou 2008). Thus, corotation torque may be important to the M-P relation in certain regimes of planetary mass and disk heating, as noted above.
Our mechanism has a movable barrier that shrinks from larger disk radii on the 10
Myr (for CTTSs) viscous timescale of the dead zone. As a concrete example, this shrinkage of the dead zone could explain the recent detected Super-Earth at 2 AU around an M star (Beaulieu et al. 2006). This is because a Super-Earth is likely to be the most massive planet that would surely form in a low mass disk and is therefore most likely to have been formed beyond the outer dead zone radius and left behind as the dead zone shrinks away.
We conclude that the stopping mechanisms arising from dead zones -via thermal and density gradients -are important barriers to rapid planetary migration. We have shown that the thermal barrier that arises from disk irradiation by a heated dust wall is robust and may be most important in the earlier phases of disk evolution and for the evolution of low mass systems as found around M stars as an example. Dead zones may provide the most significant slowing mechanism of type I migration that is needed to explain the longer period of population of the M-P relation, which will be checked in future population synthesis models. M ⊙ is one solar mass, and R ⊙ is one solar radius.
-17 - Fig. 1.-The density and temperature structures of disks with a 5M ⊕ mass planet located at 8 AU. For both panels, the density is denoted by colors [g cm −3 ] as shown in colorbar, and the disk temperature is denoted by contours [K] (both panels show the identical temperature since the disk temperature is defined by taking the mass-average of the dust temperatures).
The size of a dead zone is 6 AU. Top: the dust density. Bottom: the gas density. The thick black circle denotes the Hill sphere r H = r p (M p /3M * ) 1/3 , where r p and M p is the position and mass of a planet, respectively, and M * is the stellar mass. A wall-like structure appears at the boundary between the active and dead zones in the dust distribution while it is not in the gas.
Fig. 2 .
2-The disk temperature in the disk mid-plane. The solid line is for the case of dead zone, the dashed line is the analytical model of disk temperature(Chiang & Goldreich 1997), and the dashed-dot line is for analytical fit to a positive temperature gradient. The size of a dead zone, which is 6 AU, is indicated by the vertical solid line.
Fig. 3 .
3-Migration rate and time for a 5 mass M ⊕ planet at various orbital radii on the top and bottom panels, respectively. For both panels, the solid line denotes the case of a dead zone which is 6 AU in size whose position is indicated by the vertical solid line, the red dashed line is for the well-mixed case, and the blue dashed-dot line is for the dust settling case. The negative migration time region in the bottom panel (outward migration) corresponds to the region with a positive temperature gradient (seeFigure 2).
The authors thank Kees Dullemond, Thomas Henning, Hubert Klahr, Kristen Menou, Soko Matsumura and Richard Nelson for stimulating discussions. We also thank an anonymous referee for a useful report. Our simulations were carried out on computer clusters of the SHARCNET HPC Consortium at McMaster University. YH is supported by McMaster University, as well as by Graduate Fellowships from SHARCNET and the Canadian Astrobiology Training Program (CATP). REP is supported by Discovery Grant from NSERC.
Table 1 :
1Summary of parameters & symbolsSymbol
Meaning
Value
M *
Stellar mass
0.1 M ⊙
R *
Stellar radius
0.4R ⊙
T *
Stellar effective temperature
2850 K
Σ
The surface density of (gas + dust)
∝ r −1
M d
The total disc mass (gas + dust)
4.5 × 10 −3 M ⊙
gas-to-dust ratio
100
α
Parameter for turbulence (active/dead)
10 −2 /10 −5
M p
Planetary mass
5 M ⊕
See the website: http://exoplanet.eu.
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| []
|
[
"Properties of scalar wave emission in a scalar-tensor theory with kinetic screening",
"Properties of scalar wave emission in a scalar-tensor theory with kinetic screening"
]
| [
"Masaru Shibata \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany\n\nCenter for Gravitational Physics and Quantum-Information\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n",
"Dina Traykova \nMax Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany\n"
]
| [
"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany",
"Center for Gravitational Physics and Quantum-Information\nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan",
"Max Planck Institute for Gravitational Physics (Albert Einstein Institute)\nAm Mühlenberg 114476Potsdam-GolmGermany"
]
| []
| We study numerically the scalar wave emission by a non-spherical oscillation of neutron stars in a scalartensor theory of gravity with kinetic screening, considering both the monopole and quadrupole mode emission. In agreement with previous results in the literature, we find that the monopole is always suppressed by the screening effect, regardless of the size of the screening radius, r sc . For the quadrupole mode, however, our analysis shows that the suppression only occurs for screening radius larger than the wavelength of scalar waves, λ wave , but not for r sc < λ wave . This demonstrates that to fully understand the nature of this theory, it is necessary to study other more complex systems, such as neutron star binaries, considering a wide range of r sc values. | 10.1103/physrevd.107.044068 | [
"https://export.arxiv.org/pdf/2210.12139v2.pdf"
]
| 253,080,679 | 2210.12139 | 6a50d0841be7a919d2e9155d8716086fa960584a |
Properties of scalar wave emission in a scalar-tensor theory with kinetic screening
Masaru Shibata
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 114476Potsdam-GolmGermany
Center for Gravitational Physics and Quantum-Information
Yukawa Institute for Theoretical Physics
Kyoto University
606-8502KyotoJapan
Dina Traykova
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
Am Mühlenberg 114476Potsdam-GolmGermany
Properties of scalar wave emission in a scalar-tensor theory with kinetic screening
We study numerically the scalar wave emission by a non-spherical oscillation of neutron stars in a scalartensor theory of gravity with kinetic screening, considering both the monopole and quadrupole mode emission. In agreement with previous results in the literature, we find that the monopole is always suppressed by the screening effect, regardless of the size of the screening radius, r sc . For the quadrupole mode, however, our analysis shows that the suppression only occurs for screening radius larger than the wavelength of scalar waves, λ wave , but not for r sc < λ wave . This demonstrates that to fully understand the nature of this theory, it is necessary to study other more complex systems, such as neutron star binaries, considering a wide range of r sc values.
I. INTRODUCTION
The ample evidence for the current accelerated expansion of the Universe has hinted at the existence of some new physics at cosmological scales [1][2][3][4][5][6][7][8]. One of the simplest modifications to general relativity (GR), which can provide a possible explanation of this phenomenon, is the so-called scalar-tensor theories, where an additional scalar degree of freedom is minimally (e.g. quintessence [9][10][11][12]; see also Refs. [13,14] for reviews) or non-minimally coupled to the gravitational metric (see Refs. [15][16][17][18] for a review on scalartensor gravity). On cosmological scales, it is possible to measure and constrain physical parameters that capture this novel behaviour [19][20][21][22][23], showing that modifications to GR that can account for the observed accelerated expansion of the Universe on these scales with the dark sector whose density is of the order of the critical density, ρ c . This means that we can expect similar deviations on small scales too. However, Solar System [24,25] and binary pulsar [26][27][28][29][30] tests show no violations of the predictions of GR there. In addition, radio observations of pulsars (neutron stars) accompanying white dwarfs constrain the emissivity of scalar-type gravitational waves (hereafter referred to simply as scalar waves), and thus, the parameter space for some scalar-tensor theories has been significantly limited [30][31][32]. More recently, consistency with GR has also been shown by null tests with gravitational-wave observations [33][34][35][36][37].
One possible solution to this problem is employing an appropriate screening mechanism, by which the effects of the scalar field are suppressed on local scales so that GR phenomena can be reproduced, while on cosmological scales, modifications to GR remain appreciable. Some well-studied examples of this behaviour are the chameleon [38], symmetron [39], and Vainshtein [40][41][42] screening (see also Refs. [43][44][45] for reviews). Even though screening effects have been studied extensively in a range of simplified scenarios, such as weakgravity and spherical symmetry approximations (see, e.g., Refs. [46][47][48][49][50][51][52][53][54][55][56]), they are not so well-understood in strongly self-gravitating and dynamical environments, such as the dynamical neutron star spacetime. For example, the emission mechanism of scalar waves has not been yet well-understood. In order to fully characterise the screening effect in dynamical spacetimes, for which no linearization or symmetry of the system can be employed, numerical relativity (NR), by which the solution of the fully non-linear systems can be obtained, is needed.
NR simulations of compact objects in scalar-tensor theories with a kinetic screening effect have been performed in a few recent studies [57][58][59][60][61][62], some of which report a non-trivial nature of the scalar-wave emission. In particular, in Ref. [59], the authors find that the quadrupole scalar wave emission may not be screened in the case of a binary neutron star inspiral. This study focuses on the cases with a small screening radius ( 140 km), which is smaller than the wavelength of gravitational and scalar waves. We argue here that, in a such setting, the screening effect may not be significant and one could expect different behaviour when larger screening radii, which are more realistic, are considered.
In this paper, we study numerically the emission of scalar waves from non-spherically oscillating neutron stars in the same scalar-tensor theory employed in Ref. [59]. It has been shown in Ref. [63] that scalar waves in a scalar-tensor theory of gravity can be detected by interferometers in the same way as gravitational waves. Their analysis, done in the framework of the Brans-Dicke theory shows that, for a simple Michelson interferometer, the antenna sensitivity pattern depends strongly on the frequency of the scalar gravitational waves, with essentially the same features as those of the tesnosr mode of GWs. Thus showing that as long as the dependence of the antenna sensitivity pattern on the wave length of scalar waves is taken into account in the same way as for the tensor modes, scalar waves would be detectable in the case of a scalar-tensor theory. Therefore in this work we treat both scalar and tensor modes as gravitational waves.
Our NR simulation is performed in the Jordan frame in contrast to previous works [58,59], which employ the Einstein frame instead. Doing this has three advantages: (i) the equations for hydrodynamics are not changed and have a conservative form, same as in GR; (ii) the gravitational and scalar waves are extracted independently from the spacetime metric and the scalar field, respectively; and (iii) unlike the Einstein arXiv:2210.12139v2 [gr-qc] 21 Feb 2023 frame case, the Jordan frame metric couples universally to the matter fields and so observables can also be computed in the same way as one does in GR. We will show that if the screening radius is larger than the wavelength of scalar waves, the screening effect on the scalar waves (i.e., the suppression of the scalar wave emission) is always significant irrespective of the multipoles considered.
The paper is organised as follows. In Sec. II we summarise the basic equations that we employ. Section III presents a formulation for computing equilibrium and quasi-equilibrium states, necessary for the initial conditions in NR simulation. Section IV presents numerical solutions of 1.4M spherical neutron stars and summarises the properties of a neutron star spacetime in the presence of the kinetic screening. In Sec. V we explore the non-spherical oscillation of neutron stars obtained in Sec. IV, in particular focusing on the generation and propagation of quadrupole scalar waves. Finally, we discuss our results and summarise our conclusions in Sec. VI.
Throughout this paper, we use the units of c = 1 =h, where c andh denote the speed of light and the reduced Planck constant, respectively. In these units the Planck length, p := G 1/2 = 1.616 × 10 −33 cm and the Planck mass, M p := G −1/2 = 2.176 × 10 −5 g. The subscripts a and b denote the spacetime tensor components, and i, j, and k denote the spatial components.
II. BASIC EQUATIONS
In this work we consider a scalar-tensor theory with kinetic screening, in which the action in the so-called Jordan frame is given by [24,[64][65][66],
S = 1 16πG d 4 x √ −gφ R + 3 2 +K α 2 s g ab ∇ a φ ∇ b φ φ 2 +S matter (χ matter , g ab ) .
(1)
The corresponding action in the Einstein frame can be found in, e.g., Refs. [59,67]. Here R and ∇ a are the Ricci scalar and covariant derivative associated with the spacetime metric g ab , φ (> 0) is the gravitational scalar field andK is a function of the canonical kinetic term of the scalar field, X. S matter is the action of the perfect fluid, with χ matter representing the matter fields. The kinetic term of the scalar field is defined as,
X =ḡ ab∇ aφ∇bφ = φ −1 g ab ∇ aφ ∇ bφ ,(2)
whereḡ ab is the spacetime metric in the Einstein frame,∇ a is its covariant derivative,φ = ln φ / 16πGα 2 s , and α s is a coupling constant. Following Ref. [59], we consider the case,
K(X) = − 1 2 + γ 1 4Λ 4 X − γ 2 8Λ 8 X 2 · · · ,(3)
where Λ is the strong-coupling scale (i.e., λ := Λ −1 determines the length scale of screening), and γ 1 and γ 2 are constants of order unity. Here we choose γ 1 = 0 and γ 2 = 1 as it has been shown (see Refs. [68,69]) that this is a necessary condition for having a well-posed initial value formulation, as well as screening static solutions. Screening is expected to occur in the strong field zone, where X > Λ 4 is satisfied. We suppose that φ → 1 (i.e.,φ → 0 and X → 0) for r → ∞. For γ 1 = γ 2 = 0 this theory is equivalent to the Fierz-Jordan-Brans-Dicke (FJBD) theory [64][65][66], with Brans-Dicke parameter of the form,
ω(X) := − 3 2 −K (X) α 2 s ,(4)
with X = 0, so that ω(X) = − 1 2 3 − α s −2 . Then the basic equations for the geometry, scalar field, energy momentum tensor, T ab , and rest-mass continuity are as follows,
G ab = 8πGφ −1 T ab − 3 2 +K α 2 s φ −2 (∇ a φ )∇ b φ − 1 2 g ab (∇ c φ )∇ c φ − X α 2 s φ 2 ∂K ∂ X ∇ a φ ∇ b φ + φ −1 (∇ a ∇ b φ − g ab 2 g φ ), (5) ∇ a (F∇ a φ ) = 8πGα 2 s T,(6)∇ a T a b = 0,(7)
∇ a (ρu a ) = 0,
where G ab is the Einstein tensor associated with g ab , T = T a a , u a is the fluid four velocity, ρ is the rest-mass density, and
F := −2 ∂ (XK) ∂ X = 1 − γ 1 X Λ 4 + 3γ 2 4 X 2 Λ 8 + · · · .(9)
To derive Eq. (6), we used the trace of Eq. (5),
−R = 8πGφ −1 T + 3 2 +K α 2 s − X α 2 s ∂K ∂ X (∇ a φ )∇ a φ φ 2 − 3 φ 2 g φ ,(10)where 2 g = ∇ a ∇ a .
For T ab , we consider the stress-energy tensor for a perfect fluid,
T ab = (ρ + ρε + P)u a u b + Pg ab ,(11)
where ε and P are the specific internal energy and pressure of the fluid. In the Jordan frame the fluid matter is coupled only to the gravitational field, as seen in Eq. (7). Hence, the equations for the perfect fluid are the same as those in GR in this frame. The basic equations in the 3+1 formulation for the gravitational field are derived simply by contracting n a n b , n a γ b i , and γ a i γ b j with Eq. (5). Here, γ ab = g ab + n a n b denotes the spatial metric, and n a is the unit normal to the spatial hypersurfaces. The 3+1 form of the scalar field equation is derived from Eq. (6) by defining Π := −n a ∇ a φ orΠ := −Fn a ∇ a φ .
The evolution of the scalar field and its conjugate momentum have the following form,
(∂ t − β k ∂ k )φ = −αΠ,(12)(∂ t − β k ∂ k )Π = −D i αFD i φ + αKΠ +8πGαα 2 s T ,(13)
where D i is the covariant derivative with respect to γ i j . In terms of Π and φ , X can be written as,
X = 1 16πGα 2 s φ 3 (D k φ )D k φ − Π 2 .(14)
From these one can also obtain an algebraic equation for X,
f (X) := X − 1 16πGα 2 s φ 3 (D k φ )D k φ −Π 2 F(X) 2 = 0 . (15)
For a detailed description of the 3+1 equations of the system, we refer the reader to Appendix A. The evolution equations for the gravitational fields are solved numerically in the Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formalism [70,71] with the movingpuncture gauge [72,73], as done in Ref. [74]. In particular, we evolve the conformal factor W := ψ −2 (with ψ := (detγ i j ) 1/12 ), the conformal metricγ i j := ψ −4 γ i j , the trace part of the extrinsic curvature K, the conformally weighted tracefree part of the extrinsic curvatureà i j := ψ −4 (K i j − Kγ i j /3) (with K i j -the extrinsic curvature), and the auxiliary variablẽ Γ i := −∂ jγ i j . Introducing the auxiliary variable B i and a parameter η s , which is typically set to be ∼ M −1 , M being the total mass of the system 1 , we employ the moving-puncture gauge in the form of [75],
(∂ t − β j ∂ j )α = −2αK,(16)(∂ t − β j ∂ j )β i = (3/4)B i ,(17)(∂ t − β j ∂ j )B i = (∂ t − β j ∂ j )Γ i − η s B i ,(18)
where α and β i are the lapse function and shift vector, respectively.
The spatial derivative is evaluated by a fourth-order central finite difference scheme, except for the advection terms, which are evaluated by a fourth-order non-centred finite difference. For the time evolution, we employ a fourth-order Runge-Kutta method (see Ref. [76]). We use the same scheme for the evolution of the scalar field as for the tensor, because the structure of the equations is essentially the same.
To solve the hydrodynamics equations, we evolve ρ * := ραu t W −3 ,û i := hu i , and e * := hαu t − P/(ραu t ), with h being the specific enthalpy. The advection terms are handled with a high-resolution shock capturing scheme of a third-order piecewise parabolic interpolation for the cell reconstruction. For the equation of state (EOS), we decompose the pressure and the specific internal energy into cold and thermal parts as,
P = P cold + P th , ε = ε cold + ε th .(19)
Here, P cold and ε cold are functions of ρ, and their forms are determined by nuclear-theory-based zero-temperature EOSs. Specifically, the cold part of both variables are determined using the piecewise polytropic version (see, e.g., Ref. [77]) 1 We note that the total mass includes a contribution both from the ADM mass and the scalar charge (see the tensor mass in Sec. III).
of the APR4 EOS [78], for which the maximum mass of the neutron stars in GR is ≈ 2.2M . Then the thermal part of the specific internal energy is defined from ε as ε th := ε − ε cold . Because ε th vanishes in the absence of shock heating, ε th is regarded as the finitetemperature part (and thus, this part is minor in the present study). The thermal pressure is determined by a Γ-law EOS,
P th = (Γ th − 1)ρε th ,(20)
and we choose Γ th equal to 1.8, following Refs. [74,77].
III. FORMULATION FOR INITIAL CONDITIONS
Here we outline the formulation for computing quasiequilibrium configurations for a binary in a circular orbit with angular velocity Ω following Refs. [79][80][81]. This description is also valid for computing static spherical stars with Ω = 0.
To derive quasi-equilibrium configurations, for simplicity, we assume the conformal flatness of the three metric, such that
γ i j = ψ 4 f i j ,(21)
where f i j is the flat spatial metric, and employ the conformal thin-sandwich prescription. We also impose the maximal slicing K = 0. For integrating the hydrodynamics equations, we assume the presence of a helical Killing vector, ξ a = (∂ t + Ω∂ ϕ ) a . For the fluid part, the basic equations in the Jordan frame are the same as those in GR. Thus, assuming that the velocity field is irrotational, the first integral of the hydrodynamics equations is readily determined in the same manner as those in GR [82,83]. The basic equations for the tensor field are obtained from the Hamiltonian and momentum constraints, together with the evolution equation for K (see Appendix A) under the maximal slicing condition, K = 0 = ∂ t K. Except for the modifications introduced by the presence of the scalar field, φ , the equations are again the same as in GR. The Hamiltonian and momentum constraints are written as,
(0) ∆ ψ = −2πGφ −1 ρ h ψ 5 − 1 8à i jà i j ψ 5 − ψ 5 8 ω φ 2 Π 2 + (D i φ )D i φ + 2φ −1 D i D i φ − 2Π 2 α 2 s φ 2 X ∂K ∂ X ,(22)and (0) D i (ψ 6Ãi j ) = ψ 6 8πGφ −1 J j + ω − X α 2 s ∂K ∂ X φ −2 Π (0) D j φ + φ −1 ( (0) D j Π −à i j (0) D i φ ) ,(23)
respectively. Here D i are the Laplacian and covariant derivatives with respect to f i j , ρ h := T ab n a n b , and J i := −T ab n a γ b i .à i j is the trace-free conformal extrinsic curvature, satisfying K j i =à j i for K = 0. The equation forà i j can be obtained from the evolution equation for γ i j with Eq. (21) and has the form,
A i j = 1 2α f ik (0) D j β k + f jk (0) D i β k − 2 3 f i j (0) D k β k ,(24)
where indices ofà i j ,à i j , and
D i are raised and lowered by f i j and f i j . The condition K = 0 = ∂ t K yields the equation for α, which leads to the equation for χ := αψ in the form,
(0) ∆ χ = χψ 4 2πGφ −1 (2S + ρ h ) + 7 8à i jà i j + 1 8 ωφ −2 7Π 2 − (D i φ )D i φ − 1 4α 2 s φ 2 X ∂K ∂ X 2(D k φ )D k φ + Π 2 + 3 4φ (D i D i φ − 22 g φ ) ,(25)
where S := T ab γ ab . Note that we replace 2 g φ using
2 g φ = 1 F 8πGα 2 s T − (∇ a X)(∇ a φ ) ∂ F ∂ X = 1 F 8πGα 2 s T − (D k X)D k φ + (n a ∇ a X)Π ∂ F ∂ X ,(26)
and will replace the Laplacian term of D i D i φ using the equation for φ , as defined below.
For the scalar field, if we simply set Π = 0, Eq. (13) (with K = 0) leads to an elliptic equation for φ ,
D i D i φ = ψ −4 (0) ∆ φ + 2ψ −1 ( (0) D i ψ) (0) D i φ = −(D i ln α)D i φ +F −1 8πGα 2 s T − (D k φ )(D k X) ∂ F ∂ X ,(27)with X = (D k φ )D k φ /(16πGα 2 s φ 3 ).
The treatment with Π = 0 is justified in the case where the gravitational radiation reaction timescale is much longer than the orbital period, 2π/Ω. With the choice of Π = 0, the equation for 2 g φ simplifies to
2 g φ = F −1 8πGα 2 s T − (D k φ )(D k X) ∂ F ∂ X .(28)
Furthermore, Eqs. (22), (23), and (25) are also simplified given the choice of Π = 0.
To obtain the solution for spherical stars in exact equilibrium, we set Ω = 0, β k = 0,Ã i j = 0, and solve the elliptic equations only for ψ, χ, and φ with appropriate boundary conditions at r = 0 and r → ∞. The hydrostatic equation has the form,
αh = const.(29)
The asymptotic behaviour of ψ, χ, and φ for r → ∞ is given by
ψ → 1 + M ADM 2r ,(30)χ → 1 − 2M K − M ADM 2r ,(31)φ → 1 + 2M S r ,(32)
where M ADM , M K , and M S are the ADM mass, Komar mass, and scalar charge. The tensor mass, which is a conserved quantity in scalar-tensor theories and the ADM mass in the Einstein frame, is defined from M ADM and M S by [84]
M T = M ADM + M S .(33)
The virial relation, which is satisfied in stationary and quasiequilibrium solutions, is written as [85]
M K = M ADM + 2M S = M T + M S .(34)
Equation (27) indicates that in the far zone, for which X < Λ 4 is satisfied, |φ − 1| is of the same order of magnitude as α 2 s GM/r, where M denotes the mass of the system. Using the definition of X in Eq. (14), the magnitude of X/Λ 4 is written as
∼ α 4 s λ 4 16π 2 p r g r 2 2 ,(35)
where r g = GM/c 2 is the gravitational radius. Thus the screening effect occurs for r r sc := α s λ (r g / p ) 1/2 . Here λ ≈ 1.97 × 10 −11 cm (Λ/1 MeV) −1 . In the following, we specify the strength of the screening by the dimensionless parameter
β := λ 8 4 p r 4 g, ≈ 1.20 × 10 28 λ 5 × 10 −11 cm 8 ≈ 1.08 × 10 28 Λ 0.4 MeV −8 ,(36)
where r g, = GM /c 2 . Using this parameter, the radius of the screening region can be expressed as
r sc = α s β 1/8 (r g r g, ) 1/2 = 5.53 × 10 2 km α s 0.1 β 10 28 1/8 r g 1.4r g, 1/2 = 5.58 × 10 2 km α s 0.1 Λ 0.4 MeV −1 r g 1.4r g, 1/2 .(37)
Any object that has a screening radius larger than it's physical size would screen modifications to gravity within this region.
IV. SPHERICAL NEUTRON STARS
In this section we summarise how the screening effect appears in static spacetimes by showing solutions of spherical neutron stars of M T = 1.4M for a wide range of β , defined in Sec. III. We find that the qualitative behaviour of φ , F(X), and geometric quantities is essentially the same for other values of M T , and thus, we focus only on this specific mass case. We fix α s = 0.1. For the 1.4M neutron star, the stellar radius (circumferential radius) is ≈ 11.1 km and the scalar charge is ≈ 0.018M irrespective of the value of β . The validity of the numerical equilibrium profile is confirmed by the fact that the virial relation is satisfied within a relative error < 10 −4 . Figure 1 plots the profiles of φ − 1 (in the left panel) and F(X) (right panel) as functions of the coordinate radius r (in isotropic coordinates) for β = 1 and 10 16 -10 36 . Note that for β = 1, F(X) ≈ 1 for the entire region, and hence, the solution may be considered as that in the FJBD theory. It is found that the central value of φ − 1, φ c − 1, decreases with the increase of β , reflecting the screening effect. The value of φ c − 1 is approximately proportional to β 1/8 , i.e., proportional to the screening radius, r sc , for β ≥ 10 16 .
The right panel of Fig. 1 demonstrates that Eq. (37) approximately indicates the screening region of F(X) 2. For the larger values of β , we find a wider screening region, whereas for β 10 16 , the screening region disappears. Around the stellar centre, F(X) approaches unity because D j φ = 0 = Π in such a region, and thus, the screening is absent near the stellar centre. Note that the peak of F(X) (and thus X in our present choice) always appears near the stellar surface (which is located at r ≈ 8.9 km). Outside the stellar surface, F(X) decreases approximately proportional to r −n , where n ≈ 1.6 (denoted by the red dashed line on the plot). The reason for this is explained by the following analysis. Outside the neutron star, Eq. (6) is integrated to give (in the present case),
αψ 2 r 2 F∂ r φ = 8πGα 2 s T αψ 6 r 2 dr = 2M T .(38)
Assuming that F ∝ r −n and φ ∝ r −p , the left-hand side is approximately proportional to r 1−p−n , resulting in n = 1 − p. On the other hand, X is approximately proportional to (∂ r φ ) 2 ∝ r −2p−2 , and for X 1, F(X) ∝ X 2 ∝ r −4p−4 , resulting in n = 4p + 4. Thus we obtain p = −3/5 and n = 8/5. 2 For X 1, it scales as X ∝ r −4 , and thus, F(X) steeply approaches unity. Inside the stellar surface, F(X) increases with the radius for M T = 1.4M . However, this is not always the case for high-mass neutron stars (M T 2M for the APR4 EOS), for which T (= −ρ(1 + ε) + 3P) can be positive for a very high-density region. For such a star, F(X) becomes unity not only at r = 0 but also at an stellar interior; thus, F(X) does not increase monotonically inside the star. However, outside such a radius, F(X) starts to increase again until the stellar surface. (34)). On the other hand, in the presence of the scalar charge, it goes as M S /r (purple dashed line). This plot shows that in the presence of screening, 1 − αψ 2 ∝ r −2 , while outside the screening region it behaves approximately as M S /r. As already mentioned, the scalar charge depends only weakly on the value of β , and hence, in the far region, the profile of αψ 2 is essentially the same for any value of β 3 .
V. NON-SPHERICAL OSCILLATION OF SPHERICAL NEUTRON STARS
Here, we explore the emission of scalar and gravitational waves from oscillating neutron stars 4 . As a zeroth-order solution, we take the M T = 1.4M neutron stars from Sec. IV. We also perform simulations for a high-mass neutron stars with M T = 1.9M and find very similar results to the 1.4M case. Thus, in the following, we present only the results for M T = 1.4M . All the simulations are performed for β ≤ 10 32 , i.e., Λ 0.1 MeV.
To excite a small quadrupole oscillation we superimpose
u x = σ x and u y = −σ y ,(39)
where we set σ = 1.0 × 10 3 s −1 . The oscillation velocity is at most 3% of the speed of light near the stellar surface, and hence, the density and pressure profiles remain close to the spherical ones. However, the quadrupole mode, l = |m| = 2, 3 We note that outside the screening region, the geometrical profile is the same as that in the FJBD theory with a Brans-Dicke parameter, as defined in Eq. (4), ω = (−3 + α −2 s )/2, irrespective of the value of β . 4 We note that in scalar-tensor theory, Birkoff's theorem is not valid. of scalar and gravitational waves is still appreciably excited so, in the following, we pay particular attention to this mode.
The numerical simulations are performed using a fixedmesh refinement code, SACRA [76], covering the radius of spherical neutron stars by N = 45 and 55 grid points in the finest computational domain. We find that the dependence of the numerical results on the grid resolution is very weak in the present problem, and we always show the result for N = 55 in the following. For scalar waves we directly analyse φ − 1 in the far region of r λ wave . For gravitational waves, we extract the outgoing component of the complex Weyl scalar (the so-called Ψ 4 ). For more details, see Appendix B. Our simulations are performed at longest for 15 ms. For high values of β , we find that it is in fact not trivial to perform a long-term simulation (with duration longer than 10 ms) as a small numerical error often emerges in the primitive recovery process of determining X from Eq. (15) and in some cases leads to a pathological solution (see Appendix A for details). However, it is still possible to draw an important conclusion from relatively short-term simulations as we will show. We leave developing an implementation for a long-term simulation (with duration of 10 ms) for future work.
We perform simulations for β = 1, 10 16 , 10 20 , 10 22 , 10 24 , 10 26 , 10 28 , 10 30 , 10 32 , and 10 36 (the corresponding Λ for which are Λ ∼ {1.28 × 10 3 , 12.8, 4.04, 2.27, 1.27, 0.718, 0.404, 0.227, 0.128, 4.04 × 10 −2 } MeV, respectively), as well as in GR (i.e., in the absence of the scalar field or φ = 1). When β = 1, F(X) ≈ 1 in the entire region, and hence, the results are essentially the same as those in the FJBD theory. Figure 3 shows the evolution of the central density for β = 1, 10 20 , 10 24 , and 10 28 as well as in GR. Due to the input perturbation, the star oscillates with time not only nonspherically but also spherically, and as a result, the central density also varies with time. In this figure we can see that the oscillation pattern and amplitude depend very weakly on the value of β , although the ones with larger screening effect (i.e. β ≥ 10 24 ) appear to agree best with GR. Since the oscillation pattern is approximately identical for all the models, we may consider that the source of the scalar and gravitational wave emission is approximately identical in the present setting.
We indeed find that the gravitational waveforms depend only very weakly on the value of the β parameter (see Fig. 7 in Appendix B). In particular, for β 10 24 , i.e., where the screening effect to the scalar wave generation becomes noticeable, the gravitational waveforms are in a good agreement with those in GR (although about 10% level disagreement is found irrespective of β values presumably due to the numerical error). For β = 1 (approximately same as the FJBD case), the amplitude of gravitational waves is slightly higher than those in GR, reflecting a significant contribution of the scalar field in determining the stellar profile.
By contrast, the amplitude of scalar waves depends strongly on the β parameter in spite of approximately the same emission source, although the frequency is always identical in all cases. Figure 4 shows the quadrupole mode of scalar waves as a function of t − r for β = 1, 10 16 , 10 20 , 10 24 , and 10 28 , and and for the monopole (l = m = 0) mode (filled squares). For the monopole mode the asymptotic amplitude of ∂ t φ 00 r ex is plotted. The red dotted line denotes ∝ β −1/8 , which indicates that the asymptotic amplitude of scalar waves decreases approximately as r −1 sc for the parameter space of r sc > λ wave irrespective of the modes considered. The black vertical dashed line denotes the value of β which satisfies r sc = λ wave for the quadrupole mode. For the monopole mode, r sc = λ wave is satisfied at β 1/8 ∼ 460 in the present case. β 1/8 (see the hollow squares). These plots show that, for r sc < λ wave , scalar waves are emitted to the far zone broadly in the same manner as in the FJBD case, in which screening is absent. Interestingly, we find that for r sc > λ wave , where the screening effect plays an important role, the amplitude of scalar waves is suppressed 5 . The reason for the suppression 5 Besides the amplitude dependence on β , a phase misalignment among the can be seen in the large value of F(X) inside the screening radius. By rewriting Eq. (6) as
∇ a ∇ a φ + (∇ a ln F)∇ a φ = 8πGα 2 s T F −1 ,(40)
we can see that the factor F −1 suppresses the scalar wave generation associated with the matter motion by T . One point to be added is that the suppression in the wave amplitude is not as large as the one by the F factor. For example, for β = 10 28 , F > 10 2 for r = 1-20 km, while the suppression fraction in the wave amplitude is ∼ 1/10. The reason for this is that the wave amplitude, defined by φ 22 (r/M), increases during the outward propagation inside the screening radius, i.e., for r < r sc , by F −η (see Appendix B for details).
As we can see in Fig. 5, the amplitude of quadrupole scalar waves depends only weakly on β for r sc λ wave /3 (i.e., β 10 20 ), with the steep decline starting only at r sc ∼ λ wave This suggests that the suppression effect by F −1 in the wave generation and the amplification effect during the propagation of waves in the region of F > 1 is likely to be balanced for the quadrupole mode.
On the contrary, for the monopole we find that the steep decline does not start at the point of r sc = λ wave , where the wavelength of the monopole mode is ∼ 80 km, and thus, r sc = λ wave (l = m = 0) is satisfied at β ≈ 2 × 10 21 . The decrease of the asymptotic amplitude is again approximately proportional to β −1/8 , satisfied for a wide range of β values, as can be seen in the filled squares of Fig. 5. This can also be seen clearly in Fig. 6, which shows the monopole waveforms for β = 1, 10 16 , 10 20 , 10 24 , and 10 28 , extracted at r = 591 km. (We should note that in this case, we analyse ∂ t φ 00 simply because it is clearer to see the oscillation mode.) This feature is in agreement with the results found in Ref. [58], in which the scalar waves is found. The reasons for this are discussed in more detail in Appendix B.
authors analyse the amplitude of l = m = 0 scalar waves emitted by the spherical oscillation of a neutron star. Therefore, we can conclude that while the steep decline of the amplitude is always found irrespective of the modes for r sc > λ wave , in the case of r sc λ wave , the emergence of the screening effect on the scalar wave emission depends on the modes considered, presumably reflecting the generation mechanism (e.g., the main generation location) of each mode.
Finally, we consider the results of Ref. [59], in which the authors explored scalar and gravitational waves from the late inspiral phase of binary neutron stars, in the case where λ wave ( 300 km) > r sc ≈ 140 km 6 . From our present analysis our suspicion is that one cannot expect the screening effect to appear in the quadrupole mode during the inspiral in that setting. We argue that to fully understand the screening effect on the quadrupole mode, one should consider parameters, for which r sc > 300 km. As we discussed above, the screening effect may appear in the low-multipole mode even for the case of r sc < λ wave , which means one can expect to find screening in the dipole mode even for small screening radii, as they report.
VI. DISCUSSION
By analysing an oscillating spherical neutron star, we have confirmed that in a scalar-tensor theory with kinetic screening, the scalar wave emission is suppressed for a screening radius, r sc , larger than the wavelength of the emitted waves, λ wave , irrespective of multipole modes considered. Therefore, inside the screening radius satisfying the condition r sc > λ wave , both the matter motion and wave emission are essentially the same as those in GR.
However, for a screening radius r sc λ wave , we have found emission of quadrupole scalar waves with a large amplitude, comparable to that in FJBD theory and additionally that the amplitude depends only weakly on r sc . Therefore, if the analysis were to be restricted to small values of r sc , this could have lead to the conclusion that no screening effect is present in these theories. We argue that to fully understand the nature of this theory it is necessary to perform the analysis at a wide range of r sc values, including r sc > λ wave .
For the monopole mode, we have confirmed that the screening effect appears even for the case of r sc < λ wave as was also found previously in Ref. [58]. Furthermore, we have found that, irrespective of the modes considered, the asymptotic scalar wave amplitude decreases roughly as r −1 sc when r sc > λ wave . For ground-based gravitational wave detectors, such as advanced LIGO and advanced Virgo, the lower limit of the frequency in the sensitive band of gravitational waves is about 10 Hz, and thus, the upper limit of the observable wavelength is ≈ 3 × 10 4 km. Therefore, if r sc > 3 × 10 4 km, it would be difficult to detect scalar-type gravitational waves due to the screening in this kind of scalar-tensor theories. A number of previous solar system experiments have reported no evidence for the presence of a scalar field effect, which implies that r sc has to be larger than the solar radius (≈ 7 × 10 5 km). Thus, the detection of scalar waves, e.g., from neutron-star oscillations and inspiraling binary neutron stars, by the groundbased gravitational wave detectors might be unlikely in kinetic screening theories. 7 Our analysis in this paper has focused only on scalar and gravitational waves from oscillating neutron stars. To fully understand the emission mechanism of scalar waves in screened modified gravity theories, we should also perform simulations for other systems, such as binary neutron stars for a wide range of r sc . As we have pointed out here, the emissivity of scalar waves is determined by the profile of F(X), and if the profile for other systems is similar to that of single neutron stars, we can expect the conclusion to be the same; i.e., that the scalar wave emission is suppressed in the presence of screening with r sc > λ wave irrespective of the multipole modes. Thus, the question is what the profile of F(X) is for other systems. We leave this further investigation for subsequent work.
And contracting Eq. (5) with n a γ b i gives the momentum constraint,
D i K i j − D j K = 8πGφ −1 J j + ω − X α 2 s ∂K ∂ X φ −2 Π D j φ + φ −1 (D j Π − K i j D i φ ) ,(A2)
and so the evolution equation can be obtained by contracting Eq. (5) with γ a i γ b j ,
∂ t K i j = αR i j − 8πGαφ −1 S i j − 1 2 γ i j (S − ρ h ) + α(−2K ik K k j + KK i j ) − D i D j α + β k D k K i j + K ik D j β k + K jk D i β k − α ω − X α 2 s ∂K ∂ X φ −2 (D i φ )D j φ − αφ −1 (D i D j φ − K i j Π) − α 2φ γ i j 2 g φ − α 2φ 2 γ i j (D k φ )D k φ − Π 2 X α 2 s ∂K ∂ X ,(A3)
where R i j is the spatial Ricci tensor and S i j := T ab γ a i γ b j with S its trace. Equation (A3) together with the Hamiltonian constraint yields the following evolution equation for K,
(∂ t − β k ∂ k )K = 4πGαφ −1 (S + ρ h ) + αK i j K i j − D i D i α + αωφ −2 Π 2 + αφ −1 D i D i φ − KΠ − αX 2α 2 s φ 2 ∂K ∂ X (D k φ )D k φ + Π 2 − 3α 2φ 2 g φ ,(A4)
and thus, the evolution equation forà i j = ψ −4 (K i j − Kγ i j /3), where ψ = (det γ i j ) 1/12 is the conformal factor, is written in the form
(∂ t − β k ∂ k )à i j = α ψ 4 R i j − 1 3 γ i j R k k − ψ −4 D i D j α − 1 3 γ i j D k D k α +à ik ∂ j β k +à jk ∂ i β k − 2 3à i j ∂ k β k + α Kà i j − 2à ikà k j − 8πG α ψ 4 φ S i j − 1 3 γ i j S − α ψ 4 φ 2 ω − X α 2 s ∂K ∂ X (D i φ )D j φ − 1 3 γ i j (D k φ )D k φ − α ψ 4 φ D i D j φ − 1 3 γ i j D k D k φ − ψ 4à i j Π .(A5)
The term, 2 g φ , in the right-hand side of Eq. (A4) is undesirable in numerical evolution because of the presence of the time derivative of Π. Therefore, to handle this term, we use the following expression of 2 g φ ,
2 g φ = D k D k φ + D k α α D k φ − KΠ + n a ∇ a Π = D k D k φ + D k α α D k φ − KΠ + φ α (∂ t − β k ∂ k ) Π φ − Π 2 φ ,(A6)
and redefine the evolution equation forK := K + 3Π/(2φ ) as
(∂ t − β k ∂ k )K = 4πGαφ −1 (S + ρ h ) + αK i j K i j − D i D i α +α ω + 3 2 φ −2 Π 2 − 1 2 αφ −1 D i D i φ − KΠ − αX 2α 2 s φ 2 ∂K ∂ X (D k φ )D k φ + Π 2 − 3 2φ (D k α)D k φ ,(A7)
which guarantees the hyperbolicity of the geometric equations. Equation (6) is rewritten into a set of equations, (12) and (13), which are first-order in the time derivatives. Once φ andΠ(= F(X)Π) are determined from these equations, X (as well as F(X) and Π) are obtained from Eq. (14), which is considered to be an algebraic equation for X (see Eq. (15)). For the present choice ofK(X) (and F(X)), Eq. (15) has one or two or three real solutions for X. For a small value ofΠ 2 , there is only one real solution. However, for a value ofΠ 2 larger than a critical value, there are more than two real solutions. For the case that there are two real solutions, one should be a multiple solution. In this case, the solution satisfies not only Eq. (15) but also the following,
d f (X) dX = 1 −Π 2 8πGα 2 s φ 3 F(X) 3 dF dX = 0 .(A8)
This solution (d f (X)/dX = 0) has a pathology, and hence, in its presence the computation breaks down (see below). Therefore, for a problem in whichΠ is initially small everywhere (i.e., f (X) > 0), but later it increases significantly leading to f (X) ≤ 0 at points, it is possible that the computation breaks down. If X is determined, K is obtained from K =K − 3Π/(2φ ). We also note that D j X, which appears in the computation of D j F = (dF/dX)D j X, is calculated as
D j X = − 3X φ D j φ + 1 8πGα 2 s φ 3 (D j D k φ )D k φ −Π D jΠ F(X) 2 +Π 2 F(X) 3 dF dX D j X ,(A9)
and hence,
D j X = − 3X φ D j φ + 1 8πGα 2 s φ 3 (D j D k φ )D k φ −Π D jΠ F(X) 2 × 1 − 1 8πGα 2 s φ 3Π 2 F(X) 3 dF dX −1 .(A10)
This shows that 1−Π 2 /(8πGα 2 s φ 3 F 3 )(dF/dX) (i.e., d f /dX) has to be non zero in general. This implies that if the solution of X is a multiple root of Eq. (15), a discontinuity appears in the scalar field and the computation in general breaks down in the present formulation. In this work, we present results for which such a pathology is not encountered.
During the numerical simulation, we examine the violation of the Hamiltonian constraint by monitoring the following quantity:
H = 1 M * |H| ∑ l |H l | ρ * d 3 x,(A11)
where H is defined by the left-hand side minus the right-hand side of Eq. (A1), H l denotes each individual term in Eq. (A1) so that H = ∑ l H l , and M * is the rest mass of the system defined by
M * = ρ * d 3 x.(A12)
We find that H remains to be always of order 10 −4 during the simulation time in our present grid resolution if the simulation is successful; no indication of the growth of the constraint violation is found. For higher values of β , the magnitude of H is larger; e.g., for β = 10 32 it is 10 −3 . For β > 10 32 with which stable evolution is not successful, it can quickly grow when the code crashes. This suggests that for such cases, higher grid resolution might be necessary for the successful simulation. ) Note that the high amplitude waves found at t − r ≈ 0 are the junk radiation numerically induced during the relaxation of the given initial data to those fitted to the computational setting. Note that the vertical scale is the same for all the panels.
outside the neutron star with r < r sc . Thus, the analysis shown here is likely to be valid for any neutron star. The above analysis also shows that for n < 0, the amplitude defined by φ lm (r ex /M) decreases with the radius. Thus, if a wave is generated in r 8 km, the wave amplitude is suppressed, and hence, the wave amplitude should depend strongly on the wave generation region.
FIG. 1 .
1φ − 1 (left) and F(X) (right) as functions of the radius in isotropic coordinates for spherical neutron stars of mass, M T = 1.4M . The dashed slope line in the right panel indicates that F(X) outside the stellar surface is approximately proportional to r −1.6 .
FIG. 2. 1
1− αψ 2 as a function of the coordinate radius for spherical neutron stars of M T = 1.4M . The dashed lines denote the slope of r −1 and r −2 .
FIG. 3 .
3Evolution of the central density for β = 1, 10 20 , 10 24 , and 10 28 as well as in GR. All the curves approximately overlap with each other.
Figure 2 plots 1
1− αψ 2 = 1 − χψ as a function of the coordinate radius, r. In GR, where M S = 0, this quantity falls off as r −2 in isotropic coordinates (as shown by the green dashed line) due to the presence of the virial relation (see Eqs. (30), (31), and
FIG. 4 .
4Fig. 5summarises the wave amplitude as a function of Waveforms of the quadrupole mode for scalar waves as functions of t − r for β = 1, 10 16 , 10 20 , 10 24 , 10 28 , and 10 32 . The waveforms extracted at r = 591 km are shown together. For β = 10 28 and 10 32 , correction factors of F 0.6 and F 0.5 are multiplied, respectively (see Appendix B on the correction factor).
FIG. 5 .
5Asymptotic amplitudes of scalar waves as functions of β 1/8 (∝ r sc ) for the quadrupole (l = m = 2) mode (hollow squares)
FIG. 6 .
6Waveforms (∂ t φ 00 r) of the monopole mode for scalar waves as functions of t − r for β = 1, 10 16 , 10 20 , 10 24 , and 10 28 . The waveforms extracted at r = 591 km are shown together.
FIG. 7 .
7Gravitational and scalar waveforms of the (l = m = 2) quadrupole mode as functions of the retarded time, t − r ex , in GR (upper left), for β = 1 (upper right), 10 16 (middle left), 10 20 (middle right), 10 24 (lower left), and 10 28 (lower right). For gravitational waves, we plot the real part of the complex Weyl scalar, Ψ 22 . For each panel, the waveforms are plotted with several extraction radius, r ex ≈ 236 (magenta), 354 (green), 472 (blue), 591 (orange), and 709 km (yellow). (For the upper and middle panels as well as for gravitational waves, all the curves approximately overlap with each other.
FIG. 8 .
8The scalar waveform of φ 22 (r ext /M)F η as a function of the retarded time for β = 10 28 with η = 0.6 (upper) and β = 10 32 with η = 0.5 (lower). The extraction radius and the meaning of the colour are same as inFig. 7. To align the waveforms, the plots for r ex ≈ 354 (green), 472 (blue), 591 (orange), and 709 km (yellow) are shifted to the positive time direction.
We note that this relation should be satisfied sufficiently outside the matter source even for stationary and quasi-stationary spacetime (but the powers, n and p, depend on the chosen function of F(X)) and that grasping the behaviour of F(X) plays an important role for understanding the propagation property of scalar waves (see Sec. V).
Note that for typical binary neutron stars, the orbital period at their innermost stable circular orbits is ∼ 2 ms, and thus, the wavelength of the quadrupole mode is 300 km.
Note, however, that when a black hole is formed dynamically, scalar waves of a characteristic wave shape with an appreciable amplitude can be emitted even in the presence of the screening effect irrespective of r sc because the non-uniform scalar field disappears after the formation of the black hole (e.g., Refs.[58,86,87]).
Appendix A: 3+1 formulationHere we describe the 3+1 form of the gravitational and scalar fields equations. By contracting Eq. (5) with n a n b , the Hamiltonian constraint is derived aswhere R k k is the three-dimensional Ricci scalar.Appendix B: Extraction methodHere we analyse multipole components of scalar and gravitational waves. For the scalar waves, we definewhere Y lm is the spherical harmonics, and pay attention to the l = m = 2 mode. Gravitational waveforms are analysed by first extracting the outgoing component of the complex Weyl scalar and by decomposed into multipole modes[76].Since the waves are approximately monochromatic, the gravitational wave amplitude, h lm , may be calculated from each multipole mode of the complex Weyl scalar, Ψ lm , bywhere ω w is the angular velocity of gravitational waves and in the present case GMω w ≈ 0.087 with M = 1.4M . Thus, h lm (r/M) ≈ 260|Ψ lm |(rM).Figure 7plots the quadrupole waveforms of gravitational and scalar waves for β = 1, 10 16 , 10 20 , 10 24 , and 10 28 , as well as in GR. The amplitude of Ψ 22 (r ex M) is ∼ 4 × 10 −5 irrespective of the value of β . For scalar waves, if the condition, r sc λ wave , is satisfied, the asymptotic amplitude isThe order of magnitude for this agrees approximately with the expected value calculated bywith R the stellar radius and M S /M ∼ α 2 s = 10 −2 . In addition, the waveforms with different extraction radii as functions of the retarded time, t − r ex , approximately align with each other for the case where r sc λ wave . This behaviour is always found for gravitational waves irrespective of the screening effect.By contrast, for r sc λ wave , the amplitude defined by φ 22 (r ex /M) increases with the extraction radius whenever r ex r sc . Moreover, the waveforms with different extraction radii as functions of the retarded time, t − r ex , do not overlap for this case, because of the presence of a large factor of F(X) 1 in the screening region (see the scalar waveforms for β = 10 24 and 10 28 ). To determine the asymptotic amplitude of scalar waves, we have to extract them in a far zone, in which F(X) ≈ 1 or we perform an extrapolation. In the present work, we consider the latter possibility for high values of β ≥ 10 28 .Since F(X) decreases approximately proportional to r −n with n ≈ 1.6 outside the neutron stars (seeFig. 1), it is possible to predict the behaviour of the amplitude for λ wave r r sc using the following method. Neglecting the curvature effect (i.e., assuming the flat spacetime), approximating F as a fixed background and setting T = 0, the equation of φ lm can be written asIn addition, we assume that φ lm ∝ exp(iω sw t). The general solution of Eq. (B3) is written in terms of the outgoing component of the modified Bessel function, Z ν ,where ν = l(l + 1) + (n − 1) 2 /4. Since the amplitude of Z ν is proportional to r −1/2 for ω sw r 1 irrespective of ν, the wave amplitude of φ lm is proportional to r n/2−1 . Thus for n = 1.6, φ lm ∝ r −0.2 , which implies that φ lm (r ex /M) ∝ r 0.8 ex ∝ F −1/2 ; i.e., the amplitude defined by φ lm (r ex /M) increases with radius. For β = 10 28 , we find that φ 22 (r ex /M) is approximately proportional to F(X) −0.6 of r = r ex because for this case, the screening region with r > λ wave is rather narrow. However, for β = 10 30 and β = 10 32 , we confirm that the relation of φ 22 (r ex /M) ∝ F −1/2 is satisfied well. Thus, for determining the asymptotic amplitude in these cases, we utilise this approximate relation. InFig. 8, we plot φ lm (r ex /M)F η as a function of the retarded time with η = 0.6 and 0.5 for β = 10 28 and 10 32 , respectively, showing that this extraction method works well. The analysis performed here also gives the reason that a phase misalignment is found among the scalar waves of different values of β plotted inFig. 4and among those with the different extraction radii for β ≥ 10 24 plotted inFig. 7. As described in Eq. (B4), the wave phase is determined by the functional form of Z ν . Thus during the propagation of scalar waves in a region of F > 1, the wave phase is changed and this is reflected in the asymptotic wave phase.We note that not only for M T = 1.4M but also for other values of M T we find that F ∝ r −1.6 is approximately satisfied
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| []
|
[
"HAWKING MASS MONOTONICITY FOR INITIAL DATA SETS",
"HAWKING MASS MONOTONICITY FOR INITIAL DATA SETS"
]
| [
"Sven Hirsch "
]
| []
| []
| We introduce new systems of PDE on initial data sets (M, g, k) whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications. | null | [
"https://export.arxiv.org/pdf/2210.12237v2.pdf"
]
| 253,098,553 | 2210.12237 | 280af2795b519ee1dba298022a3fe0eb7e15b6b2 |
HAWKING MASS MONOTONICITY FOR INITIAL DATA SETS
6 Mar 2023
Sven Hirsch
HAWKING MASS MONOTONICITY FOR INITIAL DATA SETS
6 Mar 2023
We introduce new systems of PDE on initial data sets (M, g, k) whose solutions model double-null foliations. This allows us to generalize Geroch's monotonicity formula for the Hawking mass under inverse mean curvature flow to initial data sets satisfying the dominant energy condition. We study the existence theory of these systems and give geometric applications.
Introduction
In the pioneering work [27] G. Huisken and T. Ilmanen proved the Riemannian (that is k = 0) Penrose inequality for connected horizons. Their proof relies on the monotonicity of the Hawking mass under inverse mean curvature flow (IMCF) which has been first discovered by R. Geroch, P. Jang and R. Wald in [18,30]. Due to IMCF's crucial role in G. Huisken and T. Ilmanen's proof, it has been of great interest to generalize IMCF to the spacetime setting (that is k = 0) in order to prove the general Penrose conjecture which remains open for nearly half a century. There have been several approaches in this direction, including by H. Bray, S. Hayard, M. Mars and W. Simon [5], H. Bray and M. Khuri [9], K. Moore [38], J. Frauendiener [17], and G. Huisken and M. Wolf [28]. However, the existence theory for the flows in [9,5,17] appears to be out of reach, while for the flows in [38,28] there appears to be no analogue to the Hawking mass monotonicity. In this manuscript we suggest a new approach to generalize IMCF to the spacetime setting and introduce systems of PDE which model double null foliations. This leads simultaneously to a generalization of the Hawking mass monotonicity, cf. Corollary 1.2, and to existence results outside spherical symmetry, cf. Theorem 1.7. To motivate the statement of our main result, we begin with reviewing the Hawking mass monotonicity in the Riemannian setting.
Let (M, g) be an asymptotically flat 1 , 3-dimensional, complete manifold with non-negative scalar curvature R. Such manifolds arise naturally in General Relativity (GR) where they are used to model isolated gravitational systems such as stars, galaxies and black holes. The apparent horizon of the latter can be modeled by a connected, outermost, minimal surface Σ 0 ⊂ M. We denote with Σ t the IMCF starting from Σ 0 , i.e. unless there are jumps, Σ t flows in outward normal direction with speed 1 Ht where H t is the mean curvature of Σ t . The famous monotonicity formula for IMCF [18,30,27] states that ∂ t m H (Σ t ) ≥ 0 where the Hawking mass m H is defined by
m H (Σ) = |Σ| 16π 1 − 1 16π Σ H 2 dA .
An important ingredient in G. Huisken and T. Ilmanen's proof of the Riemannian Penrose inequality is to recognize that there is a level-set formulation of IMCF for which one can find weak solutions. More precisely, by defining the function U via Σ t = ∂{x ∈ M : U(x) < t}, we see that U satisfies the degenerate elliptic equation
div ∇U |∇U| = |∇U|
where we note that the term on the left hand side is the mean curvature of the level-sets Σ t . Reparametrizing u = e 1 2 U , we obtain the homogeneous equation
∆u = ∇ 2 νν u + 2 |∇u| 2 u (1)
where ν is the outer normal to the level sets Σ t . In this context, we can rephrase the Hawking mass monotonicity formula m H (Σ t ) − m H (Σ 0 ) ≥ 0, t ≥ 0, as integral formula
m H (Σ t ) − m H (Σ 0 ) = 1 16π Ωt R|∇u| + |H 2 u| 2 − (H 2 νν u) 2 |∇u| dV.(2)
Here Ω t is the region bounded by Σ 0 and Σ t , and H is a symmetric 2-tensor defined by
H ij u = ∇ ij u − |∇u| 2 u g ij + ∇ i u∇ j u u .(3)
The RHS of equation (2) is non-negative in case R ≥ 0.
To formulate such a monotonicity formula in the spacetime setting, we will take a more general point of view. In case we do not integrate the integrand on the RHS of equation (2) over a domain Ω, we obtain R|∇u| + |H 2 u| 2 − (H 2 νν u) 2 |∇u| − 2K u |∇u| = 2 div ∇|∇u| + |∇u| u ∇u − ∆u ∇u |∇u| (4) where K u is the Gaussian curvature of Σ t = {u(x) = t}. The above version of the Hawking mass monotonicity generalizes to the spacetime setting:
Suppose that (M, g, k) is an initial data set, i.e. (M, g) is a complete, asymptotically flat 2 , 3-dimensional manifold, and k is a symmetric 2-tensor on M. If (M, g, k) is embedded in a Lorentzian 4-manifold (M ,ḡ), k corresponds to the second fundamental form of M within M. To each initial data set we associate the energy density µ and the momentum density J given by 2µ =R + (tr g (k)) 2 − |k| 2 , J = div(k − tr g (k)g).
We say (M, g, k) satisfies the dominant energy condition (DEC), in case µ ≥ |J| everywhere on M. Our main result generalizes the Hawking mass monotonicity formula (4) to initial data sets: Theorem 1.1. Let a ∈ [0, 1] and suppose u, v ∈ C 2,α (M) are positive solutions of the system ∆u = − tr g (k)|∇u| + ak ηη |∇u| + a∇ 2 ηη u + 3|∇u||∇v| + ∇u, ∇v u + v , ∆v = tr g (k)|∇v| − ak ηη |∇v| + a∇ 2 ηη v + 3|∇u||∇v| + ∇u, ∇v u + v (5) with |∇u|, |∇v| = 0, where η = ∇u|∇v|+∇v|∇u| |∇u|∇v|+∇v|∇u|| . Then
div Y = |H 2 + u| 2 − (a(H 2 + ) ηη u) 2 |∇u| + |H 2 − v| 2 − (a(H 2 − ) ηη v) 2 |∇v| + 2µ(|∇u| + |∇v|) + 2 J, ∇u − ∇v − 2K u |∇u| − 2K v |∇v| (6)
where K u , K v are the Gaussian curvatures of the level sets of u, v,
Y =2∇(|∇u| + |∇v|) + 2k(∇(u − v), ·) + 4(|∇u|∇v + |∇v|∇u) 1 u + v − 2∆u ∇u |∇u| − 2∆v ∇v |∇v| − 2 tr g (k)∇(u − v)
and
(H 2 + ) ij u = ∇ ij u + k ij |∇u| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v , (H 2 − ) ij v = ∇ ij v − k ij |∇v| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v .
Observe that the first line in (6) is always non-negative, and the second line (6) is nonnegative in case the DEC is satisfied. In Section 4 we see that the above formula implies upon integrating:
(u + v)| ∂ + M 1 − 1 8π ∂ + M 2θ + |∇u| u + v + 2θ − |∇v| u + v − 8 |∇u||∇v| (u + v) 2 dA ≥(u + v)| ∂ − M 1 − 1 8π ∂ − M 2θ + |∇u| u + v + 2θ − |∇v| u + v − 8 |∇u||∇v| (u + v) 2 dA
where θ ± = H ± (tr g (k) − k ηη ) are the null expansions. In the case a = 1 we furthermore have |∇u| = 1 4 θ − (u + v) and |∇v| = 1 4 θ + (u + v) on ∂ ± M which implies the generalized Hawking mass monotonicity
(u + v)| ∂ + M 1 − 1 16π ∂ + M θ − θ + dA ≥(u + v)| ∂ − M 1 − 1 16π ∂ − M θ − θ + dA .
For k = 0, system (5) decouples if u, v have the same boundary data, and we recover several important monotonicity formulas: For k = 0 and a = 1, the function u = v is rescaled IMCF (as in equation (1)), and we obtain the Hawking mass monotonicity formula (4). For k = 0 and 0 ≤ a < 1, the function u = v solves the rescaled 3 p-Laplacian equation
∆u = a∇ 2 νν u + 2 |∇u| 2 u with monotonicity formula R|∇u| + |H 2 u| 2 − (aH 2 νν u) 2 |∇u| − 2K u |∇u| = 2 div ∇|∇u| + |∇u| u ∇u − ∆u ∇u |∇u| .
This formula has been first discovered by V. Agostiniani, L. Mazzieri and F. Oronzio in [2] which enabled them to give a new proof of the Riemannian Positive Mass Theorem [2], and, together with C. Mantegazza, the Riemannian Penrose inequality [1]. However, even in the special case k = 0, the above formula has some new contents since we can prescribe different boundary conditions for u and v, such that u = v and the system does not decouple.
Another special case is given by v = 0. Then u is a spacetime harmonic function. i.e. u solves the PDE ∆u = − tr g (k)|∇u|, and we recover the main integral formula of [23], Proposition 3.2. Moreover, we will see in Theorem 1.6 that (6) recovers the monotonicity formula of the spacetime Hawking energy [21] m
H (Σ) = |Σ| 16π 1 − 1 16π Σ θ + θ − dA .
under IMCF in spherical symmetry which implies the Penrose inequality in this setting. . Hence, in the case of equality of the spacetime PMT the level-sets Σ t of u can be obtained by intersecting null planes with the initial data set (M, g, k) ⊂ R 3,1 .
A similar situation occurs for any a ∈ [0, 1] for the system
∆u = − tr g (k)|∇u| + ak ηη |∇u| + a∇ 2 ηη u + 3|∇u||∇v| + ∇u, ∇v u + v , ∆v = tr g (k)|∇v| − ak ηη |∇v| + a∇ 2 ηη v + 3|∇u||∇v| + ∇u, ∇v u + v .
However, instead of leading to a single null foliation, the level sets Σ u , Σ v of u, v lead to a double null foliation. More precisely, we have the following: Theorem 1.3. Let u = r + t and v = r − t where r, t are the radial and the time coordinate functions of Minkowski space R 3,1 . Then the restrictions of u, v to any initial data set (M, g, k) ⊂ R 3,1 solve system (5) for any a ∈ [0, 1]. In fact, we have (H 2 + ) ij u = 0 and (H 2 − ) ij v = 0, and have equality in Corollary 1.2.
As in [26], the reverse statement is also true. If u, v solve the overdetermined systems (H 2 + ) ij u = 0 and (H 2 − ) ij v = 0 on the initial data set (M, g, k), and if (M, g, k) is vacuum, then (M, g, k) arises as subset of Minkowski space with second fundamental form k and with optical functions u = r + t and v = r − t. Figure 1. The double null foliation (Σ u ,Σ v ) for the initial data set (M,g,k) ⊂ R 3,1 is obtained by intersecting past and future directed lightcones in R 3,1 with (M,g,k). We would like to highlight that the individual null foliations Σ u and Σ v differ. This implies that an integral formula as in Theorem 1.1 is a more general concept than a monotonicity formula such as the one for the Hawking mass under IMCF. Furthermore, we can interpret system (5) for a = 1 as coupled inverse null mean curvature flow and for a = 0 as coupled spacetime harmonic functions. Given an initial data set (M, g, k) and a surface Σ ⊂ M, we can define the future and past null expansions θ + and θ − by
(M,g,k) ⊂ R 3,1 with Σ u Foliation (M,g,k) ⊂ R 3,1 with Σ v Foliationθ + = H + tr g (k) − k νν , θ − = H − tr g (k) + k νν
where ν is the outer normal to Σ. A generalization of IMCF to initial data sets is given by flows with speeds 1 θ + and 1 θ − in the outward normal direction. These so called inverse null mean curvature flows have been studied K. Moore in [38] where an existence theory under the assumptions tr g (k) ≥ 0 has been developed, also see [28]. Inverse null mean curvature flows A, B have like regular IMCF level-set formulations which after rescaling α = e become ∆α = − tr g (k)|∇α| + ∇ 2 νν α + k νν |∇α| + 2 |∇α| 2 α for the speed 1 θ + , and ∆β = − tr g (k)|∇β| + ∇ 2 νν β + k νν |∇β| + 2 |∇β| 2 β for the speed 1 θ − . We emphasize the similarities of these above equations with our system (5) for a = 1. More rigorously, we observe in Section 4 that the solutions (u, v) to our system (5) are in spherical symmetry rescalings of 1 θ − and 1 θ + flows. The rescaling factor is given by the usual IMCF.
Finally, we would like to give another example where the double null foliation concept is useful. In [23] the spacetime PMT has been proven via spacetime harmonic functions, and in [6] the PMT with charge has been proven via charged harmonic functions, i.e. functions solving ∆u = E, ∇u where E is the electrical field. Given an initial data set (M, g, k) equipped with an electrical field E, we need to combine both approaches and use double null foiliations: Theorem 1.4. Let E be a divergence-free vector field on (M, g, k). Suppose u, v solve the system ∆u = ξE η − tr g (k)|∇u| ∆v = ξE η + tr g (k)|∇v| (7) with |∇u|, |∇v| = 0, where ξ = |∇v||∇u| and η = ∇u|∇v|+∇v|∇u| |∇u|∇v|+∇v|∇u|| . Then we have
div(Z) = 1 2|∇u| (|E 2 + u| 2 + |∇u| 2 (2µ − 2K u − 2|E| 2 ) + 2|∇u| J, ∇u ) + 1 2|∇v| (|E 2 − v| 2 + |∇v| 2 (2µ − 2K v − 2|E| 2 ) − 2|∇v| J, ∇v ).(8)
where K u , K v are the Gaussian curvatures of the level-sets of u, v, Z =∇|∇u| − ∆u ∇u |∇u| + ∇|∇v| − ∆v ∇v |∇v| + 2ξ −1 (|∇u||∇v| + ∇u, ∇v )E − tr g (k)∇u + tr g (k)∇v + k(∇u, ·) − k(∇v, ·),
and
(E 2 + ) ij u =∇ 2 ij u + ξη i E j + ξη j E i − ξE η g ij + k ij |∇u|, (E 2 − ) ij v =∇ 2 ij v + ξη i E j + ξη j E i − ξE η g ij − k ij |∇v|.
We remark the important role the vector field η plays in both integral formulas. Observe that the above formula recovers Proposition 3.2 of [23] in case E = 0 which has been the main ingredient to prove the spacetime PMT, and equation (8.7) of [6] in case k = 0 which has been the main ingredient to prove PMT with charge.
1.2. The Penrose Conjecture. GR is concerned with the study of Lorentzian manifolds (M 4 , g) satisfying the Einstein equations Ric − 1 2 Rg = 8π where Ric, R are the Ricci and scalar curvature of g, and T is the stress-energy-momentum tensor. We refer to the books of D. Lee and R. Wald [33,46] for a detailed introduction to this topic. An interesting feature of GR is the existence of singularities which can arise even in elementary examples such as the Schwarzschild spacetime. However, in Schwarzschild the singularity is hidden behind the event horizon and is believed that this is generically the case 5 which is known as the Cosmic Censorship Conjecture. To verify this conjecture, R. Penrose proposed in 1973 a test: Assuming the Cosmic Censorship Conjecture holds, R. Penrose combined S. Hawking's area theorem [20,46], the fact that apparent horizons lie within the event horizon [40,46], and the final state conjecture to obtain the following: 6 Conjecture 1.5. Let (M, g, k) be an initial data set satisfying the DEC. Let Σ 0 be a MOTS in (M, g, k), and let Σ be the minimal area enclosure of Σ 0 . Then the mass m = E 2 − |P | 2 of (M, g, k) is bounded from below by m ≥ |Σ| 16π .
Moreover, we have equality if and only if (M, g, k) is a slice in Schwarzschild spacetime.
Here a marginally outer trapped surfaces (MOTS) is a surface Σ satisfying θ + = 0 and models an apparent horizons. Moreover, the ADM energy and momentum of (M, g, k) are defined by
E = lim r→∞ 1 16π Sr i (g ij,i − g ii,j ) υ j dA, P i = lim r→∞ 1 8π Sr (k ij − (tr g k)g ij ) υ j dA.
A counter example to the Penrose conjecture would pose a serious challenge to the Cosmic Censorship Conjecture which is considered to be the weakest link the above heuristic argument. Besides its physical significance, the Penrose conjecture also presents a strengthening of the famous positive mass theorem [2,8,15,16,23,27,34,37,44,45]. By time-reversal, i.e. by replacing k with −k, one also expects Conjecture 1.5 hold also for marginally inner trapped surfaces (MITS), i.e surfaces satisfying θ − = 0.
The conjecture has been established in the case k = 0 by G. Huisken and T. Ilmanen [27] (for connected horizons), and by H. Bray [4] (for arbitrary horizons). H. Bray's proof employs Bray's conformal flow and has also been generalized up to dimension 7 by H. Bray and D. Lee in [11] and to the electrostatic setting by M. Khuri, G. Weinstein and S. Yamada in [31].
In the general case k = 0 the conjecture is wild open outside spherical symmetry [9,10,21,22,29,35] and H. Roesch' result on certain null cones [42]. In the pioneering work [9] H. Bray and M. Khuri proposed a method to couple IMCF and Jang's equation to solve the conjecture. This leads to a complicated system of PDE which (if it can be solved) implies the Penrose conjecture for initial data sets which are asymptotic to the Riemannian Schwarzschild manifold. In fact, this system would even imply the Penrose conjecture for generalized horizons, i.e. surfaces satisfying θ + θ − = 0. Thus, there have to arise some complications in the existence theory in view of A. Carrasco and M. Mars' counter example [12]. For more information we refer to the survey [36] by M. Mars and the references therein.
We remark that in the statement of the Penrose inequality it is necessary to consider the minimal area enclosure Σ instead of the MOTS Σ 0 . It is easy to construct counterexamples to m ≥ |Σ 0 | 16π , see for instance Figure 1 in [27], and even the assumption of Σ 0 being an outermost MOTS is insufficient as demonstrated by I. Ben-Dov in [3].
Given that system (5) with a = 1 and integral formula (6) generalizes IMCF including the Hawking mass monotonicity formula, it is natural to ask whether there are applications towards the Penrose conjecture. Theorem 1.6. Let (M, g, k) be a spherically symmetric initial data set satisfying the DEC, and let a = 1. Then system (5) can be solved, and the integral formula (6) reduces to the monotonicity formula of the spacetime Hawking energy
m H (Σ) = |Σ| 16π 1 − 1 16π Σ θ + θ − dA .
It is well-known that the monotonicity of spacetime Hawking energy on spherically symmetric initial data sets satisfying the DEC leads to the Penrose inequlity, see for instance [21]. Therefore, Theorem 1.6 (and thus Theorem 1.1) implies the Penrose inequality in spherical symmetry.
One difficulty most approaches towards the Penrose conjecture face, is to solve certain PDE. For instance, P. Jang and R. Wald already showed in [30] that R. Geroch's monotonicity formula [18] implies the Riemannian Penrose inequality if an existence theory for IMCF can be established. This has also been observed in the spacetime setting for Inverse Mean Curvature Vector Flow by J. Frauendiener [17]. In the Riemannian case this has been resolved in [1,27,39], but the spacetime case this is still completely open. Similarly, the pioneering approach by H. Bray and M. Khuri [9] did not yield a proof of the Penrose conjecture due to the difficulties of solving the underlying PDE systems outside spherical symmetry.
Our systems (5) have the advantage that there are no second-order coupling terms, and there is a simple expression for ∆(v − u). This allows us to obtain an existence theory for system (5) with a = 0 in full generality without having to assume any symmetry: Theorem 1.7. Let (M, g, k) be a compact 3-dimensional Riemannian manifold equipped with symmetric 2-tensor k. Suppose that the boundary of M has two connected components ∂ − M and ∂ + M. Then we can solve system (5) for a = 0, i.e. there exist functions u, v ∈ C 2,α (M) solving
∆u = − tr g (k)|∇u| + 3|∇u||∇v| + ∇u, ∇v u + v , ∆v = tr g (k)|∇v| + 3|∇u||∇v| + ∇u, ∇v u + v on M, with Dirichlet boundary data u = c ± , v = d ± on ∂ ± M for positive constants c ± , d ± .
Acknowledgements. The author would like to express his gratitude to Hubert Bray, Simon Brendle, Demetre Kazaras, Marcus Khuri and Yiyue Zhang for many insightful discussions. Moreover, he would like to thank Xiaoxiang Chai, Gerhard Huisken, Florian Johne, Marc Mars, Lorenzo Mazzieri, Pengzi Miao, Francesca Oronzio and Alec Payne for their interest in this work.
Proof of the integral formulas
In this section we compute the integral formulas (6) and (8). They in particular generalize the Hawking mass monotonicity formula for IMCF [18,27,30], the spacetime Hawking energy monotonicity in spherical symmetry [21], the integral formula for harmonic and p-harmonic functions [1,2,8], for spacetime harmonic functions [23,7] and for charged harmonic functions [6]. We remark that the aforementioned formulas led to proofs of the Riemannian Penrose inequality [1,27], the spacetime Penrose inequality in spherical symmetry [21], the Riemannian [2,8], spacetime [23] and hyperbolic PMT [7], as well as the PMT with charge [6] and corners [25].
2.1. Spacetime IMCF and spacetime p-harmonic functions. We denote with ν u = ∇u |∇u| and ν v = ∇v |∇v| the unit normals to the level sets of u and v. Throughout this section we assume that both ν u and ν v are well-defined, i.e. |∇u|, |∇v| = 0. We expect that the cases where ∇u, ∇v are allowed to vanish can be treated in a similar fashion as in [1,23,43].
Next, we define η = νu+νv |νu+νv| in case ν u = −ν v , and η = 0 in case ν u = −ν v . Similarly, we define f = νu−νv |νu−νv| in case ν u = ν v and in case ν u = ν v we set f = 0
(which is the case in spherical symmetry). It is convenient to compute formula (6) in this frame. We remark that ν u = −ν v for any initial data set contained in Minkowski space.
We start with collecting several elementary properties about η and f :
Lemma 2.1. We have ∇ η u∇ f v = −∇ f u∇ η v.
In particular, for any symmetric 2-tensor A ij
A ij (∇ i u∇ j v + ∇ j u∇ i v) = 2A ηη ∇ η u∇ η v + 2A f f ∇ f u∇ f v.
Proof. To prove the first identity we can assume without loss of generality that ν u = ν v and ν u = −ν v . We compute
∇ η u∇ f v = 1 |ν u + ν v ||ν u − ν v ||∇u||∇v| (|∇u||∇v| + ∇u, ∇v )(−|∇u||∇v| + ∇u, ∇v )
and
∇ η v∇ f u = 1 |ν u + ν v ||ν u − ν v ||∇u||∇v| (|∇u||∇v| + ∇u, ∇v )(|∇u||∇v| − ∇u, ∇v ).
The second identity directly follows from the first one.
Lemma 2.2. We have ∇u, ∇v = ∇ η u∇ η v + ∇ f u∇ f v and |∇u||∇v| = ∇ η u∇ η v − ∇ f u∇ f v.
Proof. The first identity is trivial, so it suffices to show the second one. Observe that
|∇u| 2 = (∇ η u) 2 + (∇ f u) 2 which holds also in case ν u = ν v or ν u = −ν v . We compute using Lemma 2.1 |∇u| 2 |∇v| 2 =((∇ η u) 2 + (∇ f u) 2 )((∇ η v) 2 + (∇ f v) 2 ) =(∇ η u∇ η v) 2 + (∇ f u∇ f v) 2 + (∇ η u) 2 (∇ f v) 2 + (∇ η v) 2 (∇ f u) 2 =(∇ η u∇ η v) 2 + (∇ f u∇ f v) 2 − 2∇ η u∇ η v∇ f u∇ f v =(∇ η u∇ η v − ∇ f u∇ f v) 2 .
Taking the square root on both sides yields
|∇u||∇v| = |∇ η u∇ η v − ∇ f u∇ f v|. We clearly have ∇ η u∇ η v − ∇ f u∇ f v ≥ 0 in case ν u = ν v or ν u = −ν v . In case ν u = ν v and ν u = −ν v we have |∇u| 2 |∇v| 2 (∇ η u∇ η v − ∇ f u∇ f v) = 1 |ν u + ν v | 2 (|∇u||∇v| + ∇u, ∇v ) 2 + 1 |ν u − ν v | 2 (|∇u||∇v| − ∇u, ∇v ) 2 ≥0
which finishes the proof.
Lemma 2.3. We have |ν u + ν v | = 2 ν u , η = 2 ν v , η as well as |ν u − ν v | = 2 ν u , f = −2 ν v , f .
Proof. Recall that ν u = ∇u |∇u| and ν v = ∇v |∇v| . We compute
|ν u + ν v | 2 = ∇u|∇v| + ∇v|∇u| |∇u||∇v| 2 = 2 (|∇u||∇v| + ∇u, ∇v ) |∇u||∇v| and |ν u + ν v | ν u , η = ν u , ν u + ν v = 1 |∇u||∇v| (|∇u||∇v| + ∇u, ∇v ) which implies |ν u + ν v | = 2 ν u , η . In case ν u = −ν v , we clearly have ν u , η = ν v , η . Moreover, observe in case ν u = −ν v ν u , η = 1 + ν u , ν v |ν u + ν v | = ν v , η
which implies the first identity. Replacing v by −v, the second identity follows.
Recall that
(H 2 + ) ij u = ∇ ij u + k ij |∇u| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v , (H 2 − ) ij v = ∇ ij v − k ij |∇v| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v .
The proof of Theorem 1.1 is implied by the following two proposition regarding H 2 ± :
Proposition 2.4. Let a ∈ [0, 1] and suppose u, v solve system (5). Then we have
|H 2 + u| 2 |∇u| − (a(H 2 + ) ηη u) 2 |∇u| =R|∇u| − 2K u |∇u| + |k| 2 |∇u| − tr g (k) 2 |∇u| + 2|∇u|∇ νu tr g (k) − 2|∇u|∇ i k iνu + div −2∇|∇u| + 2∆u ∇u |∇u| + 2k(∇u, ·) − 2(∇u tr g (k)) + 4|∇u| ∇v u + v − 4a∇ ηη v ν u , f 2 |∇u| u + v + 4a∇ ηη u ν u , f 2 |∇v| u + v + 4(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 + 2k ij ∇ i u∇ j v + ∇ j u∇ i v u + v − 4|∇u||∇v| tr g (k) u + v − 8|∇u||∇v| k ηη u + v .(9)
Here R is the scalar curvature of g, and K u , K v are the Gaussian curvatures of the level sets of u, v.
Proposition 2.5. Let a ∈ [0, 1] and suppose u, v solve system (5). Then we have
|H 2 − v| 2 |∇v| − (a(H 2 − ) ηη v) 2 |∇v| =R|∇v| − 2K v |∇v| + |k| 2 |∇v| − tr g (k) 2 |∇v| − 2|∇v|∇ νv tr g (k) + 2|∇v|∇ i k iνv + div −2∇|∇v| + 2∆v ∇v |∇v| − 2k(∇v, ·) + 2(∇v tr g (k)) + 4|∇v| ∇u u + v − 4a∇ ηη u ν u , f 2 |∇u| u + v + 4a∇ ηη v ν u , f 2 |∇v| u + v + 4(|∇v| − |∇u|) |∇u||∇v| (u + v) 2 − 2k ij ∇ i u∇ j v + ∇ j u∇ i v u + v + 4|∇u||∇v| tr g (k) u + v + 8|∇u||∇v| k ηη u + v .(10)
Proof of Theorem 1.1. This follows immediately from adding equation (9) to equation (10).
Observe how the last two lines of both (9) and (10) cancel.
To prove Proposition 2.4 and Proposition 2.5 we will make use of several auxiliary lemma:
Lemma 2.6. Let a ∈ [0, 1] and suppose u, v solve system (5). Then we have
|H 2 + u| 2 + (∆u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 8 (∇ η u∇ η v) 2 (u + v) 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v + (4 + 4a)∇ ηη u ∇ η u∇ η v u + v + 4∇ f f u ∇ f u∇ f v u + v + (tr g (k) − ak ηη ) 2 |∇u| 2 + a 2 (∇ ηη u) 2 − 2a∇ ηη u(tr g (k) − ak νν )|∇u| − 4∆u ∇ f u∇ f v u + v − 4 ∇ η u∇ η v u + v (tr g (k) − ak νν )|∇u|.
Proof. Using Lemma 2.1 several times we obtain
(∇ i u∇ j v + ∇ j u∇ i v)(∇ i u∇ j v + ∇ j u∇ i v) =4(∇ η u∇ η v) 2 + 4(∇ f u∇ f v) 2 and ∇ ij u(∇ i u∇ j v + ∇ j u∇ i v) = 2∇ ηη u∇ η u∇ η v + 2∇ f f u∇ f ∇ f v as well as g ij (∇ i u∇ j v + ∇ j u∇ i v) = 2∇ η u∇ η v + 2∇ f u∇ f v.
This allows us to compute
|H 2 + u| 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 12 (∇ η u∇ η v) 2 (u + v) 2 + 4 (∇ η u∇ η v) 2 + (∇ f u∇ f v) 2 (u + v) 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v − 4∆u ∇ η u∇ η v u + v − 8∇ η u∇ η v ∇ η u∇ η v + ∇ f u∇ f v (u + v) 2 + 4∇ ηη u ∇ η u∇ η v u + v + 4∇ f f u ∇ f u∇ f v u + v .
Grouping together terms, we obtain
|H 2 + u| 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 8 (∇ η u∇ η v) 2 (u + v) 2 + 4 (∇ f u∇ f v) 2 (u + v) 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v − 4∆u ∇ η u∇ η v u + v − 8∇ η u∇ η v ∇ f u∇ f v (u + v) 2 + 4∇ ηη u ∇ η u∇ η v u + v + 4∇ f f u ∇ f u∇ f v u + v .(11)
Next, we recall that the PDE (5) for u states ∆u = − tr g (k)|∇u| + ak νν |∇u| + a∇ 2 ηη u + 3|∇u||∇v| + ∇u, ∇v u + v .
Moreover, we note that Lemma 2.2 implies
3|∇u||∇v| + ∇u, ∇v = 4∇ η u∇ η v − 2∇ f u∇ f v.
Thus, we are able to calculate
(∆u) 2 − 4∆u ∇ η u∇ η v u + v =∆u a∇ ηη u − (tr g (k) − k ηη )|∇u| − 2 ∇ f u∇ f v u + v =(tr g (k) − ak ηη ) 2 |∇u| 2 + a 2 (∇ ηη u) 2 − 2a∇ ηη u(tr g (k) − ak νν )|∇u| − 2∆u ∇ f u∇ f v u + v + 4 ∇ η u∇ η v u + v (a∇ ηη u − (tr g (k) − k ηη )|∇u|) − 2 ∇ f u∇ f v u + v (a∇ ηη u − (tr g (k) − k ηη )|∇u|) =(tr g (k) − ak ηη ) 2 |∇u| 2 + a 2 (∇ ηη u) 2 − 2a∇ ηη u(tr g (k) − ak νν )|∇u| − 4∆u ∇ f u∇ f v u + v − 4 (∇ f u∇ f v) 2 (u + v) 2 − 4 ∇ η u∇ η v u + v (tr g (k) − ak νν )|∇u| + 4a∇ ηη u ∇ η u∇ η v u + v + 8∇ η u∇ η v ∇ f u∇ f v (u + v) 2 .
Combining the above identity with equation (11), we obtain
|H 2 + u| 2 + (∆u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 8 (∇ η u∇ η v) 2 (u + v) 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v + (4 + 4a)∇ ηη u ∇ η u∇ η v u + v + 4∇ f f u ∇ f u∇ f v u + v + (tr g (k) − ak ηη ) 2 |∇u| 2 + a 2 (∇ ηη u) 2 − 2a∇ ηη u(tr g (k) − ak νν )|∇u| − 4∆u ∇ f u∇ f v u + v − 4 ∇ η u∇ η v u + v (tr g (k) − ak νν )|∇u|
which finishes the proof.
Lemma 2.7. Let a ∈ [0, 1] and suppose u, v solve system (5). Then we have
4 |∇u| (1 + a)∇ ηη u ∇ η u∇ η v u + v =4 ν u , f 2 ∇ f f u |∇v| u + v − 4a∇ ηη u ν u , f 2 |∇u| u + v + 4a∇ ηη (u − v) |∇u| u + v + 4 div |∇u| ∇v u + v + 4|∇u||∇v| |∇v| − 3|∇u| (u + v) 2 − 4|∇u||∇v| tr g (k) − ak ηη u + v Proof. Recall that ∆v = tr g (k)|∇v| − ak νν |∇v| + a∇ 2 ηη v + 3|∇u||∇v| + ∇u, ∇v u + v .
Using this equation we compute
4∇ νvνu u |∇v| u + v =4 ∇|∇u|, ∇v u + v =4 div |∇u| ∇v u + v − 4|∇u| ∆v u + v + 4|∇u| |∇v| 2 + ∇u, ∇v (u + v) 2 =4 div |∇u| ∇v u + v − 4a|∇u| ∇ ηη v u + v + 4|∇u||∇v| |∇v| − 3|∇u| (u + v) 2 − 4|∇u||∇v| tr g (k) − ak ηη u + v .(12)
Next, observe by Lemma 2.3
ν u , η 2 ∇ ηη u = ν u , η 2 1 |ν u + ν v | 2 ∇ (νu+νv)(νu+νv) u = 1 4 (2∇ νuνv u + ∇ νuνu u + ∇ νvνv u) = 1 4 (2∇ νuνv u + ∇ νu(f +νv ) u + ∇ νv(νu−f ) u) =∇ νuνv u + 1 4 ∇ (νu−νv)(νu−νv) u =∇ νuνv u + ν u , f 2 ∇ f f u. Hence, 4 |∇u| (1 + a)∇ ηη u ∇ η u∇ η v u + v = 4 |∇u| ∇ ηη u ∇ η u∇ η v u + v + 4a |∇u| ∇ ηη u ∇ η u∇ η v u + v =4∇ νvνu u |∇v| u + v + 4 ν u , f 2 ∇ f f u |∇v| u + v + 4a∇ ηη u |∇u| u + v − 4a∇ ηη u ν u , f 2 |∇u| u + v .
Combining this with equation (12) yields
4 |∇u| (1 + a)∇ ηη u ∇ η u∇ η v u + v =4 ν u , f 2 ∇ f f u |∇v| u + v − 4a∇ ηη u ν u , f 2 |∇u| u + v + 4a∇ ηη (u − v) |∇u| u + v + 4 div |∇u| ∇v u + v + 4|∇u||∇v| |∇v| − 3|∇u| (u + v) 2 − 4|∇u||∇v| tr g (k) − ak ηη u + v as desired.
Lemma 2.8. Let a ∈ [0, 1] and suppose u, v solve system (5). Then we have −2a tr g (k)∇ ηη u = − 2 div(∇u tr g (k)) + 2(− tr g (k)|∇u| + ak ηη |∇u|) tr g (k)
+ 2 tr g (k) 4∇ η u∇ η v − 2∇ f u∇ f v u + v + 2|∇u|∇ νu tr g (k).
Proof. Using the PDE for u (5), we compute − 2a tr g (k)∇ νuνu u = − 2a ∇|∇u|, ∇u tr g (k) |∇u| = − 2a div(∇u tr g (k)) + 2a(− tr g (k)|∇u| + ak ηη |∇u| + a∇ ηη u) tr g (k)
+ 2a tr g (k) 4∇ η u∇ η v − 2∇ f u∇ f v u + v − 2a∇ νuνu u tr g (k) + 2a|∇u|∇ νu tr g (k).
Thus, we obtain −2a tr g (k)∇ ηη u = − 2 div(∇u tr g (k)) + 2(− tr g (k)|∇u| + ak ηη |∇u|) tr g (k)
+ 2 tr g (k) 4∇ η u∇ η v − 2∇ f u∇ f v u + v + 2|∇u|∇ νu tr g (k)
which finishes the proof. Lemma 2.9. For any twice-differentiable function u we have div ∇|∇u| − ∆u ∇u |∇u|
= 1 2|∇u| (|∇ 2 u| 2 + |∇u| 2 (R − 2K u ) − (∆u) 2 ).
Proof. This formula has already been established in equation (4.8) of [6], also see the article of D. Stern [43]. We nonetheless include a proof to make this manuscript more self-contained. We compute using Bochner's identity and the Gauss equations
2∆|∇u| =2|∇u| −1 (Ric(∇u, ∇u) + |∇ 2 u| 2 + ∇∆u, ∇u − |∇|∇u|| 2 ) =2|∇u| −1 (|∇ 2 u| 2 + ∇∆u, ∇u − |∇|∇u|| 2 ) + (R − 2K u )|∇u| + |∇ 2 u| 2 + |∇u| −1 ((∆u − ∇ νν u) 2 − |∇ 2 u| 2 + 2|∇|∇u|| 2 − (∇ νν u) 2 )
Rewriting the term 2|∇u| −1 ∇∆u, ∇u , the result follows.
Now we have all the auxiliary ingredients to proceed with the proof of Proposition 2.4. The proof of Proposition 2.5 is identical so we will omit it.
Proof of Proposition 2.4. Combining Lemma 2.6 and Lemma 2.7, we obtain
|H 2 + u| 2 + (∆u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 8 (∇ η u∇ η v) 2 (u + v) 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v + (tr g (k) − ak ηη ) 2 |∇u| 2 + a 2 (∇ ηη u) 2 − 2a∇ ηη u(tr g (k) − ak νν )|∇u| − 4∆u ∇ f u∇ f v u + v − 4 ∇ η u∇ η v u + v (tr g (k) − ak νν )|∇u| − 4a∇ ηη v ν u , f 2 |∇u| 2 u + v + 4|∇u| div |∇u| ∇v u + v − 8|∇u||∇v| |∇u||∇v| (u + v) 2 + 4|∇u|(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 − 4|∇u| 2 |∇v| tr g (k) − ak ηη u + v .
Observe how the ∇ f f u terms cancel. Next, we calculate using Lemma 2.2
8 (∇ η u∇ η v) 2 (u + v) 2 − 4∆u ∇ f u∇ f v u + v − 8|∇u||∇v| |∇u||∇v| (u + v) 2 = − 4∆u ∇ f u∇ f v u + v − 8 (∇ f u∇ f v) 2 (u + v) 2 + 16 ∇ f u∇ f v∇ η u∇ η v (u + v) 2 = − 4a∇ ηη u ∇ f u∇ f v u + v + 4(tr g (k) − ak νν )|∇u| ∇ f u∇ f v u + v .
Moreover,
((H 2 + ) ηη u) 2 =(∇ ηη u + k ηη u|∇u|) 2 = (∇ ηη u) 2 + 2k ηη |∇u|∇ ηη u + k 2 ηη |∇u| 2 . Hence, we obtain
|H 2 + u| 2 + (∆u) 2 − (a(H 2 + ) ηη u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 2∇ ij uk ij |∇u| − 4 tr g (k)|∇u| ∇ η u∇ η v u + v + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v + (tr g (k) 2 − 2a tr g (k)k ηη )|∇u| 2 − 2a∇ ηη u tr g (k)|∇u| − 4 ∇ η u∇ η v u + v (tr g (k) − ak νν )|∇u| − 4a∇ ηη v ν u , f 2 |∇u| 2 u + v − 4a∇ ηη u ∇ f u∇ f v u + v + 4|∇u| div |∇u| ∇v u + v + 4|∇u|(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 − 4|∇u| 2 |∇v| tr g (k) − ak ηη u + v + 4(tr g (k) − ak νν )|∇u| ∇ f u∇ f v u + v .
Next, we use the divergence identity ∇ ij uk ij = div k(∇u, ·) − |∇u|∇ i k iνu and Lemma 2.8 which results in
|H 2 + u| 2 + (∆u) 2 − (a(H 2 + ) ηη u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 + 2|∇u| div k(∇u, ·) − 2|∇u| 2 ∇ i k iνu + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v − tr g (k) 2 |∇u| 2 − 4 ∇ η u∇ η v u + v (−ak νν )|∇u| − 4a∇ ηη v ν u , f 2 |∇u| 2 u + v − 4a∇ ηη u ∇ f u∇ f v u + v + 4|∇u| div |∇u| ∇v u + v + 4|∇u|(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 − 4|∇u| 2 |∇v| tr g (k) − ak ηη u + v − 4ak νν |∇u| ∇ f u∇ f v u + v − 2|∇u| div(∇u tr g (k)) + 2|∇u| 2 ∇ νu tr g (k).
By collecting terms which are homogeneous of degree 1 in k (though note they will cancel anyways with the corresponding terms from Proposition 2.5), this simplifies further to
|H 2 + u| 2 + (∆u) 2 − (a(H 2 + ) ηη u) 2 =|∇ 2 u| 2 + |k| 2 |∇u| 2 − tr g (k) 2 |∇u| 2 + 2|∇u| 2 ∇ νu tr g (k) − 2|∇u| 2 ∇ i k iνu + 2|∇u| div k(∇u, ·) − 2|∇u| div(∇u tr g (k)) + 4|∇u| div |∇u| ∇v u + v − 4a∇ ηη v ν u , f 2 |∇u| 2 u + v − 4a∇ ηη u ∇ f u∇ f v u + v + 4|∇u|(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 + 2k ij |∇u| ∇ i u∇ j v + ∇ j u∇ i v u + v − 4|∇u| 2 |∇v| tr g (k) u + v − 8a|∇u| 2 |∇v| k ηη u + v
Next, we use Lemma (2.9) to obtain
|H 2 + u| 2 |∇u| − (a(H 2 + ) ηη u) 2 |∇u| =R|∇u| − 2K u |∇u| + |k| 2 |∇u| − tr g (k) 2 |∇u| + 2|∇u|∇ νu tr g (k) − 2|∇u|∇ i k iνu + div −2∇|∇u| + 2∆u ∇u |∇u| + 2k(∇u, ·) − 2(∇u tr g (k)) + 4|∇u| ∇v u + v − 4a∇ ηη v ν u , f 2 |∇u| u + v + 4a∇ ηη u ν u , f 2 |∇v| u + v + 4(|∇u| − |∇v|) |∇u||∇v| (u + v) 2 + 2k ij ∇ i u∇ j v + ∇ j u∇ i v u + v − 4|∇u||∇v| tr g (k) u + v − 8|∇u||∇v| k ηη u + v
which finishes the proof.
2.2.
Spacetime charged harmonic functions. Again, we set η = νu+νv |νu+νv| in case ν u = −ν v , and η = 0 in case ν u = −ν v . Note that the integral formula (8) in Theorem 1.4 reduces to the integral formula for spacetime harmonic functions in case η = 0, cf. Proposition 3.2 in [23]. Therefore, we assume without loss of generality that ν u = −ν v in the proof below.
Proof of Theorem 1.4. Recall that the charged spacetime Hessians are given by
(E 2 + ) ij u =∇ 2 ij u + ξη i E j + ξη j E i − ξE η g ij + k ij |∇u|, (E 2 − ) ij v =∇ 2 ij v + ξη i E j + ξη j E i − ξE η g ij − k ij |∇v| where ξ = |∇u||∇v|. We compute |E 2 + u| 2 =|∇ 2 u| 2 + 2ξ 2 |E| 2 + 3ξ 2 E 2 η + 4ξ∇ ij uη i E j − 2a∆uE η + 2a 2 E 2 η − 4a 2 E 2 η + 2(ξη i E j + ξη j E i − ξE η g ij )k ij |∇u| + |k| 2 |∇u| 2 + 2∇ 2 ij uk ij |∇u| =|∇ 2 u| 2 + 2ξ 2 |E| 2 + 4ξ∇ ij uη i E j − ξ 2 E 2 η + 2(ξη i E j + ξη j E i − ξE η g ij )k ij |∇u| + |k| 2 |∇u| 2 + 2∇ 2 ij uk ij |∇u| + 2 tr g (k)|∇u|ξE η . Similarly, we obtain |E 2 − v| 2 =|∇ 2 v| 2 + 2ξ 2 |E| 2 + 4ξ∇ ij vη i E j − ξ 2 E 2 η − 2(ξη i E j + ξη j E i − ξE η g ij )k ij |∇v| + |k| 2 |∇u| 2 − 2∇ 2 ij uk ij |∇u| − 2 tr g (k)|∇u|ξE η . Using Lemma 2.9, we obtain div ∇|∇u| − ∆u ∇u |∇u| + ∇|∇v| − ∆v ∇v |∇v| = 1 2|∇u| (|E 2 + u| 2 − 2ξ 2 |E| 2 + ξ 2 E 2 η − 4ξ∇ 2 ij uη i E j ) + 1 2|∇u| (|∇u| 2 (R M − 2K u ) − (ξE η − tr g (k)|∇u|) 2 − |k| 2 |∇u| 2 − 2∇ 2 ij uk ij |∇u|) + 1 2|∇v| (|E 2 − v| 2 − 2ξ 2 |E| 2 + ξ 2 E 2 η − 4ξ∇ 2 ij vη i E j ) + 1 2|∇v| (|∇v| 2 (R M − 2K v ) − (ξE η + tr g (k)|∇v|) 2 − |k| 2 |∇v| 2 + 2∇ 2 ij vk ij |∇v|).
Simplifying yields div ∇|∇u| − ∆u ∇u |∇u| + ∇|∇v| − ∆v ∇v |∇v|
= 1 2|∇u| (|E 2 + u| 2 − 2ξ 2 |E| 2 − 4ξ∇ 2 ij uη i E j ) + 1 2|∇u| (|∇u| 2 (R M − 2K u ) + 2 tr g (k)∆u|∇u| + (tr g (k) 2 − |k| 2 )|∇u| 2 − 2∇ 2 ij uk ij |∇u|) + 1 2|∇v| (|E 2 − v| 2 − 2ξ 2 |E| 2 − 4ξ∇ 2 ij vη i E j ) + 1 2|∇v|
(|∇v| 2 (R M − 2K v ) − 2 tr g (k)∆v|∇v| + (tr g (k) 2 − |k| 2 )|∇v| 2 + 2∇ 2 ij vk ij |∇v|).
Next, we compute
1 |∇u| ξ∇ 2 ij uη i E j = div |∇v| |∇u| ∇ i uη i E − ∇ j |∇v| |∇u| E j ∇ η u − |∇v| |∇u| ∇ i u∇ j η i E j and similarly 1 |∇v| ξ∇ 2 ij vη i E j = div |∇u| |∇v| ∇ i vη i E − ∇ j |∇u| |∇v| E j ∇ η v − |∇u| |∇v| ∇ i v∇ j η i E j .
Observe that
2∇ j |∇v| |∇u| E j ∇ η u + 2∇ j |∇u| |∇v| E j ∇ η v =ξ −1 ∇ j |∇v|E j ∇ η u − ξ −1 |∇v||∇u| −1 ∇ j |∇u|∇ η uE j + ξ −1 ∇ j |∇u|E j ∇ η v − ξ −1 |∇u||∇v| −1 ∇ j |∇v|∇ η vE j = 0
where we used Lemma 2.3 in combination with |∇v|∇ η u = |∇v||∇u| ν u , η . Moreover,
|∇v| |∇u| ∇ i u∇ j η i E j + |∇u| |∇v| ∇ i v∇ j η i E j =ξ −1 ∇ E η, ∇u|∇v| + ∇v|∇u| =ξ −1 ||∇u|∇v + ∇u|∇v|| ∇ E η, η = 0
where we used that η, η = 1 and ∇ E η, η = 0. Hence, we obtain div ∇|∇u| − ∆u ∇u |∇u|
+ ∇|∇v| − ∆v ∇v |∇v| + 2 |∇u| |∇v| ∇ η vE + 2 |∇v| |∇u| ∇ η uE = 1 2|∇u| (|E 2 + u| 2 − 2aξ 2 |E| 2 ) + 1 2|∇u| (|∇u| 2 (R M − 2K u ) + 2 tr g (k)∆u|∇u| + (tr g (k) 2 − |k| 2 )|∇u| 2 − 2∇ 2 ij uk ij |∇u|) + 1 2|∇v| (|E 2 − v| 2 − 2ξ 2 |E| 2 ) + 1 2|∇v| (|∇v| 2 (R M − 2K v ) − 2 tr g (k)∆v|∇v| + (tr g (k) 2 − |k| 2 )|∇v| 2 + 2∇ 2 ij vk ij |∇v|).
Integrating by parts, we find div(Z) = 1 2|∇u|
(|E 2 + u| 2 − 2|∇v||∇u||E| 2 )
+ 1 2|∇u| (|∇u| 2 (2µ − 2K u ) + 2|∇u| J, ∇u ) + 1 2|∇v| (|E 2 − v| 2 − 2|∇u||∇v||E| 2 ) + 1 2|∇v| (|∇v| 2 (2µ − 2K v ) − 2|∇v| J, ∇v ). where Z =∇|∇u| − ∆u ∇u |∇u| + ∇|∇v| − ∆v ∇v |∇v| + 2ξ −1 (|∇u||∇v| + ∇u, ∇v )E − tr g (k)∇u + tr g (k)∇v + k(∇u, ·) − k(∇v, ·).
This finishes the proof.
Classifying Minkowski space
In this section we show that for any initial data set (M, g, k) contained in Minkowski space (R 3,1 ,ḡ) the functions u = r + t and v = r − t solve system (5) and satisfy H 2
+ u = 0, H 2 − v = 0 where (H 2 + ) ij u = ∇ ij u + k ij |∇u| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v , (H 2 − ) ij v = ∇ ij v − k ij |∇v| − 2 ∇ η u∇ η v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v .
Moreover, we analyze what the existence of functions satisfying the conditions H 2 + u = 0, H 2 − v = 0 implies for general initial data sets.
Proof of Theorem 1.3. We begin with observing that∇ 2 (v − u) = 0 and∇ 2 (uv) = 2ḡ wherē g is the metric of Minkowski space. This implies
2ḡ =v∇ 2 u + u∇ 2 v +∇u ⊗∇v +∇v ⊗∇u =(u + v)∇ 2 u +∇u ⊗∇v +∇v ⊗∇u.
Similarly,
2ḡ = (u + v)∇ 2 v +∇u ⊗∇vv +∇v ⊗∇u.
Restricting the above two equalities onto T * M ⊗ T * M we obtain
2g =(u + v)∇ 2 | T * M ⊗T * M u + ∇u ⊗ ∇v + ∇v ⊗ ∇u, 2g =(u + v)∇ 2 | T * M ⊗T * M v + ∇u ⊗ ∇v + ∇v ⊗ ∇u.
Next, let us denote with N the future pointing unit normal of M ⊂ R 3,1 . It is well-known that∇ 2 | T * M ⊗T * M u =∇ 2 u + kN(u),
∇ 2 | T * M ⊗T * M v =∇ 2 v + kN(v).
Since∇u and∇v are null, we havē
∇ 2 | T * M ⊗T * M u =∇ 2 u + k|∇u|, ∇ 2 | T * M ⊗T * M v =∇ 2 v − k|∇u|.
Combining everything yields
2g =(u + v)(∇ 2 u + k|∇u|) + ∇u ⊗ ∇v + ∇v ⊗ ∇u, 2g =(u + v)(∇ 2 v − k|∇u|) + ∇u ⊗ ∇v + ∇v ⊗ ∇u.
Observe that Lemma 2.2 implies
|∇u||∇v| + ∇u, ∇v = 2∇ η u∇ η v.(13)
Therefore, it suffices to show that 2 = |∇u||∇v| + ∇u, ∇v for any initial data set (M, g, k) ⊂ R 3,1 . To do so, we observe that We remark that in Minkowski space we can also solve for any c > 0 the boosted system
0 = ∇ ij u + k ij |∇u| − 2 ∇ η u∇ η v 1 c 2 u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v 1 c 2 u + v , 0 = ∇ ij v − k ij |∇v| − 2 ∇ η u∇ η v u + c 2 v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + c 2 v .
where u = c(r + t) and v = 1 c (r − t). The vector field η appears in both the integral formulas (6) and (8). Next, we describe η for IDS (M, g, k) in Minkowski space. In the following proposition we equip (R 3,1 ,ḡ) with spherical coordinates (∂ r , ∂ t , ∂ φ , ∂ θ ). Therefore, η||∇u|∇v + |∇v|∇u| = |∇v|∇u + |∇u|∇v = ∂ r (|∇u| + |∇v|) + ∂ t (|∇u| − |∇v|), and the result follows.
Next, let us comment on the reverse statement of Theorem 1.1. More precisely, we want to analyze whether the existence of functions u, v on (M, g, k) with H 2 + u = 0 and H 2 − v = 0 together with the vacuum conditions µ = 0, J = 0 imply that (M, g, k) ⊂ R 3,1 . In [26] Y. Zhang and the author established the case of equality of the spacetime PMT via spacetime harmonic functions. The argument goes as follows: In [23] D. Kazaras, M. Khuri and the author established the integral formula
E − |P | ≥ 1 16π Mext |∇ 2 u + k|∇u|| 2 |∇u| + 2µ|∇u| + 2 J, ∇u .
Here u is a spacetime harmonic function whose gradient is asymptotic to − P |P | and M ext is the exterior region 7 of M. In case E = |P | the above integral formula implies
∇ 2 u = − k|∇u|, µ|∇u| = − J, ∇u .
Using these identities one can verify that (M, g, k) satisfies the Gauss and Codazzi equations
R ijkl + k il k jk − k ik k jl =0, ∇ i k jk − ∇ j k ik =0.
This implies that (M, g, k) ⊂ R 3,1 in view of the Lorentzian version of the fundamental theorem of hypersurfaces.
Similar computations are possible when the identity ∇ 2 u = −k|∇u| is replaced by H 2 + u = 0 and H 2 − v = 0. For instance, denoting as in [26] with e 1 , e 2 vectors tangential to Σ u and with e 3 the vector ν u , we obtain from the identity H 2 + u = 0
∇ 1 k 23 − ∇ 2 k 13 = − ∇ 1 ∇ 23 u |∇u| + 1 |∇u| ∇ 2 u∇ 3 v + ∇ 3 u∇ 2 v u + v + ∇ 2 ∇ 13 u |∇u| + 1 |∇u| ∇ 1 u∇ 3 v + ∇ 3 u∇ 1 v u + v = − ∇ 12 log |∇u| − ∇ 1 1 |∇u| 0 + |∇u|∇ 2 v u + v + ∇ 12 log |∇u| + ∇ 2 1 |∇u| 0 + |∇u|∇ 1 v u + v =0
which establishes one Codazzi equation. Details will appear in a forthcoming paper. 7 See [23,26] for precise statements and definitions
The Penrose inequality in spherical symmetry
We begin with the proof of Corollary 1.5 before proceeding with the Penrose inequality in spherical symmetry 1.6 and studying arbitrary slices of Schwarzschild.
Proof of Corollary 1.2. Recall from Theorem 1.1 that
div Y = |H 2 + u| 2 − ((H 2 + ) ηη u) 2 |∇u| + |H 2 − v| 2 − ((H 2 − ) ηη v) 2 |∇v| + 2µ(|∇u| + |∇v|) + 2 J, ∇u − ∇v − 2K u |∇u| − 2K v |∇v| ≥ − 2K u |∇u| − 2K v |∇v|.(14)
where we used that (M, g, k) satisfies the DEC, and where
Y =2∇(|∇u| + |∇v|) + 2k(∇(u − v), ·) + 4(|∇u|∇v + |∇v|∇u) 1 u + v − 2∆u ∇u |∇u| − 2∆v ∇v |∇v| − 2 tr g (k)∇(u − v).
Next, recall that each boundary component ∂ ± M is a level set for both u and v which can be interpreted as ∂ − M and ∂ + M being unboosted with respect to each other. Hence ν := η = ν u = ν v on ∂ ± M and H|∇u| + ∇ νν u = ∆u = − tr k |∇u| + ∇ νν u + k νν |∇u| + 4 |∇u||∇v| u + v , H|∇v| + ∇ νν v = ∆v = tr k |∇v| + ∇ νν v − k νν |∇v| + 4 |∇u||∇v| u + v on ∂ ± M. Since |∇u|, |∇v| are non-zero, this implies
|∇v| = 1 4 θ + (u + v), |∇u| = 1 4 θ − (u + v).(15)
Combining these equations with the identity ∆u = ∇ νν u + H∇ ν u, we obtain
Y ν =2∇ νν (u + v) + 2k νν (|∇u| − |∇v|) + 8|∇u||∇v| u + v − 2∆(u + v) − 2 tr g (k)(|∇u| − |∇v|) = − 2H(|∇u| + |∇v|) + 2k νν (|∇u| − |∇v|) + 8|∇u||∇v| u + v − 2 tr g (k)(|∇u| − |∇v|) = − 2θ + |∇u| − 2θ − |∇v| + 8|∇u||∇v| u + v = − 1 2 θ + θ − (u + v).
Combining this with equation (14) yields after integration
− (u + v)| ∂ + M 16π ∂ + M θ + θ − dA ≥ − (u + v)| ∂ − M 16π ∂ − M θ + θ − dA + 1 4π M (K u |∇u| + K v |∇v|)dV
Next, we use twice the coarea formula and Gauss-Bonnet's theorem to obtain
M (K u |∇u| + K v |∇v|)dV = u| ∂ + M u| ∂ − M 4πdt + v| ∂ + M v| ∂ − M 4πdt = 4π(u + v)| ∂ − M − 4π(u + v)| ∂ + M .
Hence, we have
(u + v)| ∂ + M 1 − 1 16π ∂ + M θ + θ − dA ≥ (u + v)| ∂ − M 1 − 1 16π ∂ − M θ + θ − dA(16)
which finishes the proof.
In order to prove Theorem 1.6 which implies the Penrose inequality in spherical symmetry, we first establish the following Lemma.
w(r) = r 0 1 2 (tr g (k) − k νν )sdρ
where r is the distance to Σ 0 . Then u, v, implicitly defined by
u + v = s, v − u = w,
solve system (5) for a = 1. Moreover, we have
|∇v| = 1 4 θ + s, |∇u| = 1 4 θ − s.(17)
Here ν is the unit normal to the spherically symmetric surfaces and we note that ν = ν u = ν v = η.
Proof. Since s solves rescaled IMCF, and using ∆s = ∇ νν s + H∇ ν s, we deduce that |∇s| = 1 2 Hs. Moreover, we have |∇w| = 1 2 | tr g (k) − k νν |s. Since Σ 0 is the outermost horizon and is therefore not enclosed by any MITS or MOTS, we also obtain that θ + , θ − > 0 for all spherically symmetric surfaces outside Σ 0 . This implies H > | tr g (k) − k νν | for all spherically symmetric surfaces outside Σ 0 , and since u = 1 2 (s − w) and v = 1 2 (s + w) we obtain
|∇v| = 1 4 θ + s, |∇u| = 1 4 θ − s.
Note that this in particular implies ∇u, ∇v = 0 outside Σ 0 as well as ∇ r u, ∇ r v > 0. Multiplying the above identities by |∇u|, |∇v|, we obtain in the same fashion as in the computation of equation (15) that (u, v) solve system (5) with a = 1. This finishes the proof.
Observe that (17) implies that the level sets of u move by rescaled 1 θ − flow, and the level sets of v move by rescaled 1 θ + flow. The rescaling factor is in both cases given by 1 4 s where s is rescaled IMCF.
The above lemma immediately yields Corollary 4.2. We can solve the system (5) for a = 1 in spherical symmetry.
We remark that although system (5) is in many ways the most complicated for a = 1 due to its degenerate elliptic character, the existence theory for a = 1 is substantially simpler than for a ∈ [0, 1) in spherical symmetry. This contrasts the Riemannian (i.e. k = 0 case) where the existence theory for harmonic functions is elementary compared to the sophisticated existence theory for IMCF [27,39]. The reason for this reverse behavior stems from the fact that the system decouples for a = 1 in spherical symmetry as demonstrated in Lemma 4.1. However, the system appears not to decouple in spherical symmetry for a = 1, and the function u + v is not the rescaling of a p-harmonic function.
Proof of Theorem 1.6. Let (M, g, k) be a spherically symmetric initial data set satisfying the DEC, and let u, v be solutions to system (5) for a = 1 outside the horizon Σ 0 as described in Lemma 4.1. As in the proof of Corollary 1.2 above, we obtain
(u + v)| Σ 2 1 − 1 16π Σ 2 θ + θ − dA ≥ (u + v)| Σ 1 1 − 1 16π Σ 1 θ + θ − dA(18)
for any spherically symmetric surface Σ 2 enclosing Σ 1 enclosing Σ 1 . Since u + v = s solves rescaled IMCF ∆s = ∇ νν s + 2 |∇s| 2 s with s(Σ 0 ) = |Σ 0 | 16π , and because IMCF is uniformly area expanding, we obtain
|Σ 2 | 16π 1 − 1 16π Σ 2 θ + θ − dA ≥ |Σ 1 | 16π 1 − 1 16π Σ 1 θ + θ − dA
Hence, the spacetime Hawking energy
m H (Σ t ) = |Σ t | 16π 1 − 1 16π Σt θ − θ + dA(19)
is monotonically increasing for spherically symmetric initial data sets satisfying the DEC.
In Minkowski space we can obtain for any initial data set (M, g, k) ⊂ R 3,1 solutions to system
2 − v| 2 −((H 2 − ) ηη v) 2 = 0. Moreover, ∇ η X η = ∇ η Y η = m r 2 .
We would like to remark that X + Y = 2∇r and X − Y = 2T where T is the time-like Killing vector field T = (1 − 2m r )∇t.
Proof. Recall that we have in view of equation (13) (
H 2 + ) ij u = ∇ ij u + k ij |∇u| − |∇u||∇v| + ∇u, ∇v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v , (H 2 − ) ij v = ∇ ij v − k ij |∇v| − |∇u||∇v| + ∇u, ∇v u + v g ij + ∇ i u∇ j v + ∇ j u∇ i v u + v .
Next, observe that we have ∇ i X j = ∇ j X i unless (i, j) = (r, t), (t, r). Thus, X, Y are integrable on each spherically symmetric IDS. Moreover,∇X = ∇X + k N, X ḡ = ∇X + |X|k and∇Y = ∇Y − k|∇Y | since X, Y are null vectors. This implies
|X||Y | + X, Y =ḡ(N(u)N, N(v)N) + X, Y =ḡ(X, Y ) = 2φ.
To prove the above proposition, it thus suffices to show on (M ,ḡ)
∇ α X β = φ rḡ αβ − X α Y β + X β Y α 2r ,∇ α Y β = φ rḡ αβ − X α Y β + X β Y α 2r(20)
for all α, β apart form (α, β) = (r, r), (r, t), (t, r), (t, t). We merely perform the computation for∇ α X β since the ones for∇ α Y β are analogous.
Denoting with A = ∂ φ , ∂ θ the standard spherical coordinates, we compute for α = Ā
∇ A X α =∇ α X A = 0 and φ rḡ Aα − X α Y A + X A Y α 2r = 0.
Moreover, we have∇
A X A =Γ r AA X r = φ rḡ AA − X α Y A + X A Y α 2r .
Next, observe that η has only components in ∇r and ∇t direction, i.e. η = a∇ r + b∇ t . We calculate
∇ 2 rr r = φ m r 2 , ∇ 2 tt r = −φ −1 m r 2 .
This leads to
∇ η X η = ∇ 2 ηη r = m r 2ḡ (η, η) = m r 2 which finishes the proof.
We would like to remark that in case m = 0, the Killing vector field T = 1 2 (X − Y ) does not satisfy ∇ r T t = 0, ∇ t T r = 0. Hence, equation (20) is not satisfied for (α, β) = (r, t), and X, Y are not integrable on the entire spacetime (M ,ḡ). Thus, in contrast to Minkowski space, there are no globally defined functions u, v such that when they are restricted to an IDS, they solve system (5) for a = 1. However, there are other such globally defined null functions u = r * + t and v = r * − t which do satisfy another nice set of equations 8 . More precisely, we have: which implies on any initial data set (M, g, k)
∇ 2 u = − k|∇u| + g r − φ 2r
(∇u ⊗ ∇v + ∇v ⊗ ∇u) − m r 2 ∇u ⊗ ∇u. Next, we observe that on each initial data set |∇u||∇v| + ∇u, ∇v =ḡ(N(u)N, N(v)N) + ∇u, ∇v = 2φ −1 .
Therefore, we are lead to the equation
∇ 2 u = − k|∇u| + g 2r φ(|∇u||∇v| + ∇u, ∇v ) − φ 2r (∇u ⊗ ∇v + ∇v ⊗ ∇u) − m r 2 ∇u ⊗ ∇u.
Taking the trace, we obtain
∆u = − tr g k|∇u| + φ 3|∇u||∇v| + ∇u, ∇v 2r − m r 2 |∇u| 2 . Moreover, ∇ 2 νuνu u = −k νuνu |∇u| − m r 2 |∇u| 2 + φ |∇u||∇v| − ∇u, ∇v 2r and ∇ 2 ηη u = −k ηη |∇u| − m r 2 (∇ η u) 2 .
Thus, the result follows.
Note that by relating u and v to r via the identity u + v = 2r * , systems (21) and (22) can also be studied for an arbitrary initial data set which does not arise as slice in Schwarzschild. Moreover, system (22) reduces to system (5) with a = 1 in case m = 0.
Finally, we would like to point out the importance of equations such as (23) lies in the fact that they can be used to characterize slices in certain spacetimes. See for instance [26,23] for slices in Minkowski space and Proposition 2 in J. Krohn's paper [32] for slices of Schwarzschild.
Existence theory outside spherical symmetry
To solve the system
∆u = − tr g (k)|∇u| + 3|∇u||∇v| + ∇u, ∇v u + v , ∆v = tr g (k)|∇v| + 3|∇u||∇v| + ∇u, ∇v u + v(24)
on M with u = c ± on ∂ ± M and v = d ± on ∂ ± M, we will first obtain uniform estimates for the system ∆u σ,ε = − σ tr g (k)|∇u σ,ε | + 3|∇u σ,ε ||∇v σ,ε | + ∇u σ,ε , ∇v σ,ε |u σ,ε + v σ,ε | + ε , ∆v σ,ε =σ tr g (k)|∇v σ,ε | + 3|∇u σ,ε ||∇v σ,ε | + ∇u σ,ε , ∇v σ,ε |u σ,ε + v σ,ε | + ε .
Here σ ∈ [0, 1], ε > 0, and we consider the boundary data u σ,ε = c ± on ∂ ± M and v σ,ε = σd ± + (1 − σ)c ± − ε on ∂ ± M. We assume ε to be sufficiently small such that σd ± + (1 − σ)c ± − ε > 0 for all σ ∈ [0, 1]. Without loss of generality we assume that c − < c + and d − < d + .
Lemma 5.1. Suppose u σ,ε ∈ C 2,α (M) and v σ,ε ∈ C 2,α (M) solve the system (25). Then we have
c − ≤ u σ,ε ≤ c + , σd − + (1 − σ)c − − ε ≤ v σ,ε ≤ σd + + (1 − σ)c + − ε on M.
Proof. This follows immediately from the maximum principle.
Lemma 5.2. Suppose u σ,ε and v σ,ε solve system (25). Then there exists a constant C independent of σ, u ε , v ε , ε such that
u σ,ε W 2,p (M ) + v σ,ε W 2,p (M ) ≤ C.
Proof. To prove this proposition, it will be helpful to rewrite the above system in terms of
w σ,ε = v σ,ε − u σ,ε , h σ,ε = 1 u σ,ε + v σ,ε + ǫ .
We compute for w σ,ε ∆w σ,ε = σ tr g (k)(|∇u σ,ε | + |∇v σ,ε |), and for h σ,ε
1 2 h −2 σ,ε ∆h σ,ε = − 1 2 ∆(u σ,ε + v σ,ε + ǫ) + |∇(u σ,ε + v σ,ε + ǫ)| 2 u σ,ε + v σ,ε + ǫ = 1 u σ,ε + v σ,ε + ǫ (3|∇u σ,ε | 2 − 3|∇u σ,ε ||∇u σ,ε + w σ,ε | + 3 ∇u σ,ε , ∇w σ,ε + |∇w σ,ε | 2 ) + σ tr g (k) 2 |∇u σ,ε | − σ tr g (k) 2 |∇u σ,ε + w σ,ε |.
Using the identity |∇(u σ,ε + w σ,ε )| − |∇u σ,ε | = 1 |∇(u σ,ε + w σ,ε )| + |∇u σ,ε | (|∇w σ,ε | 2 + 2 ∇u σ,ε , ∇w σ,ε ), we obtain 1 2 h −2 σ,ε ∆h σ,ε = − 1 u σ,ε + v σ,ε + ǫ 3|∇u σ,ε | |∇(u σ,ε + v σ,ε )| + |∇u σ,ε | (|∇w σ,ε | 2 + 2 ∇u σ,ε , ∇w σ,ε ) + 1 u σ,ε + v σ,ε + ǫ 3 ∇u σ,ε , ∇w σ,ε + |∇w σ,ε | 2 − σ tr g (k) 2 1 |∇(u σ,ε + w σ,ε )| + |∇u σ,ε | (|∇w σ,ε | 2 + 2 ∇u σ,ε , ∇w σ,ε ) = 1 u σ,ε + v σ,ε + ǫ −3|∇u σ,ε | 1 |∇(u σ,ε + w σ,ε )| + |∇u σ,ε | |∇w σ,ε | 2 + |∇w σ,ε | 2 + 1 u σ,ε + v σ,ε + ǫ 3 ∇u σ,ε , ∇w σ,ε (|∇(u σ,ε + w σ,ε )| + |∇u σ,ε |) 2 (|∇w σ,ε | 2 + 2 ∇u σ,ε , ∇w σ,ε ) − σ tr g (k) 2 1 |∇(u σ,ε + w σ,ε )| + |∇u σ,ε | (|∇w σ,ε | 2 + 2 ∇u σ,ε , ∇w σ,ε ).
Having established our identities for ∆w σ,ε and ∆h σ,ε we proceed with estimating the above terms. We have |∆w σ,ε | ≤C(|∇(u σ,ε + v σ,ε )| + |∇(u σ,ε − v σ,ε )| ≤C(|∇h σ,ε |h −2 σ,ε + |∇w σ,ε |)
where C is depending on M, k, c ± , d ± whose value may change from line to line. Moreover, |∆h σ,ε | ≤Ch 3 σ,ε |∇w σ,ε | 2 + Ch 2 σ,ε |∇w σ,ε |
where we used |∇w σ,ε | |∇(u σ,ε + w σ,ε )| + |∇u σ,ε | ≤ 1.
The W 2,p estimate for solutions of elliptic equations states w σ,ε W 2,p (M ) ≤ C( w σ,ε L p (M ) + h σ,ε W 1,p (M ) ) ≤ C + C h σ,ε W 1,p (M ) and h σ,ε W 2,p (M ) ≤ C( h σ,ε L p (M ) ) + w σ,ε Hence, we are lead to h σ,ε W 2,p (M ) ≤C + C w σ,ε W 2,p (M ) ≤ C + C h σ,ε W 1,p (M )
≤C + C h σ,ε α W 2,p (M ) ≤ + 1 2 h σ,ε W 2,p (M ) .
Thus, we have h σ,ε W 2,p (M ) ≤ C which implies w σ,ε W 2,p (M ) ≤ C. Reconstructing u σ,ε , v σ,ε from h σ,ε , w σ,ε , we also obtain u σ,ε W 2,p (M ) + v σ,ε W 2,p (M ) ≤ C which finishes the proof..
We can use the Sobolev inequality and Schauder estimates to improve the above estimate to C 2,α . More precisely, we obtain: Lemma 5.3. Suppose u σ,ε , v σ,ε solve system (25). Then there exists a constant C independent of σ, u σ,ε , v σ,ε , ε such that u σ,ε C 2,α (M ) + v σ,ε C 2,α (M ) ≤ C.
Having obtained uniform estimates for system (25) we will use Leray-Schauder's fixed point theorem below to obtain solutions of (25) for σ = 1. Passing to a limit ε → 0 then gives a solution to (24) establishing Theorem 1.7.
Proof of Theorem 1.7. Let φ σ,ε ± be two functions on M with φ σ,ε ± = c ± , φ σ,ε + = σd ± + (1 − σ)c ± − ε on ∂ ± M. We denote with C 2,α 0 (M) the set C 2,α -functions on M which vanish on ∂ ± M. Observe that C 2,α 0 (M) is a Banach space. We define a family of maps F σ,ǫ : C 2,α 0 (M) ⊕ C 2,α 0 (M) → C 2,α 0 (M) ⊕ C 2,α 0 (M) via F σ,ε (u, v) = ∆ −1 0 (G σ,ε − (u, v)) , ∆ −1 0 (G σ,ε + (u, v))
where G σ,ε ± (u, v) = ± σ tr g (k)|∇(u + φ − )|
+ 3|∇(u + φ − )||∇(v + φ + )| + ∇(u + φ − ), ∇(v + φ + ) |u + φ − + v + φ + | + ε − ∆φ σ,ε ±
and where ∆ −1 0 maps a function f to the solution ψ of ∆ψ = f on M with vanishing Dirichlet boundary data. By standard elliptic theory, F σ,ε is indeed a map into C 2,α 0 (M) ⊕ C 2,α 0 (M). Moreover, F σ,ε is a compact operator since the image of a bounded sequence {(u i , v i )} has a convergent subsequence. Observe that if F σ,ε (u, v) = (u, v), then (u + φ − , v + φ + ) solve system (25). Hence, we can use our uniform estimates, Lemma 5.3, and Leray-Schauder's fixed point theorem, see for instance Theorem 11.6 in [19], to deduce that there exists a solution of F 1,ε (u 1,ε , v 1,ε ) = (u 1,ε , v 1,ε ) if there exists a solution of F 0,ε (u 0,ε , v 0,ε ) = (u 0,ε , v 0,ε ). Let U 0,ε be the harmonic function with U 0,ε = 1 c ± on ∂ ± M. Then u 0,ε = 1 U 0,ε satisfies ∆u 0,ε = 2 |∇u 0,ε | 2 u 0,ε with u 0,ε = c ± on ∂ ± M. Next, let v 0,ε = u 0,ε − ε. Note that v 0,ε = c ± − ε on ∂ ± M and v 0,ε > 0 on M. Then ∆u 0,ε = 3|∇u 0,ε ||∇v 0,ε | + ∇u 0,ε , ∇v 0,ε |u 0,ε + v 0,ε | + ε , ∆v 0,ε = 3|∇u 0,ε ||∇v 0,ε | + ∇u 0,ε , ∇v 0,ε |u 0,ε + v 0,ε | + ε .
Thus, we may find solutions (u 1,ε , v 1,ε ) to system (25) for σ = 1. Since u 1,ε , v 1,ε are uniformly bounded away from zero, and we have uniform C 2,α -estimates for (u 1,ε , v 1,ε ) in terms of ε, we can take the limit ε → 0 to obtain solutions (u, v) to system (24).
A crucial ingredient of the above proof is that the system (5) takes a simpler form for w = u−v and h = 1 u+v as in (26) and (27). We remark that for the p-harmonic system, i.e. system (5) for a ∈ (0, 1), can be rewritten in a very similar form to (26) and (27) though we have to re-define h = (u + v) − 1+a 1−a . Note that the radial function r − 1+a 1−a is p-harmonic in R 3 for p = 2 − a. We also expect that the solutions of system (5) for a = 0 established in this section can be used to give a new proof for the spacetime PMT.
Corollary 1. 2 .
2Let (M, g) be an annulus satisfying the dominant energy condition with spherical boundary components ∂ − M and ∂ + M. Suppose u, v are constant on both ∂ − M and ∂ + M and that u| ∂ − M < u| ∂ + M , v| ∂ − M < v| ∂ + M . Then we have under the same assumptions as in Theorem 1.1
Figure 2 .
2There are no monotone quantities associated with the level sets Σ u and Σ v individually.
2 =ḡ(∇u,∇v) = ∇u, ∇v +ḡ(N(u)N, N(v)N). Sinceḡ (N(u)N, N(v)N) = −N(u)N(v) = |∇u||∇v|, the result follows.
Proposition 3 . 1 .
31Let (M, g, k) ⊂ (R 3,1 ,ḡ) be an IDS contained in Minkowski space. Then g(η, ∂ θ ) = 0 andḡ(η, ∂ φ ) = 0.Proof. We have ∂ r + ∂ t =∇u = ∇u + N(u)N and ∂ r − ∂ t = ∇v + N(v)N.
Lemma 4 . 1 .
41Let (M, g, k) be a spherically symmetric initial data set and let Σ 0 ⊂ M be the outermost horizon. Let s be a smooth solution of rescaled IMCF starting from Σ 0 , i.e. ∆s = ∇ νν s + 2 |∇s| 2 s with s(Σ 0 ) = |Σ 0 | 16π . Outside Σ 0 we define the spherically symmetric function w = w(r) via
( 5 )
5by restricting the optical functions u = r + t and v = r − t to M. Moreover (H 2+ ) ij u = 0 and (H 2 − ) ij v = 0.For Schwarzschild the situation is similar, though the underlying objects are null vector fields rather than null functions:Proposition 4.3. Let (M,ḡ) be the Schwarzschild spacetime of mass m ≥ 0 in static coordinates, i.e.ḡ = −φdt 2 + φ −1 dr 2 + r 2 g S 2 where φ = (1− 2m r ) On (M ,ḡ)we define the null vector fields X = φ∇(r * +t) and Y = φ∇(r * − t) where r * = r +2m ln( r 2m −1) is the tortoise coordinate. Then on each spherically symmetric initial data set (M, g, k) in (M,ḡ) the vector fields X| T M and Y | T M are integrable, i.e. there are functions u, v on M with ∇u = X| T M and ∇v = Y | T M . These functions u, v solve the system (5) for a = 1 and we have |H 2 + u| 2 −((H 2 + ) ηη u) 2 = 0 as well as |H
2 W 1 ,
212p (M ) ) ≤ C + C w σ,ε 2 W 1,2p (M ) .By the Gagliardo-Nirenberg interpolation inequality, we have∇w σ,ε 2 L 2p (M ) ≤ C ∇ 2 w σ,ε L p (M ) w σ,ε L ∞ (M ) and ∇h σ,ε L p (M ) ≤ C ∇ 2 h σ
1.1. Double Null Foliations. To give a new proof of the spacetime PMT, D. Kazaras, M. Khuri and the author introduced in [23] spacetime harmonic 4 functions which are functions satisfying the PDE ∆u = − tr g (k)|∇u|. In case (M, g, k) arises as subset of Minkowski space R 3,1 , the spacetime harmonic function u can be obtained by restricting a null coordinate function of Minkowski space such as x + t, to (M, g, k)
See[23] for a precise definition.
i.e. U = u − 1+a 1−a is (2 − a)-harmonic4 We refer to[6,24] for a more detailed overview on spacetime harmonic functions.
The additional assumption of genericity is necessary as Christodoulou demonstrated in[13,14].6 We refer to M. Mars' survey[36] for a detailed explanation of R. Penrose's heuristic argument and more information about the current status of the Penrose conjecture.
The author would like to thank Yiyue Zhang for helpful conversations on this topic
Proposition 4.4. Consider the functions u = r * +t and v = r * −t in the Schwarzschild spacetime (M ,ḡ). Then on any IDS (M, g, k) ⊆ (M ,ḡ) (not necessarily spherically symmetric) the restrictions of the functions u, v onto M satisfyandas well asProof. We only show the computation for u since the one for v is analogous. We havēθφ r * = 0, ∇ 2 At r * = 0 and∇ 2 tr r * = 0. Hence,(∇u ⊗∇v +∇v ⊗∇u) − m r 2∇ u ⊗∇u
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"Spatiotemporally Discriminative Video-Language Pre-Training with Text Grounding",
"Spatiotemporally Discriminative Video-Language Pre-Training with Text Grounding"
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| [
"Yuanhao Xiong \nGoogle Research\n\n\nUCLA\n\n",
"Long Zhao \nGoogle Research\n\n",
"Boqing Gong \nGoogle Research\n\n",
"Ming-Hsuan Yang \nGoogle Research\n\n",
"Florian Schroff \nGoogle Research\n\n",
"Ting Liu \nGoogle Research\n\n",
"Cho-Jui Hsieh \nUCLA\n\n",
"Liangzhe Yuan \nGoogle Research\n\n"
]
| [
"Google Research\n",
"UCLA\n",
"Google Research\n",
"Google Research\n",
"Google Research\n",
"Google Research\n",
"Google Research\n",
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| Most of existing video-language pre-training methods focus on instance-level alignment between video clips and captions via global contrastive learning but neglect rich fine-grained local information, which is of importance to downstream tasks requiring temporal localization and semantic reasoning. In this work, we propose a simple yet effective video-language pre-training framework, namely G-ViLM, to learn discriminative spatiotemporal features. Two novel designs involving spatiotemporal grounding and temporal grouping promote learning local region-noun alignment and temporal-aware features simultaneously. Specifically, spatiotemporal grounding aggregates semantically similar video tokens and aligns them with noun phrases extracted from the caption to promote local region-noun correspondences. Moreover, temporal grouping leverages cutand-paste to manually create temporal scene changes and then learns distinguishable features from different scenes. Comprehensive evaluations demonstrate that G-ViLM performs favorably against existing approaches on four representative downstream tasks, covering text-video retrieval, video question answering, video action recognition and temporal action localization. G-ViLM performs competitively on all evaluated tasks and in particular achieves R@10 of 65.1 on zero-shot MSR-VTT retrieval, over 9% higher than the state-of-the-art method. | 10.48550/arxiv.2303.16341 | [
"https://export.arxiv.org/pdf/2303.16341v1.pdf"
]
| 257,804,907 | 2303.16341 | 0a0a67796ac4feff84ae6a91d158fd92c3d81b60 |
Spatiotemporally Discriminative Video-Language Pre-Training with Text Grounding
Yuanhao Xiong
Google Research
UCLA
Long Zhao
Google Research
Boqing Gong
Google Research
Ming-Hsuan Yang
Google Research
Florian Schroff
Google Research
Ting Liu
Google Research
Cho-Jui Hsieh
UCLA
Liangzhe Yuan
Google Research
Spatiotemporally Discriminative Video-Language Pre-Training with Text Grounding
Most of existing video-language pre-training methods focus on instance-level alignment between video clips and captions via global contrastive learning but neglect rich fine-grained local information, which is of importance to downstream tasks requiring temporal localization and semantic reasoning. In this work, we propose a simple yet effective video-language pre-training framework, namely G-ViLM, to learn discriminative spatiotemporal features. Two novel designs involving spatiotemporal grounding and temporal grouping promote learning local region-noun alignment and temporal-aware features simultaneously. Specifically, spatiotemporal grounding aggregates semantically similar video tokens and aligns them with noun phrases extracted from the caption to promote local region-noun correspondences. Moreover, temporal grouping leverages cutand-paste to manually create temporal scene changes and then learns distinguishable features from different scenes. Comprehensive evaluations demonstrate that G-ViLM performs favorably against existing approaches on four representative downstream tasks, covering text-video retrieval, video question answering, video action recognition and temporal action localization. G-ViLM performs competitively on all evaluated tasks and in particular achieves R@10 of 65.1 on zero-shot MSR-VTT retrieval, over 9% higher than the state-of-the-art method.
Introduction
Video-language pre-training with the goal of learning transferable multi-modal representations has recently attracted much attention that finds numerous applications [1,7,23,39,40,41,44,47,64]. Such a model trained on those web-crawled unlabeled noisy data achieves promising performance on various downstream tasks, ranging from single-modal video action recognition to multi-modal text-* Work done as a student researcher at Google Caption: After squeezing the bottle, the woman then applied facial cream to her face.
Similarity scores of baseline features
Similarity scores of temporal-aware features Figure 1. A video example from the pre-training dataset with scene shift and region-noun correspondences. Matrices below show frame-wise similarity scores computed from baseline and temporal-aware features respectively. video retrieval. Video-language pre-training typically follows the pipeline: a) encoding video and text pairs into latent representations; b) modality fusion and interaction; c) pre-training on specific objectives.
Existing methods typically optimize these three components in the pre-training pipeline by designing expressive encoders [7,23,40,45,48], fusing two modalities via a crossencoder [23,39,40,41,44,63,77], or adopting a combination of various pre-training tasks such as contrastive learning and masked modeling [11,19,23,24,40,75]. While these modifications benefit the pre-trained model, they lack local discriminative capabilities. Instead, we approach the videolanguage pre-training task from a different perspective with a focus on local fine-grained information.
It has been shown that most video-language pre-training methods merely perform well on learning holistic representations to match a video, caption pair and neglect fine-grained information (e.g., region-noun correspondences, or scene/action changes along the time in a video) [1,7,39,41,44,47,48,64]. However, such regional or temporal fine-grained information has been demonstrated to play an important role in localization and reasoning tasks [23,40,45,73,76]. For example, in Figure 1, given a video of a woman applying facial cream, per-frame features from the model pre-trained with only global contrastive loss are relatively hard to distinguish, which makes it challenging to find boundaries in tasks like temporal action localization. Furthermore, region-noun alignment frequently appears in a video and caption pair as shown in Figure 1, where we highlight objects and their associated noun phrases with the same color. Few methods take such alignment into consideration.
In this work, we incorporate fine-grained video-caption interactions into the pre-training stage and propose a novel framework, Text-Grounded Video-Language Modeling, named G-ViLM. G-ViLM encourages instance-level videocaption alignment, fine-grained region-noun alignment, and temporal-aware video representations simultaneously. As Figure 2 shows, G-ViLM consists of three primary training objectives: spatiotemporal grounding, temporal grouping, and global contrastive learning. With the video-caption pair as the input, a classical dual-encoder is leveraged to extract the representation for each modality respectively. Note that videos are pre-processed with the cut-and-paste operation [74,76], i.e., randomly selecting a clip in one video as the foreground and pasting it onto the other background video, to explicitly introduce temporal scene changes, as shown in Figure 2. We first adopt the structure of grouping blocks [65,72] to aggregate semantically similar video patches to represent objects without off-the-shelf detectors. In spatiotemporal grounding, we align grouped video tokens with noun concepts extracted from the caption via our designed grounding loss. Furthermore, we design a temporal grouping objective to capture temporal information by distinguishing foreground and background representations. Finally, the model is trained by a global video-caption contrastive loss to match instance-level video-caption pairs.
Our key contributions are summarized as follows:
• We design a spatiotemporal grounding module to capture fine-grained correspondences by aligning nouns from the caption and regions from the video in a selfsupervised manner. • We leverage a cut-and-paste operation to introduce temporal scene changes into videos during pretraining, and propose the temporal grouping module to learn more temporal-aware features. • G-ViLM is evaluated comprehensively on four representative downstream tasks, including text-video retrieval, video question answering, video action recognition and temporal action localization.
• Experimental results have shown the effectiveness of G-ViLM and in particular, it outperforms SOTA by 9% in R@10 in zero-shot text-video retrieval and performs competitive on other downstream tasks.
Related Work
Video-language pre-training. Video-language pretraining is an emerging research area that aims to develop machine learning models capable of jointly understanding visual and textual content. Representations learned from large scale noisy datasets such as HowTo100M [47], Web-Vid [7] and VideoCC [48] have demonstrated great potentials in adapting to downstream tasks, including but not limited to text-video retrieval, video question answering and video captioning. Elaborately designed pre-training objectives ranging from generative [15,19,42,43] to discriminative [1,7,23,39,40] have been proposed, among which contrastive learning is prevalent and widely adopted to attract paired video-caption instances and repelling unpaired ones. Early approaches [41,44,46,47,59,64,77] merely leverage offline video features extracted from frozen backbone models, and are less effective in adaptation to various domains. Recently, end-to-end training [1,7,10,23,24,39,40,70] enables video and language features to be learned from raw pixels and captions, respectively. For instance, Frozen [7] adopts a vision transformer as the visual encoder taking both raw images and videos as input and updates the visual and text encoder via contrastive learning. In addition, some methods [10,19,70] attempt to make encoders more expressive by introducing richer information from raw data like region features, adding a multi-modal fusion encoder to facilitate modality interaction, or adopting a combination of contrastive learning and masked modeling. However, their primary focus is still on learning holistic global representations to align instance-level video, caption pairs.
Recently, some approaches have been proposed to leverage finer-grained information such as nouns/verbs phrases from a caption. ALPRO [40] extracts pseudo entity labels by feeding noun prompts into a frozen model and use contrastive objective to align cropped visual regions and the corresponding textual labels. In [23], MCQ recovers randomly masked noun/verbs tokens via resorting to global video features, which implicitly improves text entity association in visual encoding. Despite these efforts, correspondences between visual regions and noun concepts in captions and temporal scene shifts in a video, is still neglected and not modeled explicitly in existing video-language pretraining methods. In this work, we propose two novel designs, spatiotemporal grounding and temporal grouping, to leverage fine-grained information in pre-training stage. Vision language grounding. The goal of Visual Grounding (VG) is to locate the most relevant object or region in a visual input based on a natural language query [17, 20,
U z E i a a C T D 8 K E o 5 0 h C Z z o x 6 T l G g + M o C J Z K Z X R A Z Y Y q L N d v J m C c 7 s y P P Q P C o 7 l f L x p V O s 1 m 4 h U w 7 2 4 Q B K 4 M A J V K E G d W g A g T 7 c w S M 8 W d x 6 s J 6 t l 2 n p g j W 9 Y Q / + y H r 9 B p R G k j E = < / l a t e x i t > e = 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " E l K + j l N s M G h X p m k T J u q P W k r j E y I = " > A A A B 6 n i c b Z D L S g M x F I b P e K 3 1 V n X p J l i E r s q M Y H U j F t x 0 W d F e o D O U T J p p Q z O Z I c k I Z S j 4 A i K 4 U M S t T + I L C O 5 8 E P e m U x f a + k P I x / + f k H O O H 3 O m t G 1 / W g u L S 8 s r q 7 m 1 / P r G 5 t Z 2 Y W e 3 q a J E E t o g E Y 9 k 2 8 e K c i Z o Q z P N a T u W F I c + p y 1 / e D H J W z d U K h a J a z 2 K q R f i v m A B I 1 g b 6 4 q e V b q F o l 2 2 M 6 F 5 c H 6 g e P 7 u l r 7 e 7 t 1 6 t / D h 9 i K S h F R o w r F S H c e O t Z d i q R n h d J x 3 E 0 V j T I a 4 T z s G B Q 6 p 8 t K s 1 T E 6 N E 4 P B Z E 0 R 2 i U u b 9 f p D h U a h T 6 p j L E e q B m s 4 n 5 X 9 Z J d H D q p U z E i a a C T D 8 K E o 5 0 h C Z z o x 6 T l G g + M o C J Z K Z X R A Z Y Y q L N d v J m C c 7 s y P P Q P C o 7 l f L x p V O s 1 m 4 h U w 7 2 4 Q B K 4 M A J V K E G d W g A g T 7 c w S M 8 W d x 6
s J 6 t l 2 n p g j W 9 Y Q / + y H r 9 B p R G k j E = < / l a t e x i t > e = 6 Figure 2. Framework of G-ViLM with three training objectives: temporal grouping, spatiotemporal grounding, and global contrastive learning. Cut-and-paste is leveraged to introduce temporal changes manually. The example involves two videos with 8 frames with s = 1 and e = 6, leading to the masking as (0, 1, 1, 1, 1, 1, 1, 0). These objectives promote modality interaction from both local and global perspectives: (1) spatiotemporal grounding focuses on local correspondences between regions and nouns; (2) temporal grouping learns temporal-aware features by distinguishing whether clips are from background or foreground; (3) global contrastive learning matches instance-level video, caption pairs. 25,26,53]. Recently, visual grounding has been adapted to a pre-training task in a self-supervised manner for openvocabulary image segmentation [25,65]. For example, OpenSeg [25] semantically aligns a caption with extracted image regions via a grounding loss. Moreover, without the off-the-shelf object detectors, GroupViT [65] learns to group together semantic regions from text supervision by contrastive learning. Note that visual grounding is mostly discussed in the image domain and its success motivates us to extend visual-semantic alignment to video-language pre-training. We integrate a novel spatiotemporal grounding module in our framework to promote visual and textual entity correspondences in a self-supervised manner. Video temporal modeling. In contrast to images, videos contain a sequence of dynamic frames and how to model temporal information is critical in video understanding [3,9,18,52,60,76]. Specifically, TSP [3] learns temporal information via predicting clips inside or outside the action with substantial annotations while PAL [76] aligns features of pasted pseudo action regions from two synthetic videos. These techniques are elaborately designed for training models on long videos such as movies or TV dramas, which contains natural scene changes. However, few of them have been considered in video-language pre-training since the majority of the dataset contains short videos which feature repeated frames and are lacking in temporal differences.
In this work, we develop a temporal grouping method to learn temporal-aware clip features in video-language self-supervised learning. We show that features extracted from explicitly temporal modeling achieve significant improvements in not only temporal action localization tasks, but also coarse-grained reasoning and understanding tasks such as video question answering and video action recognition.
Method
Overview
The framework of G-ViLM is presented in Figure 2. We adopt the dual encoder architecture for video-language pretraining, and there are three primary objectives used in the pre-training stage: 1) spatiotemporal grounding, 2) temporal grouping, and 3) global contrastive learning.
As shown in Figure 2, temporal changes are first artificially introduced into training examples through cut-andpaste, and then a set of spatiotemporal video tokens are obtained given the patch size and a linear projection layer. Video tokens are processed in two branches: (1) group tokens aggregate semantically similar video tokens via grouping blocks to promote region-noun groundingness; (2) the output tokens from the last layer of the video encoder are utilized in temporal grouping to improve temporal discriminativeness. In contrast to previous methods where regions are extracted with pre-trained object detectors [10,40,70], we leverage the learned group tokens to cluster and organize semantically similar regions in a self-supervised manner, which is more effective and reduces the artifacts of any detectors. For the language branch, the original captions are tokenized into a sequence of text tokens, which are then fed into a text encoder to extract the corresponding representation from the preceding [CLS] token. Noun tokens are extracted in the same way given a set of noun prompts.
We model the interaction between region features and noun tokens using spatiotemporal grounding loss. To further promote temporal awareness, we use masks derived from the cut-and-paste operations as the ground-truth for temporal grouping. Finally, a global contrastive loss is computed between the video and the caption representations to match the instance-level video, caption pair.
Grounding with Group Tokens
Observing the correspondences between visual regions in a video and noun phrases in a caption, as demonstrated in Figure 1, we model such fine-grained alignment for more expressive encoders. In practice, it is infeasible to pool tokens of interest as cluster centers since we do not have ground-truth spatiotemporal segmentation. Thus, we adopt M learnable group tokens to cluster semantic similar regions in a self-supervised manner. Note that group tokens are randomly initialized and shared among different videos. The detailed structure of a grouping block is presented in Appendix A and multiple grouping blocks are placed at different layers of the video encoder to update group tokens progressively. Final group tokens denoted as G = {g m i } M m=1 aggregate semantically similar voxels and represent different regions in the video v i .
We obtain noun tokens as follows: for each caption c i of a video, we extract K noun phrases using noun chunking in spaCy 1 and prompt each of them with a set of handcrafted sentence templates, e.g., "A photo of a {noun}". Such prompted noun phrases are fed into the text encoder to extract noun tokens {n k i } K k=1 . We define the notation for softmax on a vector x at the ith element as: σ(x) i = exp(xi)/τ j exp(xj )/τ , where τ is the temperature to scale logits. The similarity of all group tokens G to a noun token n k is s(G, n k ) = [ g 1 , n k , . . . , g M , n k ] ∈ R M , where ·, · is the cosine similarity. Since ground-truth correspondences between regions and nouns are inaccessible, we compute the grounding similarity between all group and noun tokens by:
G(v, c) = 1 K K k=1 n k , M m=1 σ s(G, n k ) m · g m . (1)
G(v, c) encourages each noun to be grounded to one or a few regions and avoids penalizing regions that cannot find any relevant nouns.
1 https://spacy.io/ Similarity scores over a batch of size B are computed as:
G(V, c i ) = [G(v 1 , c i ), . . . , G(v B , c i )] ∈ R B and G(v i , C) = [G(v i , c 1 ), . . . , G(v i , c B ] ∈ R B , where V = {v i } B i=1 and C = {c i } B i=1
denote the set of videos and captions in a batch respectively. Spatiotemporal grounding loss L g is then defined to enable nouns to be matched with regions for each positive video, caption pair. L g = L v→c g +L c→v g , consists of a video-to-caption grounding loss
L v→c g = − 1 B B i=1 log σ (G (v i , C)) i ,(2)
and a caption-to-video grounding loss
L c→v g = − 1 B B i=1 log σ (G (V, c i )) i .(3)
Temporal Grouping with Cut-and-Paste
Training data in pre-training stage are usually short video clips with repetitive scenes. To simulate scene shifts, we design a cut-and-paste operation inspired from image augmentations [74,76] to introduce temporal changes manually as augmentation to further improve video representations.
Given a target video v i with T frames as the foreground and a randomly sampled video v pi with the index p i as the background from the same batch of size B, we divide each video into N t = T /t clips with the temporal window size t. We then sample the start and end clip indices s and e from (0, N t ), and paste the corresponding region from v i into the background video v pi to form a blended videov i . We define the foreground-background mask as m i ∈ R Nt = {1(j ∈ [s, e])|j ∈ [0, N t )}, where 1(·) is the indicator function. This operation is illustrated in Figure 2.
A video is first flattened into N non-overlapping voxels. After projected by a linear layer, these voxel tokens are fed into the transformer encoder to obtain transformed tokens z v i ∈ R N ×d , where d is the feature dimension. To obtain clip-level representations z clip i ∈ R Nt×d , we average-pool over z v i along spatial dimension after recovering the feature map's 3D shape. Two cluster centers, z b i for the background and z f i for the foreground, are further computed by averaging features from z v i on the corresponding position based on the mask m i . To assign each clip to background or foreground, we compute a i via dot product with an elementwise softmax function applying on the last dimension:
a i = Softmax(z clip i · [z b i ; z f i ] T ) ∈ R Nt×2 .(4)
Finally, the temporal grouping loss can be computed within a batch between a i and the ground-truth masking m i of onehot version using mean squared error as
L t = 1 B B i MSE (a i , One-hot(m i )).(5)
In addition, the cut-and-paste operation indicates thatv i has another positive caption c pi apart from its original c i , and the positive indices of captions with weights become
W v ∈ R B×B = {w v i,j } satisfying w v i,j = β i , j = i 1 − β i , j = p i 0, otherwise ,(6)
where β i = (e − s)/N t is the ratio of the foreground in the cut-and-paste videov i . From the perspective of captions, we can obtain W c = (W v ) . We can derive the augmented grounding loss L g with the video-to-caption loss as
L v→c g = − 1 B B i=1 B j=1 w v i,j log σ (G (v i , C)) j ,(7)
and the caption-to-video loss withV
= {v i } B i=1 as L c→v g = − 1 B B i=1 B j=1 w c i,j log σ G V , c i j .(8)
Overall Pre-training Objective
We also include a global contrastive learning objective for instance-level video-caption alignment. f v i , the video representation ofv i , is extracted from average-pooled group tokens and f c i , the caption representation c i , is selected as [CLS] token from the original caption. Instance similarity scores are defined as:
s(V, c i ) = [ f v 1 , f c i , . . . , f v B , f c i ] ∈ R B and s(v i , C) = [ f v i , f c 1 , . . . , f v i , f c B ] ∈ R B .
The global contrastive loss is defined as L contrast = L v→c contrast + L c→v contrast , with the video-to-caption view of
L v→c contrast = − 1 B B i=1 B j=1 w v i,j log σ(s(v i , C)) j ,(9)
and the caption-to-video view of
L c→v contrast = − 1 B B i=1 B j=1 w c i,j log σ(s(V, c i )) j .(10)
The overall pre-training objective is a combination of weighted sum of grouping loss, grounding loss, and global contrastive loss: L = ω 1 L t + ω 2 L g + ω 3 L contrast . We set three weights equal to one in our experiments for brevity.
Experiments
We conduct comprehensive evaluations of G-ViLM against the state-of-the-art methods. First, we introduce the pre-training datasets used in our method and four selected downstream tasks. Next, the implementation details of both pre-training and fine-tuning procedures are presented. We compare the performance of G-ViLM with existing methods and demonstrate the effectiveness of incorporating finegrained information. In addition, we present ablation results on choices of pre-training datasets and training objectives.
Pre-training Datasets
We pre-train G-ViLM with VideoCC [48] dataset, which contains about 3.3M video-caption pairs. Specifically, VideoCC is mined online using the Conceptual Captions [56] as a seed dataset, and has shown to be more effective in retrieval and captioning tasks than commonlyadopted datasets such as HowTo100M [47] and WebVid-2M [7]. In addition, we include ActivityNet-Caption [36] with 20K well-aligned pairs into the pre-training corpus. Note that for temporal action localization, the model is pretrained on HowTo100M only, which is observed to benefit to TAL compared with VideoCC + ActivityNet.
Downstream Tasks
Text-Video Retrieval. We adopt the widely used text-video retrieval benchmark MSR-VTT [66] for evaluation. It consists of 10K YouTube video clips with 200K captions. Similar to existing methods [7,23], we train and test the model on the split of 9K and 1K videos. Video Question Answering (VQA). We consider openended VQA settings with two representative datasets: 1) MSRVTT-QA [62] with 1500 answer candidates; 2) MSVD-QA [62] with 2423 answer candidates. To comply with the data policy, the size of these two datasets that we are using are smaller than the original ones and detailed statistics could be found in Table 4. Video Action Recognition. We select HMDB51 [37] with 6,766 videos from 51 categories and UCF101 [57] with 13,320 videos from 101 categories. Both linear probing and fine-tuning the whole model are explored. Temporal Action Localization (TAL). TAL aims for predicting the temporal extent and the class labels of action instances. We evaluate the performance on ActivityNet [29], an action understanding dataset of 19,994 temporally annotated untrimmed videos with 200 action categories.
Implementation Details
Input. We sample 32 frames for each video and resize them into 224×224 as input with the same augmentations in [48]. Each caption is tokenized into 32 tokens including [CLS] during training. K = 2 noun phrases are extracted for each caption and then prompted with a set of manually designed templates such as "It is a video of {noun}". Model architecture. We use a 12-layer ViT-base model with the patch size of 2 × 16 × 16 as the video encoder and initialize it with weights pre-trained on Kinetics-400. We adopt 32 learnable group tokens and 3 grouping blocks featuring K-means attention [65,72]. Grouping blocks are inserted at the 6th, 9th and last layers of the video encoder, following GroupViT [65] and kMaX-DeepLab [72]. The text encoder is initialized from the pre-trained BERT-base model. All representations are projected into the common space with the dimension of 256.
Pre-training and fine-tuning setups. We implement G-ViLM in JAX and train all models on TPU accelerators. During pre-training, a synchronous SGD with momentum 0.9 and initial learning rate 0.1 is used for optimization. We train G-ViLM for 10 epochs with a batch size 1024 and adopt a cosine learning rate decay schedule with a warmup ratio 0.05. It takes about one day for the whole pre-training stage. We use the same text templates as in [40] to generate text prompts. In terms of fine-tuning, different tasks are trained independently with their own set of hyperparameters on the target dataset and more details can be found in Appendix B. For temporal action localization, we fix weights of the pre-trained video encoder and its grouping blocks to extract video features, which are then evaluated by G-TAD [69], a commonly used method for TAL.
Evaluation Results
Text-Video Retrieval
We evaluate G-ViLM for the task of text-video retrieval on MSR-VTT under both zero-shot and fine-tuning settings, and present detailed results in Table 1 and 2. G-ViLM outperform other methods significantly for zero-shot evaluation with R@10 of 65.1, yielding approximately 9% improvement over the best-performing baseline MCQ. The superior results demonstrate that our pre-trained model builds up a good alignment between video and language and has great generalization to unseen datasets. G-ViLM also achieves performance gain when the model is fine-tuned on the target MSR-VTT dataset, which further validates advantages of the pre-trained model. Note that G-ViLM performs favorably against existing methods despite the much smaller size of the pre-training data used in G-ViLM than those in baselines, such as HowTo100M and WebVid-2M. These results are consistent with the findings in [48] and demonstrate the importance of high-quality video-caption pairs in retrieval tasks. G-ViLM leverages a dual-encoder design and does not include a fusion encoder during the pretraining stage which saves much computation cost. On the other hand, retrieval can be achieved efficiently by computing dot-product between video and caption features without feeding every combination into the fusion model, compared with models such as ClipBERT [39].
Video Question Answering
VQA results on two open-ended datasets MSRVTT-QA and MSVD-QA are shown in Table 3. To enable G-ViLM to deal with the VQA task, we add a fusion head adapted from BUTD [6] by integrating video and text features with simple linear layers. Then a classifier is inserted after the fusion module to perform question answering as a classification problem. Compared with previous methods which leverage particular architectures for VQA or include a complicated fusion encoder, G-ViLM is the most efficient and flexible for various vision-language tasks. Meanwhile, G-ViLM achieves on-par or even better performance with selected baselines, with the accuracy of 43.5% (1.4% lift) and 45.2% (0.7% drop) on MSRVTT-QA and MSVD-QA respectively. Note that due to data restrictions shown in Table 4, the training sets are not complete as original ones and we believe the performance of our method can be further improved if trained on more VQA pairs.
Video Action Recognition
For video action recognition, we only keep the video encoder together with its grouping blocks to extract singlemodality video representations for evaluation. Two evaluation settings are considered: (1) linear probing where the backbone encoder is frozen and only the last linear classifier is trained and (2) end-to-end fine-tuning where both the backbone and the classifier are trained. Top-1 accuracy on UCF101 and HMDB51 is reported in Table 5. We can observe that in linear probing, G-ViLM performs well against the other methods, with 3.0% and 2.9% higher than current SOTA, MMV with audio and text on UCF101 and HMDB51. G-ViLM also achieves consistently superior performance under fine-tuning evaluation. Outstanding performance of G-ViLM demonstrates that learning the alignment between videos and captions with fine-grained information
Temporal Action Localization
We report mean Average Precision (mAP) values under different temporal Intersection over Union (tIoU) thresholds on ActivityNet in Table 7. As mentioned in Section 4.3, we directly use pre-trained models to extract the video features as the input to G-TAD and do not further train the encoder. G-ViLM consistently exceeds other self-supervised competitors and even fully supervised approaches such as LoFi [68] and BSP [67] in three tIoU thresholds. This observation again consolidates the conclusion that visionlanguage pre-training can not only be applied to specific VL problems like text-video retrieval and VQA, but also benefit single-modal downstream tasks.
Ablation Study
We analyze the effects of the choice of pre-training datasets and different combinations of three pre-training objectives of G-ViLM in this section. Pre-training datasets.
To analyze of effect of the pre-training datasets, we evaluate the performance on selected downstream tasks and present detailed results in Ta Table 8. Ablation study on pre-training datasets. We observe that the combination of VideoCC and ActivityNet yields the best performance on retrieval task, while using HowTo100M achieves the best results on temporal action localization.
Howto100M. On the other hand, G-ViLM performs best on the same task when pre-trained with the combination of VideoCC and ActivityNet. These empirical results coincide with the findings in [48], in which HowTo100M has been pointed out not appropriate for such vision-language tasks requiring strong alignment. It is also worth noting that pre-training on ActivityNet and VideoCC performs consistently better than using only one dataset, which validates our choice for PT dataset. On the other hand, HowTo100M contains a large number of action-related videos, which contributes to learning temporal-aware features and demonstrates better capacities in temporal action localization and thus is used for all TAL experiments. Training objective. We analyze the role of our proposed training objectives and present results of this ablation study in Table 6. We evaluate four scenarios and find that in video-language tasks like zero-shot text-video retrieval and VQA, both temporal grouping loss L t and spatiotemporal grounding loss L g contribute to performance gain. And the combination of L t and L g can further improve the performance to 28.6 for R@1 on MSRVTT-ZS and 45.2 accuracy on MSVD-QA. For the temporal action localization task, L g would not significant achieve performance gain in mAP compared to scenario 2 and 4. We hypothesize that HowTo100M is a noisy dataset with weakly-aligned videocaption pairs and grounding in a self-supervised manner might not benefit representation learning on such data.
Conclusion
In this paper, we present a novel video-language pretraining framework, named G-ViLM, that aims to utilize fine-grained local information to capture region-noun correspondences and temporal-aware features at the same time.
A. Implementation Details
A.1. Text Prompt Templates
As mentioned in Section 3.2, we extract noun tokens by prompting noun phrases with pre-defined templates. We randomly select one from 12 templates to generate the prompt and details are presented in Table 9.
A.2. Structure of Grouping Blcok
We demonstrate the structure of a grouping block in Figure 3. It features a K-means clustering attention layer, in which attention scores are computed between group tokens as query and video tokens as value. The cluster assignment is computed via gumbel softmax over group tokens and converted into a one-hot hard assignment. Figure 3. The structure of a grouping block. It is inserted at different layers of the video encoder to update group tokens by merging semantically similar video tokens.
A.3. Downstream tasks
Implementation details of fine-tuning the pre-trained model on downstream tasks are described in this section. During fine-tuning, we resize video frames to 224×224 and sample 32 frames for each video. The maximum length for each caption is 32 by default, the same as the value in the pre-training stage. Specific optimization settings for each dataset are shown in Table 10 Table 12. End-to-end fine-tuning configurations for UCF101 and HMDB51 of video action recognition.
config ActivityNet optimizer Adam base learning rate 4e-3 weight decay 1e-4 optimizer momentum β 1 =0.9, β 2 =0.999 learning rate schedule decay by γ=0.1 every 5 epochs batch size 16 training epochs 10 Table 13. End-to-end fine-tuning configurations for ActivityNet of temporal action localization.
B. Visualization
We present the visualization of spatiotemporal grounding in Figure 4. For each example, we choose the group token which has the maximum similarity score of the target noun phrase, and compute the attention heatmap based on corresponding video tokens assigned to that group token. It can be observed that the alignment between the region and the noun phrase has been learned during the pretraining stage without any fine-grained annotations. In addition, more comparisons between similarity scores of baseline features and temporal-aware features are provided in Figure 5. With temporal grouping, features from different scenes are much easier to distinguish Caption: There is one blue pencil, one green pencil, and an eraser on the table.
Similarity scores of baseline features
Similarity scores of temporal-aware features Caption: A whale tail is seen out of water with hills in the background.
Similarity scores of baseline features
Similarity scores of temporal-aware features Caption: A man was driving a tractor for land reclamation.
Similarity scores of baseline features
Similarity scores of temporal-aware features Caption: A person cut the paper to make a gift box.
Similarity scores of baseline features
Similarity scores of temporal-aware features
Figure 4 .
4Visualization of spatiotemporal grounding. The attention feature map of each example is computed from corresponding regions assigned to the group token which achieves the highest similarity score to the target noun phrase.
Figure 5 .
5Visualization of temporal grouping.
Table 2. Fine-tuning performance of text-video retrieval on MSR-VTT test set with 1K videos, where higher R@k and lower MedR (Median Rank) indicate better performance.Method
Video Encoder Input
PT Dataset
#Pairs PT R@1 R@5 R@10 MedR
ActBERT [77]
ResNet-3D
HowTo100M
120M
8.6
23.4
33.1
36.0
MMV [2]
Raw Videos
HowTo100M, AudioSet
138M
9.3
23.0
31.1
38.0
MIL-NCE [46]
Raw Videos
HowTo100M
120M
9.9
24.0
32.4
29.6
VATT [1]
Raw Videos
HowTo100M, AudioSet
138M
-
-
29.7
49.0
NoiseEst [5]
ResNeXt-101
HowTo100M
110M
8.0
21.3
29.3
33.0
TACo [71]
I3D, S3D
HowTo100M
120M
9.8
25.0
33.4
29.0
VideoCLIP [64]
S3D
HowTo100M
110M
10.4
22.2
30.0
-
MCN [12]
ResNeXt-101
HowTo100M
120M
10.5
25.2
33.8
-
SupportSet [50]
R(2+1)D-34
HowTo100M
120M
12.7
27.5
36.2
24.0
Frozen [7]
Raw Videos
CC3M, WebVid-2M
5.5M
18.7
39.5
51.6
10.0
AVLnet [54]
ResNeXt-101
HowTo100M
120M
19.6
40.8
50.7
9.0
DemoVLP [10]
Raw Videos
CC3M, WebVid-2M
5.5M
24.0
44.0
52.6
-
ALPRO [40]
Raw Videos
CC3M, WebVid-2M
5.5M
24.1
44.7
55.4
8.0
MCQ [23]
Raw Videos
CC3M, WebVid-2M
5.5M
26.0
46.4
56.4
7.0
G-ViLM
Raw Videos
VideoCC, ActivityNet
3.3M
28.6
53.6
65.1
5.0
Table 1. Zero-shot performance of text-video retrieval on MSR-VTT test set with 1K videos, where higher R@k and lower MedR (Median
Rank) indicate better performance.
Method
Video Encoder Input
PT Dataset
#Pairs PT R@1 R@5 R@10 MedR
ActBERT [77]
ResNet-3D
HowTo100M
120M
16.3
42.8
56.9
10.0
UniVL [44]
S3D
HowTo100M
110M
21.2
49.6
63.1
6.0
MMT [21]
S3D
HowTo100M
120M
26.6
57.1
69.6
4.0
HERO [41]
SlowFast
TV and HowTo100M
120M
16.8
43.4
57.7
-
NoiseEst [5]
ResNeXt-101
HowTo100M
110M
17.4
41.6
53.6
8.0
ClipBert [39]
Raw Videos
COCO, VisGenome
5.6M
22.0
46.8
59.9
6.0
AVLnet [54]
ResNeXt-101
HowTo100M
120M
27.1
55.6
66.6
4.0
VLM [63]
S3D
HowTo100M
110M
28.1
55.5
67.4
4.0
TACo [71]
I3D, S3D
HowTo100M
120M
28.4
57.8
71.2
4.0
SupportSet [50]
R(2+1)D-34
HowTo100M
120M
30.1
58.5
69.3
3.0
VideoCLIP [64]
S3D
HowTo100M
110M
30.9
55.4
66.8
-
Frozen [7]
Raw Videos
CC3M, WebVid-2M
5.5M
31.0
59.5
70.5
3.0
DemoVLP [10]
Raw Videos
CC3M, WebVid-2M
5.5M
24.0
44.0
52.6
-
ALPRO [40]
Raw Videos
CC3M, WebVid-2M
5.5M
24.1
44.7
55.4
8.0
MCQ [23]
Raw Videos
CC3M, WebVid-2M
5.5M
37.6
64.8
75.1
3.0
G-ViLM
Raw Videos
VideoCC, ActivityNet
3.3M
38.4
65.7
76.3
2.0
Table 3 .Table 4 .
34Statistics of training examples in MSRVTT-QA and MSVD-QA. Due to the data policy, we use reduced number of QA pairs in our experiments.Experiment results on Video Question Answering on
MSRVTT-QA and MSVD-QA datasets in top-1 accuracy (%).
Method
MSRVTT-QA MSVD-QA
Original
158,581
30,933
Ours
117,210
25,115
Reduction ratio
26.1%
18.8%
Method
Modal
UCF101
HMDB51
Lin
FT
Lin
FT
CCL [35]
-
54.0 69.4 29.5 37.8
CBT [58]
-
54.0 79.5 29.5 44.5
MemDPC [27]
OF
54.1 86.1 30.5 54.5
CoCLR [28]
OF
77.8 90.6 52.4 62.9
MVCGC [31]
MV
78.0 90.8 53.0 63.4
XDC R [4]
A
80.7 88.8 49.9 61.2
XDC K [4]
A
85.3 91.5 56.0 63.1
MIL-NCE [46]
T
83.4 89.1 54.8 59.2
Frozen [7]
T
87.8 89.8 61.3 66.3
VATT [1]
A, T
89.2
-
63.3
-
ELO [51]
A, OF
-
93.8 64.5 67.4
MMV [2]
A
77.1
-
53.6
-
MMV [2]
T
86.8
-
55.1
-
MMV [2]
A, T
91.8 95.2 67.1 75.0
MCQ [23]
T
89.1 92.3 65.8 69.8
G-ViLM
T
94.8 96.5 70.0 73.9
Table 5. Experiments of action recognition on UCF101 and
HMDB51 with linear evaluation (Lin) and fully fine-tuning eval-
uation (FT). "Modal" denotes the modality used for pre-training
in addition to videos, i.e., optical flow (OF), motion vector (MV),
audio (A), text (T).
contributes to meaningful video representations for tasks in-
volved in a single modality.
ble8. Four choices of PT datasets are considered, including HowTo100M, ActivityNet, VideoCC, and VideoCC + Ac-tivityNet. G-ViLM performs performs poorly in the zeroshot text-video settings on MSR-VTT when pre-trained onPT Dataset
MSRVTT-ZS
TAL
R@1 R@5 R@10 [email protected] 0.75 0.95 Avg
HowTo100M
9.4
22.9
31.3
51.7
36.4 9.7 35.6
ActivityNet
14.4
33.5
44.0
50.5
35.3 8.7 34.5
VideoCC
24.7
47.4
59.0
50.5
35.0 9.2 34.2
VideoCC, ActivityNet 28.6
53.6
65.1
50.8
35.6 9.3 34.7
Template Prompts for noun phrases1
A footage of a {}.
2
A footage of the {}.
3
A footage of one {}.
4
A video of a {}.
5
A video of the {}.
6
A video of one {}.
7
A portrait of a {}.
8
A portrait of the {}.
9
A portrait of one {}.
10
A video footage of a {}.
11
A video footage of the {}.
12
A video footage of one {}.
Table 9 .
9Prompt templates used for generating noun tokens.
-13.Table 10. End-to-end fine-tuning configurations for MSR-VTT of text-to-video retrieval.Table 11. End-to-end fine-tuning configurations for MSRVTT-QA and MSVD-QA of video question answering.config
MSR-VTT
optimizer
SGD
base learning rate
2.5e-1
optimizer momentum 0.9
learning rate schedule cosine decay
batch size
512
warmup ratio
0.1
training epochs
20
config
MSRVTT-QA MSVD-QA
optimizer
SGD
SGD
base learning rate
1e-1
5e-5
optimizer momentum 0.9
0.9
learning rate schedule cosine decay
cosine decay
batch size
64
64
warmup ratio
0.1
0.1
training epochs
30
30
config
UCF101
HMDB51
optimizer
SGD
SGD
base learning rate
1e-1
1e-1
optimizer momentum 0.9
0.9
learning rate schedule cosine decay cosine decay
batch size
64
64
warmup ratio
0.1
0.1
training epochs
30
60
Spatiotemporal grounding and temporal grouping are introduced to achieve the goal of local region-noun alignment and temporal distinguishment in a self-supervised manner. The proposed framework outperforms existing methods significantly on several downstream tasks, such as text-video retrieval, video question answering, video action recognition, and temporal action localization.
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| []
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[]
| [
"\nGraduate School of Mathematics\nDepartment of Mathematics\nCollege of Humanities and Sciences\nNagoya University\n464-8602NagoyaJapan\n",
"\nNihon University\nSetagaya-ku156-8550TokyoJapan\n"
]
| [
"Graduate School of Mathematics\nDepartment of Mathematics\nCollege of Humanities and Sciences\nNagoya University\n464-8602NagoyaJapan",
"Nihon University\nSetagaya-ku156-8550TokyoJapan"
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| To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants and finiteness properties. Using this, we also provide a comparison result onétale cohomology groups under the tilting. As an application, we prove finiteness of the prime-to-p-torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.Contents | null | [
"https://export.arxiv.org/pdf/2203.16400v3.pdf"
]
| 247,794,175 | 2203.16400 | b8b4f27abcbdd15ac29d15485ab031d002316028 |
10 Mar 2023
Graduate School of Mathematics
Department of Mathematics
College of Humanities and Sciences
Nagoya University
464-8602NagoyaJapan
Nihon University
Setagaya-ku156-8550TokyoJapan
10 Mar 2023arXiv:2203.16400v3 [math.AC] PERFECTOID TOWERS AND THEIR TILTS : WITH AN APPLICATION TO THEÉTALE COHOMOLOGY GROUPS OF LOCAL LOG-REGULAR RINGS SHINNOSUKE ISHIRO, KEI NAKAZATO, AND KAZUMA SHIMOMOTOand phrases Perfectoid towertiltingFrobenius mapsétale cohomologylog-regularity 2020 Mathematics Subject Classification:13B0213B4013F3514A2114G45
To initiate a systematic study on the applications of perfectoid methods to Noetherian rings, we introduce the notions of perfectoid towers and their tilts. We mainly show that the tilting operation preserves several homological invariants and finiteness properties. Using this, we also provide a comparison result onétale cohomology groups under the tilting. As an application, we prove finiteness of the prime-to-p-torsion subgroup of the divisor class group of a local log-regular ring that appears in logarithmic geometry in the mixed characteristic case.Contents
Introduction
In recent years, the perfectoid technique is one of the most effective tools to commutative ring theory and singularity theory in mixed characteristic. The tilting operation S S ♭ for a perfectoid ring S is a central notion in this method, which makes a bridge between objects in mixed characteristic and objects in positive characteristic. However, perfectoid rings themselves are too big to fit into Noetherian ring theory. Hence, for applications, one often requires distinguished Noetherian ring extensions approximate to perfectoids. Indeed, in many of earlier works (such as [9], [10] and [23]), one constructs a highly ramified tower of regular local rings or local log-regular rings:
R 0 ⊆ R 1 ⊆ R 2 ⊆ · · ·
that converges to a (pre)perfectoid ring. Our purposes in this paper are to axiomatize the above towers and establish a kind of Noetherization of perfectoid theory. As an application, we show a finiteness result on the divisor class groups of local log-regular rings.
Fix a prime p. We first notice that the highly ramified towers in the positive characteristic case are of the form:
R ⊆ R 1/p ⊆ R 1/p 2 ⊆ · · · .
This type of towers also appears when one considers the perfect closure of a reduced F p -algebra. Thus we formulate this class as a tower-theoretic analogue of perfect F p -algebras, and call them perfect towers (Definition 3.2). Next, we introduce perfectoid towers as a generalization of perfect towers, which includes the towers applied so far (cf. Proposition 3.60 and Example 3.64). A perfectoid tower is given as a direct system of rings R 0 t 0 − → R 1 t 1 − → · · · satisfying seven axioms in Definition 3.6 and Definition 3.22. If we assume that each R i is Noetherian, then these axioms are essential to cope with two main difficulties which we explain below (although the axioms depend on what adic topology is considered, here we deal with only p-adic one for simplicity).
The first difficulty is that the residue ring R i /pR i on each layer is not necessarily semiperfect. We overcome it by the axioms (b), (c), and (d); these ensure the existence of a surjective ring map F i : R i+1 /pR i+1 → R i /pR i which gives a decomposition of the Frobenius endomorphism. We call F i the i-th Frobenius projection, and define a ring R s.♭ j (j ≥ 0) as the inverse limit of Frobenius projections starting at R j /pR j . Then the resulting tower R s.♭ − − → · · · is perfect, and thus we obtain the tilting operation (
{R i } i≥0 , {t i } i≥0 ) ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 )
. We remark that this strategy is an axiomatization of the principal arguments in [46].
The second one is that each R s.♭ i could be imperfect. Because of this, the Witt ring W (R s.♭ i ) is often uncontrollable. On the other hand, the definition of Bhatt-Morrow-Scholze's perfectoid rings ( [5]) contains an axiom involving Fontaine's theta map θ S : W (S ♭ ) → S (see Definition 3.51 (3)), where perfectness of S ♭ is quite effective. Our axioms (f) and (g) are the substitutes for it; these require the Frobenius projections to behave well, especially on the p-torsion parts. This idea is closely related to Gabber and Ramero's characterization of perfectoid rings ( [23,Corollary 16.3.75]; see also Theorem 3.52). Indeed, we apply it to deduce that the completed direct limit of a perfectoid tower is a perfectoid ring (Corollary 3.54).
We then verify fundamental properties of the tilting operation for towers. For example, the tilt ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) is a perfectoid tower with respect to an ideal I s.♭ 0 ⊆ R s.♭ 0 which is the kernel of the 0-th projection R s.♭ 0 → R 0 /pR 0 (Proposition 3.43). It induces an isomorphism between two perfectoid objects of different characteristics modulo the defining ideals (Lemma 3.41). Moreover, this operation preserves several finiteness properties such as Noetherianness on each layer (Proposition 3.44). A key to deducing these statements is the following result (see Remark 3.42
for homological interpretation).
Main Theorem 1 (Theorem 3.37). I s.♭ 0 is a principal ideal. Moreover, we have isomorphisms of (possibly) non-unital rings (R s.♭ i ) I s.♭ 0 -tor ∼ = (R i ) p-tor (i ≥ 0) that are compatible with {t s.♭ i } i≥0 and {t i } i≥0 .
notions of perfect towers, perfectoid towers, and their tilts. The most part of this section is devoted to studying fundamental properties of them; in particular, §3.4 deals with Main Theorem 1. In the last subsection §3.6, we provide explicit examples of perfectoid towers consisting of local logregular rings, and compute their tilts. In §4, we give a proof for Main Theorem 2 using the tilting operation, which is an application of §2 and §3. In Appendix, we review the notion of maximal sequences associated to certain differential modules due to Gabber and Ramero [23]. This plays an important role in the construction of perfectoid towers of local log-regular rings (Construction 3.58).
Convention: Throughout this paper, we follow the convention stated below.
• We consistently fix a prime p > 0. If we need to refer to another prime different from p, we denote it by ℓ. • All rings are assumed to be commutative with a unity (unless otherwise stated; cf. Theorem 3.37 (2)). We mean by a ring map a unital ring homomorphism. • A local ring is a (not necessarily Noetherian) ring with a unique maximal ideal. When a ring R is local, then we use m R (or simply m if no confusion is likely) to denote its unique maximal ideal. We say that a ring map f : R → S is local if R and S are local rings and f −1 (m S ) = m R . • Unless otherwise stated, we mean by a pair a pair (A, I) consisting of a ring A and an ideal I ⊆ A. • The Frobenius endomorphism on an F p -algebra R is denoted by F R . If there is no confusion, we denote it by Frob.
Acknowledgement. The authors are grateful to Professor K. Fujiwara for his continued support and comments. Our gratitude also goes to Professor S. Takagi, Shunya Saito, and Ryo Ishizuka for their advice on the content of this paper.
Log-regularity
In this section, we discuss several properties of monoids and local log-regular rings. In §2.1, we review basic terms on monoids, and examine the behavior of p-times maps which are effectively used in Gabber-Ramero's treatment of perfectoid towers (see Construction 3.58). In §2.2, we review the definition of local log-regular rings and crucial results by K. Kato, and study the relationship with strong F -regularity. In §2.3, we recall Gabber-Ramero's result which claims that any local log-regular ring is a splinter (Theorem 2.27), and give an alternative proof for it using the Direct Summand Theorem.
Preliminaries on monoids.
2.1.1. Basic terms. Here we review the definition of several notions on monoids.
Definition 2.1. A monoid is a semigroup with a unity. A homomorphism of monoids is a semigroup homomorphism between monoids that sends a unity to a unity.
Throughout this paper, all monoids are assumed to be commutative. We denote by Mnd the category whose objects are (commutative) monoids and whose morphisms are homomorphisms of monoids.
We denote a unity by 0. Let Q be a monoid and Q * denote the set of all p ∈ Q such that there exists q ∈ Q such that p + q = 0. Q gp denote the set of the form of a − b for all a, b ∈ Q where a − b = a ′ − b ′ if and only if there exists c ∈ Q such that a + b ′ + c = a ′ + b + c. By definition, Q gp is an abelian group. The following conditions yield good classes of monoids.
Definition 2.2. Let Q be a monoid.
(1) Q is called integral if for x, x ′ and y ∈ Q, x + y = x ′ + y implies x = x ′ .
(2) Q is called fine if it is finitely generated and integral.
(3) Q is called sharp if Q * = 0.
(4) Q is called saturated if the following conditions hold.
(a) Q is integral.
(b) For any x ∈ Q gp , if nx ∈ Q for some n ≥ 1, then x ∈ Q.
For an integral monoid Q, the map ι Q : Q → Q gp ; q → q − 0 is injective (see [41, Chapter I, Proposition 1.3.3]). In Definition 2.2 (4), we identify Q with its image in Q gp .
Next we recall the definition of a module over a monoid. 3 (b) (p + q) + x = p + (q + x) for any p, q ∈ Q and x ∈ M . (2) A homomorphism of Q-modules is a (set-theoretic) map f : M → N between Q-modules such that f (q + x) = q + f (x) for any q ∈ Q and x ∈ M . We denote by Q-Mod the category of Q-modules and homomorphisms of Q-modules.
We refer the reader to the definition of a monoid algebra R[Q] to [41]. We denote by e q (resp. e Q ) the image of an element q of Q (resp. the monoid Q) in R[Q]. For a monoid Q, one obtains the functor
(2.1) Q-Mod → R[Q]-Mod ; M → R[M ],
which is a left adjoint of the forgetful functor R[Q]-Mnd → Q-Mod. Notice that (2.1) preserves coproducts (we use this property to prove Proposition 2.7). Like ideals (resp. prime ideals, the Krull dimension) of a ring, an ideal (resp. prime ideals, the dimension) of a monoid is defined as follows.
Definition 2.4. Let Q be a monoid.
(1) A Q-submodule of Q is called an ideal of Q.
(2) An ideal I is called prime if I = Q and p + q ∈ I implies p ∈ I or q ∈ I. Remark that the empty set ∅ is a prime ideal of Q. (3) The dimension of a monoid Q is the maximal length d of the ascending chain 4 of prime ideals ∅ = q 0 ⊂ q 1 ⊂ · · · ⊂ q d = Q + , where Q + is the set of non-unit elements of Q (i.e. Q + = Q \ Q * ). We also denote it by dim Q.
3 This is called a Q-set in [41]. We call it as above to follow the convention of the terminology in commutative ring theory. 4 In this paper, the symbol ⊂ is used to indicate proper inclusion for making an analogy to the inequality symbols as in [41].
Next we review a good class of homomorphisms of monoids, called exact homomorphisms.
Definition 2.5 (Exact homomorphisms). Let P and Q be monoids.
(1) A homomorphism of monoids ϕ : P → Q is said to be exact if the diagram of monoids:
P ϕ / / Q P gp ϕ gp / / Q gp is cartesian. (2) An exact submonoid of Q is a submonoid Q ′ of Q such that the inclusion map Q ′ ֒→ Q is exact (in other words, (Q ′ ) gp ∩ Q = Q ′ ).
There is a quite useful characterization of exact submonoids (Proposition 2.7). To see this, we recall a graded decomposition of a Q-module attached to a submonoid. For a monoid Q and a submonoid Q ′ ⊆ Q, we denote by Q → Q/Q ′ the cokernel of the inclusion map Q ′ ֒→ Q.
Definition 2.6. Let Q be an integral monoid, and let Q ′ ⊆ Q be a submonoid. Then for any g ∈ Q/Q ′ , we denote by Q g a Q ′ -module defined as follows.
• As a set, Q g is the inverse image of g ∈ Q/Q ′ under the cokernel Q → Q/Q ′ of Q ′ ֒→ Q.
• The operation Q ′ × Q g → Q g is defined by the rule: (q, x) → q + x (where q + x denotes the sum of q and x in Q).
By definition, Q = g∈Q/Q ′ Q g as sets. The right-hand side is viewed as the coproduct of Q ′ -modules {Q g } g∈Q/Q ′ , and hence a Q/Q ′ -graded decomposition of the Q ′ -module Q. Using this notion, one can refine a characterization of exact embeddings described in [ (
1) The Z[Q ′ ]-module Z[Q] admits a G-graded decomposition Z[Q] = g∈G Z[Q g ].
(2) The following conditions are equivalent. (a) The inclusion map θ : Q ′ ֒→ Q is exact. In other words,
(Q ′ ) gp ∩ Q = Q ′ . (b) Q 0 = Q ′ . (c) Z[θ]Q Q := {x ⊗ r ∈ Q gp ⊗ Z Q | x ∈ Q, r ∈ Q ≥0 }.
Using this, one can define the following monoid which plays a central role in Gabber-Ramero's construction of perfectoid towers consisting of local log-regular rings.
Definition 2.10. Let Q be an integral sharp monoid. Let c and i be non-negative integers with c > 0.
(1) We denote by Q
(i) c a submonoid of Q Q defined as Q (i) c := {γ ∈ Q Q | c i γ ∈ Q}. (2) We denote by ι (i) c : Q (i) c ֒→ Q (i+1) c the inclusion map, and by Z[ι (i) c ] : Z[Q (i) c ] → Z[Q (i+1) c ]
the induced ring map.
To prove several properties of Q (i) c , the following one is important as a starting point. Lemma 2.11. Let Q be an integral sharp monoid. Then for every c > 0 and every i ≥ 0, the following assertions hold.
(1) Q (i) c is integral and sharp. (2) Q (i+1) c = (Q (i) c ) (1) c . (3)
The c-times map on Q Q restricts to an isomorphism of monoids:
f c : Q (i+1) c ∼ = − → Q (i) c ; γ → cγ. Proof. (1): Since Q gp ⊗ Z Q is an integral monoid, so is Q (i) c . Let us show that Q (i) c is sharp. Pick x, y ∈ Q (i) c such that x + y = 0. Then c i x = 0 because Q is sharp. Thus, since Q (i)
c is a submonoid of the torsion-free group Q gp ⊗ Z Q, we have x = 0, as desired.
(2): Since any g ∈ (Q
(i) c ) gp satisfies c i g ∈ Q gp , the inclusion map Q gp ֒→ (Q (i) c ) gp becomes an isomorphism ϕ : Q gp ⊗ Z Q ∼ = − → (Q (i)
c ) gp ⊗ Z Q by extension of scalars along the flat ring map Z → Q. The restriction ϕ : Q Q ֒→ (Q (i) c ) Q of ϕ is also an isomorphism, and one can easily check that ϕ restricts to the desired canonical isomorphism Q
(i+1) c ∼ = − → (Q (i) c )(1)
c .
(3): It is easy to see that the c-times map on Q Q restricts to the homomorphism of monoids f c . Since the abelian group Q Q = Q gp ⊗ Z Q is torsion-free, f c is injective. Moreover, any element γ in Q (i) c is of the form x ⊗ r for some x ∈ Q gp and r ∈ Q, which satisfy that c(x ⊗ r c ) = γ and c i+1 (x ⊗ r c ) ∈ Q. Hence f c is also surjective, as desired.
1 p x 1 + · · · + n r 1 p x r mod Q gp , we have p r = |(Q (1) p ) gp /Q gp || Ker(f )|. Hence |(Q(1)
p ) gp /Q gp | = p s for some s ≥ 0. Thus the assertion follows from (3).
By assuming saturatedness, one finds the exactness of ι
(i) c : Q (i) c ֒→ Q (i+1) c .
Lemma 2.13. Let Q be an integral sharp saturated monoid. Then for every c > 0 and every i ≥ 0, the following assertions hold.
(1) Q (i)
c is integral, sharp, and saturated. . Let R be a ring and let Q be a monoid with a homomorphism α : Q → R of monoids, where the monoid structure of R is multiplicative. Then we say that the triple (R, Q, α) is a log ring. Moreover, we say that (R, Q, α) is a local log ring if (R, Q, α) is a log ring, where R is a local ring and α −1 (R × ) = Q * .
(2) ι (i) c : Q (i) c ֒→ Q (i+1) c is exact (i.e. Q (i+1) c ∩ (Q (i) c ) gp = Q (i) c ).
Unless otherwise stated, we always assume that the monoid structure of a commutative ring is specified by the multiplicative structure.
In order to preserve the locality of a log structure, we need the locality of a ring map.
Lemma 2.16. Let (R, Q, α) be a local log ring and let (S, m S ) be a local ring with a local ring map φ : R → S. Then (S, Q, φ • α) is also a local log ring.
Proof. By the locality of φ, we obtain the equality (φ • α) −1 (S × ) = Q * , as desired.
Now we define log-regular rings according to [41].
Definition 2.17. Let (R, Q, α) be a local log ring, where R is Noetherian and Q := Q/Q * is fine and saturated. Let I α be the ideal of R generated by the set α(Q + ). Then (R, Q, α) is called a log-regular ring if the following conditions hold.
(1) R/I α is a regular local ring.
(2) dim R = dim R/I α + dim Q.
Remark 2.18. Note that for a monoid Q such that Q is fine and saturated, the natural projection π : Q ։ Q splits (see [23,Lemma 6.2.10]). Thus, in the situation of Definition 2.17, α extends to the homomorphism of monoids α : Q → R along π. Namely, we obtain another local log-regular ring (R, Q, α) with the same underlying ring, where Q is fine, sharp, and saturated.
In his monumental paper [31], Kato considered log structures of schemes on theétale sites, and he then considered them on the Zariski sites [32]. However, we do not need any deep part of logarithmic geometry and the present paper focuses on the local study of schemes with log structures. We should remark that if k is any fixed field and Q ⊆ N d is a fine and saturated monoid, then the monoid algebra k[Q] is known as an affine normal semigroup ring which is actively studied in the research of combinatorial commutative algebra (see the book [35]). The following theorem is a natural extension of the Cohen-Macaulay property for the classical toric singularities over a field proved by Hochster [28].
Theorem 2.19 (Kato). Every local log-regular ring is Cohen-Macaulay and normal.
Let R be a ring and let Q be a fine sharp monoid. We denote by R[Q + ] the ideal of R[Q] generated by elements q∈Q + a q e q , where a q is an element of R. Then we denote by R Q the adic completion of R[Q] with respect to the ideal R[Q + ].
As to the structure of complete local log-regular rings, we have the following result analogous to the classical Cohen's structure theorem, originally proved in [32]. We borrow the presentation from [41, Chapter III, Theorem 1.11.2].
Theorem 2.20 (Kato). Let (R, Q, α) be a local log ring such that R is Noetherian and Q is fine, sharp, and saturated. Let k be the residue field of R and m R its maximal ideal. Let r be the dimension of R/I α . Then the following assertions hold.
(1) Suppose that R contains a field. Then (R, Q, α) is log-regular if and only if there exists a commutative diagram:
Q − −−− → k Q ⊕ N r α ψ R − −−− → R
where R is the completion along the maximal ideal and ψ is an isomorphism of rings.
(2) Assume that R is of mixed characteristic p > 0. Let C(k) be a Cohen ring of k. Then (R, Q, α) is log-regular if and only if there exists a commutative diagram:
Q − −−− → C(k) Q ⊕ N r α ψ R − −−− → R
where R is the completion along the maximal ideal and ψ is a surjective ring map with Ker(ψ) = (θ) for some element θ ∈ m R whose constant term is p. Moreover, for any element θ ′ ∈ Ker(ψ) whose constant term is p, Ker(ψ) = (θ ′ ) holds.
Proof. The assertion (1) and the first part of (2) are [41, Chapter III, Theorem 1.11.2]. Pick an element θ ′ ∈ Ker(ψ) whose constant term is p. Note that θ ′ is a regular element that is not invertible. By [41, Chapter III, Proposition 1.10
.13], C(k) Q ⊕ N r /(θ ′ ) is a domain of dim Q + r = dim R = dim R. Thus Ker(ψ) = (θ ′ ) holds. 5
The completion of a normal affine semigroup ring with respect to the ideal generated by elements of the semigroup is a typical example of local log-regular rings:
Lemma 2.21. Let Q be a fine, sharp and saturated monoid and let k be a field. Then (k Q , Q, ι) is a local log-regular ring, where ι : Q ֒→ k Q is the natural injection.
Proof. By [41, Chapter I, Proposition 3.6.1], (k Q , Q, ι) is a local log ring. Now applying Theorem 2.20, it is a local log-regular ring.
2.2.2.
Log-regularity and strong F -regularity. Strongly F -regular rings are one of the important classes appearing in the study of F -singularities. Let us recall the definition. Definition 2.22 (Strong F -regularity). Let R be a Noetherian reduced F p -algebra that is Ffinite. Let F e * R be the same as R as its underlying abelian groups with its R-module structure via restrictions of scalars via the e-th iterated Frobenius endomorphism F e R on R. Then we say that R is strongly F -regular, if for any element c ∈ R that is not in any minimal prime of R, there exist an e > 0 and a map φ ∈ Hom R (F e * R, R) such that φ(F e * c) = 1. It is known that strongly F -regular rings are excellent, normal, and Cohen-Macaulay. Let us show that log-regularity implies strong F -regurality (in positive characteristic cases).
Lemma 2.23. Let (R, Q, α) be a local log-regular ring of characteristic p > 0 such that R is F -finite. Then R is strongly F -regular.
Proof. The completion of R with respect to its maximal ideal is isomorphic to the completion of k[Q ⊕ N r ], and Q is a fine, sharp and saturated monoid by Theorem Under the hypothesis in the following proposition, one can easily establish the finiteness of the torsion part of the divisor class group, which is the first assertion of Theorem 4.13.
Proposition 2.24. Assume that R ∼ = C(k) Q , where C(k) is a Cohen ring with F -finite residue field k and Q is a fine, sharp, and saturated monoid. Let Cl(R) tor be the torsion subgroup of Cl(R). Then Cl(R) tor ⊗ Z (ℓ) is finite for all ℓ = p, and vanishes for almost all ℓ = p.
Proof. Since R ∼ = C(k) Q , we have R/pR ∼ = k Q , which is a local F -finite log-regular ring. There is an induced map Cl(R) → Cl(R/pR). By restriction, we have Cl(R) tor → Cl(R/pR) tor . Then Lemma 2.23 together with Polstra's result [42] says that Cl(R/pR) tor is finite. Let C ℓ be the maximal ℓ-subgroup of Cl(R) tor . Since ℓ = p, we find that the map Cl(R) tor → Cl(R/pR) tor restricted to C ℓ is injective in view of [24,Theorem 1.2]. In conclusion, C ℓ is finite for all ℓ = p, and C ℓ vanishes for almost all ℓ = p, as desired. [38] for the study of purity of ring extensions.
In order to prove the splinter property, we need a lemma on splitting a map under completion. Proof. First, we prove the theorem when R is complete. By Remark 2.18, we may assume that Q is fine, sharp, and saturated. By Theorem 2.20, we have
R ∼ = k Q ⊕ N r , or R ∼ = C(k) Q ⊕ N r /(p − f ),
depending on whether R contains a field or not. Let us consider the mixed characteristic case. By Lemma 2.9, there is a split injection C(k)[Q ⊕ N r ] → C(k)[N d ] for some d > 0, which comes from an injection δ : Q ⊕ N r → N d that realizes δ(Q ⊕ N r ) as an exact submonoid of N d . After dividing out by the ideal (p − f ), we find that the map
C(k) Q ⊕ N r /(p − f ) → C(k) N d /(p − f )
splits as a C(k) Q ⊕ N r /(p − f )-linear map by Remark 2.8 and Lemma 2.26. Hence, R becomes a direct summand of the complete regular local ring A := C(k) x 1 , . . . , x d /(p − f ). Pick a map α : A → R that splits R → A. Consider a module-finite extension R → S such that S is a domain. We want to show that this map splits. Now there is a commutative diagram:
R + − −−− → A + S γ − −−− → B R − −−− → A
where R + (resp. A + ) is the absolute integral closure of R (resp. A), and B is a subring of A + that is constructed as the chain of S and A, thus being finite over A. By the Direct Summand Theorem
[2], there is a map β : B → A that splits A → B. Therefore, the composite map S γ − → B β − → A α − → R splits R → S, as desired. The case where R = k Q ⊕ N r can be treated similarly.
Next let us consider the general case. Let R → S be a module-finite extension with S being a domain, and let R be as in Theorem 2.20. By applying the functor ( ) ⊗ R R to the exact sequence 0 → R → S → S/R → 0, we get an exact sequence: 0 → R → S ⊗ R R → S/R ⊗ R R → 0. We have proved that R is a splinter, so the induced sequence
0 → Hom R (S/R ⊗ R R, R) → Hom R (S ⊗ R R, R) → Hom R ( R, R) → 0
is exact. By the faithful flatness of R over R, the above exact sequence induces the exact sequence: 0 → Hom R (S/R, R) → Hom R (S, R) → Hom R (R, R) → 0, and we conclude.
Perfectoid towers and small tilts
In this section, we establish a tower-theoretic framework to deal with perfectoid objects using the notion of perfectoid towers. We first introduce the class of perfect towers (Definition 3.2) in §3.1, and then define inverse perfection of towers (Definition 3.9) in §3.2. These notions are tower-theoretic variants of perfect F p -algebras and inverse perfection of rings, respectively. In §3.3, we give a set of axioms for perfectoid towers. In §3.4, we adopt the process of inverse perfection for perfectoid towers as a new tilting operation. Indeed, we verify the invariance of several good properties under the tilting; Main Theorem 1 is discussed here. In §3.5, we describe the relationship between perfectoid towers and perfectoid rings. This subsection also includes an alternative characterization of perfectoid rings without A inf . In §3.6, we calculate the tilts of perfectoid towers consisting of local log-regular rings.
3.1. Perfect towers. First of all, we consider the category of towers of rings. (1) A tower of rings is a direct system of rings of the form
R 0 t 0 / / R 1 t 1 / / R 2 t 2 / / · · · t i−1 / / R i t i / / · · · , and we denote it by ({R i } i≥0 , {t i } i≥0 ) or {R 0 t 0 − → R 1 t 1 − → · · · }. (2) A morphism of towers of rings f : ({R i } i≥0 , {t i } i≥0 ) → ({R ′ i } i≥0 , {t ′ i } i≥0 ) is defined as a collection of ring maps {f i : R i → R ′ i } i≥0
that is compatible with the transition maps; in other words, f represents the commutative diagram
R 0 / / f 0 R 1 / / f 1 R 2 / / f 2 · · · / / R i / / f i · · · R ′ 0 / / R ′ 1 / / R ′ 2 / / · · · / / R ′ i / / · · · .
For a tower of rings
({R i } i≥0 , {t i } i≥0 ), we often denote by R ∞ an inductive limit lim − →i≥0 R i . Clearly, an isomorphism of towers of rings ({R i } i≥0 , {t i } i≥0 ) → ({R ′ i } i≥0 , {t ′ i } i≥0 ) induces the iso- morphism of rings R ∞ ∼ = − → R ′ ∞ . For every i ≥ 0, we regard R i+1 as an R i -algebra via the transition map t i .
Let us define perfect towers. This type of tower naturally appears when one considers the perfect closure of a reduced F p -algebra.
Definition 3.2 (Perfect towers).
A perfect F p -tower (or, perfect tower as an abbreviated form) is a tower that is isomorphic to a tower (
{R 1/p i } i≥0 , {ι i } i≥0 ) of the following form.
• There exists a reduced F p -algebra R such that
R 1/p i := lim − → {R F R − − → R F R − − → · · · F R − − → i Frobenius endomorphisms R}
for every i ≥ 0. • For every i ≥ 0, the transition map ι i : R 1/p i → R 1/p i+1 is the map between direct limits induced by the commutative diagram:
(3.1) R id R F R / / R id R F R / / · · · F R / / R id R F R ❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ R F R / / R F R / / · · · F R / / R F R / / R.
Remark 3.3. R 1/p i is isomorphic to the ring of p i -th roots of elements of R. Indeed, let R 1/p j be the ring of p j -th roots of elements of R for every j ≥ 0. 7 Then we have the isomorphism
F j : R 1/p j+1 → R 1/p j ; x → x p . By putting F 0,j+1 := F 0 • · · · • F j ,
we obtain the following commutative ladder: 7 For more details of the ring of p-th roots of elements of a reduced ring, we refer to [33] where the top horizontal arrows are the natural inclusions. Since all the vertical arrows are isomorphisms, we obtain the isomorphism
R / / id R R 1/p / / F 0,1 · · · / / R 1/p i−1 / / F 0,i−1 R 1/p i F 0,i R F R / / R F R / / · · · F R / / R F R / / R,R 1/p i ∼ = R 1/p i .
To study perfect towers, we first investigate the ones of the form (
{R 1/p i } i≥0 , {ι i } i≥0 ). Lemma 3.4. Let ({R 1/p i } i≥0 , {ι i } i≥0 )
be the perfect tower defined in Definition 3.2. Let us define F i : R 1/p i+1 → R 1/p i as the ring map between direct limits induced by the commutative diagram:
(3.2) R F R F R / / R F R F R / / · · · F R / / R F R / / F R R id R6 ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ R F R / / R F R / / · · · F R / / R.
Then, for any i ≥ 0, the following assertions hold.
(1) F i is an isomorphism. (2) F i • ι i = F R 1/p i . (3) ι i • F i = F R 1/p i+1 . Proof. (1): Since the rightest (diagonal) arrow in (3.2) is the identity map, F i is an isomorphism.
(2) and (3): According to the diagrams (3.1) and
(3.2), F i • ι i (resp. ι i • F i ) is none other than the Frobenius endomorphism on R 1/p i (resp. R 1/p i+1 ).
Let us describe the relationship between perfect towers and direct perfection.
Corollary 3.5. Let R be a reduced F p -algebra. Then the direct system {R F R − − → R F R − − → · · · } (
whose direct limit is the direct perfection) is a perfect tower. In particular, the direct limit of a perfect tower
({R i } i≥0 , {t i } i≥0 ) is isomorphic to the direct perfection of R 0 . Proof. Put F 0,i := F 0 • · · · • F i , where F 0,0 = id R . Since F i • ι i = F R 1/p i by Lemma 3.4 (2), we obtain the morphism of towers {F 0,i } i≥0 : {R ι 0 − → R 1/p ι 1 − → · · · } → {R F R − − → R F R − − → · · · }. Moreover, F 0,i is an isomorphism by Lemma 3.4 (1). Hence {F 0,i } i≥0 is an isomorphism. 3.2.
Purely inseparable towers and inverse perfection. In this subsection, we define inverse perfection for towers, which assigns a perfect tower to a tower by arranging a certain type of inverse limits of rings. For this, we introduce the following class of towers that admit distinguished inverse systems of rings. Definition 3.6 (Purely inseparable towers). Let R be a ring, and let I ⊆ R be an ideal.
(1) A tower ({R i } i≥0 , {t i } i≥0 ) is called a p-purely inseparable tower arising from (R, I) if it satisfies the following axioms. (a) R 0 = R and p ∈ I. (b) For any i ≥ 0, the ring map t i : R i /IR i → R i+1 /IR i+1 induced by t i is injective. (c) For any i ≥ 0, the image of the Frobenius endomorphism on R i+1 /IR i+1 is contained in the image of t i : R i /IR i → R i+1 /IR i+1 . (2) Let ({R i } i≥0 , {t i } i≥0
) be a p-purely inseparable tower arising from (R, I). For any i ≥ 0, we denote by F i : R i+1 /IR i+1 → R i /IR i the ring map (which uniquely exists by the axioms (b) and (c)) such that the following diagram commutes:
(3.3) R i+1 /IR i+1 F i ) ) ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ F R i+1 /IR i+1 / / R i+1 /IR i+1 R i /IR i . t i O O We call F i the i-th Frobenius projection (of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I)).
Hereafter, we leave out 'p-' from 'p-purely inseparable towers' if no confusion occurs (i.e. we call them simply 'purely inseparable towers'). Similarly, we omit the phrase in parentheses subsequent to 'the i-th Frobenius projection' (but we should be careful in some situations; cf. Remark 3.40).
Example 3.7. Any perfect tower is a purely inseparable tower. More precisely,
({R 1/p i } i≥0 , {ι i } i≥0 )
appearing in Definition 3.2 is a purely inseparable tower arising from (R, (0)). Indeed, the axioms (a) and (b) are obvious. Moreover, for any i ≥ 0, the ring map (3) of the lemma. Hence the axiom (c) is also satisfied, and F i is the i-th Frobenius projection.
F i : R 1/p i+1 → R 1/p i defined in Lemma 3.4 satisfies ι i • F i = F R 1/p i+1 by the assertion
To develop the theory of perfectoid towers, we often use a combination of the diagram (3.3) in Definition 3.6 and the diagram (3.4) in the following lemma.
Lemma 3.8. Let ({R i } i≥0 , {t i } i≥0
) be a purely inseparable tower arising from some pair (R, I).
Then for every i ≥ 0, the following assertions hold.
(
1) Ker(F i ) = Ker(F R i+1 /IR i+1 ).
(2) Any element of R i+1 /IR i+1 is a root of a polynomial of the form X p −t i (a) with a ∈ R i /IR i . In particular, the ring map t i :
R i /IR i ֒→ R i+1 /IR i+1 is integral. (3)
The following diagram commutes:
(3.4) R i+1 /IR i+1 F i ) ) ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ R i /IR i t i O O F R i /IR i / / R i /IR i .
Proof. Since t i is injective, the commutative diagram (3.3) yields the assertion (1). Moreover, (3.3) also yields the equality x p −t i (F i (x)) = 0 for every x ∈ R i+1 /IR i+1 . Hence the assertion (2) follows. To prove (3), let us recall the following equalities
t i • F R i /IR i = F R i+1 /IR i+1 • t i = t i • F i • t i ,
where the second one follows from the commutative diagram (3.3). Since t i is injective, we obtain the equality
F R i /IR i = F i • t i , as desired.
Now we can introduce the notion of inverse perfection for towers.
Definition 3.9 (Inverse perfection of towers). Let ({R i } i≥0 , {t i } i≥0
) be a (p-)purely inseparable tower arising from some pair (R, I).
(1) For any j ≥ 0, we define the j-th inverse quasi-perfection of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I) as a limit:
(R j ) q.frep I := lim ← − {· · · → (R j+i+1 /IR j+i+1 ) F j+i − −− → (R j+i /IR j+i ) → · · · F j − → (R j /IR j }.
(2) For any j ≥ 0, we define an injective ring map (t j ) q.frep
I : (R j ) q.frep I ֒→ (R j+1 ) q.frep I by the rule: (t j ) q.frep I ((a i ) i≥0 ) := (t j+i (a i )) i≥0 .
Moreover, we call the resulting tower (
{(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0 ) the inverse perfection of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I). (3) For any j ≥ 0, we define a ring map (F j ) q.frep I : (R j+1 ) q.frep I → (R j ) q.frep I by the rule: (3.5) (F j ) q.frep I ((a i ) i≥0 ) := (F j+i (a i )) i≥0 .
(4) For any j ≥ 0 and for any m ≥ 0, we denote by Φ (j) m the m-th projection map:
(R j ) q.frep I → (R j+m /IR j+m ) ; (a i ) i≥0 → a m .
If no confusion occurs, we also denote by R q.frep
j (resp. t q.frep j , resp. F q.frep j ) the symbol (R j ) q.frep I (resp. (t j ) q.frep I , resp. (F j ) q.frep I
) as an abbreviated form.
Example 3.10. Let R be an F p -algebra. Set R i := R and t i := id R for every i ≥ 0. Then the tower ({R i } i≥0 , {t i } i≥0 )
is a purely inseparable tower arising from (R, (0)). Moreover, for every j ≥ 0, the attached j-th inverse quasi-perfection is a limit
R q.frep j = lim ← − {· · · F R − − → R F R − − → R F R − − → R},
which is none other than the inverse perfection of R.
In the situation of Definition 3.9, we have the commutative diagram:
(3.6) (R j+1 ) q.frep I (F j ) q.frep I ) ) | | | | | | | | | | | | | | F (R j+1 ) q.frep I / / (R j+1 ) q.frep I (R j ) q.frep I . (t j ) q.frep I O O Hence the tower ({(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0 )
is also a purely inseparable tower associated to ((R 0 ) q.frep I , (0)). Some basic properties of inverse quasi-perfection are contained in the following proposition.
Proposition 3.11. Let ({R i } i≥0 , {t i } i≥0
) be a purely inseparable tower arising from some pair (R, I). Then for any j ≥ 0, the following assertions hold.
(1) Let J ⊆ (R j ) q.frep I be a finitely generated ideal such that J k ⊆ Ker(Φ (j) 0 ) for some k > 0 (see Definition 3.9 (4) for Φ (j) 0 ). Then (R j ) q.frep I is J-adically complete and separated. (2) Let x = (x i ) i≥0 be an element of (R j ) q.frep I . Then x is a unit if and only if x 0 ∈ R j /IR j is a unit. (3) The ring map (F j ) q.frep I (see Definition 3.9 (3)) is an isomorphism. (4) (R j ) q.frep I is reduced. Moreover, the tower ({(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0 ) (see Definition 3.9(2)) is a perfect tower. Proof. Since ({(R j+i ) q.frep } i≥0 , {(t j+i ) q.frep I } i≥0 ) is the inverse perfection of ({R j+i } i≥0 , {t j+i } i≥0 ),
we are reduced to showing the assertions in the case when j = 0.
(1): By definition, (R 0 ) q.frep I is complete and separated with respect to the linear topology induced by the descending filtration
Ker(Φ (0) 0 ) ⊇ Ker(Φ (0) 1 ) ⊇ Ker(Φ (0) 2 ) ⊇ · · · . Moreover, since J k ⊆ Ker(Φ (0) 0 ), we have (J k ) [p i ] ⊆ Ker(Φ (0) i ) for every i ≥ 0 by the commutative diagram (3.3). 8 On the other hand, since J k is finitely generated, (J k ) p i r ⊆ (J k ) [p i ] for some r > 0.
Thus the assertion follows from [19, Lemma 2.1.1].
(2): It is obvious that
x 0 ∈ R 0 /IR 0 is a unit if x ∈ (R 0 ) q.frep I
is a unit. Conversely, assume that
x 0 ∈ R 0 /IR 0 is a unit. Then for every i ≥ 0, x p i i is a unit because it is the image of x 0 in R i /IR i . Hence x i is also a unit. Therefore, we have isomorphisms R i /IR i ×x i − − → R i /IR i (i ≥ 0) that(R 0 ) q.frep I → (R 1 ) q.frep I defined by the rule s 0 ((a i ) i≥0 ) := (a i+1 ) i≥0 . Then one can easily check that s 0 is the inverse map of (F 0 ) q.frep I . (4): Since (t 0 ) q.frep I is injective and (F 0 ) q.frep I • (t 0 ) q.frep I = F (R 0 ) q.frep I , the assertion follows from (3). Finally, let us show that ({(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0 ) is a perfect tower. Define F q.frep 0,i : (R i ) q.frep I → (R 0 ) q.frep I as the composite map (F 0 ) q.frep I • · · · • (F i−1 ) q.frep I (if i ≥ 1) or the iden- tity map (if i = 0). Then by (3), the collection {F q.frep 0,i } i≥0 gives an isomorphism of towers from ({(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0 ) to (R 0 ) q.frep I F (R 0 ) q.frep I −−−−−−→ (R 0 ) q.frep I F (R 0 ) q.frep I −−−−−−→ · · · .
Hence by Corollary 3.5, the assertion follows.
The operation of inverse quasi-perfection preserves the locality of rings and ring maps.
Lemma 3.12. Let ({R i } i≥0 , {t i } i≥0
) be a purely inseparable tower of local rings arising from some pair (R, I). Assume that I = R 0 . Then for any j ≥ 0, the following assertions hold.
(1) The ring maps t j , t j , and F j are local.
(2) (R j ) q.frep I is a local ring. (3) The ring map (t j ) q.frep I : (R j ) q.frep I → (R j+1 ) q.frep I is local.
Proof. Like Proposition 3.11, it suffices to show the assertions in the case when j = 0.
(1): Since the diagrams (3.3) and (3.4) are commutative, F 0 • t 0 and t 0 • F 0 are local. Hence t 0 and F 0 are local. In particular, the composition
R 0 ։ R 0 /I t 0 − → R 1 /IR 1 is local.
Since this map factors through t 0 , t 0 is also local, as desired.
(2): Let m 0 be the maximal ideal of R 0 . Consider the ideal (m 0 ) q.frep
I = {(x i ) i≥0 ∈ (R 0 ) q.frep I | x 0 ∈ m 0 /IR 0 }, where m 0 /IR 0 is the maximal ideal of R 0 /IR 0 . Then by Proposition 3.11 (2), (m 0 ) q.frep I is a unique maximal ideal of (R 0 ) q.frep I .
Hence the assertion follows.
R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0
) is a purely inseparable tower of local rings. Hence by the assertion (1), (t 0 ) q.frep I is local.
A purely inseparable tower also satisfies the following amusing property. This is well-known in positive characteristic, in which case R i → R i+1 is a universal homeomorphism (see also Proposition 3.47).
Lemma 3.13. Let ({R i } i≥0 , {t i } i≥0
) be a purely inseparable tower arising from some pair (R, I).
For every i ≥ 0, assume that R i is I-adically Henselian. 9 Then the ring map t i induces an equivalence of categories:
F.Ét(R i ) ∼ = − → F.Ét(R i+1 ),
where F.Ét(A) is the category of finiteétale A-algebras for a ring A.
Proof. Consider the commutative diagram
(3.7) R i ϕ i t i / / R i+1 ϕ i+1 R i /IR i φ i t i / / R i+1 /IR i+1 φ i+1 R i / √ IR i (t i ) red / / R i+1 / √ IR i+1 where ϕ i , ϕ i+1 , φ i and φ i+1 areIR i = R i ∩ IR i+1 . Hence √ IR i = R i ∩ √ IR i+1 and (t i ) red is injective. Moreover by Lemma 3.8 (2), the image of (t i ) red contains {x p | x ∈ R i+1 / √ IR i+1 }. So [47, Tag 0BRA] shows that (t i ) red is a universal
homeomorphism. Finally, as for ϕ i and ϕ i+1 , these maps induce an equivalence of categories of finiteétale algebras over respective rings by [47,Tag 09ZL]. By going around the diagram (3.7), we finish the proof.
Axioms for perfectoid towers.
3.3.1. Remarks on torsionness. In the subsequent §3.3.2, we introduce the class of perfectoid towers as a generalization of perfect towers. For this purpose, we need to deal with a purely inseparable tower arising from (R, I) in the case when I = (0) at least, and hence plenty of I-torsion elements. Thus we begin with giving several preliminary lemmas on torsionness of modules over rings.
Definition 3.14. Let R be a ring, and let M be an R-module.
(1) Let x ∈ R be an element. We say that an element m ∈ M is x-torsion if x n m = 0 for some n > 0. We denote by M x-tor the R-submodule of M consisting of all x-torsion elements in M .
(2) Let I ⊆ R be an ideal. We say that an element m ∈ M is I-torsion if m is x-torsion for every x ∈ I. We denote by M I-tor the R-submodule of M consisting of all I-torsion elements in M . Note that M x-tor = M (x)-tor . (3)
For an element x ∈ R (resp. an ideal I ⊆ R), we say that M has bounded x-torsion (resp.
bounded I-torsion) if there exists some l > 0 such that x l M x-tor = (0) (I l M I-tor = (0)). First we record the following fundamental lemma.
Lemma 3.15. Let R be a ring, and let M be an R-module. Let x ∈ R be an element. Then for every n > 0, we have
M x-tor ∩ x n M = x n M x-tor .
Proof. Pick an element m ∈ M x-tor ∩ x n M . Then m = x n m 0 for some m 0 ∈ M , and x l m = 0 for some l > 0. Hence x l+n m 0 = 0, which implies that m 0 ∈ M x-tor and thus m ∈ x n M x-tor . The containment Proof. By assumption, there exists some l > 0 such that x l M x-tor = (0). On the other hand,
x n M x-tor ⊆ M x-tor ∩ x n M is clear.Ker(ψ tor ) = M x-tor ∩ ∞ n=0 x n M is contained in M x-tor ∩ x l M , which is equal to x l M x-tor by Lemma 3.15.
Thus the assertion follows.
The following lemma is used for proving Main Theorem 1 (cf. Lemma 3.50).
Lemma 3.18. Let R be a ring, and let M be an R-module. Let x ∈ R be an element. Then for every n > 0, we have
(3.9) Ann M/x n M (x) ⊆ Im(ϕ (x n ),M ) + x n−1 (M/x n M ). Proof. Pick an element m ∈ M such that xm ∈ x n M . Then x(m − x n−1 m ′ ) = 0 for some m ′ ∈ M .
In particular, m − x n−1 m ′ ∈ M x n -tor . Hence m mod x n M lies in the right-hand side of (3.9), as desired.
In the case when M = R, we can regard M I-tor as a (possibly) non-unital subring of R. This point of view provides valuable insights. For example, "reducedness" for R I-tor induces a good property on boundedness of torsions. Proof. It suffices to show that xR I-tor = 0 for every x ∈ I. Pick an element a ∈ R I-tor . Then for a sufficiently large n > 0, x n a = 0. Hence (xa) n = x n a · a n−1 = 0. Thus we have xa = 0 by assumption, as desired. Furthermore, we can treat R I-tor as a positive characteristic object (in the situation of our interest), even if R is not an F p -algebra.
Lemma-Definition 3.21. Let (R, I) be a pair such that p ∈ I and IR I-tor = (0). Then the multiplicative map:
(3.10)
R I-tor → R I-tor ;
x → x p is also additive. We denote by F R I-tor the map (3.10).
Proof. It immediately follows from the binomial theorem.
3.3.2.
Perfectoid towers and pillars. Now, we define perfectoid towers.
Definition 3.22. (Perfectoid towers) Let R be a ring, and let I 0 ⊆ R be an ideal. A tower
({R i } i≥0 , {t i } i≥0 )
is called a (p-)perfectoid tower arising from (R, I 0 ) if it is a p-purely inseparable tower arising from (R, I 0 ) (cf. Definition 3.6 (1)) and satisfies the following additional axioms.
(d) For every i ≥ 0, the i-th Frobenius projection (2)) is surjective. (e) For every i ≥ 0, R i is an I 0 -adically Zariskian ring. (f) I 0 is a principal ideal, and R 1 contains a principal ideal I 1 that satisfies the following axioms.
F i : R i+1 /I 0 R i+1 → R i /I 0 R i (cf. Definition 3.6
(f-1)
I p 1 = I 0 R 1 . (f-2) For every i ≥ 0, Ker(F i ) = I 1 (R i+1 /I 0 R i+1 ). (g) For every i ≥ 0, I 0 (R i ) I 0 -tor = (0). Moreover, there exists a (unique) bijective map (F i ) tor : (R i+1 ) I 0 -tor → (R i ) I 0 -tor such that the diagram: (R i+1 ) I 0 -tor (F i )tor ϕ I 0 ,R i+1 / / R i+1 /I 0 R i+1 F i (R i ) I 0 -tor ϕ I 0 ,R i / / R i /I 0 R i
commutes (see Definition 3.14 for the notation; see also Corollary 3.16).
Remark 3.23. If I 0 is generated by a regular element in R ∞ := lim − →i≥0 R i , then the axiom (g) is satisfied automatically. If I 0 = (0), then the axiom (g) follows from the axioms (c) and (f). Consequently, the axiom (g) is satisfied if R ∞ is a domain.
We have some examples of perfectoid towers. (1) (cf. [46,Definition 4.4]) Let (R, m, k) be a d-dimensional unramified regular local ring of mixed characteristic p > 0 whose residue field is perfect. Then we have
R ∼ = W (k) x 2 , . . . , x d . For every i ≥ 0, set R i := R[p 1/p i , x 1/p i 2 , . . . , x 1/p i d ]
, and let t i : R i → R i+1 be the inclusion map. Then the tower ({R i } i≥0 , {t i } i≥0 ) is a perfectoid tower arising from (R, (p)). Indeed, the Frobenius projection F i : R i+1 /pR i+1 → R i /pR i is given as the p-th power map. 10 (2) For some generalization of (1), one can build a perfectoid tower arising from a complete local log-regular ring. For details, see §3.6. (3) We note that t i (resp. F i ) of a perfectoid tower is not necessarily the inclusion map (resp. the p-th power map). For instance, let R be a reduced F p -algebra. Set R i := R, t i := F R , and
F i := id R for every i ≥ 0. Then ({R i } i≥0 , {t i } i≥0 )
is a perfectoid tower arising from (R, (0)).
The class of perfectoid towers is a generalization of perfect towers.
Lemma 3.25. Let ({R i } i≥0 , {t i } i≥0
) be a tower of F p -algebras. Then the following conditions are equivalent.
(1) ({R i } i≥0 , {t i } i≥0 ) is a perfect F p -tower (cf. Definition 3.2). (2) ({R i } i≥0 , {t i } i≥0 )
is a p-perfectoid tower arising from (R 0 , (0)).
Proof. First we verify the implication (1) ⇒ (2). For this, we may assume that (
{R i } i≥0 , {t i } i≥0 ) is of the form ({R 1/p i } i≥0 , {ι i } i≥0 ) ({R i } i≥0 , {t i } i≥0 )
is a perfectoid tower arising from (R 0 , (0)). By the axioms (d) and (f-2) in Definition 3.22 and Lemma 3.8 (1), F i is an isomorphism for any i ≥ 0. Moreover, we have the following commutative ladder:
R 0 id R 0 t 0 / / R 1 F 0 t 1 / / R 2 F 0 •F 1 t 2 / / R 3 F 0 •F 1 •F 2 t 3 / / · · · R 0 F R 0 / / R 0 F R 0 / / R 0 F R 0 / / R 0 F R 0 / / · · · . Hence ({R i } i≥0 , {t i } i≥0 ) is isomorphic to {R 0 F R 0 −−→ R 0 F R 0 −−→ · · · },
which is a perfect tower by Corollary 3.5. Thus the implication (2) ⇒ (1) holds.
Let us verify the uniqueness of I 1 ⊆ R 1 appearing in the axiom (f). We carry out this in more general situations for later application.
Lemma 3.26. Let ({R i } i≥0 , {t i } i≥0 ) be a perfectoid tower arising from some pair (R 0 , I 0 ). Fix an integer i ≥ 0 and an ideal J i ⊆ R i containing I 0 R i . Suppose that R i+1 contains an ideal J i+1 such that J p i+1 = J i R i+1 and F −1 i (J i (R i /I 0 R i )) = J i+1 (R i+1 /I 0 R i+1 ). Then J i+1 is unique. Proof.
The assumption on J i+1 induces the following implication for every r ∈ R i+1 :
r p ∈ J i R i+1 ⇒ r mod I 0 R i+1 ∈ J i+1 (R i+1 /I 0 R i+1 ) ⇒ r ∈ J i+1 + I 0 R i+1 ⇒ r ∈ J i+1 .
Hence for every ideal J ′ i+1 ⊆ R i+1 with the same assumption (that is,
(J ′ i+1 ) p = J i R i+1 and F −1 i (J i (R i /I 0 R i )) = J ′ i+1 (R i+1 /I 0 R i+1 )), we have J ′ i+1 ⊆ J i+1 and J i+1 ⊆ J ′ i+1
. Therefore, the assertion follows.
Definition 3.27. We call I 1 the first perfectoid pillar of ({R i } i≥0 , {t i } i≥0 ) arising from (R 0 , I 0 ).
The relationship between I 0 and I 1 can be observed also in higher layers (see Proposition 3.28 below). In the rest of this section, we fix a perfectoid tower ({R i } i≥0 , {t i } i≥0 ) arising from some pair (R 0 , I 0 ), and let I 1 denote the first perfectoid pillar.
Proposition 3.28. Set R i := R i /I 0 R i for every i ≥ 0. Then the following assertions hold.
(1) For a sequence of principal ideals {I i ⊆ R i } i≥2 , the following conditions are equivalent.
(a) F −1 i (I i R i ) = I i+1 R i+1 for every i ≥ 0. (b) F i (I i+1 R i+1 ) = I i R i for every i ≥ 0.
(2) Each one of the equivalent conditions in (1) implies that I p i+1 = I i R i+1 for every i ≥ 0. (3) There exists a unique sequence of principal ideals {I i ⊆ R i } i≥0 that satisfies one of the equivalent conditions in (1). Moreover, there exists a sequence of elements
{f i ∈ R i /I 0 R i } i≥0 such that I i R i = (f i ) and F i (f i+1 ) = f i for every i ≥ 0.
Proof.
i • F i = F R i+1 implies (3.11) I p i+1 R i+1 = I i R i+1 because I i+1 is principal. In particular, Ker(F i ) = I 1 R i+1 ⊆ I i+1 R i+1 (cf.
the axiom (f-2)). On the other hand, by the surjectivity of F i and the assumption again, we have
F i (F −1 i (I i R i )) = I i R i = F i (I i+1 R i+1 ). Hence F −1 i (I i R i ) ⊆ I i+1 R i+1 + Ker(F i ) ⊆ I i+1 R i+1 ⊆ F −1 i (I i R i )
, which yields the assertion.
(2): Let us deduce the assertion from (3.11) by induction. By definition, I p 1 = I 0 R 1 . We then fix some i ≥ 1. Suppose that for every 1 ≤ k ≤ i, I p k = I k−1 R k . Then, I 0 R i ⊆ I i . Hence by (3.11),
we have I p i+1 + I 0 R i+1 = I i R i+1 .
It yields the first one of the following isomorphisms: (3), we call I i the i-th perfectoid pillar of
I i R i+1 /I p i+1 ∼ = (I p i+1 + I 0 R i+1 )/I p i+1 ∼ = I p i+1 /(I p i+1 ∩ I 0 R i+1 ). Therefore, I 0 (I i R i+1 /I p i+1 ) = (0({R i } i≥0 , {t i } i≥0 ) arising from (R 0 , I 0 ).
The following property of perfectoid pillars is applied to prove our main result.
Lemma 3.30. Let {I i } i≥0 denote the system of perfectoid pillars of ({R i } i≥0 , {t i } i≥0 ), and let π i : R i /I 0 R i → R i /I i R i (i ≥ 0)
be the natural projections. Then for every i ≥ 0, there exists a unique isomorphism of rings:
F ′ i : R i+1 /I i+1 R i+1 ∼ = − → R i /I i R i such that π i • F i = F ′ i • π i+1 .
Proof. Since F i and π i are surjective, the assertion immediately follows from Ker(π
i • F i ) = F −1 i (I i (R i /I 0 R i )) = I i+1 (R i+1 /I 0 R i+1 ).
Tilts of perfectoid towers.
3.4.1. Invariance of some properties. Here we establish tilting operation for perfectoid towers. For this, we first introduce the notion of small tilt, which originates in [46]. (1) For any j ≥ 0, the j-th inverse quasi-perfection of (
{R i } i≥0 , {t i } i≥0 ) associated to (R, I 0 ) is called the j-th small tilt of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I 0 ) and is denoted by (R j ) s.♭ I 0 in distinction from (R j ) q.frep I 0 .
(2) Let the notation be as in Lemma 3.30. Then we define I s.♭ i := Ker(π i • Φ (i) 0 ) for every i ≥ 0. Note that the ideal I s.♭ i ⊆ R s.♭ i has the following property. Lemma 3.32. Keep the notation as in Definition 3.31. Then for every i ≥ 0 and j ≥ 0, we have Φ Proof. Since Φ (j) 0 is surjective, we have Φ (j) 0 (I s.♭ j ) = I j (R j /I 0 R j ). On the other hand, since Φ
(j) 0 = F j • Φ (j) 1 , we have F −1 j (Φ (j) 0 (I s.♭ j )) = Φ (j) 1 (I s.♭ j ) + Ker(F j ) = Φ (j) 1 (I s.♭ j ).
Hence by the condition (a) in Proposition 3.28 (1), Φ (j) 1 (I s.♭ j ) = I j+1 (R j+1 /I 0 R j+1 ). By repeating this procedure recursively, we obtain the assertion.
The next lemma provides some completeness of the small tilts attached to a perfectoid tower of characteristic p > 0 (see also Remark 3.35).
Lemma 3.33. Let ({R i } i≥0 , {t i } i≥0
) be a perfectoid tower arising from (R, (0)). Then, for any element f ∈ R and any j ≥ 0, the inverse limit lim
← − {· · · F j+1 − −− → R j+1 /f R j+1 F j − → R j /f R j } is isomorphic to the f -adic completion of R j .
Proof. It suffices to show the assertion when j = 0. Since ({R i } i≥0 , {t i } i≥0 ) is a perfectoid tower arising from (R, (0)), each Frobenius projection F i : R i+1 → R i is an isomorphism. In particular, the 0-th projection map on (R 0 ) s.♭ (0) is an isomorphism
(3.12) (R 0 ) s.♭ (0) = lim ← − {· · · → R 1 → R 0 } ∼ = − → R 0 .
Set the element f := (. . . ,
(F 0 • F 1 ) −1 (f ), F −1 0 (f ), f ) ∈ (R 0 ) s.♭ (0)
. Then for any i ≥ 0, we obtain the following diagram
(R 0 ) s.♭ (0) ×f p i+1 −−−−→ (R 0 ) s.♭ (0) φ i+1 − −−− → R i+1 /f R i+1 − −−− → 0 ×f p i (p−1) id R s.♭ 0 F i (R 0 ) s.♭ (0) ×f p i − −−− → (R 0 ) s.♭ (0) φ i − −−− → R i /f R i − −−− → 0
where φ i is the composite map of the i-th projection (R 0 ) s.♭ (0) → R i and the natural surjection R i → R i /f R i . Then taking the inverse limits for the above diagrams, we obtain the isomorphism
lim ← − i≥0 (R 0 ) s.♭ (0) /f p i (R 0 ) s.♭ (0) ∼ = − → lim ← − {· · · → R 1 /f R 1 → R 0 /f R 0 }.
On the other hand, (3.12) induces the isomorphism lim
← −i≥0 (R 0 ) s.♭ (0) /f p i (R 0 ) s.♭ (0) ∼ = − → lim ← −i≥0 R 0 /f p i R 0 .
Hence the assertion follows.
Example 3.34. Let S be a perfect F p -algebra. Pick an arbitrary f ∈ S, and let S denote the f -adic completion. Applying the argument of the above proof to the tower
S id S − − → S id S − − → S id S − − → · · · ,
we obtain a canonical isomorphism S
∼ = − → lim ← −Frob S/f S.
Remark 3.35. In Lemma 3.33, if we add the assumption that t i :
R i /f 0 R i → R i+1 /f 0 R i+1 is injective, then the tower ({R i } i≥0 , {t i } i≥0 )
becomes a purely inseparable tower arising from
(R, (f 0 )), and the inverse limit lim
← − {· · · F j+1 − −− → R j+1 /f 0 R j+1 F j − → R j /f 0 R j } is the j-th inverse quasi- perfection (R j ) q.frep I .
Now we define tilts of perfectoid towers. After discussing several basic properties of this tilting operation, we illustrate how to compute the tilts of perfectoid towers in some specific cases; when they consist of log-regular rings (see Theorem 3.63 and Example 3.64).
We should remark that all results on the perfection of purely inseparable towers (established in §3.2) can be applied to the tilts of perfectoid towers.
Let us state Main Theorem 1 in a more refined form. This is an important tool when one wants to see that a certain correspondence holds between Noetherian rings of mixed characteristic and those of positive characteristic. (1) For every j ≥ 0 and every element f s.♭ j ∈ R s.♭ j , the following conditions are equivalent.
(a) f s.♭ j is a generator of I s.♭ j . (b) For every i ≥ 0, Φ (j) i (f s.♭ j )
is a generator of I j+i (R j+i /I 0 R j+i ). In particular, I s.♭ j is a principal ideal, and (I s.♭ j+1 ) p = I s.♭ j R s.♭ j+1 .
(2) We have isomorphisms of (possibly) non-unital rings (R s.♭ j ) I s.♭ 0 -tor ∼ = (R j ) I 0 -tor that are compatible with {t j } j≥0 and {t s.♭ j } j≥0 . We give its proof in the subsequent §3.4.2. Before that, let us observe that this theorem induces many good properties of tilting. In the rest of this subsection, we keep the notation as in Theorem 3.37.
F i ) s.♭ I 0 : (R i+1 ) s.♭ I 0 /I s.♭ 0 (R i+1 ) s.♭ I 0 → (R i ) s.♭ I 0 /I s.♭ 0 (R i ) s.♭ I 0 by the rule: (F i ) s.♭ I 0 (α i+1 mod I s.♭ 0 (R i+1 ) s.♭ I 0 ) = (F i ) q.frep I 0 (α i+1 ) mod I s.♭ 0 (R i ) s.♭ I 0 where α i+1 ∈ (R i+1 ) s.♭ I 0 .
Remark 3.40. Although the symbols ( · ) s.♭ and ( · ) q.frep had been used interchangeably before Definition 3.39,
(F i ) s.♭ I 0 differs from (F i ) q.frep I 0 in general.
The following lemma is an immediate consequence of Theorem 3.37 (1), but quite useful.
0 : R s.♭ j /I s.♭ 0 R s.♭ j ∼ = − → R j /I 0 R j ; a mod I s.♭ 0 R s.♭ j → Φ (j) 0 (a).
Moreover, {Φ
(i) 0 } i≥0 is compatible with {t i } i≥0 (resp. {F R s.♭ i /I s.♭ 0 R s.♭ i } i≥0 , resp. {F s.♭ i } i≥0 ) and {t s.♭ i } i≥0 (resp. {F R i /I 0 R i } i≥0 , resp. {F i } i≥0 ).
Proof. By the axiom (d) in Definition 3.22, (3.13) is surjective. Let us check the injectivity. By Theorem 3.37 (1), I s.♭ 0 is generated by an element f s.♭ 0 ∈ R s.♭ 0 such that Φ
(0) i (f s.♭ 0 ) is a generator of I i (R i /I 0 R i ) (i ≥ 0). Note that ({R j+i } i≥0 , {t j+i } i≥0 )
is a perfectoid tower arising from (R j , I 0 R j ). Moreover, {I i R j+i } i≥0 is the system of perfectoid pillars associated to (R j , I 0 R j ) (cf. the condition (b) in Proposition 3.28 (1)). Put J 0 := I 0 R j . Then by Theorem 3.37 (1) again, we find that
J s.♭ 0 = f s.♭ 0 R s.♭ j = I s.♭ 0 R s.♭ j . Since J s.♭ 0 = Ker Φ (j)
0 , we obtain the first assertion. One can deduce that {Φ (i) 0 } i≥0 is compatible with the Frobenius projections from the commutativity of (3.3), because the other compatibility assertions immediately follow from the construction.
Remark 3.42. Theorem 3.37 (2) and Lemma 3.41 can be interpreted as a correspondence of homological invariants between R i and R s.♭ i by using Koszul homologies. Indeed, for any generator f 0 (resp. f s.♭ 0 ) of I 0 (resp. I s.♭ 0 ), the Koszul homology H q (f s.♭ 0 ; R s.♭ i ) is isomorphic to H q (f 0 ; R i ) for any q ≥ 0 as an abelian group. 11 Now we can show the invariance of several properties of perfectoid towers under tilting. The first one is perfectoidness, which is most important in our framework.
Proposition 3.43. ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) is a perfectoid tower arising from (R s.♭ 0 , I s.♭ 0 ). Proof. Set R i := R i /I 0 R i for every i ≥ 0. By Lemma 3.41, we find that ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 )Ker(F s.♭ i ) = (Φ (i+1) 0 ) −1 (Ker(F i )) = (Φ (i+1) 0 ) −1 (I 1 R i+1 ) = (Φ (1) 0 ) −1 (I 1 R 1 )R s.♭ i+1 = I s.♭ 1 R s.♭ i+1 .t s.♭ i •ϕ I s.♭ 0 ,R s.♭ i •(F s.♭ i ) tor = ϕ I s.♭ 0 ,R s.♭ i+1 •(t s.♭ i ) tor •(F s.♭ i ) tor = ϕ I s.♭ 0 ,R s.♭ i+1 •F (R s.♭ i+1 ) I s.♭ 0 -tor = t s.♭ i •F s.♭ i •ϕ I s.♭ 0 ,R s.♭ i+1 .
Hence the injectivity of t s.♭ i yields the assertion. Next we focus on finiteness properties. "Small" in the name of small tilts comes from the following fact.
Proposition 3.44. For every j ≥ 0, the following assertions hold.
(1) If t j : R j → R j+1 is module-finite, then so is t s.♭ j : R s.♭ j → R s.♭ j+1 . Moreover, the converse holds true when R j is I 0 -adically complete and separated.
(2) If R j is a Noetherian ring, then so is R s.♭ j . Moreover, the converse holds true when R j is I 0 -adically complete and separated.
(3) Assume that R j is a Noetherian local ring, and a generator of I 0 R j is regular. Then the dimension of R j is equal to that of R s.♭ j . 11 Note that (Ri)I 0 -tor = AnnR i (I0) by the axiom ( (3): By Theorem 3.37, I s.♭ 0 R s.♭ j is also generated by a regular element. Thus we obtain the equalities dim R j = dim R j /I 0 R j + 1 and dim R s.♭ j = dim R s.♭ j /I s.♭ 0 R s.♭ j + 1. By combining these equalities with Lemma 3.41, we deduce assertion. Proposition 3.44 (2) says that "Noetherianness" for a perfectoid tower is preserved under tilting.
Definition 3.45. We say that ({R i } i≥0 , {t i } i≥0 ) is Noetherian if R i is Noetherian for each i ≥ 0. Corollary 3.46. If ({R i } i≥0 , {t i } i≥0 ) is Noetherian, then so is the tilt ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ).
Moreover, the converse holds true when R i is I 0 -adically complete and separated for each i ≥ 0.
Proof. It immediately follows from Proposition 3.44 (2).
Finally, let us consider perfectoid towers of henselian rings. Then we obtain the equivalence of categories of finiteétale algebras over each layer.
Proposition 3.47. Assume that R i is I 0 -adically Henselian for any i ≥ 0. Then we obtain the following equivalences of categories:
F.Ét(R s.♭ i ) ∼ = − → F.Ét(R i ).
Proof. This follows from Lemma 3.38, Lemma 3.41 and [47, Tag 09ZL].
3.4.2.
Proof of Main Theorem 1. We keep the notation as above. Furthermore, we set R i := R i /I 0 R i and I i := I i R i for every i ≥ 0. To prove Theorem 3.37, we investigate some relationship between (R i ) I 0 -tor and Ann R i (I i ). First recall that we can regard (R i ) I 0 -tor as a non-unital subring of R i by Corollary 3.16. Moreover, the map t i naturally restricts to (R i ) I 0 -tor ֒→ (R i+1 ) I 0 -tor , as follows.
Lemma 3.48. For every i ≥ 0, let (t i ) tor : (R i ) I 0 -tor → (R i+1 ) I 0 -tor be the restriction of t i . Then the following assertions hold.
(1) (t i ) tor is the unique map such that
ϕ I 0 ,R i+1 • (t i ) tor = t i • ϕ I 0 ,R i . (2) (t i ) tor • (F i ) tor = (F i+1 ) tor • (t i+1 ) tor = F (R i+1 ) I 0 -tor .
Proof. Since ϕ I 0 ,R i is injective by Corollary 3.16, the assertion (1) is clear from the construction. Hence we can regard (t i ) tor and (F i ) tor as the restrictions of t i and F i , respectively. Thus the assertion (2) follows from the compatibility
t i • F i = F i+1 • t i+1 = F R i+1 induced by Lemma 3.8 (3). The map ϕ I 0 ,R i : (R i ) I 0 -tor ֒→ R i /I 0 R i restricts to Ann R i (I i ) ֒→ Ann R i (I i )
. On the other hand, Ann R i (I i ) turns out to be equal to (R i ) I 0 -tor by the following lemma.
ϕ I 0 ,R i ) ⊆ Ann R i (I i ).
Proof. By Lemma 3.48 (2) and the axiom (g) in Definition 3.22, we find that F (R i ) I 0 -tor is injective. In other words, (R i ) I 0 -tor does not contain any non-zero nilpotent element. Moreover, (R i ) I 0 -tor = (R i ) I i -tor . Hence the assertion follows from Lemma 3.19.
The following lemma is essential for proving Theorem 3.37.
Lemma 3.50. For every i ≥ 0, F i restricts to a Z-linear map Ann R i+1 (I i+1 ) → Ann R i (I i ). Moreover, the resulting inverse system {Ann R i (I i )} i≥0 has the following properties.
(1) For every j ≥ 0, lim (3.14)
← − 1 i≥0 Ann R j+i (I j+i ) = (0). (2) There are isomorphisms of Z-linear maps lim ← −i≥0 Ann R j+i (I j+i ) ∼ = (R j ) I 0 -tor (j ≥ 0) that are multiplicative, and compatible with {t s,♭ j } j≥0 and {t j } j≥0 . Proof. Since F i (I i+1 ) = I i , F i restricts to a Z-linear map (F i ) ann : Ann R i+1 (I i+1 ) → Ann R i (I i ). Let ϕ i : (R i ) I 0 -tor ֒→ Ann R i (I i )0 / / (R i+1 ) I 0 -tor ϕ i+1 / / (F i )tor Ann R i+1 (I i+1 ) / / I i+1 p i+1 −1 / / 0 0 / / (R i ) I 0 -tor ϕ i / / Ann R i (I i ) / / I i p i −1 / / 0
where the second and third vertical maps are the restrictions of
F i . Since F i (I i+1 p i+1 −1 ) = 0, the both functors lim ← −i≥0 and lim ← − 1 i≥0 assign (0) to the inverse system {I j+i p j+i −1 } i≥0 . Moreover, since (F i ) tor is bijective, lim ← −i≥0 (R j+i ) I 0 -tor ∼ = (R j+i ) I 0 -tor and lim ← − 1 i≥0 (R j+i ) I 0 -tor = (0). Hence we find that lim ← − 1 i≥0
Ann R j+i (I j+i ) = (0), which is the assertion (1). Furthermore, we obtain the isomorphisms of Z-modules: i (f s.♭ j ), and let π i and F ′ i be as in Lemma 3.30. Then, by the assumption, we have the following commutative ladder with exact rows:
(3.15) (R j ) I 0 -tor (Φ (j) 0 )tor ← −−−− − lim ← − i≥0 (R j+i ) I 0 -tor lim ← −i≥0 ϕ j+i − −−−−−− → lim ← − i≥0 R j+i (where (Φ0 / / (f i+1 ) ι i+1 / / R i+1 π i+1 / / F i R i+1 /I i+1 / / F ′ i 0 0 / / (f i ) ι i / / R i π i / / R i /I i / / 0
where ι i is the inclusion map. Let us consider the exact sequence obtained by taking inverse limits for all columns of the above ladder. Then, since each F ′ i is an isomorphism, the map lim ← −i≥0 π j+i :
R s.♭ j → lim ← −i≥0 R j+i /I j+i is isomorphic to π j • Φ (j)
0 . Thus we find that I s.♭ j = Im(lim ← −i≥0 ι j+i ). Let us show that the ideal Im(lim ← −i≥0 ι j+i ) ⊆ R s.♭ j is generated by f s.♭ j . For i ≥ 0, let µ i : R i → (f i ) be the R i -linear map induced by multiplication by f i . Then we obtain the commutative ladder:
R i+1 F i µ i+1 / / (f i+1 ) ι i+1 / / R i+1 F i R i µ i / / (f i ) ι i / / R i .
Then, since Ker µ i = Ann R i (I i ) for every i ≥ 0, lim ← −i≥0 µ j+i is surjective by Lemma 3.50 (1). Hence (1),
we have Im(lim ← −i≥0 ι j+i ) = Im(lim ← −i≥0 (ι j+i • µ j+i )),(R s.♭ j ) I s.♭ 0 -tor = Ann R s.♭ j (I s.♭ 0 ) = Ker(lim ← − i≥0 µ j+i ) = lim ← − i≥0 Ann R j+i (I j+i ).
Thus by Lemma 3.50 (2), we obtain an isomorphism (R s.♭ j ) I s.♭ 0 -tor ∼ = (R j ) I 0 -tor with the desired property.
3.5. Relation with perfectoid rings. In the rest of this paper, for a ring R, we use the following notation. Set the inverse limit
R ♭ := lim ← − {· · · → R/pR → R/pR → · · · → R/pR},
where each transition map is the Frobenius endomorphism on R/pR. It is called the tilt (or tilting) of R. Moreover, we denote by W (R) the ring of p-typical Witt vectors over R. If R is p-adically complete and separated, we denote by θ R : W (R ♭ ) → R the ring map such that the diagram: (1) S is ̟-adically complete and separated for some element ̟ ∈ S such that ̟ p divides p.
(3.16) W (R ♭ ) θ R / / R R ♭ / / R/pR
(2) The Frobenius endomorphism on S/pS is surjective.
(3) The kernel of θ S : W (S ♭ ) → S is principal.
We have a connection between perfectoid towers and perfectoid rings. To see this, we use the following characterization of perfectoid rings.
S π 2 / / π 1 (S/̟S) red π 4 S π 3 / / ( S/̟ S) red
(where π i is the canonical projection map for i = 1, 2, 3, 4) is cartesian. Hence S ̟-tor (= Ker(π 1 )) is isomorphic to Ker(π 4 ) as a (possibly) non-unital ring. Since (S/̟S) red is a perfect F p -algebra, it admits the Frobenius endomorphism and the inverse Frobenius. Moreover, Ker(π 4 ) is closed under these operations because ( S/̟ S) red is reduced. Consequently, it follows that one has a bijection (3.17). Hence ̟ has the property (3), as desired.
Remark 3.53. In view of the above proof, the "only if" part of Theorem 3.52 can be refined as follows. For a perfectoid ring S, an element ̟ ∈ S such that p ∈ ̟ p S and S is ̟-adically complete and separated satisfies the properties (2) and (3) in Theorem 3.52.
Corollary 3.54. Let ({R i } i≥0 , {t i } i≥0
) be a perfectoid tower arising from some pair (R 0 , I 0 ). Let R ∞ denote the I 1 -adic completion of R ∞ . Then R ∞ is a perfectoid ring.
Proof. Since we have lim − →i≥0
F R i /I 0 R i = (lim − →i≥0 t i ) • (lim − →i≥0 F i ) and lim − →i≥0
t i is a canonical isomorphism, the Frobenius endomorphism on R ∞ can be identified with lim − →i≥0 F i . Hence one can immediately deduce from the axioms in Definition 3.22 that any generator of I 1 R ∞ has the all properties assumed on ̟ in Theorem 3.52.
In view of Theorem 3.52, one can regard perfectoid rings as a special class of perfectoid towers.
Example 3.55. Let S be a perfectoid ring. Let ̟ ∈ S be such that p ∈ ̟ p S and S is ̟-adically complete and separated. Set S i = S and t i = id S for every i ≥ 0, and I 0 = ̟ p S. Then by Remark 3.53, the tower (
{S i } i≥0 , {t i } i≥0 )
is a perfectoid tower arising from (S, I 0 ). In particular, I 0 S I 0 -tor = (0), and F S I 0 -tor is bijective.
Moreover, we can treat more general rings in a tower-theoretic way.
Example 3.56 (Zariskian preperfectoid rings). Let R be a ring that contains an element ̟ such that p ∈ ̟ p R, R is ̟-adically Zariskian, and R has bounded ̟-torsion. Assume that the ̟-adic completion R is a perfectoid ring. Set R i = R and t i = id R for every i ≥ 0, and I 0 = ̟ p R. Then the tower ({R i } i≥0 , {t i } i≥0 ) is a perfectoid tower arising from (R, I 0 ). Indeed, the axioms (a) and (e) are clear from the assumption. Moreover, since R is perfectoid and R/̟ p R ∼ = R/̟ p R, the axioms (b), (c), (d) and (f) hold by Example 3.55. In view of Lemma 3.19, for proving that the axiom (g) holds, it suffices to show that the map:
(3.18) R I 0 -tor → R I 0 -tor ; x → x p
is bijective. By Corollary 3.17, the natural map ψ tor : R I 0 -tor → ( R) I 0 -tor is injective. Hence so is F ( R) I 0 -tor • ψ tor , which factors through (3.18). Therefore, (3.18) is injective (in particular, I 0 R I 0 -tor = (0)). To check the surjectivity, we pick an element x ∈ R I 0 -tor . Then ψ tor (x) = η p for some η ∈ ( R) I 0 -tor . Let y ∈ R be such that y ≡ η mod I 2 0 R. Then y p ≡ x mod I 2 0 and I 0 y ⊆ I 2 0 . By Lemma 3.18, the second property implies that y ≡ z mod I 0 for some z ∈ R I 0 -tor . Hence by the binomial theorem, we have x ≡ y p ≡ z p mod I 2 0 . On the other hand, R I 0 -tor = R I 2 0 -tor , and hence ϕ I 2 0 ,R is injective by Corollary 3.16. Thus we have x = z p , as desired. Recall that we have two types of tilting operation at present; one is defined for perfectoid rings, and the other is for perfectoid towers. The following result asserts that they are compatible.
Lemma 3.57. Let ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) be the tilt of ({R i } i≥0 , {t i } i≥0 ) associated to (R 0 , I 0 ). Let R s.♭ ∞ be the I s.♭ 0 -adic completion of R s.♭ ∞ . Let I ♭ 0 ⊆ R ♭ ∞ be the ideal that is the inverse image of I 0 R ∞ mod pR ∞ via the first projection. Then there exists a canonical isomorphism R s.♭ ∞ ∼ = − → R ♭ ∞ that sends I s.♭ 0 R s.♭ ∞ onto I ♭ 0 . Proof. Since R s.♭
∞ is perfect, one can deduce the following isomorphism from Example 3.34:
R s.♭ ∞ ∼ = − → lim ← − Frob R s.♭ ∞ /I s.♭ 0 R s.♭ ∞ ; (s i mod (I s.♭ 0 ) p i R s.♭ ∞ ) i≥0 → (s 1/p i i mod I s.♭ 0 R s.♭ ∞ ) i≥0 .
On the other hand, (3.13) in Lemma 3.41 induces a canonical isomorphism lim 3.6.1. Calculation of the tilts. As an example of tilts of Noetherian perfectoid towers, we calculate them for certain towers of local log-regular rings. Firstly, we review a perfectoid tower constructed in [23].
← −Frob R s.♭ ∞ /I s.♭ 0 R s.♭ ∞ ∼ = − → lim ← −Frob R ∞ /I 0 R ∞ . Moreover,
Construction 3.58. Let (R, Q, α) be a complete local log-regular ring with perfect residue field of characteristic p > 0. Assume that Q is fine, sharp, and saturated (see Remark 2.18). Set A := R/I α . Let (f 1 , . . . , f r ) be a sequence of elements of R whose image in A is maximal (see Definition 5.3). Since the residue field of R is perfect, r is the dimension of A (see §5). For every i ≥ 0, we consider the ring
A i := A[T 1 , . . . , T r ]/(T p i 1 − f 1 , . . . , T p i r − f r )
, where each f j denotes the image of f j in A (j = 1, . . . , r). Notice that A i is regular by Theorem 5.2. Moreover, we set Q (i) := Q
R i := R ′ i ⊗ R R ′′ i .
Let t i : R i → R i+1 be the ring map that is naturally induced by the inclusion map ι (i) : Q (i) ֒→ Q (i+1) . Since R ′′ i+1 is a free R ′′ i -module, t i is universally injective by Lemma 2.13 (2) and the condition (e) in Proposition 2.7 (2). Proposition 3.59. Keep the notation as in Construction 3.58. Let α i : Q (i) → R i be the natural map. Then (R i , Q (i) , α i ) is a local log-regular ring.
Proof. We refer the reader to [23, 17.2.5].
By the construction, we obtain the tower of rings (
{R i } i≥0 , {t i } i≥0 ) (see Definition 3.1).
Proposition 3.60. Keep the notation as in Construction 3.58. Then the tower ({R i } i≥0 , {t i } i≥0 ) of local log-regular rings defined above is a perfectoid tower arising from (R, (p)).
Proof. We verify (a)-(g) in Definition 3.6 and Definition 3.22. The axiom (a) is trivial. Since t i is universally injective, the axiom (b) follows. The axioms (c) and (d) follow from [23, (17.2.10) and Lemma 17.2.11]. Since R is of residual characteristic p, the axiom (e) follows from the locality. Since R i is a domain for any i ≥ 0, the axiom (g) holds by Remark 3.23. Finally, let us check that the axiom (f) holds. In the case when p = 0, it follows from [23, Theorem 17.2.14 (i)]. Otherwise, there exists an element ̟ ∈ R 1 that satisfies ̟ p = pu for some unit u ∈ R 1 by [23, Theorem 17.2.14 (ii)]. Set I 1 := (̟). Then the axiom (f-1) holds. Moreover, the axiom (f-2) follows from [23, Theorem 17.2.14 (iii)]. Thus the assertion follows.
For calculating the tilt of the perfectoid tower constructed above, the following lemma is quite useful.
Lemma 3.61. Keep the notation as in Proposition 3.59. Let k be the residue field of R. Then there exists a family of ring maps {φ i : C(k) Q (i) ⊕ (N r ) (i) → R i } i≥0 which is compatible with the log structures of {(R i , Q (i) , α i )} i≥0 such that the following diagram commutes for every i ≥ 0:
(3.21) C(k) Q (i) ⊕ (N r ) (i) / / φ i C(k) Q (i+1) ⊕ (N r ) (i+1) φ i+1 R i t i / / R i+1
(where the top arrow is the natural inclusion). Moreover, there exists an element θ ∈ C(k) Q ⊕ N r whose constant term is p such that the kernel of φ i is generated by θ for every i ≥ 0.
Proof. First we remark the following. Let k i be the residue field of R i . Then by Lemma 3.12 (1) and Lemma 3.8 (2), the transition maps induce a purely inseparable extension k ֒→ k i . Moreover, this extension is trivial because k is perfect. Therefore, we can identify k i (resp. the Cohen ring of R i ) with k (resp. C(k)).
Next, let us show the existence of a family of ring maps {φ i } i≥0 with the desired compatibility. Since (R i , Q (i) , α i ) is a complete local log-regular ring, we can take a surjective ring map ψ i : C(k) Q (i) ⊕ N r → R i as in Theorem 2.20; its kernel is generated by an element θ i whose constant term is p, and the diagram:
Q (i) / / α i ' ' ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ C(k) Q (i) ⊕ N r ψ i R i
commutes. For j = 1, . . . , r, let us denote by f 1/p i j the image of T j ∈ R[T 1 , . . . , T r ] in R i (see (3.19) and (3.20)). Note that the sequence f 1/p i 1 , . . . , f 1/p i r in R i becomes a regular system of parameters of R i /I α i by the reduction modulo I α i (see [23, 17.2.3] and [23, 17.2.5]). Thus, for the set of the canonical basis {e 1 , . . . , e r } of N r , we may assume ψ i (e e j ) = f 1/p i j by the construction of ψ i (see the proof of [41, Chapter III, Theorem 1.11.2]). Hence we can choose {ψ i } i≥0 so that the diagram: Lemma 2.11 (3) and ψ i . Finally, note that the image of θ 0 ∈ Ker(ψ 0 ) in C(k) Q (i) ⊕ N r is contained in Ker(ψ i ), and its constant term is still p. Thus, by the latter assertion of Theorem 2.20 (2), Ker(ψ i ) is generated by θ 0 . Hence by taking θ 0 as θ, we complete the proof.
(3.22) C(k) Q (i) ⊕ N r / / ψ i C(k) Q (i+1) ⊕ N r ψ i+1 R i t i / / R i+1 commutes. Thus it suffices to define φ i : C(k) Q (i) ⊕ (N r ) (i) → R i as the composite map of the isomorphism C(k) Q (i) ⊕ (N r ) (i) ∼ = − → C(k) Q (i) ⊕ N r obtained by
Let us consider the monoids Q (i) for an integral sharp monoid Q. Since there is the natural inclusion ι (i) : Q (i) ֒→ Q (i+1) for any i ≥ 0, we obtain a direct system of monoids (
{Q (i) } i≥0 , {ι (i) } i≥0 ).
Moreover, the p-times map on Q (i+1) gives a factorization:
Q (i+1) ×p / / ×p $ $ $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ Q (i+1) Q (i) . ? ι (i) O O
From this discussion, we define the small tilt of {Q (i) } i≥0 . Definition 3.62. Let Q be an integral sharp monoid, and let ({Q (i) } i≥0 , {ι (i) } i≥0 ) be as above. Then for an integer j ≥ 0, we define the j-th small tilt of ({Q (i) } i≥0 , {ι (i) } i≥0 ) as the inverse limit
(3.23) Q s.♭ j := lim ← − {· · · → Q (j+1) → Q (j) },
where the transition map Q (i+1) → Q (i) is the p-times map of monoids.
Now we can derive important properties of the tilt of the perfectoid tower given in Construction 3.58.
Theorem 3.63. Keep the notation as in Lemma 3.61. Then the following assertions hold.
(1) The tower
({(R i ) s.♭ (p) } i≥0 , {(t i ) s.♭ (p) } i≥0 ) is isomorphic to ({k Q (i) ⊕(N r ) (i) } i≥0 , {u i } i≥0 ), where u i is the ring map induced by the natural inclusion Q (i) ⊕ (N r ) (i) ֒→ Q (i+1) ⊕ (N r ) (i+1) .
(2) For every j ≥ 0, there exists a homomorphism of monoids α s.♭ j :
Q s.♭ j → (R j ) s.♭ (p) such that ((R j ) s.♭ (p) , Q s.♭ j , α s.♭ j ) is a local log-regular ring. (3) For every j ≥ 0, (t j ) s.♭ (p) : (R j ) s.♭ (p) → (R j+1 ) s.♭ (p) is module-finite and (R j ) s.♭ (p) is F -finite. Proof. (1): By Lemma 3.61, each R i is isomorphic to C(k) Q (i) ⊕(N r ) (i) /(p−f )C(k) Q (i) ⊕(N r ) (i)
where f is an element of C(k) Q ⊕ N r which has no constant term. Set S i := k Q (i) ⊕ (N r ) (i) for any i ≥ 0 and let u i : S i ֒→ S i+1 be the inclusion map induced by the natural inclusion (3), the Frobenius endomorphism on S i+1 factors through a surjection G i : S i+1 → S i . In particular,
Q (i) ⊕ (N r ) (i) ֒→ Q (i+1) ⊕ (N r ) (({S i } i≥0 , {u i } i≥0 )
is a perfectoid tower arising from (S 0 , (0)) and G i is the i-th Frobenius projection (cf. Lemma 3.25).
Put f := f mod pC(k) Q ⊕ N r ∈ S 0 . Then each S i is f -adically complete and separated by [19,Lemma 2.1.1]. Moreover, the commutative diagram (3.21) yields the commutative squares (i ≥ 0):
S i+1 /f S i+1 G i ∼ = / / R i+1 /pR i+1 F i S i /f S i ∼ = / / R i /pR i that are compatible with {u i : S i /f S i → S i+1 /f S i+1 } i≥0 and {t i } i≥0 .
Hence by Lemma 3.33, we obtain the isomorphisms
(3.24) (R j ) s.♭ (p) ∼ = ← − lim ← − {· · · G j+1 − −− → S j+1 /f S j+1 G j − − → S j /f S j } ∼ = − → S j (j ≥ 0)
that are compatible with the transition maps of the towers. Thus the assertion follows.
(2): Considering the inverse limit of the composite maps
Q (j+i) α j+i − −− → R j+i ։ R j+i /pR j+i (i ≥ 0), we obtain a homomorphism of monoids α s.♭ j : Q s.♭ j → (R j ) s.♭ (p)
. On the other hand, let α j : Q (j) → S j be the natural inclusion. Then, since S j is canonically isomorphic to k Q (j) ⊕ N r , (S j , Q (j) , α j ) is a local log-regular ring by Theorem 2.20 (1). Thus it suffices to show that ((R j ) s.♭ (p) , Q s.♭ j , α s.♭ j ) is isomorphic to (S j , Q (j) , α j ) as a log ring. Since the transition maps in (3.23) are isomorphisms by Lemma 2.11 (3), we obtain the isomorphisms of monoids
(3.25) Q s.♭ j id Q s.♭ j ← −−− − Q s.♭ j ∼ = − → Q (j) (j ≥ 0).
Then one can connect (3.25) to (3.24) to construct a commutative diagram using α s.♭ j and α j . Hence the assertion follows.
(3): By Lemma 2.12 (2), t j : R j → R j+1 is module-finite. Hence by Proposition 3.44 (1),
(t j ) s.♭ (p) : (R j ) s.♭ (p) → (R j+1 ) s.♭ (p)
is also module-finite. Finally let us show that (R j ) s.♭ (p) is F -finite. By the assertion (2), (R j ) s.♭ (p) is a complete Noetherian local ring, and the residue field is F -finite because it is perfect. Thus the assertion follows from [34,Theorem 8.4].
Example 3.64.
(1) A tower of regular local rings which is treated in [9] and [10] is a perfectoid tower in our sense. Let (R, m, k) be a d-dimensional regular local ring whose residue field k is perfect and let x 1 , . . . , x d be a regular sequence of parameters. Let e 1 , . . . , e d be the canonical basis of N d . Then (R, N d , α) is a local log-regular ring where α : N d → R is a homomorphism of monoids which maps e i to x i . Furthermore, assume that R is m-adically complete. Then, by Cohen's structure theorem, R is isomorphic to
W (k) x 1 , . . . , x d /(p − f )
where f = x 1 or f ∈ (p, x 1 , . . . , x d ) 2 (the former case is called unramified, and the latter is called ramified). Let us construct a perfectoid tower arising from (R, (p)) along Construction 3.58. Since k is perfect, Ω k is zero by the short exact sequences (5.4)
R i = R ′ i = Z[(N d ) (i) ]⊗ Z[N d ] R ∼ = R[T 1 , . . . , T d ]/(T p i 1 −x 1 , . . . , T p i d −x d ) ∼ = W (k) x 1/p i 1 , . . . , x 1/p i d /(p−f ).
Set the natural injection t i : R i → R i+1 for any i ≥ 0. Then, by Proposition 3.60,
({R i } i≥0 , {t i } i≥0 )
is a perfectoid tower arising from (R, (p)). By Theorem 3.63, its tilt (2) ֒→ · · · , which can be written as
({(R i ) s.♭ (p) } i≥0 , {(t i ) s.♭ (p) } i≥0 ) is isomorphic to the tower k N d ֒→ k (N d ) (1) ֒→ k (N d )k x 1 , . . . , x d ֒→ k x 1/p 1 , . . . , x 1/p d ֒→ k x 1/p 2 1 , . . . , x 1/p 2 d ֒→ · · · .
(2) Consider the surjection:
S := W (k) x, y, z, w /(xy−zw) ։ R := W (k) x, y, z, w /(xy−zw, p−w) = W (k) x, y, z /(xy−pz).
where k is a perfect field. Let Q ⊆ N 4 be a saturated submonoid generated by (1, 1, 0, 0), (0, 0, 1, 1), (1, 0, 0, 1), and (0, 1, 1, 0).
Then S admits a homomorphism of monoids α S : Q → S by letting (1, 1, 0, 0) → x, (0, 0, 1, 1) → y, (1, 0, 0, 1) → z and (0, 1, 1, 0) → w. With this, (S, Q, α S ) is a local log-regular ring. The composite map α R : Q → S → R makes R into a local log ring. Indeed, we can write R ∼ = W (k) Q /(p − e (0,1,1,0) ), hence (R, Q, α R ) is log-regular by Theorem 2.20.
Next, note that R/I α R ∼ = k. Then, for the same reason in (1),
R ′′ i is equal to R. Moreover, Q (i) is generated by 1 p i , 1 p i , 0, 0 , 0, 0, 1 p i , 1 p i , 1 p i , 0, 0 1 p i , 0, 1 p i , 1 p i , 0 .
Thus, applying Construction 3.58, we obtain
R i = R Q (i) ∼ = W (k) Q (i) /(p − e (0,1,1,0) ) ∼ = W (k) x 1/p i , y 1/p i , z 1/p i , w 1/p i /(x 1/p i y 1/p i − z 1/p i w 1/p i , p − w).
Set a natural injection t i : R i → R i+1 . Then, by Proposition 3.60, ({R i } i≥0 , {t i } i≥0 ) is a perfectoid tower arising from (R, (p)). Hence
R ∞ = lim − → i≥0 R i ∼ = i≥0 W (k) x 1/p i , y 1/p i , z 1/p i , w 1/p i /(x 1/p i y 1/p i − z 1/p i w 1/p i , p − w),
and its p-adic completion is perfectoid. Moreover, one can calculate the tilt (2) ֒→ · · · by Theorem 3.63, or, more explicitly, k x, y, z, w /(xy − zw) ֒→ k x 1/p , y 1/p , z 1/p , w 1/p /(x 1/p y 1/p − z 1/p w 1/p ) ֒→ · · · .
({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) to be k Q ֒→ k Q (1) ֒→ k Q
3.6.2. Towers of split maps and sousperfectoid rings. Let us recall that Hansen and Kedlaya introduced a new class of topological rings that guarantees sheafiness on the associated adic spectra (see [27,Definition 7.1]). Definition 3.65. Let A be a complete and separated Tate ring such that a prime p ∈ A is topologically nilpotent. We say that A is sousperfectoid, if there exists a perfectoid ring B in the sense of Fontaine (see [27,Definition 2.13]) with a continuous A-linear map f : A → B that splits in the category of topological A-modules. That is, there is a continuous A-linear map σ :
B → A such that σ • f = id A .
Let us show that a perfectoid tower consisting of split maps induces sousperfectoid rings. In view of Theorem 2.27, one can apply this result to the towers discussed above. See [39] for detailed studies on algebraic aspects of Tate rings.
Proposition 3.66. Let ({R i } i≥0 , {t i } i≥0
) be a perfectoid tower arising from some pair (R, (f 0 )). Assume that f 0 is regular, R is f 0 -adically complete and separated, and t i splits as an R i -linear map for every i ≥ 0. We equip R[ 1 f 0 ] with the linear topology in such a way that {f n 0 R} n≥1 defines a fundamental system of open neighborhoods at 0 ∈ R[ 1 f 0 ]. Then R[ 1 f 0 ] is a sousperfectoid Tate ring. In particular, it is stably uniform.
In order to prove this, we need the following lemma. Proof. We use the fact that each t i : R i → R i+1 splits as an R i -linear map by assumption. This implies that the short exact sequence of R-modules
0 → R 0 → R i → R i /R → 0 splits for any i ≥ 0. It induces a commutative diagram of R-modules 0 − −−− → Hom R 0 (R i+1 /R 0 , R 0 ) − −−− → Hom R 0 (R i+1 , R 0 ) − −−− → Hom R 0 (R 0 , R 0 ) − −−− → 0 α i β i 0 − −−− → Hom R 0 (R i /R 0 , R 0 ) − −−− → Hom R 0 (R i , R 0 ) − −−− → Hom R 0 (R 0 , R 0 ) − −−− → 0
where each horizontal sequence is split exact, and each vertical map forms an inverse system induced by t i : R i → R i+1 . In particular, β i is surjective and it thus follows from the snake lemma that α i is surjective as well. By taking inverse limits, we obtain the short exact sequence:
0 → lim ← − i≥0 Hom R 0 (R i /R 0 , R 0 ) → lim ← − i≥0 Hom R 0 (R i , R 0 ) h − → Hom R 0 (R 0 , R 0 ) → 0.
It follows from [43,Lemma 4.1] that h is the canonical surjection Hom R 0 (R ∞ , R 0 ) ։ Hom R 0 (R 0 , R 0 ). Then choosing an inverse image of id R 0 ∈ Hom R 0 (R 0 , R 0 ) gives a splitting of R 0 → R ∞ .
Applications toétale cohomology of Noetherian rings
In this section, we establish several results onétale cohomology of Noetherian rings, as applications of the theory of perfectoid towers developed in §3. In §4.1, for a ring that admits a certain type of perfectoid tower, we prove that finiteness ofétale cohomology groups on the positive characteristic side carries over to the mixed characteristic side (Proposition 4.7). In §4.2, we apply this result to a problem on divisor class groups of log-regular rings.
We prepare some notation. Let X be a scheme and let Xé t denote the category of schemes that areétale over X, and for anyétale X-scheme Y , we specify the covering {Y i → Y } i∈I so that Y i isétale over Y and the family {Y i } i∈I covers surjectively Y . For an abelian sheaf F on Xé t , we denote by H i (Xé t , F) the value of the i-th derived functor of U ∈ Xé t → Γ(U, F). For the most part of applications, we consider torsion sheaves, such as Z/nZ and µ n for n ∈ N. However, for the multiplicative group scheme G m , we often use the following isomorphism:
H 1 (Xé t , G m ) ∼ = Pic(X).
For the basics onétale cohomology, we often use [16] or [36] as references. (1) For any abelian torsion sheaf F on Xé t , we have RΓ(Spec(A)é t , F ) ≃ RΓ(Spec(A/J)é t , F | Spec(A/J) ).
(2) Assume that J is finitely generated. Then for any abelian torsion sheaf F on Xé t and any open subset U ⊆ X such that X \ V (J) ⊆ U , we have RΓ(Ué t , F ) ≃ RΓ( Ué t , F ).
Proof. The first statement is known as Affine analog of proper base change in [21], while the second one is known as Formal base change theorem which is [17, Theorem 7.1.1] in the Noetherian case, and [30,XX,4.4] in the non-Noetherian case.
We will need the tilting invariance of (local)étale cohomology from [10, Theorem 2.2.7]. To state the theorem and establish a variant of it, we give some notations. We also denote U s.♭ I 0 by U s.♭ as an abbreviated form.
Note that by the compatibility described in Lemma 3.41, the operation U U s.♭ is compatible with the base extension along the transition maps of a perfectoid tower.
Let us give some examples of U s.♭ .
Example 4.4 (Punctured spectra of regular local rings). Keep the notation as in Example 3.64 (1).
In this situation, the isomorphism Φ
(4.2) θ R : ( R) ♭ /I ♭ 0 ( R) ♭ ∼ = − → R/I 0 R
which is induced by the bottom map in the diagram (3.16). In this case, we denote F
R ♭ ,Φ (0) 0 (U ) by U ♭ in distinction from U s.♭ .
The comparison theorem we need, due toČesnavičius and Scholze [10], is stated as follows. (1) For every torsion abelian group G, we have RΓ(Ué t , G) ∼ = RΓ(U ♭ et , G) in a functorial manner with respect to A, U , and G.
(2) Let Z be the complement of U ⊆ Spec(A). Then for a torsion abelian group G, we have RΓ Z (Spec(A)é t , G) ∼ = RΓ Z (Spec(A ♭ )é t , G).
Now we come to the main result on tiltingétale cohomology groups. Recall that we have fixed a prime p > 0.
Proposition 4.7. Let ({R j } j≥0 , {t j } j≥0
) be a perfectoid tower arising from some pair (R, I 0 ). Suppose that R j is I 0 -adically Henselian for every j ≥ 0. Let ℓ be a prime different from p. Suppose further that for every j ≥ 0, t j : R j → R j+1 is a module-finite extension of Noetherian normal domains whose generic extension is of p-power degree. 12 (1) {U i } i∈I forms a filter base. In particular, one can define a partial order on I so that it is a directed set and {U i } i∈I together with the inclusion maps forms an inverse system. (1) and (2), then the natural map Pic(U i ) → Cl(X) is injective for any i ∈ I. If {U i } i∈I satisfies (1), (2) and (3), then lim − →i∈I Pic(U i ) ∼ = Cl(X). In particular, if U ⊆ X is any open subset that is locally factorial with codim X (X \ U ) ≥ 2, then Pic(U ) ∼ = Cl(X).
(2) Let V i := X \ U i . Then codim X (V i ) ≥ 2. (3) For any x ∈ i∈I U i , the local ring O X,x is factorial. If {U i } i∈I satisfies
Next we establish the following two results on the torsion part of the divisor class group of a (Noetherian) normal domain; these are a part of numerous applications of Theorem 4.1 of independent interest. (1) For any prime ℓ = p,
Proof. Since R → R sh is a local ring map, (R sh , Q, α sh ) is a local log ring by Lemma 2.16. Note that we have the equality I α sh = I α R sh . Since we have the isomorphism R sh /I α sh ∼ = (R/I α ) sh by [47, Tag 05WS] and (R/I α ) sh is a regular local ring by [47, Tag 06LN], R sh /I α sh is a regular local ring. Moreover, since the dimension of R is equal to the dimension of a strict henselization R sh , we obtain the following equalities:
dim R sh − dim(R sh /I α sh ) = dim R sh − dim(R/I α ) sh = dim R − dim(R/I α ) = dim Q.
So the local log ring (R sh , Q, α sh ) is log-regular.
Now we can prove the following result on the divisor class groups of local log-regular rings, as an application of the theory of perfectoid towers.
Theorem 4.13. Let (R, Q, α) be a local log-regular ring of mixed characteristic with perfect residue field k of characteristic p > 0, and denote by Cl(R) the divisor class group with its torsion subgroup Cl(R) tor . Then the following assertions hold.
(1) Assume that R ∼ = W (k) Q for a fine, sharp, and saturated monoid Q, where W (k) is the ring of Witt vectors over k. Then Cl(R) tor ⊗ Z[ 1 p ] is a finite group. In other words, the ℓ-primary subgroup of Cl(R) tor is finite for all primes ℓ = p and vanishes for almost all primes ℓ = p.
(2) Assume that R sh [ 1 p ] is locally factorial, where R sh is the completion of the strict Henselization R sh . Then Cl(R) tor ⊗ Z[ 1 p ] is a finite group. In other words, the ℓ-primary subgroup of Cl(R) tor is finite for all primes ℓ = p and vanishes for almost all primes ℓ = p.
Proof. The assertion (1) was already proved in Proposition 2.24. So let us prove the assertion (2). We may assume that Q is fine, sharp, and saturated by Remark 2.18. The proof given below works for the first case under the assumption of local factoriality of R sh [ 1 p ]. Since R → R sh is a local flat ring map, the induced map Cl(R) → Cl( R sh ) is injective by Mori's theorem (c.f. [15,Corollary 6.5.2]). Thus, it suffices to prove the theorem for R sh . Moreover, R sh is log-regular with respect to the induced log ring structure α : Q → R → R sh by Lemma 4.12. So without loss of generality, we may assume that the residue field of R is separably closed (hence algebraically closed in our case).
Henceforth, we denote R sh by R for brevity and fix a prime ℓ that is different from p. By Lemma 4.9 and the local factoriality of R[ 1 p ], we claim that there is an open subset U ⊆ X := Spec(R) such that the following holds:
• Pic(U ) ∼ = Cl(X), X \ V (pR) ⊆ U and codim X (X \ U ) ≥ 2.
Indeed, note that X is a normal integral scheme by Kato's theorem (Theorem 2.19) and let U be the union of the regular locus of X and the open Spec(R[ 1 p ]) ⊆ X. Then by Serre's normality criterion, we see that codim X (X \ U ) ≥ 2. We fix such an open U ⊆ X once and for all. Taking the cohomology sequence associated to the exact sequence 0 → Z/ℓ n Z → G m which is isomorphic to the regular ring
W (k) s, t, u [ 1 p ]/(st − u) ∼ = W (k) s, t [ 1 p ]
which is a UFD. Moreover, p ∈ R is irreducible. Indeed, assume that f, g ∈ W (k) x, y, z are non-unit elements such that p − f g ∈ (xy − pz). Then we have p − f g = (xy − pz)h for some h ∈ W (k) x, y, z and so p(1 + zh) = f g + xyh. This gives p = (1 + zh) −1 (f g + xyh), which is impossible. The n-th symbolic power (p) (n) never becomes principal for n ≥ 1. Therefore, Cl(R) ∼ = Z and Cl(R) tor = 0. The log-regular ring R does not arise from the classical case, i.e., it is not of the form W (k) Q for a fine, sharp and saturated monoid Q, because p ∈ W (k) Q is a prime element.
Appendix: Construction of differential modules and maximality
The content of this appendix is taken from Gabber-Ramero's treatise [23] whose purpose is to supply the corrected version of Grothendieck's original presentation in EGA. So we give only a sketch of the constructions of relevant modules and maps. The readers are encouraged to look into [23] for more details as well as proofs. We are motivated by the following specific problem. Problem 1. Let (A, m A ) be a Noetherian regular local ring and fix a system of elements f 1 , . . . , f n ∈ A and a system of integers e 1 , . . . , e n with e i > 1 for every i = 1, . . . , n. We set B := A[T 1 , . . . , T n ]/(T e 1 1 − f 1 , . . . , T en d − f n ). Then find a sufficient condition that ensures that the localization B with respect to a maximal ideal n with m A = A ∩ n is regular.
From the construction, it is obvious that the induced ring map A → B is a flat finite injective extension. Let now (A, m A , k) be a Noetherian local ring with residue field k A := A/m A of characteristic p > 0. Following the presentation in [23, (9.6.15)], we define a certain k 1/p A -vector space Ω A together with a map d A : A → Ω A as follows. Case I: (p / ∈ m 2 A ) Let W 2 (k A ) denote the p-typical ring of length 2 Witt vectors over k A . Then there is the ghost component map ω 0 : W 2 (k A ) → k A , and set V 1 (k A ) := Ker(ω 0 ). More specifically, we have W 2 (k A ) = k A × k A as sets with addition and multiplication given respectively by Using this structure, we see that V 1 (k A ) = 0 × k A as sets, which is an ideal of W 2 (k A ) and V 1 (k A ) 2 = 0. This makes V 1 (k A ) equipped with the structure as a k A -vector space by letting x(0, a) := (x, 0)(0, a) for x ∈ k A . One can define the map of k A -vector spaces (5.1) k 1/p A → V 1 (k A ) ; a → (0, a p ), which is a bijection. With this isomorphism, we may view V 1 (k A ) as a k 1/p A -vector space. Next we form the fiber product ring:
A 2 := A × k A W 2 (k A ).
It gives rise to a short exact sequence of A 2 -modules
(5.2) 0 → V 1 (k A ) → A 2 → A → 0,
where A 2 → A is the natural projection, and the A 2 -module structure of V 1 (k A ) is via the restriction of rings A 2 → W 2 (k A ). From (5.2), we obtain an exact sequence of A-modules:
V 1 (k A ) → Ω A → Ω 1 A/Z → 0, where we put Ω A = Ω 1 A 2 /Z ⊗ A 2 A.
After applying ( )⊗ A k A to this sequence, we have another sequence of k A -vector spaces:
(5.3) 0 → V 1 (k A ) j A − → Ω A ⊗ A k A → Ω 1 A/Z ⊗ A k A → 0.
Then this is right exact. Moreover, (5.1) yields a unique k A -linear map ψ A :
V 1 (k A ) ⊗ k A k 1/p A → V 1 (k A ).
Define Ω A as the push-out of the diagram:
V 1 (k A ) ψ A ← − − V 1 (k A ) ⊗ k A k 1/p A j A ⊗k 1/p A − −−−− → Ω A ⊗ A k 1/p A .
More concretely, we have
Ω A = V 1 (k A ) ⊕ (Ω A ⊗ A k 1/p A ) T , where T = (ψ(x), −(j A ⊗k 1/p A )(x)) x ∈ V 1 (k A )⊗ k A k 1/p A .
By the universality of push-outs, we get the commutative diagram:
0 − −−− → V 1 (k A ) ⊗ k A k 1/p A − −−− → Ω A ⊗ A k 1/p A − −−− → Ω 1 A/Z ⊗ A k 1/p A − −−− → 0 ψ A ψ A 0 − −−− → V 1 (k A ) − −−− → Ω A − −−− → Ω 1 A/Z ⊗ A k 1/p A − −−− → 0
We define the map d A : A → Ω A as the composite mapping
A 1×τ k A − −−− → A 2 = A × k A W 2 (k A ) d − → Ω 1 A 2 /Z id ⊗1 − −− → Ω A = Ω 1 A 2 /Z ⊗ A k 1/p A ψ A − − → Ω A .
Here, d : A 2 → Ω 1 A 2 /Z is the universal derivation and τ k A : A → k A → W 2 (k A ), where the first map is the natural projection and the second one is the Teichmüller map. Case II: (p ∈ m 2 A ) We just set Ω A := Ω 1 A/Z ⊗ A k (1) Suppose that φ is formally smooth for the m A -adic topology on A and the m B -adic topology on B. Then the maps induced by φ and Ω φ respectively
(m A /m 2 A ) ⊗ k A k B → m B /m 2 B , Ω A ⊗ K 1/p A k 1/p B → Ω B are injective. (2) Suppose that (a) m A B = m B . (b)
The residue filed extension k A → k B is separable algebraic.
(c) φ is flat.
Then Ω φ induces an isomorphism of k 1/p A -vector spaces:
Ω A ⊗ A B ∼ = Ω B .
(3) If B = A/m 2 A and φ : A → B is the natural map, then Ω φ is an isomorphism. (4) The functor Ω • and the natural transformation d • commute with filtered colimits.
We provide an answer to Problem 1 as follows. (1) A is a regular local ring, and dim k 1/p A E = n.
(2) B is a regular local ring.
In particular, in the situation of the above theorem, B is a regular local ring if A is a regular local ring and f 1 , . . . , f n is maximal in the sense of the following definition. In general, we have the following fact.
Lemma 5.4. Let (A, m A , k A ) be a regular local ring of mixed characteristic and assume that f 1 , . . . , f d is a regular system of parameters of A. Then the following hold:
(1) f 1 , . . . , f d satisfies the condition (1) of Theorem 5.2.
(2) If the residue field k A of A is perfect, then the sequence f 1 , . . . , f d is maximal.
Proof. (1): In the case that p / ∈ m 2 A , [23, Proposition 9.6.17] gives a short exact sequence:
(5.4) 0 → m A /m 2 A ⊗ k A k 1/p A → Ω A → Ω 1 k A /Z ⊗ k A k 1/p A → 0.
Then the images f 1 , . . . , f d form a basis of the k
1/p A -vector space m A /m 2 A ⊗ k A k 1/p A .
The desired claim follows from the left exactness of (5.4).
In the case that p ∈ m 2 A , [23, Lemma 9.6.6] gives a short exact sequence (5.5) 0 → m A /(m 2 A + pm A ) → Ω A → Ω 1 k A /Z → 0. and we can argue as in the case p / ∈ m 2 A . (2): If k A is perfect, then Ω 1 k A /Z = 0. Therefore, (5.4) and (5.5) (in the latter case, one tensors it with k 1/p A over k A ) gives the desired conclusion.
Definition 2. 3
3(Q-module). Let Q be a monoid. (1) A Q-module is a set M equipped with a binary operation Q × M → M ; (q, x) → q + x having the following properties: (a) 0 + x = x for any x ∈ M ;
Lemma 2 . 26 .
226Let R be a ring and let f : M → N be an R-linear map. Consider a decreasing filtration of R-submodules {M λ } λ∈Λ of M and a decreasing filtration of R-submodules {N λ } λ∈Λ of N such that f (M λ ) ⊆ N λ for each λ ∈ Λ. Set M := lim ← − λ∈Λ M/M λ and N := lim ← − λ∈Λ N/N λ , respectively. Finally, assume that f is a split injection that admits an R-linear map g : N → M such that g • f = id M , g(N λ ) ⊆ M λ for each λ ∈ Λ. Then f extends to a split injection M → N . Proof. By assumption, there is an induced map an identity on M/M λ . Taking inverse limits, we get an identity map M → N → M , which proves the lemma.The next result is originally due to Gabber and Ramero [23, Theorem 17.3.12], 6 and we give an alternative and short proof, using the Direct Summand Theorem.
Theorem 2.27. A local log-regular ring (R, Q, α) is a splinter.
Definition 3.1 (Towers of rings).
are compatible with the Frobenius projections. Thus we obtain the isomorphism between inverse limits (R 0 ) the shifting map s 0 :
( 4 )
4For an ideal I ⊆ R, we denote by ϕ I,M : M I-tor → M/IM the composition of natural R-linear maps: (3.8) M I-tor ֒→ M ։ M/IM.
Corollary 3. 16 .
16Keep the notation as in Lemma 3.15, and suppose further that xM x-tor = (0). Then the map ϕ (x),M : M x-tor → M/xM (see Definition 3.14 (4)) is injective.Proof. It is clear from Lemma 3.15.
Corollary 3 . 17 .
317Keep the notation as in Lemma 3.15, and suppose further that M has bounded x-torsion. Let M be the x-adic completion of M , and let ψ : M → M be the natural map. Then the restriction ψ tor : M x-tor → ( M ) x-tor of ψ is injective.
Lemma 3 . 19 .
319Let (R, I) be a pair such that R I-tor does not contain any non-zero nilpotent element of R. Then IR I-tor = (0).
Corollary 3. 20 .
20Let ({R i } i≥0 , {t i } i≥0 ) bea purely inseparable tower arising from some pair (R, I). Then for every i ≥ 0 and every ideal J ⊆ (R i ) q.frep I , we have J((R i ) q.frep I ) J-tor = (0).Proof. Since (R i ) q.frep I is reduced by Proposition 3.11 (4), the assertion follows from Lemma 3.19.
( 1 )
1: Since the implication (a) ⇒ (b) follows from the axiom (d) in Definition 3.22, it suffices to show the converse. Assume that the condition (b) is satisfied. Then for every i ≥ 0, the compatibility t
Definition 3.31 (Small tilts). Let ({R i } i≥0 , {t i } i≥0) be a perfectoid tower arising from some pair (R, I 0 ).
= I j+i (R j+i /I 0 R j+i ).
Definition 3 . 36 (
336Tilts of perfectoid towers). Let ({R i } i≥0 , {t i } i≥0 ) be a perfectoid tower arising from some pair (R, I). Then the inverse perfection of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I) is called the tilt of ({R i } i≥0 , {t i } i≥0 ) associated to (R, I), and is denoted by({(R i ) s.♭ I } i≥0 , {(t i ) s.♭ I } i≥0 ) in distinction from ({(R i ) q.frep I } i≥0 , {(t i ) q.frep I } i≥0). Moreover, we set (R ∞ ) s.♭ I := lim − →i≥0 (R i ) s.♭ I .
Theorem 3 . 37 .
337Let ({R i } i≥0 , {t i } i≥0) be a perfectoid tower arising from some pair (R, I 0 ), and let {I i } i≥0 be the system of perfectoid pillars. Let ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) denote the tilt associated to (R, I 0 ). Then the following assertions hold.
Lemma
Lemma 3 . 49 .
349For every i ≥ 0, I i (R i ) I 0 -tor = 0. In particular, Im(
be the restriction of ϕ I 0 ,R i . By Lemma 3.18 and Lemma 3.49, we canwrite Ann R i (I i ) = Im(ϕ i ) + I i p i −1 . Moreover, Im(ϕ i ) ∩ I i p i −1 = (0) byLemma 3.15 and Lemma 3.49. Hence we have the following ladder with exact rows:
tor denotes the 0-th projection map), which are also multiplicative. Let us deduce (2) from it. Since we have t s.♭ j = lim ← −i≥0 t j+i by definition, the maps lim← −i≥0 ϕ j+i (j ≥ 0) are compatible with {lim ← −i≥0 (t j+i ) tor } j≥0 (induced by Lemma 3.48(2)) and {t s.♭ j } j≥0 by Lemma 3.48 (1). On the other hand, the projections (Φ(j) 0 ) tor (j ≥ 0) are compatible with {lim ← −i≥0 (t j+i ) tor } j≥0 and {(t j ) tor } j≥0 . Hence the assertion follows. Let us complete the proof of Theorem 3.37. Proof of Theorem 3.37. (1): The implication (a) ⇒ (b) follows from Lemma 3.32. Let us show the converse (b)⇒(a). For every i ≥ 0, put f j+i := Φ (j)
(
where the vertical maps are induced by reduction modulo p and the bottom map is the first projection) commutes.Recall the definition of perfectoid rings.Definition 3.51. ([5, Definition 3.5]) A ring S is perfectoid if the following conditions hold.
Theorem 3.52 (cf.[23, Corollary 16.3.75]). Let S be a ring. Then S is a perfectoid ring if and only if S contains an element ̟ with the following properties.
( 1 )
1̟ p divides p, and S is ̟-adically complete and separated.(2) The ring map S/̟S → S/̟ p S induced by the Frobenius endomorphism on S/̟ p S is an isomorphism. (3) The multiplicative map(3.17) S ̟-tor → S ̟-tor ; s → s p is bijective. Proof. ("if" part): It follows from [23, Corollary 16.3.75]. ("only if" part): Let ̟ ∈ S be as in Definition 3.51. Then, such ̟ clearly has the property (1) (in Theorem 3.52), and also has the property (2) by [5, Lemma 3.10 (i)]. To show the remaining part, we set S := S/S ̟-tor . By [10, §2.1.3], the diagram of rings:
by [ 5 ,
5Lemma 3.2 (i)], we can identify R ♭ ∞ with lim ← −Frob R ∞ /I 0 R ∞ , and the ideal I ♭ 0 ⊆ R ♭ ∞ corresponds to the kernel of the first projection map on lim ← −Frob R ∞ /I 0 R ∞ . Thus the resulting composite map R s.♭ ∞ ∼ = − → R ♭ ∞ has the desired property. 3.6. Examples: complete local log-regular rings.
i := Z[Q (i) ] ⊗ Z[Q] R, R ′′ i := R[T 1 , . . . , T r ]/(T p i 1 − f 1 , . . . , T p i r − f r ), and (3.20)
Lemma 3 . 67 .
367Keep the notations and assumptions as in Proposition 3.66. Then the natural map R 0 → lim − →i≥0 R i splits as an R 0 -linear map.
4. 1 .
1Tiltingétale cohomology groups. Let A be a ring with an ideal J and let U ⊆ Spec(A) be an open subset. Then we define the J-adic completion of U to be the open subset U ⊆ Spec( A), which is the inverse image of U via Spec( A) → Spec(A). We will use the following result for deriving results on the behavior ofétale cohomology under the tilting operation as well as some interesting results on the divisor class groups of Noetherian normal domains (see Proposition 4.10 and Proposition 4.11).
Theorem 4 . 1 (
41Fujiwara-Gabber). Let (A, J) be a Henselian pair with X := Spec(A) and let A be the J-adic completion of A. Then the following assertions hold.
Definition 4. 2 .
2Let (A, I) and (B, J) be pairs such that there exists an isomorphism of rings Φ : A/I ∼ = − → B/J. Then for any open subset U ⊆ Spec(B) containing Spec(B) \ V (J), we define an open subset F A,Φ (U ) ⊆ Spec(A) as the complement of the closed subset Spec(Φ) Spec(B) \ U ⊆ Spec(A).One can define small tilts of Zariski-open subsets.
Definition 4 . 3 .
43Let ({R i } i≥0 , {t i } i≥0 ) be a perfectoid tower arising from some pair (R, I 0 ), and let ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ) be the tilt associated to (R, I 0 ). Recall that we then have an isomorphism of rings Φ R i /I 0 R i for every i ≥ 0. For every i ≥ 0 and every open subset U ⊆ Spec(R i ) containing Spec(R i ) \ V (I 0 R i ), we define U s.♭ I 0 := F R s.♭ i ,Φ (i) 0 (U ).
1 , . . . , x d /(p s.♭ ) ∼ = − → R/pR, where p s.♭ ∈ k x 1 , . . . , x d is some element. Set U := Spec(R) \ V (m). Then, since the maximal ideal m ⊆ R/pR corresponds to the (unique) maximal ideal of k x 1 , . . . , x d /(p s.♭ ), we have U s.♭ ∼ = Spec(k x 1 , . . . , x d ) \ V ((x 1 , . . . , x d )).
Example 4. 5 (
5Tilting for preperfectoid rings). Keep the notation as in Example 3.56. R 0 /I 0 is identified with the isomorphism:
Theorem 4. 6 (
6Česnavičius-Scholze). Let A be a ̟-adically Henselian ring with bounded ̟-torsion for an element ̟ ∈ A such that p ∈ ̟ p A. Assume that the ̟-adic completion of A is perfectoid.Let U ⊆ Spec(A) be a Zariski-open subset such that Spec(A) \ V (̟A) ⊆ U , and let U ♭ ⊆ Spec(A ♭ ) be its tilt (see Example 4.5).
Proposition 4 . 10 .
410Let (R, m, k) be a strictly Henselian Noetherian local normal F p -domain of dimension ≥ 2, let X := Spec(R) and fix an ideal J ⊆ m. Let {U i } i∈I be any family of open subsets of X satisfying (1), (2) and (3) as in the hypothesis of Lemma 4.9 and let U ∞ i be the F p -scheme which is the perfection of U i .
strict local scheme X and arguing as in the proof of Proposition 4.10, we have an isomorphism: (4.8) H 1 (Ué t , Z/ℓ n Z) ∼ = Pic(U )[ℓ n ] ∼ = Cl(X)[ℓ n ].
(a, b) + (c, d) = a + c, b + d + a p + c p − (a + c) p p and (a, b)(c, d) = (ac, a p d + c p b).
A
, and define d A : A → Ω A as the map induced by the universal derivation d A : A → Ω 1 A/Z . Combining both Case I and Case II together, we have a map d A :A → Ω A . Moreover, if φ : (A, m A ) → (B, m B ) isa local ring map of local rings, it gives rise to the following commutative diagram: With this in mind, one can consider the functor A → Ω A from the category of local rings (A, m A ) of residual characteristic p > 0 to the category of the k 1/p A -vector spaces Ω A . Some distinguished features in the construction above are expressed by the following proposition.
Let φ : (A, m A ) → (B, m B ) be a local ring map of Noetherian local rings such that the residual characteristic of A is p > 0. Then
Theorem 5.2 ([23, Corollary 9.6.34]). Let f 1 , . . . , f n be a sequence of elements in A, and let e 1 , . . . , e n be a system of integers with e i > 1 for every i = 1, . . . , n. Set C := A[T 1 , . . . , T n ]/(T e 1 1 − f 1 , . . . , T e n n − f n ). Fix a prime ideal n ⊆ C such that n ∩ A = m A , and let B := C n . Let E ⊆ Ω A be the k 1/p A -vector space spanned by d A f 1 , . . . , d A f n . Then the following conditions are equivalent.
Definition 5 . 3 .
53Let (A, m A , k A ) be a local ring with residual characteristic p > 0. Then we say that a sequence of elements f 1 , . . . , f n in A is maximal if d A f 1 , . . . , d A f n forms a basis of the k1/p A -vector space Ω A .
Proposition 2.7 (cf. [41, Chapter I, Proposition 4.2.7]). Let Q be an integral monoid, and let Q ′ ⊆ Q be a submonoid. Let θ : Q ′ ֒→ Q be the inclusion map, and let Z[θ] : Z[Q ′ ] → Z[Q] be the induced ring map. Set G := Q/Q ′ . Then the following assertions hold.41, Chapter I,
Proposition 4.2.7].
splits as a Z[Q ′ ]-linear map. (d) Z[θ] is equal to the canonical embedding Z[Q 0 ] ֒→ g∈G Z[Q g ]. is none other than [41, Chapter I, Proposition 4.2.7]. Moreover, (d) implies (c) obviously. Thus it suffices to show the implication (b)⇒(d). Assume that (b) is satisfied. Then one can decompose Q into the direct sum of Q ′ -modules g∈G Q g with Q 0 = Q ′ . Hence the inclusion map Q ′ ֒→ Q is equal to the canonical embedding Q 0 ֒→ g∈G Q g . Thus the induced homomorphism Z[θ] : Z[Q 0 ] ֒→ Z[ g∈G Q g ] satisfies (d), as desired. Remark 2.8. In the situation of Proposition 2.7, assume that the condition (d) is satisfied. Then the split surjection π : Z[Q] → Z[Q ′ ] has the property that π(e Q ) = e Q ′ by the construction of theG-graded decomposition Z[Q] = g∈G Z[Q g ]. Moreover, π(e Q + ) ⊆ e (Q ′ ) + because Q + ∩Q ′ ⊆ (Q ′ ) + .We use this fact in our proof for Theorem 2.27. Proposition 2.7 implies the following useful lemma. Lemma 2.9. Let Q be a fine, sharp, and saturated monoid. Let A be a ring. Then there is an embedding of monoids Q ֒→ N d such that the induced map of monoid algebrasA[Q] → A[N d ]splits as an A[Q]-linear map.Proof. Since Q is saturated, there exists an embedding Q into some N d as an exact submonoid in view of [41, Chapter I, Corollary 2.2.7]. Then by Proposition 2.7, the associated map of monoid algebras Integral sharp monoids and c-times maps. For an integral monoid Q, we denote by Q Q the submonoid of Q gp ⊗ Z Q defined as(e) Z[θ] is universally injective.
Proof. (1): By applying the functor (2.1) (that admits a right adjoint) to the decomposition Q =
g∈G Q g , we find that the assertion follows.
(2): Since Q 0 = (Q ′ ) gp ∩ Q as sets by definition, the equivalence (a)⇔(b) follows. The assertion
(a)⇔(c)⇔(e) (2.2)
Z[Q] → Z[N d ]
splits as a Z[Q]-linear map. By tensoring (2.2) with A, we get the desired split map.
2.1.2.
Proof. (1): By Lemma 2.11, it suffices to show that Q (1) c is saturated. Pick an element x of (Q c ) g ]. Thus it suffices to show that Coker(Z[ι g ]) ⊗ Z[Q] K 0 = (0), i.e., Coker(Z[ι g ]) is a torsion Z[Q]-module. On the other hand, we also have a homomorphism of Q-modules (Q (1) c ) g → Q gp ; y → y − y g , which induces an embedding of Z[Q]-modules Coker(Z[ι g ]) ֒→ Z[Q gp ]/Z[Q]. Since Z[Q gp ]/Z[Q] is Z[Q]-torsion, the assertion follows. Local log-regular rings. 2.2.1. Definition of local log-regular rings. We review the definition and fundamental properties of local log-regular rings.Definition 2.15 ([41, Chapter III, Definition 1.2.3])(1)
c ) gp
such that nx ∈ Q
(2.4)
ι g : Q ֒→ (Q (1)
c ) g ; x → x + y g ,
which induces an injective Z[Q]-linear map Z[ι g ] : Z[Q] ֒→ Z[(Q
(1)
(2): It immediately follows from the combination of Lemma 2.13 (2), Proposition 2.7 (2), and
the assertion (1) of this lemma.
2.2.
2.20 and [41, Chapter I, Proposition 3.4.1]. Then it follows from Lemma 2.9 that Q⊕N r can be embedded into N d for d > 0, and k[Q⊕N r ] → k[N d ] ∼ = k[x 1 , . . . , x d ] splits as a k[Q⊕N r ]-linear map. Applying [29, Theorem 3.1], we see that k[Q ⊕ N r] is strongly F -regular. After completion, the complete local ring k Q ⊕ N r is strongly F -regular in view of[1, Theorem 3.6]. Then by faithful flatness of R → k Q ⊕ N r , [29, Theorem 3.1] applies to yield strong F -regularity of R.
2.3.Log-regularity and splinters. Local log-regular rings have another notable property; they are splinters. Let us recall the definition of splinters.Definition 2.25. A Noetherian ring A is a splinter if every finite ring map f : A → B such that Spec(B) → Spec(A) is surjective admits an A-linear map h : B → A such that h • f = id A .In general, it is not easy to see under what algebraic operations being a splinter is preserved. For instance,Datta and Tucker proved remarkable results ([13, Theorem B], [13, Theorem C], or [13, Example 3.2.1]). See also Murayama's work
the natural quotient maps, and (t i ) red is obtained from t i by killing out the nilradical part. Recall that a universal homeomorphism induces an equivalence of respective categories of finiteétale algebras in view of [47, Tag 0BQN]. By [47, Tag 054M], the maps φ i and φ i+1 are universal homeomorphisms. By the axiom (b) in Definition 3.6,
Definition 3.29.In the situation of Proposition 3.28). Thus, by the axioms (e) and (f) in Definition 3.22 and Nakayama's
lemma, we obtain I p
i+1 = I i R i+1 as desired.
(3): By the axiom of (dependent) choice, the existence follows from the axiom (d) in Definition
3.22. The uniqueness is due to Lemma 3.26.
3.38. For every i ≥ 0, R s.♭ i is I s.♭ 0 -adically complete and separated.To discuss perfectoidness for the tilt ({R s.♭ i } i≥0 , {t s.♭ i } i≥0 ), we introduce the following maps.Proof. By Theorem 3.37, the ideal I s.♭
0 R s.♭
i ⊆ R s.♭
i is principal. Hence one can apply Proposition
3.11 (1) to deduce the assertion.
Definition 3.39. For every i ≥ 0, we define a ring map (
is a purely inseparable tower arising from (R s.♭ 0 , I s.♭ 0 ) with Frobenius projections {F s.♭ i } i≥0 , and satisfies the the axiom (d) in Definition 3.22. Moreover, Lemma 3.41 also implies that
Hence the axiom (f) follows from Theorem 3.37(1). The axiom (e) holds by Lemma 3.38. Let us check that the axiom (g) holds. By Corollary 3.20, I s.♭0 (R s.♭
i ) I s.♭
0 -tor = (0). Let (t s.♭
i ) tor :
(R s.♭
i ) I s.♭
0 -tor → (R s.♭
i+1 ) I s.♭
0 -tor be the restriction of t s.♭
i . Then by Theorem 3.37 (2), there exists a
bijection (F s.♭
i ) tor : (R s.♭
i+1 ) I s.♭
0 -tor → (R s.♭
i ) I s.♭
0 -tor such that (t s.♭
i ) tor • (F s.♭
i ) tor = F (R s.♭
i+1 ) I s.♭
0 -tor
(cf.
Lemma-Definition 3.21). Thus we have
where the right hand side is the ideal of R s.♭j
generated by f s.♭
j . Thus we obtain the desired implication. Finally, note that by Proposition
3.28 (3), we can take a system of elements {f s.♭
j ∈ R s.♭
j } j≥0 satisfying the condition (b) such that
(f s.♭
j+1 ) p = f s.♭
j (j ≥ 0).
(2): We have I s.♭
0 (R s.♭
j ) I s.♭
0 -tor = (0) by Corollary 3.20. Hence by the assertion
i+1) . Then the tower ({S i } i≥0 , {u i } i≥0 ) is a perfect tower. Indeed, each S i is reduced by Theorem 2.19; moreover, by the perfectness of k and Lemma 2.11
and the definition of itself. This implies that the image of the empty subset of R in k forms a maximalsequence. Hence R ′′
i in Construction 3.58 is equal to R. Moreover, (N d ) (i) is generated by
1
p i e 1 , . . . , 1
p i e d . Thus, applying Construction 3.58, we obtain
Fix a Zariski-open subset U ⊆ Spec(R) such that Spec(R) \ V (pR) ⊆ U and the corresponding open subset U s.♭ ⊆ Spec(R s.♭ ) (cf. Definition 4.3). Then, for any fixedi, n ≥ 0 such that |H i (U s.♭ et , Z/ℓ n Z)| < ∞, one has |H i (Ué t , Z/ℓ n Z)| ≤ |H i (U s.♭ et , Z/ℓ n Z)|. In particular, if H i (U s.♭ et , Z/ℓ n Z) = 0, then H i (Ué t , Z/ℓ n Z) = 0.Remark 4.8. One can formulate and prove the version of Proposition 4.7 for theétale cohomology with support in a closed subscheme of Spec(R), using Theorem 4.6. Then the resulting assertion gives a generalization ofČesnavičius-Scholze's argument in [9, Theorem 3.1.3] which is a key part of their proof for the absolute cohomological purity theorem. One of the advantages of Proposition 4.7 is that it can be used to answer some cohomological questions on possibly singular Noetherian schemes (e.g. log-regular schemes) in mixed characteristic.4.2.Tilting the divisor class groups of local log-regular rings. We need a lemma of Grothendieck on the relationship between the divisor class group and the Picard group via direct limit. Its proof is found in[25, Proposition (21.6.12)] or [26, XI Proposition 3.7.1]. Lemma 4.9. Let X be an integral Noetherian normal scheme, and let {U i } i∈I be a family of open subsets of X. Consider the following conditions.
This argument is due to Ogus. See the proof of [41, Chapter III, Theorem 1.11.2 (2)].
One notices that the treatment of logarithmic geometry in[23] is topos-theoretic, while[32] considers mostly the Zariski sites.
The symbol I [p n ] for an ideal I in an Fp-algebra A is the ideal generated by the elements x p n for x ∈ I.
This condition is realized if R0 is I-adically Henselian and each ti : Ri → Ri+1 is integral.
The axiom (f-2) follows from the normality of Ri. The other axioms are clearly satisfied.
The existence of such towers is quite essential for applications toétale cohomology, because the extension degree of each Rj → Rj+1 is controlled in such a way that the p-adic completion of its colimit is a perfectoid ring.
It is not obvious whether R s.♭ j is normal. However, the normality was used only in the trace argument and we do not need it in the following argument.
Proof of Proposition 3.66. We have constructed an infinite extension R → R ∞ such that if R ∞ is the f 0 -adic completion, then the associated Tate ring R ∞ [ 1 f 0 ] is a perfectoid ring in the sense of Fontaine by Theorem 3.54 and[5,Lemma 3.21].By Lemma 2.26 and Lemma 3.67, it follows that the map R[ 1splits in the category of topological R[ 1 f 0 ]-modules (notice that R is f 0 -adically complete and separated). Thus, R[ 1 f 0 ] is a sousperfectoid Tate ring. The combination of[27,Corollary 8.10],[27,Proposition 11.3]and[27,Lemma 11.9] allows us to conclude that R[ 1 f 0 ] is stably uniform. As a corollary, one can obtain the stable uniformity for complete local log-regular rings (see also Construction 3.58 and Theorem 2.27). Proof. Since each R j is a p-adically Henselian normal domain, so is R ∞ = lim − →j≥0 R j . Moreover, every prime ℓ different from p is a unit in R j and R ∞ . Attached to the tower ({R j } i≥0 , {t j } j≥0 ), we get a tower of finite (not necessarily flat) maps of normal schemes:More precisely, let h j : Spec(R j+1 ) → Spec(R j ) be the associated scheme map. Then the open set U j+1 is defined as the inverse image h −1 j (U j ), thus defining the map U j+1 → U j in the tower (4.3). Since h j is a finite morphism of normal schemes,[3,Lemma 3.4]applies to yield a well-defined trace map: Tr :is multiplication by the generic degree of h j (=p-power order). Then this is bijective, as the multiplication map by p on Z/ℓ n Z is bijective. We have the natural map:. Composing these maps, the composite map (4.4) inducesand the composition is bijective. Since h * j Z/ℓ n Z ∼ = Z/ℓ n Z, we get an injection (4.5)Since each morphism U j+1 → U j is affine, by using (4.5) and[47,Tag 09YQ], we haveThus, it suffices to show thatHence by tiltingétale cohomology using Theorem 4.6, we are reduced to showingOn the other hand, considering the tilt of (. Now the combination of Lemma 3.57 and Theorem 4.1 (2) together with the assumption finishes the proof of the theorem.(2) Let R 1/p ∞ denote the J-adic completion of R 1/p ∞ . If moreover each U i has the property that X \ V (J) ⊆ U i , then for any prime ℓ = p,Proof. Let us begin with a remark on the direct limit ofétale cohomology groups. Note that for the transition morphism g :Z/ℓ n Z , which defines the direct system of cohomology groups.(1): First we prove the following claim:• There is an injection of abelian groups:for any n ∈ N, where U ⊆ X is an open subset whose complement is of codimension ≥ 2.To prove this, consider the Kummer exact sequencewhere the identification ofétale sheaves µ ℓ n ∼ = Z/ℓ n Z follows from the fact that R is strict Henselian (one simply sends 1 ∈ Z/ℓ n Z to the primitive ℓ n -th root of unity in R). Let U ⊆ X be an open subset with its complement V = X \ U having codimension ≥ 2. Then we have an exact sequence ([36, Proposition 4.9; Chapter III]):. Since R is strict local and ℓ = p, Hensel's lemma yields that R × = (R × ) ℓ n . Moreover, since codim X (V ) ≥ 2 and X is normal, we have Γ(Ué t , G m ) = R × . Thus, H 1 (Ué t , Z/ℓ n Z) ∼ = Pic(U )[ℓ n ]. Note that Pic(U ) ֒→ Cl(U ) restricts to Pic(U )[ℓ n ] ֒→ Cl(U )[ℓ n ]. Moreover, the natural homomorphism Cl(X) → Cl(U ) is an isomorphism, thanks to codim X (V ) ≥ 2. Hence H 1 (Ué t , Z/ℓ n Z) ∼ = Pic(U )[ℓ n ] ⊆ Cl(X)[ℓ n ], which proves the claim.Since R is normal, the regular locus has complement with codimension ≥ 2. Using this fact, we can apply Lemma 4.9 to get an isomorphism Cl(X)[ℓ n ] ∼ = lim − →i∈I H 1 (U i )é t , Z/ℓ n Z . Byétale invariance of cohomology under taking perfection of F p -schemes ([47, Tag 03SI]), we getas desired.(2): Since R is Henselian along m and J ⊆ m, it is Henselian along J by [47, Tag 0DYD]. Moreover, the perfect closure of R still preserves Henselian property along J. Theorem 4.1 yieldsZ/ℓ n Z and the conclusion follows from (1). Proposition 4.11. Let A be a Noetherian ring with a regular element t ∈ A such that A is tadically Henselian and A → A/tA is the natural surjection between locally factorial domains. Pick an integer n > 0 that is invertible on A. Then if Cl(A) has no torsion element of order n, the same holds for Cl(A/tA). If moreover A is a Q-algebra and Cl(A) is torsion-free, then so is Cl(A/tA).Proof. The Kummer exact sequence 0 → µ n → G m ( ) n −−→ G m → 0 induces the following commutative diagram:− −−− → Pic(A/tA) By Theorem 4.1, the map α is an isomorphism. Then if Pic(A) has no torsion element of order n, δ 1 is the zero map. This implies that δ 2 is also the zero map and hence, Pic(A/tA) has no element of order n. Since both A and A/tA are locally factorial by assumption, we have Cl(A) ∼ = Pic(A) and Cl(A/tA) ∼ = Pic(A/tA). So the assertion follows.It is not necessarily true that δ 1 (resp. δ 2 ) is injective, because we do not assume A to be strictly Henselian.Lemma 4.12. Let (R, Q, α) be a log-regular ring. Then a strict Henselization (R sh , Q, α sh ) is also a log-regular ring where α sh : Q → R → R sh is the composition of homomorphisms.On the other hand, there is a perfectoid tower of module-finite extensions of local log-regular rings arising from (R, (p)):Notice that each map is generically of p-power rank in view of Lemma 2.14 (2) and Lemma 2.12(4). Moreover, the tilt of (4.9) (associated to (R, (p))) is given by Remark 4.14. One can also deduce a special case of Polstra's result on the divisor class group of a strongly F -regular local F p -domain R, usingétale cohomology and the main result of[8]. Recall that for a connected (separated) Noetherian scheme X, any integer n > 0 and a finite abelian group G, there are isomorphisms:H 1 (Xé t , G) ∼ = Hom cont (πé t 1 (X), G) ∼ = Hom cont (π ab 1 (X), G), where π ab 1 (X) is the maximal abelian quotient of theétale fundamental group πé t 1 (X). (4.12) is found in[16,Proposition 5.7.20]via an interpretation of classifying G-torsors over X. Let us replace R with R sh by [1, Theorem 3.6]. Since R is normal, the non-singular locus U ⊆ X := Spec(R) is of codimension ≥ 2, and Lemma 4.9 gives an isomorphism: Cl(X) ∼ = Pic(U ). On the other hand, we know |πé t 1 (U )| < ∞ by[8,Theorem 5.1]. For any prime ℓ = p, (4.13)Cl(X)[ℓ n ] ∼ = H 1 (Ué t , µ ℓ n ) ∼ = Hom cont (πé t 1 (U ), µ ℓ n ) by(4.12). Then the finiteness of πé t 1 (U ) implies that (4.13) vanishes for almost all ℓ = p, while the right-hand side of (4.13) is bounded for a fixed ℓ and varying n by Pontryagin duality for finite abelian groups (one notices that the sheaf µ ℓ n is constant, because we are assuming that R is strictly Henselian). In conclusion, Cl(R) tor ⊗ Z[ 1 p ] is finite. We should note that Polstra proved that Cl(R) tor is indeed finite and his proof is more elementary.Example 4.15. It is usually hard to compute the divisor class group of a complete local normal domain. We construct an example that fits into the setting of Theorem 4.13. Now, let us consider the local log-regular ring (R, Q, α R ) defined as in Example 3.64(2). Assume that k is an algebraically closed field of characteristic p > 0. Then we have
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| []
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[
"NETWORK PERFORMANCE ESTIMATOR WITH APPLICATIONS TO ROUTE SELECTION FOR IOT MULTIMEDIA APPLICATIONS",
"NETWORK PERFORMANCE ESTIMATOR WITH APPLICATIONS TO ROUTE SELECTION FOR IOT MULTIMEDIA APPLICATIONS"
]
| [
"Fabiano Bhering \nUniversidade Federal Fluminense (UFF) Niterói -RJ\nBrazil\n",
"Diego Passos \nUniversidade Federal Fluminense (UFF) Niterói -RJ\nBrazil\n",
"Célio Albuquerque \nUniversidade Federal Fluminense (UFF) Niterói -RJ\nBrazil\n",
"Katia Obraczka \nUnivesity of California\nSanta Cruz (UCSC) Santa Cruz -CAUSA\n"
]
| [
"Universidade Federal Fluminense (UFF) Niterói -RJ\nBrazil",
"Universidade Federal Fluminense (UFF) Niterói -RJ\nBrazil",
"Universidade Federal Fluminense (UFF) Niterói -RJ\nBrazil",
"Univesity of California\nSanta Cruz (UCSC) Santa Cruz -CAUSA"
]
| []
| Estimating the performance of multimedia traffic is important in numerous contexts, including routing and forwarding, QoS provisioning, and adaptive video streaming. This paper proposes a network performance estimator which aims at providing, in quasi real-time, network performance estimates for IoT multimedia traffic in IEEE 802.11 multihop wireless networks. To our knowledge, the proposed multimedia-aware performance estimator, or MAPE, is the first deterministic simulation-based estimator that provides real-time per-flow throughput, packet loss and delay estimates while considering inter-flow interference and multi-rate flows, typical of multimedia traffic. Our experimental results indicate that MAPE is able to provide network performance estimates that can be used by IoT multimedia services, notably to inform real-time route selection in IoT video transmission, at a fraction of the execution time when compared to stochastic network simulators. When compared to existing deterministic simulators, MAPE yields higher accuracy at comparable execution times due to its ability to consider multi-rate flows.As will be discussed in more detail in Section 2, different performance estimators have been proposed but do not fulfill the needs of IoT multimedia applications which require estimators to account for multi-rate flows as well as inter-flow interference, while being able to provide their estimates in a timely and resource-efficient manner.In this paper we propose the Multimedia-Aware Performance Estimator, or MAPE for short, which estimates network performance for multi-rate multimedia flows using their video coding rate as input. To the best of our knowledge, MAPE is the first estimator that is able to provide throughput, packet loss and delay estimates in real time considering rate-heterogeneous flows and accounting for inter-flow interference.Experiments using different IoT multimedia application scenarios demonstrate that MAPE is able to provide real time network performance estimates, i.e., throughput, delay, and packet loss, with savings of over two orders of magnitude in execution time when compared to the ns-3 [8] network simulator. Furthermore, we show how MAPE can be used to improve video transmission quality by guiding route selection on a per-flow basis.The remainder of this paper is organized as follows: Section 2 reviews related work on network performance estimation in IEEE 802.11 networks. Section 3 describes MAPE's design and operation in detail. Our experimental methodology, and results from our evaluation of MAPE's accuracy are reported in Sections 4 and 5, respectively. Section 6 shows how MAPE can be used to guide route selection in order to improve video transmission quality. Finally, Section 7 concludes the paper and presents directions for future work. | 10.1177/00375497231156618 | [
"https://arxiv.org/pdf/2203.15126v1.pdf"
]
| 247,779,209 | 2203.15126 | 01a38a7a25cca13023aa397416dd8107405108ac |
NETWORK PERFORMANCE ESTIMATOR WITH APPLICATIONS TO ROUTE SELECTION FOR IOT MULTIMEDIA APPLICATIONS
Fabiano Bhering
Universidade Federal Fluminense (UFF) Niterói -RJ
Brazil
Diego Passos
Universidade Federal Fluminense (UFF) Niterói -RJ
Brazil
Célio Albuquerque
Universidade Federal Fluminense (UFF) Niterói -RJ
Brazil
Katia Obraczka
Univesity of California
Santa Cruz (UCSC) Santa Cruz -CAUSA
NETWORK PERFORMANCE ESTIMATOR WITH APPLICATIONS TO ROUTE SELECTION FOR IOT MULTIMEDIA APPLICATIONS
Network Performance EstimatorIoT Multimedia ApplicationsPerformance PredictionsWireless Route Selection
Estimating the performance of multimedia traffic is important in numerous contexts, including routing and forwarding, QoS provisioning, and adaptive video streaming. This paper proposes a network performance estimator which aims at providing, in quasi real-time, network performance estimates for IoT multimedia traffic in IEEE 802.11 multihop wireless networks. To our knowledge, the proposed multimedia-aware performance estimator, or MAPE, is the first deterministic simulation-based estimator that provides real-time per-flow throughput, packet loss and delay estimates while considering inter-flow interference and multi-rate flows, typical of multimedia traffic. Our experimental results indicate that MAPE is able to provide network performance estimates that can be used by IoT multimedia services, notably to inform real-time route selection in IoT video transmission, at a fraction of the execution time when compared to stochastic network simulators. When compared to existing deterministic simulators, MAPE yields higher accuracy at comparable execution times due to its ability to consider multi-rate flows.As will be discussed in more detail in Section 2, different performance estimators have been proposed but do not fulfill the needs of IoT multimedia applications which require estimators to account for multi-rate flows as well as inter-flow interference, while being able to provide their estimates in a timely and resource-efficient manner.In this paper we propose the Multimedia-Aware Performance Estimator, or MAPE for short, which estimates network performance for multi-rate multimedia flows using their video coding rate as input. To the best of our knowledge, MAPE is the first estimator that is able to provide throughput, packet loss and delay estimates in real time considering rate-heterogeneous flows and accounting for inter-flow interference.Experiments using different IoT multimedia application scenarios demonstrate that MAPE is able to provide real time network performance estimates, i.e., throughput, delay, and packet loss, with savings of over two orders of magnitude in execution time when compared to the ns-3 [8] network simulator. Furthermore, we show how MAPE can be used to improve video transmission quality by guiding route selection on a per-flow basis.The remainder of this paper is organized as follows: Section 2 reviews related work on network performance estimation in IEEE 802.11 networks. Section 3 describes MAPE's design and operation in detail. Our experimental methodology, and results from our evaluation of MAPE's accuracy are reported in Sections 4 and 5, respectively. Section 6 shows how MAPE can be used to guide route selection in order to improve video transmission quality. Finally, Section 7 concludes the paper and presents directions for future work.
Introduction
Efficient transmission of multimedia traffic in multihop wireless networks poses significant challenges mainly due to their more stringent Quality of Service (QoS) requirements (e.g., throughput and delay), especially in the case of real-time applications [1]. Additionally, multihop wireless communication is inherently more prone to losses and congestion; for instance, the performance of a single wireless link can vary due to factors such as link-layer transmission rate, its signal-to-noise-ratio (SNR), and complex propagation phenomena. Furthermore, transmission of multiple flows that are not limited by rate control mechanisms can also cause congestion, as well as inter-flow interference, medium access contention, and collisions [2]. And, in the specific case of multimedia traffic, even though compression techniques use a pre-defined average data rate as a target, the actual data rate of the compressed flow may vary considerably depending on scene complexity, flow resolution, and the different types of frames [3].
Estimating network performance is an effective mechanism to address the challenges raised by multimedia traffic as a way to achieve QoS-aware admission control, resource provisioning and allocation in multihop wireless networks [4]. It allows estimating current available network capacity as well as deciding whether the network can fulfill each flow's requirements. In addition, accurate multimedia performance estimates are useful for routing and video coding decisions [5,1,6].
There is a wide variety of IoT (Internet of Things) multimedia applications that can benefit from a real-time network performance estimator to route selection [5], such as surveillance systems for outdoor or indoor spaces in smart cities that require multiple video sources transmitting simultaneously to the monitoring center [7]. Note that the performance of these application scenarios can vary according to the selected route for each video flow, as this may cause inter-flow interference.
Related Work
IEEE 802.11 networks have offered several attractive rate-capable amendments that serve various multimedia application scenarios [1]. Providing performance estimates is critical to meet QoS guarantees in such networks. Existing approaches to network performance estimation in IEEE 802.11 networks can be classified in three main categories, namely: mathematical models, online estimators, and discrete-event simulators.
Mathematical models Estimators based on mathematical models typically make simplifying assumptions to make modeling tractable. For instance, most existing proposals target one-hop flows [9,10,11]. Moreover, they make additional simplifications, such as perfect links and identical transmission rates for all nodes. In the context of per-flow performance estimation, Laufer and Kleinrock [12] present a more complete model for analyzing the throughput of CSMA/CA networks. This model estimates the maximum throughput for each flow by modeling the network behavior as a system of non-linear equations and solving the resulting optimization problem. That approach can become prohibitively expensive for larger networks, as the size of the system of equations grows exponentially with the number of network nodes.
Online Estimators While mathematical models for performance estimation are useful to understand the limits of contention-based medium access protocols, approaches that can be operated online are required in practice, e.g., for real-time applications such as adaptive video streaming [13,14,15,16,17,18,19,6,20] and routing protocols [21]. In particular, performance estimation for adaptive video streaming is discussed in [16,18]. These studies also consider buffer occupancy information for predicting performance to improve video streaming quality of experience (QoE). The work reported in [20] proposes a method to reduce the impact of inaccurate throughput prediction on QoE by controlling the buffer occupancy within a safe range. In turn, routing metrics provide indirect information that is expected to correlate well with throughput [21], but they usually fail to evaluate the interference between flows.
Discrete-Event Simulators Discrete event simulators can be stochastic or deterministic. Stochastic simulators use pseudo-random number generators to determine the outcomes of events that have some level of randomness (e.g., the choice of backoff intervals for medium access), while deterministic simulators replace pseudo-random generation with deterministic values (e.g., a fixed average backoff interval).
Network performance estimation performed by stochastic simulators like ns-3 [8] and OMNET++ [22] is commonly used to either conduct an a-priori evaluation of a certain network and its protocols, guide network provisioning, deployment or operational tasks. Because of their random nature, they usually require a large enough number of runs for every experimental configuration in order to obtain statistically meaningful results, which adds to their inherent scalability limitations, long execution times, and high computational resource needs. On the other hand, deterministic estimators provide an adequate accuracy with identical results no matter how many times they are run. However, they must be designed to perform in real-time while the network operates to help dynamically adjust operational parameters.
One notable example of this latter class of performance estimators is AFTER [23]. It was proposed to tackle the problem of real-time throughput estimation for multihop IEEE 802.11 networks. AFTER simulates the behavior of the link-and network layers to quickly converge to steady state behavior that allows it to estimate the long term average throughput of each flow for a given set of application flows and corresponding routes. To this end, it maintains in memory a complete view of the network topology and performs a deterministic simulation of the network dynamics, generating simulated virtual packets (v_packets) for each flow at their respective virtual source nodes, triggering a number of other relevant simulation events, such as wireless medium access, queue management (v_packets being added, removed and discarded from buffers) and, eventually, the delivery of v_packets to their virtual destination nodes. In particular, AFTER takes into account inter-flow interference, employing a set of deterministic rules to deal with nodes competing to access the wireless medium. However, AFTER cannot handle arbitrary traffic models because it seeks to estimate the maximum achievable network throughput by considering each flow to have an infinite backlog at the source. This means that AFTER provides no support for scenarios in which multimedia applications themselves limit the transmission rate of each flow.
In summary, to our knowledge, MAPE is the first deterministic performance estimator that takes into account both inter-flow interference and heterogeneous flows, i.e., flows with different data rates, while being able to be executed in real-time.
Multimedia-Aware Performance Estimator
As discussed in Section 2, although a number of performance estimation approaches have been proposed, none of them is able to provide real-time performance estimates that account for both inter-flow interference and rate-heterogeneous flows. The proposed Multimedia-Aware Performance Estimator, or MAPE, tries to fill this gap and uses a deterministic simulation-based approach to estimate the long term average throughput, packet loss and end-to-end delay for all (multi-rate) flows considering inter-flow interference. Note that considering multi-rate flows is essential to more realistically reproduce the behavior of multimedia applications. For instance, in video applications, transmission rates are determined by video coding at each source and, therefore, each flow can be transmitted at different rates.
MAPE -Design and Operation
Algorithm 1 illustrates MAPE's overall operation, which is divided in three steps:
Step 1 -MAPE starts with a complete snapshot of the current network state as input consisting of a representation of the network topology that includes link quality estimates (i.e., link frame delivery probability), list of currently active flows along with the respective paths, and each flow's data rate;
Step 2 -MAPE then uses the initial network snapshot to simulate the network as it operates until reaching steady state, which is used to compute long term throughput, packet loss, and end-to-end delay estimates in Step 3. Note that we employ the term steady state in the same sense as in [23], i.e., as a finite cycle of states that repeats themselves. At the end of each iteration, MAPE stores a snapshot of the current network state, which consists of currently ongoing transmissions with their respective remaining times, the content of the queues and the backoff counter of the wireless medium access for all nodes that are traversed by any flow on the evaluated flow set, and the current medium access priority list. To decide whether the steady state has been achieved, the current state is compared to all previous ones. Whenever a duplicate state is found, MAPE declares that steady state has been reached and computes the average throughput, packet loss rate and end-to-end delay for each flow. A heuristic stop criterion is also used to guarantee low execution time and adequate real-time performance independent of application scenarios. When duplicated states are not found, MAPE computes the average cycle performance of events within which at least one packet from each flow has been delivered to its final destination as an attempt to approximate steady state performance.
Unlike stochastic simulators that study network behavior over a predefined period of time, MAPE aims at estimating the performance of the network, e.g., throughput, packet loss, and end-to-end delay at steady-state. This can be especially useful for QoS provisioning and, as previously noted, for route selection in real-time multimedia applications. Additionally, as discussed in Section 2, deterministic simulators that assume rate-homogeneous flows may result in severely inaccurate estimates for a number of reasons. First of all, the performance of a flow is necessarily limited by its transmission rate. Thus, such simulators may grossly overestimate performance in scenarios where network capacity is much larger than the aggregate demand of the active flows. Furthermore, severe underestimates may also occur for individual flows because interfering flows may be transmitted at a higher rate, consuming more network resources than they would in reality, reducing the achievable performance of other flows. MAPE overcomes these limitations by explicitly accounting for both multi-rate flows and inter-flow interference and thus attains more accurate performance estimates in more realistic multimedia application scenarios.
While MAPE builds on "traditional" deterministic estimators such as AFTER [23], unlike these estimators, MAPE relaxes the assumption that all flows have infinite backlogs and instead generates v_packets according to the rate of each flow -which can be specified as an input, based on the flows' video coding rate, for instance. Whenever invoked, MAPE receives flow rate arguments as input and uses them deterministically to simulate the network dynamics by: (1) generating simulated v_packets for each flow at their respective source nodes, (2) triggering a number of other relevant simulation events, such as wireless medium access transmission, queue management (v_packets being added, removed and discarded from buffers), and, (3) eventually, delivering v_packets to their destination nodes. As such, inter-flow interference happens as a result of buffer overflow, link-layer transmission losses, and medium access conflicts.
MAPE -Implementation
MAPE's current implementation uses AFTER [23] as the underlying deterministic performance estimator. As shown in Figure 1, MAPE starts by initializing the simulation state with its input arguments. In this phase, the first packet of each flow is added to the queue of the respective source node and the simulation time is initialized to keep track of the events that are used to generate scheduled v_packets. Thus, the main loop of the simulation starts with the advance of the simulation according to the time of next possible events. This loop also handles packet receptions and eventually generates new transmission events until it detects that the network has reached a steady state -a state when the same sequence of events starts to repeat itself -which informs MAPE that it can then compute the estimated performance of each flow. MAPE uses a representation of the current simulation state, which includes information about all received v_packets. Furthermore, the SPR module implements a procedure to schedule the next packet generation for each flow according to the specified rate and keeps track of the number of v_packets received per flow, which is used to calculate performance estimates for each flow. The number of received v_packets is also used to determine the steady state cycle -i.e. the shortest sequence of network states that repeats itself indefinitely on the steady state of the simulation. While in the original AFTER the network state is updated when an v_packet from any flow arrives at the destination, in the SPR module the condition was modified to update it only when all flows receive at least one v_packet.
To simulate v_packet transmissions, MAPE starts by placing the initial v_packet of each flow on the queue of the respective source node. It then iterates through all nodes that have at least one v_packet on their queues and triggers events for dequeuing a v_packet and adding this v_packet to a transmission buffer, where the v_packet is stored while waiting for an opportunity to be transmitted.
Note that MAPE's SPR Module introduces a mechanism to schedule the next v_packet generation for each flow according to the specified rate. Once per-flow rates have been specified, the scheduler uses them to place new v_packets in each source node's queue until the steady state is detected. Thus, v_packets of each flow are generated according to the intervals of the simulation time. To keep track of the simulated time, MAPE uses a time variable that is updated according to the end time of a link layer transmission attempt and the backoff procedure.
Once the simulation reaches steady state, MAPE computes the average throughput of each flow as the ratio between the total number of v_packets delivered within the last steady state cycle -i.e., the period between two repeating simulation events -the steady state cycle length. In addition, the SPR Module computes packet loss and end-to-end delay by tracking all v_packets from the instant when they are generated at their source nodes until they are received at their destinations. MAPE is then able to estimate the average per-flow packet loss rate and end-to-end delay. Such metrics account for the data link layer transmission attempts and queuing delays.
Discussion
While MAPE makes assumptions about network events and convergence to steady state, our experimental evaluation (see Section 5) shows that MAPE is still able to estimate per-flow performance with adequate accuracy in quasi real time.
Note that MAPE uses information about the topology of the network and the driving application (e.g., multimedia sources, flow rates), and, in the application scenarios envisioned (e.g., Smart Cities, Industrial Automation), nodes are typically stationary and have access to continuous power sources. As such, frequent topology changes (and energy limitations) are not expected to play a significant role. In scenarios where topology changes need to be considered, topology updates can be conveyed by proactive routing protocols.
Route selection is an example of how MAPE can be used in practice. The routing protocol would invoke MAPE with an up-to-date network snapshot as input. Then, based on MAPE's performance estimates, it would perform route selection accordingly. For instance, a proactive link-state algorithm (e.g., OLSR [24]) can periodically discover topology changes and disseminate this information through link state updates that MAPE can use to adjust its estimates. Network topology information would be updated whenever a node identifies "significant" changes in the network topology, e.g., link failures, new nodes/links or changes in link quality. As part of our experimental evaluation (see Section 6), we show how MAPE can guide route selection and, as a result, improve video transmission quality.
In its current implementation, MAPE assumes that flows are transmitted at constant bitrate. However, multimedia applications typically employ variable bitrate transmission. One way to address this is to simply have MAPE use the flow's average bitrate, which can be determined during transmission. Another approach to handle dynamic traffic patterns is to provide MAPE with updated data rate information whenever significant transmission rate changes are detected in the video coding process. In this work, we use the average bitrate of each video trace as input to MAPE. As part of future work, We plan to add support to variable bit rate flows.
Evaluation Methodology
We evaluated MAPE against two types of discrete event simulators, stochastic (ns-3) and deterministic (AFTER). We chose ns-3 because it is widely used by the network researchers and practitioners since it provides an adequate model of the network and, thus, provides reliable estimates of network performance. We use AFTER as the example of a deterministic simulator and demonstrate that MAPE can achieve better accuracy by being able to model specific per-flow rates, i.e., it simulates each flow transmitting at specified multimedia bitrates.
In this section, we describe the experimental methodology we use to evaluate MAPE, including the topologies and traffic models considered, as well as how the experiments were carried out.
Experimental Topologies
We evaluate MAPE using two different IoT wireless network topologies akin of IoT scenarios and whose parameters are summarized in Table 1. The Random Indoor topology aims to replicate smart building scenarios and was generated by placing nodes randomly within an indoor environment. The Grid Outdoor topology tries to simulate smart city scenarios of neighborhood blocks and streets in an urban region, represented by a grid. More specifically, we reproduced a region of the city of Niterói, in the state of Rio de Janeiro, Brazil using an 8 × 7 grid of nodes in which two consecutive nodes are placed 60m and 70m apart on a given line and column of the grid, respectively. For a fair comparison between the two simulators, we set up the same link speeds, queue sizes and packet lifetime policy on ns-3 and MAPE. The Shadowing and Cost231 propagation models [25] were chosen to more realistically reproduce indoor and urban environments. All simulations use the same MAC and PHY technology and the same link speed, which was chosen to support multimedia application scenarios.
In order to estimate link quality (an information that is required by MAPE), a series of preliminary simulations were performed using the ns-3 simulator. For all nodes in each topology, we executed a simulation transmitting 20, 000 packets to extract the long term quality of each link.
Traffic Models
In addition to link quality, MAPE requires per-flow transmission rate information. In our experiments, we use a mix of three different rates (as shown in Table 2) to represent different levels of video quality. The EvalVid framework [26] was used to generate traces of the same video clip with these three rates, and the resulting average bitrate of each video trace was used as input to MAPE. Additional traffic generation parameters and their values used in our simulation experiments are listed in Table 2. Experiments which used multimedia (MM) traffic employ a publicly available and commonly used video clip, namely "Hall Monitor" [3], which was converted to H.264 format with a rate of 30 frames per second. Considering real-time transmission delay and human tolerance, the play-out buffer is set to 300ms to mitigate potential out-of-order packets; packets with delay longer than 300ms are discarded at the decoder.
In video traffic, transmission rates may vary according to the coding technique used. For example, more important video frames (e.g., MPEG I-frames) are often transmitted at higher rates than the target compression bitrate, while less important frames (e.g., MPEG P-frames and B-frames) are transmitted at lower rates. In our experiments, multimedia (MM) traffic target bitrates used by MAPE are based on long-term average bitrates calculated at the video source encoder. Because MAPE currently models variable bit rate flows using their long-term average rates, we also ran experiments with CBR traffic in our ns-3 simulations in order to assess how short-term fluctuations of the video traffic bitrate affect MAPE's estimates. In those experiments, we adopt the same bitrates used as input for AFTER and MAPE as listed in Table 2. As part of our future work (see Section 7), we will modify MAPE's current variable bit rate traffic model to be able to account for shorter-term transmission rate variations.
Experiments
Simulation experiments were conducted as follows. For each topology, we computed the 5 best paths (based on the quality of their links) for 500 source-destination pairs generated randomly. Selecting one path for each pair, out of their 5 best, we generated random instances for scenarios with 3, 6, 9, and 12 pairs (or flows), which are used to transmit concurrent video flows with 3 different levels of quality -a third of the flows use each of the three transmission rates listed in Table 2. For instance, in a scenario with 6 flows, we have 2 sources transmitting at 256 kb/s, 2 sources transmitting at 512 kb/s, and 2 sources transmitting at 1024 kb/s. We left out the evaluations of scenarios with more than 12 flows because the networks become saturated. These scenarios do not provide satisfactory support for video applications, so they are not relevant for this work.
Finally, all scenarios were also executed in the ns-3 simulator for both the CBR and MM traffic models using a simulation time of 120s. For each scenario, execution time, per-flow throughput, end-to-end delay, and packet loss were computed by averaging results over all runs.
Evaluation Metrics
We evaluate MAPE's performance according to execution time and prediction accuracy. Since ns-3 is a packet-level simulation platform, we are using its throughput, packet loss and end-to-end delay statistics as the ground truth in our performance study. Throughput is calculated as the ratio between the number of packets delivered to the destination and simulation time. End-to-end delay is the time interval between when a packet is transmitted by the source node and when that packet is delivered at the destination, averaged over all packets received, and packet loss is calculated as the percentage of packets transmitted that were not delivered to the destination.
We expect MAPE to achieve predictions close to those of ns-3, but in a reproducible manner and at a fraction of the required execution time. We also use SSIM [27] and another metric called classification inversions -as defined in [23], and further explained in Subsection 6.2 -to evaluate video quality and demonstrate the practical suitability of MAPE to the problem of route selection for multimedia applications.
MAPE's Accuracy Evaluation
Our experimental evaluation aims at demonstrating MAPE's ability to accurately estimate per-flow throughput, delay and packet loss in a timely manner when compared to estimates provided by existing stochastic and deterministic simulators. To this end, we compare MAPE against ns-3 [8] and AFTER [23] by considering the trade-off between execution time and throughput, delay and packet loss estimate accuracy.
Execution Time
We measure average execution time for AFTER, MAPE, and ns-3 for each scenario considering the 95% confidence intervals. All mean times are in milliseconds and simulations were performed on a dedicated server with an Intel i7-860 processor running at 2.8 GHz and 32 GB of RAM. As shown in Figure 2, MAPE and AFTER report execution times that are at least 2 orders of magnitude lower than those of ns-3 for different scenarios.
Note that execution times for ns-3 vary from tens to hundreds of seconds for the scenarios considered. While we observe a slight increase in MAPE's time complexity when compared to AFTER's for scenarios with only a few flows, that difference becomes negligible when the number of flows increases. It demonstrate that MAPE is able to compute per-flow network performance estimates in real time which can be used to inform core network services such as routing. MAPE and AFTER are fast because, unlike stochastic simulators, they do not need to simulate nearly as many events to reach steady state.
As expected, execution times increase with the number of flows. However, AFTER and MAPE's execution times increase more significantly with the number of path hops because that increases the number of transmission and
Estimated Throughput
We measure throughput estimate accuracy as the ratio between the per-flow estimate returned by AFTER or MAPE and the per-flow throughput obtained by ns-3. Differently from other common ways to measure accuracy, such as the mean squared error, the way we evaluate accuracy conveys whether the estimate is an underestimate or overestimate of the reference value, which is the value reported by ns-3. Figures 3 and 4 show MAPE's and AFTER's throughput estimate accuracy for CBR and multimedia traffic in both the Random Indoor and the Grid Outdoor topologies, respectively. The red line represents the "ideal" ratio of 1, i.e., a perfect match between the estimates and ns-3's measured throughput. We observe that AFTER's throughput estimates are significantly less accurate when compared to MAPE because AFTER's simulated flows attempt to transmit at the highest supported rate, typically resulting in overestimates. This is particularly pronounced for scenarios with few flows in which there is low inter-flow interference and, consequently, more residual network capacity to support higher transmission rates. As more flows are added, AFTER's prediction improves because, with more flows sharing the network's capacity, there is less room for each flow's transmission rate to increase beyond the real transmission rate.
This prediction discrepancy between AFTER and MAPE also quantitatively demonstrates the impact that not accounting for specific flow transmission rates may have. It also illustrates that MAPE is able to significantly improve prediction accuracy for scenarios with few flows (in our experiments, 3and 6-flow scenarios). MAPE's accuracy decreases in scenarios with more flows -with a bias toward overestimates due to some simplifications inherited from AFTER. For instance, AFTER does not take into account packet losses due to collision, which may influence network throughput when there are more flows transmitting simultaneously. Instead, in its inter-flow interference model, AFTER implements a medium access scheduler based on an interference graph of the topology. In future work, we plan to address this issue by improving how flow interference is modeled.
Note that MAPE yields higher accuracy for CBR traffic (Figures 3a and 4a). That is because its scheduler also generates v_packets at constant rates. For multimedia (MM) traffic scenarios (Figures 3b and 4b), however, transmission rate variations cause MAPE to overestimate the throughput. This is because bursts of the more important video packets cause losses due to buffer overflow and packet collisions, while less important video packets which have lower transmission rates are delivered more reliably.
We also evaluate the per-flow throughput prediction accuracy considering the different classes of flows based on their transmission rates. Figure 5 summarizes the results for 6 flows using CBR and MM traffic in both the indoor and outdoor topologies. We also ran these experiments for 3, 9 and 12 flows, but we omit those results since they show similar trends. The red reference lines represent the ideal throughput based on the average bitrate generated for each video trace.
The figure shows that AFTER tends to overestimate all three classes of flows. Moreover, as the source-destination pair is chosen randomly regardless of the transmission rate of the flow, the average throughput estimated by AFTER tends to be roughly the same for all three classes. Conversely, by knowing the transmission rate of each flow, MAPE is able to more accurately estimate per-flow throughput. Note, however, that it slightly overestimates MM's throughput. This issue, which is more pronounced in the outdoor topology due to its higher link reliability, is due to the fact that MAPE's current implementation uses AFTER, and thus it inherits the mechanism used by AFTER to estimate packet loss. It considers two possible sources of packet loss: buffer overflow and link-layer transmission losses. If all links that compose a path have perfect delivery rates, losses computed by AFTER are only due to buffer overflow. In practice, however, there are other sources of losses, such as collisions, and as a result, MAPE and AFTER tend to overestimate flows' throughputs.
Estimated Delay and Packet Loss
We also evaluate MAPE's delay and packet loss estimates. Figure 6 shows the average end-to-end delay, considering the 95% confidence intervals for different number of flows in both experimental topologies. When compared to the results obtained by ns-3, MAPE shows similar delay increase trend as the number of flows increases. Note that MAPE overestimates the end-to-end delay for scenarios with 12 flows in both topologies. This is due to MAC layer congestion as more flows share the same nodes/links increasing contention and consequently increasing MAPE's time to reach steady state, which, in turn, may cause MAPE's execution to end before reaching steady state. Although MAPE's estimate is less accurate compared to ns-3 when it does not reach steady state, we will demonstrate in the Section 6 that these results are still useful to inform the route selection process ahead of multimedia flow transmission. Figure 7 plots the average packet loss rate. It also shows a discrepancy between MAPE's and ns-3's estimates in both topologies. But here, instead of overestimating, losses are generally underestimated by MAPE. The culprit is the absence of a collision packet loss counter in MAPE, which causes it to be more prone to estimate lower overall loss rates. These results also help explain the reason for instances in which MAPE overestimates the throughput -a consequence of fewer packets being discarded at the MAC layer. Furthermore, as expected, packet losses for MM traffic were even more impacted by the bursty nature of the video packet flows. As part of our future work, we plan to improve how MAPE models packet losses due to collision.
Despite those discrepancies, the results shown in Figures 5, 6 and 7 demonstrate MAPE's ability to capture the overall trend in throughput, delay and packet loss for multimedia flows in different application scenarios. Furthermore, we note that the discrepancies for 9 and 12 flows are mostly caused by network congestion and MAPE estimates being generated before steady state is achieved.
In order to confirm this hypothesis, in Figure 8 we show a scatter plot for the 9-flow runs using the Random Indoor topology representing which instances did and did not reach the steady state and their respective delays discrepancies when comparing MAPE to ns-3 -i.e., the difference between MAPE's and ns-3's average delay estimates. Note that we show results for the 9-flow Random Indoor topology experiments because, with 9 flows (and above), the network gets more congested and consequently the number of instances that do not reach steady state increases, which, as previously discussed, results in higher delay and packet loss discrepancies.
As the plot shows, when steady state is reached, MAPE yields adequate estimation accuracy, with discrepancies concentrating around less than 100ms. However, MAPE tends to overestimate end-to-end delay for instances that do not reach steady state, causing higher discrepancies. Figure 9 confirms this observation -it shows a scatter plot of the path average throughput according to ns-3 for instances that reached the steady state and those that did not as a function of the delay estimate discrepancies for the 12-flow experiments in the Random Indoor Topology (since they showed the highest discrepancies). As the plot shows, higher throughput paths are concentrated around the smallest discrepancies, while the largest discrepancies happen with paths that present poor network performance and are, thus, not suitable for multimedia flows. Note that instances with higher throughput were those in which MAPE was able to reach steady state, unlike runs that exhibit higher discrepancies, which, again, are the ones where steady state was not reached.
As a tool to guide real-time route selection decisions for IoT multimedia applications, low throughput routes -likely because of congestion -are generally undesirable, as they are often unable to meet the requirements of multimedia flows. As such, overestimating delay for those paths should not negatively impact path selection. That is, MAPE's delay overestimates when compared to ns-3's correspond to paths that are undesirable for video traffic anyway and therefore should not be selected by routing.
Video Quality Evaluation
The ability to estimate network performance is essential to ensure adequate network support for many IoT multimedia applications. In the case of applications involving video transmission, for instance, timely and fresh estimates of the current state of the network can significantly help routing protocols to rapidly identify paths that satisfy QoS constraints, as well as promote load balancing and network resource utilization. To examine how MAPE's performance estimates can be used to improve overall video quality, we use a well-known Quality-of-Experience (QoE) metric called Structural Similarity Index Measure (SSIM) [27] measured by the EvalVid video transmission and quality evaluation framework [26]. In the second part of this section, we evaluate the quality of the video transmitted using the route selected based on MAPE's estimates.
Video Structural Similarity
The Structural Similarity Index Measure, or SSIM, measures video structural distortion which is known to correlate with video quality as perceived by the end user [27]. This metric combines luminance, contrast, and structural similarity of the frames to compute the correlation between the original frame and the (possibly distorted) displayed one. SSIM values vary between 0 and 1, with higher values meaning better quality.
To show how MAPE estimates can be used to improve video quality, we run experiments transmitting the "Hall Monitor" video clip (as described in Section 4.2) and compute the SSIM by comparing all transmitted and received video frames. Figure 10 plots the average SSIM of the instances for different numbers of flows in the Random Indoor topology. According to the delay discrepancy ranges observed in Figure 8, we group experimental run instances in two classes, where the first class exhibits delay discrepancies greater than 100ms (called lager delay discrepancy) and the second class exhibits delay discrepancies equal to or less than 100ms (called smaller delay discrepancy) when compared to the results obtained with ns-3. From Figure 10, we observe that larger delay discrepancy instances were not found in scenarios with 3 flows; however, for the 6-, 9-, and 12-flow experiments, larger delay discrepancy instances resulted in lower video quality. Consequently, as previously discussed in Section 5, since MAPE's larger delay discrepancies correlate with poorer video quality, MAPE estimates could be used to discard paths which would result in inadequate performance for video transmission as they do not currently match the video transmission requirements. For example, in real-time video applications, MAPE could be used to identify paths that provide a minimum threshold latency that preserves adequate video quality.
Classification Inversions
To evaluate how MAPE can be used to inform path selection for video transmission, we use the concept of classification inversions [23] defined as follows. Consider two different paths a and b. Suppose that video transmissions using path a yield higher SSIM than if path b was used. If MAPE's throughput estimate indicates that path b will outperform path a, that constitutes a classification inversion using throughput as metric; otherwise, if path a is selected, there is no inversion.
To better understand how classification inversions can be used in practice, let us consider the route selection problem in multipath forwarding, where a set of paths needs to be selected for the transmission of multiple flows based on low transmission rate. In this example, the most important aspect is to get the relative ranking of the paths correctly in order to make adequate flow-to-path assignments. Figure 11 shows a comparison between MAPE and AFTER in terms of classification inversions for both the Random Indoor and the Grid Outdoor topologies as a function of the number of flows. We evaluate the quality of paths using SSIM metric. For MAPE, we consider three possible scenarios: using the average throughput, packet loss or end-to-end delay as metrics to compare the set of paths of all instances. Since AFTER does not estimate delay or packet loss, we only show results when throughput is used to calculate classification inversions based on AFTER estimates. MAPE results in lower percentages of classification inversions (lower than 20%) in all scenarios. AFTER, on the other hand, yields 3 to 6 times higher classification inversion rates, as it ignores flow transmission rates.
Note that MAPE's packet loss and delay estimates result in lower classification inversions for 9 and 12 flows when compared to classification inversions based on throughput. This demonstrates that both delay and loss should be considered when selecting paths for video transmission, especially when the network becomes saturated. These results are relevant because they confirm that MAPE's estimates, which can be computed in quasi real time, can be used to select paths that improve user QoE in terms of perceived video quality.
Conclusion
This paper introduced the Multimedia-Aware Performance Estimator, or MAPE for short, a per-flow estimator based on a deterministic discrete event simulation approach. MAPE estimates the throughput, packet loss and end-to-end delay of individual flows using their average transmission rate as input. To the best of our knowledge, MAPE is the first performance estimator that is able to both account for inter-flow interference and accommodate rate-heterogeneous flows, which is essential to more realistically model the behavior of multimedia traffic.
We evaluated MAPE in terms of execution time, prediction accuracy and ability to classify sets of paths according to the video quality at the receiver. Our results indicate that MAPE yields comparable throughput, packet loss and delay estimate accuracy when compared to stochastic network simulators such as ns-3 at a fraction of the execution time. When compared to AFTER, through its ability to consider specific per-flow rates, MAPE yields higher accuracy at comparable execution times. We also show in practice that by adopting video coding rates as input, MAPE is able to obtain estimates similar to the ones obtained by ns-3 when driven by multimedia (MM) traffic. We also demonstrate how MAPE's accurate real-time throughput and delay predictions can be used to make routing decisions for multimedia applications. In particular, we show that MAPE makes correct path selection decisions for over 80% of the cases, including saturated network scenarios.
As part of future work, we plan to refine MAPE's packet loss and delay models which will help improve its estimation accuracy. While our current implementation uses IEEE 802.11, we also plan to extend MAPE so that it can also be used with other IoT communication technologies such as IEEE 802.15.4 networks. We also intend to further explore the correlation between routing metrics and video quality (e.g., based on the SSIM) and to incorporate into MAPE alternate ways to model variable bitrate streams, including traffic patterns representative of prominent adaptive bitrate streaming traffic, e.g., by simulating video frame packets bursts. Our overarching goal is to propose a cross-layer framework that integrates MAPE with video coding for improved QoE.
Algorithm 1 :
1MAPE's pseudo-code. {Step 1: Initialization} networkT opology ←graph representing the network f lowsP ath ← list of paths of all flows f lowsRate ← list of bitrate of all flows {Step 2: Simulation} while no Steady State do foreach flow f ∈ f lowsP ath do Update the number of v_packets received by flow f ; Schedule the queuing of new packet of flow f to its queue according to f lowsRate; end Next network state; end {Step 3: Estimation} Compute long term per-flow performance.
Figure 1 :
1MAPE's implemention MAPE's functionality is implemented as a module (called SPR for Specific Per-flow Rates) that interfaces with the deterministic simulation engine to (1) provide flow rate information as part of simulation initialization, (2) update each flow when their v_packets are received, (3) generate new v_packets according to the stipulated flow rates, and (4) provide per-flow performance measurements.
Random Indoor topology.
Grid Outdoor topology.
Figure 2 :
2Execution time (log scale) for different number of flows reception events needed to deliver the flows' v_packets to the destination node. This explains the slightly higher times measured with the Grid Outdoor topology, which typically requires paths with more hops because of the greater distances between nodes.
Figure 3 :
3Estimated throughput accuracy relative to ns-3 in Indoor Random topology.
Figure 4 :
4Estimated throughput accuracy relative to ns-3 in the Outdoor Grid topology.
Grid Outdoor Topology.
Figure 5 :
5Average throughput for scenarios with 6 flows.
Grid Outdoor Topology.
Figure 6 :
6Average end-to-end delay for different number of flows. Grid Outdoor Topology.
Figure 7 :
7Average packet loss for different number of flows.
Figure 8 :
8Difference between MAPE's and ns-3's average delay predictions considering MAPE's steady and non-steady instances for scenarios with 9 flows in the Random Indoor Topology. discrepancy (P rediction − Ref erence) ms reference throughput (kb/s) reached steady state did not reach steady state
Figure 9 :
9Path average throughput (according to ns-3) as a function of the difference between MAPE's and ns-3's average delay predictions for scenarios with 12 flows in the Random Indoor Topology.
Figure 10 :
10Average SSIM according to the instances with larger and smaller delay discrepancies for scenarios with different number of flows in Random Indoor Topology.
Figure 11 :
11Percentage of classification inversions in terms of SSIM for different numbers of flows.
Table 1 :
1Simulation scenariosParameters
Topologies
Random Indoor Grid Outdoor
Deployment area (m 2 ):
100 x 100
360 x 490
Number of nodes:
30
55
PHY/MAC technology:
802.11g
802.11g
Link speed:
18 M b/s
18 M b/s
Mac Queue Size:
10p
10p
Packet Lifetime:
1000ms
1000ms
Traffic Control Queue Size:
1p
1p
Network Queue Size:
1p
1p
Propagation Model:
Shadowing
Cost231
Table 2 :
2MM and CBR traffic parametersParameters
Values
MM-Traffic
Video:
Hall Monitor
Encoding:
H.264/MPEG-4 AVC
Frame rate:
30 Hz
Format:
YUV CIF, 352 x 288
Number of frames: 3600
Target Bitrate:
256, 512, 1024 kb/s
Packet size:
1024 bytes
CBR-Traffic
Bitrate:
261, 485 and 836 kb/s
Packet size:
1024 bytes
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| []
|
[
"Harrow-Hassidim-Lloyd algorithm without ancilla postselection",
"Harrow-Hassidim-Lloyd algorithm without ancilla postselection"
]
| [
"D V Babukhin \nDukhov Research Institute of Automatics (VNIIA)\n127055MoscowRussia\n"
]
| [
"Dukhov Research Institute of Automatics (VNIIA)\n127055MoscowRussia"
]
| []
| Harrow-Hassidim-Lloyd algorithm (HHL) allows for the exponentially faster solution of a system of linear equations. However, this algorithm requires the postselection of an ancilla qubit to obtain the solution. This postselection makes the algorithm result probabilistic. Here we show conditions when the HHL algorithm can work without postselection of ancilla qubit. We derive expressions for expectation values for an observable M on the HHL outcome state when ancilla qubit is measured in |0 and |1 . When a commutator of an input matrix and an observable matrix is zero, the HHL algorithm can give correct expectation values for any outcome of ancilla measurement. We demonstrate this result on a toy 2 by 2 matrix example. We further provide more general examples of matrices A and observables M , which allow the postselection-free running of the algorithm. Our work can improve the performance of the HHL-based algorithms. | 10.1103/physreva.107.042408 | [
"https://export.arxiv.org/pdf/2208.02200v1.pdf"
]
| 251,280,254 | 2208.02200 | e751636362b6b3e588f24bcfe1ea2962bc1d057a |
Harrow-Hassidim-Lloyd algorithm without ancilla postselection
D V Babukhin
Dukhov Research Institute of Automatics (VNIIA)
127055MoscowRussia
Harrow-Hassidim-Lloyd algorithm without ancilla postselection
Harrow-Hassidim-Lloyd algorithm (HHL) allows for the exponentially faster solution of a system of linear equations. However, this algorithm requires the postselection of an ancilla qubit to obtain the solution. This postselection makes the algorithm result probabilistic. Here we show conditions when the HHL algorithm can work without postselection of ancilla qubit. We derive expressions for expectation values for an observable M on the HHL outcome state when ancilla qubit is measured in |0 and |1 . When a commutator of an input matrix and an observable matrix is zero, the HHL algorithm can give correct expectation values for any outcome of ancilla measurement. We demonstrate this result on a toy 2 by 2 matrix example. We further provide more general examples of matrices A and observables M , which allow the postselection-free running of the algorithm. Our work can improve the performance of the HHL-based algorithms.
I. INTRODUCTION
The HHL algorithm [1] is a quantum algorithm that provides a solution for a linear system with an exponential speed up. While having several caveats [2], this algorithm is an important subroutine in quantum algorithms. Since its invention, numerous applications of this algorithm to practical problems have been demonstrated: solving linear systems is used for differential equations [3,4], calculating scattering cross-sections [5], and building quantum machine learning algorithms [6][7][8]. Progress in quantum computing devices in the last decades allowed conducting low-dimensional experiments, which investigate practical opportunities and caveats of the HHL algorithm [9][10][11].
The HHL algorithm requires postselection of an ancilla qubit in a quantum state |1 to produce a solution. Ancilla measurement in the |1 state has a non-unity probability, which leads to discarding part of the algorithm runs on a quantum processor. Consequently, discarding results leads to an increase in quantum processor running time. An amplitude amplification algorithm [12] is usually used after the HHL circuit to increase the probability of measuring ancilla in |1 . This step requires O(κ) repetitions of the amplitude amplification to make a success probability sufficiently high. Here κ is a conditional number of the input matrix A, where A represents a system of linear equations we want to solve. A running time of the HHL is O(log(N )s 2 κ 2 / ) [1], where one κ comes from the amplitude amplification step. Although adding only a polynomial (in κ) complexity overhead, the amplitude amplification step increases algorithm running time and introduces gate errors into computation when the algorithm is running on a NISQ device [13]. Until fault-tolerant quantum computation is available, the postselection step decreases the efficiency of the HHL algorithm.
In this paper, we demonstrate conditions for running the HHL algorithm without postselection of the ancilla qubit. This is possible for input matrices A of linear equation systems and for measurement matrices M , which satisfy the commutator identity [[M, A], A] = 0. When this relation is satisfied, the algorithm produces quantum states for two ancilla measurement outcomes (|0 or |1 ), in which expectation values deviate from each other only by a constant. This connection of expectation values allows using both output states to obtain an expectation value of M on the solution of the linear system. This reduction of postselection leads to the economy of O(κ) operations of amplitude amplification, otherwise used to amplify the success probability of ancilla measurement. This paper is organized as follows. In Sec. II, we provide a formulation of the HHL protocol and derive the postselection-free condition. In Sec. III, we provide an explicit toy example, which illustrates the main result with a 2-dimension input matrix. In Sec IV, we provide more general examples of input matrices and observables, which satisfy the postselection-free condition. Finally, we conclude the result of this paper and discuss perspectives in Sec. V. arXiv:2208.02200v1 [quant-ph] 3 Aug 2022
II. DERIVATION OF THE POSTSELECTION-FREE CONDITION
For an invertable matrix A, a vector x of unknown variables and a vector b of known values, a system of linear equations has a form
A x = b (1)
with a solution of this system
x = A −1 b.(2)
In quantum formalism, this solution has a form
|x = A −1 |b = N j=1 β j λ j |u j(3)
for an input matrix A
A = N j=1 λ j |u j u j |(4)
and an input state b
|b = N j=1 β j |u j(5)
Here λ j and |u j are eigenvalues and eigenvectors of the input matrix A. The HHL algorithm produces a following quantum state
|Ψ = N j=1 β j |u j 1 − C 2 λ 2 j |0 a + C λ j |1 a(6)
After measurement of ancilla qubit, this state transforms into one of two quantum states
|x 0 = N j=1 β j 1 − C 2 λ 2 j
|u j , when we measure a = 0 (7)
|x 1 = N j=1 β j C λ j |u j , when we measure a = 1(8)
which we provide unnormalized here for simplicity (we will restore state norms further). We see, that the state (8) is a solution of the linear system up to a constant C (see (3)), and this is a main result of running the HHL algorithm. At the same time, (7) is the outcome when the algorithm fails to solve the linear system problem. Let us rewrite the (7) in a form of a linear system solution:
|x 0 =Ã −1 |b(9)
whereà −1 denotes an inverse matrix of some linear system. The form of this matrix can be derived from (7):
A −1 |b = N j=1 β j 1 − C 2 λ 2 j |u j = k N j=1 β j 1 − C 2 λ 2 k |u k u k |u j = k 1 − C 2 λ 2 k |u k u k | j β j |u j(10)
so a matrix form ofà −1 isÃ
−1 = k 1 − C 2 λ 2 k |u k u k |(11)
We can connect this matrix with the inverse matrix of the initial system as following:
A −1 = k C λ k λ 2 k C 2 − 1 |u k u k | = j C λ j |u j u j | k λ 2 k C 2 − 1 |u k u k | = A −1 C D(12)
where A −1 C = j C λj |u j u j | is a normalized inverse of the initial linear system matrix and
D = k λ 2 k C 2 − 1 |u k u k |(13)
is some additional transform. Let us measure an observable M on states (7) and (8):
x 1 | M |x 1 = b| A −1 † C M A −1 C |b (14) x 0 | M |x 0 = b| (A −1 C D) † M A −1 C D |b = b| D † A −1 † C M A −1 C D |b = x 1 | D † M D |x 1(15)
where the last equality is valid, because [A −1 C , D] = 0 as both matrices are diagonal at the same basis. The equation (15) shows that the result of measurement the observable M on the quantum state (7) is equal to measurement of another observable D † M D on the correct answer (8). Let us rewrite matrix D in a form
D = j 1 − (1 − λ 2 k C 2 − 1) |u j u j | = I − ∆(16)
Then, (15) transforms to the form
x 1 | D † M D |x 1 = x 1 | M |x 1 − x 1 | M ∆ |x 1 − x 1 | ∆M |x 1 + x 1 | ∆M ∆ |x 1 = x 1 | M |x 1 + x 1 | δM |x 1 (17)
where we separated error operator
To proceed, we need to prove an equality
∆ 2 = A 2 C − 2D(20)
It follows from a sequence of equations
∆ 2 = j (1 − λ 2 j C 2 − 1) 2 |u j u j | = j 1 − 2 λ 2 j C 2 − 1 + ( λ 2 j C 2 − 1) |u j u j | (21) = j λ 2 j C 2 |u j u j | − 2 j λ 2 j C 2 − 1 |u j u j | = A 2 C − 2D(22)
Next, we rewrite the error operator (18) in two forms:
1. First form:
∆M ∆ = (M ∆ − R)∆ = M ∆ 2 − R∆ (23) [M, ∆] + = M ∆ + ∆M = 2M ∆ − R (24) δM = M ∆ 2 − R∆ − 2M ∆ + R = M ∆ 2 − 2M ∆ + R(I − ∆) = M (A 2 C − 2D) − 2M (I − D) + RD = M A 2 C − 2M + RD(25)
2. Second form:
∆M ∆ = ∆(M ∆ + R) = ∆ 2 M + ∆R (26) [M, ∆] + = M ∆ + ∆M = 2∆M + R (27) δM = ∆ 2 M + ∆R − 2∆M − R = ∆ 2 M − 2∆M − R(I − ∆) = (A 2 C − 2D)M − 2(I − D)M − RD = A 2 C M − 2M − RD(28)
Next we transform a square A 2 C terms of (25) and (28) equations:
M A 2 C = M A 2 C − A C M A C + A C M A C = (M A C − A C M )A C + A C M A C = A C M A C + [M, A C ]A C (29) A 2 C M = A 2 C M + A C M A C − A C M A C = A C (A C M − M A C ) + A C M A C = A C M A C − A C [M, A C ](30)
Substituting (29) in (25) and (30) in (28) and summing results, we obtain
δM = A C M A C − 2M + 1 2 [M, A C ]A C − A C [M, A C ] = A C M A C − 2M + 1 2 [[M, A C ]A C ](31)
Measuring δM in |x 0 state, we substitute (31) into (17) and (15) and obtain
x 0 | M |x 0 = x 1 | M |x 1 + x 1 | δM |x 1 = x 1 | A † C M A C |x 1 − x 1 | M |x 1 + 1 2 x 1 | [[M, A C ]A C ] |x 1(32)
Finally recalling than |x 1 = A −1 C |b , we obtain
x 0 | M |x 0 = b| M |b − x 1 | M |x 1 + x 1 | K |x 1(33)
where we denote a commutator
K = 1 2 [[M, A C ]A C ](34)
If the commutator (34) is 0, then (35) simplifies to a form
x 0 | M |x 0 = b| M |b − x 1 | M |x 1(35)
where x 0 | M |x 0 and x 1 | M |x 1 are now connected through a term b| M |b . As vector |b is a given vector by assumption of the HHL algorithm ( [1,2]), the expectation value of M on |b can be obtained at will. Thus, we can obtain an expectation value x 1 | M |x 1 on a correct solution of linear system even if we measured |0 on the ancilla and obtained an output vector |x 0 . This allows to state, that if the commutator (34) is zero, than the HHL algorithm is postselection-free in a sense, that any outcome of ancilla qubit measurement leads to the correct expectation value. Previously we worked with unnormalized state for simplicity. If we restore norms of quantum state, which occur after ancilla measurement, we obtain a following connection of expectation values
x 1 | M |x 1 = 1 P r(a = 1) b| M |b − P r(a = 0) x 0 | M |x 0(36)
where |x 0 and |x 1 are now normalized quantum states. This identity follows from (35) with use of a fact, that after ancilla qubit measurement a state of the input register normalizes as |x i / P r(a = i) with i = 0, 1.
|0a
Ry( 2π 2 r ) Ry( π 2 r ) |m |0c,0 H • QF T † • U † |01,c H • • |b exp(iA t 0 4 ) exp(iA t 0 2 )
FIG. 1: A quantum circuit implementing the HHL algorithm for a system of 2 linear equations for matrix (37)
III. TOY 2 BY 2 MATRIX EXAMPLE
To demonstrate the idea of the postselection-free HHL, we explicitly calculate a simple realization of the HHL algorithm, introduced in [14]. Here, the HHL algorithm is used to solve a system of 2 linear equations, represented by a matrix
A = 1.5 0.5 0.5 1.5(37)
In Fig.1, we provide a quantum circuit of the toy-example algorithm. Here H is a Hadamard gate, QF T is a quantum Fourier transform gate, exp(iAt 0 ) is a unitary evolution for matrix A, R y is a Y -rotation of a qubit state and U † denotes an uncomputing gate for a sequence of gates before controlled ancilla rotations. In this example, for an
observable M = 0 1 1 0 = X a commutator with input matrix A is [M, A] = 0, which is seen from decomposition A = 1.5I + 0.5X.
A. Statevector simulator
We run the algorithm for 100 random initial vectors |b = cos θ 2 |0 + sin θ 2 |1 and obtained solution vectors |x 0 for ancilla measurement in state |0 and |x 1 for ancilla measurement in state |1 . In Fig. 2 we provide expectation values
M 1 = x 1 | M |x 1 and 1 P r(a=1) (M b − P r(a = 0)M 0 ), where M 0 = x 0 | M |x 0 and M b = b| M |b .
We compared results with a classical solution for the system of linear equations x, in particular, with a value x T M x. From Fig.2, we see perfect coincidence with classical results and results obtained from the HHL algorithm outcomes for ancilla values 0 and 1. In Fig.3 we provide dependencies of probability to measure the ancilla in a state 1, fidelity of the |x 1 with respect to classical solution and an error on observable M between classical and quantum solutions on a parameter r, which governs a rotation constant C = 2π 2 r in the HHL algorithm (refer to (6)). We can see that increasing r (lowering C) leads to better precision of the HHL performance while decreasing the probability of measuring the ancilla qubit in a state |1 . A low probability of ancilla measurement (thus, a low probability of the HHL correct outcome) is usually compensated with an amplitude amplification [12]. Here we demonstrate that using both output states for two ancilla measurement outcomes can give a correct expectation value of M with only slight postprocessing (subtracting a constant value). This extraction of correct expectation values is possible if a postselection-free condition (34) is satisfied.
B. QASM simulator
Measurement of an observable value in quantum computing requires gathering statistics on a computational basis. Thus, every observable value has a statistical error, which depends on the size of the sample. Here we analyze how statistical error influences the observable value of M , obtained from two cases of ancilla qubit measurement.
In the HHL algorithm, an important value for the algorithm result is the probability to measure ancilla in a state -for both cases of ancilla measurement we need to divide the result on an estimate of P r(a = 1). In the previous section, we have seen that P r(a = 1) decreases as a constant C decreases while the algorithm precision increases. The smaller the P r(a = 1) value, the harder it is to estimate this value with gathered statistics. For example, if P r(a = 1) is of order 10 −3 , it is required to make about 10 3 algorithm runs in average to obtain one measurement of ancilla in state |1 . In Fig. 4 we provide a variance-value ration of P r(a = 1) for different number of algorithm runs (shots). We can see that we need to make more runs of the algorithm to estimate P r(a = 1) as algorithm precision increases (in a sense of result fidelity, provided in Fig. ?? (B)). That means that a number of shots N shots and a constant value C have a trade-off, and we need to choose them concerning requirements of algorithm results fidelity and time-consumption (if any). For a fixed number of algorithm runs N shots , a plot of resulting observable M values for two cases of ancilla measurement is provided in Fig. 5. Here are provided estimates of observable values with standard deviations for different values of algorithm constant C (remember that C is parametrized with a parameter r in the considered toy problem). First, we can see that for r ≥ 3 both estimates (dots on plots) converge to a classical value x T M x. That means that, for a particular problem, there is a sufficient value of C, which gives adequate precision to the answer, and taking C lower does not increase the estimate precision significantly. Second, we can see that the standard deviation (error bars on plots) is different for two ancilla measurement cases. If we measure ancilla qubit in a state |0 and construct an estimate of observable M with equation (36), we obtain a correct estimate with a standard deviation larger than in the case of the straightforward HHL algorithm use, when ancilla is measured in state |1 . The estimate (36) uses estimated values of b| M |b , x 0 | M |x 0 and P r(a = 1), each of which has statistical error. In the numerator of (36), we have two estimated values with non-zero variance, hence we have two sources of uncertainty instead of one in the case when ancilla is measured in state |1 . As a result, the method to estimate observable M value for the state |x 0 , which we propose in this paper, provides a correct estimate with a price of a higher statistical error. Nonetheless, this increased statistical error is not dramatic, and, with a proper choice of a constant C and a number of algorithm runs N shots , we can obtain estimates with comparable precision. For a fixed value of parameter r = 4, a plot of the standard error of observable value estimates is provided in Fig. 6. For a number of algorithm runs more than 10 4 , standard errors of two estimates (for ancilla measured in state |1 and |0 ) are of comparable value.
IV. MORE GENERAL EXAMPLES
In this section, we provide explicit examples of input matrices A and M , which satisfy the postselection-free condition (equality of a commutator (34) to 0).
Here we consider a measurement observable to be a string of Pauli operators of the form
M = P k ⊗ P k ⊗ · · · ⊗ P k(38)
where k = 1, 2, 3. The Pauli observable is common in quantum computation. In particular, a computation basis measurement is a Pauli observable of the form Z ⊗N , and any other observable (including other Pauli) deviates only with a proper basis rotation. The Pauli observable is suitable for benchmarking current quantum processors for its simplicity and availability.
A. Matrix A is an even Pauli polynomial
The first case is an input matrix, which is a sum of even polynomials of Pauli matrices
A = p=1,2,3 i1,i2 J i1,i2 p P i1 p ⊗ P i2 p + i1,i2,i3,i4 J i1,i2,i3,i4 p P i1 p ⊗ P i2 p ⊗ P i3 p ⊗ P i4 p + . . .(39)
where J i1,i2,... p are arbitrary complex coefficients. Although simple, even polynomials of Pauli matrices can be used to construct approximations of other input matrices.
To prove that this matrix commutes with the Pauli observable of the form (38), we need to show that every term in the sum (39) commutes with (38). Pauli matrices P 1 , P 2 , P 3 are matrices which satisfy relations P 1 P 2 = iP 3 , P 2 P 1 = −iP 3 , P 2 1 = P 2 2 = P 2 3 = I
Using these properties, now we calculate a commutator of K-terms of Pauli matrices [P ⊗K 1 , P ⊗K 2 ]:
[P ⊗K 1 , P ⊗K 2 ] = P ⊗K 1 P ⊗K 2 − P ⊗K 2 P ⊗K 1 = (P 1 P 2 ) ⊗K − (P 2 P 1 ) ⊗K = (iP 3 ) ⊗K − (−iP 3 ) ⊗K = (i K − (−i) K )P ⊗K 3(41)
or, in short,
[P ⊗K 1 , P ⊗K 2 ] = (i K − (−i) K )P ⊗K 3(42)
We see, than for even K = 2k, k = 1, 2, ... we have [P ⊗K 1 , P ⊗K 2 ] = 0 since i 2k − (−i) 2k = 0. Next, we prove that if a commuting K-term contains identity matrices on some positions, they factor out from the commutator. Consider a commutator of N-terms, where one of the terms has K < N non-identity Pauli operators, while the other has N Pauli terms:
[P ⊗K 1 ⊗ I ⊗(N −K) , P ⊗K 2 ⊗ P ⊗(N −K) 2 ] = (P ⊗K 1 ⊗ I ⊗(N −K) )(P ⊗K 2 ⊗ P ⊗(N −K) 2 ) − (P ⊗K 2 ⊗ P ⊗(N −K) 2 )(P ⊗K 1 ⊗ I ⊗(N −K) ) = (P 1 P 2 ) K ⊗ P N −K 2 − (P 2 P 1 ) K ⊗ P N −K 2 = (P 1 P 2 ) K − (P 2 P 1 ) K ⊗P N −K 2 = (i K − (−i) K )P K 3 ⊗ P N −K 2(43)
or, in short
[P ⊗K 1 ⊗ I ⊗(N −K) , P ⊗K 2 ⊗ P ⊗(N −K) 2 ] = (i K − (−i) K )P K 3 ⊗ P N −K + · · · = 0 or, in short [A, M ] = 0
The postselection-free condition is not satisfied for the odd Pauli polynomial, which follows from (42).
It is an open question, how large is an error from odd Pauli polynomials would be for a particular input matrix. Pauli matrices (with an identity matrix) form a basis of hermitian matrices space. Thus any hermitian matrix A can be expanded into polynomials of Pauli matrices, even half of which satisfy the postselection-free condition.
B. Matrix A is a 2nd order derivative matrix
The second case is an input matrix of the form
A = a b 0 . . .0 . . . 0 b a = a N −1 k=0 |k k| + b N −2 k=0 (|k + 1 k| + |k k + 1|)(45)
This matrix arises in a finite-difference method of solving differential equations when the equation has terms up to the second-order derivative. We prove here that this matrix commutes with an observable of the form M = X ⊗X ⊗· · ·⊗X.
To prove this statement, we rewrite the observable in the following form
M = X ⊗ X ⊗ · · · ⊗ X = N −1 k=0 |k N − 1 − k|(46)
The commutator then takes the form
[A, M ] = a[I, M ] + b N −2 k1=0 N −1 k2=0 |k 1 k 1 + 1|k 2 N − 1 − k 2 | − |k 2 N − 1 − k 2 |k 1 k 1 + 1| + (47) |k 1 + 1 k 1 |k 2 N − 1 − k 2 | − |k 2 N − 1 − k 2 |k 1 + 1 k 1 |(48)
Using that [I, M ] = 0 and orthogonality of quantum basis states k i |k j = δ ij , we obtain
[A, M ] = b N −2 k=0 |k N − 2 − k| − |N − 2 − k k| +b N −2 k=0 |k + 1 N − 1 − k| − |N − 1 − k k + 1|(49)
Each of the two sums is equal to zero. To show this, we rearrange the summation index in every second term of each sum as follows
N −2 k=0 |k N − 2 − k| − N −2 k=0 |N − 2 − k k| = k new = N − 2 − k = N −2 k=0 |k N − 2 − k| − N −2 knew=0 |k new N − 2 − k new | = 0 (50) N −2 k=0 |k + 1 N − 1 − k| − N −2 k=0 |N − 1 − k k + 1| = k new + 1 = N − 1 − k = N −2 k=0 |k + 1 N − 1 − k| − N −2 knew=0 |k new + 1 N − 1 − k new | = 0(51)
C. Change of basis for matrix A
Now we consider how a change of basis affects the postselection-free condition. Suppose a matrix A is easy to exponentiate in some basis, represented with matrix U (e.g., a Fourier basis with U QF T ). Then, before applying the HHL algorithm to the input state |b , we transform the input register from computational to U basis and proceed with the algorithm, where the input matrix has a form
A U = U AU †(52)
Let us consider how commutator (34) transforms under the change of basis. First, we transform the inner commutator:
[M, A] = [M, U † A U U ] = M U † A U U − U † A U U M = U † U M U † A U U − U † A U U M U † U = U † U M U † A − AU M U † U = U † [M U , A U ]U(53)
where M U = U M U † . Next, we transform full commutator (34):
[[M, A], A] = [U † [M U , A U ]U, U † A U U ] = U † [[M U , A U ], A U ]U(54)
From the last equation, we can conclude that
[[M U , A U ], A U ] = 0 =⇒ [[M, A], A] = 0(55)
so the postselection-free condition holds on any specific basis U . As unitary transform preserves expectation values as follows
ψ U | M U |ψ U = ψ| U † U M U † U |ψ = ψ| M |ψ(56)
we can work in a basis U to simplify exponentiation of A if it is easier to make a unitary operator from A U .
V. CONCLUSION AND OUTLOOK
We demonstrated, that for an input matrix A and an observable M , which satisfy [[M, A], A] = 0, the HHL can work postselection-free. We derived the outcome state of the HHL algorithm for two possible ancilla measurement results and calculated their expectation values on an observable matrix M . We showed that these expectation values deviate only by a constant value, when the postselection-free condition is met. Thus, we can extract a correct value of linear system solution from both algorithm outcomes. To illustrate our result, we provided a 2 by 2 matrix toy example along with more general examples of input matrices A and observables M , which allow running the HHL without postselection.
Our result can improve algorithms that use the HHL algorithm as a subroutine. The HHL algorithm is efficient when an output state is used to measure an expectation value of some observable instead of measuring output vector components (which takes O(N ) steps). For this application of the HHL outcome state, we demonstrated that it is possible to get the correct output without postselection of the ancillary qubit. The other way to use HHL efficiently is to use the output state as an input to another quantum algorithm. For example, the HHL algorithm allows solving differential equations [3,4,15] and numerous machine learning problems [16]. It is an open question if the postselection-free condition translates to algorithms based on the HHL. An explicit demonstration of such translation is a subject of future research.
Another question is finding more problems, which satisfy the postselection-free condition. The commutator relation (34) provides a recipe for construction an input matrix A, given a measurement matrix M and vice versa. For instance, with a fixed measurement matrix M , one can look for an input matrix A, which solves a particular problem and can be effectively simulated (in the sense of Hamiltonian simulation problem [17,18]). Contrary, with a fixed input matrix A, one can look for a measurement M , which satisfies the postselection-free condition, is efficiently realized on the quantum device, and provides a solution for the problem under the scope. Building explicit examples of such procedures is another subject of future research.
δM = ∆M ∆ − [M, ∆] + , [M, ∆] + = M ∆ + ∆M (18) Let us denote a commutator [M, ∆] = M ∆ − ∆M = R
FIG. 2 :
2|1 . This value is needed to introduce a proper normalization of < M > and make it equal to a classical value of x T M x Expectation values of M on the HHL algorithm outcomes |x 1 (left plot), |x 0 (right plot), compared to x T M x value (blue solid curve), where x is a classical vector of the linear system solution. Classical solution values for different values of a parameter θ are connected with a curve to guide an eye.
FIG. 3 :
3Dependencies of a probability to measure the ancilla in a state |1 (A), an average absolute difference between classical and quantum solutions (B), and a difference between observable values on classical and quantum solutions (C)
FIG. 4 :
4Dependence (on parameter r) of a standard deviation to value relation of an estimated probability to measure ancilla in a state |1 .
FIG. 5 :
5Dependencies (on parameter r) of estimated M value on resulting HHL vectors with ancilla measured in state |0 (left), and ancilla measured in state |1 . Every point is an average of 100 values, each of which is calculated by gathering statistics of 10 6 shots.
FIG. 6 :
6Dependence (on a number of shots) of standard deviations of observable value estimates for ancilla measured in a state |0 (M x1estimated ), and for ancilla measured in a state |1 (M x1 ). Both dependencies are provided for a parameter value r = 4.
which proves that [A, M ] = 0 for the matrix (45) and the observable (46). It is an open question, what other matrices of finite-difference derivatives satisfy the postselection-free condition (34).
ACKNOWLEDGEMENTSThe author would like to thank W.V. Pogosov and A.V. Lebedev for useful discussions of the manuscript.(44)We see, that for even K = 2k, k = 1, 2, ... we have [P ⊗KFinally, we use (42) and (44) and linearity of commutator operation to prove, that for every matrix A of the form (39) and an observable (38) [A, M ] = 0:
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| []
|
[
"The Contribution of Low Surface-Brightness Galaxies to Faint Galaxy Counts",
"The Contribution of Low Surface-Brightness Galaxies to Faint Galaxy Counts"
]
| [
"Henry C Ferguson [email protected] ",
"Stacy S Mcgaugh ",
"\nSpace Telescope Science Institute\n\n",
"\nInstitute of Astronomy\nUniversity of Cambridge\n3700 San Martin Drive, The Observatories, Madingley Road21218, CB3 0HABaltimore, CambridgeMDEngland\n"
]
| [
"Space Telescope Science Institute\n",
"Institute of Astronomy\nUniversity of Cambridge\n3700 San Martin Drive, The Observatories, Madingley Road21218, CB3 0HABaltimore, CambridgeMDEngland"
]
| []
| Recent observations have revealed a population of blue galaxies at intermediate redshift with a space density well in excess of expectations from the local luminosity function and standard cosmology. The colors and luminosities of these faint blue galaxies are similar to nearby low surface-brightness (LSB) galaxies. Such galaxies are severely underrepresented in surveys used to de ne the local luminosity function, but could in principle be detected in deep surveys. If LSB galaxy density is high enough, the faint-galaxy counts could be explained without requiring rapid galaxy evolution.To explore the consequences of including LSB galaxies, we construct catalogs of simulated non-evolving galaxies drawn from a multivariate distribution of galaxy luminosities, central surface brightnesses, bulge/disk ratios and spectral-energy distributions. We compare two models dominated by LSB galaxies to a \standard" non-evolving model. Model galaxies are convolved with seeing and selected in a manner that closely matches real surveys. For each model we compute the local B J band luminosity function, HI mass function, number counts in the B J , I, and K bands, redshift distributions, and color distributions.We nd it possible to include a large population of LSB galaxies and incorporate a steep faint-end slope of the luminosity function in our simulations without violating the constraints on the local eld-galaxy luminosity function or the HI mass function. For q 0 = 0:5, the most favorable model matches the counts to B = 23, but falls short of the observations at fainter magnitudes. The discrepancy at faint magnitudes is smaller in the I and K bands. The colors and redshift distributions remain roughly consistent with observations to B = 24. The most serious discrepancy with observations is in the distribution of r e at faint magnitudes, suggesting that the model contains too many LSB galaxies.Nevertheless, the results suggest that LSB galaxies could be a signi cant contributor to faint-galaxy counts, reducing the need for such extreme models of galaxy evolution as rapid merging or bursting dwarf galaxies. We propose several tests to assess the contribution of LSB galaxies to faint galaxy counts and to di erentiate models involving moderate galaxy evolution from models involving rapid merging or starbursts. | 10.1086/175289 | [
"https://export.arxiv.org/pdf/astro-ph/9409071v1.pdf"
]
| 16,353,682 | astro-ph/9409071 | 27d71a5999ddf521b6d8ecaed25ac8b2eeb46342 |
The Contribution of Low Surface-Brightness Galaxies to Faint Galaxy Counts
Henry C Ferguson [email protected]
Stacy S Mcgaugh
Space Telescope Science Institute
Institute of Astronomy
University of Cambridge
3700 San Martin Drive, The Observatories, Madingley Road21218, CB3 0HABaltimore, CambridgeMDEngland
The Contribution of Low Surface-Brightness Galaxies to Faint Galaxy Counts
astro-ph/9409071 27 Sep 94
Recent observations have revealed a population of blue galaxies at intermediate redshift with a space density well in excess of expectations from the local luminosity function and standard cosmology. The colors and luminosities of these faint blue galaxies are similar to nearby low surface-brightness (LSB) galaxies. Such galaxies are severely underrepresented in surveys used to de ne the local luminosity function, but could in principle be detected in deep surveys. If LSB galaxy density is high enough, the faint-galaxy counts could be explained without requiring rapid galaxy evolution.To explore the consequences of including LSB galaxies, we construct catalogs of simulated non-evolving galaxies drawn from a multivariate distribution of galaxy luminosities, central surface brightnesses, bulge/disk ratios and spectral-energy distributions. We compare two models dominated by LSB galaxies to a \standard" non-evolving model. Model galaxies are convolved with seeing and selected in a manner that closely matches real surveys. For each model we compute the local B J band luminosity function, HI mass function, number counts in the B J , I, and K bands, redshift distributions, and color distributions.We nd it possible to include a large population of LSB galaxies and incorporate a steep faint-end slope of the luminosity function in our simulations without violating the constraints on the local eld-galaxy luminosity function or the HI mass function. For q 0 = 0:5, the most favorable model matches the counts to B = 23, but falls short of the observations at fainter magnitudes. The discrepancy at faint magnitudes is smaller in the I and K bands. The colors and redshift distributions remain roughly consistent with observations to B = 24. The most serious discrepancy with observations is in the distribution of r e at faint magnitudes, suggesting that the model contains too many LSB galaxies.Nevertheless, the results suggest that LSB galaxies could be a signi cant contributor to faint-galaxy counts, reducing the need for such extreme models of galaxy evolution as rapid merging or bursting dwarf galaxies. We propose several tests to assess the contribution of LSB galaxies to faint galaxy counts and to di erentiate models involving moderate galaxy evolution from models involving rapid merging or starbursts.
Introduction
Counts of very faint galaxies o er a simultaneous probe of the curvature of the universe and the evolution of its contents. The classic number{magnitude relation set out by Sandage (1961) as a test of the cosmological model has now been measured to the limits of 4 meter telescopes and the results are di cult to interpret. The counts of faint galaxies in the B and I bands show numbers are well in excess of cosmological models that do not include galaxy evolution, for any value of the deceleration parameter q 0 (Tyson & Jarvis 1979;Tyson 1988;Lilly, Cowie, & Gardner 1991;Metcalfe et al. 1991). Early attempts to reconcile the observations to the standard Friedmann-Lemaître cosmological model postulated that the faint-galaxy excess was primarily due to luminosity evolution: galaxies were brighter in the past because they were forming more stars (Tinsley 1980;Bruzual & Kron 1980;Yoshii & Takahara 1988;Guiderdoni & Rocca-Volmerange 1990). A robust prediction of such models is that the redshift distribution of faint galaxies should peak at higher z than in non-evolving models. However, deep redshift surveys show a distribution that appears consistent with the no-evolution prediction to B = 24 (Broadhurst, Ellis, & Shanks 1988;Colless et al. 1990;Cowie, Songaila, & Hu 1991;Colless et al. 1993). Furthermore, while the B counts show a strong excess over non-evolving models, counts in the K band show no excess (Gardner, Cowie, & Wainscoat 1993).
Explanations o ered for this discrepancy involve altering cosmology, modifying the standard picture of galaxy evolution, or altering properties of the local galaxy population. Fukugita et al. (1990) attempt to t the count and redshift data with standard ) galaxy evolution models, and nd that models with a sizable ( 0 0:5) cosmological constant are favored. However, gravitational lensing statistics and Ly -cloud statistics (Fukugita & Turner 1991;Maoz & Rix 1993;Fukugita & Lahav 1991) favor a small or vanishing cosmological constant. Modi cations of galaxy evolution include either rapid density evolution through merging or selective luminosity evolution. To explain the excess in the B band, merger models require triggered bursts of star formation while the galaxies are still widely separated (Guiderdoni & Rocca-Volmerange 1991;Broadhurst, Ellis, & Glazebrook 1992;Lacey & Silk 1991;Lacey et al. 1993). The required merger rate is di cult to reconcile with constraints on the fraction of stars that could have formed in elliptical galaxies in the last 3 Gyr, the thinness of spiral galaxy disks, or the weak angular correlation among galaxies at B = 25 (T oth & Ostriker 1992;Dalcanton 1993;Efstathiou et al. 1991;Roche et al. 1993). Models involving selective luminosity evolution propose that dwarf galaxies have faded more than giants in the last 3 Gyr (Broadhurst, Ellis, & Shanks 1988;Cole, Treyer, & Silk 1992;Lilly 1993). The most extreme models suppose that dwarf galaxy evolution is halted until redshifts z < 1, whereupon dwarfs form their stars in rapid bursts and subsequently fade beyond detectability (Babul & Rees 1992). For such a high space density of dwarf galaxies to have been missed locally requires that they have extremely low surface brightnesses, either because of expansion after expelling their gas in supernovae (Dekel & Silk 1986), or because of a top-heavy initial mass function.
A more conservative suggestion is that plausible modi cations to the local luminosity function and distribution of galaxy spectral types can bring no-evolution (NE) models much closer to the data. Driver et al. (1994) present the limiting case of a dwarf-dominated noevolution model, while Koo, Gronwall, & Bruzual (1993) primarily modify galaxy spectralenergy distributions. Neither model fully succeeds. Driver et al. (1994) attempt to match the counts by adopting a luminosity function with a very steep faint-end slope = 1:8. The resulting model is able to reproduce the faint-galaxy count and color data but predicts a median redshift well below that observed at B = 22 (0.18 for the model compared to 0.3 from the observations of Colless et al. 1993). Incompleteness in the combined LDSS survey (Colless et al. 1990;Colless et al. 1993) cannot be the cause of the discrepancy, as the survey is more than 95% complete. Koo et al. (1993) adopt a set of eleven plausible galaxy spectral types from the evolutionary models of Bruzual & Charlot (1993), then adjust the luminosity functions and space densities of the di erent types to provide the best simultaneous t to the faint-galaxy counts, colors and redshift distributions. Post-facto comparison to the local eld-galaxy luminosity function shows reasonable agreement to the limits of the Loveday et al. (1992) survey. The success of the model is due to the assumption of an open universe (q 0 = 0:05) and the inclusion of a much larger proportion of blue galaxies than typically seen in low-redshift surveys (see Fig. 8). Indeed, the model contains no galaxies redder than B V = 0:85 with absolute magnitudes M B J < 21. This is equivalent to removing all ellipticals and S0 galaxies brighter than L from the standard NE model. Of course, within the standard cosmological framework, all galaxies must evolve. NE models are clearly unphysical and are intended only to provide a baseline to isolate the e ects of the cosmological curvature (or selection e ects) from the e ects of galaxy evolution. Philosophically, we nd the Koo et al. approach, which gives large weight to observations high-z galaxies, a bit less useful than the standard approach of xing the distribution of galaxy properties to match observations of nearby galaxies. In the standard NE model, it is easy to \turn on" evolution, adjusting the redshifts of formation, star-formation timescales, dust content, etc. to try to match the faint-galaxy observations for an assumed q 0 and 0 , while still reproducing the properties of nearby galaxies. If this does not work (as many argue), then more exotic solutions are required. In the Koo et al. approach, it is not clear what form the evolution should take. There is no physically acceptable way to add \mild" evolution to their model. If one simply \turns on" evolution of galaxies at some high-z with the star-formation timescales given in their Table 1, the luminosity function and color distribution will be a strong function of z, and the resulting distribution of nearby galaxy properties is unlikely to be acceptable. If one instead adopts their luminosity function and color distribution at z = 0 and evolves the models backwards with the assumed starformation timescales, the predictions for faint galaxy counts and redshift distributions will change drastically; the three bluest classes of galaxies would form at redshifts z < 0:5, and hence disappear at the faint magnitudes where k-corrections would otherwise make them the most numerous population.
The model presented here is more in the spirit of standard NE models, with the exception that we propose a speci c selection e ect that could account for the discrepancy between the local galaxy properties and the deep counts. The crux of our model is the observation that the rest-frame isophotal limits of deep CCD surveys are fainter than the isophotal limits of the photographic surveys used to de ne the local luminosity function (McGaugh 1994). This is illustrated in Fig. 1, where we show the local and deep survey limits for three galaxies with the same total magnitude but with di erent scale lengths. Hence a large population of low-surface-brightness galaxies could contribute to the counts at faint magnitudes without violating current limits on the local luminosity function. While their local space density is not well known, examples of nearby LSB galaxies have been found in deep photographic surveys (Bothun et al. 1987;Davies, Phillipps, & Disney 1988;Schombert et al. 1990;Schombert et al. 1992). These galaxies are typically very blue , and appear to be weakly clustered (Bothun et al. 1993;Mo, McGaugh, & Bothun 1994), properties reminiscent of faint blue galaxies. LSB galaxies found in the eld are typically HI-rich disk galaxies with low star-formation rates . The inferred color and luminosity evolution of such galaxies is slow enough that a non-evolving model is a reasonable approximation for their appearance at moderate redshifts.
The presentation is as follows. In x2 we brie y review the canonical wisdom on disk-galaxy Figure 1: This gure shows simulated radial surface-brightness pro les for three exponential disk galaxies with (M B J = 19). Galaxy A has the canonical Freeman central surface brightness 0 (B J ) = 21:6 at z = 0. Galaxies B and C have scale lengths a factor of 2 and 4 larger, respectively. The left panel shows the galaxies viewed in 3 00 (FWHM) seeing at z = 0:01. The horizontal line shows the isophotal threshold of APM survey (Loveday et al. 1992). Galaxy C would be missed entirely by the survey, while galaxy B would have a measured \total" magnitude too faint by 0.7 mag, if the isophotal-to-total correction were based on galaxy A. The right panel shows the same three galaxies shifted to z = 0:4 and viewed in 1 00 seeing. A type II (see text) SED was assumed, giving 1.07 mag of k dimming. The horizontal line in this panel is Tyson's (1988) isophotal threshold B J = 28:7. All three galaxies would be detected above this threshold. Isophotal-to-total magnitude corrections are 0.1, 0.2, and 1.0 mag for galaxies A,B, and C, respectively. surface brightnesses, and show that existing constraints do not rule out the possibility of a large population of LSB galaxies. Because the intrinsic distribution of surface brightness of disks is not well constrained, in subsequent sections we adopt three di erent distributions and compare their predictions for counts, redshift distributions, colors, and the HI mass function. The modeling is done via a Monte-Carlo technique described in x3. Comparisons to the observations are presented in x4. In x5, we discuss tests that can be carried out with high-resolution data to assess whether LSB galaxies are indeed a signi cant contributor to faint galaxy counts.
The Intrinsic Distribution of Central Surface Brightness of Galaxy Disks
Most determinations of the galaxy luminosity function and number counts predicted therefrom implicitly assume that the distribution of galaxy surface brightnesses is afunction. This is an important simpli cation, as it allows one to make use of easily measured isophotal magnitudes without the need for detailed surface photometry, and suppresses one dimension of integration. It is observationally supported by Freeman (1970), who found that all spiral galaxies had central surface brightnesses 0 = 21:65 0:35B mag arcsec 2 , the scatter being consistent with observational error. Disney (1976) pointed out that selection e ects could cause the apparent distribution to appear sharp even if the intrinsic distribution were broad. Allen & Shu (1979) concurred that such selection e ects could act against surface brightnesses fainter than the Freeman value, but argued that higher surface brightness objects would not be missed. Disney & Phillipps (1983) developed a formalism to correct the apparent distribution in which a particular value of the central surface brightness is favored, and both high and low surface brightness objects can be missed.
That the apparent distribution of central surface brightnesses peaks at the Freeman value has been con rmed by Phillipps et al. (1987) and van der Kruit (1987) who found 0 = 21:75 and 21.5 (in B J ), respectively. These authors employed di erent methods to recover the intrinsic distribution. Phillipps et al. (1987) applied the method of Disney & Phillipps (1983), and found a distribution which was broad and asymmetric. Davies (1990) modeled the way in which central surface brightnesses were measured by Phillipps et al. (1987) and concluded that deviations from pure exponential pro les caused by even modest bulge components would broaden the distribution still further, implying large numbers of LSB galaxies. The V/V max method (Schmidt 1968) was used by van der Kruit (1987), who found that a narrow scatter of 0.4 mag. about the Freeman value was recovered if the sample was restricted to large, early type galaxies. However, the distribution did not approximate a -function for dwarf galaxies.
The results of these studies are mutually inconsistent, and the situation remains confused. What is really needed is the bivariate distribution of luminosity and surface brightness. Since the surface brightness portion of the distribution is not well constrained, we adopt two very di erent forms of the bivariate distribution in order to demonstrate the e ects on the predicted counts. The rst (model A) increases the normalization of the luminosity function, while the second (model B) steepens the slope of the faint end. This is accomplished with-out violating constraints on the local luminosity function when the procedures of isophotal measurement and selection are taken into account.
In model A, galaxies exist with equal numbers per unit magnitude over the range of central surface brightness 21:6 0 25 in B J . This is motivated by the existence of large, low surface brightness galaxies (Bothun et al. 1987;Schombert & Bothun 1988;Impey & Bothun 1989;Schombert et al. 1992). While the space density of such galaxies is not well known, it no doubt exceeds that expected from a Gaussian distrubution of central surface brightness with a dispersion = 0:4 mag about the Freeman value. In order to isolate the e ect of this surface brightness distribution on one parameter of the luminosity function (the normalization) it is assumed that there is no correlation between surface brightness and luminosity (i. e., the shape of the luminosity function is the same for every surface brightness). This is consistent with, though not demanded by, the similarity between the HI rotation velocity distributions of LSB and other eld galaxies, which suggests that they all have comparable masses (Schombert et al. 1992).
The range of surface brightnesses included in the at distribution of the model is dictated at the faint end by the fact that galaxies fainter than 0 25 will not contribute signi cantly to the counts even if large and luminous. At the bright end the distribution is truncated at the Freeman value following the results of Allen & Shu (1979) that very few high surface brightness galaxies exist. If there are many of these, then of course cosmological dimming will bias faint galaxy samples towards high surface brightnesses (Phillipps, Davies, & Disney 1990).
In model B, we assume that the narrow distribution of central surface brightnesses found by van der Kruit (1987) e ectively holds for giant (L > L ) galaxies, but that there is a systematic trend between luminosity and surface brightness for fainter galaxies. This strongly a ects the slope of the faint end of the luminosity function when isophotal rather than total measures are used because intrinsically faint galaxies have their luminosities signi cantly underestimated. To illustrate the severity of this a ect, we assume the following: 0 = 21:6 0:4 for L L (1) and 0 (L) = 21:6 2:5 log(L=L ) 0:4 for L < L :
(2) The equation for sub-L galaxies is essentially a constant size relation, with some (Gaussian) scatter.
Finally, for comparison, we compute a \standard" NE model using a galaxy mix similar to that adopted by Yoshii & Takahara (1988), but incorporating isophotal selection with disk central surface brightnesses set to the Freeman value with 0.4 mag Gaussian scatter. These distributions are illustrated schematically in Fig. 2.
Models A and B are clearly ad hoc, and are intended not to represent reality but to illustrate the importance of galaxy selection criteria and magnitude estimation techniques in both the local and deep surveys. Model B is perhaps closer to the truth in that it includes the general trend of decreasing surface brightness with decreasing luminosity observed for Virgo and Fornax Cluster dwarf galaxies (Binggeli, Sandage, & Tarenghi 1984;Ferguson Figure 2: Schematic illustration of the central-surface-brightness distributions assumed for galaxy disks in our models. In model A, we assume that surface-brightness is independent of luminosity. In model B, surface brightness depends on luminosity as described in the text, with 0:4 mag scatter about the mean relation. The Monte-Carlo NE model assumes Freeman disks with 0.4 mag scatter. The analytic NE model assumes no scatter. & Sandage 1988). The surface-brightness{luminosity relation in model B is much steeper than observed, but is di cult to rule out as galaxies with low surface brightness for their luminosity will always be preferentially missed in real surveys. An intermediate case might be one with a shallower trend of surface brightness with luminosity but with broader scatter in central surface brightness at xed luminosity (as found by Malin 1988 andMalin 1991). Model B is also similar to the model of Lacey et al. (1993), which included isophotal selection with a surface-brightness{luminosity relation similar to that seen in Virgo and a steep luminosity function. These properties were predicted from their model of tidally triggered galaxy formation that included halo formation, galactic winds, spectral evolution, and extinction. The e ects of surface brightness are thus di cult to disentangle from other aspects of the model.
Construction of Monte-Carlo Models
The simplest way to include intrinsic scatter into the galaxy distribution functions is to construct simulated galaxies using Monte-Carlo techniques. For the three models described above galaxy parameters are chosen at random from the distribution functions describing space density (constant), luminosity, surface brightness, morphological type, and bulge/disk ratio. Catalogs of simulated galaxies are constructed and are observed (i.e. they are selected and their magnitudes are measured) in a way that closely matches real surveys.
For comparison, and as a check on our Monte-Carlo technique, we compute a second 0.0 III NE model with total magnitude selection, using the analytic form of the luminosity function and numerically integrating over the luminosity function to compute N(m) and N(z) distributions. The techniques are identical to those used by Yoshii & Takahara (1988), but the luminosity function and type distributions are slightly di erent. For our comparison we use luminosity functions identical to those for the isophotal NE model. The distribution of spectral-energy distributions (SED's) is slightly di erent, however, as the Yoshii & Takahara models do not include separate bulge and disk components. Galaxies are divided into ve broad Hubble types, each characterized by the following: 1. a Schechter (1976) luminosity function
(L)dL = (L=L ) e L=L d(L=L );(3)
characterized by a space density , a faint-end slope ; and a characteristic luminosity L ; 2. a ratio of bulge/total luminosity in the B band (Simien & de Vaucouleurs 1986); and
Gaussian scatter about this ratio characterized by a dispersion (di erent for each type) and constrained such that 0 L B (bulge)=L B (total) 1; 3. surface brightness pro les g(r) given by g(r) = g 0 exp(a n (r=r e ) 1=n );
(4) with n = 1 for galaxy disks and n = 4 for galaxy bulges and coe cients a 1 = 1:68 and a 4 = 7:67; and 4. separate (luminosity independent) spectral-energy distributions (Coleman, Wu, & Weedman 1980) for the bulge and disk components. The parameters describing the di erent models are shown in Table 1.
For the Monte-Carlo models, we construct simulated catalogs of galaxies for each morphological type, selecting the parameters for each galaxy at random from the distribution functions of bulge/total ratio, central surface-brightness, and luminosity. The code uses a double-precision random number generator, and has been extensively tested to ensure that input distribution functions are properly reproduced by the random selection, even in the tails of the distribution. Redshifts of the galaxies are selected so as to produce a uniform co-moving density using the rejection method (Press et al. 1992) with the Euclidean volume element as the comparison function to the cosmological volume element for q 0 = 0:5. Magnitudes are then computed for speci ed bandpasses by integrating the redshifted spectralenergy distributions through the lter bandpasses. Depending on the survey we are trying to simulate, these catalogs contain anywhere from 10 3 to 10 5 galaxies, and list for each galaxy the redshift, luminosity in the rest-frame B J band, apparent magnitudes in various bands, and scale lengths of the bulge and disk components. These galaxy catalogs are then fed to separate programs that \observe" the galaxies using seeing and selection criteria that closely match real surveys. Distributions of apparent magnitude, redshift, color, etc. are then compiled from galaxies that pass the selection criteria, using isophotal, aperture, or total magnitudes as appropriate. The resulting distributions are normalized to represent the correct volume densities of each galaxy type. Note that our simulations do not explicitly include the noise present in any real observations. To the extent that the algorthims used in the deep surveys for detecting galaxies and measuring their magnitudes are unbiased, the e ect of noise is simply to increase the scatter in the measured magnitudes. Scatter of a few tenths of a magnitude is unimportant over the many decades of the N(m) diagram and over the relatively wide magnitude intervals used for N(z). In any case, the best way to assess the impact of noise would be to construct simulated images with noise and analyze them in the exact same way as the observations. While such an experiment would be worthwhile, it is beyond the scope of this paper. Assumed complete to D 28:6 > 2 00 . Isophotal magnitudes used. 5.
Isophotal magnitudes for galaxies with D 28 > 2 00 ; otherwise, used 2 00 aperture. 6.
Assumed complete to D 28 > 2 00 . 3 00 aperture magnitudes used. 7.
Assumed complete to D 25:4 > 1 00 . Isophotal magnitudes used.
The surveys we have chosen to simulate are listed in Table 2. Each is described by the estimated FWHM of the seeing disk, a type of selection (diameter or magnitude), and a type of magnitude (aperture, isophotal, hybrid, or total). We tune the model to match roughly the canonical distribution of morphologies and total luminosity function in bright surveys (Table 2 lines 1 & 2). This involves adjusting , L , and separately for the di erent Hubble types. As an independent test of the properties of local galaxies, in x4.2 we predict the color distributions for a sample of bright galaxies and compare to the observed distribution for a subsample of the RC3 catalog (de Vaucouleurs et al. 1991). We then compute the faint galaxy counts, redshifts, and colors for the rest of the surveys listed in Table 2. 3.1. Cosmological Model Our models are constructed in the framework of the standard Friedman-Lemaître model with zero cosmological constant. The relevant formulae are set out in Yoshii & Takahara (1988) and Sandage (1988). Brie y, the apparent magnitudes of sources in a bandpass are related to their absolute magnitudes M by m = M + k (z) + 5 log(d L ) + 25;
(5) where k is the standard k-correction that incorporates the frequency shift and the bandpass dilation due to redshift and d L is the luminosity distance in Mpc. The luminosity distance is d L = c H 0 q 2 0 fq 0 z + (q 0 1) q 1 + 2q 0 z 1]g;
where H 0 , q 0 , and c are the Hubble constant, the deceleration parameter, and the velocity of light, respectively. As our aim in this paper is to elucidate the e ects of surface-brightness selection, rather than to test a particular cosmology, we adopt H 0 = 50 km s 1 Mpc 1 and q 0 = 0:5 throughout. The co-moving density of galaxies is conserved in our model. The number of galaxies per unit redshift per steradian depends only on the volume element dV dz = cd 2 L H 0 (1 + z) 3 p 1 + 2q 0 z :
Galaxy angular sizes are computed using the angular-diameter distance
d A = d L (1 + z) 2 :(8)
3.2. Point-Spread Function Convolution Atmospheric distortions (seeing) can have a signi cant e ect on galaxy counts at faint magnitudes. This e ect was not included in McGaugh's (1994) initial consideration of the counts of LSB galaxies, but has been considered in other contexts by Pritchet & Kline (1981) and Yoshii (1993). To compare our results to the counts from deep surveys, we convolve our model galaxy pro les with point-spread functions that closely match the conditions in the real surveys. Assuming a circular galaxy with a surface brightness pro le g( ), where = r=r e , and assuming a circular Gaussian point-spread function with dispersion in units of r e , the convolved one-dimensional pro le has the form g( ; ) = 2 f( ) 1 Z 0 g( )I 0 ( = 2 )f( ) d : (9) I 0 (x) is the modi ed Bessel function of the rst kind and f(x) = exp( x 2 =2 2 ):
To speed our calculations we have computed the surface-brightness pro leg( ; ) and the integrated pro leG
( ; ) = 2 Z 0g ( ; ) d(11)
separately for bulge and disk pro les for a grid of 0 10 and 0 10 in steps of 0.1 in and . For each model galaxy we compute r e in arcsec for the disk and bulge components using the angular-diameter distance, and compute g(0) through the lter bandpass using the appropriate k corrections. To compute an isophotal radius, we step through the gridg, using a value of that approximates real observing conditions, and summing bulge and disk components until the ux drops below that corresponding to the limiting isophote.
The Distribution of Morphological Types
When isophotal selection is not considered, the actual morphologies of galaxies are irrelevant and all that is important is their spectral energy distributions. A typical assumed mix of Hubble types is 30% E/S0, 50% Sa-Sb, and 20% Sc or later (Tinsley 1980;Shanks et al. 1984;Yoshii & Takahara 1988). This morphological mix is justi ed on the basis of wide-area surveys of bright galaxies such as those of Tammann, Yahil, & Sandage (1979); Kirshner, Oemler, & Schechter (1978); and (Peterson et al. 1986).
As our models include isophotal selection, both spectral energy distributions and galaxy pro les are important. Galaxies in our models are simple entities composed of pure r 1=4law bulges and pure exponential disks. We have separated galaxies into ve morphological classes. For each type, we x the ratio of bulge to total light in the B J band to mean values found by Simien & de Vaucouleurs (1986), but allow Gaussian scatter about this mean. The adopted bulge/total parameters are shown in Table 1. Spectral energy distributions are speci ed separately for bulge and disk components (see below), so the ratio of bulge/total light will vary with bandpass.
The key new feature of our model is the inclusion of LSB disk galaxies. Matching the distribution of morphological types in wide-area surveys introduces an additional uncertainty in our models, as the selection criteria of those surveys are not well quanti ed. In particular the fraction of late-type galaxies (especially in model B) is critically dependent on the limiting isophote of the surveys. As these limits have never been explicitly quanti ed for surveys with good morphological resolution, we have simply chosen a limit that appears to us to be a good approximation of the survey material used by Kirshner et al. (1978) and Shanks et al. (1984). Speci cally, we use our code to simulate a survey in 3 00 seeing complete to B J = 16 for galaxies with isophotal diameters D 24:5 > 20 00 . We adjust the space densities of the di erent types to produce the proportions E:S0:Sab:Sbc:Sdm = 10:20:25:25:20 in the nal survey.
3.4. The Luminosity Function While each morphological type is characterized by its own luminosity function, for the overall normalization we require that the total luminosity function match that observed locally (Kirshner, Oemler, & Schechter 1979;Efstathiou, Ellis, & Peterson 1988;Loveday et al. 1992). Our goal here is to demonstrate that when a realistic model for isophotal selection is included, existing observations still allow room for a large population of LSB galaxies. To model the local luminosity function (LF), we try to mimic the selection criteria of Loveday et al. (1992;herafter LPEM). However, for comparison to previous faint-galaxy modeling, we have adopted the Yoshii & Takahara (1988) LF slope ( = 1:1) and normalization ( = 2:3 10 3 gal Mpc 3 ) in preference to the best t found by LPEM. Our adopted local LF is consistent to their best t to within their 1 errors; adopting the LPEM best t would increase the discrepancy in N(m B J ) for all models, but would not a ect the comparison of the di erent models.
The LPEM eld-galaxy luminosity function was derived from a redshift survey of galaxies detected on IIIaJ Schmidt plates by the Automatic Plate Measuring (APM) machine (Kibblewhite et al. 1984). An automated star{galaxy separation algorithm was used, combined with visual inspection of the images, to decide whether or not to include an object in the survey. Sources were detected by the APM above an isophote of B J 24:5 mag arcsec 2 (Loveday 1989). APM magnitudes are simply a logarithmic scaling of the sum of the linearized pixel intensities above this threshold. They therefore closely correspond to isophotal magnitudes, with the exception that saturation at surface brightnesses B J < 22 may arti cially increase the magnitudes of high-surface-brightness galaxies. Loveday (1989) used CCD images of selected galaxies to compute a constant conversion from APM magnitudes to total B J magnitudes. This process makes no correction for the di erent proportions of the total galaxy light encompassed for high and low surface-brightness galaxies above the APM detection threshold 3 . The magnitudes are therefore essentially isophotal, with a constant o set.
To simulate this process, we compute isophotal magnitudes at B J = 24:5 for each of our simulated galaxies (assuming 3 00 FWHM seeing), but then add a constant o set of m = 0:27 mag to simulate the conversion from APM to \total" magnitudes. This is the conversion appropriate for pure Freeman disks, which presumably dominate the calibration sample. We select galaxies from our simulated catalogs with these \corrected" magnitudes in the range 15 < B J < 17:15, and use the standard V=V max technique (Schmidt 1968) to reconstruct the luminosity function. We tune the shapes of the type-speci c luminosity functions to produce an \observed" luminosity function that matches a Schechter function with = 2:3 10 3 gal Mpc 3 and = 1:1. Figures 3-5 show the comparison of the models to our adopted eld-galaxy LF. The rather steep intrinsic faint end slope of the model B luminosity function is consistent with that determined by Bothun, Impey, & Malin (1991) from the distribution of scale lengths and surface brightnesses found in the Fornax Cluster (not shown in the gure).
Spectral Energy Distributions
We use standard spectral-energy distributions (SED's) determined from observations of nearby galaxies as the basis for our magnitude computations (Pence 1976;Coleman, Wu, & Weedman 1980). While these SED's undoubtedly do not cover the full range exhibited by real galaxies, we take them as a conservative starting point. In this respect, our model is distinctly di erent from that of Koo et al. (1993), where much of the increase in counts comes from galaxies that are bluer than commonly found in local galaxy samples. Our ability to match the counts without producing excess blue galaxies locally illustrates that surface-brightness selection e ects are at least as important as k-corrections in governing the 3 Though Loveday (1989) searched for the possibility a surface brightness e ect, the bright isophotal limit does not allow su cient dynamic range for this e ect to be readily apparent, especially when there is large scatter in the calibration data. Indeed, much of the scatter in the raw APM magnitudes could be due to uctuations in the isophotal limit from plate to plate. In an ideal situation, correction from isophotal to total magnitudes should be made on an individual basis for each galaxy based on pro le shape. When a single average correction is applied instead, the most seriously a ected objects are those associated with large volume corrections, producing a pronounced e ect on the luminosity function. Loveday et al. (1992). For comparison, the dashed line shows a luminosity function with the same , but with = 1:5. The open symbols show the luminosity function that would be recovered from our simulated galaxy catalogs if galaxies could be selected by total magnitude. The solid symbols show the luminosity function recovered when the APM selection criteria and magnitude estimation scheme are adopted as described in the text. The luminosity functions of the individual morphological types have been tuned to match the ducial total luminosity function, and the three panels here show the extent to which this tuning has been successful. This panel shows the NE model. The SED's adopted are shown in Fig. 6. For 1400 10000 A we use the compilation of Coleman, Wu, & Weedman (1980). We adopt three of their SED's, labeling them I, II, and III. Type I is used for ellipticals, bulges and S0 disks, and is taken from their Table 2 tabulation of the mean M31 + M81 SED. Type II is is taken from their Table 3 and is used for Sab disks. Type III comes from their Table 5 SED for Im galaxies. These SED's are extrapolated to long wavelengths following Yoshii & Takahara (1988) and linearly extrapolated to short wavelengths as shown in Fig. 6. The SED's chosen for the bulge and disk components of each galaxy type are shown in Table 1. We normalize the ux in each SED to produce an absolute magnitude M BJ = 21:1 in the rest-frame, the multiply by the appropriate factor for the absolute magnitude of the disk or bulge. We then redshift the galaxy and compute a weighted mean ux over the lter bandpass to account for k-dimming.
To convert to magnitudes we use the zero points shown in Table 3.
HI Mass Function
A standard argument against a high space-density of LSB galaxies is that blind HI surveys do not nd many optically invisible HI clouds (Ferguson & Sandage 1988;Briggs 1990).
We regard this as a serious constraint, and devote x4.1 to a detailed discussion of the HI constraints. To compare our luminosity function to HI surveys, we need a relation between optical luminosities and HI masses. We assume E and S0 galaxies have no HI, and adopt for later types the conversion formula of Briggs (1990): M HI = 3:2 10 9 M (L=L 0 ) 0:9 ;
where we adopt L=L 0 = 10 0:4(M B J +21:1) for all types regardless of the value of L used for the optical luminosity function. Comparison to the observed HI mass function is shown in Fig. 7 and discussed in x4.1.
Galaxy Surface-Brightness{Luminosity Relations
The most important departure of our model from others that have considered isophotal selection is that we adopt non-standard models for the distribution of central surface brightness of spiral galaxy disks (for galaxies of type Sa and later). Our assumptions are described in x2 and illustrated in Fig. 1.
The disks of S0 galaxies are assumed to have 0 = 21:6 0:4, regardless of luminosity. Because we are ignoring evolution, the properties of S0 disks are not important for our models, as the k corrections preferentially remove early-type galaxies from high-redshift samples.
For elliptical galaxies and bulges, we use the relation from Sandage & Perelmuter (1990) e = 0:48 M B T + 11:02;
which translates to e = 1:20 log(L=L ) + 21:16 (14) for M B T = 21. In this model, surface brightness increases with decreasing luminosity for elliptical galaxies. While this appears to hold for r 1=4 -law ellipticals seen in nearby clusters, it does not hold for the dwarf-ellipticals that dominate the counts in those clusters (Binggeli, Sandage, & Tarenghi 1984;Ferguson & Sandage 1988). However, dE galaxies do not appear to be abundant in the eld, and are in any case bluer than giant E's and have exponential surface-brightness pro les. Thus, to the extent that dE galaxies are included, they are grouped implicitly with low-luminosity LSB galaxies in our models. The E and S0 galaxy luminosity functions in our models are either at or declining at faint magnitudes, and hence do not predict large numbers of compact high-surface brightness galaxies at low luminosities. Isophotal selection makes very little di erence to the counts of early-type galaxies in our models.
Galaxy Selection Functions in Deep Surveys
The details of object selection and photometry in faint galaxy surveys are di cult to model precisely and are sometimes not completely speci ed in the published reports. Automated galaxy detection algorithms typically catalog pixels above a certain S/N threshold, then assign adjacent detected pixels to a single \object." To increase the detection probability for faint galaxies, the images are usually convolved with a kernel of 1-2 00 FWHM before running the detection algorithm. Various star-galaxy separation procedures are used to remove unresolved objects, but the details of these are unimportant at faint magnitudes, where stars are a negligible fraction of the total source counts. For the detected objects, some surveys report isophotal magnitudes, others aperture magnitudes, and still others report a hybrid of the two.
The ultimate test of the models would be to construct simulated images with signalto-noise ratios and PSF's that match the real surveys, then analyze them with the same software that was used on the real images. This task is beyond the scope of our paper, and in any case is best done in collaboration with the actual observers. For illustrative purposes, we present in x5.4 simulated deep HST WFPC-2 images, but for comparison to existing observations we use the approximations to the surveys described below.
For the B J -band counts, we compare to Tyson (1988). His isophotal threshold was B J = 28:7 and he reports isophotal magnitudes above that threshold. The seeing FWHM was about 1:7 00 . We use the Lilly et al. (1991) survey for the I band. Magnitudes used for the N(m) diagram in that study were in the Oke (1974) AB system (I AB = I + 0:48) and were isophotal above a threshold of I AB = 28 for galaxies with D 28 > 2 00 , but were aperture magnitudes through a 2 00 aperture for smaller galaxies. Images were convolved to an e ective seeing of 1:2 00 . The K-counts are taken from Gardner, Cowie, & Wainscoat (1993). Details of the galaxy selection and photometry for the deepest K counts have not been published. We assume, following Yoshii (1993), that the survey is complete for galaxies with K AB (= K + 1:8) isophotal diameters D 25:4 > 1 00 . The seeing FWHM was taken to be 1.7 00 .
At the faintest levels, the published counts include completeness corrections for the number of overlapping sources. We have not simulated this process, as our model galaxies are all counted if they meet the survey selection criteria.
We compare our redshift distributions to three published samples (Cowie, Songaila, & Hu 1991;Lilly 1993;Colless et al. 1990Colless et al. , 1993. Deep Anglo-Australian Telescope (AAT) prime-focus plates were used to select galaxies for the LDSS redshift survey (Colless et al. 1990(Colless et al. , 1993. The magnitudes measured were essentially isophotal above B J 26:5; with a constant (unspeci ed) shift to bring them into agreement with a standard sequence of galaxies measured through a 10 00 aperture with a CCD. An Sbc Freeman disk observed at z = 0:4 would have a central surface brightness of 0 (B J ) = 24:12 In 2 00 seeing the correction from m 26:5 to m(10 00 ) would be 0:58 mag, which we adopt for all galaxies in the sample. The Lilly (1993) sample was compiled from deep CCD imaging with the Canada-France-Hawaii Telescope (CFHT) in 0:7 00 seeing. The isophotal selection limits are not speci ed. For our models, we assume that the sample is complete for galaxies with D 28 > 2 00 . Magnitudes for the detected galaxies are computed through a 3 00 aperture, and galaxies are required to have aperture magnitudes 21:0 < I AB < 22:5. The deepest redshift survey is that of Cowie, Songaila, & Hu (1991), who published redshifts for a small sample of galaxies to B = 24 selected from the Lilly, Cowie, & Gardner (1991) survey. The selection criteria of the nal sample of 21 objects are not well de ned, as only 13 were from the original input sample. In our models, we assume the survey was diameter-limited at D 28:63 (B AB ) > 2 00 , and use uncorrected isophotal magnitudes to decide whether the galaxies are brighter than the magnitude limit.
While the galaxy counts are very sensitive to selection criteria and magnitude schemes in our models, the redshift distributions are not. In the NE model and model A the luminosity function is not steeply rising, so a change in the limiting isophote does not bring in many lowredshift LSB galaxies. The steeply rising luminosity function in model B, however, causes the redshift distribution to be progressively skewed toward lower redshifts for fainter limiting isophotes.
Comparison to Observations
The rst test of any model that claims to require no evolution is that it match the observed properties of local galaxies. Complete samples of local galaxies, with HI uxes, colors, scale-lengths, and bulge/disk ratios are non-existent. We are therefore forced to use incomplete samples and to approximate their selection criteria. We have tuned our type-speci c luminosity functions as described above to match the locally observed total luminosity function and to be in rough agreement with proportions of morphological types determined by Pence (1976), Tinsley (1980), and Shanks et al. (1984). It is useful to have an independent test that the models reproduce known properties of local galaxies. To this end, in x4.1 and x4.2, we compare our model predictions to local estimates of the HI mass function and galaxy B V color distribution.
The HI Mass Function
The LSB galaxies that populate our models are for the most part visible on sky-survey plates (that is how many of the known examples have been found), and are especially easy to nd if you know where to look (i.e. from an HI position). The impressive constraints on the space density of isolated HI clouds (Fisher & Tully 1981;Briggs 1990) are essentially useless for constraining our model, as the isophotal limits of the searches for optical counterparts have not been quanti ed. Detections of HI emission in the \o " beams are not uncommon in these surveys (Briggs 1990), but are almost always associated with optically visible galaxies. These galaxies (even if they are LSB) are removed from the samples before the space density of \intergalactic HI clouds" is computed. This HI cloud density therefore does not constrain the LSB galaxy density. An HI mass function derived purely from the o -beam detections without reference to optical counterparts would provide strong constraints on our model, but nothing along these lines has been published. The most relevant surveys, therefore, are those that allow some comparison between the frequency of serendipitous detections and the galaxy luminosity function (Kerr & Henning 1987;Weinberg et al. 1991). (The surveys of Schneider et al. (1989) andHo man et al. (1989) are in high-density regions, and are Figure 7: Constraints on the HI mass-function of galaxies. The solid curve shows the HI massfunction for normal galaxies derived from the NE model as described in the text. The dotted and dashed curves show the predictions for models A and B, respectively. Squares show the HI mass function from Kerr & Henning (1987), with dotted lines showing the behavior if the mass-function is normalized to the normal-galaxy HI mass function at 4 10 10 M . The triangles are the Weinberg et al. (1991) mass function, normalized to that computed from the canonical optical luminosity function M HI = 10 10:1 M . The Arecibo HI cloud limits are the limits on the space-density of isolated HI clouds computed by Briggs (1990). therefore di cult to compare to the expectations from the eld-galaxy luminosity function.) Kerr & Henning (1987) carried out a blind HI search with the 300 0 Greenbank telescope. They observed 1900 test directions in the galactic plane and 860 directions out of the plane, detecting a total of 28 objects. Briggs (1990) converts their numbers to an HI mass function, which we show in Fig. 7 (converted to proper units for H 0 = 50). Comparison of the Kerr & Henning (1987) mass function to the optical luminosity function is highly uncertain, as we do not know the density of galaxies behind the plane of the Milky Way. With our adopted L B to M HI conversion, objects with masses M HI > 1 10 9 M appear de cient by a factor of 7 compared to the expected HI mass function for normal galaxies. This de cit was also noted by Briggs (1990) and attributed to the di culty of detecting broad line-width galaxies. As this di culty is not quanti ed, we do not know how many large line-width LSB galaxies could be lurking in the night sky. However, the de cit may also be due in part to a di erence in mean densities of galaxies behind the plane. If we normalize the Kerr & Henning HI mass function to the eld-galaxy Schechter function at M HI = 10 10 M (as shown by the dotted lines in Fig. 7), our dwarf dominated model B is still marginally consistent with the data. Weinberg et al. (1991) conducted another blind HI survey using the VLA. They chose elds both within the Perseus-Pisces supercluster and in a foreground void. Seventeen objects were detected in the supercluster and none in the void. Once again, normalization is a problem and we have chosen simply to normalize the counts in the mass range 10 9 < M HI < 10 10 to the eld galaxy M HI mass function computed from our ducial local luminosity function. This is not strictly proper, as some of the Weinberg et al. elds were centered on bright galaxies. If anything, however, that should lead us to underestimate the relative proportion of low M HI galaxies. The Weinberg et al. mass function is shown as triangles in Fig. 7. Weinberg et al. nd that their HI mass function is consistent with a at luminosity function = 1 in the mass range 10 8 < M HI < 10 9 . However, with only 13 galaxies in this mass range, the constraints on the slope are not particularly good. Furthermore, their velocity resolution, 40 km s 1 , was not optimal for detecting narrow line-width galaxies, and may lead to some incompleteness at the low-mass limits of their mass function.
We conclude that our models are neither supported nor ruled out by existing limits on the HI mass function. The steep HI mass function of model B shows the largest discrepancy with the data, but is still within the uncertainties of the overall normalization of the observed HI mass function and the conversion between blue luminosity and HI mass. As most known LSB eld galaxies are rich in HI, a more complete HI survey would provide a stringent test of our proposal.
Local B V Colors
The choice of SED's for the bulge and disk components in our model was largely dictated by the availability of the Coleman et al. (1980) templates. Our procedure for tuning the luminosity type-speci c functions provides no guarantee that the color distribution of galaxies will match that seen in local galaxy samples. Other investigators (Tinsley 1980) have used the B J R F colors distribution of Kirshner et al. (1978) to x the distribution of morphological types. However, there are di culties in reproducing the Kirshner et al. (1978) passbands (Bruzual 1981). Instead, we selected all galaxies from the RC3 (de Vaucouleurs et al. 1991) satisfying the following criteria: B T < 14; D 25 > 120 00 ; jbj > 30 ; and v GSR < 4000 km s 1 :
To correct for extinction we use the catalogued A B values and assume A B = 4E(B V ). We suspect that the RC3 is reasonably complete to this limit. Of those galaxies satisfying the above criteria, 92% have measured B V , su cient for the color distribution to be representative of the sample as a whole.
We apply the same diameter, magnitude, and velocity selection criteria to our models and compare the resulting color distribution to the RC3 sample in Fig. 8. For comparison, we also show the local B-V distribution that would be inferred from the (Koo et al. 1993) model if galaxies were selected purely by total magnitude. As our models were not tuned to match the observed color distribution, we regard the agreement for models A and B as satisfactory. Model B shows the best agreement (and also the best agreement with the deep survey data | see below). As our NE model is very similar to other NE models found in the literature, they would presumably show the same surfeit of red galaxies if subjected to the same test. Part of the failure of NE models to match the counts may therefore be due to poor assumptions about the distribution of z = 0 SED's, as Koo et al. (1993). The comparison sample is selected from the RC3 catalog as described in the text. The mean B V for the di erent samples are shown in the upper left. The mean for the RC3 sample is B V = 0:72: The models have been normalized to match the number of galaxies in the RC3 sample.
suggested by Koo et al. (1993). However, the Koo et al. (1993) counter-evolution model appears to skew the distribution too far to the blue.
Number Counts
Having shown that our models are at least consistent with low-redshift observations, we now turn to a comparison with estimates of N(m) from deep surveys. Our assumptions about the surveys are found in Table 2. The results are shown in Fig. 9. The NE model with isophotal selection follows quite closely the NE model computed with total magnitude selection. In both cases our NE model agrees closely with that of Yoshii & Takahara (1988), and falls short of the observed counts by a factor of four at B J = 24. Model A is a slight improvement, but still falls short by a factor of three at B J = 24. Model B begins to depart from the observed counts at B J = 23, and is a factor of two short at B J = 24.
The trends are similar in the I band, with model B providing an acceptable t down to I AB 23. In the K band, the agreement between the standard NE model and the observations becomes worse when isophotal selection is included. Once again, model B provides the best t.
The turnover at faint magnitudes is of course in part due to the adopted (q 0 = 0:5) cosmology. For q 0 = 0:05 (not shown in the gures), model B agrees with the observed counts to within a factor of two down to B = 25 and to within the observational uncertainties down to the limits of the I and K band observations. Fig. 12. The observed median redshifts are z = 0:32; 0:30 and 0:38 for the Colless, Cowie, and Lilly samples, respectively. The median redshifts predicted by our models are shown in Table 4. The models predict a slightly higher proportion of low redshift galaxies than observed by Colless et al. (1993), and slightly more high redshift galaxies than Cowie sees. The most serious discrepancy is in the I band, where our models predict a median redshift 25% higher than observed. As the models underpredict the counts in the magnitude ranges sampled by the Lilly (I-band) and Cowie (B-band) surveys, detailed agreement with the observed redshift distributions is not expected. The most important point is that none of the models, when properly normalized, overpredict the numbers of galaxies seen at low-z. This was certainly not a foregone conclusion for model B, which is dominated by dwarfs, and illustrates the importance of isophotal selection. The median redshifts are signi cantly lower for this model when galaxies are selected by total magnitude. Lilly et al. (1991). The models were binned in 0.2 mag intervals, then smoothed with a boxcar lter over 3 bins to simulate photometric errors near the survey limits.
Color Distributions
Color distributions can provide an additional constraint on the models. With only three SED's we cannot hope to match the color distribution in detail, however we can at least test whether the mean and dispersion of the predicted colors are close to those observed. In Fig. 15 we compare the model colors to those observed by Lilly, Cowie, & Gardner (1991). Galaxies were required to have 20 < I AB < 25 and B AB < 27 through a 3 00 aperture in 1:2 00 seeing for both the models and the data. The model color distributions plotted in the gure are are binned in 0.2 mag intervals, and smoothed with a boxcar lter over three bins to simulate photometric errors. The means and standard deviations of the B AB I AB colors are shown in Table 5.
Diagnostics for High-Resolution Imaging
One of the major goals of high resolution imaging with the refurbished Hubble Space Telescope (HST) is to reveal the morphologies of high-redshift galaxies. In this section we illustrate the di erences between models dominated by LSB galaxies and the standard NE model when the faint galaxies are well resolved. As neither the standard NE model nor the models dominated by LSB galaxies have been tuned (e.g. by adjusting q 0 ) to match the counts at faint magitudes, detailed agreement is not expected. However, neither changing cosmology nor including mild evolution is likely to change the relative distributions of the \normal galaxy" models and LSB galaxy dominated models in the diagrams we present.
As we have not simulated models involving evolution, we cannot compare our predictions for faint-galaxy morphology in detail to those of other models. The only other model to make quantitative predictions for faint galaxy morphology is that of Lacey et al. (1993). However, Figure 16: Distribution of intensity-weighted axial ratios (b=a). Galaxies for this sample were selected in the I AB band to have diameters D 28 > 1 00 and magnitudes 23 < m iso < 25. The axial ratio is the weighted mean of the bulge and disk axial ratios, using the ux above the limiting isophote as the weighting factor. The instrumental PSF FWHM is assumed to be 0.1 00 . Smearing by this PSF is included in the modeling the selection of galaxies, but not in the computation of their axial ratios.
a few very general considerations may serve to illustrate the expected di erences between a universe dominated by slowly-evolving LSB galaxies and one dominated by merging galaxies or star-forming dwarfs. In a model dominated by bursting dwarf galaxies, for example, those galaxies that have completed their starburst but have not yet faded from view will be redder than their bursting counterparts. The model therefore predicts that low-surface brightness galaxies will be redder than high-surface brightness galaxies. Our LSB galaxy models predict the opposite trend. Models involving merging and triggered star formation predict that the bluest galaxies will have close neighbors, or will be in the process of merging and therefore have disturbed morphologies. On the other hand, if they are similar to local LSB galaxies, the bluest galaxies in deep surveys should have a dearth of close companions and should display the morphologies of late-type spirals or irregulars.
As most of the galaxies of interest are near the WFPC-2 detection limit for reasonable exposure times (< 10 ks), the task of measuring faint-galaxy morphologies may not be entirely straightforward. Bulge/disk decomposition at low S/N, for example, may not provide stable estimates of the central surface brightnesses of disks. In the remainder of this section we explore several parameters (axial ratios, e ective radii, and isophotal magnitudes) that can be measured in WFPC-2 images without detailed pro le tting and that provide a robust distinction between models that are dominated by LSB galaxies and models that are not. In the LSB galaxy models the counts are dominated by disk galaxies. A simple way of testing whether disk galaxies dominate the counts is to look at the distribution of axial ratios (b=a). Figure 16 shows the distribution our three models. Model B is clearly the most dominated by disk galaxies and would be easily distinguished from the no-evolution case in the absence of competing e ects. However, there are fairly serious competing e ects. Dust in the disks of LSB galaxies could make edge-on galaxies appear fainter than their face-on counterparts. We have included no extinction in our models. Perhaps equally important is the e ect of inclination on the selection algorithm. Galaxy detection routines such as FOCAS detect objects by recording the number of connected pixels above a xed threshold. In the absence of dust, an edge-on LSB galaxy will contain a smaller number of pixels, but each pixel will have a higher ux than the detected pixels in the same galaxy seen face on. Face-on galaxies near the survey limit will tend to break up into many small regions of unconnected pixels, while their edge-on counterparts might not. Finally, in dwarf galaxies the stellar velocity dispersion is a much larger fraction of the rotation velocity than in L galaxies, suggesting that the dwarfs might not be as at as we have assumed. This could be an important e ect in the dwarf-dominated model B.
Axial Ratio Distribution
E ective Radii
The distribution of half-light radii provides a much more robust test of models involving LSB galaxies. We have computed r e from the one-dimensional pro les in our catalogs, including the e ects of seeing. Figure 17 shows the predicted r e distribution for the three models considered here. Lilly et al. (1991) nd a distribution of r e in V images of an I AB selected sample that peaks 0:5 00 (close to the limit imposed by seeing). The distribution Figure 18: The variation in the number of galaxies detected brighter than an isophotal magnitude of I AB = 25 with the limiting isophote. The instrumental PSF is assumed to be 0:1 00 FWHM. No diameter limit is imposed on the galaxies.
for Model B to roughly the same I AB limit peaks at 1:3 00 , suggesting that a constant-size relation for galaxies with L < L is probably too extreme.
The E ect of Isophotal Thresholds
Another way to test for LSB galaxies is to examine the e ect of imposing di erent isophotal thresholds on the number counts. In the standard NE model the number of galaxies seen to a xed isophotal limiting magnitude is not a strong function of the isophotal limit chosen (so long as it is well below the Freeman value). On the other hand, the number of galaxies detected in LSB galaxy models is quite sensitive to the limiting isophote. Figure 18 shows the surface-density of galaxies in our models brighter than an isophotal magnitude I AB = 25 as a function of the limiting isophote. The expected number of galaxies rises much more steeply in model B than in the other models.
Simulated Images
The tests described above provide a quantitative means of establishing whether LSB galaxies are a signi cant contributor faint galaxy counts. The simulated images shown in Figs. 19-21 provide a more qualitative impression of the approximate morphologies expected in HST images. To produce the images, galaxies were selected at random from the simulated catalogs, with the appropriate relative proportions of the di erent morphological types. Image parameters were were fed to the IRAF 4 \mkobjects" task and the resulting images 4 IRAF is distributed by Kitt Peak National Observatory, National Optical Astronomy Observatories, operated by the Association of Universities for Research in Astronomy for the Figure 19: Simulated WFPC-2 (WF3) image of a standard NE model, with galaxy densities increased by a factor of two. The area covered is 80 80 00 , with a pixel size of 0:1 00 . In this and the following images, exposure times were assumed to be 1 10 4 s through the F814W lters. A count rate of 0:039counts s 1 pixel 1 for the sky background, estimated from the WFPC-2 instrument handbook for high ecliptic latitudes. Images were convolved with a model WFPC-2 PSF before adding noise with the IRAF`mknoise' task. Read noise was taken to be 7e pixel 1 , and dark count was assumed to be 0:015 counts s 1 pixel 1 . Comparison to gure 21 illustrates the dramatic di erence in morphologies expected if LSB galaxies are the dominant contributor to faint galaxy counts. To give an idea of the depth of the simulated images, two galaxies are marked. I AB magnitudes are 23.8 and 25.1 for galaxies 1 and 2, respectively.
were convolved with a model WFPC-2 point-spread function. The NE model is shown in Fig. 19, scaled up by a factor of two in density to provide a reasonable number of galaxies in the image. Even at HST resolution, many of the galaxies are barely resolved. In contrast, model B (Fig. 21) shows a larger number of galaxies (as it must to match the counts), and a much larger fraction of resolved galaxies. These appear as as fuzzy patches, which, although lacking detailed structures like spiral arms not attempted in the simulations, are clearly distinct from the majority of galaxies seen in the standard NE image. The larger fraction of edge-on systems is also readily apparent. The distinctions between Model A (Fig. 20) and the NE model are more subtle, but still readily apparent to the eye. Ultimately, the most robust test of the models and assessment the potential biases in of various detection algorithms would be to take such simulations and analyze them in the same way as the real observations.
Conclusions
Our goal in this paper has been to construct a set of alternative models of non-evolving galaxy distributions to test whether LSB galaxies could be a signi cant contributor to faint galaxy counts, and to illustrate the importance of galaxy selection and magnitude measurements schemes in intrepreting faint-galaxy counts.
We have compared two rather extreme models for the surface-brightness distribution of galaxies to a standard non-evolving model. In model A galaxy disks have central surface brightnesses ranging from the Freeman value of 0 (B J ) = 21:6 to 0 (B J ) = 25, with constant numbers per unit magnitude independent of luminosity. In contrast, model B incorporates a steep surface-brightness{luminosity relation. Disk galaxies brighter than L have 0 (B J ) 21:6 ( 0:4 mag), while those fainter than L follow a constant size relation (with the same scatter). By tuning the type-speci c luminosity functions, we nd that we can include a large population of LSB galaxies in either model A or model B without violating the constraints on the local eld-galaxy luminosity function (e.g. Loveday et al. 1992). The HI mass function is not in con ict with observations, but could be easily tested with a larger purely HI survey, or with more attention to the selection criteria for identifying the optical counterparts of existing HI detections.
The inclusion of LSB galaxies reduces the discrepancy between non-evolving models and faint-galaxy counts for q 0 = 0:5. For the parameters of Tyson's (1988) survey, the predicted surface-density of galaxies at B J = 24 for model A is only a factor of 1.2 higher than the NE model. Model B increases the counts by a factor of 2.2, but is still a factor of two shy of the observations at B J = 24. The redshift and color distributions for both models match the observations reasonably well. Agreement between the LSB galaxy models and the faint galaxy data would improve for lower values of q 0 , or for bluer LSB galaxy colors.
These results indicate that LSB galaxies could in principle be an important component of the observed faint-blue-galaxy population. Local examples of LSB galaxies have modest star formation rates, and have probably not drastically changed their luminosities or colors in National Science Foundation. the last 5 Gyr. For these galaxies, the assumption of no evolution in our models is probably a reasonable approximation. The assumed surface brightness distributions, on the other hand, are completely arbitrary. The distribution of r e predicted by model B is probably incompatible with ground-based observations, while model A is consistent with the available constraints. The true surface brightness distribution probably lies somewhere between the extremes of our two models.
High-resolution imaging with HST will allow a quantitative estimate of the contribution of LSB galaxies. Sensitive tests for LSB galaxies include measuring the distribution of r e within a limited magnitude range, and measuring the variation of N(m) as a function of limiting isophote (see x5). Qualitatively, our models di er from models involving merging and/or triggered star formation (Guiderdoni & Rocca-Volmerange 1991;Lacey & Silk 1991;Broadhurst, Ellis, & Glazebrook 1992) in predicting that the bluest galaxies within a xed magnitude will be the most isolated, and have the lowest surface brightnesses. Models involving the late formation of dwarf galaxies (Babul & Rees 1992) also predict that the bluest galaxies will be the most isolated. However, in these models low-surface-brightness galaxies will be redder on average than high-surface-brightness galaxies, while our models predict the opposite trend.
If LSB galaxies are a signi cant contributor to the counts of faint galaxies, then the requirements on galaxy evolution models become much less extreme. Standard models of high-surface-brightness galaxy evolution (Bruzual 1983;Rocca-Volmerange & Guiderdoni 1988;Bruzual & Charlot 1993) predict some color and luminosity evolution that must be included in any self-consistent model. We suspect that such modest evolution, coupled with a more realistic surface-brightness distribution and surface-brightness{luminosity relation could restore the agreement between galaxy counts and q 0 = 0:5 models, at least to the limits of current redshift surveys.
Figure 3 :
3This and the next two gures show the luminosity functions for our models. The solid line shows our ducial \observed" total luminosity function, consistent with the APM survey of
Figure 4 :Figure 5 :
45Model A.Selected by total mag.; measured total mag Selected by iso. mag.; measured iso. Model B.
Figure 6 :
6Adopted spectral energy distributions.
Figure 8 :
8Local B V color distribution for our models, and for that of
Figure 9 :Figure 10 :Figure 11 :
91011This and the next two gures show number counts in the B AB ; I AB ; and K AB bands for our models compared to the data from various sources. Solid lines show the expected N(m) for each of our models. The dashed lines show the standard (non-Monte-Carlo) noevolution model based on total-magnitude selection as described in the text. This gure shows the NE Model. Model Model B.
Figure 12 :
12This and the next two gures show redshift distributions in the B J , B AB , and I AB bands. The histograms show the data. The solid curve shows the results of the Monte-Carlo models that simulate the selection criteria of the surveys, and the dotted lines show the result of the standard NE model.
Figure 13 :
13Redshift distributions for model A.
Figure 14 :
14Redshift distributions for model B.
Figure 17 :
17Distribution of half-light radii r e for the three models. Selection criteria are the same as the previous gure. The computation of r e includes the e ect of smearing by the 0:1 00 PSF.
Figure 20 :
20Simulated WFPC-2 image of model A.
Figure 21 :
21Simulated WFPC-2 image of model B.
Table 1 :
1Galaxy ParametersNE model
bulge to
Bulge Disk
Type M B
Table 2 :
2Assumed Survey Parameters
Band Observer Magnitude Seeing ( 00 ) D min ( 00 ) lim
B J Loveday Isophotal
3.0
|
24.5
1
B J KOS78
Total
3.0
20
24.5
2
B J Tyson
Isophotal
1.7
|
28.7
B J Colless
Isophotal
1.0
|
26.5
3
B AB Cowie
Isophotal
1.0
1.0
28.6
4
I AB Lilly N(m) Hybrid
1.2
2.0
28.0
5
I AB Lilly N(z) Aperture
0.7
3.0
28.0
6
K AB Cowie
Aperture
1.0
3.0
25.4
7
Notes:
1.
Correction -0.27 added to isophotal magnitudes.
2.
Rough estimates. Used to set type fractions.
3.
Correction -0.58 added to isophotal magnitudes.
4.
Table 3 :
3Photometric Zero Points 10 9 f erg cm 2 s 1 A 1 for m = 0Band
Zeropoint
B J
5.27
B
6.29
V
3.64
B AB
5.61
I AB
1.34
K AB
0.22
counts.
Table 4 :
4Median Redshifts
Magnitude
Survey Band Range
NE model A model B Observed
Colless B J
21-22.5 0.30 0.27
0.27
0.32
Cowie B AB 21-24.0 0.42 0.41
0.36
0.30
Lilly
I A B
21-22.5 0.52 0.51
0.54
0.38
4.4. Redshift Distributions
Our model redshift distributions are compared to the observations in
Table 5 :
5B AB I AB Colors
Sample
Mean
Lilly (observed) 1.52 0.85
NE model
1.98 1.05
Model A
1.90 1.09
Model B
1.51 0.90
This work was supported by the Science and Engineering Research Council, and by NASA through grant #HF-1043 awarded by the Space Telescope Science Institute which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA under contract NAS5-26555. We thank Arif Babul for stimulating the Monte-Carlo modeling.
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| []
|
[
"G eneration of spin-w ave dark solitons w ith phase engineering",
"G eneration of spin-w ave dark solitons w ith phase engineering",
"G eneration of spin-w ave dark solitons w ith phase engineering",
"G eneration of spin-w ave dark solitons w ith phase engineering"
]
| [
"B Engt B I Schof \nInstitut f ur Festk orperphysik\nTechnische\n",
"A Ndrein Sl Avi N \nInstitut f ur Festk orperphysik\nTechnische\n",
"H Artm Ut B Enner \nInstitut f ur Festk orperphysik\nTechnische\n",
"Yurik I Vshar \nInstitut f ur Festk orperphysik\nTechnische\n",
"B Engt B I Schof \nInstitut f ur Festk orperphysik\nTechnische\n",
"A Ndrein Sl Avi N \nInstitut f ur Festk orperphysik\nTechnische\n",
"H Artm Ut B Enner \nInstitut f ur Festk orperphysik\nTechnische\n",
"Yurik I Vshar \nInstitut f ur Festk orperphysik\nTechnische\n"
]
| [
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische",
"Institut f ur Festk orperphysik\nTechnische"
]
| []
| U niversit at D arm stadt, D -64289 D arm stadt, G erm any 2 D epartm ent of Physics, O akl and U niversity, Rochester, M ichigan 48309 3 N onl inear Physics C entre, Research School of Physical Sciences and Engineering, A ustral ian N ational U niversity, C anberra, A C T 0200, A ustral ia W e generate experi m ental l y spi n-w ave envel ope dark sol i tons from rectangul ar hi gh-frequency dark i nput pul ses w i th externall y introduced phase shi fts i n yttri um -i ron garnet m agneti c l m s. W e observe the generati on ofboth odd and even num bers ofm agneti c dark sol i tons w hen the external phase shi ft vari es. T he experi m entalresul ts are i n a good qual i tati ve agreem ent w i th the theory of the dark-sol i ton generati on i n m agneti c l m s devel oped earl i er [ Phys. R ev. Lett. 82,2583] . | 10.1103/physrevb.71.104424 | [
"https://export.arxiv.org/pdf/nlin/0411056v1.pdf"
]
| 16,462,792 | nlin/0411056 | 61b6093f80bacf977c3242f4af8606e9808506dc |
G eneration of spin-w ave dark solitons w ith phase engineering
28 Nov 2004
B Engt B I Schof
Institut f ur Festk orperphysik
Technische
A Ndrein Sl Avi N
Institut f ur Festk orperphysik
Technische
H Artm Ut B Enner
Institut f ur Festk orperphysik
Technische
Yurik I Vshar
Institut f ur Festk orperphysik
Technische
G eneration of spin-w ave dark solitons w ith phase engineering
28 Nov 2004arXiv:nlin/0411056v1 [nlin.PS]
U niversit at D arm stadt, D -64289 D arm stadt, G erm any 2 D epartm ent of Physics, O akl and U niversity, Rochester, M ichigan 48309 3 N onl inear Physics C entre, Research School of Physical Sciences and Engineering, A ustral ian N ational U niversity, C anberra, A C T 0200, A ustral ia W e generate experi m ental l y spi n-w ave envel ope dark sol i tons from rectangul ar hi gh-frequency dark i nput pul ses w i th externall y introduced phase shi fts i n yttri um -i ron garnet m agneti c l m s. W e observe the generati on ofboth odd and even num bers ofm agneti c dark sol i tons w hen the external phase shi ft vari es. T he experi m entalresul ts are i n a good qual i tati ve agreem ent w i th the theory of the dark-sol i ton generati on i n m agneti c l m s devel oped earl i er [ Phys. R ev. Lett. 82,2583] .
D ark sol i tonshavebeen predi cted theoreti cal l y and observed experi m ental l y i n di erent types ofdi spersi ve or di racti ve nonl i near m edi a, i ncl udi ng opti caland m agneti c system s [ 1] . R ecentl y, dark sol i tons have been generated i n B ose-Ei nstei n condensates ofdi l ute atom i c gases [ 2] by engi neeri ng the phase of the m acroscopi c wave functi on w i th a techni que know n asphase im printing,earl i er devel oped to create opti caland m atter-wave vorti ces [ 1] .Phasei m pri nti ng i sa new toolofm ani pul ati ng coherentm atterwaves,and i ti sdescri bed asshi ni ng an o -resonance l aser on a B ose-Ei nstei n condensate i n orderto create phase stepsbetween i tsdi erentparts.A gi ven phase step de nes the param eters ofdark sol i tons travel l i ng i n the sam e or opposi te di recti ons,and the total num ber of the generated m atter-wave dark sol i tons w hi ch can be either odd or even [ 3] .
A sa m atteroffact,these recentresul tson the generati on ofdark sol i tonsi n B ose-Ei nstei n condensatescan be com pared w i th and l i nked to m uch earl i er experi m ental studi es of m i crowave m agneti c-envel ope spi n-wave sol itons i n yttri um -i ron garnet (Y IG ) m agneti c l m s [4] . Indeed, the rst observati on of dark sol itons i n Y IG m agneti c l m s [5]reveal ed unusualfeatures ofthe darksol i ton generati on observed asa change ofthe totalnumberofgenerated spi n-wavedark sol i tonsfrom even to odd w i th the grow th ofthe i nput power. T hese observati ons have l aterbeen expl ai ned theoreti cal l y by em pl oyi ng the concept of the so-cal l ed induced spatial phase shift [ 6] , w hi ch i s cl osel y rel ated to the concept ofphase i m pri nti ng,as was al so di scussed i n R ef. [ 3] . R ecentl y,a further attem ptto em pl oy the phase m ani pul ati on techni que for generati ng si ngl eand m ul ti pl e dark m agneti csol i tonshas been undertaken i n R ef. [ 7] .
T he purpose of thi s paper i s twofol d. Fi rst, we extend the concept of phase i m pri nti ng i m pl em ented for dark sol i tons and vorti ces i n B ose-Ei nstei n condensates to the el d of m agneti c sol i tons, and generate experim ental l y si ngl e and m ul ti pl e spi n-wave envel ope m agneti c dark sol i tons from rectangul ar dark i nput hi ghfrequency pul ses w i th externall y introduced phase shifts i n Y IG m agneti c l m s. Second,we provi de a di rect veri cati on ofthe theory devel oped earl i er i n R ef. [6] .
T he paper i s organi zed as fol l ow s. In Secti on II we i ntroduce our m odelw hi ch was rst suggested i n the pioneeri ng papers [ 4] and i s descri bed by the cubi c nonl i nearSchr odi nger(N LS) equati on forthe m agneti c el d envel ope. Secti on III sum m ari zes the basi c theoreti cal resul ts for the generati on of dark sol i tons by an i nput pul sew i th a jum p acrossthel ow -i ntensi ty regi on [ 6] .T he m ai n Sec.IV presentsourexperi m entalresul tsw hi ch are show n to be i n a good agreem entw i th the basi c theoreti calpredi cti ons.
II. M O D E L
W e consi der the evol uti on ofa spi n wavepacketi n the form (x;t)= u(x;t)expfi(k 0 x ! 0 t)g,w herethesl ow l y varyi ng com pl ex envel ope u(t;x) i s descri bed by the dim ensi onl ess nonl i near Schr odi nger (N LS) equati on [ 4] ,
i @u @t + v g @u @x + 1 2 D @ 2 u @x 2 N j uj 2 u = 0;(1)
w here v g = @!=@k i s the group vel oci ty eval uated from the spi n-wave nonl i near di spersi on !(k;j uj 2 ) at the carri er wavenum ber k 0 = k(! 0 ), D = @ 2 !=@k 2 i s the coe ci ent of l i near di spersi on, and N = @!=@j uj 2 i s the coe ci ent of nonl i neari ty. In the deri vati on of Eq. ( 1) di ssi pati on i s negl ected [ 4] . In the case D N > 0,Eq.(1)hasa sol uti on i n the form ofa dark sol iton that can be w ri tten as fol l ow s [ 8] :
u(x;t)= u 0 tanh expfi(K x t)g;(2)
w here = (t t 0 x=v s )= 0 . Sol uti on (2) descri bes a l ocal i zed di p i n the conti nuous wave (C W ) background w i th the hal f-w i dth 0 , the center t 0 , and the vel oci ty v s = v g + D K . In the phase factor of the dark sol itons, K i s the sol i ton wave num ber and = v g K + (1=2)D K 2 + N j u 0 j 2 i s the sol i ton frequency shi ft,w hi ch arethe nonl i neari ty-i nduced correcti onsto the wavenumber k 0 and carri erfrequency ! 0 . Sol uti on (2) i s a speci al case of a dark sol i ton w i th a m odul ati on factor A = 1, i . e. w hen the dark sol i ton has a m i ni m um am pl i tude ofu = 0 (the so-cal l ed \bl ack sol i ton"). D ue to the tanh-functi on,thi s sol i ton has an overal lphaseshi ftof = .H owever,such pul sesusual l y do notappeari n experi m entalsi tuati ons.Instead,a m ore generalform ofa dark sol i ton [ 8]shoul d be used,
u(x;t)= u 0 (1 A 2 sech 2 ) 1=2 expfi[K x ~ t+ ( )] g; (3)
w here the phase factor has the form :
( )= si n 1 A tanh p 1 A 2 sech 2 ! ;
(4)
K = K p 1 A 2 v s 0 A ;~ = p 1 A 2 0 A :
Sol uti on (3)descri bes a si ngl e \grey sol i ton" w i th an arbi trary val ue ofthe m odul ati on depth A . For A = 1, Eq.(3) transform s i nto the "bl ack" sol i ton (2). T he i m portant condi ti on for such dark sol i tons to exi st i s the phase shi ft gi ven by Eq. (4) that has to be present i n the carri er wave [ 1] . In the experi m entalsi tuati on, there i s no such total phase shi ft and, therefore, onl y even-num bered sym m etri c pai rs of dark sol itonsw i th equalm odul ati on j A j< 1 can be generated.In these pai rs,the phase shi fts ofthe i ndi vi dualdark sol itons have opposi te si gns and they com pensate one another,so the totalphase shi ft adds up to vani sh, = 0. U ndersuch condi ti ons,exci tati on ofodd num bersofdark sol i tons seem s,i n general ,i m possi bl e [ 1] .
III. T H E O R E T IC A L B A C K G R O U N D
R ecentl y, i t was show n anal yti cal l y and num eri cal l y that w hen an i nput pul se withoutany i ni ti alphase m odul ati on enters a nonl i near di spersi ve m edi um , the generated l ocal i zed wave acqui res an induced spatial phase shiftaccum ul ated duri ng i tsgenerati on [ 6] . Such a phase shi fti s negl i gi bl e forl arge group vel oci ti es,e. g. ,foropticalsol i tons i n bers,but i t becom es i m portant for spi n waves i n m agneti c l m s. M oreover,i f the i ni ti alphase shi ft (Fi g.1) i s i ncl uded i n the i nverse scatteri ng transform [ 9] ,the resul ts can be em pl oyed to expl ai n [ 6]the speci c features ofthe generati on ofboth odd and even num bers ofdark sol i tons observed i n experi m ent [ 5] .
In order to dem onstrate the rol e of the i ni ti al phase shi ft,we sum m ari ze the resul ts for the dark-sol i ton generati on i n thefram ework ofthenorm al i zed N LS equati on, i @u @t + @ 2 u @x 2 nj uj 2 u = 0
Equati on (5) fol l ow s from Eq.(1) after rescal i ng,by assum i ng the reference fram e m ovi ng w i th the group vel oci ty v g ,and renorm al i zi ng thevari abl estand x.T hei nput pul sei scharacteri zed by thephaseshi ft and thedi m ensi onl essval ue S 0 ,w hi ch i sde ned asS 0 = j u 0 j L p (N =D ) and i s usual l y cal l ed the i nput pul se area as i t i s proporti onal to the carri er wave am pl i tude u 0 m ul ti pl i ed by (spati al )pul se l ength L = v g T . T he sol i ton wavenum ber shi ft can then be presented as K = 2 =L.
A ccordi ng to R ef. [ 6] , the num ber and sym m etry of the exci ted sequence ofdark sol i tons are de ned by the transcendentalequati on for the sol i ton ei genval ue ,
tan( )= p S 2 0 2 :(6)
Every sol uti on ofthi s equati on forthe realei genval ue n correspondsto a dark sol i ton w i th the m odul ati on depth n = p S 2 0 2 n =L,w here n has the di m ensi on of the renorm al i zed am pl i tude, j u 0 j p (N =D ). T he ri ght-hand si de ofEq.(6)i sal wayspoi nt-sym m etri c around the origi n w hereasforthetan-functi on thi si sonl y thecasew hen = =2 i s zero [11] . In thi s l atter case,Eq.(6) yi el ds pai rsofei genval uesw i th thesam eabsol uteval uebutw i th opposi te si gns. T hi s expl ai ns w hy onl y even-num bered sym m etri c pai rs of dark sol i tons are predi cted by the earl i er theory [ 9] . N ote that a dark sol i ton w i th zero am pl i tude m i ni m um i s de ned by n = j u 0 j (N =D ) 1=2 , i . e. at n = 0. T hi s i sonl y the case for = =2 and odd m ul ti pl es thereof.
A ni te i ni ti al val ue of the phase shi ft 6 = 0 al l ow s to predi ct the generati on of asym m etri c or sym m etri c, odd-or even-num bered dark sol i ton pul ses. T he typi cal structure ofan N = 3-sol i ton state and the correspondi ng vari ati on ofi ts phase are schem ati cal l y show n i n Fi g. the totalphase shi ft . T hi s suggests [ 6]the use ofthe param eter pl ane (S 0 ; ) as presented i n Fi g.3. C hangi ng the pul se l ength and/ori nputam pl i tude i sequi val ent to m ovi ng on the hori zontalaxi s. A change i n the total phase shi fti sdenoted by a shi fton the verti calaxi s.Every poi nti n thatparam eterpl ane can be assi gned a val ue ofS 0 and ,therefore i t i s di rectl y connected to a dark sol i ton ofdi sti nct ei genval ues n .
In R ef. [ 6] ,i twasshow n thati n physi calsystem s w i th l ow val uesofthe group vel oci ty,an addi ti onalphase shi ft i s acqui red duri ng the process ofpul se exci tati on. H owever,i n the generalcase,the totalphase shifti n the i nput pul se carri er wave m ay ori gi nate ei ther from the pul se form i ng setup,i . e. i t i s provi ded external l y,or from an i ntri nsi cm echani sm i n thephysi calsystem .In thi spaper, we present a system ati c study ofthe form ati on ofm ag-neti c dark sol i tons w i th a vari abl e externalphase shi ft ext . U si ng these data,we veri fy,i n parti cul ar,w hether an acqui red phase shi ft acq i spresentatal land w hether i t ts the theoreti cal l y expected val ues.
In essence,the resul ts of R ef. [ 6]predi ct that an addi ti onalphase shi ft acq acqui red duri ng the process of dark-sol i ton generati on can be w ri tten i n the form ,
acq = bT T D " 1 + 1 2 S 0 2 # ;(7)
w here T D = (v 2 g T 2 = 2 j D j ) i s the (posi ti ve) di spersi on ti m e. T he phenom enol ogi calfactor b (i ntroduced i n the theory [ 6]and eval uated there to be cl ose to 2)takesi nto account the i n uence of the ni te l ength of the background C W si gnal .
U sual l y,the si gn of acq woul d not be i m portant due to the sym m etry ofEq. (6). H owever,as we are usi ng a phase shi fter i n the experi m ent to produce the external phase ext ,the rel ati ve si gn ofthese two phase shi fts i s i m portant. In the experi m ents descri bed bel ow ,the si gn ofthe acqui red phase shi ft acq was opposi te to that of the externalphase shi ft ext . T hus,the totalphase shi ft ofthe i nput pul se i s expressed as total = ext acq ; If the i nput pul se param eters are i n a range that T D T ,the acqui red phase shi ft m ay m ani fest i tsel fby produci ng asym m etri c and/or odd-num bered dark sol itons,and the poi nts correspondi ng to these sol i tons on the param eterpl ane ofFi g.3 w i l lbel ong to the parabol i c curve de ned by Eq.(7) and havi ng a m i ni m um at the poi nt S 0 = 0.T hi s case was real i zed i n the pi oneeri ng experi m ents perform ed by Patton' s group [ 5] .
O n the other hand,i fthe constant and posi ti ve externalphase shi ft ext > 0 i s i ntroduced external l y,w hi l e durati on T of the i nput pul se i s kept constant and the i nput wave am pl i tude (and, therefore,S 0 ) i s i ncreased, the dependence total (S 0 ) has the form of a quadrati c parabol a w i th a m axi m um atS 0 = 0 [ see Eq.(8)] . A s w i l l be seen bel ow ,i t i s thi s case that i s real i zed i n our currentexperi m ents.T herefore,the poi ntscorrespondi ng to di erent output sol i toni c pul ses obtai ned i n our experim entsw i th a posi ti ve externalphase shi ftand i ncreasi ng i nputwaveam pl i tude bel ong to parabol ashavi ng a m axi m um at S 0 = 0 (see Fi g.6 bel ow ).
IV . E X P E R IM E N T A L R E SU LT S
To study the e ects produced by an external phase shi fton thegenerati on ofm agneti cdark sol i tons,weneed to desi gn an experi m entalsetup w i th the fol l ow i ng characteri sti cs: (i ) the setup shoul d i ncl ude a nonl i near di spersi ve wavegui de capabl e ofsupporti ng spi n-wave dark sol i tons,i . e. the necessary condi ti on forthe form ati on of dark sol i tons,N D > 0,shoul d beful l l ed;(i i )theparameters ofthe pul ses propagati ng i n the wavegui de shoul d t the range w here the i nduced phase shi ft de ned by the Eq.(7) has a l arge enough val ue and coul d be detected experi m ental l y; (i i i ) there shoul d be a techni cal possi bi l i ty to i ntroduce an externalphase shift between the l eadi ng and trai l i ng fronts ofthe i nput pul se,si m il arto the phase shi fti ntroduced by the m ethod ofphase i m pri nti ng i n B ose-Ei nstei n condensatesornonl i nearopti cs [ 1] .
For our experi m ents, we have chosen m agnetostati c spi n waves propagati ng i n a thi n quasi -one-di m ensi onal wavegui de m ade of a si ngl e-crystal Y IG l m . N onl i near properti es of these waves are wel l understood, and the form ati on ofboth dark and bri ght sol i tons has been observed [ 4,5] . M oreover,the m easurem ent techni ques i ncorporati ng the so-cal l ed del ay l ine setup are wel lestabl i shed. In ourcase,i ti sespeci al l y i m portantthatone can easi l y i ntroduce an externalphase shi ft i n the i nput pul se si m pl y usi ng a m i crowave phase shi fter. T he m ost i m portantfeature ofthe m agnetostati c wavesi n the contextofthi sstudy i sthei rcom parati vel y l ow group vel ocity that shoul d resul t i n a substanti alval ue ofthe i nduced phase shi ft de ned by Eq. In the experi m entalschem e descri bed above,the trai l - i ng edge of the bri ght rectangul ar pul se created i n the rstm odul ati on channelprovi desthe l eadi ng edge ofthe i nput dark pul se, w hi l e the l eadi ng edge of the bri ght rectangul arpul secreated i n thesecond m odul ati on channel provi des the trai l i ng edge of the i nput dark pul se. T he phase shi ft ext i s i ntroduced by the vari abl e phase shi fteri nserted i n the second m odul ati on channel .T hus, thedurati on T oftheform ed dark i nputpul se(seeFi g.5) i s determ i ned by the di erence ofthe ti m e del ay t between the l eadi ng edgesofboth bri ghtrectangul arpul ses and the durati on t 1 ofthe bri ghtpul se i n the rstm odul ati on channelT = t t 1 . T he durati on T ofthe i nput dark pul ses i n our experi m ent coul d be changed i n 1 ns steps,and thedurati on oftheC W background beforeand after the i nput dark pul se i s control l ed by changi ng the durati ons t 1 and t 2 i n the rst and second m odul ati on channel s,respecti vel y. In our experi m ents,the val ues of t 1 and t 2 are chosen to be constant and equal , t 1 = t 2 = 1 s. T he i nput pul se sequences ofthe form show n i n Fi g.5 are produced w i th a repeti ti on rate ofabout 100 kH z.
T he m agni tude ofthe externalphase shi ft i ntroduced i nto the dark i nput pul se i s control l ed by the vari abl e phaseshi fteri n the second m odul ati on channel .T hezero poi nt and the scal e cal i brati on of the phase shi fter are establ i shed by adjusti ng i tto the poi ntsofm i ni m um and m axi m um i nterference of two overl appi ng pul ses. T he si gn ofthe phase shi ft i s establ i shed by cal i brati ng i t at di erent frequenci es.
T he si gnali s recorded w i th a di gi talosci l l oscope and a transi ent recorder (500 M H z) through a di ode detector. T he si gnalrecei ved by the output antenna i s ampl i ed usi ng a l ow -noi se sm al l -si gnal am pl i er to dri ve the detector di ode i n i ts opti m um sensi ti vi ty range and to achi eve the hi ghestsi gnal -to-noi se rati o possi bl e. T he outputvol tage i n thi srange i sproporti onalto the power at the i nput ofthe detector di ode.
In our experi m ent the vari abl e param eters are: (i ) i nput power P in ,(i i ) durati on ofthe dark i nput pul se T , and (i i i ) externalphase shi ft ext . T he i nput power i s vari ed i n the i nterval P in = 1: 6:::100 m W , w hi l e the dark pul se durati on i svari ed i n the i ntervalT = 10:::30 nsi n stepsof5 ns. W e use the fol l ow i ng val uesofthe externalphase shi ft: 0. 17 ,0. 42 ,0. 6 ,0. 72 ,and 0. 95 .
T heexperi m entalresul tsaresum m ari zed usi ng theparam eterpl ane ofFi g.3.T he pro l esofthe outputpul ses correspondi ng to the xed val uesofthe i nputpowerP in , dark i nputpul sedurati on T ,and externalphaseshi ft ext are recorded,and the correspondi ng poi ntsare pl aced on the param eterpl ane Fi g.3. C ounti ng the num ber ofthe m i ni m a i n the output pul se pro l e correspondi ng to a parti cul arsetofi nputparam eters,we can determ i ne the num ber ofsol i tons and com pare i t w i th the theory.
H owever,to com parethe experi m entalresul tsw i th the theory,the di spersi on and nonl i neari ty param etersofthe m agnetostati c waves propagati ng i n the Y IG l m wavegui de shoul d be ei ther m easured or cal cul ated. Fi rst, the group vel oci ty atthe worki ng poi nti sdeterm i ned by m easuri ng the ti m e ofthe pul se propagati on between the i nput and output antennae of the setup. T hi s vel oci ty i s found to be equal to 5: 5 10 6 cm /s ( 2% ) w hi ch i s cl ose to the theoreti calesti m ate ofthe group vel oci ty of 5: 3 10 6 cm /s ( 2% ) obtai ned from Eq.(55)i n R ef. [ 12] .
T he val ue ofdi spersi on D was establ i shed by com pari ng v g at severalfrequenci es 50M H z above and bel ow the carri erfrequency ! 0 . U si ng a pol ynom i al tfuncti on ofv g (!),the val ue ofD at the worki ng poi nt was determ i ned to be 7: 2 10 3 cm 2 =s ( 20% ). T he l arge error i s due to the sm al ldi erences of v g at sm al lfrequency i nterval s. T he theoreti cal esti m ati on of the di spersi on coe ci entD done usi ng Eq.(55)from R ef. [12]gi vesthe val ue of 7: 1 10 3 cm 2 =s 20% .T hus,i n the cal cul ati ons bel ow we use the val ues: v g = 5: 5 10 6 cm /s ( 2% ) and D = 7: 2 10 3 cm 2 =s ( 20% ).
T he val ue ofthe nonl i neari ty coe ci ent N coul d not be m easured di rectl y. T hus,i t was cal cul ated usi ng Eq. (52) from R ef. [ 12]to gi ve N = 9: 5 10 9 s 1 .
T he resul ts ofour experi m ents for the pul se durati on of T = 20 ns are presented i n Fi g. 6. T wo adjustabl e param etersare used to pl ace the experi m entalpoi nts on thi s graph. T he rst param eter i s the coe ci ent rel ati ng the i nput power P in to the norm al i zed am pl i tude of the spi n wave u 0 , i . e. j u 0 j 2 = B P in . It de nes the rel ati on between the i nput power P in and the param eter S 0 used i n the theoreti calform ul a (7) and,therefore,i t determ i nes the scal e of the graph al ong the hori zontal axi s. Si m i l ar to R ef. [ 6] , we chose B assum i ng that at the i nput power ofP in = 175m W ,w hen therm ale ects begi n to m ani fest them sel ves, we have a typi cal val ue of the spi n wave am pl i tude i n the l m equalto j u 0 j = 0. 07 (see R ef. [ 13] ). For the condi ti ons of our experim entthi syi el dsB = 27 10 3 W 1 ,w hi ch i saboutthree ti m esl argerthan the si m i l arval ue i n R ef. [ 5] .T he di erence m ay be attri buted to a betterm atchi ng between the i m pedances ofthe suppl y l i ne and the i nput m i cro-stri p antenna. T hesecond adjustabl eparam eteri sthephenom enol ogi calcoe ci entb i n Eq.( 7)w hi ch descri besthe e ectofa ni te C W background ofthe i nputdark pul se and determ i nes the scal e ofthe graph i n Fi g.6 al ong the verti cal axi s. Si m i l ar to R ef. [ 6] ,we take b = 2.
T he parabol aspresented i n Fi g.6 are com puted usi ng Eq.(8),and they correspond to di erentval uesofthe externalphase shi ft 0. 17 ,0. 42 ,0. 60 ,0. 72 ,and 0. 95 , so that the l owest parabol a corresponds to the sm al l est externalphase shi ft of 0. 17 . T he val ues of the i ni ti al (l i near) phase shi fts correspondi ng to the i ni ti al poi nt S 0 = 0 are com puted as
( total ) 0 = ext 2T=T D :(9)
T he poi nts on the parabol as correspond to the poi nts w here the experi m entalosci l l ogram softhe outputsi gnal are recorded.Sam pl es ofsuch experi m entalosci l l ogram s correspondi ng to the poi nts denoted by l etters "a","b", "c","d" and "e" i n Fi g.6 arepresented i n Fi g.7.T hel eft row ofosci l l ogram sdem onstratesthat,w hen the external phaseshi ftgrow sfrom 0. 17 to 0. 95 butthei nputpower i s constant (i . e. , S 0 = 0: 75 ),the output si gnalpro l e w i th (a)two dark sol i tonstransform si nto (b)three dark sol i tons,and then agai n i nto (c) two dark sol i tons.
T he ri ght row ofosci l l ogram s i n Fi g.7 dem onstrates thatw i th an i ncreaseofthei nputpowerata xed val ueof the externalphaseshi ft0. 6 ,the num berofdark sol i tons i s i ncreasi ng as the output osci l l ogram s contai n ei ther one,three,or ve dark sol i tons. T hese resul ts appear to be i n a good qual i tati ve agreem entw i th the theory [ 6] .
W e shoul d m enti on that the num ber of dark sol i tons observed i n the experi m ental pro l es does not al ways m atch the num ber predi cted by the theory [ 6] . T hi s i s especi al l y true for very sm al l and very l arge val ues of the i nput power. For sm al l val ues of P in , correspond-i ng to the val ues S 0 < 0: 5 , the dark sol i ton i s not properl y form ed as the ti m e of the si gnal propagati on between the antennae i ssm al l erthan the so-cal l ed "nonl i nearti m e" T N = 1=(N j uj 2 )duri ng w hi ch the spi n-wave nonl i neari ty coul d si gni cantl y a ect the pul se pro l e. O n the other hand,for l arge val ues ofP in correspondi ng to S 0 > 1: 5 ,nonl i near di ssi pati on and other nonl i near e ects,not taken i nto account i n our si m pl e theoreti cal m odel ,m ake the dependence ofj u 0 j 2 on the i nputpower nonl i near, w hi ch prevents us from m aki ng a quanti tati ve com pari son between theory and experi m ent. A t the sam e ti m e, i t i s cl ear from the experi m ental data presented above thati n the range ofthe i nterm edi ate i nput powers,0. 5 < S 0 < 1: 5 ,the theory gi ves a good quali tati ve and even reasonabl e quanti tati ve expl anati on of the experi m entaldata.
In concl usi on,we have extended the concept ofphase engi neeri ng,dem onstrated earl i er for m atter-wave dark sol i tonsi n B ose-Ei nstei n condensates,to spi n-wave m agneti c sol i tons.W e have dem onstrated experi m ental l y the cruci alrol epl ayed by an external l y i ntroduced phaseshi ft i n the i nput pul se for the process and outcom e of the dark-sol i ton generati on. O ur experi m entalresul ts are i n a good qual i tati ve agreem entw i th the anal yti caland num eri calpredi cti ons ofdark-sol i ton generati on,and they provi de m ore di rect veri cati on ofthe theory.
A cknow ledgem ents W e thank C arl Patton and El ena O strovskaya for useful di scussi ons. T hi s project was supported by the D eutsche Forschungsgem ei nschaft, the A l exander von H um bol dt Foundati on,the M U R I grant W 911N F-04-1-0247 ofthe U S A rm y R esearch O ce,and the A ustral i an R esearch C ounci l .
FIG . 1 :
1Top: D ark i nput pul se w i th am pl i tude u0,l ength L, and area S0 (shaded). B ottom : C orrespondi ng vari ati on of the external l y i ntroduced i nput phase (not to scal e)[ 10] .
FIG . 2 :
22. Such a sol i ton sequence i s characteri zed by the overal l num ber ofdark sol i tons and the m odul ati on ofeach i ndi vi dualdark sol i ton. A ccordi ng to Eq.(6),these val ues arecom pl etel y determ i ned by thei nputpul searea S 0 and Top: Schem ati c for the generati on ofN = 3 dark sol itons. B ottom : C orrespondi ng si m pl i ed phase structure.
FIG . 3 :
3N um ber of dark sol i tons i n the pl ane (S0; ). T he ci rcl e i ndi cates the N = 3-sol i ton show n i n Fi g. 2. T he separati on l i nes betw een di erent num bers of sol i tons are at = M S0, M = 0; 1;::: , the l i nes of exact sym m etry are hori zontall i nes at = N =2. For even N ,there w i l l be pai rs ofsym m etri c sol i tons;for uneven N ,a si ngl e sol i ton of100% m odul ati on depth surrounded by pai rs ofsym m etri c sol i tons.
w here acq i s determ i ned by Eq.(7). N ote that there i s al ways a xed ' o set' phase shi ft, even i n the l i m i t S 0 = 0. T hi s m eans that al ldark sol i tons i n m agneti c system s acqui re a phase shi ft duri ng the process ofthei r generati on.H owever,i fthegroup vel oci ty and,therefore, T D i s l arge,as we have for the case of dark sol i tons i n opti cs[ 1] ,the phase acq i s sm al land w i l lrem ai n uni mportant.In thi scase,the dark sol i tonsare al wayspl aced on a hori zontall i ne at = 0 on the (S 0 ; )-pl ane w hi ch i s equi val ent to the producti on of even-num bered symm etri c pai rs ofdark sol i tons.
(7)(see R ef.[ 6]for detai l s). T he experi m entsare perform ed on a 1. 5 m m w i de and 7 m thi ck Y IG l m wavegui de w hi ch, due to i ts relati vel y sm al l w i dth, coul d be consi dered as quasi onedi m ensi onal . T he l m wavegui de i s xed i n a standard del ay l i ne setup consi sti ng oftwo stri p-l i ne antennas of 45 m w i dth and 9.2 m m separati on. A m i crowave frequency of 5. 065 G H z i s used. A m agneti c bi as el d of 1107. 5 O e strength i sappl i ed tangenti al l y to the l m surface and perpendi cul arl y to the propagati on di recti on of spi n waves and,therefore,the condi ti ons for the exci tati ons of m agnetostati c surface waves are ful l l ed. T he val ue of the m agneti c el d i s chosen so as to pl ace the worki ng poi nt frequency i n the m i ddl e ofthe spi n-wave spectrum , l etti ng al l Fouri er com ponents of the darkpul se si gnalpass through. A vari abl e phase shi ft i n the i nput pul se i s created i n the fol l ow i ng way: a conti nuous m i crowave si gnal produced by a sweep-generator i s di vi ded i nto two parts, w hi ch are then i ndi vi dual l y control l ed by m i crowave m odul ators connected to the pul se generators. A standard m i crowave phase shi fter i s i ntroduced i n one ofthe m odul ati on channel s. T hen, the two resul ti ng pul sem odul ated si gnal sare com bi ned togetheri n a m i crowave m i xer devi ce, and are suppl i ed to the i nput antenna of the experi m entaldel ay l i ne (see Fi g.4).
FIG . 4 :
4Schem ati c di agram ofthe experi m entalsetup forgenerati ng dark pul ses. T he tw o pul se generators tri gger the tw o m i crow ave m odul ators w i th a xed del ay. Vari ati on of the l ength ofthe rstpul se provi desthe dark-pul sedurati on. T he phase shi fter i ntroduces the externalphase shi ft ext.
schem e ofthe dark-pul se generati on setup;t1 i s vari ed,thus changi ng the val ue ofthe param eter T .
FIG . 6 :
6Experi m entalresul ts for the param eters ofdark sol itonsw i th T = 20ns on the pl ane of total vs. S0. T he m arked poi nts correspond to the sol i ton pro l es show n i n Fi g.7.
FIG . 7 :
7Exam pl e of a seri es of m easured dark sol i tons w i th equalpul se durati on of 20 ns. Left row : equali nput pow er Pin = 9dB m (equi val entto S0 = 0: 75 ),butdi erentexternalphase shi fts ext = 0: 17 (a),0. 60 (b)and 0. 95 (c). R i ght row : equalexternalphase shi ft ext = 0: 60 ,butdi erent i nputpow erPin = 18dB m (d),-9dB m (b)and 0dB m (e). T he sol i ton pro l es correspond to the verti cal l y and hori zontal l y al i gned poi nts i n Fi g.6.
A graw al , O ptical Sol itons: From Fibers to Photonic C rystal s (A cadem i c,San-D i ego. Yu S , G P , 540 pp;and references therei nYu. S. K i vshar and G . P. A graw al , O ptical Sol itons: From Fibers to Photonic C rystal s (A cadem i c,San-D i ego, 2003),540 pp;and references therei n.
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] the de ni ti on of the phase shi ft i s sl i ghtl y di erent. H ere, onl y i nput pul ses w i th zero i ntensi ty i n the dark part are consi dered,and si m pl. In R Ef, d confusi onIn R ef. [ 6] the de ni ti on of the phase shi ft i s sl i ghtl y di erent. H ere, onl y i nput pul ses w i th zero i ntensi ty i n the dark part are consi dered,and si m pl i ed phase jum p w as chosen i n order to avoi d confusi on.
2" betw een theoreti caland experi m ental phase shi fts kept to avoi d confusi on w i th. N Ote, Factor, 6N ote the factor"2" betw een theoreti caland experi m ental phase shi fts kept to avoi d confusi on w i th [ 6] .
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[
"DETERMINANT BUNDLE OVER THE UNIVERSAL MODULI SPACE OF PRINCIPAL BUNDLES OVER THE TEICHMÜLLER SPACE",
"DETERMINANT BUNDLE OVER THE UNIVERSAL MODULI SPACE OF PRINCIPAL BUNDLES OVER THE TEICHMÜLLER SPACE"
]
| [
"Arideep Saha "
]
| []
| []
| For a Riemann surface X and the moduli of regularly stable G-bundles M , there is a naturally occuring "adjoint" vector bundle over X × M . One can take the determinant of this vector bundle with respect to the projection map onto M . Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmüller space.2010 Mathematics Subject Classification. 32G13, 14D20. | 10.1007/s13226-018-0251-1 | [
"https://arxiv.org/pdf/1704.00558v1.pdf"
]
| 119,176,760 | 1704.00558 | d5f8029c2289242addfa7b351abe727302bac90a |
DETERMINANT BUNDLE OVER THE UNIVERSAL MODULI SPACE OF PRINCIPAL BUNDLES OVER THE TEICHMÜLLER SPACE
3 Apr 2017
Arideep Saha
DETERMINANT BUNDLE OVER THE UNIVERSAL MODULI SPACE OF PRINCIPAL BUNDLES OVER THE TEICHMÜLLER SPACE
3 Apr 2017arXiv:1704.00558v1 [math.DG]
For a Riemann surface X and the moduli of regularly stable G-bundles M , there is a naturally occuring "adjoint" vector bundle over X × M . One can take the determinant of this vector bundle with respect to the projection map onto M . Our aim here is to study the curvature of the determinant bundle as the conformal structure on X varies over the Teichmüller space.2010 Mathematics Subject Classification. 32G13, 14D20.
Introduction
Let X be a Riemann surface of genus g ≥ 3 with a Riemannian metric. Let G be a semisimple linear algebraic group defined over C. Let M denote the moduli of regularly stable principal G−bundles over X. Let T g denote the Teichmüller space, the space of all conformal structures on a genus g smooth curve. Let C g denote the universal family of Riemann surfaces of genus g over the Teichmüller space T g . Let M denote the "universal moduli space" over T g , which we would construct here. M is such that the fibre over t ∈ T g is the moduli of regularly stable principal G-bundles over the Riemann surface represented by the point t.
Our aim here is to first construct the "universal adjoint bundle" ad(P) over C g × Tg M. Let p 2 : C g × Tg M −→ M be the projection map. Denote by Θ the top exterior product of the first direct image of ad(P), i.e. Θ := top (R 1 p 2 * ad(P)). By a general construction of Bismut, Gillet and Soulé [BGS1,BGS2,BGS3] we have a hermitian metric on the line bundle Θ. The above mentioned authors also give a general formula for the curvature of the connection corresponding to the constructed hermitian metric on Θ.
We show here that in our particular situation, the curvature form of Θ conincides with a natural (1, 1) form on M. We prove that the curvature form K(Θ) has the expression:
interchange between algebraic and analytic category of principal G-bundles on X and call it a G-bundle without specifying whether it belongs to algebraic category or analytic category.
Next we define the notion of stability (semistability) as introduced by Ramanathan [R1]. Recall that a subgroup P of G is called a parabolic subgroup of G if G/P is a complete variety.
Definition 2.2. A G-bundle E −→ X is called stable (semistable) if for any reduction of structure group to any maximal parabolic subgroup P , i.e. σ : X −→ E/P , we have deg σ * (T E/P ) > 0 (≥ 0), where T E/P is the tangent bundle along the fibres of E/P −→ X. Definition 2.3. A G-bundle E −→ X is called regularly stable if the automorphism group of E is equal to Z, the center of G.
Next we recall the definitions of connection and curvature on a G-bundle.
Definition 2.4. (Connection) Let E over X be a G-bundle, then a G-connection on E is a sectionh : T X −→ T E /G. [X, Y ], where X, Y ∈ T X and [, ] is the lie algebra structure on the respective tangent spaces.
Definition 2.5. (Curvature) Let h be a connection on a principal G-bundle E over X. Then the curvature K(h) of the connection h is defined by
K(h)(X, Y ) = [h(X), h(Y )] − h
We now recall and discuss briefly the results of Ramanathan [R1] which relates moduli of semistable G-bundles on a Riemann surface X and certain representations of π 1 (X), the fundamental group of X.
Definition 2.6. Let H ⊂ K be a subgroup of a maximal compact subgroup K. H is said to be irreducible if g ∈ G|hgh −1 = g, ∀h ∈ H =center of G.
Definition 2.7. A representation ρ : π −→ K is said to be irreducible if the image subgroup is irreducible.
Proposition 7.7 of [R1] coupled with Theorem 7.1 of [R1] says that, for G semisimple, there is a 1-1 correspondence between regularly stable G-bundles and the irreducible representations of π 1 (X), modulo equivalences.
Note: The definition of irreducibility, given in this article, differs slightly from the definition given in [R1]. Hence, although irreducible representations (as defined in [R1]) correspond to stable G-bundles, we would be considering the subset of stable G-bundles, namely the regularly stable G-bundles, as they precisely correspond to representations irreducible in our sense. Also, when the genus of X is ≥ 3, the smooth locus in the moduli of G-bundles coincides with the regularly stable G-bundles.
2.2. Teichmüller space. In this subsection we recall some known facts about the Teichmüller spaces. Let S be a compact connected oriented C ∞ surface of genus g ≥ 2. The space of all complex structures on S compatible with the orientation of S, denoted by Com(S), has a natural structure of an infinite dimensional complex Fréchet manifold.
Let Dif f + (S) be the group of all orientation preserving diffeomorphism of S. Dif f + (S) has a natural action on Com(S), given by the pushforward of a complex structure by a diffeomorphism. Let Dif f 0 (S) ⊂ Dif f + (S) be the subgroup consisting of all the diffeomorphisms of S which are homotopic to the identity map.The Teichmüller space for S, denoted by T g , is defined as
T g = Com(S)/Dif f 0 (S)
For f ∈ Dif f + (S) we know that f preserves the complex structure of Com (S). Hence there is a natural induced complex structure on T g .
To give a more elaborate description of T g , consider pairs of the form (X, f ) where X is a Riemann surface and f : X −→ S is a diffeomorphism. Identify two pairs (X, f ) and (Y, g) if there is a biholomorphic map h : X −→ Y satifying the commuting diagram
X Y S S f h g h ′ with h ′ ∈ Dif f 0 (S).
The Teichmüller space T g is the moduli space of equivalence classes of such pairs. For any point t(= (X, f )) ∈ T g , the holomorphic tangent space T Tg ,t can be identified with H 1 (X, T X ). By Serre duality we have T * Tg,t = H 0 (X, K 2 X ), where, K X is the holomorphic cotangent bundle on X. We also know that X has a unique metric with curvature -1, known as the Poincaré metric, which we denote by g X . Using the Poincaré metric g X , we give a bilinear pairing Q on H 0 (X, K 2 X ) defined by
Q(α, β) = X α ⊗β ⊗ g −1 X
The above pairing Q defines a hermitian metric. In this way we get a Riemannian metric on T g . It can be shown that this metric, known as the Weil-Petersson metric, is a Kähler metric.
Universal moduli space
Henceforth we fix the notations and conventions introduced in the introduction.
Let K be a maximal compact subgroup of G. By Ramanathan's result (Proposition 7.7 [R1]) we know, M= Hom ir (π 1 (X), K)/K where Hom ir means the irreducible representations and the action of K on Hom ir (π 1 (X), K) comes via its conjugation action on K. We also know that there is a complex structure on M (infact an algebraic structure) since M is the moduli of G-bundles over the algebraic curve X.
Our aim in this section is to construct a complex manifold M with a holomorphic map φ:M → T g such that fibre over t ∈ T g is the moduli of regularly stable G-bundles over X t . We denote by M t the complex manifold of moduli of regularly stable Gbundles on X t . We denote Hom ir (π 1 (X t ), K)/K by R which is the underlying smooth manifold for M t for all t ∈ T g . Let us denote by S the underlying Riemann surface of X t . Note that X t is diffeomorphic to S for all t ∈ T g .
To construct M as a complex manifold we give a complex structure on T g × R such that φ becomes holomorphic. So first, we shall give an almost complex structure and then show it is integrable.
Let (t, m) ∈ T g × R.
Since T g is a complex manifold it has an (integrable) almost complex structure J 1 , say. Similarly, since t × R has the structure of the complex manifold M t , t × R has an (integrable) almost complex structure, J t say. The two almost complex structures, J 1 and J t give an almost complex structure on T (t,m) (T g × R) given by J 1 ⊕ J t . This also shows T
(1,0) (t,m) (T g × R) = T (1,0) (t) (T g ) ⊕ T (1,0)
t,m (R). Note that the almost complex structure on T To show integrability we use Newlander-Nirenberg criterion. We show that if χ, ψ ∈ T (1,0) (t,m) (T g ×R), then [χ, ψ] ∈ T (1,0) (t,m) (T g ×R). Although the integrability can be shown by a direct calculation, we use construction of moduli spaces by Faltings' (see [F]) to quickly understand the integrability of the almost complex structure on T g × R.
We first recall the definition of the mapping class group, denoted by MCG. For a C ∞ Riemann surface S the mapping class group is defined as:
MCG = Dif f + (S)/Dif f 0 (S).
where Dif f + (S) and Dif f 0 (S) are, as defined at the beginning of the subsection 2.2, orientation preserving diffeomorphisms of S and diffeomorphisms homotopic to the identity morphism of S respectively.
We have an action of MCG on T g and R which preserves the respective almost complex structures. The action of MCG on T g is the natural action which is induced from the action which pulls back complex structures on S via diffeomorphisms. MCG acts on the representation space R via the action of diffeomorphisms on π 1 (S). Here we note that, although for each t ∈ T g , R has a different almost complex structure, the action of MCG preserves all such complex structures.
Let U ⊂ T g /MCG be the set of curves with no non-trivial (holomorphic) automorphisms. U is zariski open in T g /MCG. Using the constructions of moduli of G-bundles over curves parametrized by varieties as in [F], we have universal moduli of G-bundles,M, over curves parametrized by U. The algebraic structure ofM gives a complex structure, and hence, an integrable almost complex structure onM defined over the complex manifold U.
Let,
q : T g −→ T g /MCG be the quotient map. Both U and q −1 (U) are dense in T g /MCG and T g respectively. We know that the action of MCG on q −1 (U) is discrete and q −1 (U) is a covering space over U.
We note that, q * M , which is the moduli of G-bundles over curves parametrized by q −1 (U), is canonically diffeomorphic to q −1 (U) × R. The diffeomorphism is given by identifying R with the fibre q * M u over u ∈ q −1 (U) using the results of Ramanathan which identifies moduli of G-bundles and the representation space. Here, to see that q * M is the moduli of G-bundles parametrized by curves over q −1 (U) we use that q −1 (U) −→ U is a covering space.
The integrable almost complex structure onM gives an integrable almost complex structure on q * M , which eventually gives an integrable almost complex structure on q −1 (U) × R. Now using that q −1 (U) is dense in T g , if we have an integrable almost complex structure on q −1 (U) × R which extends to an almost complex structure on T g × R, we can say the extended almost complex structure is integrable.
Now what remains to show is that, the almost complex structure on T g × R we started with, coincides with this integrable almost complex structure we get from the moduli constructions. We shall denote q −1 (U)×R with the integrable almost complex structure as M ′ .
Lemma 3.1. Fix a representation ρ ∈ R. Then we have a holomorphic section
s : q −1 (U) −→ M ′ , such that, s(q −1 (U)) = q −1 (U) × ρ .
Proof. Since T g is simply connected, the fundamental group of C g , curves parametrized by T g , is same as π 1 (S). Using the representation ρ, we can construct (construction is similar as for a single curve S) a holomorphic principal G-bundle parametrized by T g . This gives a holomorphic principal G bundle parametrized by q −1 (U). Hence, by the construction of moduli of Gbundles, we get a holomorphic map s : q −1 (U) −→ M ′ such that the image gives the G-bundles coming from the fixed representation ρ.
Using the above lemma we conclude that i : q −1 (U)×ρ −→ M ′ is a holomorphic map ∀ρ ∈ R. Also for t ∈ T g , the almost complex structures of the fibres M ′ t and M t come from the complex structure of M t (moduli of G-bundles on the Riemann surface X t ). Hence we conclude that the almost complex structures of M ′ and M coincide, which finally gives the integrability of the almost complex structure on M, as q −1 (U) is dense in T g .
The holomorphicity of the map φ follows because it is a projection onto the first n coordinates of M.
Construction of universal adjoint bundle and determinant bundle
After constructing M, the universal moduli of G−bundles over T g , in the previous section, our aim in this section is to construct the universal projective bundle over C g × Tg M with a complex structure. We have, as introduced earlier, G a semisimple linear algebraic group and K ⊂ G to be a maximal compact subgroup. Let Z denote the centre of G and K respectively. The centre of both G and K are same because K is a maximal compact subgroup of G.
We also fix the following notations: X will denote a Riemann surface of genus g ≥ 3 with S the underlying smooth surface. M will denote the moduli of regularly stable principal G−bundles on X, R=Hom ir (π 1 (X), K)/K and R = Hom ir (π 1 (X), K).
First we construct a smooth K/Z bundle on X × M such that by extending the structure group we have a holomorphic G/Z bundle. Then we have a unique connection ∇(t) on the K/Z bundle corresponding to the holomorphic structure of the G/Z bundle. Here the letter t is used in ∇(t) to denote that X and M have the complex structure corresponding to the point t ∈ T g , the Teichmüller space.
Lemma 4.1. There exist a smooth principal K/Z bundle P over X × R.
Proof. LetX be the universal cover of X. ThenX ×R×K is a trivial principal bundle overX ×R. We have an action of π 1 (X) onX ×R given by a trivial action onR and usual action onX. This action of π 1 (X) can be lifted to give an action of π 1 (X) onX ×R ×K. To see this we choose σ ∈ π 1 (X), f ∈R and k ∈ K. Then σ takes (f, k) to (f, f (σ)k). Let the mapp :X ×R×K −→X ×R be the projection map . Going modulo the action of π 1 (X) onX ×R × K and X ×R we have a map p :P −→ X ×R, whereP is a principal K bundle over X ×R and the map p is induced fromp.
P is a smooth K-bundle because the action of π 1 (X) is properly discontinuous. Recall that the representation space R is the quotient ofR by K, where the action of K onR is via the conjugation action of K on itself. NOTE that the center Z, of K, is precisely the subgroup acting trivially because we are looking at irreducible representations of π 1 (X).
Recall that the action of K onX ×R × K is via translation on K, hence the action does not have a fixed point. So the induced action of K onP also does not have any fixed point. We have already shown that Z acts trivially on X ×R. Hence taking quotient ofP and X ×R by K with respect to the respective actions we have a mapp :
P −→ X × R where P (=P /K) is a K/Z bundle.
In the following lemma we discuss a natural holomorphic structure on P (G/Z), the extension of the structure group of P by G/Z, which is induced from the holomorphic structures of X, M and G/Z. Before we discuss the proof of the natural holomorphic structure on P (G/Z), we state a lemma which would be needed in the construction of such holomorphic structure.
Lemma 4.2. Let G, H be two linear algebraic groups. Let X be a holomorphic manifold. Let P G denote a principal G-bundle on X and P H denote a principal H-bundle on P G such that the action of G on P H commutes with the action
of H on P H . Let D G : T P G −→ T P H /H denote a G-invariant connection and h : T X −→ T P G /G be another connection. Then D G induces a connection D : T X −→ T P H /G /H. Proof. We have the following diagram P H P G P H /G X q
where q is the quotient map. Let dq : T P H −→ T P H /G denote the corresponding map at the level of tangent spaces. Since the action of G and H commute, dq is H equivariant, and hence we have a mapdq :
T P H /G/H −→ T P H /G /H. Since, by assumption, D G is G-invariant, we haveD G : T P G /G −→ T P H /H/G. Then D can be defined by D =dq •D G • h.
Thus giving the desired connection.
Note. Sometimes we can construct the connection D G , as mentioned in the above lemma, as a G-invariant connection. Even in that case the existence of section like h is needed to construct connections on spaces we get through taking quotient by G .
We recall that giving a complex structure on P (G/Z) is equivalent to giving a connenction on P (K/Z) with a (1, 1) curvature form. Proof. Let E be the trivial principal G-bundle over the Riemann surface X. Let Conn be the space of all irreducible flat connections on E giving E the structure of a regularly stable principal bundle. Conn is an infinite dimensional analytic space with a vector space structure. Let Aut(E) denote the C ∞ automorphisms of E. We know that M = Conn/Aut(E) (see [AB]) as holomorphic manifolds where M denotes the moduli of regularly stable G-bundles.
Consider the following diagram
X × Conn × K X × Conn P (K/Z) X × M q
where the vertical maps are obtained by taking quotients by Aut(E) action.
Note that for k ∈ Z, multiplication by k induces an automorphism of E. Since E is regularly stable for connections belonging to Conn, the subgroup of G inducing automorphisms of E is precisely Z. Also X × Conn is fixed by elements of Z. Hence taking a quotient of X × Conn × K by Aut(E) we have a principal K/Z bundle over X × M.
There is the canonical connection D : T X×Conn −→ T X×Conn×K /K which can be described as follows. For a fixed point t ∈ Conn D is defined by t and D is defined as identity for a fixed x ∈ X. Then D, by definition, becomes Aut(E)-equivariant.
Also recall that there is a inclusion T M ֒→ T Conn /Aut(E) where T M is the space of all Harmonic forms. Hence we get a connection h : T X×M −→ T X×Conn /Aut(E). Then using Lemma 4.2. we have a connection on the K/Zbundle P (K/Z). The curvature of this connection is a (1, 1)-form. This is because we can extend the structure group of the principal bundle using the adjoint map G −→ Aut(g) and the curvature for the corresponding connection is known to be a (1, 1)-form, as in the case of vector bundles.
The complex structure constructed above is the unique complex structure on the projective bundle P (G/Z) over X ×M, satisfying the following conditions: For any complex manifold S and a map f : S −→ P (G/Z) we have 1.
S P (G/Z) X × M ff q f has to be holomorphic, wheneverf is holomorphic.
2. The map f S : S −→ P ′ (G/Z), induced from f via the pull-back diagram
P ′ (G/Z) P (G/Z) S × X × M X × M and S × X × M X × M S pt
has to be holomorphic.
We also have that X × ρ ֒→ X × M sits as a complex submanifold; as the inclusion, which is apriori just a smooth map, preserves the almost complex structures. The unique flat connection corresponding to the holomorphic structure of P (G/Z) restricted to X ×ρ is the same as ∇(t) restricted to X ×ρ.
So far we have fixed a holomorphic structure on the Riemann surface and have constructed a holomorphic principal G/Z bundle P (G/Z) over X × M. Now our aim is to construct a holomorphic principal G/Z bundle P on C g × Tg M and study the properties of the curvature of its holomorphic connection. Let S be the underlying C ∞ surface of the Riemann surface X. Using the previous arguments we have a C ∞ projective bundle P ∞ (the underlying C ∞ surface of P is P ∞ ) on S × R with a partial flat connection along the tangent space of S. The construction of this partial flat connection is as follows: Fix a representation class ρ ∈ R. We can have a connection h for the projection map pr :
T P ∞ −→ T S defined by h(v) = (v, 0) where v ∈ T S .
The reason of this construction is that it would be independent of any choice of complex structure on S. We denote the projective bundle along with the partial flat connection as (P ∞ , ∇(F )).
Recall Dif f 0 (S) to be the orientation preserving diffeomorphisms of S which are homotopic to identity. Dif f 0 (S) acts on S × R via the usual action on S and the trivial action on R. In the next paragraph we shall describe an action of Dif f 0 (S) on P ∞ which is a lift of the above action on S × R.
Choose f ∈ Dif f 0 (S) and fix ρ ∈ R. Let (P ∞ (ρ), ∇(ρ)) be the projective bundle with the unique connection which comes as a restriction of P on S × ρ. Consider the pullback bundle f * P ∞ (ρ) equipped with the pullback connection f * ∇(ρ). (Note that the diffeomorphism f of S induces a diffeomorphism of S × ρ.) Since f ∈ Dif f 0 (S), (P ∞ (ρ), ∇(ρ)) and (f * P ∞ (ρ), f * ∇(ρ)) are isomorphic. They are isomorphic upto a unique isomorphism because automorphisms of (P ∞ (ρ), ∇(ρ)) is trivial (The automorphism group is trivial because the structure group has trivial centre). Hence f induces an action on P ∞ (ρ).
The explicit description of the action of f ∈ Dif f 0 (S) on P ∞ (ρ) can be given as follows: Let φ f : f * P ∞ (ρ) −→ P ∞ (ρ) be the unique isomorphism.
Then the action of f on P ∞ (ρ) takes a point (s, ρ, k) to φ f (f −1 (s), ρ, k). Clearly such action of Dif f 0 (S) on P ∞ (ρ) is a group action because φ f •g = φ f • φ g for f, g ∈ Dif f 0 (S).
For f ∈ Dif f 0 (S) we have a unique isomorphism I(f ) : (f × Id) * P ∞ −→ P ∞ such that over S × ρ it is the same isomorphism φ f mentioned in the previous paragraph. We conclude here similarly as the previous paragraph that we have a goup action i.e. if g ∈ Dif f 0 (S) we have the equality I(f ) • I(g) = I(f • g). Hence we have a lift of the action of Dif f 0 (S) on S × R to the pair (P ∞ , ∇(F )).
Let Com(S) denote the space of all complex structures on S. Let p 12 denote the projection of S × R × Com(S) onto S × R. Consider the projective bundle with a partial connection (p * 12 P ∞ , p * 12 ∇(F )) −→ S × R × Com (S). The group Dif f 0 (S) acts on S × R × Com (S). Dif f 0 (S) acts on S × R by the previous action and acts on Com(S) by the push-forward of a complex structure on S by a diffeomorphism. Also the acton of Dif f 0 (S) on (P ∞ , ∇(F )) induces action of Dif f 0 (S) on (p * 12 P ∞ , p * 12 ∇(F )). Consider the projection (S))/Dif f 0 (S) We know that (S ×R×Com(S))/Dif f 0 (S) = C g ×R = C g × Tg M. Hence we get a K/Z-bundle with a partial connection over C g × Tg M. Let (P, ∇(par)) denote the projective K/Z bundle with the partial connection ∇(par) over C g × Tg M constructed above.
p * 12 (P ∞ )/Dif f 0 (S) −→ (S × R × Com
Let P(G/Z) be the G/Z bundle obtained from P by extending the structure group. In the next lemma we discuss the existence of complex structure on P(G/Z).
Lemma 4.4. P(G/Z) has a complex structure.
Proof. Our aim is to construct a connection on P with a (1, 1) curvature form which will give a corresponding unique complex structure on P(G/Z).The proof would be similar as in Lemma 4.3.
Let E be the trivial C ∞ principal K-bundle over the Riemann surface S. Let Conn be the space of all irreducible flat connections on E such that E becomes regularly stable. Let Aut(E) denote the C ∞ automorphisms of E. We know that M = Conn/Aut(E) (see [AB]) as C ∞ manifolds where M denotes the moduli of regularly stable G-bundles.
Consider the following diagram
C g × Conn × K C g × Conn P C g × Tg M q
where the vertical maps are obtained by taking quotients by Aut(E) action.
Note that for k ∈ Z, multiplication by k induces an automorphism of E. Since E is regularly stable the subgroup of G inducing automorphisms of E is precisely Z. Also C g ×Conn is fixed by elements of Z. Hence taking a quotient of C g × Conn × K by Aut(E) we have a principal K/Z bundle over C g × Tg M.
We have the canonical connection D : T Cg ×Conn −→ T Cg×Conn×K /K which is defined similarly as in Lemma 4.3.
Using Lemma 4.3, we see that for a fixed t ∈ T g we have inclusion T Xt×Mt ֒→ T Xt×Conn /Aut(E). Hence we get a connection h : T Cg × Tg M −→ T Cg ×Conn /Aut(E). Then using Lemma 4.2. we have a connection on the K/Z-bundle P. The curvature of this connection is a (1, 1)-form because of the same reasons as in Lemma 4.3.
Similar to the case of X × M as before, we know that P(G/Z) has a unique complex structure satisfying the following conditions:
For any complex manifold S and a map f : S −→ P(G/Z) we have : 1.
S P(G/Z) C g × Tg M f f q f has to be holomorphic wheneverf is holomorphic. 2. The map f S : S −→ P(G/Z) × Tg S, induced from f via the pull-back diagram P(G/Z) × Tg S P(G/Z) S T g
has to be a holomorphic map.
Summing up the constructions , so far, we have obtained a holomorphic projective G/Z-bundle with the holomorphic connection (P(G/Z), ∇).
Let ad(P) be the corresponding adjoint vector bundle ,of the G/Z bundle P(G/Z), over C g × Tg M. Let p 2 : C g × Tg M −→ M be the projection map. Let Θ be the determinant bundle on M of ad(P) with respect to the projection map, defined as Θ := top (R 1 p 2 * ad(P)).
The existence of the above defined determinant bundle, as a holomorphic line bundle, follows from the following general result which has been studied by Bismut, Gillet and Soulé:
Curvature of the determinant line bundle
We have constructed the holomorphic determinant bundle Θ in the previous section. Due to its holomorphic structure Θ has a holomorphic connection. We denote by K(Θ) the curvature of the holomorphic connection.
We know that the holomorphic tangent space of M at a point ρ ∈ M can be identified as the space of deformations of the G-bundle ρ and hence can be identified with H 1 (X, ad(ρ)), where ad(ρ) is the adjoint bundle of the principal bundle corresponding to the point ρ. Ω is the 2-form corresponding to the natural metric on M which we get by identifying the tangent space of M with H 1 (X, ad(ρ)) and using the cup product for H 1 (X, ad(ρ)).
In this section we prove the following statement:
Theorem 5.1. K(Θ) = −2πι.q * Ω + (r/6π)ι.σ * ω
Let Φ denote the R.H.S. of the above equation. The notations involved in Φ are as follows: r = rank(ad(P)) (which is the same as dim(Lie(G))). ι = √ −1. ω is a Kähler (1, 1) form on T g corresponding to the W eil − P etersson metric. The maps q and σ are obvious projections of M onto R and T g respectively.
Before beginning the proof of the theorem we wish to recall a few results involving chern class and T odd class which would be used in the proof. See [Ko] for details.
Let Y be a complex manifold and E be a holomorphic vector bundle on Y . We shall use the following notations: Ch i will denote the i − th chern form which is a differential 2i-form on Y , ch i will denote the i−th chern class which is the class of Ch i in the deRham cohomology H 2i (Y, Z). Moreover Ch will denote the chern character for the vector bundle E. Similarly T d i will denote the i − th Todd form and td i will denote the i − th Todd class. We note down a couple of lemmas (proved in Kobayashi [Ko]) involving chern forms and chern character.
Lemma 5.2. Let E, F be two holomorphic vector bundles on Y . Then Ch(E⊗ F ) = Ch(E).Ch(F ).
Lemma 5.3. For a holomorphic vector bundle E of rank r, Ch(E) = r + Ch 1 (E) + 1 2 (Ch 1 (E) 2 − 2.Ch 2 (E)) + ... .
Next we state the result (Theorem 0.1 [BGS1]), due to Bismut-Gillet-Soulé, which we would be using to calculate K(H Q ) . First we give a brief outline of a proof of a special case of the theorem, due to Quillen [Q]. Let X be a Riemann surface of genus g. E be a smooth vector bundle of rank r and degree d over X. Let B = Ω 0,1 (End(E)) and A be the space of∂ operators which is an affine space relative to B. Let B ∈ B be of the form α(z)dz. Let α(z) * be the adjoint of α(z) relative to the local orthonormal frame of E. Using the adjoint α(z) * of α(z) we construct an element of Ω 1,0 (End(E), denoted by B * := α(z) * dz. Then tr(B * .B) is a (1, 1) form on X and hence we get a metric on A defined by |B| 2 = X ι 2π tr(B * .B). Fixing the notations as above we give a brief outline of a proof for the following result:
Theorem 5.5. (Quillen, [Q]) The curvature of the determinant line bundle (constructed by Quillen) is equal to the Kähler form obtained from the above mentioned metric on A.
Proof. We know that for a holomorphic line bundle the curvature form of the holomorphic connection is given by∂∂log|σ| 2 where σ is any local holomorphic section. Before calculating the curvature form we recall a few identities (see [Q] for details). In terms of local orthonormal frame of E we have the following local equations:
• ds 2 = ρ(z)|dz| 2 ; • D = dz(∂z + α), where D is a∂ operator; • ∇ = dz(∂ z − α * ) + dz(∂z + α); • G(z, z ′ ) = ιdz ′ 2π(z−z ′ ) [1 + (z − z ′ )β(z ′ ) − (z −z ′ )β(z ′ ) + ...];
Let J = ι 2π dz(β − α * − 1 2 ∂ z log(ρ)). In [Q] Quillen gives a different definition for J and then uses the above local equations to get this expression. We would need only this expression for the calculation of the curvature form. We fix a one-parameter family D w of invertible∂ operators depending holomorphically on the complex variable w. Since invertible operators are dense in End(E), enough to show that the curvature form and the Kähler form coincide over such a family.
We have the curvature form∂∂log|σ|
2 = dwdw ∂ 2 ∂w∂w ζ ′ (0), where ζ(s) = tr(∆ −s ) and ∆ = D * .D. Now, −∂ w ζ(s) = s.tr(∆ −s−1 ∂ w ∆) = s.tr(∆ −s D −1 ∂ w D) (∂ w D * = 0, as D depends holomorphically on w) = s Γ(s) ∞ 0 tr(exp(−t∆)D −1 ∂ w D)t s−1 dt = s.[ M tr(J∂ w D) + O(s)] as s → 0.
The last equality follows from Theorem 2 of [Q], which states that for B ∈ B we have the equality lim t→0 tr(exp(−t∆)D −1 B) = M tr(JB).
Hence it follows that −∂ w ζ ′ (0) = tr(J∂ w D). We also have the equalities:
∂wJ = − ι 2π dz∂wα * = − ι 2π (∂ w D) + .
The first equality holds because only α * is not holomorphic in w. The second equality holds because ∂ w D = ∂ w αdz.
Hence finally we have:
∂ 2 ∂w∂w ζ ′ (0) = − ∂wtr(J∂ w D) = ι 2π tr[(∂ w D) + ∂ w D]
. Identifying the vector space B with its tangent space we get the desired result.
Next we deduce the following result from the theorem which we would be using later. For a Riemann surface X, let M V be the moduli of stable vector bundle of rank = r and degree = d on X. Since a vector bundle can be seen as a principal GL n -bundle, we get a universal projective bundle P V over X ×M V . Let ad(P V ) be the adjoint bundle of the P GL n bundle P V .
Corollary 5.6. The curvature K(ad(P V )) of the holomorphic determinant bundle with respect to the projection p :
X × M V −→ M V is equal to 4πιrΩ V ,
where Ω V is a (1, 1) form on M V defined as in the beginning of the section.
Proof. For any vector bundle E over X, let A s be the collection of stable holomorphic structures on E. Then A s would be an open subset of A, which is the collection of all complex structures on E. There is a natural action of the complex gauge group PG = Aut(E)/C * on A, which, when restricted to A s is a free action. Here Aut(E) is the group of smooth automorphisms of E. Note that Aut hol (E), the holomorphic automorphisms of E with a fixed stable structure is precisely C * . The quotient space of A s by the action PG is M V .
The Quillen determinant line bundle on A s descends down to a holomorphic line bundle L on M V . This is true because the action of Aut(E)/C * on A s lifts to give an action on the Quillen determinant line bundle. The line bundle L inherits a natural Quillen metric from the Quillen metric of the determinant bundle over A s . Using the results of Bismut-Gillet-Soulé we know that the construction of the determinant bundle by Quillen and the determinant bundle of the direct image of ad(P V ) are isomorphic as holomorphic line bundles. Using Theorem 5.5 we see that the curvature of the holomorphic line bundle equals to a particular Kähler form on M V . One can calculate (Proposition 2.9, [B]) that the metric induced on M V from the Quillen metric on A s is the same as 4πιrΩ V . Hence we have K(ad(P V )) = 4πιrΩ V .
We note down a simple lemma which will used in the proof of Theorem 4.1 .
Lemma 5.7. Let X be a complex manifold. G, H be linear algebraic groups and f : G −→ H be a map of linear algebraic groups. Let P be a principal G−bundle on X. Let g, h be the lie algebras of G, H respectively. Then the following diagram commutes:
H i (X, P (g)) × H i (X, P (g)) H 2i (X, P (g)) H i (X, P (h)) × H i (X, P (h)) H 2i (X, P (h)) ∪ f * f * ∪ Proof.
Follows from the functoriality of the cup product.
Proof. (of Theorem 4.1) The result of Bismut-Gille-Soule applied to our situation says K(Θ) = 2πι.p 2 * (T d 2 .Ch 0 + T d 1 .Ch 1 + T d 0 .Ch 2 ), where p 2 * denotes integrating along the fibres which are curves. Note that since we are integrating along a curve , which has real dimension=2, the degree 2 compo-
nent of 2πι Z T d(−R Z /2πι)T r[exp(−L ζ /2πι)] is precisely 2πι.p 2 * (T d 2 .Ch 0 + T d 1 .Ch 1 + T d 0 .Ch 2 ).
We recall that Ch i are polynomials in the curvature form corresponding to the holomorphic connection on ad(P). Also recall that T d i are polynomials in the curvature of the hermitian connection on T (1,0) Z.
Since ad(P) is an adjoint bundle, Ch 1 = 0. Also we have T d 0 = 1 and Ch 0 = r. We observe that, since T d i depends on the curvature of the hermitian connection on T (1,0) Z, T d 2 depends only on the fibre and not on the holomorphic vector bundle ad(P).
Now with the aim of calculating T d 2 and Ch 2 we note a few more properties of them. Let α = (t, ρ)
∈ T g × R. Let v, w ∈ T t T g , T ρ R respectively. Let v ′ , w ′ be the image of v, w respectively in T α M. Recall that T α M = T t T g ⊕ T ρ M t ,
where M t is the moduli of G-bundles on X corresponding to the complex structure t ∈ T g . Then we have :
Lemma 5.8. p 2 * T d 2 (w ′ ) = 0.
Proof. The 2-form p 2 * T d 2 doesn't depend on the coordinates of R. This is because the coordinates of the fibre are only dependent on the coordinates of S and T g and hence , in particular, the curvature of the hermitian connection on T (1,0) (Z) is independent of the coordinates of R. Hence we have that p 2 * T d 2 is a pullback of a 2-form on T g . This implies that p 2 * T d 2 (w ′ ) = 0.
Lemma 5.9. p 2 * Ch 2 (v ′ ) = 0.
Proof. Let i : C g × ρ −→ C g × Tg M be the inclusion. Since the connection to the projective bundle, constructed earlier, is flat when restricted to C g × ρ we have
Ch 2 (v ′ ) = 0
where v ′ by abuse of notation will also denote the image of v in the tangent space of C g × ρ. Since Ch 2 (v ′ ) = 0 for all tangent vectors of T g , we conclude that Ch 2 is a pullback of a 4-form on S × R. Hence p 2 * Ch 2 is a pullback of a 2-form on R. Hence we conclude that p 2 * Ch 2 (v ′ ) = 0 for all v ′ ∈ T t T g .
We conclude from these observations that K(Θ)(v ′ , w ′ ) = 0 where v ′ and w ′ are as described above in the conditions of the above two lemmas. We also conclude that:
• K(Θ) and 2πιp 2 * .rT d 2 coincide on T g × ρ.
• K(Θ) and 2πιp 2 * .Ch 2 coincide on t × R.
To calculate the first case we observe that T d 2 does not depend on the adjoint bundle. Hence if we replace the adjoint bundle by the trivial bundle with the trivial metric, T d 2 remains the same. The only difference which occurs while caculating K(Θ)is that Ch 0 = 1 for the trivial bundle and Ch 0 = r for the adjoint bundle. Hence K(Θ) | Tg = r.K(Θ) triv .
K(Θ) triv is the curvature of the determinant line bundle on T g for the trivial bundle on C g . We know that K(Θ) triv = (ι/6π)ω (see [ZT]). Recall that ω is the Kähler (1, 1) form corresponding to the Weil-Petersson metric on T g . Hence K(Θ) = (r.ι/6π).σ * ω when restricted to T g × ρ.
To get a complete expression of K(Θ), what remains is, to calculate K(Θ) when restricted to t×R ,where t ∈ T g . As defined earlier, let X, M, M V denote the Riemann surface (corresponding to the complex structure t), moduli of stable G−bundles on X and the moduli of vector bundles of rank= r on X respectively. Let f : M G −→ M V be the map which takes a principal bundle ρ to the adjoint vector bundle ad(ρ).
To calculate K(Θ) restricted to t × M we need to calculate p 2 * Ch 2 (ad(P )). Recall that P is the universal principal G/Z-bundle which had been constucted over X × M. To calculate p 2 * Ch 2 (ad(P )) we observe that:
• Ch 2 (ad(P ) ⊗ ad(P )) = −2r.Ch 2 (ad(P )). Using lemma 5.2 and lemma 5.3. • Since we are integrating the chern form along the fibre we can as well take the corresponding chern class. Hence q 2 * Ch 2 (.) would not depend on the choice of a metric, and hence not on the choice of a connection on the vector bundle as well.
Next we consider the vector bundle End, the endomorphism bundle over X × M g,V . We know that f * (End) = ad(P ) 2 in the sense that the Einstein-Hermitian connection pulls back to the Einstein-Hermitian connection. Where we recall f : M −→ M V to be the adjoint map i.e. the map takes a principal bundle to its adjoint bundle. Let Det(End) be the determinant bundle on M V corresponding to the vector bundle End and K(End) be its curvature.
We have f * (Det(End)) = Det(ad(P )) ⊗ Det(ad(P )), since we know that Det(.) behaves well with respect to pull back. More precisely, suppose we have a commutative diagram:
X 1 X 2 Y 1 Y 2 f X f Y
and let E be a vector bundle on X 2 . Then f * Y (Det(E)) = Det(f * X (E)). Hence we have f * (K(End)) = −2r.K(Θ)| t×R as 2-forms over M.
So we reduce the problem to calculating K(End). We observe that
End = ad(P V ) ⊕ L,
where L is the trivial line bundle X × C and P V is the universal projective bundle.
On End we give the connection D induced from the Einstein-Hermitian connection D 1 on P V and the connection D 2 on L defined by D 2 (f.c) = df.c, where f is a smooth function on X and c is a constant section. Since D and D 1 have the same curvature form, End and P V have the same chern form Ch 2 . The corresponding chern class does not depend on the choice of the connection. Hence using the formula of Bismut-Gillet-Soulé, for calculating the curvature of the determinant bundle, we conclude that K(End) = K(ad(P V )) = 4πrι.Ω V .
We have f * (Ω V ) = Ω which follows from lemma 5.7 by taking H = Aut(g), where the map from G to Aut(g) is given by adjoint action. Hence we finally have K(H Q ) = −2πι.q * Ω + r 6π ι.σ * ω.
m (R) is dependent on t ∈ T g .
Lemma 4.3. P (G/Z) exists as a holomorphic G/Z bundle over X × M.
Theorem 4.5. (Theorem 0.1[BGS1])Let π : Y −→ B be a proper holomorphic map of complex analytic manifolds and let ζ be a complex holomorphic vector bundle on Y . Then the image of ζ by π has a determinant say λ, which is a holomorphic line bundle detRπ * ζ on B.
Theorem 5 . 4 .
54Let π : Y −→ B be a proper holomorphic map of complex analytic manifolds and let ζ be a complex holomorphic vector bundle on Y . Let λ be the determinant of ζ by π. For b ∈ B let Z denote the fibre. Let R Z , L ζ be the curvatures of the holomorphic hermitian connections on T (1,0) Z and ζ. Then the curvature of the holomorphic connection on λ is the component of degree 2 in the folowing form on B: 2πι Z T d(−R Z /2πι)T r[exp(−L ζ /2πι)].
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| []
|
[
"Improving Neural Conversational Models with Entropy-Based Data Filtering",
"Improving Neural Conversational Models with Entropy-Based Data Filtering"
]
| [
"Richard Csaky \nDepartment of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n\n",
"Patrik Purgai [email protected] \nDepartment of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n\n",
"Gabor Recski \nDepartment of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n\n",
"Apollo Ai \nDepartment of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n\n"
]
| [
"Department of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n",
"Department of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n",
"Department of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n",
"Department of Automation and Applied Informatics\nDepartment of Automation and Applied Informatics Budapest University of Technology and Economics\nBudapest University of Technology and Economics\n"
]
| []
| Current neural network-based conversational models lack diversity and generate boring responses to open-ended utterances. Priors such as persona, emotion, or topic provide additional information to dialog models to aid response generation, but annotating a dataset with priors is expensive and such annotations are rarely available. While previous methods for improving the quality of open-domain response generation focused on either the underlying model or the training objective, we present a method of filtering dialog datasets by removing generic utterances from training data using a simple entropy-based approach that does not require human supervision. We conduct extensive experiments with different variations of our method, and compare dialog models across 17 evaluation metrics to show that training on datasets filtered this way results in better conversational quality as chatbots learn to output more diverse responses. | 10.18653/v1/p19-1567 | [
"https://arxiv.org/pdf/1905.05471v3.pdf"
]
| 153,312,586 | 1905.05471 | 21da033c32b635877c15ea4c4d56e7ebf44d50c1 |
Improving Neural Conversational Models with Entropy-Based Data Filtering
Richard Csaky
Department of Automation and Applied Informatics
Department of Automation and Applied Informatics Budapest University of Technology and Economics
Budapest University of Technology and Economics
Patrik Purgai [email protected]
Department of Automation and Applied Informatics
Department of Automation and Applied Informatics Budapest University of Technology and Economics
Budapest University of Technology and Economics
Gabor Recski
Department of Automation and Applied Informatics
Department of Automation and Applied Informatics Budapest University of Technology and Economics
Budapest University of Technology and Economics
Apollo Ai
Department of Automation and Applied Informatics
Department of Automation and Applied Informatics Budapest University of Technology and Economics
Budapest University of Technology and Economics
Improving Neural Conversational Models with Entropy-Based Data Filtering
Current neural network-based conversational models lack diversity and generate boring responses to open-ended utterances. Priors such as persona, emotion, or topic provide additional information to dialog models to aid response generation, but annotating a dataset with priors is expensive and such annotations are rarely available. While previous methods for improving the quality of open-domain response generation focused on either the underlying model or the training objective, we present a method of filtering dialog datasets by removing generic utterances from training data using a simple entropy-based approach that does not require human supervision. We conduct extensive experiments with different variations of our method, and compare dialog models across 17 evaluation metrics to show that training on datasets filtered this way results in better conversational quality as chatbots learn to output more diverse responses.
Introduction
Current open-domain neural conversational models (NCM) are trained on pairs of source and target utterances in an effort to maximize the likelihood of each target given the source (Vinyals and Le, 2015). However, real-world conversations are much more complex, and a plethora of suitable targets (responses) can be adequate for a given input. We propose a data filtering approach where the "most open-ended" inputs -determined by calculating the entropy of the distribution over target utterances -are excluded from the training set. We show that dialog models can be improved using this simple unsupervised method which can be applied to any conversational dataset. We conduct several experiments to uncover how some of the current open-domain dialog evaluation methods behave with respect to overfitting and random data. Our software for filtering dialog data and automatic evaluation using 17 metrics is released on GitHub under an MIT license 12 . This paper exists in poster 3 and blog post 4 form as well.
Background
Most open-domain NCMs are based on neural network architectures developed for machine translation (MT, Sutskever et al. (2014); Cho et al. (2014); Vaswani et al. (2017)). Conversational data differs from MT data in that targets to the same source may vary not only grammatically but also semantically Tandon et al., 2017): consider plausible replies to the question What did you do today?. Dialog datasets also contain generic responses, e.g. yes, no and i don't know, that appear in a large and diverse set of contexts (Mou et al., 2016;. Following the approach of modeling conversation as a sequence to sequence (seq2seq, Sutskever et al. (2014)) transduction of single dialog turns, these issues can be referred to as the one-to-many, and many-to-one problem. seq2seq architectures are not suited to deal with the ambiguous nature of dialogs since they are inherently deterministic, meaning that once trained they cannot output different sequences to the same input. Consequently they tend to produce boring and generic responses (Li et al., 2016a;Shao et al., 2017;Zhang et al., 2018a;.
Previous approaches to the one-to-many, manyto-one problem can be grouped into three categories. One approach involves feeding extra information to the dialog model such as dialog history , categorical information like persona (Li et al., 2016b;Joshi et al., 2017;Zhang et al., 2018b), mood/emotion Li et al., 2017c), and topic (Xing et al., 2017;Baheti et al., 2018), or through knowledge-bases (Dinan et al., 2019;Ghazvininejad et al., 2018;Zhu et al., 2017;Moghe et al., 2018). A downside to these approaches is that they require annotated datasets which are not always available, or might be smaller in size. Augmenting the model itself, with e.g. latent variable sampling (Serban et al., 2017b;Gu et al., 2019;Park et al., 2018;Shen et al., 2018b;Gao et al., 2019), or improving the decoding process (Shao et al., 2017;Kulikov et al., 2018; is also a popular approach. Sampling provides a way to generate more diverse responses, however such models are more likely to output ungrammatical or irrelevant responses. Finally, directly modifying the loss function (Li et al., 2016a), or training by reinforcement (Li et al., 2016d;Serban et al., 2017a;Li et al., 2016c;Lipton et al., 2018;Lewis et al., 2017) or adversarial learning (Li et al., 2017b;Ludwig, 2017;Olabiyi et al., 2018;Zhang et al., 2018c) has also been proposed, but this is still an open research problem, as it is far from trivial to construct objective functions that capture conversational goals better than cross-entropy loss.
Improving dataset quality through filtering is frequently used in the machine learning literature (Sedoc et al., 2018;Ghazvininejad et al., 2018;Wojciechowski and Zakrzewicz, 2002) and data distillation methods in general are used both in machine translation and dialog systems (Axelrod et al., 2011;Li et al., 2017a). Xu et al. (2018b) introduced coherence for measuring the similarity between contexts and responses, and then filtered out pairs with low coherence. This improves datasets from a different aspect and could be combined with our present approach. However, natural conversations allow many adequate responses that are not similar to the context, thus it is not intu-itively clear why filtering these should improve dialog models. Our experiments also further support that cross-entropy is not an adequate loss function (shown qualitatively by Csaky (2019) and Tandon et al. (2017)), by showing that many automatic metrics continue to improve after the validation loss reaches its minimum and starts increasing. However, we found that the metrics steadily improve even after we can be certain that the model overfitted (not just according to the loss function). Further research is required, to determine whether this indicates that overfitted model responses are truly better or if it's a shortcoming of the metrics that they prefer such models.
Currently, there is no well-defined automatic evaluation method (Liu et al., 2016), and while some metrics that correlate more with human judgment have been proposed recently (Li et al., 2017b;Lowe et al., 2017;Tao et al., 2018), they are harder to measure than simpler automatic metrics like perplexity or BLEU (Papineni et al., 2002). Furthermore, even human evaluation has its downsides, like high variance, high cost, and difficulty of replicating experimental setups (Zhang et al., 2018b;Tao et al., 2018). Some works resort to human evaluations (Krause et al., 2017;Fang et al., 2018), others use automatic metrics only (Olabiyi et al., 2018;Xing and Fernández, 2018;Kandasamy et al., 2017;Shalyminov et al., 2018;Xu et al., 2018b), and some use both (Shen et al., 2018a;Baheti et al., 2018;Ram et al., 2018). While extensive human evaluation of the methods presented here is left for future work, we do conduct an especially thorough automatic evaluation both at the validation loss minimum and of overfitted models. We believe our experiments also shed light on the limitations of frequently used automatic metrics.
Methods
Intuition
We approach the one-to-many, many-to-one problem from a relatively new perspective: instead of adding more complexity to NCMs, we reduce the complexity of the dataset by filtering out a fraction of utterance pairs that we assume are primarily responsible for generic/uninteresting responses. Of the 72 000 unique source utterances in the Dai-lyDialog dataset (see Section 4.1 for details), 60 000 occur with a single target only. For these it seems straightforward to maximize the conditional probability P (T |S), S and T denoting a specific source and target utterance. However, in the case of sources that appear with multiple targets (oneto-many), models are forced to learn some "average" of observed responses .
The entropy of response distribution of an utterance s is a natural measure of the amount of "confusion" introduced by s. For example, the context What did you do today? has high entropy, since it is paired with many different responses in the data, but What color is the sky? has low entropy since it's observed with few responses. The many-toone scenario can be similarly formulated, where a diverse set of source utterances are observed with the same target (e.g. I don't know has high entropy). While this may be a less prominent issue in training NCMs, we shall still experiment with excluding such generic targets, as dialog models tend to generate them frequently (see Section 2).
Clustering Methods and Filtering
We refer with IDENTITY to the following entropy computation method. For each source utterance s in the dataset we calculate the entropy of the conditional distribution T |S = s, i.e. given a dataset D of source-target pairs, we define the target entropy of s as
H tgt (s, D) = − (s,t i )∈D p(t i |s) log 2 p(t i |s) (1)
Similarly, source entropy of a target utterance is
H src (t, D) = − (s i ,t)∈D p(s i |t) log 2 p(s i |t) (2)
The probabilities are based on the observed relative frequency of utterance pairs in the data.
For the purposes of this entropy-based filtering, we considered the possibility of also including some form of similarity measure between utterances that would allow us to detect whether a set of responses is truly diverse, as in the case of a question like What did you do today?, or diverse only on the surface, such as in the case of a question like How old are you? (since answers to the latter are semantically close). Measuring the entropy of semantic clusters as opposed to individual utterances may improve our method by reducing data sparsity. For example How are you? can appear in many forms, like How are you <name>? (see Section 4.2). While the individual forms have low entropy (because they have low frequency), we may decide to filter them all if together they form a high-entropy cluster.
To this end we performed the filtering based not only on the set of all utterances, as in the case of IDENTITY, but also on clusters of utterances established by clustering their vector representations using the Mean Shift algorithm (Fukunaga and Hostetler, 1975). Source and target utterances are clustered separately. In the AVG-EMBEDDING setup the representation R(U ) of utterance U is computed by taking the average word embedding weighted by the smooth inverse frequency
R(U ) = 1 |U | w∈U E(w)·0.001
0.001+p(w) of words (Arora et al., 2017), where E(w) and p(w) are the embedding and the probability 5 of word w respectively. We also experiment with SENT2VEC 6 , a more sophisticated sentence embedding approach, which can be thought of as an extension of word2vec to sentences (Pagliardini et al., 2018).
The target entropy of a source cluster c s is
H tgt (c s , C) = − c i ∈C p(c i |c s ) log 2 p(c i |c s ) (3)
where C is the set of all clusters and p(c i |c s ) is the conditional probability of observing an utterance from cluster i after an utterance from cluster s. In the context of these methods, the entropy of an utterance will mean the entropy of its cluster. Note that IDENTITY is a special case of this cluster-based entropy computation method, since in IDENTITY a "cluster" is comprised of multiple examples of one unique utterance. Thus a target cluster's entropy is computed similarly to Equation 2, but using clusters as in Equation 3. Entropy values obtained with each of these methods were used to filter dialog data in three ways. The SOURCE approach filters utterance pairs in which the source utterance has high entropy, TARGET filters those with a high entropy target, and finally the BOTH strategy filters all utterance pairs that are filtered by either SOURCE or TARGET. Some additional techniques did not yield meaningful improvement and were excluded from further evaluation. Clustering based on the Jaccard similarity of the bag of words of utterances only added noise to IDENTITY and resulted in much worse clusters than SENT2VEC. Clustering single occurrences of each unique utterance (as opposed to datasets with multiplicity) lead to less useful clusters than when clustering the whole dataset, probably because it resulted in less weight being given to the frequent utterances that we want to filter out. K-means proved inferior to the Mean Shift algorithm, which is a density-based clustering algorithm and seems to work better for clustering vectors of sentences. Filtering stop words before clustering did not improve the quality of clusters, probably because many utterances that we want to filter out contain a large number of stop words.
Data Analysis
Dataset
With 90 000 utterances in 13 000 dialogs, Dai-lyDialog (Li et al., 2017c), our primary dataset, is comparable in size with the Cornell Movie-Dialogs Corpus (Danescu-Niculescu-Mizil and Lee, 2011), but contains real-world conversations. Using the IDENTITY approach, about 87% of utterances have 0 entropy (i.e. they do not appear with more than one target), 5% have an entropy of 1 (e.g. they appear twice, with different targets), remaining values rise sharply to 7. This distribution is similar for source and target utterances. Entropy is clearly proportional to utterance frequency ( Figure 1), but has a wide range of values among utterances of equal frequency. For example, utterances with a frequency of 3 can have entropies ranging from 0 to log 2 3 ≈ 1.58, the latter of which would be over our filtering threshold of 1 (see Section 5.1 for details on selecting thresholds). Since high-entropy utterances are relatively short, we also examined the relationship between entropy and utterance length ( Figure 2). Given the relationship between frequency and entropy, it comes as no surprise that longer utterances have lower entropy.
Clustering Results
Compared to IDENTITY, both SENT2VEC and AVG-EMBEDDING produce a much lower number of clusters with 0 entropy, but also a huge cluster with more than 5000 elements (the size of the second largest cluster is below 500), which we didn't filter since it clearly doesn't group utterances with similar meaning. Generally, clusters were formed of similar utterances with the occasional exception of longer outlier utterances clustered together (instead of creating a separate cluster for each outlier), which can be attributed to the nature of the clustering algorithm. Overall, SENT2VEC appeared to produce better clusters than AVG-EMBEDDING, as reflected in the evaluation in Section 5.
We experimented with different bandwidth values 7 for the Mean Shift algorithm to produce clusters with as many elements as possible while also keeping the elements semantically similar. In an example cluster ( Figure 3) we can see that the clustering was able to group together several variants of How are you?, in particular, those with different names. In general, we noticed that both in the case of IDENTITY and the clustering methods, utterances labeled with the highest entropy are indeed those generic sources and replies which we hoped to eliminate. See Appendix A.1 for a selection of high entropy utterances and clusters.
Experiments
In this section the model and parameter setups are presented along with 17 evaluation metrics. Limitations of these metrics are discussed and a comparison between our filtering methods is presented on DailyDialog (Section 5.3), and other datasets (Section 5.4). We use transformer (Vaswani et al., 2017) as our dialog model, an encoder-decoder architecture relying solely on attention mechanisms (Bahdanau et al., 2015). transformer has already been applied to a plethora of natural language processing tasks, including dialog modeling (Dinan et al., 2019;Mazare et al., 2018;Devlin et al., 2018). We used the official implementation 8 (see Appendix A.2 for a report of hyperparameters). 8 https://github.com/tensorflow/ tensor2tensor
Model and Parameters
The vocabulary for DailyDialog was limited to the most frequent 16 384 words, and train / validation / test splits contained 71 517 / 9 027 / 9 318 examples, respectively.
Clustering and
Filtering. For AVG-EMBEDDING fastText 9 embeddings were used. The bandwidth of Mean Shift was set to 0.7 and 3.5 for AVG-EMBEDDING and SENT2VEC, which produced 40 135 and 23 616 clusters, respectively. Entropy thresholds and amount of data filtered can be found in Table 1. Generally we set the threshold so that filtered data amount is similar to the DailyDialog IDENTITY scenario. We also set a threshold for the maximum average utterance length (15 and 20 for AVG-EMBEDDING and SENT2VEC) in clusters that we considered for filtering, excluding outliers from the filtering process (see Section 4.2).
Training and Decoding. Word embeddings of size 512 were randomly initialized, batch size was set to 2048 tokens, and we used the Adam optimizer (Kingma and Ba, 2014). We experimented with various beam sizes (Graves, 2012), but greedy decoding performed better according to all metrics, also observed previously (Asghar et al., 2017;Shao et al., 2017;Tandon et al., 2017).
Evaluation Metrics
As mentioned in Section 2, automatic evaluation of chatbots is an open research problem. In order to get as complete a picture as possible, we use 17 metrics that have been applied to dialog models over the past years, briefly described below. These metrics assess different aspects of response quality, thus models should be compared on the whole set of metrics.
Response length. Widely used as a simple engagement indicator (Serban et al., 2017b;Tandon et al., 2017;Baheti et al., 2018).
Word and utterance entropy. The per-word entropy H w = − 1 |U | w∈U log 2 p(w) of responses is measured to determine their non-genericness (Serban et al., 2017b). Probabilities are calculated based on frequencies observed in the training data. We introduce the bigram version of this metric, to measure diversity at the bigram level as well. Utterance entropy is the product of H w and |U |, also reported at the bigram level.
KL divergence. We use the KL divergence between model and ground truth (GT) response sets to measure how well a model can approximate the GT distribution of words. Specifically, we define distributions p gt and p m based on each set of responses and calculate the KL divergence D kl = 1 |Ugt| w∈Ugt log 2 pgt(w) pm(w) for each GT response. The bigram version of this metric is also reported.
Embedding metrics. Embedding average, extrema, and greedy are widely used metrics (Liu et al., 2016;Serban et al., 2017b;Zhang et al., 2018c). average measures the cosine similarity between the averages of word vectors of response and target utterances. extrema constructs a representation by taking the greatest absolute value for each dimension among the word vectors in the response and target utterances and measures the cosine similarity between them. Finally, greedy matches each response token to a target token (and vice versa) based on the cosine similarity between their embeddings and averages the total score across all words. For word embeddings and average word embedding representations, we used the same setup as in AVG-EMBEDDING. Coherence. We measure the cosine similarity between pairs of input and response (Xu et al., 2018b). Although a coherence value of 1 would indicate that input and response are the same, generally a higher value seems better as model responses tend to have lower coherence than targets.
Distinct metrics. Distinct-1 and distinct-2 are widely used in the literature (Li et al., 2016a;Shen et al., 2018a;Xu et al., 2018b), measuring the ratio of unique unigrams/bigrams to the total number of unigrams/bigrams in a set of responses. However, they are very sensitive to the test data size, since increasing the number of examples in itself lowers their value. While the number of total words increases linearly, the number of unique words is limited by the vocabulary, and we found that the ratio decreases even in human data (see Appendix A.3 for details). It is therefore important to only compare distinct metrics computed on the same test data.
Bleu. Measuring n-gram overlap between response and target is widely used in the machine learning and dialog literature (Shen et al., 2018a;Xu et al., 2018b). We report BLEU-1, BLUE-2, BLEU-3, and BLEU-4 computed with the 4th smoothing algorithm described in Chen and Cherry (2014). Normally metrics are computed at the validation loss minimum of a model, however in the case of chatbot models loss may not be a good indicator of response quality (Section 2), thus we also looked at how our metrics progress during training. Figure 4 shows how coherence and the 3 embedding metrics saturate after about 80-100k steps, and never decrease (we ran the training for 300k steps, roughly 640 epochs). Most metrics show a similar trend of increasing until 100k steps, and then stagnating (see Appendix A.3 for more figures).
In contrast, validation loss for the same training reaches its minimum after about 10-20k steps ( Figure 5). This again suggests the inadequacy of the loss function, but it also questions the validity of these metrics, as they seem to favor a model that overfitted the training data, which we can assume after 640 epochs. This could be due to the many identical inputs in train and test splits, because of the nature of dialog data. Most interesting are embedding metrics and BLEU scores (Section 5.3), since they show that even after overfitting responses do not get farther from targets. This is in line with other findings reporting that qualitatively responses are better after overfitting (Csaky, 2019; Tandon et al., 2017), however occasionally they also tend to be too specific and irrelevant. We leave it for future work to conduct human evaluation between non-overfitted and overfitted models to solidify these claims. In light of these issues, we compare trainings on the DailyDialog dataset both at the validation loss minimum and at an overfitted point (150 epochs).
|U | H u w H b w H u u H b u D u kl D b kl
DailyDialog Results
We compute metrics on the unfiltered test set to show that filtered trainings perform better even on utterances that would have been filtered from the training data. TRF, the baseline transformer model trained on unfiltered data is compared to the 9 trainings on filtered data. In all tables the 17 metrics from left to right are: response length, unigram and bigram entropy, unigram and bigram utterance entropy, unigram and bigram KL divergence, embedding average, extrema and greedy, coherence, distinct-1 and distinct-2, and finally, BLEU-1, BLEU-2, BLEU-3 and BLEU-4 (see Section 5.2). Evaluating at the minimum validation loss (Ta-Input Response your starting salary is 2500 yuan a month and after you become a permanent employee it will be higher . ble 2) clearly shows that models trained on data filtered by IDENTITY and SENT2VEC are better than the baseline. IDENTITY performs best among the three methods, surpassing the baseline on all but the distinct-1 metric. SENT2VEC is a close second, getting higher values on fewer metrics than IDENTITY, but mostly improving on the baseline. Finally, AVG-EMBEDDING is inferior to the baseline, as it didn't produce clusters as meaningful as SENT2VEC, and thus produced a lower quality training set. It seems like filtering high entropy targets (both in the case of IDENTITY and SENT2VEC) is more beneficial than filtering sources, and BOTH falls mostly in the middle as expected, since it combines the two filtering types. By removing example responses that are boring and generic from the dataset the model learns to improve response quality. Finding such utterances is useful for a number of purposes, but earlier it has been done mainly manually (Li et al., 2016d;Shen et al., 2017), whereas we provide an automatic, unsupervised method of detecting them based on entropy. Every value is higher after 150 epochs of training than at the validation loss minimum (Table 3). The most striking change is in the unigram KL divergence, which is now an order of magnitude lower. IDENTITY still performs best, falling behind the baseline on only the two distinct metrics. Interestingly this time BOTH filtering was better than the TARGET filtering. SENT2VEC still mostly improves the baseline and AVG-EMBEDDING now also performs better or at least as good as the baseline on most metrics. In some cases the best performing model gets quite close to the ground truth performance. On metrics that evaluate utterances without context (i.e. entropy, divergence, dis-tinct), randomly selected responses achieve similar values as the ground truth, which is expected. However, on embedding metrics, coherence, and BLEU, random responses are significantly worse than those of any model evaluated.
Computing the unigram and bigram KL divergence with a uniform distribution instead of the model yields a value of 4.35 and 1.87, respectively. Thus, all models learned a much better distribution, suggesting that this is indeed a useful metric. We believe the main reason that clustering methods perform worse than IDENTITY is that clustering adds some noise to the filtering process. Conducting a good clustering of sentence vectors is a hard task. This could be remedied by filtering only utterances instead of whole clusters, thus combining IDENTITY and the clustering methods. In this scenario, the entropy of individual utterances is computed based on the clustered data. The intuition behind this approach would be that the noise in the clusters based on which we compute entropy is less harmful than the noise in clusters which we consider for filtering. Finally, Table 4 shows responses from the baseline and the best performing model to 3 randomly selected inputs from the test set (which we made sure are not present in the training set) to show that training on filtered data does not degrade response quality. We show more example responses in Appendix A.3.
Cornell and Twitter Results
To further solidify our claims we tested the two best performing variants of IDENTITY (BOTH and TARGET) on the Cornell Movie-Dialogs Corpus and on a subset of 220k examples from the Twit- Table 5 and Table 6, respectively. On these noisier datasets our simple IDENTITY method still managed to improve over the baseline, but the impact is not as pronounced and in contrast to DailyDialog, BOTH and TAR-GET perform best on nearly the same number of metrics. On these noisier datasets the clustering methods might work better, this is left for future work. Compared to DailyDialog there are some important distinctions that also underline that these datasets are of lesser quality. The CO-HERENCE metric is worse on the ground truth responses than on model responses (Table 5, and some embedding metrics and BLEU scores are better on randomly selected responses than on model responses (Table 6).
|U | H u w H b w H u u H b u D u kl D b kl
Conclusion
We proposed a simple unsupervised entropy-based approach that can be applied to any conversational dataset for filtering generic sources/targets that cause "confusion" during the training of opendomain dialog models. We compared various setups in an extensive quantitative evaluation, and showed that the best approach is measuring the 10 https://github.com/Marsan-Ma/chat_ corpus/ entropy of individual utterances and filtering pairs based on the entropy of target, but not source utterances. Some limitations of current automatic metrics and the loss function have also been shown, by examining their behavior on random data and with overfitting.
In the future, we plan to explore several additional ideas. As mentioned in Section 5.3, we want to extend our clustering experiments combining the ideas behind IDENTITY and the clustering methods to make them more robust to noise. We wish to conduct clustering experiments on noisier datasets and try other sentence representations (Devlin et al., 2018). We also plan to combine our method with coherence-based filtering (Xu et al., 2018b). Furthermore, we intend to perform a direct quantitative evaluation of our method based on human evaluation. Finally, we believe our method is general enough that it could also be applied to datasets in other similar NLP tasks, such as machine translation, which could open another interesting line of future research. you might want to change your plans . with the cold and wet weather that we re supposed to have tomorrow the roads are going to be rather slippery . Table 9: Responses to randomly selected test inputs which we made sure were not in the training data (DailyDialog). Unfiltered is the model trained on unfiltered data, and IDENTITY TARGET is the model trained on IDENTITY, TARGET filtered data. Overfitted means that the respective model is evaluated at an overfitted point. ok . i will take it .
A.1.2 High Entropy Clusters
A.3 Evaluation Metrics and Examples
thank you very much .
we will try our best to do our best to help you .
i understand . i understand that you are right .
ok i will . i look forward to it .
ok . any time on friday will be ok with me .
ok . see you then . i think so . i 'll be expecting you then .
great ! no wonder you can control your voice so well . you are a professional singer .
i am not a singer . i would n't mind a position that would be great .
i am a singer . i think i can be a singer in the lead .
i would n't like to be a singer but i love to sing that dream .
when can i get high speed internet installed ?
you can make a list of the company and make sure that you do n't have to use the machine .
you 'll need to make an appointment before the company starts .
you can get up at the airport and you 'll have to make an appointment .
you 'll have to make an appointment before we can come in and install it . i like those kinds of programmes too . they re very informative . i think that many people underrate the education value of tv . log). Unfiltered is the model trained on unfiltered data, and IDENTITY TARGET is the model trained on IDENTITY, TARGET filtered data. Overfitted means that the respective model is evaluated at an overfitted point.
Figure 1 :
1Entropy of source utterances (computed with IDENTITY) with respect to utterance frequency.
Figure 2 :
2Entropy of source utterances (computed with IDENTITY) with respect to utterance length.
Figure 3 :
3A cluster produced by SENT2VEC.
Figure 4 :
4Embedding metrics and coherence (on validation data) as a function of the training evolution of transformer on unfiltered data (DailyDialog).
Figure 5 :
5Training (bottom) and validation (top) loss with respect to training steps of transformer trained on unfiltered data (DailyDialog).
Figure 6 :
6A high entropy cluster from DailyDialog.
Figure 7 :
7A high entropy cluster from DailyDialog.
Figure 8 :
8A high entropy cluster from DailyDialog.
Figure 9 :
9Distinct-1 metric with respect to number of test examples (on DailyDialog). Model responses were evaluated on 9000 examples only, since the rest were training examples.
Figure 10 :
10Distinct-2 metric with respect to number of test examples (on DailyDialog). Model responses were evaluated on 9000 examples only, since the rest were training examples.
Figure 11 :
11Average length of responses (computed on the validation set) with respect to the number of training steps of the transformer trained on unfiltered data (DailyDialog).
Figure 12 :
12Word entropy of responses (computed on the validation set) with respect to the number of training steps of the transformer trained on unfiltered data (DailyDialog).
Figure 13 :
13Utterance entropy of responses (computed on the validation set) with respect to the number of training steps of the transformer trained on unfiltered data (DailyDialog).
Figure 14 :
14KL divergence of responses (computed on the validation set) with respect to the number of training steps of the transformer trained on unfiltered data (DailyDialog).
Figure 15 :
15Distinct-1 and distinct-2 metrics (computed on the validation set) with respect to the number of training steps of the transformer trained on unfiltered data (DailyDialog).
Table 2 :
2Metrics computed at the minimum of the validation loss on the unfiltered test set (DailyDialog). TRF refers to transformer, ID to IDENTITY, AE to AVG-EMBEDDING, and SC to SENT2VEC. SOURCE-side, TARGET-side, and filtering BOTH sides are denoted by initials. Best results are highlighted with bold and best results separately for each entropy computing method are in italic (and those within a 95% confidence interval).|U |
H u
w
H b
w
H u
u
H b
u
D u
kl
D b
kl
AVG
EXT GRE COH d1
d2
b1
b2
b3
b4
TRF
11.5 7.98 13.4 95 142 .0360 .182 .655 .607 .640 .567 .0465 .297 .333 .333 .328 .315
ID
B 13.1 8.08 13.6 107 162 .0473 .210 .668 .608 .638 .598 .0410 .275 .334 .340 .339 .328
T 12.2 8.04 13.6 100 150 .0335 .181 .665 .610 .640 .589 .0438 .289 .338 .341 .339 .328
S 12.3 7.99 13.5 101 153 .0406 .187 .662 .610 .641 .578 .0444 .286 .339 .342 .338 .326
AE
B 11.9 7.98 13.5 98 147 .0395 .197 .649 .600 .628 .574 .0434 .286 .318 .321 .318 .306
T 12.5 7.99 13.5 102 155 .0436 .204 .656 .602 .634 .580 .0423 .279 .324 .327 .325 .313
S 12.1 7.93 13.4 99 148 .0368 .186 .658 .605 .636 .578 .0425 .278 .325 .328 .324 .311
SC
B 12.8 8.07 13.6 105 159 .0461 .209 .655 .600 .629 .583 .0435 .282 .322 .328 .327 .316
T 13.0 8.06 13.6 107 162 .0477 .215 .657 .602 .632 .585 .0425 .279 .324 .330 .329 .318
S 12.1 7.96 13.4 100 150 .0353 .183 .657 .606 .638 .576 .0443 .286 .331 .333 .329 .317
RT
13.5 8.40 14.2 116 177 .0300 .151 .531 .452 .481 .530 .0577 .379 .090 .121 .130 .125
GT
14.1 8.39 13.9 122 165 0
0
1
1
1
.602 .0488 .362 1
1
1
1
Table 3 :
3Metrics computed on the unfiltered test set (DailyDialog) after 150 epochs of training. TRF refers to transformer, ID to IDENTITY, AE to AVG-EMBEDDING, and SC to SENT2VEC. SOURCE-side, TARGET-side, and filtering BOTH sides are denoted by initials. Best results are highlighted with bold and best results separately for each entropy computing method are in italic (and those within a 95% confidence interval). GT refers to ground truth responses and RT refers to randomly selected responses from the training set.
BASE: i can tell you what is the best way to find a job . BASE-O: do you know what it is ? TARGET: i 'm very interested in the position . TARGET-O: that 's very nice of you . i 'll have to think about it . you can greatly improve your grades by always reflecting on what you can improve on .BASE: i want to be a great singer .
BASE-O: i really appreciate it . thanks .
TARGET: i think i am experienced in that position . i think i would
like to make a good impression .
TARGET-O: i always liked it . thank you .
umm i think i would be hard to find something
in that range in this area . you know the
environment in the south district is the nicest .
BASE: what about the kitchen ?
BASE-O: what about the kitchen ?
TARGET: what about the kitchen ?
TARGET-O: what about the kitchen ?
Table 4 :
4Example inputs and responses from DailyDialog. BASE is trained on unfiltered data, and TARGET is the model trained on IDENTITY, TARGET filtered data. Models marked with O are evaluated at an overfitted point.
Table 5 :
5Metrics on the unfiltered test set (Cornell) at the validation loss minimum. TRF refers to transformer, ID to IDENTITY. TARGET-side, and filtering BOTH sides are denoted by initials. Best results are highlighted with bold. GT refers to ground truth responses and RT refers to randomly selected responses from the training set.|U | H u
w
H b
w H u
u H b
u
D u
kl
D b
kl
AVG EXT GRE COH d1
d2
b1
b2
b3
b4
TRF
20.6 6.89 11.4 121 177 2.28 3.40 .643 .395 .591 .659 2.1e-3 6.2e-3 .0519 .0666 .0715 .0693
ID
B 20.3 6.95 11.4 119 171 2.36 3.41 .657 .394 .595 .673 1.2e-3 3.4e-3 .0563 .0736 .0795 .0774
T 29.0 6.48 10.7 157 226 2.68 3.69 .644 .403 .602 .660 1.4e-3 4.6e-3 .0550 .0740 .0819 .0810
RT
14.0 9.81 15.9 136 171 .05
.19 .681 .334 .543 .695 8.5e-2 5.4e-1 .0444 .0751 .0852 .0840
GT
14.0 9.78 15.8 135 167
0
0 1
1
1
.734 8.1e-2 5.3e-1 1
1
1
1
Table 6 :
6Metrics on the unfiltered test set (Twitter) at the validation loss minimum. TRF refers to transformer, ID to IDENTITY. TARGET-side, and filtering BOTH sides are denoted by initials. Best results are highlighted with bold. GT refers to ground truth responses and RT refers to randomly selected responses from the training set.ter corpus 10 . Entropy thresholds were selected
to be similar to the DailyDialog experiments (Ta-
ble 1). Evaluation results at the validation loss
minimum on the Cornell corpus and the Twitter
dataset are presented in
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Table 7 :
7Top 20 source utterances (from DailyDialog)
sorted by entropy. The entropy was calculated with
IDENTITY.
Table 8 :
8Transformer hyperparameters.
about twelve hours . it 's about fortyfive minutes to an hour . it 's about 20 minutes . it 's only about fortyfive minutes to an hour . is that yellow bus it ? no it 's not far . it 's a bit cold . yeah . there s a new one in america . no it 's a yellow one . no it 's not very expensive these days . hi mary ! what do you like to do in your spare time ? i like playing basketball . well i like to play the piano . i like playing basketball . i like playing chess . let me see . we can visit the country . we have n't done that for such a long time . yes it is . yes it 's fully furnished . yes the furniture is right over there . yes it 's a furnished eatin kitchen . the furniture and the icebox are all pretty outdated . where do they go ? they 're on the bottom floor . i have no idea . they re in the cabinet next to the library . go to them department please . let me pay for it . ok . here you are . cash credit card or debit card ? ok . here is the money . cash credit card or debit card ? he never does and in fact he seldom communicates with him . he is a hard job . he can also learn a lot from the government . that 's right . he has no children and his wife experience too . he is a very kind of man . yeah . he 's a very easy teacher . so you must have had a busy and enjoyable day yesterday . yes i am . i really enjoyed it . yes . i am very sorry i can t make it . i will . thanks for coming . yes . let 's go . yes . the reservation under mr . watson . ok . would you please fill out the table ? there are so many to choose from . i did n't know you could do all of this using a card . i see . what 's the best part ? i 'm not sure . i 'll go to the doctor and get my card right away . it was great . i 'm glad i did n't like it . it was ok . there was a lot of turbulence though . the worst thing was n't too good ! tomorrow ? no i am afraid i won t be able to . i m going on business to birmingham . i think you are looking for a .<unk><unk>in the future . the best reason why are you want to see the results .well there is a problem with the exchange rate . it s the same size that you have in the u . s . you should go to the dentist .Input
Unfiltered
Unfiltered-Overfitted
IDENTITY TARGET fil-
tered
IDENTITY
TARGET
filtered-overfitted
some high speed chase
scene i think .
i 'm not sure .
but many things are
went wrong !
i 'm so sorry .
enough !
how long is this bus ride
?
i hope so .
yes there are many fa-
mous paintings on show
. i hope it 's the nicest
city in the united states .
i 'm glad to hear that .
where are your parents ?
thirty u . s . dollars . . .
and it 's only been used
for six months .
that 's fine .
i see . is it a long time
to come to the station ?
how many months did
you get there ?
what 're two weeks ?
five hundred dollars al-
together .
here you are .
here you are .
here is your bill .
here you are . thank you
byebye .
look at this bridge . we
will soon be crossing
the thames .
yes i agree .
oh yeah .
how far is it ?
well there 's a spanish
teacher . so i 'll often go
back to the spanish mu-
seum .
is the furniture crafted
well ?
ok .
could you spell that
please ?
i 'm afraid there 's no
seat in the middle of the
table .
you mean you do n't
have to tell me the truth
?
it was great ! how was
your year ?
it was great . i had a
great time .
my mother had a birth-
day party for me .
filtered-overfitted you 're flattering me . no . no really . no but you can t . no really . i can only hear a little accent . three bags and a suitcase . this is my luggage to check . ok . i will take one . what 's this ? essential oil ? wow ! this is a small bag . i 'll put you in the bag . ok . is this your luggage properly labelled sir ? oh shut up ! she 's not what you are thinking about . if you give us your inquiry i shall go very carefully into the price and try my best to put you on the best of the terms .Input
Unfiltered
Unfiltered-Overfitted
IDENTITY TARGET fil-
tered
IDENTITY
TARGET
i like orange better .
what 's your favorite fla-
vor ?
how about drinks ?
i 'll tell you what you 're
doing .
could i have some water
?
i like to watch barbie
sticking her tongue out
.
what 's her figure ?
oh she 's a pretty
woman .
you are not supposed to
be serious .
i agree . people often criticize tv for showing too much sex and violence . i agree . people often criticize tv for showing too much sex and violence . i think that teenagers should be often become addicted and violence . i agree . people often criticize tv for showing too much sex and violence . can you tell that i m excited ? i m afraid i m not a child . i m a fashion designer . next month i have a fashion show . i m not sure . what do you want to know ? i m sure that i ll cook the same for you . would you like to have a shampoo sir ? yes i 'd like a glass of iced tea . no thank you . no i 'd rather have it with me . yes with the lux dandruff shampoo . it helps a lot to get rid of my dandruff . what else would i prepare sir ? you can bring your beef with your chicken and a bottle of wine . oh good ! can you make some recommendation for me ? well the interviewers are not interested in the position . let me see . everything looks fine . he always says i am a hard worker with consciousness of responsibility sufficient education and enough experience . i think that maybe you are right . what s the matter with you ? how many years of experience will you earn ? would you like to work in the finance department ? what made you think that ? i like it a lot . i love that shirt on you . i think it 's great . i think i ve learned a lot from different countries . i will care for it . i can tell you what bus to catch but you have to walk a little bit .i do n't know .
tell me how to get a
ticket and a student bus
station .
i 'm sorry but i do n't
have to wait .
you 're going to have a
car .
Table 10 :
10Responses to randomly selected test inputs which we made sure were not in the training data (DailyDia
Based on the observed relative frequency in the data. 6 https://github.com/epfml/sent2vec
Bandwidth is like a radius in the latent space of utterance representations(Fukunaga and Hostetler, 1975).
https://fasttext.cc/
AcknowledgmentsWe wish to thank EvelinÁcs, Péter Ihász, Márton Makrai, Luca Szegletes, and all anonymous reviewers for their thoughtful feedback. Work partially supported by Project FIEK 16-1-2016-0007, financed by the FIEK 16 funding scheme of the Hungarian National Research, Development and Innovation Office (NKFIH).A Appendix
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| Though machine translation errors caused by the lack of context beyond one sentence have long been acknowledged, the development of context-aware NMT systems is hampered by several problems. Firstly, standard metrics are not sensitive to improvements in consistency in document-level translations. Secondly, previous work on context-aware NMT assumed that the sentence-aligned parallel data consisted of complete documents while in most practical scenarios such document-level data constitutes only a fraction of the available parallel data. To address the first issue, we perform a human study on an English-Russian subtitles dataset and identify deixis, ellipsis and lexical cohesion as three main sources of inconsistency. We then create test sets targeting these phenomena. To address the second shortcoming, we consider a set-up in which a much larger amount of sentence-level data is available compared to that aligned at the document level. We introduce a model that is suitable for this scenario and demonstrate major gains over a context-agnostic baseline on our new benchmarks without sacrificing performance as measured with BLEU. 1 | 10.18653/v1/p19-1116 | [
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When a Good Translation is Wrong in Context: Context-Aware Machine Translation Improves on Deixis, Ellipsis, and Lexical Cohesion
Association for Computational LinguisticsCopyright Association for Computational LinguisticsJuly 28 -August 2, 2019. 2019
Elena Voita [email protected]@[email protected]
YandexRussia
University of Amsterdam
Netherlands
Rico Sennrich
University of Edinburgh
Scotland
University of Zurich
Switzerland
Ivan Titov
University of Amsterdam
Netherlands
University of Edinburgh
Scotland
When a Good Translation is Wrong in Context: Context-Aware Machine Translation Improves on Deixis, Ellipsis, and Lexical Cohesion
Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics
the 57th Annual Meeting of the Association for Computational LinguisticsFlorence, ItalyAssociation for Computational LinguisticsJuly 28 -August 2, 2019. 2019
Though machine translation errors caused by the lack of context beyond one sentence have long been acknowledged, the development of context-aware NMT systems is hampered by several problems. Firstly, standard metrics are not sensitive to improvements in consistency in document-level translations. Secondly, previous work on context-aware NMT assumed that the sentence-aligned parallel data consisted of complete documents while in most practical scenarios such document-level data constitutes only a fraction of the available parallel data. To address the first issue, we perform a human study on an English-Russian subtitles dataset and identify deixis, ellipsis and lexical cohesion as three main sources of inconsistency. We then create test sets targeting these phenomena. To address the second shortcoming, we consider a set-up in which a much larger amount of sentence-level data is available compared to that aligned at the document level. We introduce a model that is suitable for this scenario and demonstrate major gains over a context-agnostic baseline on our new benchmarks without sacrificing performance as measured with BLEU. 1
Introduction
With the recent rapid progress of neural machine translation (NMT), translation mistakes and inconsistencies due to the lack of extra-sentential context are becoming more and more noticeable among otherwise adequate translations produced by standard context-agnostic NMT systems (Läubli et al., 2018). Though this problem has recently triggered a lot of attention to contextaware translation (Jean et al., 2017a;Wang et al., 2017;Tiedemann and Scherrer, 2017; Bawden 1 We release code and data sets at https://github.com/lena-voita/ good-translation-wrong-in-context. Voita et al., 2018;Maruf and Haffari, 2018;Agrawal et al., 2018;Miculicich et al., 2018;Zhang et al., 2018), the progress and widespread adoption of the new paradigm is hampered by several important problems. Firstly, it is highly non-trivial to design metrics which would reliably trace the progress and guide model design. Standard machine translation metrics (e.g., BLEU) do not appear appropriate as they do not sufficiently differentiate between consistent and inconsistent translations (Wong and Kit, 2012). 2 For example, if multiple translations of a name are possible, forcing consistency is essentially as likely to make all occurrences of the name match the reference translation as making them all different from the reference. Second, most previous work on context-aware NMT has made the assumption that all the bilingual data is available at the document level. However, isolated parallel sentences are a lot easier to acquire and hence only a fraction of the parallel data will be at the document level in any practical scenario. In other words, a context-aware model trained only on documentlevel parallel data is highly unlikely to outperform a context-agnostic model estimated from much larger sentence-level parallel corpus. This work aims to address both these shortcomings.
A context-agnostic NMT system would often produce plausible translations of isolated sentences, however, when put together in a document, these translations end up being inconsistent with each other. We investigate which linguistic phenomena cause the inconsistencies using the OpenSubtitles (Lison et al., 2018) corpus for the English-Russian language pair. We identify deixis, ellipsis and lexical cohesion as three main sources of the violations, together amounting to about 80% of the cases. We create test sets focusing specifically on the three identified phenomena (6000 examples in total). We show that by using a limited amount of document-level parallel data, we can already achieve substantial improvements on these benchmarks without negatively affecting performance as measured with BLEU. Our approach is inspired by the Deliberation Networks (Xia et al., 2017). In our method, the initial translation produced by a baseline context-agnostic model is refined by a context-aware system which is trained on a small document-level subset of parallel data.
The key contributions are as follows:
• we analyze which phenomena cause contextagnostic translations to be inconsistent with each other;
• we create test sets specifically addressing the most frequent phenomena;
• we consider a novel and realistic set-up where a much larger amount of sentencelevel data is available compared to that aligned at the document level;
• we introduce a model suitable for this scenario, and demonstrate that it is effective on our new benchmarks without sacrificing performance as measured with BLEU.
Analysis
We begin with a human study, in which we:
1. identify cases when good sentence-level translations are not good when placed in context of each other, 2. categorize these examples according to the phenomena leading to a discrepancy in translations of consecutive sentences.
The test sets introduced in Section 3 will then target the most frequent phenomena.
Human annotation
To find what makes good context-agnostic translations incorrect when placed in context of each other, we start with pairs of consecutive sentences. We gather data with context from the publicly available OpenSubtitles2018 corpus (Lison et al., all one/both bad both good bad pair good pair 2000 211 140 1649 100% 11% 7% 82% In the first stage, the annotators are instructed to mark as "good" translations which (i) are fluent sentences in the target language (in our case, Russian) (ii) can be reasonable translations of a source sentence in some context.
For the second stage we only consider pairs of sentences with good sentence-level translations. The annotators are instructed to mark translations as bad in context of each other only if there is no other possible interpretation or extra additional context which could have made them appropriate. This was made to get more robust results, avoiding the influence of personal preferences of the annotators (for example, for using formal or informal speech), and excluding ambiguous cases that can only be resolved with additional context.
The statistics of answers are provided in Table 1. We find that our annotators labelled 82% of sentence pairs as good translations. In 11% of cases, at least one translation was considered bad at the sentence level, and in another 7%, the sentences were considered individually good, but bad in context of each other. This indicates that in our setting, a substantial proportion of translation errors are only recognized as such in context.
Types of phenomena
From the results of the human annotation, we take all instances of consecutive sentences with good translations which become incorrect when placed in the context of each other. For each, we identify the language phenomenon which caused a discrepancy. The results are provided in Table 2. Below we discuss these types of phenomena, as well as problems in translation they cause, in more detail. In the scope of current work, we concentrate only on the three most frequent phenomena.
Deixis
In this category, we group several types of deictic words or phrases, i.e. referential expressions whose denotation depends on context. This includes personal deixis ("I", "you"), place deixis ("here", "there"), and discourse deixis, where parts of the discourse are referenced ("that's a good question."). Most errors in our annotated corpus are related to person deixis, specifically gender marking in the Russian translation, and the T-V distinction between informal and formal you (Latin "tu" and "vos").
In many cases, even when having access to neighboring sentences, one cannot make a confident decision which of the forms should be used, as there are no obvious markers pointing to one form or another (e.g., for the T-V distinction, words such as "officer", "mister" for formal and "honey", "dude" for informal). However, when (a) EN We haven't really spoken much since your return. Tell me, what's on your mind these days? RU Мы не разговаривали с тех пор, как вы вернулись. Скажи мне, что у тебя на уме в последнее время? RU My ne razgovarivali s tekh por, kak vy vernulis'. Skazhi mne, chto u tebya na ume v posledneye vremya?
(b) EN I didn't come to Simon's for you. I did that for me. RU Я пришла к Саймону не ради тебя. Я сделал это для себя. RU Ya prishla k Saymonu ne radi tebya. Ya sdelal eto dlya sebya. pronouns refer to the same person, the pronouns, as well as verbs that agree with them, should be translated using the same form. See Figure 1(a) for an example translation that violates T-V consistency. Figure 1(b) shows an example of inconsistent first person gender (marked on the verb), although the speaker is clearly the same. Anaphora are a form of deixis that received a lot of attention in MT research, both from the perspective of modelling (Le Nagard and Koehn, 2010;Hardmeier and Federico, 2010;Jean et al., 2017b;Bawden et al., 2018;Voita et al., 2018, among others) and targeted evaluation (Hardmeier et al., 2015;Guillou and Hardmeier, 2016;Müller et al., 2018), and we list anaphora errors separately, and will not further focus on them.
Ellipsis
Ellipsis is the omission from a clause of one or more words that are nevertheless understood in the context of the remaining elements.
In machine translation, elliptical constructions in the source language pose a problem if the target language does not allow the same types of ellipsis (requiring the elided material to be predicted from context), or if the elided material affects the syntax of the sentence; for example, the grammatical function of a noun phrase and thus its inflection in Russian may depend on the elided verb (Figure 2(a)), or the verb inflection may depend on the type of discrepancy frequency wrong morphological form 66% wrong verb (VP-ellipsis) 20% other error 14% elided subject. Our analysis focuses on ellipses that can only be understood and translated with context beyond the sentence-level. This has not been studied extensively in MT research. 3 We classified ellipsis examples which lead to errors in sentence-level translations by the type of error they cause. Results are provided in Table 4.
It can be seen that the most frequent problems related to ellipsis that we find in our annotated corpus are wrong morphological forms, followed by wrongly predicted verbs in case of verb phrase ellipsis in English, which does not exist in Russian, thus requiring the prediction of the verb in the Russian translation ( Figure 2(b)).
Lexical cohesion
Lexical cohesion has been studied previously in MT (Tiedemann, 2010;Gong et al., 2011;Wong and Kit, 2012;Kuang et al., 2018;Miculicich et al., 2018, among others).
There are various cohesion devices (Morris and Hirst, 1991), and a good translation should exhibit lexical cohesion beyond the sentence level. We focus on repetition with two frequent cases in our annotated corpus being reiteration of named entities ( Figure 3(a)) and reiteration of more general phrase types for emphasis ( Figure 3(b)) or in clarification questions.
Test Sets
For the most frequent phenomena from the above analysis we create test sets for targeted evaluation. Each test set contains contrastive examples. It is specifically designed to test the ability of a system to adapt to contextual information and handle the phenomenon under consideration. Each test instance consists of a true example (sequence of sentences and their reference translation from the data) and several contrastive translations which differ from the true one only in the considered aspect. All contrastive translations we use are correct plausible translations at a sentence level, and only context reveals the errors we introduce. All the test sets are guaranteed to have the necessary context in the provided sequence of 3 sentences. The system is asked to score each candidate example, and we compute the system accuracy as the proportion of times the true translation is preferred over the contrastive ones.
Test set statistics are shown in Table 5.
Deixis
From Table 3, we see that the most frequent error category related to deixis in our annotated corpus is the inconsistency of T-V forms when translating second person pronouns. The test set we construct for this category tests the ability of a machine translation system to produce translations with consistent level of politeness. We semi-automatically identify sets of consecutive sentences with consistent politeness markers on pronouns and verbs (but without nominal markers such as "'Mr." or "officer") and switch T and V forms. Each automatic step was followed by human postprocessing, which ensures the quality of the final test sets. 4 This gives us two sets of translations for each example, one consistently informal (T), and one consistently formal (V). For each, we create an inconsistent contrastive example by switching the formality of the last sentence. The symmetry of the test set ensures that any contextagnostic model has 50% accuracy on the test set.
Ellipsis
From Table 4, we see that the two most frequent types of ambiguity caused by the presence of an elliptical structure have different nature, hence we construct individual test sets for each of them.
Ambiguity of the first type comes from the inability to predict the correct morphological form of some words. We manually gather examples with such structures in a source sentence and change the morphological inflection of the relevant target phrase to create contrastive translation. Specifically, we focus on noun phrases where the verb is elided, and the ambiguity lies in how the noun phrase is inflected.
The second type we evaluate are verb phrase ellipses. Mostly these are sentences with an auxiliary verb "do" and omitted main verb. We manually gather such examples and replace the translation of the verb, which is only present on the target side, with other verbs with different meaning, but 4 Details are provided in the appendix. the same inflection. Verbs which are used to construct such contrastive translations are the top-10 lemmas of translations of the verb "do" which we get from the lexical table of Moses (Koehn et al., 2007) induced from the training data.
Lexical cohesion
Lexical cohesion can be established for various types of phrases and can involve reiteration or other semantic relations. In the scope of the current work, we focus on the reiteration of entities, since these tend to be non-coincidental, and can be easily detected and transformed.
We identify named entities with alternative translations into Russian, find passages where they are translated consistently, and create contrastive test examples by switching the translation of some instances of the named entity. For more details, please refer to the appendix.
Model and Setting
Setting
Previous work on context-aware neural machine translation used data where all training instances have context. This setting limits the set of available training sets one can use: in a typical scenario, we have a lot of sentence-level parallel data and only a small fraction of document-level data.
Since machine translation quality depends heavily on the amount of training data, training a contextaware model is counterproductive if this leads to ignoring the majority of available sentence-level data and sacrificing general quality. We will also show that a naive approach to combining sentencelevel and document-level data leads to a drop in performance.
In this work, we argue that it is important to consider an asymmetric setting where the amount of available document-level data is much smaller than that of sentence-level data, and propose an approach specifically targeting this scenario.
Model
We introduce a two-pass framework: first, the sentence is translated with a context-agnostic model, and then this translation is refined using context of several previous sentences (context includes source sentences as well as their translations). We expect this architecture to be suitable in the proposed setting: the baseline context-agnostic model can be trained on a large amount of sentence-level data, and the second-pass model can be estimated on a smaller subset of parallel data which includes context. As the first-pass translation is produced by a strong model, we expect no loss in general performance when training the second part on a smaller dataset.
The model is close in spirit to the Deliberation networks (Xia et al., 2017). The first part of the model is a context-agnostic model (we refer to it as the base model), and the second one is a contextaware decoder (CADec) which refines contextagnostic translations using context. The base model is trained on sentence-level data and then fixed. It is used only to sample context-agnostic translations and to get vector representations of the source and translated sentences. CADec is trained only on data with context.
Let D sent = {(x i , y i )} N i=1
denote the sentencelevel data with n paired sentences and D doc = {(x j , y j , c j )} M j=1 denote the document-level data, where (x j , y j ) is source and target sides of a sentence to be translated, c j are several preceding sentences along with their translations.
Base model For the baseline context-agnostic model we use the original Transformerbase (Vaswani et al., 2017), trained to maximize the sentence-level log-likelihood
1 N (x i ,y i )∈Dsent log P (y i |x i , θ B ).
Context-aware decoder (CADec) The contextaware decoder is trained to correct translations given by the base model using contextual infor-mation. Namely, we maximize the following document-level log-likelihood:
1 M (x j ,y j )∈D doc log E y B j ∝P (y|x j ,θ B ) P (y j |x j , y B j , c j , θ C ),
where y B j is sampled from P (y|x j , θ B ). CADec is composed of a stack of N = 6 identical layers and is similar to the decoder of the original Transformer. It has a masked self-attention layer and attention to encoder outputs, and additionally each layer has a block attending over the outputs of the base decoder ( Figure 4). We use the states from the last layer of the base model's encoder of the current source sentence and all context sentences as input to the first multi-head attention. For the second multi-head attention we input both last states of the base decoder and the target-side token embedding layer; this is done for translations of the source and also all context sentences. All sentence representations are produced by the base model. To encode the relative position of each sentence, we concatenate both the encoder and decoder states with one-hot vectors representing their position (0 for the source sentence, 1 for the immediately preceding one, etc). These distance embeddings are shown in blue in Figure 4.
Experiments
Training
At training time, we use reference translations as translations of the previous sentences. For the cur-rent sentence, we either sample a translation from the base model or use a corrupted version of the reference translation. We propose to stochastically mix objectives corresponding to these versions:
1 M (x j ,y j )∈D doc log b j · P (y j |x j ,ỹ j , c j , θ C ))+ + (1 − b j ) · P (y j |x j , y B j , c j , θ C ) ,
whereỹ j is a corrupted version of the reference translation and b j ∈ {0, 1} is drawn from Bernoulli distribution with parameter p, p = 0.5 in our experiments. Reference translations are corrupted by replacing 20% of their tokens with random tokens. We discuss the importance of the proposed training strategy, as well as the effect of varying the value of p, in Section 6.5.
Inference
As input to CADec for the current sentence, we use the translation produced by the base model. Target sides of the previous sentences are produced by our two-stage approach for those sentences which have context and with the base model for those which do not. We use beam search with a beam of 4 for all models.
Data and setting
We use the publicly available OpenSubtitles2018 corpus (Lison et al., 2018) for English and Russian. As described in detail in the appendix, we apply data cleaning after which only a fraction of data has context of several previous sentences. We use up to 3 context sentences in this work. We randomly choose 6 million training instances from the resulting data, among which 1.5m have context of three sentences. We randomly choose two subsets of 10k instances for development and testing and construct our contrastive test sets from 400k held-out instances from movies not encountered in training. The hyperparameters, preprocessing and training details are provided in the supplementary material.
Results
We evaluate in two different ways: using BLEU for general quality and the proposed contrastive test sets for consistency. We show that models indistinguishable with BLEU can be very different in terms of consistency.
We randomly choose 500 out of 2000 examples from the lexical cohesion set and 500 out of 3000 from the deixis test set for validation and leave the rest for final testing. We compute BLEU on the development set as well as scores on lexical cohesion and deixis development sets. We use convergence in both metrics to decide when to stop training. The importance of using both criteria is discussed in Section 6.4. After the convergence, we average 5 checkpoints and report scores on the final test sets.
Baselines
We consider three baselines.
baseline The context-agnostic baseline is Transformer-base trained on all sentence-level data. Recall that it is also used as the base model in our 2-stage approach.
concat The first context-aware baseline is a simple concatenation model. It is trained on 6m sentence pairs, including 1.5m having 3 context sentences. For the concatenation baseline, we use a special token separating sentences (both on the source and target side).
s-hier-to-2.tied This is the version of the model s-hier-to-2 introduced by Bawden et al. (2018), where the parameters between encoders are shared (Müller et al., 2018). The model has an additional encoder for source context, whereas the target side of the corpus is concatenated, in the same way as for the concatenation baseline. Since the model is suitable only for one context sentence, it is trained on 6m sentence pairs, including 1.5m having one context sentence. We chose s-hier-to-2.tied as our second context-aware baseline because it also uses context on the target side and performed best in a contrastive evaluation of pronoun translation (Müller et al., 2018).
General results
BLEU scores for our model and the baselines are given in Table 6. 5 For context-aware models, all sentences in a group were translated, and then only the current sentence is evaluated. We also report BLEU for the context-agnostic baseline trained only on 1.5m dataset to show how the performance is influenced by the amount of data.
We observe that our model is no worse in BLEU than the baseline despite the second-pass model being trained only on a fraction of the data. In contrast, the concatenation baseline, trained on a mixture of data with and without context is about 1 BLEU below the context-agnostic baseline and our model when using all 3 context sentences. CADec's performance remains the same independently from the number of context sentences (1, 2 or 3) as measured with BLEU. s-hier-to-2.tied performs worst in terms of BLEU, but note that this is a shallow recurrent model, while others are Transformer-based. It also suffers from the asymmetric data setting, like the concatenation baseline.
Consistency results
Scores on the deixis, cohesion and ellipsis test sets are provided in Tables 7 and 8. For all tasks, we observe a large improvement from using context. For deixis, the concatenation model (concat) and CADec improve over the baseline by 33.5 and 31.6 percentage points, respectively. On the lexical cohesion test set, CADec shows a large improvement over the context-agnostic baseline (12.2 percentage points), while concat performs similarly to the baseline. For ellipsis, both models improve substantially over the baseline (by 19-51 percentage points), with concat stronger for inflection tasks and CADec stronger for VPellipsis. Despite its low BLEU score, s-hier-to-2.tied also shows clear improvements over the context-agnostic baseline in terms of consistency, but underperforms both the concatenation model and CADec, which is unsurprising given that it uses only one context sentence. When looking only at the scores where the latest relevant context is in the model's context window (column 2 in Table 7), s-hier-to-2.tied outperforms the concatenation baseline for lexical cohesion, but remains behind the performance of CADec.
The proposed test sets let us distinguish models which are otherwise identical in terms of BLEU: the performance of the baseline and CADec is the same when measured with BLEU, but very different in terms of handling contextual phenomena.
6.4 Context-aware stopping criteria Figure 5 shows that for context-aware models, BLEU is not sufficient as a criterion for stopping: even when a model has converged in terms of BLEU, it continues to improve in terms of consistency. For CADec trained with p = 0.5, BLEU score has stabilized after 40k batches, but the lexical cohesion score continues to grow.
Ablation: using corrupted reference
At training time, CADec uses either a translation sampled from the base model or a corrupted reference translation as the first-pass translation of the current sentence. The purpose of using a corrupted reference instead of just sampling is to teach CADec to rely on the base translation and not to change it much. In this section, we discuss the importance of the proposed training strategy.
Results for different values of p are given in Table 9. All models have about the same BLEU, not statistically significantly different from the baseline, but they are quite different in terms of incorporating context. The denoising positively influences almost all tasks except for deixis, yielding the largest improvement on lexical cohesion.
Additional Related Work
In concurrent work, Xiong et al. (2018) also propose a two-pass context-aware translation model inspired by deliberation network. However, while they consider a symmetric data scenario where all available training data has document-level context, and train all components jointly on this data, we focus on an asymmetric scenario where we have a large amount of sentence-level data, used to train our first-pass model, and a smaller amount of document-level data, used to train our secondpass decoder, keeping the first-pass model fixed.
Automatic evaluation of the discourse phenomena we consider is challenging. For lexical cohesion, Wong and Kit (2012) count the ratio between the number of repeated and lexically similar content words over the total number of content words in a target document. However, Guillou (2013); Carpuat and Simard (2012) find that translations generated by a machine translation system tend to be similarly or more lexically consistent, as measured by a similar metric, than human ones. This even holds for sentence-level systems, where the increased consistency is not due to improved co-hesion, but accidental - Ott et al. (2018) show that beam search introduces a bias towards frequent words, which could be one factor explaining this finding. This means that a higher repetition rate does not mean that a translation system is in fact more cohesive, and we find that even our baseline is more repetitive than the human reference.
Conclusions
We analyze which phenomena cause otherwise good context-agnostic translations to be inconsistent when placed in the context of each other. Our human study on an English-Russian dataset identifies deixis, ellipsis and lexical cohesion as three main sources of inconsistency. We create test sets focusing specifically on the identified phenomena.
We consider a novel and realistic set-up where a much larger amount of sentence-level data is available compared to that aligned at the document level and introduce a model suitable for this scenario. We show that our model effectively handles contextual phenomena without sacrificing general quality as measured with BLEU despite using only a small amount of document-level data, while a naive approach to combining sentence-level and document-level data leads to a drop in performance. We show that the proposed test sets allow us to distinguish models (even though identical in BLEU) in terms of their consistency. To build context-aware machine translation systems, such targeted test sets should prove useful, for validation, early stopping and for model selection.
A Protocols for test sets
In this section we describe the process of constructing the test suites.
A.1 Deixis
English second person pronoun "you" may have three different interpretations important when translating into Russian: the second person singular informal (T form), the second person singular formal (V form) and second person plural (there is no T-V distinction for the plural from of second person pronouns). Morphological forms for second person singular (V form) and second person plural pronoun are the same, that is why to automatically identify examples in the second person polite form, we look for morphological forms corresponding to second person plural pronouns.
To derive morphological tags for Russian, we use publicly available pymorphy2 6 (Korobov, 2015).
Below, all the steps performed to obtain the test suite are described in detail.
A.1.1 Automatic identification of politeness
For each sentence we try to automatically find indications of using T or V form. Presence of the following words and morphological forms are used as indication of usage of T/V forms:
1. second person singular or plural pronoun, 2. verb in a form corresponding to second person singular/plural pronoun, 3. verbs in imperative form, 4. possessive forms of second person pronouns.
For 1-3 we used morphological tags predicted by pymorphy2, for 4th we used hand-crafted lists of forms of second person pronouns, because pymorphy2 fails to identify them. The first rule is needed as morphological forms for second person plural and second person singular V form pronouns and related verbs are the same, and there is no simple and reliable way to distinguish these two automatically. The second rule is to exclude cases where there is only one appropriate level of politeness according to the relation between the speaker and the listener. Such markers include "Mr.", "Mrs.", "officer", "your honour" and "sir". For the impolite form, these include terms denoting family relationship ("mom", "dad"), terms of endearment ("honey", "sweetie") and words like "dude" and "pal".
A.1.3 Automatic change of politeness
To construct contrastive examples aiming to test the ability of a system to produce translations with consistent level of politeness, we have to produce an alternative translation by switching the formality of the reference translation. First, we do it automatically:
1. change the grammatical number of second person pronouns, verbs, imperative verbs, 2. change the grammatical number of possessive pronouns.
For the first transformation we use pymorphy2, for the second use manual lists of possessive second person pronouns, because pymorphy2 can not change them automatically.
A.1.4 Human postprocessing of automatic change of politeness
We manually correct the translations from the previous step. Mistakes of the described automatic change of politeness happen because of:
1. ambiguity arising when imperative and indicative verb forms are the same,
Figure 1 :
1Examples of violation of (a) T-V form consistency, (b) speaker gender consistency. In color: (a) red -V-form, blue -T-form; (b) redfeminine, blue -masculine.
Figure 2 :
2Examples of discrepancies caused by ellipsis. (a) wrong morphological form, incorrectly marking the noun phrase as a subject. (b) correct meaning is "see", but MT produces хотели khoteli ("want").
RU Не для Джулии. Юлия умеет дразнить своих жертв. RU Ne dlya Dzhulii. Yuliya umeyet draznit' svoikh zhertv. (b) EN But that's not what I'm talking about. I'm talking about your future. RU Но я говорю не об этом. Речь о твоём будущем. RU No ya govoryu ne ob etom. Rech' o tvoyom budushchem.
Figure 3 :
3Examples of lack of lexical cohesion in MT. (a) Name translation inconsistency. (b) Inconsistent translation. Using either of the highlighted translations consistently would be good.
Figure 4 :
4Model architecture
Figure 5 :
5BLEU and lexical cohesion accuracy on the development sets during CADec training.
6
https://github.com/kmike/pymorphy2 person plural form corresponds to plural pronoun, not V form, 2. there is a clear indication of politeness.
Table 1 :
1Human annotation statistics of pairs of consecutive translation. equacy of the translations without context and in the context of each other. The whole process is two-stage:1. sentence-level evaluation: we ask if the translation of a given sentence is good, 2. evaluation in context: for pairs of consecutive good translations according to the first stage, we ask if the translations are good in context of each other.2018) for English and Russian. We train a context-
agnostic Transformer on 6m sentence pairs. Then
we translate 2000 pairs of consecutive sentences
using this model. For more details on model train-
ing and data preprocessing, see Section 5.3.
Then we use human annotation to assess the ad-
Table 2 :
2Types of phenomena causing discrepancy in
context-agnostic translation of consecutive sentences
when placed in the context of each other
type of discrepancy
frequency
T-V distinction
67%
speaker/addressee gender:
same speaker
22%
different speaker
9%
other
2%
Table 3 :
3Types of discrepancy in context-agnostic translation caused by deixis (excluding anaphora)
Table 4 :
4Types of discrepancy in context-agnostic
translation caused by ellipsis
(a) EN You call her your friend but have you been to
her home ? Her work ?
RU Ты называешь её своей подругой, но ты
был у неё дома? Её работа?
RU Ty nazyvayesh' yeyo svoyey podrugoy, no ty
byl u neye doma? Yeyo rabota?
(b) EN Veronica, thank you, but you saw what hap-
pened. We all did.
RU Вероника, спасибо, но ты видела, что
произошло. Мы все хотели.
RU Veronika, spasibo, no ty videla, chto
proizoshlo. My vse khoteli.
Table 5 :
5Size of test sets: total number of test instances and with regard to the latest context sentence with politeness indication or with the named entity under consideration. For ellipsis, we distinguish whether model has to predict correct noun phrase inflection, or correct verb sense (VP ellipsis).
Table 6 :
6BLEU scores. CADec trained with p = 0.5.
Scores for CADec are not statistically different from
the baseline (6m).
Table 7 :
7Accuracy for deixis and lexical cohesion.ellipsis (infl.) ellipsis (VP)
baseline
53.0
28.4
concat
76.2
76.6
s-hier-to-2.tied
66.4
65.6
CADec
72.2
80.0
Table 8 :
8Accuracy on ellipsis test set.
Table 9 :
9Results for different probabilities of using cor-
rupted reference at training time. BLEU for 3 context
sentences. For ellipsis, we show inflection/VP scores.
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We use the term 'inconsistency' to refer to any violations causing good translations of isolated sentences not to work together, independently of which linguistic phenomena (e.g., ellipsis or lexical cohesion) impose the violated constraints.
Exceptions include(Yamamoto and Sumita, 1998), and work on the related phenomenon of pronoun dropping(Russo et al., 2012;Wang et al., 2016;Rios and Tuggener, 2017).
We use bootstrap resampling(Koehn, 2004) for significance testing.
https://github.com/moses-smt/ mosesdecoder/tree/master/scripts/generic 9 This can be reached by using several of GPUs or by accumulating the gradients for several batches and then making an update.
AcknowledgmentsWe would like to thank the anonymous reviewers for their comments and Ekaterina Enikeeva for the help with initial phenomena classification. The authors also thank Yandex Machine Translation team for helpful discussions and inspiration. Ivan Titov acknowledges support of the European Research Council (ERC StG BroadSem 678254) and the Dutch National Science Foundation (NWO VIDI 639.022.518). Rico Sennrich acknowledges support from the Swiss National Science Foundation (105212_169888), the European Union's Horizon 2020 research and innovation programme (grant agreement no 825460), and the Royal Society (NAF\R1\180122).After the four previous steps, we have text fragments of several consecutive sentences with consistent level of politeness. Each fragment uses second person singular pronouns, either T form or V form, without nominal markers indicating which of the forms is the only one appropriate. For each group we have both the original version, and the version with the switched formality.To control for appropriateness of both levels of politeness in the context of a whole text fragment we conduct a human annotation. Namely, humans are given both versions of the same text fragment corresponding to different levels of politeness, and asked if these versions are natural. The answers they can pick are the following: The annotators are not given any specific guidelines, and asked to answer according to their intuition as a native speaker of the language (Russian).There are a small number of examples where one of the versions is not appropriate and not equally natural as the other one: 4%. Cases where annotators claimed both versions to be bad come from mistakes in target translations: OpenSubtitles data is not perfect, and target sides contain translations which are not reasonable sentences in Russian. These account for 1.5% of all examples. We do not include these 5.5% of examples in the resulting test sets.A.2 Lexical cohesionThe process of creating the lexical cohesion test set consists of several stages:A.2.1 Identification of examples with consistent translationsWe look for infrequent words that are translated consistently in a text fragment. Since the target language has rich morphology, to verify that translations are the same we have to use lemmas of the translations. More precisely, we 1. train Berkeley aligner on about 6.5m sentence pairs from both training and held-out data, 2. find lemmas of all words in the reference translations in the held-out data using pymorphy2,3. find words in the source which are not in the 5000 most frequent words in our vocabulary whose translations have the same lemma.A.2.2 Finding alternative translationsFor the words under consideration, we find alternative translations which would be (i) equally appropriate in the context of the remaining sentence and text fragment (ii) possible for the model to produce. To address the first point, we focus on named entities, and we assume that all translations of a given named entity seen in the training data are appropriate. To address the second point, we choose alternative translations from the reference translations encountered in the training data, and pick only ones with a probability at least 10%.The sequence of actions is as follows:3. group possible translations by their lemma using pymorphy2, 4. if a lemma has a probability at least 10%, we consider this lemma as possible translation for the word under consideration, 5. leave only examples with the word under consideration having several alternative translations.After that, more than 90% of examples are translations of named entities (incl. names of geographical objects). We manually filter the examples with named entities.A.2.3 Constructing a test setFrom the two previous steps, we have examples with named entities in context and source sentences and several alternative translations for each named entity. Then we 1. construct alternative translations of each example by switching the translation of instances of the named entity; since the target language has rich morphology, we do it manually, 2. for each example, construct several test instances. For each version of the translation of a named entity, we use this translation in the context, and vary the translation of the entity in the current sentence to create one consistent, and one or more inconsistent (contrastive) translation.B Experimental setup B.1 Data preprocessingWe use the publicly available OpenSubtitles2018 corpus(Lison et al., 2018)for English and Russian.7We pick sentence pairs with a relative time overlap of subtitle frames between source and target language subtitles of at least 0.9 to reduce noise in the data. As context, we take the previous sentence if its timestamp differs from the current one by no more than 7 seconds. Each long group of consecutive sentences is split into fragments of 4 sentences, with the first 3 sentences treated as context. More precisely, from a group of consecutive sentences s 1 , s 2 , . . . , s n we get (s 1 , . . . , s 4 ), (s 2 , . . . , s 5 ), . . . , (s n−3 , s n ). For CADec we also 7 http://opus.nlpl.eu/ OpenSubtitles2018.php include (s 1 , s 2 ) and (s 1 , s 2 , s 3 ) as training examples. We do not add these two groups with less context for the concatenation model, because in preliminary experiments, this performed worse both in terms of BLEU and consistency as measured on our test sets.We use the tokenization provided by the corpus and use multi-bleu.perl 8 on lowercased data to compute BLEU score. We use beam search with a beam of 4 for both base model and CADec.Sentences were encoded using byte-pair encoding(Sennrich et al., 2016), with source and target vocabularies of about 32000 tokens. Translation pairs were batched together by approximate sequence length. For the Transformer models (baselines and concatenation) each training batch contained a set of translation pairs containing approximately 16000 9 source tokens. It has been shown that Transformer's performance depends heavily on the batch size(Popel and Bojar, 2018), and we chose a large batch size to ensure that models show their best performance. For CADec, we use a batch size that contains approximately the same number of translation instances as the baseline models.B.2 Model parametersWe follow the setup of Transformer base model(Vaswani et al., 2017). More precisely, the number of layers in the base encoder, base decoder and CADed is N = 6. We employ h = 8 parallel attention layers, or heads. The dimensionality of input and output is d model = 512, and the innerlayer of a feed-forward networks has dimensionality d f f = 2048.We use regularization as described in(Vaswani et al., 2017).B.3 OptimizerThe optimizer we use is the same as in(Vaswani et al., 2017). 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. Jiacheng Zhang, Huanbo Luan, Maosong Sun, Feifei Zhai, Jingfang Xu, Min Zhang, Yang Liu, inability of pymorphy2 to inflect the singular number to some verb forms (e.g., to inflect singular number to past tense verbsJiacheng Zhang, Huanbo Luan, Maosong Sun, Feifei Zhai, Jingfang Xu, Min Zhang, and Yang Liu. 2018. 2. inability of pymorphy2 to inflect the singu- lar number to some verb forms (e.g., to inflect singular number to past tense verbs),
ambiguity arising when a plural form of a pronoun may have different singular forms. which have to agree with the pronoun, 4.presence of related adjectives, which have to agree with the pronoun, 4. ambiguity arising when a plural form of a pronoun may have different singular forms.
Human annotation: are both polite and impolite versions appropriate? 1. train Moses on the training data (6m sentence pairs), 2. for each word under consideration (from A.2.1), get possible translations from the lexical table of Moses, We use warmup_steps = 16000, scale = 4 for the models trained on 6m data (baseline (6m) and concatenation) and scale = 1 for the models trained on 1. A.1.5. 5m data (baseline (1.5m) and CADecA.1.5 Human annotation: are both polite and impolite versions appropriate? 1. train Moses on the training data (6m sentence pairs), 2. for each word under consideration (from A.2.1), get possible translations from the lex- ical table of Moses, We use warmup_steps = 16000, scale = 4 for the models trained on 6m data (baseline (6m) and concatenation) and scale = 1 for the mod- els trained on 1.5m data (baseline (1.5m) and CADec).
| [
"https://github.com/lena-voita/",
"https://github.com/kmike/pymorphy2",
"https://github.com/moses-smt/"
]
|
[
"Predictive Systems Toxicology",
"Predictive Systems Toxicology"
]
| [
"Narsis A Kiani ",
"Ming-Mei Shang ",
"Hector Zenil ",
"Jesper Tegner ",
"\nUnit of Computational Medicine\nCenter for Molecular Medicine\nDepartment of Medicine\n171 76Solna, StockholmKarolinska InstitutetSweden\n",
"\nCenter for Molecular Medicine\nAlgorithmic Dynamics Lab\nKarolinska Institutet\n171 76StockholmSweden\n",
"\nBiological and Environmental Sciences and Engineering Division\nComputer, Electrical and Mathematical Sciences and Engineering Division\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalKingdom of Saudi Arabia\n",
"\nScience for Life Laboratory\n171 21SolnaSweden\n"
]
| [
"Unit of Computational Medicine\nCenter for Molecular Medicine\nDepartment of Medicine\n171 76Solna, StockholmKarolinska InstitutetSweden",
"Center for Molecular Medicine\nAlgorithmic Dynamics Lab\nKarolinska Institutet\n171 76StockholmSweden",
"Biological and Environmental Sciences and Engineering Division\nComputer, Electrical and Mathematical Sciences and Engineering Division\nKing Abdullah University of Science and Technology (KAUST)\n23955-6900ThuwalKingdom of Saudi Arabia",
"Science for Life Laboratory\n171 21SolnaSweden"
]
| []
| In this review we address to what extent computational techniques can augment our ability to predict toxicity. The first section provides a brief history of empirical observations on toxicity dating back to the dawn of Sumerian civilization. Interestingly, the concept of dose emerged very early on, leading up to the modern emphasis on kinetic properties, which in turn encodes the insight that toxicity is not solely a property of a compound but instead depends on the interaction with the host organism. The next logical step is the current conception of evaluating drugs from a personalized medicine point-of-view. We review recent work on integrating what could be referred to as classical pharmacokinetic analysis with emerging systems biology approaches incorporating multiple omics data. These systems approaches employ advanced statistical analytical data processing complemented with machine learning techniques and use both pharmacokinetic and omics data. We find that such integrated approaches not only provide improved predictions of toxicity but also enable mechanistic interpretations of the molecular mechanisms underpinning toxicity and drug resistance. We conclude the chapter by discussing some of the main challenges, such as how to balance the inherent tension between the predicitive capacity of models, which in practice amounts to constraining the number of features in the models versus allowing for rich mechanistic interpretability, i.e. equipping models with numerous molecular features. This challenge also requires patient-specific predictions on toxicity, which in turn requires proper stratification of patients as regards how they respond, with or without adverse toxic effects. In summary, the transformation of the ancient concept of dose is currently successfully operationalized using rich integrative data encoded in patient-specific models.A brief history of Toxicologyfrom Sumerian drugs to pharmacokinetic analysis of toxicityThere are numerous examples of "drug" usage in ancient times. The first documented evidence of drug receipts is believed to be approximately 5000 years old, on a Sumerian clay slab[1]. In contrast to the long history of using substances from plants for therapeutic purposes, it was only a couple of hundred years ago that people realized the hazards of these substances. This insight can be expressed as "All substances are poisons; there is none which is not a poison. The right dose differentiates a poison and a remedy"[2]. While Paracelsus (1493-1541) had this key insight, the boundary between poison and remedy is hazy. The toxicity of individual chemicals is indeed a complex feature which itself depends on several factors, such as dose, chemistry, individual genetic make-up and exposure to environmental conditions, which all play key roles, to different degrees, in determining susceptibility to disease and adverse drug responses. In modern times it has become increasingly evident that it is not the case that each medicine works equally well, as regards both efficacy and safety, in individuals in a population-hence the rationale behind the idea of personalized medicine[3]. Following the work of Paracelsus, Mathieu Orfila (1787-1853) first described specific organ damage caused by toxins. Toxicity studies of individual substances using animals began in 1920. J.W. Trevan proposed the concept of a 50% lethal dose (LD50), defining the lethal dose of individual chemicals. As a new subject, the field of toxicology slowly developed until the occurrence of the thalidomide disaster in the early 1960s, one of the gloomiest episodes in pharmaceutical history.The drug was approved as a mild sleeping pill with a good safety profile and beneficial effects on morning sickness in pregnant women. However, this caused thousands of babies worldwide to be born with malformed limbs in less than 4 years. Since then, all 3 regulatory agencies have made it obligatory to report the toxicity profiles of Investigational New Drugs (IND). In the late 1980s, the Organization for Economic Cooperation and Development (OECD) and the International Conference on Harmonization (ICH) brought out the guidelines for the toxicity testing of pharmaceutical substances, which are still in use, supplemented with occasional amendments. In the context of regulatory guidelines, the lowest dose able to induce adverse effects (LOAEL) and the highest dose without observable adverse effects (NOAEL) must be tested to extrapolate the derived no-effect level (DNEL), which is more useful in defining the appropriate dose in clinical trials. Other conventional toxicity testing includes repeated dose toxicity testing, carcinogenicity testing, one-generation reproduction toxicity testing, and two-generation reproduction toxicity testing, et al.These depend on the formulation and indication of the drug. The toxicity testing of pharmaceuticals depends strongly on different animal models. Not surprisingly, such an evaluation is expensive (reported to cost more than $3B per year), time-consuming (two-generation reproduction toxicity testing takes around 2 years), suffers from low throughput, and in some cases raises ethical concerns relating to animal welfare [3].The low throughput of toxicity testing methods has serious consequences for public health, as 86% of chemicals (not limited to drugs) currently on the market lack the necessary toxicity data[4,5]. The most controversial issue is the translational efficiency of those compounds being tested in humans[6]. No doubt, the current toxicity model is not optimal, motivating both regulatory authorities and pharmaceutical companies to promote innovative alternatives to limit the use of animals and to better assess the risk of drug candidates as early as possible. In 2003, an EPA report proposed a computational toxicology research agenda promising several advantages, including prioritizing candidates and developing predictive models for quantitative risk 4 assessment. Yet the use of computational methods to predict toxicity has a history in toxicology. In 1962, Hansch et al. developed a Quantitative Structure-Activity Relationship (QSAR) model to estimate the concentration of chemicals using the octanol/water partition and the Hammett constant, which laid the foundation for in silico toxicity prediction [7]. Numerous tools were developed to predict carcinogenicity, mutagenicity, and developmental toxicity using pre-built QSAR models such as TopKat and METEOR, most of which have been modified and are currently deployed in academia and the pharmaceutical industry [8] . QSAR models provide a wide range of complexity for toxic endpoints, given flexible feature selection, i.e. qualitative and quantitative toxicity plus molecular descriptors can be used. Yet, QSARs require a large dataset to produce robust statistics, which makes the framework less useful in applications where data is limited. Benezra [9] used structural alerts (SAs) (also called toxicophores/toxic fragments) for skin sensitization in 1982, which was more practicable and economical with the low throughput experimental technologies available at the time. SA based models flourished in toxicity prediction in almost all types of toxic endpoint [10, 11]. Several expert systems are available for toxicity prediction based on pre-built rules and SAs, e.g. HazardExpert, Oncologic Cancer Expert System (OCES), Toxtree, et al.[12][13][14]. These models are limited to producing qualitative binary output, i.e. toxic or non-toxic. Chemical similarity cluster methods take into account the structural similarity of chemicals, physiochemical features, ADME and mechanisms of action (MoA), which in turn can provide qualitative or quantitative predictions depending on the toxicity endpoint[15]. Multiple tools implement this approach, such as AMBIT, DSSTox and Toxmatch, with applications including prediction of environmental risk, reproductive toxicity, skin sensitization and so on[16][17][18]. The statistically-derived rule-based approaches mentioned above share a 5 common limitation, namely, lack of biological insights into the mechanistic basis of toxicity. Analogous to pharmacokinetics/pharmacodynamics features indicating the mutual interaction of recipient and chemicals, toxicokinetics/toxicodynamics analysis selects the toxic response related to the chemical concentration in vivo. Importantly, measurement of the internal doses rather than administered doses and key metabolites provide a more accurate relationship to the response. In addition, it is a well-developed practice to extrapolate between various administration routes, as different species use non-identical PK/PD and ADME. However, the toxicity pathway and the MoA can only be defined with expert knowledge [19][20][21]. Drug toxicity is a complex response occurring at system, tissue, cellular and molecular levels. Classic toxicity testing and prediction methods, using either animals or in silico chemicals, similarity based or PK/PD based models, simplified complexity and left the mechanistic understanding of the chemical-induced toxicity pathways out of consideration. In 2007, the NRC released the report Toxicity testing in the 21 st century: A Vision and a Strategy, in which it addressed future directions that would take complexity and toxicity pathways into account [22].From Systems Biology to Systems ToxicologyThe revolution in biomedical science in the post genome era has made it feasible to study the effects of chemicals using cells, cellular components and tissues, preferably of human origin. High-throughput assay technologies, bioinformatics and systems biology have significantly empowered scientists to decipher how molecular components, different cells or tissues cooperate to carry out normal physiological functions that are key to maintaining health [23, 24]. Three high-throughput assays developed in recent decades have provided major impetus to the field of toxicology: omics technologies, image techniques, and automated robotic platform techniques.6The platforms enable testing of huge numbers of chemicals in a high-throughput number of samples under standardized conditions. Omics technologies collect the molecular responses to a substance while image methods decode the phenotypical and functional change of cells, organs or organisms in response to exposure to a compound. Together, these three technologies allow researchers to characterize toxicity rapidly at affordable cost [25][26][27]. As an interdisciplinary field of science, bioinformatics combines computer science, statistics, mathematics, and engineering to analyze and interpret biological data, and serves as a key tool with which to decode the enormous quantum of data generated with high-throughput assays [28]. Since 2000, Systems Biology had been used widely to "understand biology at the system level" using computational and mathematical modeling of complex biological systems [29]. The emergence of systems toxicology can be characterized as the integration of classical toxicology with the quantitative analysis of large networks of molecular and functional changes occurring across multiple levels of biological organization. This is in essence a holistic approach to deciphering the impact of environmental agents (chemicals, complex mixtures, occupational exposures, physical agents, biological agents, and lifestyle factors) on complex biological systems using an engineering approach applied to toxicological research [30]. Systems toxicology is rooted in the ongoing revolution in biology and biotechnology, and is founded on the premise that morphological and functional changes in cellular, tissue, organ, and organism levels are caused by and cause changes at the omics level. One example is the Human Toxome project launched by NIH/DDD that is intended to test the strategies that combine omics data and computational models, aiming to develop a common, community accessible framework[31]. Another is Tox-21c, which focuses on toxicity pathways, mechanisms/modes of action, and adverse outcome pathways (AOP) in 8,203E-09Development_Role of IL-8 in angiogenesis 9,139E-09Immune response_IL-4 -antiapoptotic action 9,803E-09 | 10.1007/978-1-4939-7899-1_25 | [
"https://arxiv.org/pdf/1801.05058v1.pdf"
]
| 23,154,321 | 1801.05058 | e09c154c3e244a948931a9a6bec0adfd8f707365 |
Predictive Systems Toxicology
Narsis A Kiani
Ming-Mei Shang
Hector Zenil
Jesper Tegner
Unit of Computational Medicine
Center for Molecular Medicine
Department of Medicine
171 76Solna, StockholmKarolinska InstitutetSweden
Center for Molecular Medicine
Algorithmic Dynamics Lab
Karolinska Institutet
171 76StockholmSweden
Biological and Environmental Sciences and Engineering Division
Computer, Electrical and Mathematical Sciences and Engineering Division
King Abdullah University of Science and Technology (KAUST)
23955-6900ThuwalKingdom of Saudi Arabia
Science for Life Laboratory
171 21SolnaSweden
Predictive Systems Toxicology
1 *Correspondence should be addressed to narsis.kiani @ki.se and Jesper.Tegner @ki.seToxicologySystems BiologyNetwork PharmacologyAlgorithmic Complexity
In this review we address to what extent computational techniques can augment our ability to predict toxicity. The first section provides a brief history of empirical observations on toxicity dating back to the dawn of Sumerian civilization. Interestingly, the concept of dose emerged very early on, leading up to the modern emphasis on kinetic properties, which in turn encodes the insight that toxicity is not solely a property of a compound but instead depends on the interaction with the host organism. The next logical step is the current conception of evaluating drugs from a personalized medicine point-of-view. We review recent work on integrating what could be referred to as classical pharmacokinetic analysis with emerging systems biology approaches incorporating multiple omics data. These systems approaches employ advanced statistical analytical data processing complemented with machine learning techniques and use both pharmacokinetic and omics data. We find that such integrated approaches not only provide improved predictions of toxicity but also enable mechanistic interpretations of the molecular mechanisms underpinning toxicity and drug resistance. We conclude the chapter by discussing some of the main challenges, such as how to balance the inherent tension between the predicitive capacity of models, which in practice amounts to constraining the number of features in the models versus allowing for rich mechanistic interpretability, i.e. equipping models with numerous molecular features. This challenge also requires patient-specific predictions on toxicity, which in turn requires proper stratification of patients as regards how they respond, with or without adverse toxic effects. In summary, the transformation of the ancient concept of dose is currently successfully operationalized using rich integrative data encoded in patient-specific models.A brief history of Toxicologyfrom Sumerian drugs to pharmacokinetic analysis of toxicityThere are numerous examples of "drug" usage in ancient times. The first documented evidence of drug receipts is believed to be approximately 5000 years old, on a Sumerian clay slab[1]. In contrast to the long history of using substances from plants for therapeutic purposes, it was only a couple of hundred years ago that people realized the hazards of these substances. This insight can be expressed as "All substances are poisons; there is none which is not a poison. The right dose differentiates a poison and a remedy"[2]. While Paracelsus (1493-1541) had this key insight, the boundary between poison and remedy is hazy. The toxicity of individual chemicals is indeed a complex feature which itself depends on several factors, such as dose, chemistry, individual genetic make-up and exposure to environmental conditions, which all play key roles, to different degrees, in determining susceptibility to disease and adverse drug responses. In modern times it has become increasingly evident that it is not the case that each medicine works equally well, as regards both efficacy and safety, in individuals in a population-hence the rationale behind the idea of personalized medicine[3]. Following the work of Paracelsus, Mathieu Orfila (1787-1853) first described specific organ damage caused by toxins. Toxicity studies of individual substances using animals began in 1920. J.W. Trevan proposed the concept of a 50% lethal dose (LD50), defining the lethal dose of individual chemicals. As a new subject, the field of toxicology slowly developed until the occurrence of the thalidomide disaster in the early 1960s, one of the gloomiest episodes in pharmaceutical history.The drug was approved as a mild sleeping pill with a good safety profile and beneficial effects on morning sickness in pregnant women. However, this caused thousands of babies worldwide to be born with malformed limbs in less than 4 years. Since then, all 3 regulatory agencies have made it obligatory to report the toxicity profiles of Investigational New Drugs (IND). In the late 1980s, the Organization for Economic Cooperation and Development (OECD) and the International Conference on Harmonization (ICH) brought out the guidelines for the toxicity testing of pharmaceutical substances, which are still in use, supplemented with occasional amendments. In the context of regulatory guidelines, the lowest dose able to induce adverse effects (LOAEL) and the highest dose without observable adverse effects (NOAEL) must be tested to extrapolate the derived no-effect level (DNEL), which is more useful in defining the appropriate dose in clinical trials. Other conventional toxicity testing includes repeated dose toxicity testing, carcinogenicity testing, one-generation reproduction toxicity testing, and two-generation reproduction toxicity testing, et al.These depend on the formulation and indication of the drug. The toxicity testing of pharmaceuticals depends strongly on different animal models. Not surprisingly, such an evaluation is expensive (reported to cost more than $3B per year), time-consuming (two-generation reproduction toxicity testing takes around 2 years), suffers from low throughput, and in some cases raises ethical concerns relating to animal welfare [3].The low throughput of toxicity testing methods has serious consequences for public health, as 86% of chemicals (not limited to drugs) currently on the market lack the necessary toxicity data[4,5]. The most controversial issue is the translational efficiency of those compounds being tested in humans[6]. No doubt, the current toxicity model is not optimal, motivating both regulatory authorities and pharmaceutical companies to promote innovative alternatives to limit the use of animals and to better assess the risk of drug candidates as early as possible. In 2003, an EPA report proposed a computational toxicology research agenda promising several advantages, including prioritizing candidates and developing predictive models for quantitative risk 4 assessment. Yet the use of computational methods to predict toxicity has a history in toxicology. In 1962, Hansch et al. developed a Quantitative Structure-Activity Relationship (QSAR) model to estimate the concentration of chemicals using the octanol/water partition and the Hammett constant, which laid the foundation for in silico toxicity prediction [7]. Numerous tools were developed to predict carcinogenicity, mutagenicity, and developmental toxicity using pre-built QSAR models such as TopKat and METEOR, most of which have been modified and are currently deployed in academia and the pharmaceutical industry [8] . QSAR models provide a wide range of complexity for toxic endpoints, given flexible feature selection, i.e. qualitative and quantitative toxicity plus molecular descriptors can be used. Yet, QSARs require a large dataset to produce robust statistics, which makes the framework less useful in applications where data is limited. Benezra [9] used structural alerts (SAs) (also called toxicophores/toxic fragments) for skin sensitization in 1982, which was more practicable and economical with the low throughput experimental technologies available at the time. SA based models flourished in toxicity prediction in almost all types of toxic endpoint [10, 11]. Several expert systems are available for toxicity prediction based on pre-built rules and SAs, e.g. HazardExpert, Oncologic Cancer Expert System (OCES), Toxtree, et al.[12][13][14]. These models are limited to producing qualitative binary output, i.e. toxic or non-toxic. Chemical similarity cluster methods take into account the structural similarity of chemicals, physiochemical features, ADME and mechanisms of action (MoA), which in turn can provide qualitative or quantitative predictions depending on the toxicity endpoint[15]. Multiple tools implement this approach, such as AMBIT, DSSTox and Toxmatch, with applications including prediction of environmental risk, reproductive toxicity, skin sensitization and so on[16][17][18]. The statistically-derived rule-based approaches mentioned above share a 5 common limitation, namely, lack of biological insights into the mechanistic basis of toxicity. Analogous to pharmacokinetics/pharmacodynamics features indicating the mutual interaction of recipient and chemicals, toxicokinetics/toxicodynamics analysis selects the toxic response related to the chemical concentration in vivo. Importantly, measurement of the internal doses rather than administered doses and key metabolites provide a more accurate relationship to the response. In addition, it is a well-developed practice to extrapolate between various administration routes, as different species use non-identical PK/PD and ADME. However, the toxicity pathway and the MoA can only be defined with expert knowledge [19][20][21]. Drug toxicity is a complex response occurring at system, tissue, cellular and molecular levels. Classic toxicity testing and prediction methods, using either animals or in silico chemicals, similarity based or PK/PD based models, simplified complexity and left the mechanistic understanding of the chemical-induced toxicity pathways out of consideration. In 2007, the NRC released the report Toxicity testing in the 21 st century: A Vision and a Strategy, in which it addressed future directions that would take complexity and toxicity pathways into account [22].From Systems Biology to Systems ToxicologyThe revolution in biomedical science in the post genome era has made it feasible to study the effects of chemicals using cells, cellular components and tissues, preferably of human origin. High-throughput assay technologies, bioinformatics and systems biology have significantly empowered scientists to decipher how molecular components, different cells or tissues cooperate to carry out normal physiological functions that are key to maintaining health [23, 24]. Three high-throughput assays developed in recent decades have provided major impetus to the field of toxicology: omics technologies, image techniques, and automated robotic platform techniques.6The platforms enable testing of huge numbers of chemicals in a high-throughput number of samples under standardized conditions. Omics technologies collect the molecular responses to a substance while image methods decode the phenotypical and functional change of cells, organs or organisms in response to exposure to a compound. Together, these three technologies allow researchers to characterize toxicity rapidly at affordable cost [25][26][27]. As an interdisciplinary field of science, bioinformatics combines computer science, statistics, mathematics, and engineering to analyze and interpret biological data, and serves as a key tool with which to decode the enormous quantum of data generated with high-throughput assays [28]. Since 2000, Systems Biology had been used widely to "understand biology at the system level" using computational and mathematical modeling of complex biological systems [29]. The emergence of systems toxicology can be characterized as the integration of classical toxicology with the quantitative analysis of large networks of molecular and functional changes occurring across multiple levels of biological organization. This is in essence a holistic approach to deciphering the impact of environmental agents (chemicals, complex mixtures, occupational exposures, physical agents, biological agents, and lifestyle factors) on complex biological systems using an engineering approach applied to toxicological research [30]. Systems toxicology is rooted in the ongoing revolution in biology and biotechnology, and is founded on the premise that morphological and functional changes in cellular, tissue, organ, and organism levels are caused by and cause changes at the omics level. One example is the Human Toxome project launched by NIH/DDD that is intended to test the strategies that combine omics data and computational models, aiming to develop a common, community accessible framework[31]. Another is Tox-21c, which focuses on toxicity pathways, mechanisms/modes of action, and adverse outcome pathways (AOP) in 8,203E-09Development_Role of IL-8 in angiogenesis 9,139E-09Immune response_IL-4 -antiapoptotic action 9,803E-09
7
humans. Tox-21c largely overlaps with 3Rs (replace, reduce, and refine) proposed half a century ago [32,33]. The Systems Toxicology computational challenge, sbv IMPROVER computational challenge, used crowd resourcing to demonstrate that gene expression data from blood cells are sufficiently informative to predict response to smoking in humans and across species translation [34].
Yet, a comprehensive understanding of the mechanisms of drug toxicity in specific cases requires the integration of different data modalities, from changes at the genomic, proteomic, and metabolomics level across several scales of cellular organization. In contrast to classical approaches, systems toxicology resides at the intersection of systems biology and toxicology where chemistry incorporates mechanisms into the predictive framework [35]. To understand how this complex interaction system in cells and tissues leads to toxicity requires the integration of two disciplines that have been increasingly useful in biomedical research: "Systems Biology" and "Quantitative Pharmacology". In systems biology, a system is generally described as a set of nodes (vertices) connected by edges describing functional interactions. These edges can represent physical interactions, functional interactions, and connections between data across several scales. Similarly, in systems toxicology biological networks are the basis for the prediction of drug action in complex biological systems [36].
Systems toxicology models contain expressions that characterize functional interactions within a biological network, which are very useful when drugs act at multiple targets in the network or when homeostatic feedback mechanisms are operative [37]. Therefore, these models are particularly useful in describing complex patterns of drug action such as synergies between different drugs. Although systems toxicology is still in its infancy, it has tremendous potential to change the way we approach biomedical research. It represents a movement beyond a traditional studycentric approach towards a continuous quantitative integration of data across studies and the different phases of drug development. Network-based approaches offer a wide range of possibilities for deciphering and possibly for understanding the complexity of human disease, thereby providing new tools with which to develop novel drugs. Here we review some current efforts and recent methods through the lens of quantitative systems pharmacology (QSP).
Examples of Predictive Systems Toxicology
The general notion of a network-based approach rests upon the ambition to connect several entities across the molecular, cellular pathways, organs and systems to facilitate the prediction of the effect of a drug candidate or any kind of perturbation on biological outcomes of interest [38,39]. The way in which one defines or infers a network from data is the main determining factor of the degree of reliability and applicability of network analysis in drug design. It is crucial to have a clear definition of network nodes early on, edges and edge weights in the specific application case, and in that context to consider data quality and refinements of the data based on genetic variability, aging, environmental effects. Different types of networks such as networks of chemical compounds, signaling networks, gene-gene interaction networks, proteinprotein interaction (PPI) networks or metabolic networks and disease networks can be (and have been) used in QSP models and methods [40] . Following the work on inferring a network comes the analysis of the network and its properties. In the last step, the result of analysis needs to be converted to a series of actionable hypotheses, which then need to be tested and validated or refuted (see Fig1).
Drug-target interaction is the first and most common type of network analysis that has been used in QSP models. Interactions between drugs and targets can facilitate the process of drug discovery by deciphering a drug's mechanism of action, thereby assisting researchers seeking new targets for an old (FDA approved) drug as well as new drug candidates for a known target [41][42][43][44][45]. The main source of information in reconstruction of the Drug-Target interaction network (DTN) is the Drug Bank, which is one of the major publicly available integrated sources of drugs and targets. It is a highly comprehensive database combining chemical properties and detailed clinical information about drugs and their targets. It also provides drug-related data feeds for well-known databases such as Uniprot, PubChem, PDB and KEGG [46,47].
In spite of the fact that mining drug-target interaction data is increasing at an amazing rate [57]. Another challenge when performing docking simulation is that it is computationally expensive and most of the methods must simplify the problem to make the computation feasible. The reduction of conformational space by imposing limitations on the system, such as fixed bond angles and lengths in the ligand or a simplified scoring function such as those based on empirical free energies of binding to score poses quickly at each step of the conformation search, are the most common short-cuts that are currently used in the field [52,58].
In a more recent effort, machine-learning approaches have been used for larger-scale predictions of drug-target interactions. The new interactions between drugs and targets can lead to potential insights on previously unidentified side effects for a particular drug. This idea is the basis of another category of systems toxicology methods. Machine learning-based methods mostly use structural and chemical descriptors of drugs and sequences of targets, similarity matrix or (and) any other pharmacological information about drugs as input. Then they use any machine learning method, such as support vector machines (SVMs) or kernel regression, for predicting the drug-target interactions [59][60][61][62][63]. Cobanoglu et al. used the known interactions in the Drug Bank in the form of a bipartite network to train a model that represents each drug and target as a vector of latent variables and assigns weights to drug-target interactions using probabilistic matrix factorization [64]. Approaches that use similarity scores as input are more promising than other approaches [41].
In general, the use of machine-learning algorithms is one of most promising approaches to extracting knowledge from big data using a data-driven framework.
However, the performance of machine-learning algorithms relies heavily on data representations called features, and identifying which features are more appropriate for the given task is very difficult. Deep Learning has recently emerged as a promising technique where the features do not need to be hand-crafted a priori. Recent success has been accomplished thanks to the availability of fast computations, massive (labeled) datasets and sophisticated algorithms [65]. Machine learning using deep learning is defined by neural networks with multiple hidden layers. Each layer basically constructs a feature from the preceding layers [66]. The training process allows layers deeper in the network to contribute to the refinement of earlier layers. For this reason, these algorithms can automatically engineer or discover features that are suitable for representing the data at hand. When sufficient data are available, these methods construct features attuned to a specific problem and combine those features into a predictor [67]. Deep learning algorithms have shown promise in fields as diverse as high-energy physics [68] , dermatology [69], and translation [70]. DEEPtox is one of the first methods using Deep Learning for computational toxicity prediction [65]. DeepTox normalizes the chemical representations of the compounds and computes a large number of chemical descriptors that are used as input in machine learning methods.
As a next step, DeepTox trains several models, evaluates them, and combines the best of them into ensembles. Finally, DeepTox predicts the toxicity of new compounds.
In DEEPTox SVMs, random forests, and elastic nets are used for cross-checking, supplementing the Deep Learning models, and for ensemble learning to complement Deep Neural Networks (DNNs). The networks consist of multiple layers of rectified linear units (ReLUs) to enforce sparse representations and counteract the appearance of a vanishing gradient. ReLUs are followed by a final layer of sigmoid output units, one for each task. One output unit is used for single-task learning. Stochastic gradient descent learning has been used to train the DNNs, and both dropout and L2 weight decay were implemented for the DNNs in the DeepTox pipeline for regularizing learning and avoiding overfitting. Of note is the fact that DEEPtox outperformed many other computational approaches like naive Bayes, support vector machines, and random forests in toxicity prediction of 12,000 environmental chemicals.
The output of all the above-mentioned methods is a DTN, an undirected bipartite network composed of two sets of nodes, drugs and targets. DTN have a complex topology that reflects the inherently rich polypharmacology of drugs (also known as drug repurposing) [51]. The analysis of DTN has recently emerged as an effective means to study targets and to identify new targets for known drugs. In one of the very first attempts, Ma'ayan et al. [71] reconstructed such a bipartite network, and the nodes have been connected if there is an association between a drug and a target on the basis of data from the Drug Bank. They report several classes of proteins as better targets for drugs based on network statistics and gene ontology. A decade later, Lin et al. [72] have followed the same approach to studying the drug-target interaction and could characterize the drug-target relations of different kinds of drugs. They showed that the number of multi-target new molecular entities (NME) has increased over the years, but less than single-target NMEs. In both these cases and several other cases in the literature, it has proven useful to analyze the general structure of a network in order to extract new knowledge facilitating the classification of drugs and/or their targets. Structural (graphical) analysis of a network provides insights into the organization and topology of the DTN and targets for hypothesis generation and experimental testing. As a rule this is performed through computation and analysis of network parameters-parameters that quantify different aspects of the network's internal structure, such as parameters measuring centrality, a node or more global parameters such as modularity index, network density, network entropy or network diameter [73]. Several methods have been developed and applied based on network topology, graph theory, and cluster analysis (see [8] for a recent review). Methods based on the similarity of networks is another set of techniques that have been used to uncover novel target or disease-specific changes [74,75]. A wide range of similarity measures have been used in the literature, ranging from intuitive measures such as the number of edge changes required to get one network from another or the comparison of the top-k nodes to the more complicated ones, such as using an ensemble of different model networks, and the distribution of the best-fitting ensemble.
However, it should be kept in mind that the fundamental question of checking whether two given networks have the same structure, network comparison, is computationally expensive, and despite extensive progress in the field, it remains one of the greatest challenges in the field. For example, it is still not known whether graph isomorphism is polynomial solvable or whether it is NP-complete. Therefore most of the current methods in the network comparison field are heuristic, which in turn may affect the outcome strongly, depending on which kind of prior biases exist in the particular method.
All interactions, from protein-protein interactions (PPI) to gene expression and pathways, are useful in the quest to understand the mechanism(s) of interaction between drugs and complex diseases. Remez et al. used predicted drug−protein interactions obtained with a CT-link in combination gene expression data to obtain a projected anatomical profile of a drug and use it for connecting in vitro assays with in vivo outcomes and predict potential in in vivo organ toxicities [76,77].
Kuhn et al. used a network based on drug-target interaction data and drug-ADR interaction data to systematically predict and characterize proteins that cause drug side effects. They integrated phenotypic data obtained during clinical trials with known drug-target relations to identify overrepresented protein-side effect combinations [78]. Similarly, several other approaches have been developed based on the notion of expanded drug-target interactions, combined with protein-protein interactions data, in order to develop a network-based pharmacology that could better explain the drugphenotype relationship, and this approach has been used to predict novel targets and drug repositioning [80][81][82][83][84][85]. For example, Guney et al. in [86]integrated protein-protein interaction, drug-disease association and drug-target association data. They analyzed the topological characteristics of drug targets with respect to disease proteins and showed that for a drug to be effective against a disease, it had to target proteins within or in the immediate vicinity of the corresponding disease module. Such approaches were also considered for issues related to drug safety and side effects. Cami et al.
constructed a network representation of drug-ADR associations for approximately 800 drugs and ADRs and pharmacological information for toxicity prediction. They exploited network structure to predict likely unknown adverse events using a trained logistic regression model [87]. Berger et al. used PPI networks to predict and identify drugs that likely cause Long QT Syndrome based on both a direct drug-target interaction and separate neighborhood [88] .
Complementary to protein-protein interactions, transcriptomic data and gene expression differentiation have been used in drug discovery and safety [88][89][90][91][92][93]. between drug targets to predict the pharmacodynamics of drug-drug interactions [92,93].
For the purpose of predicting drug toxicity, in most cases we require a collection of experimental data reflecting molecular changes in the context of quantifiable cellular changes across different biological scales that are linked to toxicity at the body level [35]. So in addition to all the above-mentioned data, systems toxicology depends strongly on the quality and scope of databases annotating side effects (SIDER) and drug-induced differential gene expression, or a combination thereof [94][95][96][97]. Interestingly these methods have also been used to predict drug synergies. However, most of them are limited to estimating target links on the PPI network. The advantage of using a network-based approach lies in that it helps to explain the hidden molecular mechanism of drug synergy from the interactions. Due to their effectiveness, some approaches aiming at identifying synergistic drug combinations are based on the dynamic simulation of specific subnetworks. However, these models relied on a very Using this score, we could rank all possible combinations in a reasonable amount of time. Interestingly, we learned that we should not include too many details (i.e. features or molecular components) in our network descriptions, since we may shift our description from optimal towards the 'knowledge of everything,' with the precision of the method dropping drastically as a result. This underscores the importance and challenge of pruning a large, but for the given application reasonable number of features to include in the network model.
Fig. 2 Overview of HotPPI approach
Information theoretic approach to toxicity
Both network analysis and pharmacokinetic analysis share a focus and grounding in the physical and functional interactions between molecules within the cell or tissue and the corresponding drugs. Here the overarching aim is not only to predict but also to be able to interpret the mechanism in terms of the underlying biology and chemistry. Since It is not difficult to see that complementary regions between drug and target will have a similar classical and algorithmic information content, because the structure of one is the complement of the other. Another advantage is that these measures are parameter-free and thus require no training, even though they can complement and guide machine learning approaches [101,102]. Because drug docking is not invariant to, e.g., scaling factors, but information theoretic measures are, they may fail to characterize the positive or negative docking properties of a drug. While coarsegraining techniques may be introduced, algorithmic complexity has the advantage of being able to account for scaling effects. The basic idea is the likelihood of a drug being causally generated by a mechanistic model (an algorithm). This is, in general, hard if not impossible to find (the problem is uncomputable), but approximations are possible and new numerical methods have been advanced complementary to statistical and lossless compression approaches that cannot or are very limited at accounting for causation. Drugs, and molecules in general, can be represented in many ways (see the less complex (the shorter the length of the algorithm generating it) the closer to blue, the longer (more algorithmic-random) the closer to red.
Concluding remarks
Here we have reviewed different attempts to predict toxicity from observations (i.e. the Sumerian) to classical pharmacokinetic, advancing to recent integrative systems oriented approaches taking more data into account. These systems approaches resort to performing advanced statistical analytical data processing complemented with machine learning techniques to generate paradigms attempting not only to predict toxicity but also to identify (molecular) mechanisms of toxicity. Information theoretic approaches can be situated in between, as they are as a rule less dependent upon biochemical representations in their problem formulation, while the ones presented here also aim for causal understanding of toxicity in addition to targeting prediction.
In a broader perspective, there are several immediate challenges where we need more work. These include which features to include when predicting toxicity? Minimal models may suffer from being less understandable from a mechanistic standpoint, whereas including too many features, as in the dream example above, could hamper the prediction capability of the model. Overall, a systems biology approach extends the feature space compared to classical pharmacokinetics, while an (algorithmic) information approach facilitates predictions in combination, being both scale invariant and parameter free. Hence there is a tension between predicitive capacity and mechanistic interpretability.
Furthermore, overtraining and overfitting in solving high-dimensional and complex nonlinear problems such as toxicity prediction is one of the most common problems of existing machine learning methods. This originates from the need for estimating and optimizing numerous hyper parameters. However, a method such as the relevance vector machine method solves this problem by incorporating Bayesian criteria into the learning process to reduce the irrelevant support vectors of the decision boundary in feature space, thus resulting in a sparser model [103]. Methods such as Random Forest classifiers are another category of successful methods in systems toxicology. They are one of the most robust algorithms and are able to identify the patterns important for the preferred class, even when there is a large imbalance in the class distribution within the training dataset [104]. Inspecting the results of the TOX21 data challenge demonstrates that a hybrid strategy which combines similarity scores for structural fingerprints and molecular descriptors (features) and machine-learning based prediction models can readily improve the accuracies of toxicity prediction [105]. In general, an ensemble model can be effective, since taking into account the prediction of other models can compensate for an incorrect prediction on the part of one of the individual methods. Certainly, each of the systems toxicology methods has intrinsic advantages, limitations, and practical constraints. Moreover, the performance of these methods depends on the structural diversity and representativeness of the molecules in the data set. Therefore, it is quite important to choose the most suitable machine learning method to develop the prediction model for a specific toxicity data set. Finally, the computational cost associated with each method is another practical and important factor determining the usability of a given method.
In conclusion, beyond the above challenges and considerations, the grand remaining challenge is to advance the state-of-the-art towards personalized medicine. This requires patient specific predictions on toxicity, which in turn requires proper stratification of patients with regard to how they respond or not, with or without adverse toxic effects. This most likely requires integration of multiple layers of information as a background upon which an individual has to be characterized/described, while a machinery for toxicity prediction has to be specific enough for a given patient, given the amount of (sparse) patient-specific information. This challenge and perspective will keep the field of data-driven computational toxicology busy.
Their networks have three types of nodes: drugs, targets, and side effects, and links are identified side effect causality predictors. The authors considered overrepresented protein-side effect pairs, and hypothesized that such overrepresentation could be indicative of causality. Their approach can make predictions for proteins that are the targets of a certain number of drugs. In this context, Yildirim et al. used a bipartite graph composed of FDA-approved drugs and target proteins in the context of cellular and disease networks and quantitatively demonstrated an overabundance of 'followon' drugs[79]. The authors overlaid the drug-protein network with a network of physical PPI. They demonstrated a significant increase in the number of interacting proteins as compared to the average in the PPI network. They used the distance between drugs and a drug target and the corresponding disease to show that most drug targets are not closer to the disease genes in the protein interaction network than a randomly selected group of proteins.
For example, Gottlieb et al. introduced a method for inferring drug-specific pathways [89]. They connect known drug associated genes over protein, metabolic and transcriptional interaction networks while preferring high confidence interactions participating in curated cellular processes. They use their computed pathways to suggest novel drug repositioning opportunities, gene-side effect associations, and gene-drug interactions. Huang et al. developed a new metric to measure the strength of network connection
As an example, Lounkine et al. developed an association metric asking how to prioritize those new off-targets that explained side effects better than any known target of a given drug, thereby creating a drug-target-adverse drug reaction network[43].Network-based approaches allow the generation of hypotheses about drug-targetphenotype-side effect associations but currently available interaction data are incomplete and the available parts are often non-homogeneous and biased. Thissituation results in the fact that the conclusions of such studies strongly depend not only on the quality, but importantly, on the degree of completeness of the data [98]. The other relevant point is that most of the suggested approaches in QSP are largely based upon the analysis of the structure of a network or on comparison of networks, while it has been shown[99] that network dynamics, the study of temporal changes in network structures or describing changes of phenotypes of a complex system in the state-space, is crucial to understanding the complexity of diseases and the action of drugs[39]. In this context Mucha et al. [100] developed the technique of multilayer networks, incorporating different types of nodes and edges, in order to follow the changes in module structure in a system having multiple and different types of edges.
detailed dynamical model, where the lack of information and the uncertainties involved in their kinetics parameters and lots of artificial constraints often limit the usefulness of the simulations, resulting in the model working only for a few specific pathways. Kiani et al. developed a novel integrative pipeline for systematic exploration of drug combinations as a comprehensive and flexible network-based model in the context of the DREAM challenge, a pipeline called HotPPI. Here they constructed a human protein interaction network from major PPI resources, and included both experimentally validated and computationally predicted interactions. The overall procedure resulted in a vast protein interaction network comprising 15,383 proteins and 337,413 interactions. Next, PPI was filtered based on targets of the DREAM challenge and the top 50 pathways involving these targets (table 1). The filtered PPI network comprises 6000 proteins and 16000 interactions. Molecular data are used to weigh interaction in our PPI. The main goal of HOTPPI is to find the best combination to eliminate cancer cell lines. Therefore any combination that eliminates most interactions in a network can cause network collapse followed by death of cancer cells. Thus the heat diffusion algorithm is used to predict potential synergistic drug combinations by calculating how efficient drugs are in hitting the top 200 selected nodes in a network based on their betweenness score. The Hot PPI is generally applicable to high-throughput experimental data where the challenge is to select a small number of the most promising combinations for further mechanistic studies.
the design of new drugs for new targets is difficult, and the prediction problem is easier from an inference point-of-view, compared to elucidating the mechanisms driving toxicity, complementary approaches are warranted. For example, instead of engineering a drug to target the unique pathways or mutations of a tiny subset of diseases, drug repositioning, such as the one exemplified in the DREAM challenge, involves starting with approved drugs to find combinations that can be used to treat diseases different from the ones they have been designed for, with the advantage that approved drugs can bypass much regulation if correctly controlling for the effects they can have. Thus prediction and simulation are key. This means that the whole field has to move towards causal modeling and functional inference rather than traditional statistical classification (e.g. Tanimoto coefficients) or computational simulation based on classical geometric approaches (e.g. distance between molecules, grid-based docking). To this end, information indexes can facilitate the characterization of drugs by the combinatorial and structural properties shared with or at a remove from the structural properties of the targets, because just as for any molecule, structure means function. Then all these approaches can contribute to determining drug function based on the fact that structurally similar molecules usually have similar properties (known as "neighborhood behavior"). For example, statins are associated with the heart and cholesterol, while morphine, codeine and heroin share structural properties and effects. However, algorithmic information-theoretic approaches based on both classical information and computability theory introduce predictive causal models that go beyond statistical similarities and can find, in principle, similar mechanisms shared by sets of drugs with respect to targets and functions.
Fig. 3a ,
3ab,c), some of which are natural networks or networks representing properties of the molecules. Most of these representations are lossless representations, meaning that they can reconstruct the primary representation of the molecule that they encode, e.g., the simplified molecular-input line-entry system or SMILES. The SMILES of a molecule is a string obtained by printing the symbol nodes encountered in a depth-first tree traversal of a chemical graph. SMILES can be converted back (almost) uniquely to the 2-dimensional representation of a drug.
Fig. 3
3shows some of these network (a,b) and 2-dimensional representations (c), together with 2 figures (d,e) plotting 3 information-theoretic indexes, two classical and one algorithmic based on the drugs' contact networks. While the 2 classical indexes are the ones most correlated, as one is extracted from the other with the additional information of the sequence valence, the length of the algorithmic complexity (Z-axis)represents the complexity of a hypothesized model producing the contact network.
Fig. 3
3Drug profiling by (algorithmic) information indexes: (a) The molecular (chemical) graph of Atorvastatin (C33H35FN2O5), a member of the drug class known as statins used primarily as a lipid-lowering agent for prevention associated with treatment of cardiovascular diseases. (b) In a molecular network geographical coordinates and shapes are no longer important, but rather their topology (which element is connected to which other), which can be built upon (c) the molecular contact map where grey scale (left matrix) indicates proximity between each element that can be binarized (right) using a cut-off value based on the grey scale median. (d) Algorithmic information landscape of more than 4000 drugs from the DrugBank (extracted from the Wolfram Language) constructed by taking the entropy of their SMILES codes, the valence sequences of each of the elements in their formula (from SMILES), given the importance they have for bonding (e) Algorithmic information landscape of the drugs involved in the DREAM challenge. Color is determined by the 'contact map complexity';
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ACKNOWLEDGMENTSWe thank the contributors to HotPPI, Dr. Gordon Ball , Dr. David Gomez, Alireza Mazaheri and Dr. Francesco Marabita. This study was supported by the Swedish ResearchCouncil ( J.T. & H.Z. ) and Karolinska Institutet funds (N.K. ). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
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[
"Agent Components and the Emergence of Altruism in Social Interaction Networks",
"Agent Components and the Emergence of Altruism in Social Interaction Networks"
]
| [
"Fariel Shafee \nDepartment of Physics\nPrinceton University\n08540PrincetonNJUSA\n"
]
| [
"Department of Physics\nPrinceton University\n08540PrincetonNJUSA"
]
| []
| We discuss a special aspect of agents placed in a social network. If an agent can be seen as comprising many components, the expressions and interactions among these components may be crucial. We discuss the role of patterns within the environment as a mode of expression of these components. The stability and identity of an agent is derived as a function of component and component-pattern identity. The agent is then placed in a specific social network within the environment, and the enigmatic case of altruism is explained in terms of interacting component identities. | null | [
"https://export.arxiv.org/pdf/0901.3772v1.pdf"
]
| 15,300,452 | 0901.3772 | 63749efa65dc26040a44c9f9fc9fab4fc3ebcd34 |
Agent Components and the Emergence of Altruism in Social Interaction Networks
23 Jan 2009 23 January 2009
Fariel Shafee
Department of Physics
Princeton University
08540PrincetonNJUSA
Agent Components and the Emergence of Altruism in Social Interaction Networks
23 Jan 2009 23 January 2009social networksgroup psychologyspin glass dynamicsaltruismmulti-agent interactionsagent-environment interaction
We discuss a special aspect of agents placed in a social network. If an agent can be seen as comprising many components, the expressions and interactions among these components may be crucial. We discuss the role of patterns within the environment as a mode of expression of these components. The stability and identity of an agent is derived as a function of component and component-pattern identity. The agent is then placed in a specific social network within the environment, and the enigmatic case of altruism is explained in terms of interacting component identities.
Introduction
The field of complexity has recently gained a lot of attention because of its ability to explain many complicated problems in terms of the organization of simpler units. Hierarchical structures existing to form levels of organizations were proposed (see Ahl and Allen, 1996). Interactions among units within a complicated organization was also modeled in terms of biological processes, such as the function of neural networks (see Hopfield and Herz 1995), the behavior of the stock market (Black and Scholes 1973), traffic flows (e.g. Helbing and Schreckenberg 1999) etc. Social organizations have also been studied to some extent (e.g. Newman et al 2006, Reichardt andWhite 2007). In our previous work (Shafee 2008), we have proposed a model based on agent components and interactions among agent-components, and identities of agents and social groups deriving from such interactions as minima in the interaction potential landscape. In another paper, the role of perceptions and beliefs, and interesting dynamics deriving from the connections between such beliefs within a network were discussed.
In this work, we extend some concepts associated with the model in terms of components and the perception and expression of components. We then present an example of a specific case when components of an agent and components of some coherently oriented agents interact to preserve a local potential minimum in order to explain the puzzling social phenomenon of altruism.
Interaction-based Social Model: Brief Review
Previously (Shafee 2004, Shafee 2008) an agent was modeled as an array of states, that were expressed by means of weights assigned to them. The satisfaction or utility derived was expressed as the value of the potential of interactions between the agent's states ad states found in other agents and the environment.
The model was inspired by spin glasses (Edward and Anderson 1975 , Sherrington andKirkpatrick 1975) where interactions among neighboring spins contributed to the total interaction potential of the system. The system itself evolved towards minimum potential attractors.
An agent's interactions were split into three major categories to form three components of the interaction potential:
H potential = H self + H agent−agent + H env(1)
Here the respective Hamiltonian components have interaction terms of the form
H self = −J ab i s a i s b i (2) H env = −J a i hs a i .h a(3)H agent−agent = J ab ij s i a s j b(4)
The J's are the coupling constants, representing the strength of an interaction. i is the local agent, and j is a neighboring agent. a is the label assigned for a characteristic variable.
Agent variable states are denoted with s's and in the spirit of the original spin glass model, larger environment effects are expressed with h's as environment field components.
Each agent interacts to minimize his own interaction potential to maintain his aggregate identity.
In the previous paper (Shafee 2008), the basic concepts of the agent-interaction model was discussed and social structures and dynamics derived from inter-agent interactions were studied. Here the identity and weight of agent-components are analyzed un detail, and an extreme case of altruism arising from component identity within a social, and agent identity is modeled.
Matching of Components and Symbiosis
Interaction terms may show co-operative couplings among diverse components within the agent and also among matching components of different agents. The symbiotic couplings allow the states to seek one another in a highly correlated manner, while matching components among agents find favorable interactions just like spins aligned in the same direction are energetically favored in magnetism.
Components, Patterns and Propagation
In our previous work (Shafee 2008), we have argued how inter-agent interactions giving rise to fuzzy emotions and identities at various levels are based on sums of interacting components. Here, we discuss how these components depend on coupling matrices, giving rise to certain patterns. It is known that for complex systems, the organization of the system comprising similar units yield interesting behaviors (Hopfield and Herz 1995). For example, in the complex human brain almost identical neurons are connected together. All these neurons behave according to more or less the same rules. However, their topological organization result in different personalities (Peterson and Carson 2000) that are genetically fixed, although a subset of neurons can update their physical correlation in response to agent-environment interactions, producing memories local to the agent (see Lebedev et al 2005 for a study of neural adaptivity). Hence, the agent-environment interface is included in the agent's extended identity. The preferences of agents derive from the firing patterns of the assembly of neurons, which is independent of the exact neuron producing the pattern. If these preferences are taken as variable states of an agent, it is the expression of these firing patterns that interact with the environment and not the exact neuron. Again, if the gene determines the topological organizations of certain parts of the brain associated with certain behaviors or preferences, another agent with the same gene pattern and not the same gene would produce similar behavioral patterns that the agent can identify with. In the material universe although the proportion of elementary particles (e.g. the proportion of neutrons and protons) in a molecule making a gene segment is fixed, reorganizing the same number of protons and neutrons would in general produce different patterns at the macroscopic level.
At the agent-environment interface, skin cells are constantly lost. Similar but not exact replica cells are created to replace the lost ones, though the rate of regeneration of skin cells programmed into the internal mechanism of the body may not be equal to the exact number of cells lost in a one-to-one basis.
Stable and Average-Stable Components
Neurons related to memory cannot regenerate. The exact organizations of neurons in the form of weights connecting the neurons give rise to the memories. The memories themselves depend on the agent-environment interactions, which is not determined by the genetic pattern, and hence loss and regeneration of neurons would erase any memory accumulated. The extensive number of interconnected neurons are thus preserved within a delicate environment protected within organs which themselves can regenerate continuously against certain degrees of losses, so that in an aggregate they are able to maintain an "environment field" necessary to preserve the nerve cells. Hence, as in our previous work (Shafee 2008) agents were placed with respect to environment fields, neurons can as well be said to be in connection with optimal fields produced by structures in their environment. The complexity of an agent derives from being able to protect multitudes of data and states within the many precise variables of exact organizations of neural units protected by an average constant field that is produced by an average aggregate of cells. So, it is not the total number of cells or the total number of types of cells organized into separate organs that produce the complete measure of complexity.
When a perception organ is considered, or a limb that directly interacts with the environment, the current state of the organ is dependent on the external field and the internal signals. The reorganization or disturbance of the organ allows the agent to receive information, or to send information to be stored. These organs can also reorganize into specific states such that interactions with such organs can force the environmental states to reorganize or dissociate into smaller components by means of transfer of energy. When an organ as a whole is interacting with an environment, the individual subunits are also interacting with smaller subunits of the environment. For example, friction would cause smaller units of the organ to be lost.
The organization of the organ, which is a function of each subcomponent's location and inter-relation, would depend on the internal and external environment the organ is connected with, since the resultant equilibrium of any unit derives from the balancing forces acting on it, with the magnitude, direction and interaction range of each force taken into account. The stochastic nature of many degrees of freedom associated with the environment makes it most probable that the subcomponents of the local environment would reorganize, slightly changing the balance of force at a certain point as fluctuation, and a replenished part may only have approximately the same organization.
Expression Stability and Weights
The relationship between a changing memory and, hence, beliefs derived from experience, and the interaction of a genetically defined topology contributing to certain personality traits (Toga and Thompson 2005) preserved together locally as possible output behavioral patterns was recently discussed (Shafee 2008). While the intricate connections among the brain cells and the interrelatedness of the individual cell components make it impossible to make brain transplants, many other components may be substituted within a range of fuzzy matching of components in connections with the couplings with respect to the other "individualized" organs and components. For example, blood can be transfused if the specific blood group is identified, and organs like kidneys and hearts can also be transplanted given further detailed matches. The match of livers has a high probability among close genetic relatives (see Finn 2000 for a discussion of organ matching and donation). A mismatched organ causes rejection and often death. Although the notion of cellular memory remains controversial, hormones have been found to be intricately related with personalities and preferences (see e.g. Alder 1986). Injecting hormones or transplanting or severing an organ that produces certain hormones can change the personality.
As was briefly mentioned in (Shafee 2008), the human genome has been found to consist of genes that are 99.9 percent identical within the entire genetic pool (Walton 2004). However, not all segments of the gene have the same degree of expression, and some might as well be suppressed. Only a small fraction of the human genetic material is known to code for protein synthesis (see ??????????).
Small segments of genes can also be related with high degrees of visible or expressed attribute differences. Again, a gene has been shown to express certain exons at certain times leading to the same gene coding for a large family of proteins by means of alternative splicing (Brett et al 2001). Cellular and environmental conditions such as stress, pH etc have also been seen to act as triggers in alternative splicing (Stamma et al 2005). It has also been seen that the promoter for a certain gene need not be located on the same gene, hence making it necessary for another gene to exist to trigger the action of one gene (Stamma et al 2005).
Organs specialize in carrying out the instructions of specific chunks from the gene code depending on their diversification. The development into specialized cells is again brought about by the immediate environment of the cell during growth, which might as well reflect the behavioral pattern of the mother during natation that are not coded within the gene of the child, and the location of growth (see e.g. Nicolopoulou-Stamati 2007). Different types of muscle cells have also been known to produce different types of acto-mycin because of alternative splicing triggering different expressions of the same gene in different organs (Stamma et al 2005). The concentration of certain chemicals present in within the body because of the action of genes expressed in one organ can also induce alternate splicing mechanism of another gene in another organ, hence causing one organ to effect another (Stamma et al 2005).
The expression of the internally coupled organs, and at lower hierarchical level cells that make up the organs, and even at further lower level, the genes that are expressed in the behavior of each cell that are coupled, in turn depend on the consistency, degree of coupling and coordination, and hence the fitting of components within a semi-stable compatible structure. The degree of expression of a certain gene component is dependent on the degree of coupling of that component with other components by means of the couplings of the cells that express the genes, and the couplings of the organs that make the body, and while certain couplings (e.g. those of nerve cells) are expressed intricately causing large shifts of patterns due to small changes, some only produce an aggregate behavior and may have a larger degree of robustness because of internal corrective couplings, so that any error in one is offset by another component adjusting accordingly.
Hence, the mapping of the gene, which is the blueprint of the agent, is not linearly expressed in the agent, and the expression of the agent, based on its local state is a function of both the gene and the environment, with various weights. The weight of a component (Shafee 2008) is, therefore, related to the expression of the component, which may be a function of the coupling of the component to the number of subcomponents at each level, and the coupling of the coupled subcomponent with the environment's components as well. The match of two weighted components, and hence the compatibility is expressed is given by the coupling constant, which produces an interaction energy potential. And the time averaged weight of a certain variable is related to the total expression, which takes into account the frequency of the triggering of the action as well. So, the weight of a component on an agent also reflects the ability of minuscule update of the component state to change the total interaction potential within the agent, and is a measure of the priority of the component.
Propagation of Patterns and Long Term Perpetuation
The coupled components and, hence, variables within the agent, with various degrees of stability, matching and interconnected symbiosis, thus produce a semi-stable local interaction potential minimum with respect to the environment. The idea of the semi-closed nature of such identities was discussed in (Shafee 2007). The degree of stability of such identities derive from its being able to regenerate lost components in a complex manner, and its ability to interact to reduce damage. Hence, the perpetuation of a complex agent identity is dependent on its ability to express variables to diminish the effects of unfavorable changes of the interaction potential (raising of the local minimum) by triggering expressions that "block" or flip (reorganize or align to a favorable direction) the damaging variable. This propensity of maintaining a local interaction minimum, by means of strongly coupled components that can be expressed as agent states, can thus be succinctly related to the agent's tendency to perpetuate itself, and hence the patterns that lend the agent its identity.
We consider two possibilities for expression of the genetic code. One is by means of the expression of a specific small or large chunks of genetic sequences separately, and the other is by the organization of codes so that only certain codes are triggered and expressed as intricately connected sequences, while others may remain dormant depending on the organization.
During meiosis, crossover between chromosomes make chunks of interconnected gene alleles, that can produce degrees of genetically formed skill aptitudes recombine (see Griffith et al (1993) for a discussion of linked gene mapping based on recombination). Hence, specific talents and traits that are results of intricate connections of large number of subsegment patterns are often lost. In a large population, these fragmented sections may recombine, subject to rules of probability, or similar patterns may arise because of the repetition of some genetic code sequences in different segment. If a characteristic is specific to an average of a sequence, approximate characteristics can be re-expressed even if the exact same segment of sequences is not recombined.
From the point of view of perpetuation of patterns, some of these "lost" expressions can to some order approximately reappear in a large gene pool making random genetically unrelated agents share common attributes with certain frequencies.
However, small mutations can cause specific changes in the expression of a characteristic at a large scale due to the connectedness of the specific attribute with other attributes. Such small mutations are restricted to genetic kinship (given the very small probability of the exact same point mutation happening in a separate case, because the vast number of possible locations for such a mutation within the gene). The loss and perpetuation of such characteristics are subject to the propagation frequency of the agent and the number of genetic kin emerging from the mutated agent. Hence, though the perpetuation of many components of an agent is related to maintaining a random large gene pool, the perpetuation of these point mutations is subject to maintaining a subcluster of genetically related kin.
Again, though these point mutations can continue over a very long period and many generations, some other generational fixed attributes derive from the lack of reorganization of the Y chromosome and the mother's mitochondrial DNA. If any arrangement of sequences in these chromosomes produces an expression pattern, they can thus be seen to be perpetuated from one generation to another. The weights of these specific genetically fixed expressions as opposed to the weights of the lost expressions among kin but available in random unrelated agents would come as a cohesive force of genetic kinship based clustering.
A faith, or a rule derived from perception, though stored as small weights within a set of specific neurons, can express itself as a set of highly weighted behavioral patterns. If the expression of such a faith is related to more historical experiences, the faith gains mass ("inertia") because of its inter-connections (see Shafee 2008b). Clauses of these faiths may be coupled with genetic attributes and also personal historical experiences. The fixed nature of these coupled terms can also make the faith massive and/or weighted by induction (see Shafee 2008 for such a mechanism). Some faith clauses may be genetically independent, but related to a common highly weighted massive term or a threat (see Shafee 2008b).
The faith itself may have large behavioral consequences, and programming the same faith within a large number of agents by means of common experiences and communication from credible sources (see Shafee 2008b for some possible modes of spread of axioms or faith). A highly weighted faith may exist in different conflicting states among isolated clusters because of the local origin and spread of the faith. The history experienced by the clusters separately that are coupled to their specific faiths, and are thus massive (having a significant inertia or reluctance to change) within the clusters lend the faith its own identity spreading within each cluster. Hence, within the cluster level, faith-based identities exist, and conflicting states may exist in two separate clusters.
Pattern Recognition
In neural networks, patterns can be recognized by programming attractors. Yet, the neural networks are given a capacity so that the number of patterns that can be stored is limited. The attractors allow for the input patterns to be driven to one of these attractors even if the input and the stored patterns are not identical. In a similar spirit, we can define the "matching" of states at each level, so that the agent recognizes a match or a conflict of another agent's attribute at each level of expression. The scales related to each of these levels dictate which level is expressed in an interaction. Hence, subtle differences between two states due to different lower level components existing within them may be blurred out at a different scale, where the recognition of pattern depends on stored matches or mismatches.
Pattern Component Identity within the Environment
As was explained above, while the agent attains his identity by means of the local potential minimum formed by means of interactions between coupled states, the expression of the states of the agents within the environment are by means of re-alignments or organizational changes within the environment. These changed alignments represent the relative optimality between the agent and the environment, and the stiffness to change or inertia (Shafee 2008) of the agent and the environment. The connection with the environment slowly brings about changes in an agent involving change in values, adaptation and survival of the fittest. The last mentioned depends on being able to maintain the fit components coherently, given the stochasticity and fluctuation produced by the large degrees of freedom of the environment and the diverse organization of the components. The need for symbiosis among the organs to produce coherent sustainable couplings, and hence the need to squeeze diverse states within a small space is offset by the restriction in the number of variables that can be expressed (Shafee 2008). The large scale of the environment systems to produce "fields" optimal for each agent component is offset by the restriction on the possible number of such different fields, each optimal for a different agent component to exist together within the same local environment. The connection of the local environment to a global environment, where other agents with slightly different optimality conditions are connected, causes the local fields to change, and the identity of an agent is by means of maintaining a local environment that sustains his own coupled components to create a local potential minimum. Hence, the cost of maintaining a local environment optimal to an agent's existence is in producing a change in the connected global environment if the agent were not present. The degree of change in global couplings in order to maintain a local stable pattern of states and fields is a measure of the expression of the identity of the agent within the context of the environment.
A one time change in the local environment to produce a state optimal for an agent component is interacted with by other environment systems and agent states coupled to that. So, the stability of a system depends on the degree of connections of states and components that the particular state is interacting with, and the number and coupling strengths of the states that tend to realign the optimal state in an unfavorable direction. If several agents have the same optimal environments, they will tend to orient their local environment along the direction of the same preference, and the overlapping components can be shared. The non-overlapping components can be extended to form a larger state where the same alignment exists along that preference, and the complementary skills of the multiple agents sharing the same preference can be used to realign the various types of unfavorable interacting environment states. The connection of multiple agents with the same preference enables multiple states of the agents to be expressed within that same alignment region, decreasing the constraint of the number of expressed states available to shield an agent. This large scale alignment of the environment along a particular preference thus allows multiple agents to collaborate, and if the weight of this particular preference is large, and the effect of the mismatched preferences of the collaborating agents can be kept low, a meta-stable social network can be formed. This dynamics supports the observation that networks arise partly because agents choose to associate with others who are similar to themselves in some significant respect (Lazarsfeld and Merton 1944).
The emergence of a large-scale alignment of an environment along a certain preference common to a group of agents thus expresses the identity of that particular component at a large scale, while constraining the mismatched preference of the symbiotic components of each agent to his immediate locality. Hence, an interaction minimum of a large magnitude exists with the semi-stable formation of a large scale environment alignment along that preference in a number of agents located separately but within a group identity. Such highly weighted preferences can be said to have attained its own identity at the social or group level, which is reflected in the consistent environment minimum.
A specific situation of such coherent alignment in a very highly weighted preference, and the consequent dynamics in an extreme case, is described below.
Altruism from Component Affiliation
Experiments indicate that altruism need not be always based on expectation of a return (Bowles and Gintis 2004). However, models based on genetic kinship (Dawkins 1989) do not take into account all forms of altruistic behavior, e.g. those based on values. We investigate special cases when a certain attribute/preference pattern component is in peril, and its survival or perpetuation instinct, from bonds created with matching patterns in other agents, may supersede the bond of the pattern with its own symbiotic components.
First, we look into the origin of a coherent faith based cluster so that the coherent expression of a faith among (almost) all members creates a faith identity expressed in a social level so that just as an agent has an individual identity as a component of the social cluster, the faith has a component identity within the cluster as expressed within agents coherently.
In our specific example two or more clusters with independent faith may exist in different localities, and when posed with competition (see Shafee 2008 for a discussion of competition based components among identities) over specific scarce common resources for contradictory preferences, the clusters may try to eliminate each other by realigning (converting) or eliminating agents who are members of the other clusters.
In such situations, when a cluster is in peril (i.e. losing), the rational choice of an agent is to defect, and save himself by converting to the other faith.
We present a specific scenario, when an individual agent is given the choice of saving himself (his entire locally coupled personal identity) and risking his life by not converting. Real world data show that in specific cases such irrational altruistic martyrdom exists (see ?????????????????????????) In the scenario presented below, the agent chooses to sacrifice himself in an altruistic manner. The expressions of identities and the intersection of the faith component within each individual agent's identity and the social identity plays a vital role in this scenario. Identities in each case give rise to local interaction potential minima, and the intersection of identities come into play when one possible minimum is chosen over the other if only one of the two can exist within the agent's own frame.
In (Shafee 2008), an agent was given an identity because of a strongly coupled localized group of diverse variables. The stability (meta-stability) of the identity derived from the local interaction minimum derived from was based on the symbiotic/match-based interactions among the components. These highly localized components were semi-closed in the sense that they also coupled with neighboring environment states within interaction range, and states of other agents.
In (Shafee 2009) it was shown how a rational agent could be derived when the local internal couplings are strong and the external couplings provide random matches/mismatches. However, the utility of an agent based on a preference, was also stated as modified after the zero-th rational order based on the commonality with other agents in extended group identities.
We first derive the concept of this larger identity from the point of view of couplings and stability, and then discuss a special case with coherent faith based couplings among agents within a common environment forming a strong extended identity.
The modified identity of the agent would include interaction terms in all three of his interaction Hamiltonian components, namely H self , H env and H agent−agent . Hence interacting states or variables in the local environment and other interacting agents would also give rise to second and higher order identities. However, because of the random match/mismatch among these states, and the fact that other agents would come in their own locally coupled bundle of states, many of them not optimal as interaction terms with the agent's own states, these external components of identity would be dynamic. The agent himself modifies states within his own self array in order to adapt, so that a change is necessary to maintain the entity (local minimum) as a whole. However, because of the fixed nature of many of his own states (the central dogma of molecular biology, so that information flows from genes to synthesized proteins, see Crick 1970), many of the internal states are fixed and coupled together in a massive manner, and disconnecting one component may threaten the integrity of the entire structure (see Shafee 2008).
We now consider a special case, when a group of isolated agents, connected within a local environment and experiencing very similar events, giving rise to similar sets of norms, may behave in a very different way.
A common set of experiences may give the agents a common set of norms at a time point (see e.g. Gordon 2003 for how Japan came together after World War II. If these people are genetically close, they also share a large subset of genetically fixed preferences in the form of locally preserved alleles (see Cavalli-Sforza andEdwards 1967 about genetic drifts andNei 2005 for population bottlenecks). Let us say that these agents share a common group of preferences or norms bundled as a faith that is brainwashed into a majority of them (see Shafee 2009b). Now if these induced preferences or norms are connected with the agents long enough so that the local environment is largely aligned along the faith, the coupling of each agent with the similarly aligned local environment is also large. Now we consider each agent's array split into three distinct parts based on his own genetic individuality and the effect of the faith. 1. preferences aligned along the faith :P i F , because of the connections of the faith with those preferences and the weight of the faith. As time goes by, these preferences get coupled with the local environment because of experience and alignment of the environment along the faith. Agents in the network share a large overlap of these preferences. 2. individuality components: are the agent's own set of preferences that are not related with the faith. These states may have random matches/mismatches with the local environment, network agents' states. 3. buffer variables: act as a buffer between the faith preferences and the individuality preferences. They may come in two possible states, one optimal for the faith preferences and the other optimal for the individuality components so that triggering one (because of optimality in connection) blocks the other. Just as within the agent, the three types of faith-related components are coupled to form the agent's own identity, the faith and P i F components have a large degree of match-based coupling with other agents and also the local coherent environment, which is aligned along the shared faith component. Hence, the coherence in alignment produces a network-spanning identity based on that faith.
The faith thus intersects the agent's individuality as it couples socially to form its own local interaction minimum based identity.
We now study the behavior of the buffer states, especially when the two intersecting identities come into play.
High Priority Beliefs
We start with a high priority pattern, e.g. a value that has high priority in an agent, so that an agent strives to connect to the environment with that preference with a high weight. This component may be created by repeated historical interactions within a group of agents and the common environment, or this can also be introduced or imagined initially as a virtual term to stabilize some local critical components. This belief, faith or norm component, F , may have competing components in other agents so that at the macro level, the change in the environment depends on the winning component or belief so that a certain number of agents must have this component in order for an environmental change to take place. Or else, the belief loses and no alignment of the environment takes place, so that chaos prevails.
In situations like this, in order for the environment to be aligned in a certain way, the faith component must exist in a certain number of agents so that all these agents also have a group of preferences within them shifted on account of this belief, so that they all coherently invest in aligning the environment along that belief components direction. In such a situation, the belief component might have a value such that losing the other variables in order to preserve that belief component may be favorable, since the belief component, which has a high weight in the agent and his expressions, can be perpetuated at the cost of the other variables of the agent.
Let ] when the internal couplings of the i t h agent are lost, are similar to the realignment of components of agent within the agent's identity in order to maintain the lowest aggregate local minimum. This realignment at the cost of one agent comes within the identity of the social interaction potential, that is also local to the agent because of large degrees of coherence.
Conclusion
In this paper, we have discussed the role of the expression of components in an agent within the expression of the identity of an agent. The concept of patterns and the preservation of patterns were discussed. Stability was discussed from the point of view of perpetuating pattern states, and identities evolving and perpetuating in different scales were discussed. A specific case was studied when a component identity played a game with an agent identity in terms of the preservation of the local potential minimum. The model thus extends the concepts introduced by the idea of interaction potentials in the social network to further explain some puzzling social phenomena.
[F ] be the group of belief norms originally shared by the agents. Let [P F i ] be the group of preferences or norms within the i th agent that are aligned along [F ]. Similarly, the [E F i ] are the local environment states that are aligned along [F ]. Each agent's individuality arises from [P R i ], which are the remaining preferences which are unrelated to [F ]. The buffer states as mentioned in the last section are given by [B i ], which can exist in the states [B F i ] optimal to [F ] or in [B R i ], which are optimal to the remainder states of the agent. The agents' genetic states are given by [G i ], and if the cluster is reproductively isolated, [G i ] has overlapping optimal components along [E i ](adaptation within the locality because of mutation arising within the local environment and selected), expressed as [G E i ], and also a large degree of overlap with [G j ] because of similar alleles in a small homogenous population. Let the degree of genetic overlap be [G o ]. If the agents are in a small community, [E i ] ∼ [E j ] ∼ [E]. Each agent can thus be said to be coupled to the attribute [F ], which is also coupled to [E]. The couplings of the exact matches of [F ] among the N agents in the cluster, and the couplings of these N [F ] terms and the respectively coupled [P F i ] terms with the environment [E], which is now an optimally coupled set of variables in the local environment, form an optimally interacting identity, and hence a social identity based on [F ]. Now each of these [F]s in an agent ([F i ]), are coupled to the rest of the agent's state because of the internal couplings within the agent locally. However, the introduction of a homogenous [F] term among all agents (that help the agents maintain a favorable environment together) and the difference in individuality posed by the mismatched variables of [P i ] induced along [F] within [P F i ] and the consequent remainder of individuality terms cause the internal local couplings within the agent to be slightly less optimal, in order to accommodate optimality of the environment by aligning with the other agents of the network along [F]. However, if there is a large degree of genetic overlap among agents and a large number of norms aligned in the same direction because of repeated similar social interactions and experiences, [P R i ] may become less prominent. Now the variables belonging to [B i ] attain their state by optimizing the local interaction Hamiltonian. However, these variables face the dilemma of being coupled to both the social identity and the agent's remaining individual identity. If a situation arises such that [F ] and [G R i ] and [P R i ] are competing in the sense that converting [F ] within the agent to a different state [F ′ ] is needed to maintain the agent's couplings within H self , the alignment of [B R ] will be along the future optimal Hamiltonian local to it. The couplings of the massive terms of [P F i ] and [G o ] with the social identity, which, if massive, together with the coupling of [F ] with massive terms in a network may cause the agent's local interaction Hamiltonian to become less optimal than the present case if [F ] is flipped to [F ′ ]. However, if attaching to [F ] can maintain a possible future local minimum (see Shafee 2008 for future decisions) for the social identity coupled with [B i ], the alignment of [B i ] may choose the social identity of [F] at the local identity of the agent, which may appear as irrational behavior. The loss of the variables in [B i
AcknowledgementThe author would like to thank Prof Douglas White and Prof Bertrand Roehner for their continued encouragement. She would also like to thank Prof M. Fisher for feedback on an earlier related paper.
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| []
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[
"A type of generalization error induced by initialization in deep neural networks",
"A type of generalization error induced by initialization in deep neural networks"
]
| [
"Yaoyu Zhang \nDhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n\n",
"Zhi-Qin \nDhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n\n",
"John Xu \nDhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n\n",
"Tao Luo \nDhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n\n",
"Zheng Ma \nDhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n\n"
]
| [
"Dhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n",
"Dhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n",
"Dhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n",
"Dhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n",
"Dhabi and Courant Institute of Mathematical Sciences\nDepartment of Mathematics\nDepartment of Mathematics\nNew York University Abu Dhabi and Courant Institute of Mathematical Sciences\nNew York University Abu\nPurdue University\nPurdue University\n"
]
| []
| How different initializations and loss functions affect the learning of a deep neural network (DNN), specifically its generalization error, is an important problem in practice. In this work, focusing on regression problems, we develop a kernelnorm minimization framework for the analysis of DNNs in the kernel regime in which the number of neurons in each hidden layer is sufficiently large(Jacot et al. 2018, Lee et al. 2019. We find that, in the kernel regime, for any loss in a general class of functions, e.g., any L p loss for 1 < p < ∞, the DNN finds the same global minima-the one that is nearest to the initial value in the parameter space, or equivalently, the one that is closest to the initial DNN output in the corresponding reproducing kernel Hilbert space. With this framework, we prove that a non-zero initial output increases the generalization error of DNN. We further propose an antisymmetrical initialization (ASI) trick that eliminates this type of error and accelerates the training. We also demonstrate experimentally that even for DNNs in the non-kernel regime, our theoretical analysis and the ASI trick remain effective. Overall, our work provides insight into how initialization and loss function quantitatively affect the generalization of DNNs, and also provides guidance for the training of DNNs. | null | [
"https://arxiv.org/pdf/1905.07777v1.pdf"
]
| 159,041,039 | 1905.07777 | e91b5088c5b95c4c87c74b54748bba7f1195aa22 |
A type of generalization error induced by initialization in deep neural networks
Yaoyu Zhang
Dhabi and Courant Institute of Mathematical Sciences
Department of Mathematics
Department of Mathematics
New York University Abu Dhabi and Courant Institute of Mathematical Sciences
New York University Abu
Purdue University
Purdue University
Zhi-Qin
Dhabi and Courant Institute of Mathematical Sciences
Department of Mathematics
Department of Mathematics
New York University Abu Dhabi and Courant Institute of Mathematical Sciences
New York University Abu
Purdue University
Purdue University
John Xu
Dhabi and Courant Institute of Mathematical Sciences
Department of Mathematics
Department of Mathematics
New York University Abu Dhabi and Courant Institute of Mathematical Sciences
New York University Abu
Purdue University
Purdue University
Tao Luo
Dhabi and Courant Institute of Mathematical Sciences
Department of Mathematics
Department of Mathematics
New York University Abu Dhabi and Courant Institute of Mathematical Sciences
New York University Abu
Purdue University
Purdue University
Zheng Ma
Dhabi and Courant Institute of Mathematical Sciences
Department of Mathematics
Department of Mathematics
New York University Abu Dhabi and Courant Institute of Mathematical Sciences
New York University Abu
Purdue University
Purdue University
A type of generalization error induced by initialization in deep neural networks
How different initializations and loss functions affect the learning of a deep neural network (DNN), specifically its generalization error, is an important problem in practice. In this work, focusing on regression problems, we develop a kernelnorm minimization framework for the analysis of DNNs in the kernel regime in which the number of neurons in each hidden layer is sufficiently large(Jacot et al. 2018, Lee et al. 2019. We find that, in the kernel regime, for any loss in a general class of functions, e.g., any L p loss for 1 < p < ∞, the DNN finds the same global minima-the one that is nearest to the initial value in the parameter space, or equivalently, the one that is closest to the initial DNN output in the corresponding reproducing kernel Hilbert space. With this framework, we prove that a non-zero initial output increases the generalization error of DNN. We further propose an antisymmetrical initialization (ASI) trick that eliminates this type of error and accelerates the training. We also demonstrate experimentally that even for DNNs in the non-kernel regime, our theoretical analysis and the ASI trick remain effective. Overall, our work provides insight into how initialization and loss function quantitatively affect the generalization of DNNs, and also provides guidance for the training of DNNs.
Introduction
The wide application of deep learning makes it increasingly urgent to establish quantitative theoretical understanding of the learning and generalization behaviors of deep neural networks (DNNs). In this work, we study theoretically the problem of how initialization and loss function quantitatively affect these behaviors of DNNs. Our study focuses on the regression problem, which plays a key role in many applications, e.g., simulation of physical systems (Zhang et al. 2018), prediction of time series (Qiu et al. 2014) and solving differential equations (E & Yu 2018, Xu et al. 2019. For theoretical analysis, we consider an extremely over-parameterized regime of DNN, i.e., number of neurons in each layer tends to infinity, which has attracted significant attentions recently. In this regime, the training dynamics of a DNN is found to be well approximated by the gradient flow of a linearized model of the DNN resembling kernel methods (Jacot et al. 2018, Lee et al. 2019. In our work, we refer to the regime of DNNs with this property as the kernel regime and we do not distinguish "linearized model of DNN" and "DNN in the kernel regime" for the analysis of their properties. Note that, theoretical investigation of such a regime can provide insight into the understanding of general DNNs in practice by the following facts. Heavy overparameterization is one of the key empirical tricks to overcome the learning difficulty of DNNs (Zhang et al. 2016). The DNNs in extremely over-parameterized regime preserve substantive behavior as those in mildly over-parameterized regime. For example, stochastic gradient descent (SGD) can find global minima of the training objective of DNNs which generalizes well to the unseen data (Zhang et al. 2016).
In general, the error of DNN can be classified into three general types (Poggio et al. 2018): approximation error induced by the capacity of the hypothesis set, generalization error induced by the given training data, and training error induced by the given training algorithm. By the universal approximation theorem (Cybenko 1989) and empirical experiments (Zhang et al. 2016), a large neural network often has the power to express functions of real datasets (small approximation error) and the gradient-based training often can find global minima (zero training error). Therefore, generalization error is the main source of error in applications. It can be affected by many factors, such as initialization and loss function as widely observed in experiments. Empirically, a large weight initialization often leads to a large generalization error (Xu et al. 2018(Xu et al. , 2019. However, a too small weight initialization makes the training extremely slow. Note that zero initialization leads to a saddle point of DNN which makes the training impossible. Despite above empirically observations, it remains unclear how initialization is related to the generalization error. Regarding the loss function, it is also unclear how it affects the behavior of DNNs.
Our contribution is concluded as follows. Focusing on the regression problem, we develop a kernelnorm minimization framework for the analysis of DNNs in the kernel regime based on the following theoretical results.
i) We prove that, for a general class of loss functions, e.g., any L p loss for 1 < p < ∞, the gradient flow of DNN in the kernel regime, despite trajectory difference, finds the same global minimum.
ii) Similar to Mei et al. (2019), we prove that, among the huge set of all global minima, this global minimum is the nearest to the initial value in the parameter space, or equivalently, is the closest to the initial DNN output h ini in the corresponding reproducing kernel Hilbert space (RKHS).
With the above framework, we analyze theoretically the impact of a non-zero initial DNN output h ini . i) We quantify how a h ini affects the solution of the kernel-norm minimization problem.
ii) We prove that a random h ini leads to a specific type of generalization error.
iii) We propose an AntiSymmetrical Initialization (ASI) trick which eliminates this generalization error and accelerates training while keeping the kernel of DNN unchanged.
Experimentally, we demonstrate that our theory accurately predicts the behavior of wide DNNs. Moreover, we demonstrate that the ASI trick remains effective for DNNs in the non-kernel regime as well as for the classification problem.
Related works
There are a series of works following the study of Jacot et al. (2018) on the kernel regime. For example, theoretical works provide insight into how SGD can find global minima on the training objective of DNNs (Du et al. 2018, Zou et al. 2018 ) are further studied in the kernel regime. In addition, the type of generalization error that vanishes as the width of DNN increases is analyzed (Geiger et al. 2019). Mei et al. (2019), Banburski et al. (2019) also found that the learning of DNNs in the kernel regime is associated with an optimization problem, however, they only consider the loss of mean-squared error (MSE) and a special initialization. Chizat & Bach (2018) shows that if the initial DNN output is close enough to 0 and a large factor is used to scale the DNN output, the DNN with MSE loss selects parameters that are close to the initialization. Oymak & Soltanolkotabi (2018), Jacot et al. (2018) show that in an over-parameterized regime, the convergence point in the parameter space of a DNN remains close to the initialization. However, it remains unclear that, among all global optima of the loss, whether the gradient descent (GD) algorithm converges to one with the nearest distance to the initialization.
Previous works (Xu et al. 2018, Xu 2018a,b, Xu et al. 2019, Rahaman et al. 2018) discover a Frequency-Principle (F-Principle) that DNNs prefer to learn the training data by a low-frequency function. Based on F-principle, Xu et al. (2018Xu et al. ( , 2019 point out that the final output of a DNN tends to inherit high frequencies of its initial output that can not be well constrained by the training data (Xu et al. 2018(Xu et al. , 2019. Note that this understanding is consistent with our quantitative study.
Preliminary
A summary of notations can be found in Appendix 9.
Kernel regime of DNN
In the following, we consider the regression problem of fitting the target function
f ∈ L ∞ (Ω), where Ω is a compact domain in R N I . Clearly, f ∈ L p (Ω) for 1 ≤ p ≤ ∞. Specifically, we use a DNN, h DNN (x, θ(t)) : Ω × R N P → R, to fit the training dataset {x i ; y i } M i=1 of M sampling points, where x i ∈ Ω, y i = f (x i ) for each i. For the convenience of notation, we denote X = [x 1 , · · · , x M ] T , Y = [y 1 , · · · , y M ] T ,
and g(X) := [g(x 1 ), · · · , g(x M )] T for any function g defined on Ω. It has been shown in Jacot et al. (2018), Lee et al. (2019) that, for any t ≥ 0, if the number of neurons in each hidden layer is sufficiently large, then |θ(t) − θ(0)| 1. In such cases, the following linearized model
h(x, θ) = h DNN (x, θ 0 ) + ∇ θ h DNN (x, θ) (θ − θ 0 ) .(1)
is a very good approximation of DNN output h DNN (x, θ DNN (t)) initialized with θ DNN (0) = θ 0 . Note that, we have the following requirements for h DNN which are easily satisfied for common DNNs: For any θ ∈ R N P , there exists a weak derivative of h DNN (·, θ) with respect to θ and ∇ θ h DNN (·, θ) ∈ L 2 (Ω).
For the loss function
L(θ) = D (h(X, θ),Y ),
where D is the distance function to be explained in Section 4, the gradient flow of θ(t) with respect to the linearized model h(x, θ(t)) follows
dθ(t) dt = −∇ θ h(X, θ 0 ) T ∇ h(X,θ(t)) D (h(X, θ(t)),Y ),(2)with initial value θ(0) = θ 0 , where ∇ θ h(X, θ 0 ) ∈ R M×N P , [∇ θ h(X, θ 0 )] i j = ∇ θ j h(x i , θ 0 ), ∇ h(X,θ(t)) D (h(X, θ(t)),Y ) ∈ R M .
We refer to the regime in which h(x, θ(t)) well approximate h DNN (x, θ DNN (t)) under the same loss initialized by θ DNN (0) = θ 0 for any t ≥ 0 as the kernel regime of DNN. Therefore, in the following, our analysis focuses on dynamics (2) for the analysis of the behavior of DNN in the kernel regime. Eq. (1) yields the following dynamics of h(x, t) = h(x, θ(t)),
∂ t h(x, t) = −K(x, X)∇ h(X,t) D (h(X, t),Y ),(3)
with initial value h(·, 0) = h(·, θ 0 ), where the kernel K is defined as
K(·, ·) = ∇ θ h(·, θ 0 )∇ θ h(·, θ 0 ) T , ∇ θ h(·, θ 0 ) = [∂ θ 1 h(·, θ 0 ), · · · , ∂ θ N P h(·, θ 0 )], K(x, X) ∈ R 1×M
for any x ∈ Ω. Note that Eq. (3) of h is a closed system. By Jacot et al. (2018), K is symmetric and positive semi-definite. In the following, we may denote K θ 0 (·, ·) = ∇ θ h(·, θ 0 )∇ θ h(·, θ 0 ) T when we need to differentiate kernels corresponding to different architectures or different initializations of DNNs.
Reproducing kernel Hilbert space (RKHS)
The kernel K can induce a RKHS as follows. First, we cite the Mercer's theorem (Mercer (1909)).
Theorem 1. (Mercer's theorem (Mercer (1909))) Suppose K is a continuous symmetric positive semi-definite kernel. Then there is an orthonormal basis {φ j } of L 2 (Ω) consisting of eigenfunctions of T K defined as [T K g] (·) = ∫ Ω K(·, x)g(x) dx such that the corresponding sequence of eigenvalues σ j is nonnegative. The eigenfunctions corresponding to non-zero eigenvalues are continuous on Ω and K has the representation
K(x, y) = ∞ j=1 σ j φ j (x)φ j (y),
where the convergence is absolute and uniform.
Then, we can define the RKHS as
H K (Ω) := {g ∈ L 2 (Ω) | ∞ i=1 σ −1 i g, φ i
2 L 2 (Ω) < ∞}, and the inner product in H K (Ω) is given by
f , g K = ∞ i=1 σ −1 i f , φ i L 2 (Ω) g, φ i L 2 (Ω) , where g, φ i L 2 (Ω) = ∫ Ω g(x)φ i (x)dx. Define K −1 (x, x ) = ∞ i=1 σ −1 i φ i (x)φ i (x )
, then the kernel norm of any g ∈ H K (Ω) can be expressed as
g K = g, g 1/2 K = ∫ Ω×Ω g(x)g(x )K −1 (x, x ) dx dx 1 2 . H K (Ω) satisfies Berlinet & Thomas-Agnan (2004): (i) ∀x ∈ Ω,K(·, x) ∈ H K (Ω); (ii) ∀x ∈ Ω, ∀ f ∈ H K , f (·), K(·, x) K = f (x); (iii) ∀x, y ∈ Ω, K(·, x), K(·, y) K = K(x, y).
Kernel-norm minimization framework for DNNs in the kernel regime
In this section, we introduce the kernel-norm minimization framework for the analysis of DNNs in kernel regime. As introduced in Section 3.1, for the analysis of gradient flow of h DNN (·, θ DNN (t)) in the kernel regime, we focus on the gradient flow of its linearized model h(·, θ(t)), i.e., Eqs. ((2), (3)).
We consider the gradient flow under any loss L(θ) = D (h(X, θ),Y ), where D is continuously differentiable and satisfies, for any z ∈ R M , (i) D(z, z) = 0; (ii) D(z , z) attains minimum if and only if z = z.
(iii) z = z if and only if ∇ z D(z , z) = 0. For example, D (h(X, θ),Y ) = 1 M M i=1
|h(x i , θ) − y i | p for any 1 < p < ∞. By Theorem 5 in Appendix 10, the long time solution θ(∞) = lim t→∞ θ(t) of dynamics (2) is equivalent to the solution of the optimization problem
min θ θ − θ 0 2 , s.t., h(X, θ) = Y .(4)
By Theorem 8 in Appendix 10, h(x, θ(∞)) uniquely solves the optimization problem
min h−h ini ∈H K (Ω) h − h ini K , s.t., h(X) = Y,(5)
where h ini (x) = h(x, θ 0 ) and the constraints h(X) = Y are in the sense of trace (Evans (2010), pp. 257-261). The above results hold for any initial value θ 0 . We refer to kernel-norm minimization framework as using the optimization problem (4) or (5) to analyze the long time solution of gradient flow dynamics in (2) or (3), respectively.
With this framework, we emphasize the following results. First, for a finite set of training data, given θ 0 , because D is absent in problems (4) and (5), the output function of a well-trained DNN in the kernel regime is invariant to different choices of loss functions. Note that this result is surprising in the sense that different D clearly leads to different trajectories of θ(t) and h(·, θ(t)). Based on this result, it is not necessary to stick to commonly used MSE loss for regression problems. For example, in practice, one can use
D (h(X, θ),Y ) = 1 M M i=1 |h(x i , θ) − y i | p of 1 < p < 2
to accelerate the training of DNN near convergence or 2 < p < ∞ to accelerate the training near initialization. One can even mixing different loss functions to further boost the training speed. Second, among all sets of parameters that fit the training data, a DNN in the kernel regime always finds the one closest to the initialization in the parameter space with respect to the L 2 norm. Third, in the functional space, this framework shows that DNNs always seek to learn a function that has a shortest distance (with respect to the kernel norm) to the initial output function. In the following, we denote h K (x; h ini , X,Y ) as the solution of problem (5) depending on K, h ini , X and Y .
Impact of non-zero initial output
Problems (4) and (5) explicitly incorporate the effect of initialization, thus enabling us to study quantitatively its impact to the learning of DNNs. In this section, we use the above framework to show that a random non-zero initial DNN output leads to a specific type of generalization error. We begin with a relation between the solution with zero initial output and that with non-zero initial output. Proofs of the following theorems can be found in Appendix 11. Theorem 2. For a fixed kernel function K ∈ L 2 (Ω × Ω), and training set {X;Y }, for any initial
function h ini ∈ L ∞ (Ω), h K (·; h ini , X,Y ) can be decomposed as h K (·; h ini , X,Y ) = h K (·; 0, X,Y ) + h ini − h K (·; 0, X, h ini (X)).(6)
This theorem unravels quantitatively the impact of a nonzero initialization, i.e., h ini 0, to the output function of a well-trained DNN in the kernel regime. Comparing the dynamics in (3) of zero and non-zero initialization, at the beginning, the difference of DNN output is h ini , whereas, at the end of the training, that difference shrinks to h ini − h K (·; 0, X, h ini (X)), which is the residual of fitting h ini sampled at X by the same DNN. Note that Geiger et al. (2019)figures out qualitatively that h ini , which does not vanish as the width of DNN tends to infinity, decreases during the training. However, they do not arrive at a quantitative relation as revealed by Theorem 2.
The expected generalization error of DNN with a random non-zero initial output can be estimated as follows.
Theorem 3. For a target function f ∈ L ∞ (Ω), if h ini is generated from an unbiased random function distribution P such that E h ini ∼P h ini = 0, then the generalization error of h K (·; h ini , X, f (X)) can be decomposed as follows
E h ini ∼P L (h K (·; h ini , X, f (X)), f ) = L (h K (·; 0, X, f (X)), f ) + E h ini ∼P L (h M (·; 0, X, h ini (X)), h ini ), (7) where L(h K (·; h ini , X, f (X)), f ) = h K (·; h ini , X, f (X)) − f 2 L 2 (Ω) . By above theorem, E h ini ∼P L (h M (·; 0, X, h ini (X)), h ini ) ≥ 0
is a specific type of generalization error induced by h ini . Clearly, this error decreases as the sample size M increases and as M → ∞, h ini − h K (·; 0, X, h ini (X)) → 0, which conforms with our intuition that if the optimization is sufficiently constrained by the training data, then the effect of initialization can be ignored. For real datasets of a limited number of training samples, this error is in general non-zero. By F-Principle (Xu et al. 2018(Xu et al. , 2019, DNNs tend to fit training data by low frequency functions. Therefore, qualitatively, h ini − h K (·; 0, X, h ini (X)) consists mainly of the high frequencies of h ini which cannot be well constrained at X.
AntiSymmetrical Initialization trick (ASI)
In general, from the Bayesian inference perspective, for fixed K, a random h ini introduces a prior to the inference that is irrelevant to the target function, thus should lower the accuracy of inference.
To eliminate the negative impact of non-zero initial DNN output, a naive way is to set the initial parameters sufficiently small. However, for a set of too small parameters, the kernel of DNN at initialization nearly vanishes, making the training difficult. Moreover, by problem (5), change of kernel in general leads to a different solution of DNN which may not preserve the generalization performance. Based on our above theoretical results, we design an AntiSymmetrical Initialization trick (ASI) which can fix the initial output to zero but also keep the kernel invariant. Let h [l] i be the output of the ith node of the lth layer of a H layer DNN. Then,
h l i (x) = σ [l] i (W [l] i · h [l−1] (x) + b [l] i ), for i = 1, · · · , n l . For the ith neuron of the output layer of DNN, h [H] i (x) = W [H] i · h [H−1] + b [H]
i . After initializing the DNN by any method, we obtain h [H] (x, θ(0)), where
θ(0) = [W [H] (0), b [H] (0),W [H−1] (0), b [H−1] (0), · · · , b [1] (0)].
The ASI for general loss functions is to consider a new DNN expressed as h ASI (x, Θ(t)) =
√ 2 2 h [H] (x, θ(t)) − √ 2 2 h [H] (x, θ (t)) where Θ = [θ, θ ]
, Θ is initialized such that θ (0) = θ(0). In the following, we prove a theorem that ASI trick eliminates the nonzero random prior without changing the kernel K (Proof can be found in Appendix 12). Theorem 4. For any general loss function D satisfying the conditions in Sec. 4, in the kernel regime, the gradient flow of both h(x, θ(t)) and h ASI (x, Θ(t)) follows the kernel dynamics (2018) proposes a "doubling trick" to offset the initial DNN output, that is, neurons in the last layer are duplicated, with the new neurons having the same input weights but opposite output weights. By applying the "doubling trick", h (·, 0) = 0. However, the kernel of layers H − 1 and H doubles, whereas the kernel of layers m ≤ H − 2 completely vanishes (See Appendix 13 for the proof), which could have large impact on the training dynamics as well as the generalization performance of DNNs.
∂ t h = −K(·, X)∇ h(X,t) D (h (X, t),Y ),(8)with initial value h (·, 0) = h ini = h(x, θ(0)) and h (·, 0) = 0, respectively, where {X;Y } is the training set, K(x, x ) = K θ 0 (x, x ) = ∇ θ h(x, θ 0 ) · ∇ θ h(x , θ 0 ).
Note that Chizat & Bach
Experiments
Our above theoretical results are obtained using the linearized model of DNN in Eq. (1) that well approximates the behavior of DNN in the kernel regime. In this section, we will demonstrate experimentally the accuracy of these results for very wide DNNs and the effectiveness of these results for general DNNs. First, using synthetic data, we verify the invariance of DNN output after training to different loss functions as studied in Sec. 4. Then, we verify the linear relation in Eq.
(6). Moreover, we demonstrate the effectiveness of the ASI trick on both synthetic data and the MNIST dataset. Here is a summary of the settings of DNNs in our experiments. The activation function is ReLU, parameters are initialized by a Gaussian distribution with mean 0 and standard deviation v std 2/(n in + n out ), where n in and n out are for the input and the output dimension of the neuron, respectively. For Figs. (1, 2, 3), networks are trained by full gradient descent with MSE loss and the learning rate is 10 −5 .
Invariance of DNN output to loss functions
For a DNN h(x, θ(t)) with initialization fixed at certain θ(0) = θ 0 , we consider its gradient descent training for two loss functions: the Fig. 1, as Theorems 5 and 8 predict, the well-trained DNN outputs for these two losses overlap very well not only at 4 training points, but also at all the test points.
L 2 (MSE) loss D(h(X, θ),Y ) = 1 M M i=1 (h(x i , θ) − y i ) 2 and the L 4 loss D(h(X, θ),Y ) = 1 M M i=1 (h(x i , θ) − y i ) 4 . In
Linear relation and the effectiveness of ASI trick
1-d synthetic data
In this sub-section, we use 1-d data, which is convenient for visualization, to train DNNs of a large width. As shown in Fig. 2(a), without applying any trick, the original DNN initialized with a large weight learns/interpolates the training data in a fluctuating manner (blue solid). Both the ASI trick (cyan dashed dot) and the "doubling trick" (green dashed) enable the DNN to interpolate the training data in a more "flat" way. As shown by the red dashed curve, the output computed by the right hand side (RHS) of Eq. (6) accurately predicts the final output of the original DNN on test points. In our experiments h K (x; 0, X, h ini (X)) , h K (x; 0, X,Y ), h K (x; h ini , X,Y ) are always obtained using very wide DNNs with or without the ASI trick applied. From Eq. (6), a non-zero initialization adds a prior h ini − h K (x; 0, X, h ini (X)) to the final DNN output. As shown by the cyan dashed curve in Fig. 2(b), this prior fluctuates a lot, thus, leading to an oscillatory output of DNN after training. Note that this experiment also support the prediction of F-Principle (Xu et al. 2018, 2019) that it is the high frequencies of h ini (red dashed) that remains in the final output of DNN. Concerning the training speed, as shown in Fig. 2(c), the loss function of the DNN with the ASI trick applied decreases much faster than that of the original DNN or the one with the "doubling trick" applied. For a reference, we double the original network similar to the ASI trick, then initialize it randomly following the same distribution as the original. We refer to this trick as RND. As shown by the black curve in Fig. 2(c), the loss function of the DNN with the RND trick applied also decreases much slower than the one with the ASI trick applied.
In summary, for a 1-d problem, the linear relation holds well and the ASI trick is effective in removing the artificial prior induced by h ini and accelerating the training speed. In the following, we further investigate the generalization performance of DNN with the ASI trick applied for real datasets.
Boston house price dataset
We verify our theoretical results for high dimensional regression problems using Boston house price dataset (Harrison Jr & Rubinfeld 1978), in which we normalize the value of each property and the price to [−0.5, 0.5]. We choose 400 samples as the training data, and the other 106 samples as the test data. As illustrated by the red dots concentrating near the black line of an identity relation in Fig. 3a, the RHS of Eq. (6) well predicts the final output of the original DNN without any trick, which is significant different from the final output of DNN with the ASI trick applied as shown by the blue dots deviating from the black line. As shown in 3b, similar to the experiments on 1-d synthetic data, the ASI trick accelerates the training. In addition, conforming with Theorem 3, the generalization error of the DNN with ASI trick applied is much smaller than that of the original DNN.
MNIST dataset and the non-kernel regime of DNN
Next, we use the MNIST dataset to examine the effectiveness of ASI trick in the non-kernel regime of DNNs. We use a DNN with a more realistic setting of width 784-400-400-400-400-10, cross-entropy loss, batch size 512, and Adam optimizer (Kingma & Ba 2014). In such a case, as shown in Fig. 4, the ASI trick still effectively eliminate h ini , accelerate the training speed and improve the generalization. In Fig. 4(b), with the ASI trick applied, both training and test accuracy exceeds 90% after only 1 epoch of training. This phenomenon further demonstrate that, without the interference of h ini , DNNs can capture very efficiently and accurately the behavior of the training data.
Discussion
In this work, focusing on the regression problem, we propose a kernel-norm minimization framework to study theoretically the role of loss function and initialization for DNNs in the kernel regime. We prove that, given initialization, DNNs of different loss functions in a general class find the same global minimum. Regarding initialization, we find that a non-zero initial output of DNN leads to a specific type of generalization error. We then propose the ASI trick to eliminate this error without changing the neural tangent kernel. Experimentally, we find that ASI trick significantly accelerates the training and improves the generalization performance. Moreover, ASI trick remains effective for classification problems as well as for DNNs in the non-kernel regime. Because the error of DNN output induced by random initialization shrinks during the training, the advantage of ASI trick is much more significant at the early stage of the training. Based on above results, we suggest incorporating ASI trick in the design of controlled experiments for the quantitative study of DNNs.
From the perspective of training flexibility, ASI trick can alleviate the sensitivity of generalization and training speed to different random initializations of DNNs, thus expand the range of well-generalized initializations. This property could be especially helpful for finding a well-generalized solution of a new problem when empirical guidance is not available. We also remarks that, from Eq. (6), a particular prior of h ini , such as the one from meta learning (Rabinowitz 2019), could decrease the generalization error. However, when meta learning is not available, a zero h ini is in general the best choice for generalization.
Cross-entropy loss is commonly used in classification problems, for which the DNN outputs are often transformed by a softmax function to stay in (0, 1). Theoretically, to obtain a zero cross-entropy loss given that labels of the training data take 1 or 0, weights of the DNN should approach infinity. In such a case, it is impossible for a DNN to stay in the kernel regime, which requires a small variation of weights throughout the training. However, in practice, training of a DNN often stops by meeting certain criteria of training accuracy or validation accuracy. Therefore, it is possible that weights of a sufficiently wide DNN stay in a small neighborhood of the initialization during the training. By setting a proper tolerance for the cross-entropy loss, we will analyze in the future the behavior of DNNs in kernel regime for classification problems with cross-entropy loss. 10 Theorems for the kernel-norm minimization framework Theorem 5. Let θ(t) be the solution of gradient flow dynamics
d dt θ(t) = −∇ θ h(X, θ 0 ) T ∇ h(X,θ(t)) D (h(X, θ(t)),Y )(9)
with initial value θ(0) = θ 0 , where ∇ θ h(X, θ 0 ) T is a full rank (rank M) matrix of size N P × M with N P > M. Then θ(∞) = lim t→∞ θ(t) exists and uniquely solves the constrained optimization problem
min θ θ − θ 0 2 , s.t., h(X, θ) = Y .(10)
Remark. Compared with the nonlinear gradient flow of DNN, the linearization in Eq. (9) is only performed on the hypothesis function h but not on the loss function or the gradient flow.
Proof. Gradient flow Eq. (9) can be written as
dθ(t) dt = −∇ θ D (h(X, θ(t)),Y ) . Then denote L(t) = D (h(X, θ(t)),Y ), dθ dt 2 = − d dt L(t).
Note that L(t) = D (h(X, θ(t)),Y ) ≥ 0 for any t ≥ 0. Thus
∫ ∞ 0 dθ dt 2 dt = L(0) − L(∞) ≤ L(0).
Since dθ dt is continuous,
lim t→∞ d dt θ(t) = lim t→∞ −∇ θ h(X, θ 0 ) T ∇ h(X,θ(t)) D (h(X, θ(t)),Y ) = 0. Because ∇ θ h(X, θ 0 ) T is a full rank matrix, lim t→∞ ∇ h(X,θ(t)) D (h(X, θ(t)),Y ) = 0.
Recall that ∇ z D(z , z) = 0 if and only if z = z, lim t→∞ h(X, θ(t)) = Y .
By applying singular value decomposition to ∇ θ h(X, θ 0 ) T , we obtain ∇ θ h(X, θ 0 ) T = V ΣU T , where V and U are orthonormal matrix of size N P × N P and M × M respectively, Σ =
Σ 1 0 of size N P × M,
where Σ 1 is a full rank diagonal matrix of size M × M. V can be split into two part as V = [V 1 ,V 2 ], where V 1 takes the first M columns and V 2 takes the last N P − M columns of V. Then
∇ θ h(X, θ 0 ) T = V ΣU T = [V 1 ,V 2 ] Σ 1 0 U T = V 1 Σ 1 U T , V T 2 ∇ θ h(X, θ 0 ) T = V T 2 V 1 Σ 1 U T = 0. Therefore d dt V T 2 θ(t) = −V T 2 ∇ θ h(X, θ 0 ) T ∇ h(X,θ(t)) D (h(X, θ(t)),Y ) = 0, which leads to V T 2 (θ(t) − θ 0 ) = 0, for any t ≥ 0. (11) By Eq. (1), lim t→∞ h(X, θ(t)) = Y yields lim t→∞ ∇ θ h(X, θ 0 ) (θ(t) − θ 0 ) = Y − h(X, θ 0 ),
which can be written as lim
t→∞ UΣ 1 V T 1 (θ(t) − θ 0 ) = Y − h(X, θ 0 ) hence lim t→∞ V T 1 (θ(t) − θ 0 ) = Σ −1 1 U T [Y − h(X, θ 0 )] .(12)
Combining Eq. (11) and (12), θ(∞) = lim t→∞ θ(t) exists and is uniquely determined as
V T (θ(∞) − θ 0 ) = V T 1 V T 2 (θ(∞) − θ 0 ) = Σ −1 1 U T [Y − h(X, θ 0 )] 0 , θ(∞) − θ 0 = V Σ −1 1 U T [Y − h(X, θ 0 )] 0 = V 1 Σ −1 1 U T [Y − h(X, θ 0 )], which leads to θ(∞) = V 1 Σ −1 1 U T [Y − h(X, θ 0 )] + θ 0 .
On the other hand, by the above analysis, problem (10) can be formulated as
min θ θ − θ 0 2 , s.t., V T 1 (θ − θ 0 ) = Σ −1 1 U T [Y − h(X, θ 0 )] .
Any θ satisfies above constraint can be expressed as
θ = V 1 Σ −1 1 U T [Y − h(X, θ 0 )] + V 2 ξ + θ 0 , where ξ ∈ R N P −M . Then θ − θ 0 2 2 = V 1 Σ −1 1 U T [Y − h(X, θ 0 )] 2 2 + V 2 ξ 2 2 . Clearly, θ − θ 0 2 attains minimum if and only if ξ = 0. Therefore θ(∞) = V 1 Σ −1 1 U T [Y − h(X, θ 0 )] + θ 0 uniquely solves problem (10).
For the proof of Theorem 8, we first introduce the following two lemmas.
Lemma 6. For any h ∈ H K (Ω), there exist θ = h (·), ∇ θ h(·, θ 0 ) T K such that h = ∇ θ h(x, θ 0 )θ .
Proof. For any h ∈ H K (Ω),
h (·), K(·, z) K = h (·), ∇ θ h(·, θ 0 )∇ θ h(z, θ 0 ) T K = h (·), ∇ θ h(·, θ 0 ) K ∇ θ h(z, θ 0 ) T
For θ = h (·), ∇ θ h(·, θ 0 ) T K , by the property of reproducing kernel K, h (x) = h (·), K(·, x) K = ∇ θ h(x, θ 0 )θ .
Lemma 7. For any θ ∈ R N P , ∇ θ h(·, θ 0 )θ ∈ H K (Ω).
Proof. By the Mercer's theorem,
K(x, y) = ∞ j=1 σ j φ j (x)φ j (y),
where {φ j } ∞ j=1 are orthonormal basis of L 2 (Ω). If ∇ θ h(·, θ 0 )θ H K (Ω), then there exist j 1 such that σ j 1 = 0 and ∇ θ h(·, θ 0 )θ , φ j 1 L 2 (Ω) 0. Then there exist j 2 such that ∇ θ j 2 h(·, θ 0 ), φ j 1 L 2 (Ω) 0,
where θ j 2 is the j 2 -th component of θ. Then
∫ Ω φ j 1 (x)K(x, x )φ j 1 (x )dxdx = ∫ Ω φ j 1 (x) ∇ θ h(x, θ 0 ) T ∇ θ h(x , θ 0 ) φ j 1 (x )dxdx = i ∂ θ i h(·, θ 0 ), φ j 1 2 L 2 (Ω) ≥ ∂ θ j 2 h(·, θ 0 ), φ j 1 2 L 2 (Ω) > 0.
However, on the other hand,
∫ Ω φ j 1 (x)K(x, x )φ j 1 (x )dxdx = ∫ Ω φ j 1 (x) ∞ j=1 σ j φ j (x)φ j (x )φ j 1 (x )dxdx = j σ j φ j , φ j 1 2 L 2 (Ω) = σ j 1 = 0,
which leads to an contradiction. Therefore, ∇ θ h(·, θ 0 )θ ∈ H K (Ω).
Theorem 8. Let θ be the solution of problem (10), then h(x, θ) uniquely solves the optimization problem min
h−h ini ∈H K (Ω) h − h ini K , s.t., h(X) = Y,(13)
where h ini = h(x, θ 0 ) and the constraints h(X) = Y are in the sense of trace (Evans 2010).
Proof. By Eq. (1) h(x, θ) − h ini = ∇ θ h(x, θ 0 ) (θ − θ 0 ). By Lemma 7, h(·, θ) − h ini ∈ H K (Ω). For any h − h ini ∈ H K (Ω), by lemma 6, for θ = h − h ini , ∇ θ h(·, θ 0 ) T K , h − h ini = ∇ θ h(x, θ 0 )θ . Then h − h ini K = ∇ θ h(·, θ 0 )θ K = h − h ini , ∇ θ h(·, θ 0 )θ K = h − h ini , ∇ θ h(·, θ 0 ) K θ = √ θ T θ = θ 2 .
By Problem (10), for any θ 1 θ − θ 0 that satisfies h(X, θ 1 + θ 0 ) = Y θ 1 2 > θ − θ 0 2 .
Then, for problem (13), for any h 1 satisfying h 1 − h ini ∈ H K (Ω), h 1 (X) = Y and h 1 (x) h(x, θ), let θ 1 = h 1 − h ini , ∇ θ h(·, θ 0 ) T K . Clearly, θ 1 θ − θ 0 , which leads to h 1 − h ini K = θ 1 2 > θ − θ 0 2 = h(x, θ) − h ini K .
Therefore h(x, θ) uniquely solves problem (13). Now, we obtain the equivalence between the long time solution of dynamics (14) and the solution of optimization problem (15) as follows. Corollary 9. Let h(x, t) be the solution of dynamics d dt h(x, t) = −K(x, X)∇ h(X,t) D (h(X, t),Y ),
with h(x, 0) = h(x, θ 0 ) for certain θ 0 . Then h(x, ∞) uniquely solves optimization problem min
h−h ini ∈H K (Ω) h − h ini K , s.t., h(X) = Y .(15)
Proof. Notice that dynamics (14) is the same as dynamics (3) obtained from (2). Therefore, for h(x, 0) = h(x, θ 0 ), h(x, t) = h(x, θ(t)) where θ(t) is the solution of dynamics (2) with initial condition θ(0) = θ 0 . By Theorem 5 and 8, h(x, ∞) = h(x, θ(∞)) uniquely solves dynamics (15).
Impact of non-zero initial output
In this section, we use the above framework to show that a random non-zero initial DNN output leads to a specific type of generalization error. We begin with a lemma showing the linear composition property of the final DNN outputs in the kernel regime.
Lemma 10. For a fixed kernel function K : Ω × Ω → R, for any two training sets {X;Y 1 } and {X;Y 2 }, where Y 1 = [y (1) 1 , · · · , y (1) M ] T and Y 2 = [y (2) 1 , · · · , y (2) M ] T , the following linear relation holds h K (·; 0, X,Y 1 +Y 2 ) = h K (·; 0, X,Y 1 ) + h K (·; 0, X,Y 2 ).
(16)
Proof. Let h 1 (x, t), h 2 (x, t) be the solutions of the gradient flow dynamics with respect to a MSE loss
D (h(X, t),Y ) = 1 2 M i=1 (h(x i , t) − y i ) 2 ∂ ∂t h(x, t) = −K(x, X) (h(X, t) −Y )(17)
with training labels Y = Y 1 and Y = Y 2 , respectively, and h 1 (x, 0) = h 2 (x, 0) = h ini = 0. Then ∂ t (h 1 + h 2 ) = −K(·, X) (h 1 (X, θ(t)) −Y 1 ) − K(·, X) (h 2 (X, θ(t)) −Y 2 ) . = −K(·, X) [(h 1 + h 2 ) (X, θ(t)) − (Y 1 +Y 2 )]
with initial value (h 1 + h 2 ) (·, 0) = 0. Therefore h 1 + h 2 solves dynamics (17) for Y = Y 1 + Y 2 and h ini = 0. Then, by Corollary 9, we obtain h K (·; 0, X,Y 1 +Y 2 ) = h 1 (·, ∞) + h 2 (·, ∞) = h K (·; 0, X,Y 1 ) + h K (·; 0, X,Y 2 )
Using Lemma (10), we obtain the following quantitative relation between the solution with zero initial output and that with non-zero initial output. Theorem 11. (Theorem 2 in main text) For a fixed kernel function K ∈ L 2 (Ω × Ω), and training set {X;Y }, for any initial function h ini ∈ L ∞ (Ω), h K (·; h ini , X,Y ) can be decomposed as h K (·; h ini , X,Y ) = h K (·; 0, X,Y ) + h ini − h K (·; 0, X, h ini (X)).
Proof. Because h K (·; h ini , X,Y ) is the solution of problem (5). Then h K (·; h ini , X,Y ) − h ini is the solution of problem min h ∈H K (Ω) h K , s.t., h(X) = Y − h ini (X), whose solution is denoted as h K (·; 0, X,Y − h ini (X)). By Lemma 10, h K (·; 0, X,Y − h ini (X)) = h K (·; 0, X,Y ) − h K (·; 0, X, h ini (X)).
(19) Therefore h K (·; h ini , X,Y ) = h K (·; 0, X,Y − h ini (X)) + h ini = h K (·; 0, X,Y ) + h ini − h K (·; 0, X, h ini (X)).
The generalization error of DNN contributed by a random initial output can be estimated as follows.
Theorem 12. (Theorem 3 in main text) For a target function f ∈ L ∞ (Ω), if h ini is generated from an unbiased random function distribution P such that E h ini ∼P h ini = 0, then the generalization error of h K (·; h ini , X, f (X)) can be decomposed as follows
Figure 1 :
1Invariance of DNN output to loss functions. Black stars indicate training data. Blue solid curve and the black curve indicate the outputs on the test samples of the DNN well-trained by L 2 loss and L 4 loss, respectively. The size of DNN is 1-500-500-1. v std = 5. The training and test data are randomly sampled from sin(4x) in [−1, 1] with sample size 4 and 500, respectively.
Figure 2 :
2Synthetic data. (a) Black stars indicate the training data. Other curves indicate final outputs of different DNNs evaluated at the test points. Blue solid: the original DNN (without tricks); cyan dashed dot: DNN with ASI trick applied; green dashed: DNN with "doubling trick" applied; red dashed: the RHS of Eq. (6). (b) Blue: h K (x; 0, X, h ini (X)); red: h ini ; cyan: 20|h ini − h K (x; 0, X, h ini (X))|. (c) Evolution of loss functions of different DNNs during the training. Blue: the original DNN; red: DNN with ASI trick applied; black: DNN with RND applied; yellow: DNN with the "doubling trick" applied. The width of the original DNN is 1-5000-5000-1. v std = 10. The training and test data are randomly sampled from sin(4x) in [−1, 1] with size 10 and 500, respectively.
Figure 3 :
3Boston house price dataset. (a) Each dot represents outputs evaluated at one test point. The abscissa is h K (·; h ini , X,Y ) obtained using the original DNN. The ordinate for each blue dot is h K (x; 0, X,Y ) obtained using DNN with the ASI trick applied, whereas for each red dot is the RHS of Eq. (6). The black line indicates the identity function y = x. (b) The evolution of training loss (blue solid) and test loss (red dashed) of the original DNN, and the training loss (black solid) and test loss (yellow dashed) of DNN with ASI trick applied. The width of DNN is 13-100000-1. v std = 5.
Figure 4 :
4Effectiveness of ASI trick for MNIST dataset in the non-kernel regime of DNN. (a) Evolution of loss functions with the same legend as inFig. 3(b). (b) Evolution of the corresponding accuracy. The learning rate is 2 × 10 −7 . See main text for other settings.
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Appendix 9 NotationsΩ : a compact domain of R N I ; N I : dimension of input of DNN; f : target function, f ∈ L ∞ (Ω); N P : number of parameter of DNN; M: number of training samples; X: inputs of training set [x 1 , · · · , x M ] T ∈ R M×N I ; Y : outputs of training set [y 1 , · · · , y M ] T ∈ R M ; g(X): [g(x 1 ), · · · , g(x M )] T for any function g on Ω; h DNN (x, θ DNN ): output of DNN of parameters θ DNN at x; ∇ θ (·): [∂ θ (1) (·), · · · , ∂ θ ( N P ) (·)]; h(x, θ): linearized model of DNN defined by Eq. (1); D: a general differentiable loss function satisfying conditions (i)-(iii) in Section 3.1; K(·, ·): kernel function defined asthe solution of problem (5) depending on kernel K, initial function h ini , inputs X and outputs Y of training set.By applying the "doubling trick" (Note that, inChizat & Bach (2018), there is no bias term in the last layer), we obtain a new network with network parameters θ = [W[H]In general, the kernel can be decomposed as the summation of kernels with respect the tangent space of parameters of the neural network in each layer, that isTheorem 14. For the DNN initialized by θ , by applying the "doubling trick", for any m ≤ H − 2,
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)) is offset to 0. However, the kernel of layers H − 1 and H doubles, whereas the kernel of layers m ≤ H − 2 completely vanishes. Therefore, by applying the "doubling trick. h (x. which could have large impact on the training dynamics as well as the generalization performance of DNN outputTherefore, by applying the "doubling trick", h (x, θ(0)) is offset to 0. However, the kernel of layers H − 1 and H doubles, whereas the kernel of layers m ≤ H − 2 completely vanishes, which could have large impact on the training dynamics as well as the generalization performance of DNN output.
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"Robust parameter estimation of regression models under weakened moment assumptions",
"Robust parameter estimation of regression models under weakened moment assumptions"
]
| [
"Kangqiang Li \nSchool of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina\n",
"Songqiao Tang \nSchool of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina\n",
"Lixin Zhang \nSchool of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina\n"
]
| [
"School of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina",
"School of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina",
"School of Mathematical Sciences\nZhejiang University\n310027HangzhouZhejiangChina"
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| This paper provides some extended results on estimating parameter matrix of several regression models when the covariate or response possesses weaker moment condition. We study the M -estimator of Fan et al. (Ann Stat 49(3):1239-1266, 2021) for matrix completion model with (1 + )-th moment noise. The corresponding phase transition phenomenon is observed. When 1 > > 0, the robust estimator possesses a slower convergence rate compared with previous literature. For high dimensional multiple index coefficient model, we propose an improved estimator via applying the element-wise truncation method to handle heavy-tailed data with finite fourth moment. The extensive simulation study validates our theoretical results. | 10.1016/j.spl.2022.109678 | [
"https://export.arxiv.org/pdf/2112.04358v4.pdf"
]
| 247,158,485 | 2112.04358 | 2951262a458320f8879f598511d752f701c1863d |
Robust parameter estimation of regression models under weakened moment assumptions
Kangqiang Li
School of Mathematical Sciences
Zhejiang University
310027HangzhouZhejiangChina
Songqiao Tang
School of Mathematical Sciences
Zhejiang University
310027HangzhouZhejiangChina
Lixin Zhang
School of Mathematical Sciences
Zhejiang University
310027HangzhouZhejiangChina
Robust parameter estimation of regression models under weakened moment assumptions
Linear and nonlinear statistical modelsHeavy-tailed dataElement-wise truncationRobust estimation
This paper provides some extended results on estimating parameter matrix of several regression models when the covariate or response possesses weaker moment condition. We study the M -estimator of Fan et al. (Ann Stat 49(3):1239-1266, 2021) for matrix completion model with (1 + )-th moment noise. The corresponding phase transition phenomenon is observed. When 1 > > 0, the robust estimator possesses a slower convergence rate compared with previous literature. For high dimensional multiple index coefficient model, we propose an improved estimator via applying the element-wise truncation method to handle heavy-tailed data with finite fourth moment. The extensive simulation study validates our theoretical results.
Introduction
Under the traditional settings, sub-Gaussian assumption is often required for noise and design in regression problems. Due to the heavy-tailed phenomena of real-world data, in recent years, there has been a growing body of literature on the robust regression estimation when the covariate and response are heavy-tailed. For linear-type models, Fan et al. (2017) [5] applied the Huber (1964) [11]'s loss to the sparse linear model and showed that under (2 + )th moment assumption on the noise, the proposed estimator exhibits the same statistical error rate as that of the light-tail noise case. Further, Sun et al. (2020) [21] proposed the adaptive Huber regression and extended the result of Fan et al. (2017) [5] to the case of (1 + )-th moment condition on the noise. A tight phase transition for the estimation error of the regression parameter was established which parallelled those first discovered by Bubeck et al. (2013) [2] and Devroye et al. (2016) [4] for robust mean estimation without finite variance. Motivated by Sun et al. (2020) [21], Tan et al. (2022) [22] [12] applied Huber loss to construct robust covariance and precision matrix estimators. Zhu and Zhou (2020) [26] studied the corrupted general linear model with heavy-tailed data under finite fourth moment assumption.
For robust parameter estimation of sparse non-linear regression problem, Neykov et al. (2016) [17] analyzed least squares with L 1 penalization in high-dimensional single index model (SIM) under Gaussian designs. Plan and Vershynin (2016) [18] and Plan et al. (2017) [19] considered high-dimensional SIM under Gaussian and elliptical designs. Yang et al. (2018) [24] proposed a robust estimator of high-dimensional SIM when the covariate and response only have the bounded fourth moment. The proposed estimator achieves the optimal error bound via the truncation procedure. Furthermore, Goldstein et al. (2018) [9] analyzed high-dimensional SIM under heavy-tailed elliptical distribution. Na and Kolar (2021) [15] developed an estimation procedure of the parametric components in high-dimensional index volatility models under finite moments condition. Fan et al. (2022) [7] studied implicit regularization in SIM with general heavy-tailed data. As an extension of SIM, Na et al.
(2019) [14] considered high dimensional varying index coefficient model introduced by Ma and Song (2015) [13]. For estimating sparse parameter matrix, they required the existence of bounded 6-th moment of the design and response in order to obtain the optimal rate, but whether the moment constraint can be further relaxed is unknown. Meanwhile, it is worth noting that Fan et al. (2021) [6] tested the superiority of their estimator for trace regression via selecting the scaled Cauchy noise, beyond the corresponding theoretical condition. Motivated by those, a natural question arises:
Can we further generalize their results and obtain the sharp estimation rates?
To address this problem, on the basis of Fan et al. (2021) [6]'s work, we further study matrix completion model in which the noise distribution has no finite variance. The applicable condition of their M -estimator is broadened. Simultaneously, the smooth phase transition of the convergence rate is also observed. As a generalization of matrix completion model, we also consider robust parameter estimation of high-dimensional varying index coefficient model. To handle heavy-tailed data with only finite fourth moment, we give a robust element-wise truncated estimator (see (2)) based on the research of Na et al.
(2019) [14]. It turns out that under finite fourth moment assumption, our method shows the robustness against the low order moments. The proposed estimator can achieve the same statistical error rate as that of Na et al. (2019) [14] with finite fourth moment. Meanwhile, the data-driven method facilitates to calibrate truncation parameters and is more convenient than the cross-validation method of Na et al. (2019) [14]. Note that this paper investigates the mean estimation with homoscedastic noise, since bounded moments condition has been allowed for the variance estimation.
The remainder of our paper is organized as follows. In Section 2, we analyze two specific regression problems and derive the statistical error rates of the corresponding M -estimators under weaker moment assumptions. In Section 3, some numerical simulations on synthetic data are presented and show an agreement with the theoretical results. Concluding remarks are drawn in Section 4. All the proofs are presented in the supplementary material.
Notation
For any positive integer n, we denote the set {1, 2, . . . , n} by [n]. For two matrices X, Y ∈
R d 1 ×d 2 , X, Y := tr(X T Y ). For a matrix A = (a ij ) ∈ R d 1 ×d 2 , the max norm and Frobenius norm of A are defined as A max = max i∈[d 1 ],j∈[d 2 ] |a i,j | and A F = i∈[d 1 ],j∈[d 2 ] a 2 i,j respectively. A = tr √ A T A , A op = λ max (A T A), A 1,1 = i∈[d 1 ] j∈[d 2 ] |a i,j |, A ∞ = max i∈[d 1 ] j∈[d 2 ] |a i,j | and A L 1 = max j∈[d 2 ] i∈[d 1 ] |a i,j |.
Given two positive sequences {a n } ∞ n=1 and {b n } ∞ n=1 , we use the notation a n b n , if b n a n b n where a n b n means that there exists a positive constant C such that a n ≤ Cb n for all n.
2 Parameter matrix estimation of linear and nonlinear statistical models
In this section, we analyze two types of regression models and present statistical rates of the corresponding regularized estimators under weakened moment assumptions.
Low-rank matrix completion model with weaker moment
We first consider the following matrix completion model:
y = X, Θ + ε (1) where X is uniformly sampled from { √ d 1 d 2 · e j e T k } j∈[d 1 ],Θ = argmin Θ max≤R/ √ d 1 d 2 vec(Θ) T Σ XX vec(Θ) − 2 Σ yX , Θ + λ Θ where Σ XX = 1 n n i=1 vec(X i ) vec(X i ) T and Σ yX = 1 n n i=1 sign (y i ) (|y i | ∧ τ ) X i(1) Θ * F ≤ 1, Θ * max ≤ R/ √ d 1 d 2 , Θ * max / Θ * F ≤ R/ √ d 1 d 2 and rank (Θ ) ≤ r; (2) {X i } n i=1 are i.i.d. uniformly sampled from { √ d 1 d 2 ·e j e T k } j∈[d 1 ],k∈[d 2 ] and E E(|ε i | α X i ) log(d 1 ∨d 2 ) ≤ M α < ∞ for some α ∈ (1, 2].
Then for any δ > 1, choose τ
Lαn (d 1 ∨d 2 ) log(d 1 +d 2 ) 1 α and for some constant C > 0, λ = 4C (d 1 ∨ d 2 ) log(d 1 + d 2 ) n α−1 α L 1 α α δ + Rδ + L 1 α α , there exist constants {C i } 4 i=1 such that as long as n ≥ (d 1 ∨ d 2 ) log(d 1 + d 2 ), we have with the probability at least 1 − 2(d 1 + d 2 ) 1−δ − C 1 exp (−C 2 (d 1 + d 2 )), Θ − Θ F ≤ C 3 √ r (d 1 ∨ d 2 ) log(d 1 + d 2 ) n α−1 α L 1 α α δ + Rδ + L 1 α α , Θ − Θ * ≤ C 4 r (d 1 ∨ d 2 ) log(d 1 + d 2 ) n α−1 α L 1 α α δ + Rδ + L 1 α α where L α = 2 α−1 R α + eM 1/ log(d 1 ∨d 2 ) α .
Remark 1. According to Theorem 1, we obtain that for some α > 1, choosing τ
min{L 1 α α , L 1 2 2 } n (d 1 ∨d 2 ) log(d 1 +d 2 ) max{ 1 α , 1 2 } and λ min{L 1 α α , L 1 2 2 } (d 1 ∨d 2 ) log(d 1 +d 2 ) n min{ α−1 α , 1 2 } , Θ − Θ F √ r min{L 1 α α , L 1 2 2 } (d 1 ∨ d 2 ) log(d 1 + d 2 ) n min{ α−1 α , 1 2 }
with high probability. with high probability towards matrix compressed sensing.
Our simulation study confirms the above inference and the proof is omitted for less redundancy.
High-dimensional varying index coefficient model
As a generalization of model (1), in this subsection, we concentrate on robustly estimating the direction of parameters estimation of the following varying index coefficient model:
y = d 2 i=1 z i · f i ( X, θ i ) + ε
where X ∈ R d 1 and Z = (z 1 , z 2 , . . . , z d 2 ) T ∈ R d 2 are independent covariates, and ε is the stochastic error with E[ε | X, Z] = 0. We assume that θ i 2 = 1 for model identifiability and X has the known probability density function p(X).
Further, assume that the following two conditions hold:
Assumption 1. Assume that the covariate X has the differentiable density function
E y 4 ∨ E [S(X)] 4 j ∨ E z 4 k ≤ M, ∀j ∈ [d 1 ] , k ∈ [d 2 ]
where the first-order score function S : R d 1 → R d 1 is defined as S(X) := −∇p(X)/p(X).
E y · S(X)Z T Ω = d 2 j=1 E f j θ j , X S(X) E z j · Z T Ω = d 2 j=1
µ j θ j e T j := θ 1 , . . . , θ d 2 = Θ. In order to further relax the moment condition, we consider y i S(X i )Z T i as a matrix-valued data and then use ψ τ (x) = (|x| ∧ τ ) sign(x) to truncate each entry of the matrix-variate data. Specifically, the robust element-wise truncated matrix estimator is defined as
Θ = argmin Θ∈R d 1 ×d 2 Θ 2 F − 2 1 n n i=1 ψ Γ 1 y i S(X i )Z T i Ω, Θ + λ Θ 1,1(2)
where
Γ 1 = τ (1) j,k k∈[d 2 ] j∈[d 1 ]
is a truncation parameter matrix and Ω is obtained by Cai et al.
(2011) [3]'s CLIME procedure:
Ω = argmin Ω 1,1 s.t. Σ n Ω − I d 2 max ≤ γ,(3)
where
( Σ n ) j,k = 1 n n i=1 ψ τ (2) j,k z (i) j z (i) k .
The following theorem gives the statistical error rate of the robust estimator above. Then with the probability at least
1 − 2 (d 1 d 2 ) 2 − 1 d 2 2 − 1 d 3 2 , we have Θ − Θ F ≤ 2λ sd 2 and Θ − Θ 1,1 ≤ 8λsd 2 .
Remark 3. From the above result, we have with high probability,
Θ − Θ F sd 2 log(d 1 d 2 )
Simulation Study
In this section, we provide some numerical experiments to confirm the statistical error rates of the estimators established in previous section.
In matrix completion model, let [6]. The main difference is that we adapt repetitions. In Figure 1, the slopes of the fitted lines via the robust procedure become lower as α decreases, which is in keeping with Theorem 1. Besides, when the tail of the noise distribution is heavier, the robust estimator performs better than the standard procedure in which the responses are not clipped. with computational efficiency:
Θ = V 5 V T 5 / √ 5 where V 5 isΘ n i,j = n i=1 d 1 d 2 1 {Xi= √ d 1 d 2 e i e T j } , Θ s i,j = √ d 1 d 2 n i=1 y i 1 {Xi= √ d 1 d 2 e i e T j } in their algorithm and X i is sampled from { √ d 1 d 2 · e j e T k } j∈[d 1 ],n i=1 ψ 2 τ (1) j,k y i [S(X i )] j z (i) k / τ (1) j,k 2 = 10 log(d 1 d 2 ) and n i=1 ψ 2 τ (2) k,s z (i) k z (i) s / τ (2) k,s 2 = 10 log(d 2 ).
It is noteworthy that in the presence of heavy-tailed data and outliers, the above data-driven procedure can effectively select appropriate robustification parameters to truncate data. For each s, we gather all the data points log(ρ( Θ, Θ * )), n of the robust procedure to fit the linear regression relationship (i.e. log(ρ( Θ, Θ * )) = β 0 + β 1 log(n)). The fitting results are that for s = 5, β 1 = −0.4425 with multiple R 2 = 0.9993 and for s = 10, β 1 = −0.4825 with multiple R 2 = 0.9960. Therefore, Table 1 corroborates the result of Theorem 2 and shows that our proposed estimator has smaller statistical error than the standard procedure which means that the truncation techniques in (2) and (3) are not adopted.
Concluding remarks
In this article, we extend [6] has also a tight phase transition phenomenon by following the proof of Theorem 1. This strongly implies that the theoretical rate of Theorem 1 is sharp and we leave it as future research.
established the similar phase transition results on sparse reduced rank regression. Fan et al. (2021)[6] focused on robust estimation of the trace regression and their M -estimator achieves the minimax statistical error rate under only bounded (2 + )-th moment response or both bounded fourth moment design and response. Afterwards, Han et al (2021)[10] constructed a post-selection inference procedure via the Huber loss for high-dimensional linear model and Zhang (2021)[25] investigated Huber robust estimators for high-dimensional heavy-tailed time series. Avella-Medina et al. (2018)[1] and Ke et al (2019)
k∈[d 2 ] and E(ε|X) = 0. To recover the parameter matrix Θ under near low-rank assumption, Fan et al. (2021)[6] studied the following M -estimator of Θ :
with a truncation parameter τ . Under finite (2 + )-th moment condition on the response, their robust estimator has the theoretical statistical error rate of order r(d 1 ∨ d 2 ) log(d 1 + d 2 )/n under Frobenius norm, which is the same as that of Negahban and Wainwright (2012)[16] for sub-exponential noise. The following theorem further relaxes the distributional conditions from the bounded (2 + )-th moment to (1 + )-th moment assumption to fill the gap for the robust estimator's scope of use.
Theorem 1 .
1Suppose the following conditions hold:
Compared to the result of Fan et al. (2021)[6], when α < 2, there exists a smooth phase transition phenomenon for the statistical error rate which is in line with linear regression in Sun et al. (2020)[21] and mean estimation in Bubeck et al. (2013)[2]. The truncation parameter τ and the regularized parameter λ adapt to the moment of the noise. However, this transition is observed in the low-rank matrix completion model via the shrinkage technique, which is a visible difference with previous literature.Remark 2. If vec(X) is a sub-Gaussian vector, the phase transition phenomenon still holds for matrix compressed sensing and multitask regression of Fan et al. (2021)[6]. Specifically, when E E(|ε| α X) k ≤ M α for some α ∈ (1, 2], k > 1 and d 1 + d 2 ≤ n, by choosing τ
p(X) : R d 1 → R and the link functions {f i (·)|i ∈ [d 2 ]} are differentiable such that µ i := E[f i ( X, θ i )] = 0 for ∀i ∈ [d 2 ], and E[Z] = 0 d 2 ×1. Denote Σ := E ZZ T and Ω := (Σ ) −1 . We assume Ω ∈ Ω : Ω 0, Ω L 1 ≤ , max 1≤i≤d 2 d 2 j=1 |(Ω) i,j | q ≤ K for some and q ∈ [0, 1).Assumption 2.There exists an absolute constant M > 0 such that
Based on the above assumptions and first-order Stein's identity (Stein et al. (2004)[20]), according to Na et al. (2019)[14], we have
Therefore, a feasible method to estimate the direction of{θ i } d 1 i=1 without the knowledge of the link functions {f i (·)} i∈[d 2 ] is pointed out. Given n i.i.d. samples {y i , X i , Z i } n i=1 , Na et al. (2019)[14] proposed to separately truncate the data {y i , S(X i ), Z i } n i=1 via the function x = x1 {|x|≤τ } . By assuming 6-th moments of the covariate and response exist,
Theorem 2 .
2Suppose Assumption 1 and 2 hold with θ j 0 = s for all j ∈ [d 2 ] . For j ∈ [d 1 ] and k, s ∈ [d 2 ], choose τ
that the proposed estimator possesses the same statistical error rate as that ofNa et al. (2019)[14] with bounded 6-th moment assumption.
top 5 eigenvectors of d-dimensional sample covariance matrix from 100 i.i.d. standard Gaussian random vectors. We use almost the same algorithm (the ADMM method proposed by Fang et al. (2015)[8]) as that of Fan et al. (2021)
k∈[d 2 ] . For computational convenience, Fan et al. (2021)[6] proposed a robust cross-validation procedure without adhering to the derived rates of τ and λ. In this work, to demonstrate the phase transition of the statistical rate, we select C 1 n (d 1 ∨d 2 ) log(d 1 +d 2 ) τ and λ respectively. C 1 and C 2 are the fixed constants for each line in Figure 1. Consider the scaled Student's t ν distribution with ν ∈ {1.1, 1.5, 2} as the error distribution and take α = ν − 0.01 in the simulation. The numerical results are presented based on the mean of 200 independent
Figure 1 :Figure 2 :
12Matrix completion: The x-axis and y-axis represent logarithmic sample size and log Θ − Θ * F . Next, we select the following set of functions as the link functions {f i (·) : i ∈ [9]} to verify the behavior of the robust estimator in(2) with respect to the sample size: (a) :f 1 (x) = 4x cos 2 (5x), f 2 (x) = 4x sin 2 (5x), f 3 (x) = −5x/(2 + sin(x)); (b) : f 4 (x) = 4x + exp(x) 1+exp(x) , f 5 (x) = 2x + exp(−x 2 /7), f 6 (x) = −x + 5 cos(8x) and (c) : f 7 (x) = x + 4 sin(7x), f 8 (x) = −x + cos(3x 2 /2), f 9 (x) = −2x + 4 sin(x 2 /2).We set the dimensionality d 1 = 200 and for ∈ S k , [θ* k ] = Uniform ({−1, 1})/ √ s where S k is the support of θ k chosen at random on [d 1 ] with |S k | = s.We use the distance the estimation error. Let the entries of X and ε i.i.d. follow t 5 distribution. Z follows multivariate t 5 distribution where the precision matrix Ω is defined as (Ω) i,j = 0.5 |i−j| .
Fan et al. (2021)[6]'s work to the finite mean setting for heavy-tailed noise and observe a phase transition phenomenon by theory and experiment.Moreover, for high-dimensional varying index coefficient model, our proposed estimator is superior to Na et al. (2019)[14]'s robust estimator in two aspects. First, it allows the covariate and response to have bounded fourth moment. Second, tuning parameters via the data-driven procedure offers significant advantages in convenience and computing efficiency.The numerical experiments show that the improved estimator consistently performs better than the standard procedure and has consistency with the theoretical result. It is interesting that according toSun et al. (2020)[21], the proposed estimator for part (a) of Theorem 2 inFan et al. (2021)
Inspired byWang et al. (2021)[23], we solve the following adaptive equations to obtaintruncation parameters τ
(1)
j,k , τ
(2)
k,s
k,s∈[d 2 ]
j∈[d 1 ]
Table 1 :
1However, inNa et al. (2019)[14], each truncation parameter needs to be adjusted by cross validation, which is inconvenient in practice. The numerical experiments are repeated 50 times. The logarithmic estimation error with respect to the sample size for s = 5 and 10.s
n
10000
12500
15000
17500
20000
22500
25000
30000
35000
5
0.5647
0.4662
0.3829
0.3095
0.2620
0.2100
0.1540
0.0849
0.0046
Robust
0.7570
0.7019
0.6420
0.5759
0.5004
0.4340
0.4033
0.3709
0.2569
Standard
10
0.7144
0.6293
0.5547
0.4779
0.3928
0.3458
0.2835
0.1903
0.1207
Robust
0.8801
0.7942
0.7473
0.7197
0.6428
0.5947
0.5393
0.4899
0.3800
Standard
AcknowledgementWe thank the editor and two anonymous reviewers for detailed and insight comments.
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| []
|
[
"Multifractal Spin-Glass Chaos Projection and Interrelation of Multicultural Music and Brain Signals",
"Multifractal Spin-Glass Chaos Projection and Interrelation of Multicultural Music and Brain Signals"
]
| [
"E Can Artun \nFaculty of Engineering and Natural Sciences\nKadir Has University\n34083CibaliIstanbulTurkey\n",
"Ibrahim Keçoglu \nDepartment of Physics\nBogaziçi University\n34342BebekIstanbulTurkey\n",
"Alpar Türkoglu \nDepartment of Physics\nBogaziçi University\n34342BebekIstanbulTurkey\n\nDepartment of Electrical and Electronics Engineering\nBogaziçi University\n34342BebekIstanbulTurkey\n",
"A Nihat Berker \nFaculty of Engineering and Natural Sciences\nKadir Has University\n34083CibaliIstanbulTurkey\n\nDepartment of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA\n"
]
| [
"Faculty of Engineering and Natural Sciences\nKadir Has University\n34083CibaliIstanbulTurkey",
"Department of Physics\nBogaziçi University\n34342BebekIstanbulTurkey",
"Department of Physics\nBogaziçi University\n34342BebekIstanbulTurkey",
"Department of Electrical and Electronics Engineering\nBogaziçi University\n34342BebekIstanbulTurkey",
"Faculty of Engineering and Natural Sciences\nKadir Has University\n34083CibaliIstanbulTurkey",
"Department of Physics\nMassachusetts Institute of Technology\n02139CambridgeMassachusettsUSA"
]
| []
| A complexity classification scheme is developed from the fractal spectra of spin-glass chaos and demonstrated with multigeographic multicultural music and brain electroencephalogram signals. Systematic patterns are found to emerge. Chaos under scale change is the essence of spin-glass ordering and can be obtained, continuously tailor-made, from the exact renormalization-group solution of Ising models on frustrated hierarchical lattices. The music pieces are from Turkish music, | 10.1016/j.chaos.2022.113005 | [
"https://arxiv.org/pdf/2201.10261v1.pdf"
]
| 246,275,964 | 2201.10261 | aec929450e1ff5c4930f554d1f91cae990e35376 |
Multifractal Spin-Glass Chaos Projection and Interrelation of Multicultural Music and Brain Signals
25 Jan 2022
E Can Artun
Faculty of Engineering and Natural Sciences
Kadir Has University
34083CibaliIstanbulTurkey
Ibrahim Keçoglu
Department of Physics
Bogaziçi University
34342BebekIstanbulTurkey
Alpar Türkoglu
Department of Physics
Bogaziçi University
34342BebekIstanbulTurkey
Department of Electrical and Electronics Engineering
Bogaziçi University
34342BebekIstanbulTurkey
A Nihat Berker
Faculty of Engineering and Natural Sciences
Kadir Has University
34083CibaliIstanbulTurkey
Department of Physics
Massachusetts Institute of Technology
02139CambridgeMassachusettsUSA
Multifractal Spin-Glass Chaos Projection and Interrelation of Multicultural Music and Brain Signals
25 Jan 2022namely ArabesqueRapPopClassicaland Western musicnamely BluesJazzPopClassical A surprising group defection occurs
A complexity classification scheme is developed from the fractal spectra of spin-glass chaos and demonstrated with multigeographic multicultural music and brain electroencephalogram signals. Systematic patterns are found to emerge. Chaos under scale change is the essence of spin-glass ordering and can be obtained, continuously tailor-made, from the exact renormalization-group solution of Ising models on frustrated hierarchical lattices. The music pieces are from Turkish music,
One beauty of complex systems from different sources is similar properties hidden under the complicated behaviors, waiting to be unveiled. [1,2] On the other hand, "Chaos under scale change" as the distinctive characteristic of a spin-glass phase [3][4][5][6][8][9][10][11][12][13][14] and the multifractal spectrum quantification of exceedingly complicated data can be merged to create a classification scheme for complex systems. In this scheme, the spin glasses can provide a standart metric for the wide range of complex systems. The connections of the thus classified complex systems could then be achieved, for example connecting different cultural trends, geographies, and time periods.
In the current work, we build such a complexsystem classification scheme and illustrate its application with multicultural music and brain electroencephalogram (EEG) signals.
Frozen microscopic disorder introduced into an ordering system can immediately: Completely eliminate [15] or reduce an ordered phase, change the critical exponents of a second-order phase transition [16,17], convert a first-order phase transition into a second-order phase transition [18][19][20][21]. Competing microscopic interactions, with or without frozen microscopic disorder, furthermore can introduce a new phase, namely the spin-glass phase [22]. Competing microscopic interactions have two qualitatively different effects: (1) Two competing but non-cancelling (for example, due to unequal chain lengths) chains of interactions between two spatial points repress but do not eliminate the correlations. (2) On the other hand, if the local competing interactions cancel and create a local minimum energy degeneracy, correlations occur only in the form of the local configurations that participate into the degenerate local minimum energy. This effect is called frustration [23].
Hierarchical models [24][25][26] are exactly solvable microscopic models that are widely used. [27][28][29][30][31][32][33][34][35] The construction of a hierarchical model is illustrated in Fig.1(a) [24]. The hierarchical lattice constructed in Ref. [3], for the study of competing interactions, included both the repression and frustration effects explained above. This construction is shown in Fig.1(b,c) and includes, in its basic graph, the specified indices of the number p b of frustrated units, the number p of frustrating cross bonds, the number p c of repressed units, the lengths m 1 , m 2 > m 1 of the repressed chains.
Each line segment in Fig.1 represents a nearestneighbor spin-spin interaction Js i s j , where at each site i of the lattice the Ising spin s i = ±1. The Hamiltonian of the entire hierarchical lattice is
− βH = J ij s i s j ,(1)
where β = 1/kT and ij denotes summation over all nearest-neighbor site pairs. The exact renormalizationgroup solution of the hierarchical model proceeds in the direction reverse to its construction ( Fig.1), by summing over the internal spins shown with the dark circles. This summation yields the renormalization-group recursion relation J ′ = J ′ (J), which becomes chaotic in wide ranges in the space of the indices p, p b , p c , m 1 , m 2 = m 1 + ∆m chosen in the construction of the hierarchical model. Fig.2 shows the chaotic regions in cross sections of this multidimensional space of indices. Note islands that exclude chaos, such as the crescent and star region in the upper-right panel and all the lower panels in Fig.2. As explained in Ref. [3], chaotic renormalization-group trajectories signal the spin-glass phase, strong and weak correlations occurring in a random sequence at consecutive length scales and with extreme sensitivity to infinitesimal changes in external conditions such as temperature. [36] Chaotic renormalization-group trajectories and thus the spin-glass phase have subsequently been obtained in the approximate solution of systems with competing interactions, ferromagnetic versus antiferromagnetic interactions [37][38][39] or left-chiral versus right-chiral interactions [40][41][42], in cubic-lattice systems and re-wired [38] squarelattice systems. [3]. (e) A typical chaotic renormalizationgroup trajectory for this hierarchical lattice, e.g., occurring for index values p = 4, p b = 40, pc = 1, m1 = 7, m2 = m1 + 1. Each renormalization-group rescaling transformaton is a renormalization-group iteration, here consecutively denoted by n. For a given set of indices, renormalization-group trajectories, starting at any temperature within the spin-glass phase, fall to the same chaotic trajectory. The Lyapunov exponent for the chaotic trajectory shown here is calculated (Eq.2) to be λ = 0.353. The Lyapunov exponents λ indicate the strength of chaos [7] and vary with the index values p, p b , pc, m1, m2.
II. CHAOS UNDER SCALE CHANGE IN SPIN GLASSES: A CONTROLLABLE AND WIDE-RANGED BEHAVIOR FOR PROJECTION FROM COMPLEX SYSTEMS
Multifractal spectra are calculated for chaotic renormalization-group trajectories of the standart chaotic hierarchical lattice (Fig.1), music tracks from different geographies and cultural trends (Table I), brain EEG signals under different conditions, and, for future work, any complex system data set, as explained in Appendix A. For a given set of spin-glass indices p, p b , p c , m 1 , m 2 , the starting temperature T = J −1 of the chaotic renormalization-group trajectory is immaterial, as long as it is within the spin-glass phase, since a single chaotic trajectory is the asymptotic renormalizationgroup sink of the entire phase, attracting all initial conditions within the phase. For example, numerically identical Lyapunov exponents λ [43,44], quantifying the strength of chaos, are obtained for all initial conditions inside the phase:
λ = lim n→∞ 1 n n−1 k=0 ln dJ k+1 dJ k ,(2)
where J k is the value of the interaction constant in Eq.(1) after the kth renormalization-group transformation. Thus, our chaotic trajectories are taken after throwing out the first 1000 iterations and then using the next 2 13 iterations, to eliminate transient effects of the crossover to the sink.
In the examples shown in Fig.3, e.g., we see that there is much stronger chaos in the music (λ = 0.305) than in the brain signals (λ = 0.040). As shown in Fig.3, the spin-glass multifractal spectrum points are fitted by a continuous quartic function. The root-mean-square separation between the music (or brain EEG) multifractal spectrum points and the spin-glass continuous function is minimized by varying the original spin-glass indices of frustration number p b and strand length m 1 . At the end of this procedure, the statistical correlation coefficient R s between the spin-glass multifractal spectrum points and the spin-glass continuous function and the correlation coefficient R m between the music multifractal spectrum points and spin-glass continuous function R m together give the goodness of the fit between the spin-glass multifractal spectrum points and the music (or brain EEG) multifractal spectrum points. At the optimum fit, the calculated Lyapunov exponent λ measures the strength of the fitting chaos. genres, separately listed in Table I. The genres are from Turkish music, namely Arabesque, Rap, Pop, Classical; and Western music, namely Blues, Jazz, Pop, Classical. In each panel, for the indicated values of p, p c , ∆m, the values of the frustration number p b and repressed strand length m 1 are fitted to the music tracks. Thus, each panel is an alternate attempt to resolve the same data of 80 music tracks.
Systematic patterns emerge and are quite surprising. Firstly, essentially in all panels, the indices are organized in streaks that track each other. Blues, Jazz, and Western Classical (all from Western music), but not including Western Pop, stand apart. The other grouping is Turkish music (Arabesque, Rap, Pop, Classical), but also including Western Pop, all well mixed.
IV. CHAOTIC SPIN-GLASS CLASSIFICATION OF BRAIN EEG SIGNALS FROM DIFFERENTLY RESTING STATES AND CRANULAR REGIONS
Our calculated spin-glass multifractal indices for brain EEG signals, collected at different cranular probe locations and in the different states of music listening, resting with eyes open or closed, are displayed in Fig.5. [45][46][47] Again, an even clearer systematic pattern emerges. In the front cranular region, shown in the seven F panels, the indices fall on very well defined parallel streaks. There is predominant overlap between EEG signals from music listening and resting with eyes closed, as opposed to resting with eyes open. In the back cranular region, shown in the eight C panels, the separation is even clearer. For each panel, i.e., each back cranular location, the spin-glass indices of the brain EEG data separate into two nearorthogonal branches. The horizontal branch, namely the The SG multifractal spectrum points are fitted by a continuous quartic function. The root-mean-square separation between the music or brain EEG spectrum points and the SG continuous function is minimized by varying the original spin-glass indices of frustration number p b and repressed strand length m1. At the end of the procedure, the correlation coefficient Rs between the SG multifractal spectrum points and the SG continuous function, and the correlation coefficient Rm or Re between the music spectrum or brain EEG points and SG continuous function Rm together give the goodness of the fit between the SG multifractal spectrum points and the music or brain EEG spectrum points. The calculated Lyapunov exponent λ measures the strength of the fitting chaos. In this figure, the multifractal SG music fit is illustrated with Bul Beni, Ezhel (Table I) in the left panel and the multifractal SG brain EEG fit with Music Listening, cranular location Cz (Fig.5) in the right panel.
constant (low) frustration number p b branch, has the signals from music listening. The near-vertical branch, namely the near-constant (low) strand length m 1 , we find the EEG signals from music listening in its low p b segment, then the EEG signals from resting with eyes closed in the upper segment, and the EEG signals from resting with eyes open clustered at the top p b edge of the branch.
V. CONCLUSION
We have demonstrated that a complexity classification scheme can be developed from the fractal spectra of spinglass chaos. We examplified the procedure with 80 pieces multigeographic multicultural music from 8 genres and brain electroencephalogram signals. Systematic patterns and, in music, an interesting group defection is detected.
The tailor-made breadth and exact solution of spinglass chaos makes this procedure a very widely usable classification and analysis scheme for all sorts of complex data.
ACKNOWLEDGMENTS
We thank Ratip Emin Berker and Deniz Eroglu for very useful discussions and comments. Support by the Kadir Has University Doctoral Studies Scholarship Fund and by the Academy of Sciences of Turkey (TÜBA) is gratefully acknowledged.
Appendix A: Optimized Multifractal Sectra Fits between Chaotic Spin Glasses, Music, and Brain EEG
Two types of data have been used, the first of which is the experimental data (music or brain EEG) and the other is the chaotic spin-glass data, from the exact renormalization-group solution of the frustrated Ising model on a hierarchical lattice. We first fix the p, p c , ∆m = m 2 -m 1 values and do the renormalizationgroup transformation for each p b and m 1 value between 0-200. While applying the renormalization-group transformations, we take 2 13 iterations values after discarding the first 1000 iterations as crossover to asymptotic chaos. The occurrence of chaos for the given p b and m 1 values is checked by calculating the Lyapunov exponent. For chaos, we apply Chhabra-Jensen Algorithm [49] to obtain the multifractal spectrum f (α) from the recurring J values. Thus, a repository of chaotic p b and m 1 values and their correspondent f (α) is created to fit the music and brain EEG data.
To prepare the music data for Chhabra-Jensen algorithm, firstly the audio files have been transformed into time series data. We choose the left-ear channel in the time series. Then we discard the first 50000 data, which correspond mostly to silence, and take the next 2 ⌊log 2 (N )⌋ data, where N is the number of time series data points, because a power of 2 is needed in the Chhabra-Jensen algorithm. This data is normalized as
x ′ i = (x i −x i )/σ x ,
where σ x is the root-mean-square deviation of the data points andx i = 1/(1 + e xi ).
For the brain data, firstly the EEG data FIG. 4. Scatter plots of the fitted frustrated hierarchical Ising models to multicultural music tracks. Each point represents the fitted model indices for a single music track. There are ten tracks, listed in Table I [48] Then, a bandpass filter is applied, a common procedure when dealing with EEG data. To eliminate possible artifacts caused by the filtering, the first and last 3000 data points are discarded. Again, the first 2 ⌊log 2 (N )⌋ of the data is used and is normalized as described above.
Appendix B: Calculation of Multifractal Spectra of Chaotic Renormalization Group, Music, and Brain Signals
The Chhabra-Jensen function written by França et al. [49] is used, for all time-series data A = {a 1 , a 2 , ..., a N } composed of N data points. Firstly, we split the data into disjoint sub-blocks B i = {b i1 , b i2 , ..., b il } of length l. Then we calculate the consecutive probabilities of each block:
P i (l) = l j=1 b ij N j=1 a j .(B1)
We normalize the qth power of probabilities P i and call it µ i :
µ i (q, l) = P i (l) q j P j (l) q .(B2)
Then we calculate the Hölder exponent α and the corresponding Hausdorff dimension f (α) to obtain the multifractal spectrum, by applying log-log fits to M α (l) and M f (l):
M α (l) = i=1 µ i (q, l)log 10 (P i (l)),
M f (l) = i=1
µ i (q, l)log 10 (µ i (q, l)).
The slopes found after regressing the log-log plots of M α (l) and M f (l) are α and f (α). Repeating this fit for different q values, we obtain the multifractal spectrum.
FIG. 1 .
1Construction of the chaos hierarchical lattice, from Ref.[3]. (a) Construction of a standard hierarchical lattice, as explained in Ref.[24]. A graph is self-imbedded into each bond, ad infinitum. The exact renormalization-group solution proceeds in the reverse direction, by summing over the internal spins shown with the dark circles. In(b) and (c) are shown two graphs generic to the microscopics of spin glasses. The wiggly lines are infinite antiferromagnetic bonds, which have the sole effect of reversing the signs of the interactions adjoining on one side. (b) Frustrated units: Intermediate ferromagnetic (antiferromagnetic) ordering eliminates long-range antiferromagnetic (ferromagnetic) correlations. (c) Depressed units: The correlation on the shortest path sustains, but is depressed by the competing longer path. (d) Construction of the hierarchical lattice by combining the two generic units have yielded chaos in Ref.
[ 7 ]
7III. CHAOTIC SPIN-GLASS CLASSIFICATION OF MULTICULTURAL AND MULTIGEOGRAPHIC MUSICScatter plots of the fitted frustrated hierarchical Ising models to multicultural music tracks are displayed inFig. 4. Each point represents the fitted model indices for a single music track. There are ten tracks for each of the eightFIG. 2. Index regions of the frustrated hierarchical model that admit chaotic renormalization-group trajectories and, thus, a spin-glass phase. Note islands that exclude chaos, such as the crescent and star region in the upper-right panel and all the lower panels.
FIG. 3 .
3Best fitting of SG multifractal spectrum points with the music spectrum (left panel) or brain EEG (right panel) points:
FIG. 5 .
5SG multifractal parameters for EEG signals collected at different cranular probe locations, shown by the arrows, and in the different states of music listening, resting with eyes open or closed. Again, an even clearer systematic pattern emerges.
, for each of the eight genres. In each panel, for the indicated values of p, pc, ∆m, the values of the frustration number p b and repressed strand length m1 are fitted to the music tracks. Systematic patterns emerge. is imported to MATLAB(R2021a, The Math-Works, Inc., Natick) using FieldTrip Toolbox (https://doi.org/10.1155/2011/156869).
(Fig.4)multicultural music compositions by genre, title, and artist.
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Prediction of Reaction Time and Vigilance Variability from Spatio-Spectral Features of Resting-State EEG in a Long Sustained Attention Task. M Torkamani-Azar, S D Kanik, S Aydın, M Etin, IEEE J. Biomedical and Health Informatics. 242550M. Torkamani-Azar, S. D. Kanik, S. Aydın and M. Ç etin, Prediction of Reaction Time and Vigilance Variability from Spatio-Spectral Features of Resting-State EEG in a Long Sustained Attention Task, IEEE J. Biomedical and Health Informatics, 24 2550, (2020).
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. P Kraemer, 10.5281/zenodo.392609310P. Kraemer, EEG 10, 20. Zenodo. https://doi.org/10.5281/zenodo.3926093 (2020).
R Oostenveld, P Fries, E Maris, J.-M Schoffelen, FieldTrip: Open Source Software for Advanced Analysis of MEG, EEG, and Invasive Electrophysiological Data. 156869R. Oostenveld, P. Fries, E. Maris, and J.-M. Schoffelen, FieldTrip: Open Source Software for Advanced Analysis of MEG, EEG, and Invasive Electrophysiological Data, Computational Intelligence Neuroscience 2011, e156869 (2010).
L G S França, J G V Miranda, M Leite, N K Sharma, M C Walker, L Lemieux, Y Wang, Fractal and Multifractal Properties of Electrographic Recordings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications. 91767L. G. S. França, J. G. V. Miranda, M. Leite, N. K. Sharma, M. C. Walker, L. Lemieux, and Y. Wang, Frac- tal and Multifractal Properties of Electrographic Record- ings of Human Brain Activity: Toward Its Use as a Signal Feature for Machine Learning in Clinical Applications, Frontiers in Physiology 9, 01767 (2018).
Composition, Artist Composition, Artist. Composition, Artist Composition, Artist
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[
"On Degrees in the Hasse Diagram of the Strong Bruhat Order",
"On Degrees in the Hasse Diagram of the Strong Bruhat Order"
]
| [
"Ron M Adin [email protected] \nDepartment of Mathematics and Statistics\nBar-Ilan University\nRamat-Gan 52900Israel\n",
"Yuval Roichman [email protected] \nDepartment of Mathematics and Statistics\nBar-Ilan University\nRamat-Gan 52900Israel\n"
]
| [
"Department of Mathematics and Statistics\nBar-Ilan University\nRamat-Gan 52900Israel",
"Department of Mathematics and Statistics\nBar-Ilan University\nRamat-Gan 52900Israel"
]
| []
| For a permutation π in the symmetric group S n let the total degree be its valency in the Hasse diagram of the strong Bruhat order on S n , and let the down degree be the number of permutations which are covered by π in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.The Down, Up and Total DegreesDefinition 1.1 For a permutation π ∈ S n let the down degree d − (π) be the number of permutations in S n which are covered by π in the strong Bruhat order. Let the up degree d + (π) be the number of permutations which cover π in this order. The total degree of π is the sum d(π) := d − (π) + d + (π),i.e., the valency of π in the Hasse diagram of the strong Bruhat order. | null | [
"https://arxiv.org/pdf/math/0505020v2.pdf"
]
| 16,016,935 | math/0505020 | b8d2ec4f74ca3bb83e3a5ba3968a73cbd7651217 |
On Degrees in the Hasse Diagram of the Strong Bruhat Order
12 Mar 2006 March 12, 2006
Ron M Adin [email protected]
Department of Mathematics and Statistics
Bar-Ilan University
Ramat-Gan 52900Israel
Yuval Roichman [email protected]
Department of Mathematics and Statistics
Bar-Ilan University
Ramat-Gan 52900Israel
On Degrees in the Hasse Diagram of the Strong Bruhat Order
12 Mar 2006 March 12, 2006
For a permutation π in the symmetric group S n let the total degree be its valency in the Hasse diagram of the strong Bruhat order on S n , and let the down degree be the number of permutations which are covered by π in the strong Bruhat order. The maxima of the total degree and the down degree and their values at a random permutation are computed. Proofs involve variants of a classical theorem of Turán from extremal graph theory.The Down, Up and Total DegreesDefinition 1.1 For a permutation π ∈ S n let the down degree d − (π) be the number of permutations in S n which are covered by π in the strong Bruhat order. Let the up degree d + (π) be the number of permutations which cover π in this order. The total degree of π is the sum d(π) := d − (π) + d + (π),i.e., the valency of π in the Hasse diagram of the strong Bruhat order.
Explicitly, for 1 ≤ a < b ≤ n let t a,b = t b,a ∈ S n be the transposition interchanging a and b, and for π ∈ S n let ℓ(π) := min{k | π = s i 1 s i 2 · · · s i k } be the length of π with respect to the standard Coxeter generators s i = t i,i+1 (1 ≤ i < n) of S n . Then d − (π) = #{t a,b | ℓ(t a,b π) = ℓ(π) − 1} d + (π) = #{t a,b | ℓ(t a,b π) = ℓ(π) + 1} d(π) = d − (π) + d + (π) = #{t a,b | ℓ(t a,b π) = ℓ(π) ± 1} For the general definitions and other properties of the weak and strong Bruhat orders see, e.g., [9,Ex. 3.75] and [2, § §2.1, 3.1].
We shall describe π ∈ S n by its sequence of values [π(1), . . . , π(n)].
Observation 1.2 π covers σ in the strong Bruhat order on S n if and only if there exist 1 ≤ i < k ≤ n such that 1. b := π(i) > π(k) =: a. 3. There is no i < j < k such that a < π(j) < b. Remark 1.5 The classical descent number of a permutation π in the symmetric group S n is the number of permutations in S n which are covered by π in the (right) weak Bruhat order. Thus, the down degree may be considered as a "strong descent number". Definition 1.6 For π ∈ S n denote D − (π) := {t a,b | ℓ(t a,b π) = ℓ(π) − 1}, the strong descent set of π.
Example 1.7 The strong descent set of π = [7,9,5,2,3,8,4,1,6] is D − (π) = {t 1,2 , t 1,3 , t 1,4 , t 2,5 , t 3,5 , t 4,5 , t 4,8 , t 5,7 , t 5,9 , t 6,7 , t 6,8 , t 8,9 }.
Remark 1.8 Generalized pattern avoidance, involving strong descent sets, was applied by Woo and Yong [11] to determine which Schubert varieties are Gorenstein.
Proposition 1.9 The strong descent set D − (π) uniquely determines the permutation π.
Proof. By induction on n. The claim clearly holds for n = 1. Let π be a permutation in S n , and letπ ∈ S n−1 be the permutation obtained by deleting the value n from π. Note that, by Observation 1.2,
D − (π) = D − (π) \ {t a,n | 1 ≤ a < n}.
By the induction hypothesisπ is uniquely determined by this set. Hence it suffices to determine the position of n in π. Now, if j := π −1 (n) < n then clearly t π(j+1),n ∈ D − (π). Moreover, by Observation 1.2, t a,n ∈ D − (π) =⇒ a ≥ π(j + 1). Thus D − (π) determines π(j) = π(j + 1) = min{a | t a,n ∈ D − (π)}, and therefore determines j. Note that this set of a's is empty if and only if j = n. This completes the proof. ✷
Maximal Down Degree
In this section we compute the maximal value of the down degree on S n and find all the permutations achieving the maximum. We prove Proposition 2.1 For every positive integer n max{d − (π)| π ∈ S n } = ⌊n 2 /4⌋.
Remark 2.2
The same number appears as the order dimension of the strong Bruhat poset [7]. An upper bound on the maximal down degree for finite Coxeter groups appears in [4,Prop. 3.4].
For the proof of Proposition 2.1 we need a classical theorem of Turán.
Definition 2.3
Let r ≤ n be positive integers. The Turán graph T r (n) is the complete r-partite graph with n vertices and all parts as equal in size as possible, i.e., each size is either ⌊n/r⌋ or ⌈n/r⌉. Denote by t r (n) the number of edges of T r (n). (1) Every graph of order n with more than t r (n) edges contains a complete subgraph of order r + 1.
(2) T r (n) is the unique graph of order n with t r (n) edges that does not contain a complete subgraph of order r + 1.
We shall apply the special case r = 2 (due to Mantel) of Turán's theorem to the following graph.
Definition 2.5
The strong descent graph of π ∈ S n , denoted Γ − (π), is the undirected graph whose set of vertices is {1, . . . , n} and whose set of edges is
{{a, b} | t a,b ∈ D − (π)}.
By definition, the number of edges in Γ − (π) equals d − (π).
Remark 2.6
Permutations for which the strong descent graph is connected are called indecomposable. Their enumeration was studied in [5]; see [6, pp. 7-8]. The number of components in Γ − (π) is equal to the number of global descents in πw 0 (where w 0 := [n, n − 1, . . . , 1]), which were introduced and studied in [1, Corollaries 6.3 and 6.4].
Lemma 2.7 For every π ∈ S n , the strong descent graph Γ − (π) is trianglefree.
Proof. Assume that Γ − (π) contains a triangle. Then there exist 1 ≤ a < b < c ≤ n such that t a,b , t a,c , t b,c ∈ D − (π). By Observation 1.2,
t a,b , t b,c ∈ D − (π) =⇒ π −1 (c) < π −1 (b) < π −1 (a) =⇒ t a,c ∈ D − (π).
This is a contradiction. ✷ Proof of Proposition 2.1. By Theorem 2.4(1) together with Lemma 2.7, for every π ∈ S n d − (π) ≤ t 2 (n) = ⌊n 2 /4⌋.
Equality holds since
d − ([⌊n/2⌋ + 1, ⌊n/2⌋ + 2, . . . , n, 1, 2, . . . , ⌊n/2⌋]) = ⌊n 2 /4⌋. ✷
Next we classify (and enumerate) the permutations which achieve the maximal down degree.
Lemma 2.8 Let π ∈ S n be a permutation with maximal down degree. Then π has no decreasing subsequence of length 4.
Proof. Assume that π = [. . . d . . . c . . . b . . . a . . .] with d > c > b > a and π −1 (a) − π −1 (d) minimal. Then t a,b , t b,c , t c,d ∈ D − (π) but, by Obser- vation 1.2, t a,d ∈ D − (π). It follows that Γ − (π)
is not a complete bipartite graph, since {a, b}, {b, c}, and {c, d} are edges but {a, d} is not. By Lemma 2.7, combined with Theorem 2.4(2), the number of edges in Γ − (π) is less than ⌊n 2 /4⌋. ✷ Proposition 2.9 For every positive integer n #{π ∈ S n | d − (π) = ⌊n 2 /4⌋} = n, if n is odd; n/2, if n is even.
Each such permutation has the form π = [t + m + 1, t + m + 2, . . . , n, t + 1, t + 2, . . . , t + m, 1, 2, . . . , t],
where m ∈ {⌊n/2⌋, ⌈n/2⌉} and 1 ≤ t ≤ n − m. Note that t = n − m (for m) gives the same permutation as t = 0 (for n − m instead of m).
Proof. It is easy to verify the claim for n ≤ 3. Assume n ≥ 4.
Let π ∈ S n with d − (π) = ⌊n 2 /4⌋. By Theorem 2.4(2), Γ − (π) is isomorphic to the complete bipartite graph K ⌊n/2⌋,⌈n/2⌉ . Since n ≥ 4, each side of the graph contains at least two vertices. Let 1 = a < b be two vertices on one side, and c < d two vertices on the other side of the graph. Since t b,c , t b,d ∈ D − (π), there are three possible cases:
1. b < c, and then π = [. . . c . . . d . . . b . . .] (since π = [. . . d . . . c . . . b . . .] contradicts t b,d ∈ D − (π)). 2. c < b < d, and then π = [. . . d . . . b . . . c . . .]. 3. d < b, and then π = [. . . b . . . c . . . d . . .] (since π = [. . . b . . . d . . . c . . .] contradicts t b,c ∈ D − (π)).
The same also holds for a = 1 instead of b, but then cases 2 and 3 are impossible since a = 1 < c. Thus necessarily c appears before d in π, and case 2 is therefore impossible for any b on the same side as a = 1. In other words: no vertex on the same side as a = 1 is intermediate, either in position (in π) or in value, to c and d.
Assume now that n is even. The vertices not on the side of 1 form (in π) a block of length n/2 of numbers which are consecutive in value as well in position. They also form an increasing subsequence of π, since Γ − (π) is bipartite. The numbers preceding them are all larger in value, and are increasing; the numbers succeeding them are all smaller in value, are increasing, and contain 1. It is easy to check that each permutation π of this form has maximal d − (π). Finally, π is completely determined by the length 1 ≤ t ≤ n/2 of the last increasing subsequence.
For n odd one obtains a similar classification, except that the length of the side not containing 1 is either ⌊n/2⌋ or ⌈n/2⌉. This completes the proof. ✷
Maximal Total Degree
Obviously, the maximal value of the total degree d = d − + d + cannot exceed n 2 , the total number of transpositions in S n . This is slightly better than the bound 2⌊n 2 /4⌋ obtainable from Proposition 2.1. The actual maximal value is smaller. Theorem 3.1 For n ≥ 2, the maximal total degree in the Hasse diagram of the strong Bruhat order on S n is
⌊n 2 /4⌋ + n − 2.
In order to prove this result, associate with each permutation π ∈ S n a graph Γ(π), whose set of vertices is {1, . . . , n} and whose set of edges is
{{a, b} | ℓ(t a,b π) − ℓ(π) = ±1}.
This graph has many properties; e.g., it is K 5 -free and is the edge-disjoint union of two triangle-free graphs on the same set of vertices. However, these properties are not strong enough to imply the above result. A property which does imply it is the following bound on the minimal degree. Lemma 3.2 There exists a vertex in Γ(π) with degree at most ⌊n/2⌋ + 1.
Proof. Assume, on the contrary, that each vertex in Γ(π) has at least ⌊n/2⌋ + 2 neighbors. This applies, in particular, to the vertex π(1). Being the first value of π, the neighborhood of π(1) in Γ(π), viewed as a subsequence of [π(2), . . . , π(n)], consists of a shuffle of a decreasing sequence of numbers larger than π(1) and an increasing sequence of numbers smaller than π(1). Let a be the rightmost neighbor of π(1). The intersection of the neighborhood of a with the neighborhood of π(1) is of cardinality at most two. Thus the degree of a is at most n − (⌊n/2⌋ + 2) + 2 = ⌈n/2⌉ ≤ ⌊n/2⌋ + 1, which is a contradiction. ✷ Proof of Theorem 3.1. First note that, by definition, the total degree of π ∈ S n in the Hasse diagram of the strong Bruhat order is equal to the number of edges in Γ(π). We will prove that this number e(Γ(π)) ≤ ⌊n 2 /4⌋ + n − 2, by induction on n.
The claim is clearly true for n = 2. Assume that the claim holds for n − 1, and let π ∈ S n . Let a be a vertex of Γ(π) with minimal degree, and letπ ∈ S n−1 be the permutation obtained from π by deleting the value a (and decreasing by 1 all the values larger than a). Then e(Γ(π)) ≥ e(Γ(π) \ a),
where the latter is the number of edges in Γ(π) which are not incident with the vertex a. By the induction hypothesis and Lemma 3.2, e(Γ(π)) = e(Γ(π) \ a) + d(a) ≤ e(Γ(π)) + d(a) ≤ ⌊(n − 1) 2 /4⌋ + (n − 1) − 2 + ⌊n/2⌋ + 1 = ⌊n 2 /4⌋ + n − 2. (m ∈ {⌊n/2⌋, ⌈n/2⌉}), and the permutations obtained from π 0 by one or more of the following operations:
π → π r := [π(n), π(n − 1), . . . , π(2), π(1)] (reversing π), π → π s := π · t 1,n (interchanging π(1) and π(n)), π → π t := t 1,n · π (interchanging 1 and n in π).
Proof. It is not difficult to see that all the specified permutations are indeed extremal, and their number is as claimed (for all n ≥ 2). The claim that there are no other extremal permutations will be proved by induction on n. For small values of n (say n ≤ 4) this may be verified directly. Assume now that the claim holds for some n ≥ 4, and let π ∈ S n+1 be extremal. Following the proof of Lemma 3.2, let a be a vertex of Γ(π) with degree at most ⌊(n + 1)/2⌋ + 1, which is either π(1) or its rightmost neighbor. As in the proof of Theorem 3.1, letπ ∈ S n be the permutation obtained from π by deleting the value a (and decreasing by 1 all the values larger than a). All the inequalities in the proof of Theorem 3.1 must hold as equalities, namely: e(Γ(π) \ a) = e(Γ(π)), d(a) = ⌊(n + 1)/2⌋ + 1, and π is extremal in S n . By the induction hypothesis,π must have one of the prescribed forms. In all of them, {π(1),π(n)} = {m, m + 1} is an edge of Γ(π). Therefore the corresponding edge {π(1), π(n + 1)} (or {π(2), π(n + 1)} if a = π(1), or {π(1), π(n)} if a = π(n + 1)) is an edge of Γ(π) \ a, namely of Γ(π). If a = π(1), π(n + 1) then π(n + 1) is the rightmost neighbor of π(1), contradicting the choice of a. If a = π(n + 1) we may use the operation π → π r . Thus we may assume from now on that a = π(1).
Let N (a) denote the set of neighbors of a in Γ(π). Assume first that π = π 0 = [m + 1, m + 2, . . . , n, 1, 2, . . . , m] (m ∈ {⌊n/2⌋, ⌈n/2⌉}).
Noting that ⌈n/2⌉ = ⌊(n + 1)/2⌋ and keeping in mind the decrease in certain values during the transition π →π, we have the following cases:
(1) a > m + 1 : in this case 1, . . . , m ∈ N (a), so that d(a) ≤ n − m ≤ ⌈n/2⌉ = ⌊(n + 1)/2⌋ < ⌊(n + 1)/2⌋ + 1.
Thus π is not extremal.
(2) a < m : in this case m + 3, . . . , n + 1, m + 1 ∈ N (a), so that d(a) ≤ 1 + (m − 1) ≤ ⌈n/2⌉ < ⌊(n + 1)/2⌋ + 1.
Again, π is not extremal.
(3) a ∈ {m, m + 1} : in this case d(a) = 1 + m ≤ ⌊(n + 1)/2⌋ + 1, with equality iff m = ⌊(n + 1)/2⌋. This gives π ∈ S n+1 of the required form (either π 0 or π s 0 ).
A similar analysis forπ = π s 0 gives extremal permutations only for a ∈ {m + 1, m + 2} and d(a) = 3, so that n = 4 andπ = [2413] ∈ S 4 . The permutations obtained are π = [32514] and π = [42513], which are π rt 0 , π rst 0 ∈ S 5 , respectively.
The other possible values ofπ are obtained by the π → π r and π → π t operations from the ones above, and yield analogous results. ✷
Expectation
In this subsection we prove an exact formula for the expectation of the down degree of a permutation in S n .
Theorem 4.1 For every positive integer n, the expected down degree of a random permutation in S n is
E π∈Sn [d − (π)] = n i=2 i j=2 j k=2 1 i · (k − 1) = (n + 1) n i=1 1 i − 2n.
It follows that To prove Theorem 4.1 we need some notation. For π ∈ S n and 2 ≤ i ≤ n let π |i be the permutation obtained from π by omitting all letters which are larger than or equal to i. For example, if π = [6,1,4,8,3,2,5,9,7] then π |9 = [6,1,4,8,3,2,5,7], π |7 = [6, 1, 4, 3, 2, 5], and π |4 = [1, 3,2].
Also, denote by π |j the suffix of length j of π. For example, if π = [6,1,4,8,3,2,5,9,7] then π |3 = [5,9,7] and π |2 |4 = [3,2]. Let l.t.r.m.(π) be the number of left-to-right maxima in π:
l.t.r.m.(π) := #{i | π(i) = max 1≤j≤i π(j)} Lemma 4.3 For every π ∈ S n , if π −1 |i+1 (i) = j then d − (π |i+1 ) − d − (π |i ) = l.t.r.m.(π |i−j |i ).
Proof of Theorem 4.1. Clearly, for every π ∈ S n
d − (π) = n i=2 d − (π |i+1 ) − d − (π |i ) .
Thus, by Lemma 4.3,
d − (π) = n i=2 l.t.r.m.(π |i−j i |i ),
where j i is the position of i in π |i+1 , i.e., j i := π −1 |i+1 (i). Define a random variable X to be the down degree d − (π) of a random (uniformly distributed) permutation π ∈ S n . Then, for each 2 ≤ i ≤ n, π |i+1 is a random (uniformly distributed) permutation in S i , and therefore j = π −1 |i+1 (i) is uniformly distributed in {1, . . . , i} and π |i−j |i+1 is essentially a random (uniformly distributed) permutation in S i−j (after monotonically renaming its values). Therefore, by linearity of the expectation, (q + k − 1).
E[X] = n i=2 1 i i j=1 E[X i−j ] = n i=2 1 i i−1 t=0 E[X t ],(1)
It follows that, for t ≥ 1,
E[X t ] = 1 t! σ∈St l.t.r.m.(σ) = 1 t! d dq σ∈St q l.t.r.m.(σ) q=1 = 1 t! d dq t k=1 (q + k − 1) q=1 = 1 t! t r=1 1≤k≤t k =r k = t r=1 1 r .
Of course, E[X 0 ] = 0. Substituting these values into (1) gives
E[X] = n i=2 i−1 t=1 t r=1 1 i · r
and this is equivalent (with j = t + 1 and k = r + 1) to the first formula in the statement of the theorem.
The second formula may be obtained through the following manipulations:
E[X] = n i=2 i j=2 j k=2 1 i · (k − 1) = 2≤k≤j≤i≤n 1 i · (k − 1) = 2≤k≤i≤n i − k + 1 i · (k − 1) = 2≤k≤i≤n 1 k − 1 − 1 i = 2≤k≤n n − k + 1 k − 1 − 2≤i≤n i − 1 i = n n k=2 1 k − 1 − (n − 1) − (n − 1) + n i=2 1 i = n n i=1 1 i − 2n + n i=1 1 i . ✷ Proof of Corollary 4.2. Notice that n i=1 1 i = ln n + O(1).
(The next term in the asymptotic expansion is Euler's constant.) Substitute into Theorem 4.1 to obtain the desired result. ✷
Generalized Down Degrees
Definition 5.1 For π ∈ S n and 1 ≤ r < n let
D (r) − (π) := {t a,b | ℓ(π) > ℓ(t a,b π) > ℓ(π) − 2r}
the r-th strong descent set of π.
Define the r-th down degree as − (π) = d − (π). The (n − 1)-th strong descent set is the set of inversions:
D (n−1) − (π) = {t a,b | a < b, π −1 (a) > π −1 (b)}. Thus d (n−1) − (π) = inv (π),
the inversion number of π.
Observation 5.3 For every π ∈ S n and 1 ≤ a < b ≤ n, t a,b ∈ D The r-th strong descent graph of π ∈ S n , denoted Γ (r) − (π), is the graph whose set of vertices is {1, . . . , n} and whose set of edges is {{a, b}| t a,b ∈ D (r) − (π)}.
The following lemma generalizes Lemma 2.7.
Lemma 5.7 For every π ∈ S n , the graph Γ (r) − (π) contains no subgraph isomorphic to the complete graph K r+2 .
Proof. Assume that there is a subgraph in Γ (r) − (π) isomorphic to K r+2 . Then there exists a decreasing subsequence n ≥ a 1 > a 2 > · · · > a r+2 ≥ 1 such that for all 1 ≤ i < j ≤ r + 2, t a i ,a j are r-th strong descents of π. In particular, for every 1 ≤ i < r + 2, t a i ,a i+1 are r-th strong descents of π. This implies that, for every 1 ≤ i < r + 2, a i+1 appears to the right of a i in π. Then, by Observation 5.3, t a 1 ,a r+2 is not an r-th strong descent. Contradiction. ✷ Corollary 5.8 For every 1 ≤ r < n, max{d (r) − (π) | π ∈ S n } ≤ t r+1 (n) ≤ r + 1 2 n r + 1 2 .
Proof. Combining Turán's Theorem together with Lemma 5.7. ✷ Note that for r = 1 and r = n − 1 equality holds in Corollary 5.8.
Remark 5.9
For every π ∈ S n letπ be the permutation obtained from π by omitting the value n. If j is the position of n in π then d (r)
− (π) − d (r)
− (π) equals the number of (r − 1)-th almost left-to-right minima in the (j − 1)-th suffix ofπ, see e.g. [8]. This observation may be applied to calculate the expectation of d
2. σ = t a,b π, i.e., π = [. . . , b, . . . , a, . . .] and σ = [. . . , a, . . . , b, . . .].
Corollary 1. 3
3For every π ∈ S n d − (π) = d − (π −1
Equality holds since, letting m := ⌊n/2⌋, e(Γ([m + 1, m + 2, . . . , n, 1, 2, . . . , m])) = ⌊n 2 /4⌋ + n − 2.✷ Theorem 3.3#{π ∈ S n | d(π) = ⌊n 2 /4⌋ + n − 2} = n ≥ 6 is even;16, if n ≥ 5 is odd.The extremal permutations have one of the following forms:π 0 := [m + 1, m + 2, . . . , n, 1, 2, . . . , m]
Corollary 4. 2
2As n → ∞, E π∈Sn [d − (π)] = n ln n + O(n) andE π∈Sn [d(π)] = 2n ln n + O(n).
The first strong descent set and down degree are those studied in the previous section; namely, D(1)− (π) = D − (π) and d(1)
−−−
(π) if and only if π = [. . . , b, . . . , a, . . .] and there are less than r letters between the positions of b and a in π whose value is between a and b. (π) ∪ {t 6,9 , t 1,8 , t 4,9 , t 4,7 , t 3,9 , t 3,7 , t 2,9 , t 2,7 }. Corollary 5.5 For every π ∈ S n and 1 ≤ r (π) if and only if t π −1 (a),π −1 (b) ∈ D
) .
)Example 1.4 In S 3 , d − [123] = 0, d − [132] = d − [213] = 1, and d − [321] = d − [231] = d − [312] = 2. On the other hand, d[321] = d[123] = 2 and d[213] = d[132] = d[312] = d[231] = 3.
Acknowledgements. The concept of strong descent graph came up during conversations with Francesco Brenti. Its name and certain other improvements were suggested by Christian Krattenthaler. Thanks also to Nathan Reading, Amitai Regev, Alexander Yong, and the anonymous referees.
M Aguiar, F Sottile, arXiv:math.CO/0203282Structure of the Malvenuto-Reutenauer Hopf algebra of permutations. M. Aguiar and F. Sottile, Structure of the Malvenuto-Reutenauer Hopf algebra of permutations, arXiv:math.CO/0203282.
A Björner, F Brenti, Combinatorics of Coxeter Groups, Graduate Texts in Mathematics 231. New YorkSpringerA. Björner and F. Brenti, Combinatorics of Coxeter Groups, Grad- uate Texts in Mathematics 231, Springer, New York, 2005.
Modern Graph Theory. B Bollobás, Graduate Texts in Mathematics. 184SpringerB. Bollobás, Modern Graph Theory, Graduate Texts in Mathematics 184, Springer, New York, 1998.
Upper and lower bounds for Kazhdan-Lusztig polynomials. F Brenti, Europ. J. Combin. 19F. Brenti, Upper and lower bounds for Kazhdan-Lusztig polynomials, Europ. J. Combin. 19 (1998), 283-297.
Sur les coefficients de l'inverse de la série formelle n!t n. L Comtet, Compt. Rend. Acad. Sci. Paris A-B. 275L. Comtet, Sur les coefficients de l'inverse de la série formelle n!t n , Compt. Rend. Acad. Sci. Paris A-B 275 (1972), A569-A572.
Algebraic Enumeration. I M Gessel, R P Stanley, ; R Graham, Handbook of Combinatorics. M.I.T. Press2I. M. Gessel and R. P. Stanley, Algebraic Enumeration, in: Handbook of Combinatorics, Vol. 2, Eds. R. Graham et al., M.I.T. Press, 1995.
Order dimension, strong Bruhat order and lattice properties for posets. N Reading, Order. 19N. Reading, Order dimension, strong Bruhat order and lattice prop- erties for posets, Order 19 (2002), 73-100.
Generalized statistics on S n and pattern avoidance. A Regev, Y Roichman, Europ. J. Combin. 26A. Regev and Y. Roichman, Generalized statistics on S n and pattern avoidance, Europ. J. Combin. 26 (2005), 29-57.
. R P Stanley, Enumerative Combinatorics. 1Wadsworth and Brooks/ColeR. P. Stanley, Enumerative Combinatorics, Vol. 1, Wadsworth and Brooks/Cole, Monterey, CA, 1986.
An extremal problem in graph theory (Hungarian). P Turán, Mat. Fiz. Lapok. 48P. Turán, An extremal problem in graph theory (Hungarian), Mat. Fiz. Lapok 48 (1941), 436-452.
A Woo, A Yong, arXiv:math.AG/0409490When is a Schubert variety Gorenstein?. A. Woo and A. Yong, When is a Schubert variety Gorenstein?, arXiv:math.AG/0409490.
| []
|
[
"Joint PMD Tracking and Nonlinearity Compensation with Deep Neural Networks",
"Joint PMD Tracking and Nonlinearity Compensation with Deep Neural Networks"
]
| [
"Prasham Jain ",
"Lutz Lampe ",
"Jeebak Mitra "
]
| []
| []
| Overcoming fiber nonlinearity is one of the core challenges limiting the capacity of optical fiber communication systems. Machine learning based solutions such as learned digital backpropagation (LDBP) and the recently proposed deep convolutional recurrent neural network (DCRNN) have been shown to be effective for fiber nonlinearity compensation (NLC). Incorporating distributed compensation of polarization mode dispersion (PMD) within the learned models can improve their performance even further but at the same time, it also couples the compensation of nonlinearity and PMD. Consequently, it is important to consider the time variation of PMD for such a joint compensation scheme. In this paper, we investigate the impact of PMD drift on the DCRNN model with distributed compensation of PMD. We propose a transfer learning based selective training scheme to adapt the learned neural network model to changes in PMD. We demonstrate that fine-tuning only a small subset of weights as per the proposed method is sufficient for adapting the model to PMD drift. Using decision directed feedback for online learning, we track continuous PMD drift resulting from a timevarying rotation of the state of polarization (SOP). We show that transferring knowledge from a pre-trained base model using the proposed scheme significantly reduces the re-training efforts for different PMD realizations. Applying the hinge model for SOP rotation, our simulation results show that the learned models maintain their performance gains while tracking the PMD. | 10.1109/jlt.2023.3276373 | [
"https://export.arxiv.org/pdf/2209.10085v2.pdf"
]
| 252,407,732 | 2209.10085 | 1316893f8352e4c2555adbac40b25eb0ae6f39ba |
Joint PMD Tracking and Nonlinearity Compensation with Deep Neural Networks
Prasham Jain
Lutz Lampe
Jeebak Mitra
Joint PMD Tracking and Nonlinearity Compensation with Deep Neural Networks
JAIN et al.:JOINT PMD TRACKING AND NLC WITH DEEP NEURAL NETWORKS 1Index Terms-Deep neural networksnonlinearity compensa- tionpolarization-mode dispersionstate of polarizationtransfer learningonline learningoptical fiber communications
Overcoming fiber nonlinearity is one of the core challenges limiting the capacity of optical fiber communication systems. Machine learning based solutions such as learned digital backpropagation (LDBP) and the recently proposed deep convolutional recurrent neural network (DCRNN) have been shown to be effective for fiber nonlinearity compensation (NLC). Incorporating distributed compensation of polarization mode dispersion (PMD) within the learned models can improve their performance even further but at the same time, it also couples the compensation of nonlinearity and PMD. Consequently, it is important to consider the time variation of PMD for such a joint compensation scheme. In this paper, we investigate the impact of PMD drift on the DCRNN model with distributed compensation of PMD. We propose a transfer learning based selective training scheme to adapt the learned neural network model to changes in PMD. We demonstrate that fine-tuning only a small subset of weights as per the proposed method is sufficient for adapting the model to PMD drift. Using decision directed feedback for online learning, we track continuous PMD drift resulting from a timevarying rotation of the state of polarization (SOP). We show that transferring knowledge from a pre-trained base model using the proposed scheme significantly reduces the re-training efforts for different PMD realizations. Applying the hinge model for SOP rotation, our simulation results show that the learned models maintain their performance gains while tracking the PMD.
I. INTRODUCTION
C APACITY of optical fiber communication systems is limited by the non-linear Kerr effect [1]. During signal propagation, this nonlinearity interacts with linear effects such as chromatic dispersion (CD) [2] and polarization mode dispersion (PMD) [3]. Each impairment in isolation can be compensated with relatively simple solutions by solving the associated special case of the non-linear Schrödinger equation (NLSE). The optical receiver, typically composed of a cascade of digital signal processing (DSP) blocks, compensates each effect based on such model assumptions [4]. However, it is challenging to develop a method which can effectively capture the interplay of nonlinearity with linear distortions and invert its effect. The split-step Fourier method (SSFM) [5] is a popular numerical method which solves the propagation equations by splitting the link into steps and applying the linear and non-linear distortions independently. The core assumption of SSFM being that the distortions can be decoupled for a small enough step size. Among the conventional nonlinearity compensation (NLC) techniques, digital backpropagation (DBP) [6] leverages the SSFM to solve the inverse propagation equations. However, DBP becomes prohibitively expensive for long-haul fiber due to the large number of steps required to maintain performance.
Recently, machine learning based methods have shown immense potential for NLC by learning the non-linear transfer function directly from data. It has been shown that learned NLC methods can be effective even without any a-priori knowledge of the fiber parameters [7]. Parameterization of existing models is an effective approach for developing learned NLC methods. The learned digital backpropagation (LDBP) method [8] has been developed by parameterizing the SSFM for the standard NLSE. For dual-polarized (DP) transmission, it has been extended to include distributed compensation of PMD [9] by parameterizing the SSFM for the Manakov-PMD equations. The resulting approach is referred to as LDBP-PMD. While LDBP presents a significant improvement over conventional DBP, it still shares limited capability to capture the interplay of dispersion and nonlinearity, given the large step size required to maintain feasibility. Recently, we proposed the deep convolutional recurrent neural network (DCRNN) model [10] to overcome this limitation by using a bi-directional recurrent neural network to capture the interaction of nonlinearity and dispersion based on past (and future) symbols at each NLC step. Similar to [9], its extension DCRNN-PMD incorporates the distributed compensation of PMD.
However, inclusion of distributed PMD in the learned model results in the coupling of NLC with PMD compensation and thus, requires additional considerations since in practice, PMD may drift over time [11]. Learned NLC schemes in the literature often assume that the model can be trained offline for an arbitrarily long period of time using a sufficiently large training dataset from a stationary channel/distribution [12]. But when we consider time evolution of PMD, the distribution of observed data becomes non-stationary, a phenomenon referred to as concept drift in the machine learning literature [13]. For practical application, it is therefore essential that the learned model can adapt in real-time in the presence of concept drift using an online training scheme. Unfortunately, previous studies on adaptation of joint NLC and PMD compensation schemes have only considered instantaneous uncorrelated changes in the state of PMD, wherein re-training the converged model has been found to be equivalent to learning the PMD compensation parameters from scratch or worse [9]. Transfer arXiv:2209.10085v2 [eess.SP] 7 May 2023 learning has been shown to be an effective method to reduce the re-training overhead of learned NLC models [14], [15]. Also, unsupervised learning in the form of K-means clustering has been applied to train the model online [16]. However, these studies have not investigated the impact of time-varying channel impairments. On the other hand, previous works on adapting a learned model to continuous changes in PMD have either ignored the presence of nonlinearity or assumed that it has been compensated by other means [17], thereby, limiting the investigation to a purely linear channel. In fact, real-time adaptation of learned NLC methods to time-varying channel impairments has not been investigated so far in the literature.
In this paper, we propose a transfer learning based selective online training scheme to efficiently transfer knowledge from an offline trained base model to the filter in operation. As per the proposed scheme, we only update the learned weights of the PMD compensation layers while leaving the remaining learned model unaffected. This approach builds upon the training method applied in [9], where the learned coefficients from the LDBP model are frozen and serve as the initialization for the LDBP-PMD model. By limiting the trainable weights, the proposed approach greatly reduces the training overhead. We extend the method through incorporation of online learning, based on decision directed feedback, to perform lifelong real-time adaptation to changes in PMD. For concreteness, we apply the hinge model for time-evolution of the state of polarization (SOP) to simulate continuous drift in PMD [18]. While previous attempts to adapt a pre-trained learned NLC model to a different realization of PMD found limited success [9], our results from system simulations demonstrate that knowledge can be effectively transferred and performance can be sustained, thus bridging the gap between extended offline training and real world application of learned NLC methods.
In addition to the new adaptive training scheme, combining the principles of transfer learning and online learning for continuous adaptation of the learned NLC model, this paper extends the conference version [10] by providing a detailed construction of the DCRNN-PMD model, discussing various design choices and extending the numerical results to WDM transmission systems.
The remainder of this paper is organized as follows. In Section II, we briefly review the theoretical background, including the hinge model for continuous SOP drift. The DCRNN-PMD model and the transfer learning based online model adaptation scheme for tracking PMD drift are developed in Section III. In Section IV, we discuss the methodology applied to compare the performance-complexity trade-off of learned NLC models. In Section V, we provide details of our transmission system, the hyperparameters of the various compensation schemes and the NN training routine, followed by the performance and complexity results for various NLC techniques for both static and time-varying channels. Section VI provides concluding remarks for the paper.
II. MODELING OF FIBER PROPAGATION CHANNEL
In this section, we provide a brief review of the fiber model, which forms the basis for the design of the DCRNN-PMD model and provides support for the proposed adaptive training scheme. We also describe the hinge model, which is applied to simulate continuous PMD drift.
A. SSFM Based Channel Model
The propagation of a dual-polarized signal through a singlemode optical fiber, including birefringes and rotation of the principle states of polarization (PSP), can be described using the Manakov-PMD equation [19] as
∂E(z, t) ∂z = − α 2 E(z, t) + ∆β 1 Σ(z) ∂E(z, t) ∂t − j β 2 2 ∂ 2 E(z, t) ∂t 2 + jγ 8 9 E(z, t) E(z, t) 2 ,(1)
where E(z, t) = [E x (z, t), E y (z, t)] T , and E x (z, t) and E y (z, t) are the complex baseband signals of the X and Y polarizations, respectively. The signal is a function of propagation time t and distance 0 ≤ z ≤ L, where L is the total length of the fiber. The fiber parameters include the attenuation coefficient α, the group velocity dispersion coefficient β 2 responsible for CD, and the nonlinear coefficient γ. The coefficient ∆β 1 = (β 1x − β 1y )/2 represents the differential group delay (DGD) between the two polarizations along the PSP. In relation, the PMD parameter can be described as 2 √ 2L c ∆β 1 , where L c is the correlation length of the two polarizations. The matrix Σ(z) represents the PSP rotation at distance z, which causes a linear evolution of PMD along the length of the fiber.
We solve the Manakov-PMD equation numerically using the SSFM by splitting the fiber into sections of length L c . At each step, following the SSFM assumption, we apply the linear and non-linear distortions independently [5]. For a step of length L c , the CD effect is applied as 1
E x (z + L c , f ) =Ẽ x (z, f ) exp (j2β 2 π 2 f 2 L c ),(2)
whereẼ x (z, f ) is the Fourier transform of E x (z, t), and f is the baseband frequency. The signal attenuation for each step is applied as
E x (z + L c , t) = E x (z, t) exp (−αL c /2).(3)
The Kerr nonlinearity is dependent upon the energy of the baseband signal of both polarizations [1]. For dual-polarized transmission, it can be applied in the time domain as
E x (z + L c , t) = E x (z, t) exp −j 8 9 γ(|E x (z, t)| 2 + |E y (z, t)| 2 )L eff ,(4)
where
L eff = Lc 0 exp (−αz)dz = 1 − exp (−αL c ) α(5)
is the effective nonlinear step length accounting for the attenuation of signal along the length of the step. Following the simplified polarization impairment model [20], the PMD combined with the rotation of SOP (RSOP) for the k th step is applied as the rotation of the PSP followed by the DGD operator as
E(z + L c , f ) =Ẽ(z, f )R(α (k) )D (k) (f ),(6)
where
D (k) (f ) = exp (jπf τ (k) ) 0 0 exp (−jπf τ (k) )(7)
for DGD τ (k) at the k th step, and R(α (k) ) ∈ SU (2) is the PSP rotation matrix. The rotation matrix can be formulated as
R(α (k) ) = exp (−jα (k) · − → σ )),(8)
where α (k) ∈ R 3 and − → σ = (σ 1 , σ 2 , σ 3 ) is a tensor of the three Pauli spin matrices
σ 1 = 1 0 0 −1 , σ 2 = 0 1 1 0 , σ 3 = 0 −j j 0 .(9)
Note that there can be other formulations of the PSP rotation matrix which belong to the special unitary group of degree 2. Rotation matrices for each step are generated randomly such that they are uniformly distributed over the surface of the Poincaré sphere. Different PMD realizations can be generated by choosing a different set of randomly generated DGD and PSP rotation matrices [18].
B. Hinge Model for Continuous Polarization Drift
In practical fibers, the PMD may drift over time due to environmental factors such as temperature, pressure, cable orientation, stress, cable bends, vibrations, etc [11]. As opposed to the classical PMD literature which assumes all sections to drift, field measurements from various experiments suggest that temporal changes arise from a relatively small number of "points of stress" in the fiber which may be exposed to the environment [21]. This gives rise to the hinge model, where most of the SOP rotation matrices are considered to be static while only the few polarization scramblers at the "points of stress", referred to as hinges, drift over time. Each hinge J (t) ∈ SU (2) can be formulated as a waveplate and the evolution of J (t) over time can be modelled as [18]
J (t) = J (α(t))J (t − 1),(10)
where J (α(t)) is a randomly generated innovation matrix of the same form as the PSP rotation matrix defined in equation (8). The parameters of the innovation matrix are independently drawn from the following Gaussian distribution for each time instance [18]:
α(t) ∼ N (0, σ 2 p I 3 ),(11)
where σ 2 p = 2π∆pT , ∆p is referred to as the polarization linewidth and T is the time duration between updates. To incorporate continuous PMD drift in the split-step method, we introduce a small number of time-varying hinge matrices at regular intervals across the length of the fiber. In our simulation, we add one hinge at the end of each span considering that the environmental effects would be more prominent near the optical amplifier.
III. MACHINE LEARNING FOR JOINT COMPENSATION OF NONLINEARITY AND PMD
In this section, we derive the DCRNN-PMD model with distributed compensation of PMD and present the proposed selective adaptation scheme to efficiently transfer knowledge from an offline trained model, including the incorporation of online learning to develop a real-time adaptation technique.
A. Deep Convolutional Recurrent Neural Network
For clarity, in the following we use DCRNN and DCRNN-PMD to differentiate the variants of the proposed model with lumped and distributed PMD compensation, respectively. Additionally, we use DCRNN(-PMD) when referring to common attributes of both models. Fig. 1a shows the structure of the DCRNN model for compensating the various impairments. In addition, the DCRNN-PMD model also includes the decomposed form of the modified linear filter depicted in Fig. 1b.
1) Learned Chromatic Dispersion Compensation:
In the DCRNN(-PMD) models, we follow the time domain implementation of CD compensation (CDC) using a finite impulse response (FIR) filter, since it is more efficient [8]. We implement the FIR filter using a one-dimensional (1D) complexvalued convolutional layer. Based on our understanding of the CD effect, as discussed in the previous section, we use the same weights within the convolutional layer for both polarizations. Also, to save computational complexity, we make the convolutional filters symmetrical, which is consistent with the physical model. Some previous works have considered initialization of the convolutional layer based on the associated analytical solution for either constant-linear or modlogarithmic step size [9], [12]. Although such initialization results in faster convergence, we suspect that it may have a tendency of getting trapped at a local optima during the training process. Therefore, we initialize the filter weights randomly by sampling values from a uniform distribution U(− √ 2l + 1, √ 2l + 1), where (2l + 1) is the width of the convolutional kernel. The output of the i th convolution step can be written as
z x i (k) = l n=−l a i (n)x i (k + n), z y i (k) = l n=−l a i (n)y i (k + n),(12)
where a i (n) are trainable complex-valued weights at the i th step and (x i (n), y i (n)) are the output symbols of X and Y polarizations from the (i−1) th step, respectively, starting with the received symbols at the first step. In the DCRNN model, the output of the CD compensation layerz
x/y i (k) is directly input to the RNN based NLC layer as z Fig. 1a). At the end of the DCRNN, a two-dimensional (2D) complexvalued convolutional output layer is added to mimic a 2 × 2 multiple-input multiple-output (MIMO)-FIR filter for lumped PMD compensation. 2) Distributed Compensation of PMD: In the DCRNN-PMD model, a second linear compensation is added at each step for distributed PMD compensation. Based on our discussion of the fiber channel model in the previous section, it can be decomposed into two parts, DGD and rotation of PSP. We follow the design of DGD filter proposed in [9], using short real-valued FIR filters, implemented as 1D convolutions. Based on the description of DGD in (7), we use the same weights for both polarizations but in "flipped" order. The application of the DGD filter at the i th step can be written asẑ
x/y i (k) (seex i (k) = m n=−m d i (n)z x i (k + n), z y i (k) = m n=−m d i (−n)z y i (k + n),(13)
where (2m + 1) is the width of the DGD filter and d i (n) are trainable real-valued weights. This is followed a 2 × 2 complex-valued PSP rotation matrix applied as
z x i (k) z y i (k) = p 00 i p 01 i p 10 i p 11 i ẑ x i (k) z y i (k) .(14)
Fig. 1b shows the decomposition of the 2 × 2 MIMO-FIR filter, which compensates for all linear impairments, into the component layers for CD, DGD and PSP rotation as described in (12), (13) and (14) respectively. We find that the best performance is obtained when we use free weights for the PSP matrices. However, initializing each PSP rotation matrix as identity matrix I 2 has been shown to slightly improve the convergence speed of the model [9]. Other linear impairments such as signal attenuation are assumed to be handled implicitly within the learned CD and PMD compensation layers.
While the use of free weights in the linear compensation layers provides the best performance, it should be highlighted that different random initializations may yield different performance levels from the trained NN model. In this paper, we present and compare the best performance achieved by each model over multiple random initializations.
3) RNN Based Nonlinearity Compensation: As discussed in the previous section, the nonlinear phase depends on the energy of the signal from both polarizations. However, for large step size, it is insufficient to simply consider the energy of samples at the current time-step. Rather, the energy of all samples within the dispersion spread of that step must be considered to account for the interplay of dispersion and nonlinearity [22]. In the DCRNN(-PMD) models, we introduce a novel bidirectional recurrent neural network (BiRNN) layer which efficiently processes the energy of past (and future) samples as a sequence by preserving the information in its memory. The incorporation of the BiRNN layer presents a generalized parameterization of the low-pass filtered DBP [23]. To save computational complexity, we process the received signal in large blocks and only consider the output of the center RNN cells from each direction. This cost saving strategy is similar to the center-oriented long short term memory (Co-LSTM) model [24] except that we only use ordinary RNN cells to reduce cost further by a factor of 4. The nonlinear phase corrections φ x i and φ y i , highlighted in Fig. 1a, are estimated as
φ x i (k) = (f x i ) T h f,i (k) + (b x i ) T h b,i (k), φ y i (k) = (f y i ) T h f,i (k) + (b y i ) T h b,i (k),(15)
where
h f,i (k) = tanh(W f,i [h T f,i (k−1), |z x i (k)| 2 , |z y i (k)| 2 ] T ), h b,i (k) = tanh(W b,i [h T b,i (k+1), |z x i (k)| 2 , |z y i (k)| 2 ] T ),(16)
and W ,i ∈ R n h ×(n h +2) , ∈ {f, b}, and f p i ∈ R n h , b p i ∈ R n h , p ∈ {x, y}, are trainable real-valued weights, and n h is the number of hidden units in each RNN. Each step ends with the application of NLC as
x i+1 (k) = z x i (k)e −jφ x i (k) , y i+1 (k) = z y i (k)e −jφ y i (k) .(17)
Our results in Section V suggest that ordinary RNN cells are sufficient to capture the interplay of dispersion and nonlinearity for the length of the step. As indicated in Fig. 1a, the described compensation steps are repeated N times, i.e., i = 0, 1, . . . , N − 1.
B. Transfer Learning Based Continuous Adaptation
To perform joint compensation of nonlinearity and PMD, it is essential that the learned model is able to adapt to changes in PMD. The current state of PMD in a fiber link may be different from that of the training data on which the model is initially trained.
1) Retraining: It is prohibitively expensive to recollect the required training data and rebuild the model. Since there could be infinitely many PMD realizations, even in links drawn from the same spool of fiber [21], it is essential that we minimize the retraining effort for every installation. Therefore, we turn our attention towards transfer learning (TL) [25]. The important question in TL is "what to transfer?". In the DCRNN(-PMD) models introduced above, we can clearly interpret the function of each layer based on the model design. In theory, the parameters of the CD and NLC layers should not require adaptation if only the PMD effect is different. Therefore, we can reuse (transfer) the associated learned coefficients without any adaptation. In [9], the CD parameters from the pre-trained LDBP model are similarly transferred to initialize the LDBP-PMD model. In our proposed selective training scheme, we freeze the CD and NLC layers and only re-train the parameters associated with PMD compensation. Consequently, for lumped PMD compensation, only the output layer of the model needs to be re-trained. For distributed PMD compensation schemes, i.e., DCRNN-PMD, only the DGD filter and the PSP rotation matrices within each step need to be updated. In Section V, we show that this significant reduction in trainable parameters greatly reduces the re-training efforts for knowledge transfer.
2) Tracking: In contrast to [9], we consider continuous evolution in the state of PMD, where each successive state is related to the previous state as described in (10). In order to continuously adapt the model in real-time, we combine the transfer learning solution with online learning in the proposed adaptive scheme. We divide the training process into two parts: acquisition and tracking. For acquisition, we send pilot symbols through the channel to fine-tune the model parameters until the performance of the filter stabilizes. Considering the associated overhead, during normal operation, we can only send a very limited number of pilots in each transmitted block which may be insufficient for sustained model adaptation. Therefore, after acquisition, we move into a decision directed tracking stage where we only send information symbols. The learned filter performs equalization in small blocks. At the tracking stage, the hard decoded output symbols from the previous processed block are used to update the filter in operation.
Note that the choice of block size is directly related to the delay in feedback to the model. For continuous adaptation, a small feedback delay (FD) is desired to avoid mismatch between the training and test distributions. Consequently, a trade-off emerges between the feedback delay and the computational complexity of the model for which, as discussed in Section III-A3, a larger block size is desirable. This trade-off has not been studied in the present work and would be a crucial line of inquiry for future investigations. Another key difference in the online training routine is the use of stochastic gradient descent (SGD) optimizer without momentum as opposed to the Adam optimizer employed predominantly in offline training. We find that the use of momentum negatively impacts the online training process. This is because the momentum term tries to push the gradient in the same direction as the previous iterations, which is no longer warranted since the underlying gradient surface is continuously evolving due to PMD drift. In the next section, we discuss the methods used to analyze the performance-complexity profile of the learned NLC schemes.
IV. PERFORMANCE-COMPLEXITY ANALYSIS OF NEURAL NETWORK BASED NLC SCHEMES
In order to compare the computational complexity of learned and deterministic NLC techniques, it is important to acknowledge that various learned models could have different levels of redundancy due to the presence of unimportant parameters within the neural networks. Therefore, for a fair comparison and practical application, it is essential that we reduce the complexity of each model down to its optimal level. For neural networks, model compression can be achieved by weight pruning, which has been shown to be effective for complexity reduction of NN-based NLC [26]. We perform pruning as a semi-structured iterative process. Each iteration comprises of two basic steps: pruning and retraining. In the pruning step, we remove 20% of the smallest weights within each layer of the model independently, since it is non-trivial to compare the weights of different layers. For retraining, we apply learning rate rewinding (LRR) based on the lottery ticket hypothesis (LTH) [27]. In LRR, we reset the learning rate schedule to an earlier iteration in the training process prior to fine-tuning the pruned network. This mimics the effect of rewinding the pruned model to an earlier training iteration which, as per LTH, allows the model to achieve a better optima.
In the literature, the performance complexity trade-off of various learned NLC models is predominantly examined by training differently sized variants of the model and measuring their performance [8], [12], [28]. However, in our study, we observed that model performance falls much more sharply if we train a smaller neural network from the beginning. By comparison, it is better to start with an overbuilt neural network and iteratively prune the weights to reduce its complexity. In [29], Chang et al. identify a double descent behavior, where the risk of the pruned model is consistently improved with overparameterization. Therefore, in this study, we examine the performance complexity trade-off of learned NLC models by measuring the performance of the fully parameterized base model at various iterations of the pruning process. The results obtained from this approach are discussed in Section V-C.
In the case of online learning for tracking continuous changes in PMD, it is also important to consider the complexity of the optimization step performed using backpropagation [30]. During online training, each sample is only used in a single update iteration, thus, the computational complexity of the backpropagation remains of the same order as that of the forward propagation. Therefore, reducing the number of parameters through pruning directly improves the associated training overhead. The number of computations required are further reduced since the updated weights for non-PMD related layers need not be calculated. During this study, we have assumed that the computation of updated weights during online training can be performed as a parallel process.
V. NUMERICAL RESULTS AND DISCUSSION
In this section, we present simulation results (i) to highlight the benefits of the proposed DCRNN(-PMD) models in terms of performance-complexity trade-off and (ii) to demonstrate the effectiveness of the proposed transfer and online learning for adjusting to PMD changes.
A. Simulation Setup
We test the proposed and benchmark NLC methods on two configurations of a 32 GBd dual-polarized 64-QAM transmission system over a 12 × 80 km standard singlemodel fiber (SSMF), with a single channel and with 5 WDM channels, consistent with previous works on NN based NLC [7], [10] to facilitate comparison. The detailed parameters are provided in Table I. We simulate the forward transmission using the SSFM, as described in Section II-A by dividing the fiber into 500 uniform steps per span (StPS) with an erbiumdoped fiber amplifier (EDFA) at the end of each span. As discussed in Section II-A, different choices of DGD and PSP rotation matrices generate different realizations of PMD. We generate data from 8 different PMD realizations to train and test the learned models. At the receiver, the signal is coherently detected and sampled at 2 samples per symbol. A root raised cosine (RRC) matched filter is deployed when performing CDC and its position in the transmission chain is maintained while transitioning to various NLC schemes. Fig. 2 depicts the processing at the receiver for various learned and deterministic NLC methods considered for comparison. Downsampling to symbol space is performed implicitly within the output layer of the learned NLC solution. For deterministic methods, it is performed by the lumped PMD compensation filter described in the following section. Next, we describe the implementation and (hyper)parameters of the various NLC schemes applied.
B. Setup of NLC Methods
1) Baseline Deterministic Methods:
We compare the learned model against conventional DBP [6] with uniform step size at 1, 2 and 3 StPS using 2 samples per symbol. At each step, the CDC is performed using a frequency domain equalizer followed by the NLC step applied aŝ
E x (t) =Ẽ x (t) exp (−jξγΛ(|Ẽ x (t)| 2 + |Ẽ y (t)| 2 )L eff ), E y (t) =Ẽ y (t) exp (−jξγΛ(|Ẽ x (t)| 2 + |Ẽ y (t)| 2 )L eff ),(18)
where (Ẽ x (t),Ẽ y (t)) are the output of the CDC step, Λ is the relative power scaling factor to account for signal attenuation (amplification in case of backward propagation) over the step, and ξ is a scaling coefficient numerically optimized for each launch power. To compensate for PMD, a least mean squares (LMS) based adaptive 2 × 2 MIMO-FIR filter is applied after the DBP operation with 19 taps, in accordance with the PMD parameter.
2) Baseline Learned Methods: Among the learned NLC models, we consider LDBP with both lumped and distributed compensation of PMD [8], [9]. In both cases, we consider a model with 1 StPS operating at 2 samples per symbol. All the complex-valued convolutional CD filters have the same length with 29 taps each, chosen based on the dispersion spread of the fiber. For lumped PMD compensation, we add an additional 2D complex-valued convolutional layer at the end, identical to the 2×2 MIMO-FIR filter used for deterministic methods. The complete model, including the lumped PMD compensation layer is trained together. For distributed PMD compensation, we include DGD filters and rotation matrices at each step, as described in Section III-A2. All real valued DGD filters have the same length with 3 taps each and the same weights are used for both polarizations but in inverted order.
3) Proposed DCRNN Model: For the proposed DCRNN(-PMD) models, we again consider both lumped and distributed PMD compensation to highlight the associated performance gain. For a consistent comparison, we use the same kernel width for the complex valued convolutional CD filter and the real valued DGD filter as that used in the LDBP-PMD model. For the BiRNN based NLC layer, we use just two ordinary RNN units each in both the forward and backward RNN.
4) Training of Learned Models: All learned models are implemented using PyTorch [31] and initially trained offline using the Adam optimizer [32] of stochastic gradient descent with a cosine annealed learning rate schedule [30]. The learning rate for each iteration can be defined as
η n = η min + 1 2 (η max − η min ) 1 + cos n N iter π ,(19)
where η n is the learning rate for the n th training iteration, N iter is the total number of iterations, and η min and η max are the minimum and maximum learning rates respectively. We switch to the SGD optimizer without momentum as per the proposed online training scheme discussed in Section III-B with a constant learning rate to adapt the offline trained model under PMD drift. The complete list of training parameters are reported in Table II.
C. Numerical Results
We measure the performance of each method based on the Q-factor [33] for the channel of interest (CoI) calculated from the BER as
Q[dB] = 20 log 10 √ 2 erfc −1 (2BER) .(20)
The BER is obtained by directly counting the errors. We evaluate the performance gains of each NLC scheme against that of linear compensation only using a lumped frequency domain CD equalizer followed by a lumped LMS based MIMO-FIR PMD filter. 1) Static Channel: We first look at the performance of each method in case of a static channel without considering time evolution of PMD. Fig. 3 and Fig. 4 show the Q-factor as a function of launch power for single channel and WDM The results show a strong impact of the interaction between nonlinearity and dispersion. Remarkably, using just 2 ordinary RNN cells in each BiRNN layer, the DCRNN model is able to capture and compensate for this in-teraction. 2 Furthermore, using distributed PMD compensation, the DCRNN-PMD model provides another 0.44 dB Q-factor gain. The performance of the LDBP model also improves by 0.26 dB after including distributed PMD compensation, depicted as LDBP-PMD. This demonstrates the benefit of joint distributed NLC and PMD compensation. Increasing the number of taps in the DGD filter does not result in any noticeable improvement in performance, which is expected considering the small amount of DGD per step.
The performance gains for all schemes in the WDM case are markedly slimmer than the single channel case due to the presence of cross phase modulation (XPM) [34]. Note that none of the models considered in this study explicitly mitigate the XPM effects. However, we still observe a similar trend in the relative performance of various NLC schemes. For the CoI, the DCRNN-PMD model again outperforms all other learned and deterministic methods achieving 0.71 dB Q-factor gain over linear compensation, 0.55 dB gain over DBP at 1-StPS, and 0.38 dB gain over LDBP at 1-StPS.
2) Impact of Iterative Weight Pruning: In Fig. 5, we show the performance complexity comparison for the learned NLC methods along with conventional DBP for the single channel case. The rightmost point on each curve represents the performance at the hyperparameters provided earlier. Each subsequent point to the left is obtained using the iterative pruning and fine-tuning method discussed in Section IV. For conventional DBP, different points are generated by using different number of steps per span. We use the number of real multiplications per compensated symbol as a measure of computational complexity. Using the iterative weight pruning, the computational complexity of the considered learned NLC methods can be reduced by up to 50% without incurring significant performance loss. At 1-StPS, the proposed DCRNN-PMD model continues to provide 1.3 dB Q-factor gain over DBP even after being pruned to half the computational cost.
3) Instantaneous Changes in PMD: In the next set of results, we take the first step towards adapting to changes in PMD. Until this point, the learned models were trained and tested on data collected from the same realization of PMD, i.e., the channel under test was the same as the channel under training. While we continue to assume the channel to be stationary, we now consider that the current PMD realization of the channel may be different from when the data was collected for initial offline training. For this, we collect data from 8 different PMD realizations and train the model with only the data from the first realization using the offline training routine discussed in the Section V-B. Our goal is to transfer knowledge from this trained base model such that the model can be used for other PMD realizations. As per the proposed transfer learning based adaptive scheme discussed in Section III-B, we now restrict the re-training to only the PMD compensation parameters. Fig. 6 shows the convergence of mean squared error (MSE) for initial training followed by re-training for the 7 remaining 2 We note that it may be prudent to re-examine the performance complexity trade-off of using different kinds of RNN cells for the case that the dispersion spread for the step widens due to, e.g., increasing baud rates or simply larger steps.
1000
2000 5000
Real Multiplications Per Symbol PMD realizations for the DCRNN-PMD model. It can be observed that while initial training is performed for 10 3 epochs, the MSE converges to within 1% of the minimum value in only 50 re-training epochs on average, representing a 95% reduction in required training time. More importantly, this result proves our conjecture for the proposed transfer learning solution that the model can be adapted by only re-training the parameters associated with PMD compensation. 4) Continuous Changes in PMD: Finally, we discard the stationary assumption for the channel and introduce continuous drift in PMD based on the hinge model from Section II-B. In our simulation, we add one hinge at the end of each span. We believe that this is a good choice since the environmental effects causing PMD rotation are likely to be more prominent near the optical amplifier. To set a baseline, we apply linear equalization and DBP with a decision directed LMS based MIMO-FIR filter. The lumped and distributed PMD compensation methods examined have only 152 and 132 trainable parameters respectively. We apply the proposed online training routine from Section III-B with FD = 1. We measure the tracking performance of each method at four different rates of PMD drift by choosing values of the polarization linewidth ∆p as 0 Hz, 500 Hz, 1000 Hz and 5000 Hz, respectively. The learning rate for adapting each NLC method at each value of ∆p is optimized empirically. In Fig. 7, we show the peak Q-factor achieved by each method at different values of ∆p. The Q-factor is measured at each point in time using a sliding window of 50, 000 symbols. The sliding window needs to be large enough to capture a sufficient number of errors to reliably estimate the Q-factor. The large window averaging also makes the performance curves appear much smoother. We observe that performance reduces with increasing polarization linewidth, as it is expected. However, the proposed DCRNN and DCRNN-PMD models are able to maintain their superiority over conventional DBP and linear equalization. Fig. 8 provides further insight into the interplay of adjusting to a different PMD realization, i.e., acquisition, and tracking the PMD drift due to SOP changes. We show the running performance of DCRNN-PMD model at ∆p = 500 Hz. Again, the Q-factor at each point in time, represented by the number of received symbols, is calculated using a sliding window. We use this scenario to highlight the remarkable rate at which the DCRNN-PMD model converges to the peak Q-factor using online learning.
We close our discussion by noting that in scenarios where the error rate (before error correction decoding) is higher, tracking using decision-directed feedback has the potential for error propagation from decision errors. Hence, tracking may experience a performance degradation. This could be mitigated through the use of periodic pilot signals and restricting the decision-directed feedback to only the most reliable decisions. The exact interplay of the use of decision feedback, decision errors, and tracking performance is a subject for future studies.
VI. CONCLUSION
In this paper, we have discussed the DCRNN-PMD model for NLC with distributed compensation of PMD. We have shown that the DCRNN-PMD model provides notable performance gains over previously proposed learned NLC methods as well as DBP. For example, using iterative pruning, the proposed model delivers 1.3 dB Q-factor gain over conventional DBP at only half the computational cost for the considered dual-polarized 64-QAM transmission at 32 GBd over a 12 × 80 km SSMF. We have extended the DCRNN-PMD model to facilitate adaptation to PMD changes. The proposed transfer learning based selective training scheme successfully adapts the offline trained model to the current state of PMD in only a fraction of the initial training time. The training effort has further been reduced by extending the scheme with principles of online learning. Results for a PMD drift use case highlight that combined transfer and online learning enables near real-time tracking of PMD. We thus have demonstrated that our proposed method can be used to integrate a learned NLC solution in practical optical fiber communication systems.
Fig. 1 :
1(a) Architecture of the proposed DCRNN model with N steps. The lumped PMD compensation filter (not shown) is applied to the neural network outputs (x N (k), y N (k)). (b) For DCRNN-PMD, the linear step of DCRNN is amended to include distributed PMD compensation, and the overall MIMO-FIR filter is decomposed into three steps.
Fig. 2 :
2Receiver processing schematic for various NLC schemes.
Fig. 3 :
3Performance of NLC schemes for single channel.
Fig. 4 :
4Performance of NLC schemes for 5-WDM channel. transmission, respectively. For the single channel case, the proposed DCRNN model with only lumped PMD compensation delivers state of the art performance by outperforming all baseline learned and deterministic NLC schemes. It provides a 2.43 dB gain over linear compensation, 1.42 dB gain over DBP at 1-StPS, and 1.21 dB gain over LDBP at 1-StPS. It even outperforms LDBP-PMD with distributed compensation of PMD by 0.99 dB.
Fig. 6 :
6Validation loss for re-training base model for different PMD realizations. The model is DCRNN-PMD.
Fig. 7 :
7Peak performance of each NLC method for different amount of PMD drift.
Fig. 8 :
8Q-factor trend of DCRNN-PMD model at ∆p = 500 Hz.
TABLE I :
IFiber ParametersParameter
Value
No. of channels
1, 5
Modulation
64 QAM
Transmission length
12×80 km
Baud Rate
32 GBd
Channel spacing
37.5 GHz
α
0.21 dB/km
β 2
−21.49 ps 2 /km
γ
1.14 /W-km
Noise figure
4.5 dB
PMD
0.1 ps/
√
km
SSFM slices per span
500
TABLE II :
IITraining ParametersParameter
Value
Training symbols
2 19
Optimizer
Adam
Learning rate schedule
Cosine annealing
Learning rate parameters
ηmax = 10 −3 , η min = 10 −5
Test symbols
2 19
Epochs
10 3
Batch size
10 4
Validation split
80 : 20
Fig. 5: Performance-complexity comparison of NLC schemes.0.75
1.00
1.25
1.50
1.75
2.00
2.25
2.50
2.75
Q-Factor Gain Over Linear Compensation [dB]
LDBP 1-StPS
LDBP-PMD 1-StPS
DCRNN 1-StPS
DCRNN-PMD 1-StPS
DBP
For brevity, we only show the expressions for the X polarization. The expressions for the Y polarization are analogous.
ACKNOWLEDGMENTSThis research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and Huawei Tech., Canada. It was supported in part through computational resources and services provided by Advanced Research Computing at the University of British Columbia.
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| []
|
[
"MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT",
"MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT",
"MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT",
"MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT"
]
| [
"Matt Piekenbrock ",
"Jose A Perea ",
"Matt Piekenbrock ",
"Jose A Perea "
]
| []
| []
| Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with m simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles are derived. Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are O(m 2 ) such transpositions, this maintenance procedure exhibits limited scalability and often is too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations that requires only O(m log log m) time and O(m) space to construct. By reduction to a particular longest common subsequence problem, we show the storage needed to employ this strategy is actually sublinear in expectation. Exploiting this connection, we show experimentally that the decrease in operations to compute diagrams across a family of filtrations, is proportional to the difference between the expected quadratic number of states and the proposed sublinear coarsening. Applications to video data, dynamic metric space data, and multi-parameter persistence are also presented. | null | [
"https://export.arxiv.org/pdf/2104.12285v3.pdf"
]
| 233,393,732 | 2104.12285 | 86a8d3cb8cc6b0649db167253a417f68eb11bdaf |
MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT
Matt Piekenbrock
Jose A Perea
MOVE SCHEDULES: FAST PERSISTENCE COMPUTATIONS IN COARSE DYNAMIC SETTINGS * A PREPRINT
Computational topologyPersistent homologyTopological data analysis
Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with m simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles are derived. Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are O(m 2 ) such transpositions, this maintenance procedure exhibits limited scalability and often is too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations that requires only O(m log log m) time and O(m) space to construct. By reduction to a particular longest common subsequence problem, we show the storage needed to employ this strategy is actually sublinear in expectation. Exploiting this connection, we show experimentally that the decrease in operations to compute diagrams across a family of filtrations, is proportional to the difference between the expected quadratic number of states and the proposed sublinear coarsening. Applications to video data, dynamic metric space data, and multi-parameter persistence are also presented.
Introduction
Overview
Given a triangulable topological space equipped with a tame continuous function, persistent homology captures the changes in topology across the sublevel sets of the space, and encodes them in a persistence diagram. The stability of persistence contends that if the function changes continuously, so too will the points on the persistence diagram [21,22]. This motivates the application of persistence to time-varying settings, like that of dynamic metric spaces [30]. As persistence-related computations tend to exhibit high algorithmic complexity-essentially cubic 4 in the size of the underlying filtration [39]-their adoption to dynamic settings poses a challenging computational problem. With state of the art tools, there is no recourse when faced with a time-varying complex containing millions of simplices across thousands of snapshots in time. Moreover, acquiring such a capability has far-reaching consequences: methods that vectorize persistence diagrams for machine learning purposes all immediately become computationally viable tools in dynamic settings. Such persistence summaries include adaptive template functions [43], persistence images [2], and α-smoothed Betti curves [46].
Cohen-Steiner et al. refer to a continuous 1-parameter family of persistence diagrams as a vineyard, and they give in [22] an efficient algorithm for their computation. The vineyards approach can be interpreted as an extension of the reduction algorithm [48], which computes the persistence diagrams of a filtered simplicial complex K with m simplices in O(m 3 ) time, via a particular decomposition R = DV (or RU = D) of the boundary matrix D of K. The vineyards algorithm, in turn, transforms a time-varying filtration into a certain set of permutations of the decomposition R = DV , each of which takes at most O(m) time to execute. If one is interested in understanding how the persistent homology of a continuous function changes over time, then this algorithm is sufficient, for homological critical points can only occur when the filtration order changes. The vineyards algorithm is efficient asymptotically: if there are d time-points where the filtration order changes, then vineyards takes O(m 3 + md) time; one initial O(m 3 )time reduction at time t 0 followed by one O(m) operation to update the decomposition at the remaining time points (t 1 , t 2 , . . . , t d ). When d >> m, the initial reduction cost is amortized by the cost of maintaining the decomposition, implying each diagram produced by vineyards takes just linear time per time point.
Despite its theoretical efficiency, vineyards is often not the method of choice in practical settings. While there is an increasingly rich ecosystem of software packages offering variations of the standard reduction algorithm (e.g. Ripser, PHAT, Dionysus, etc. see [41] for an overview), implementations of the vineyards algorithm are relatively uncommon. 5 The reason for this disparity is perhaps explained by Lesnick and Wright [35]: "While an update to an RU decomposition involving few transpositions is very fast in practice... many transpositions can be quite slow... it is sometimes much faster to simply recompute the RU -decomposition from scratch using the standard persistence algorithm." Indeed, they observe that maintaining the decomposition along a certain parameterized family is the most computationally demanding aspect of RIVET [44], a software for computing and visualizing two-parameter persistent homology.
The work presented here seeks to further understand and remedy this discrepancy: building on the work presented in [15], we introduce a coarser approach to the vineyards algorithm. Though the vineyards algorithm is efficient at constructing a continuous 1-parameter family of diagrams, it is not necessarily efficient when the parameter is coarsely discretized. Our methodology is based on the observation that practitioners often don't need (or want!) all of the persistence diagrams generated by a continuous 1-parameter of filtrations; usually just n << d of them suffice. By exploiting the "donor" concept introduced in [15], we are able to make a tradeoff between the number of times the decomposition is restored to a valid state and the granularity of the decomposition repair step, reducing the total number of column operations needed to apply an arbitrary permutation to the filtration. This trade off, paired with a fast greedy heuristic explained in section 3.4.2, yields an algorithm that can update a R = DV decomposition more efficiently than vineyards in coarse time-varying contexts, making dynamic persistence more computationally tractable for a wider class of use-cases. The source code containing both the algorithm we propose and the experiments performed in Section 4 is open source and available online. 6
Related Work
To the authors knowledge, work focused on ways of updating a decomposition R = DV , for all homological dimensions, is limited: there is the vineyards algorithm [22] and the moves algorithm [15], both of which are discussed extensively in section 2. At the time of writing, we were made aware of very recent work [38] that iteratively repairs a permuted decomposition via a column swapping and reduce strategy, which they call "warm starts." Though their motivation is similar to our own, their approach relies on the reduction algorithm as a subprocedure, which is quite different from the strategy we employ here.
Contrasting the dynamic setting, there is extensive work on improving the efficiency of computing a single (static) R = DV decomposition. Chen [18] proposed persistence with a twist, also called the clearing optimization, which exploits a boundary/cycle relationship to "kill" columns early in the reduction rather than reducing them. Another popular optimization is to utilize the duality between homology and cohomology [23], which dramatically improves the effectiveness of the clearing optimization [5]. There are many other optimizations on the implementation side: the use of ranking functions defined on the combinatorial number system enables implicit cofacet enumeration, removing the need to store the boundary matrix explicitly; the apparent/emergent pairs optimization identifies columns whose pivot entries are unaffected by the reduction algorithm, reducing the total number of columns which need to be reduced; sparse data structures such as bit-trees and lazy heaps allow for efficient column-wise additions with Z 2 = Z/2Z coefficients and effective O(1) pivot entry retrieval, and so on [5,6].
By making stronger assumptions on the underlying topological space, restricting the homological dimension, or targeting a weaker invariant (e.g. Betti numbers), one can usually obtain faster algorithms. For example, Attali et al. [3] give a linear time algorithm for computing persistence on graphs. In the same paper, they describe how to obtain ϵ-simplifications of 1-dimensional persistence diagrams for filtered 2-manifolds by using duality and symmetry theorems. Along a similar vein, Edelsbrunner et al. [27] give a fast incremental algorithm for computing persistent Betti numbers up to dimension 2, again by utilizing symmetry, duality, and "time-reversal" [24]. Chen & Kerber [19] give an output-sensitive method for computing persistent homology, utilizing the property that certain submatrices of D have the same rank as R, which they exploit through fast sub-cubic rank algorithms specialized for sparse-matrices.
If zeroth homology is the only dimension of interest, computing and updating both the persistence and rank information is greatly simplified. For example, if the relations are available a-priori, obtaining a tree representation fully characterizing the connectivity of the underlying space (also known as the incremental connectivity problem) takes just O(α(n)n) time using the disjoint-set data structure, where α(n) is the extremely slow-growing inverse Ackermann function. Adapting this approach to the time-varying setting, Oesterling et al. [40] give an algorithm that maintains a merge tree with e edges in O(e) time per-update. If only Betti numbers are needed, the zeroth-dimension problem reduces even further to the dynamic connectivity problem, which can be be efficiently solved in amortized O(log n) query and update times using either Link-cut trees or multi-level Euler tour trees [29].
A Motivating Example
We will use a simple experiment to illustrate why the vineyards algorithm does not always yield an efficient strategy for time-varying settings. Consider a series of grayscale images (i.e. a video) depicting a fixed-width annulus expanding about the center of a 9 × 9 grid, and its associated sublevel-set filtrations, as shown in Figure 1. Each image in the series is comprised of pixels whose intensities vary with time, upon which we build a simplicial complex using the Freudenthal triangulation of the plane. For each complex, we create a filtration of simplices whose order is determined by the lower stars of pixel values. Two events critically change the persistence diagrams: the first occurs when the central connected component splits to form a cycle, and the second when the annulus splits into four components. From left to right, the Betti numbers of the five evenly spaced 'snapshots' of the filtration shown above are: (β 0 , β 1 ) = (1, 0), (1, 1), (1, 1), (1, 1), (4,0). Thus, in this example, only a few persistence diagrams are needed to capture the major changes to the topology.
We use this data set as a baseline for comparing vineyards and the standard reduction algorithm pHcol (Algorithm 2). Suppose a practitioner wanted to know the major homological changes a time-varying filtration encounters over time. Since it is unknown a priori when the persistent pairing function changes, one solution is to do n independent persistence computations at n evenly spaced points in the time domain. An alternative approach is to construct a homotopy between a pair of filtrations (K, f ), (K, f ′ ) and then decompose this homotopy into adjacent transpositions based on the filtration order-the vineyards approach. We refer to the former as the discrete setting, which is often used in practice, and the latter as the continuous setting. Note that though the discrete setting is often more practical, it is not guaranteed to capture all homological changes in persistence that occur in simulating a continuous 1-parameter family of diagrams.
The cumulative cost (in total column operations) of these various approaches are shown in Figure 2, wherein the reduction (pHcol) and vineyard algorithms are compared. Two discrete strategies (green and purple) and two continuous strategies (black and blue) are shown.
Note that without knowing where the persistence pairing function changes, a continuous strategy must construct all ≈ 7 × 10 4 diagrams induced by the homotopy. In this setting, as shown in the figure, the vineyards approach is indeed far more efficient than naively applying the reduction algorithm independently at all time points. However, when the discretization of the time domain is coarse enough, the naive approach actually performs less column operations than vineyards, while still capturing the main events. The existence of a time discretization that is more efficient to compute than continually updating the decomposition, indicates that the vineyards framework must incur some overhead (in terms of column operations) to maintain the underlying decomposition, even when the pairing function determining the persistence diagram is unchanged. Indeed, as shown by the case where n = 10, applying pHcol independently between relatively "close" filtrations is substantially more efficient than iteratively updating the decomposition. Moreover, any optimizations to the reduction algorithm (e.g. clearing [18]) would only increase this disparity. Since persistence has found many applications in dynamic contexts [45,47,35,30], a more efficient alternative to vineyards is clearly needed.
Our approach and contributions are as follows: First, we leverage the moves framework of Busaryev et al. [15] to include coarser operations for dynamic persistence settings. By a reduction to an edit distance problem, we give a lower bound on the minimal number of moves needed to perform an arbitrary permutation to the R = DV decomposition, along with a proof of its optimality. We also give worst-case sizes of these quantities in expectation as well as efficient algorithms for constructing these operations-both of which are derived from a reduction to the Longest Increasing Subsequence (LIS) problem. These operations parameterize sequences of permutations S = (s 1 , s 2 , . . . , s d ) of minimal size d, which we call schedules. However, not all minimal size schedules incur the same cost (i.e., number of column operations). We investigate the feasibility of choosing optimal cost schedules, and show that greedy-type approaches can lead to arbitrarily bad behavior. In light of these results, we give an alternative proxy-objective for cost minimization, provide bounds justifying its relevance to the original problem, and give an efficient O(d 2 log m) algorithm for heuristically solving this proxy minimization. A performance comparison with other reduction-based persistence computations is given, wherein move schedules are demonstrated to be an order of magnitude more efficient than existing approaches at calculating persistence in dynamic settings. In particular, we illustrate the effectiveness of efficient scheduling with a variety of real-world applications, including flock analysis in dynamic metric spaces and manifold detection from image data using 2D persistence computations.
Main results
Given a simplicial complex K with filtration function f , denote by R = DV the decomposition of its corresponding boundary matrix D such that R is reduced and V is upper-triangular (see section 2.1 for details). If one has a pair of filtrations (K, f ), (K, f ′ ) of size m = |K| and R = DV has been computed for (K, f ), then it may be advantageous to use the information stored in (R, V ) to reduce the computation of R ′ = D ′ V ′ . Given a permutation P such that D ′ = P DP T , such an update scheme has the form:
( * P * R * P T * ) = (P DP T )( * P * V * P T * )
where * is substituted with elementary column operations that repair the permuted decomposition. It is known how to linearly interpolate f → f ′ using d ∼ O(m 2 ) updates to the decomposition, where each update requires at most two column operations [22]. Since each column operation takes O(m), the complexity of reindexing
f → f ′ is O(m 3 ),
which is efficient if all d decompositions are needed. Otherwise, if only (R ′ , V ′ ) is needed, updating R → R ′ using the approach from [22] matches the complexity of computing R ′ = D ′ V ′ independently.
We now summarize our main results (Theorem 1): suppose one has a schedule S = (s 1 , s 2 , . . . , s d ) yielding a corresponding sequence of decompositions:
R = R 0 = D 0 V 0 s1 → D 1 V 1 s2 → . . . s d → D d V d = R d = R ′(1)
where s k = (i k , j k ) for k = 1, . . . , d, denotes a particular type of cyclic permutation (see section 3.2). If i k < j k for all s k ∈ S, our first result extends [15] by showing that (1) can be computed using O(κ) column operations, where:
κ = d k=1 |I k | + |J k |(2)
The quantities |I k | and |J k | depend on the sparsity of the V k and R k matrices, respectively, and d ∼ O(m) is a constant that depends on how similar f and f ′ are. The advantage of this result is that it depends explicitly on the sparsity pattern of the decomposition itself and is thus output sensitive, which we leverage in Section 3.4.
Our second result turns towards lower bounding d = |S| and the complexity of constructing S itself. By reinterpreting a special set of cyclic permutations as edit operations on strings, we find that a smallest such sequence mapping f to f ′ , has size (Proposition 3) :
d = m − |LCS(f, f ′ )| (3)
where LCS(f, f ′ ) refers to the size of the longest common subsequence between the simplexwise filtrations (K, f ) and (K, f ′ ) (see section 3.2 for more details). We also show that the information needed to construct any S with optimal size can be computed in O(m log log m) preprocessing time and O(m) memory. Although d can be O(m) for pathological inputs, we provide evidence that d ∼ m − √ m in expectation for random filtrations (Corollary 1), and we give empirical results suggesting d can be much smaller for time-varying filtrations.
Outline: The paper is organized as follows: we review and establish the notations we will use to describe simplicial complexes, persistent homology, and dynamic persistence in Section 2. We also cover the reduction algorithm (designated here as pHcol), the vineyards algorithm, and the set of move-related algorithms introduced in [15], which serves as the starting point of this work. In Section 3 we introduce move schedules and provide efficient algorithms to construct them. In Section 4 we present applications of the proposed method, including the computation of crocker stacks from flock simulations and of a 2-dimensional persistence invariant on a data set of image patches derived from natural images. In Section 5 we conclude the paper by discussing other possible applications and future work.
Background
Suppose one has a family {K i } i∈I of simplicial complexes indexed by a totally ordered set I, and so that for any i < j ∈ I we have K i ⊆ K j . Such a family is called a filtration, which is deemed simplexwise if K j ∖ K i = {σ j } whenever j is the immediate successor of i in I. Any finite filtration may be trivially converted into an simplexwise filtration via a set of condensing, refining, and reindexing maps (see [5] for more details). Equivalently, a filtration can be also defined as a pair (K, f ) where K is a simplicial complex and f : K → I is a filter function satisfying f (τ ) ≤ f (σ) in I, whenever τ ⊆ σ in K. In this setting,
K i = { σ ∈ K : f (σ) ≤ i }.
Here, we consider two index sets: [m] = {1, . . . , m} and R. Without loss of generality, we exclusively consider simplexwise filtrations, but for brevity-sake refer to them simply as filtrations.
Let K be an abstract simplicial complex and F a field. A p-chain is a formal F-linear combination of p-simplices of K. The collection of p-chains under addition yields an F-vector space denoted C p (K). The p-boundary ∂ p (σ) of a p-simplex σ ∈ K is the alternating sum of its oriented co-dimension 1 faces, and the p-boundary of a p-chain is defined linearly in terms of its constitutive simplices. A p-chain with zero boundary is called a p-cycle, and together they form Z p (K) = Ker ∂ p . Similarly, the collection of p-boundaries forms B p (K) = Im ∂ p+1 . Since ∂ p • ∂ p+1 = 0 for all p ≥ 0, then the quotient space H p (K) = Z p (K)/B p (K) is well-defined, and called the p-th homology of K with coefficients in F. If f : K → [m] is a filtration, then the inclusion maps K i ⊆ K i+1 induce linear transformations at the level of homology:
H p (K 1 ) → H p (K 2 ) → · · · → H p (K m )(4)
Simplices whose inclusion in the filtration creates a new homology class are called creators, and simplices that destroy homology classes are called destroyers. The filtration indices of these creators/destroyers are referred to as birth and death times, respectively. The collection of birth/death pairs (i, j) is denoted dgm p (K, f ), and referred to as the p-th persistence diagram of (K, f ). If a homology class is born at K i and dies entering K j , the difference |i − j| is called the persistence of that class. In practice, filtrations often arise from triangulations parameterized by geometric scaling parameters, and the "persistence" of a homology class actually refers to its lifetime with respect to the scaling parameter.
Let X be a triangulable topological space; that is, so that there exists an abstract simplicial complex K whose geometric realization is homeomorphic to X. Let f : X → R be continuous and write X a = f −1 (−∞, a] to denote the sublevel sets of X defined by the value a. A homological critical value of f is any value a ∈ R such that the homology of the sublevel sets of f changes at a, i.e. if for some p the inclusion-induced homomorphism H p (X a−ϵ ) → H p (X a+ϵ ) is not an isomorphism for any small enough ϵ > 0. If there are only finitely many of these homological critical values, then f is said to be tame. The concept of homological critical points and tameness will be revisited in section 2.2.
The Reduction Algorithm
In this section we briefly recount the original reduction algorithm introduced in [48], also sometimes called the standard algorithm or more explicitly pHcol [23]. The pseudocode is outlined in Algorithm 2 in the appendix. Without optimizations, like clearing or implicit matrix reduction, the standard algorithm is very inefficient. Nonetheless, it serves as the foundation of most persistent homology implementations, and its invariants are necessary before introducing both vineyards in section 2.2 and our move schedules in section 3.
Given a filtration (K, f ) with m simplices, the main output of the reduction algorithm is a matrix decomposition R = DV , where the persistence diagrams are encoded in R and the generating cycles in the columns of V . To begin the reduction, one first assembles the elementary boundary chains ∂(σ) as columns ordered according to f into a m × m filtration boundary matrix D. Setting V = I and R = D, one proceeds by performing elementary left-to-right column operations on V and R until the following invariants are satisfied: Decomposition Invariants:
I1. R = DV where D is the boundary matrix of the filtration (K, f ) I2. V is full-rank upper-triangular l3. R is reduced: if col i (R) ̸ = 0 and col j (R) ̸ = 0, then low R (i) ̸ = low R (j)
where low R (i) denotes the largest row index of a non-zero entry in column i of R. We call the decomposition satisfying these three invariants valid. Note that though the matrices R and V are not unique, the collection of persistent pairings are [48]. The persistence diagrams of the corresponding filtration can be determined from the lowest entries in R, once it has been reduced.
It is at times more succinct to restrict to specific sub-matrices of D based on the homology dimension p, and so we write D p to represent the d p−1 × d p matrix representing ∂ p (the same notation is extended to R and V ). We illustrate the reduction algorithm with an example below.
Example 2.1: Consider a triangle with vertices u, v, w, edges a = (u, w), b = (v, w), c = (u, v), and whose filtration order is given as (u, v, w, a, b, c). Using Z 2 coefficients, the reduction proceeds to compute (R 1 , V 1 ) as follows:
D 1 a b c u 1 1 v 1 1 w 1 1 , I 1 a b c a 1 b 1 c 1 − → a b c u 1 1 1 v 1 1 w 1 , a b c a 1 1 b 1 c 1 − → R 1 a b c u 1 1 v 1 w 1 , V 1 a b c b 1 1 1 a 1 1 c 1
Since column c in R 1 is 0, the 1-chain indicated by the column c in V 1 represents a dimension 1 cycle. Similarly, the columns at u, v, w in R 0 (not shown) are all zero, indicating three 0-dimensional homology classes are born, two of which are killed by the pivot entries in columns a and b in R 1 .
Inspection of the reduction algorithm from [27] suggests that a loose upper bound for the reduction is O(m 3 ), where m is the number of simplices of the filtration-this bound is in fact tight [39]. Despite this high algorithmic complexity, the number of column operations has been observed to be super-linear in practice, due in part to the high sparsity and structure of D. Moreover, many variations and optimizations to Algorithm 2 have been proposed over the past decade, see [6,5,18] for an overview.
Vineyards
Consider a homotopy F (x, t) : X × [0, 1] → R on a triangulable topological space X, and denote its "snapshot" at a given time-point t by f t (x) = F (x, t). The snapshot f 0 denotes the initial function at time t = 0 and f 1 denotes the function at the last time step. As t varies in [0, 1], the points in dgm p (f t ) trace curves in R 3 which, by the stability of persistence, will be continuous if F is continuous and the f t 's are tame. Cohen-Steiner et al. [21] referred to these curves as vines, a collection of which forms as vineyard-the geometric analogy meant to act as a guidepost for practitioners seeking to understand the evolution of topological structure over time.
The original purpose of the vineyards algorithm, as described in [22], was to compute a continuous 1-parameter family of persistence diagrams over a time-varying filtration, detecting homological critical events along the way. As homological critical events only occur when the filtration order changes, detecting all such events may be reduced to computing valid decompositions at time points interleaving all changes in the filtration order. For simplexwise filtrations, these changes manifest as transpositions of adjacent simplices, and thus any fixed set of rules that maintains a valid R = DV decomposition under adjacent column transpositions, is sufficient to simulate persistence dynamically.
To ensure a decomposition is valid, these rules prescribe certain column and row operations to apply to a given matrix decomposition either before, during, or after each transposition. Formally, let S j i represent the upper-triangular matrix such that AS j i results in adding column i of A to column j of A, and let S j i A be the same operation on rows i and j. Similarly, let P denote the matrix so that AP T permutes the columns of A and P A permutes the rows. Since the columns of P are orthonormal, P −1 = P T , then P AP T performs the same permutation to both the columns and rows of A. In the special case where P represents a transposition, we have P = P T and may instead simply write P AP . The goal of the vineyards algorithm can now be described explicitly: to prescribe a set of rules, written as matrices S j i , such that if R = DV is a valid decomposition, then ( * P * R * P * ) = (P DP )( * P * V * P * ) is also a valid decomposition, where * is some number (possibly zero) of matrices encoding elementary column or row operations.
Example 2.2
To illustrate the basic principles of vineyards, we re-use the running example introduced in the previous section. Below, we illustrate the case of exchanging simplices a and b in the filtration order, and restoring RV to a valid decomposition.
R 1 a b c u 1 1 v 1 w 1 S 2 1 − − → a b c 1 1 1 1 P − → b a c 1 1 1 1 S 2 1 − − → b a c u 1 v 1 1 w 1 V 1 a b c a 1 1 1 b 1 1 c 1 S 2 1 − − → a b c 1 1 1 1 1 P − → b a c 1 1 1 1 1 S 2 1 − − → b a c b 1 1 1 a 1 1 c 1
Starting with a valid reduction R = DV and prior to performing the exchange, observe that that the highlighted entry in V 1 would render V 1 non-upper triangular after the exchange. This entry is removed by a left-to-right column operation, given by applying S 2 1 on the right to R 1 and V 1 . After this operation, the permutation may be safely applied to V 1 . Both before and after the permutation P , R 1 is rendered non-reduced, requiring another column operation to restore the decomposition to a valid state.
The time complexity of vineyards is determined entirely by the complexity of performing a single adjacent transposition. Formally, since column operations are the largest complexity operations needed and each column can have potentially O(m) entries, the complexity of vineyards is O(m) per transposition. Inspection of the individual cases of the algorithm from [22] shows that any single transposition requires at most two such operations on both R and V . However, several factors can affect the runtime efficiency of the vineyards algorithm. On the positive side, as both the V and R matrices tend to be quite sparse, the cost of a given column operation is proportional to the number of non-zero entries in the two columns being modified. As a rule of thumb, most transpositions require no column operations [27]. On the negative side, one needs to frequently query the non-zero status of various entries in R and V (consider evaluating e.g. Case 1.1 in [22]), which accrues a non-trivial runtime cost due to the quadratic frequency with which they are required.
Moves
Originally developed to accelerate tracking generators with temporal coherence, Busaryev et al. [15] introduced an extension of the vineyards algorithm which maintains a R = DV decomposition under move operations. A move operation Move(i, j) is a set of rules for maintaining a valid decomposition under the permutation P that moves a simplex σ i at position i to position j. If j = i ± 1, this operation is an adjacent transposition, and in this sense moves generalizes vineyards. However, the move framework presented by Busaryev is actually distinct in that it exhibits several attractive qualities not inherited by the vineyards approach that warrants further study.
For completeness, we recapitulate the motivation of the moves algorithm from [15].
Let f : K → [m] denote a filtration of size m = |K| and R = DV its decomposition, where R = [r 1 , r 2 , . . . , r m ] and V = [v 1 , v 2 , . . . , v m ]
denote the columns of R and V , respectively. By definition, if r j = Dv j = 0, then the inclusion K j−1 → K j introduced a new cycle whose generator is given by:
r j = Dv j = α (1) j · ∂(σ 1 ) + α (2) j · ∂(σ 2 ) + · · · + α (j) j · ∂(σ j ) = 0(5)
Now, consider the permutation P that moves a simplex σ i in K to position j, shifting all intermediate simplices σ i+1 , . . . , σ j down by one (i < j). To perform this shift, all of the non-zero coefficients α (i) k contributing to Dv k for k ∈ [i + 1, j] must be set to zero, as otherwise ∂(σ i ) may participate in boundary chains earlier than its own birth in the permuted filtration. Setting these coefficients to zero amounts to setting row entries v k (i) = 0 for all k ∈ [i + 1, j]-these cancellations manifest as column operations ( * ) used to ensure P V ( * ) is upper-triangular, thus maintaining invariant I2. Notice that these operations may yield an unreduced R ′ = P R( * )P T , breaking invariant I3. We could reduce R ′ with an additional k operations, with the possibility that k ∼ O(|i − j| 2 ), but then this is no more efficient than re-running the reduction algorithm on columns [i, j] in R.
To bypass this difficulty, Busaryev et al. observed that since the initial R is reduced, if it contains s pivot entries in the columns [i, j] of R, then R ′ must also have s pivots. Thus, if during the cancellation of v i+1 (i) = v i+2 (i) = 0 the column r i+2 → r ′ i+2 becomes unreduced, then the pivot low R (i + 2) becomes free, possibly becoming a pivot later in r ′ i+3 , . . . , r ′ j . If r i+2 is copied prior to modification to a donor column, it may re-use or donate its pivot entry to a later column r i+3 , . . . , r j . Repeating this process at most j − i − 1 times ensures R ′ stays reduced in all except possibly for the i-th column-and since the k-th such operation simultaneously sets v k (i) = 0 without creating non-zeros at indices v k (j) for j > k, V ′ retains its upper-triangularity.
Example 2.3:
We re-use the running example from sections 2.1 and 2.2 to illustrate moves. The donor columns of R and V are denoted as d R and d V , respectively. Consider moving edge a to the position of edge c in the filtration.
d R a u 1 v w 1 R a b c u 1 1 v 1 w 1 → b 1 1 a b c 1 1 1 1 → c a b c 1 1 1 1 1 1 P − → a b c a 1 1 1 1 1 1 d R − − → b c a 1 1 1 1 d V a a 1 b c V a b c a 1 1 1 b 1 1 c 1 → b 1 1 a b c 1 1 1 1 1 → c 1 1 1 a b c 1 1 1 P − → a 1 1 1 b c a 1 1 1 d V − − → b c a 1 1 1 1 1
Note that the equivalent permutation using vineyards requires 4 column operations on both R 1 and V 1 , respectively, whereas a single move operation accomplishes using only 2 column operations per matrix. The pseudo-code for MoveRight is given in Algorithm 3 and for MoveLeft in Algorithm 4.
Regarding the complexity of move operations, which clearly depend on the sparsity of R and V , we recall the proposition shown in [15]:
Proposition 1 (Busaryev et al. [15]). Given a filtration with n simplices of dimensions p − 1, p, and p + 1, let R = DV denote its associated decomposition. Then, the operation MoveRight(i, j) constructs a valid decomposition
R ′ = D ′ V ′ in O((|I| + |J|)n) time,
where I, J are given by:
|I| = j l=i+1 1 ( v l (i) ̸ = 0 ) , J = m l=1 1 ( low R (l) ∈ [i, j] and r l (i) ̸ = 0 )
Moreover, the quantity |I| + |J| satisfies |I| + |J| ≤ 2(j − i).
Though similar to vineyards, move operations confer additional advantages:
M1: Querying the non-zero status of entries in R or V occurs once per move.
M2: R = DV is not guaranteed to be valid during the movement of σ i to σ j .
M3: At most O(m) moves are needed to reindex f → f ′ First, consider property M1. Prior to applying any permutation P to the decomposition, it is necessary to remove non-zero entries in V which render P T V P non-upper triangular, to maintain invariant I2. Using vineyards, one must consistently perform |i − j| − 1 non-zero status queries interleaved between repairing column operation. A move operation, on the other hand, groups these status queries into a single pass prior to performing any modifying operations.
Property M2 implies that the decomposition is not fully maintained during the execution of RestoreRight and Re-storeLeft below, which starkly contrasts the vineyards algorithm. In this way, we interpret move operations as making a tradeoff in granularity: whereas a sequence of adjacent transpositions (i, i+1), (i+1, i+2), . . . , (j−1, j) generates |i − j| valid decompositions in vineyards, an equivalent move operation Move(i, j) generates only one. Indeed, Property M3 directly follows from this fact, as one may simply move each simplex σ ∈ K into its new order f ′ (σ) via insertion sort. Note that the number of valid decomposition produced by vineyards is bounded above by O(m 2 ) if each pair of simplices σ i , σ j ∈ K switches its relative ordering at most once during the interpolation from f to f ′ .
There is another aspect of the move algorithm that has a distinct benefit compared to vineyards. As shown by example 2.3, move updates can be cheaper than vineyards in terms of column operations. However, it is not immediately clear that this is always the case upon inspection of 3, as the usage of a donor column seemingly implies that many O(m) copy operations need to be performed. It turns out that we may handle all such operations except the first in O(1) time, which we formalize below.
Proposition 2. Let (K, f ) denote a filtration of size |K| = m with decomposition R = DV and let T denote the number of column operations needed by the vineyards algorithm to perform the sequence of transpositions: As a final remark, we note that the combination of MoveRight and MoveLeft enable efficient simplex additions or deletions to the underlying complex. In particular, given K and a decomposition R = DV , obtaining a valid decomposition R ′ = D ′ V ′ of K ′ = K ∪ {σ} can be achieved by appending its requisite elementary chains to D and V , reducing them, and then executing MoveLeft(m + 1, i), where i = f ′ (σ). Dually, deleting a simplex σ i may be achieved via MoveRight by moving i-th to the end of the decomposition and dropping the corresponding columns.
R = R 1 s1 → R 2 s2 → . . . s k → R k−1 = R ′ where s i denotes the transposition (i, i + 1), i < j,
Our contribution: Move Schedules
Let us begin with a brief overview of the pipeline, which we outline in Algorithm 1 below. As before, we assume as input a discrete 1-parameter family F = (f 1 , f 2 , . . . , f n ) of filtrations f i : K → [m] of a simplicial complex K with |K| = m, and the goal being to compute the persistence diagrams of each (K, f i ). Fix bijections ρ i : (2), . . . , ρ i (m)) and compute a longest increasing subsequence LIS(q); this subsequence is used to recover a longest common subsequence (LCS) between f i and f i+1 , which we denote later with LCS(f i , f i+1 ). We pass q and LIS(q) to our greedy scheduling algorithm, which returns as output an ordered set of move permutations S of minimum size, which we call a schedule (see Definition 1). Note one need not explicitly store the family of filtrations nor the schedules between them-Algorithm 1 may be easily modified to be completely online, keeping at most two filtrations and one decomposition in memory at any given time.
[m] → [m] so that f i+1 = ρ i • f i , or equivalently, permutations of the index set [m]. For each ρ i we let q = (ρ i (1), ρ i
Algorithm 1 Scheduling algorithm
Require: Ordered set of filtrations F with permutations ρ i : [m] → [m] Ensure: R = DV is computed for each (K, f 1 ), (K, f 2 ), . . . , (K, f n ) 1: procedure MOVESCHEDULE(F = (f 1 , f 2 , . . . , f n )) 2: S = ∅ 3: for i = 1 to n − 1 do 4: q ← (ρ i (1), ρ i (2), . . . , ρ i (m)) 5: lis q ← LIS(q) ▷ O(m log log m) 6: S ← S ∪ GreedySchedule(q, lis q ) ▷ O(d 2 log m) 7: (R, V ) ← REDUCTION(D = ∂(K, f 1 )) 8: for (i, j) in S do 9: (R, V ) ← if i < j MOVERIGHT(i, j) else MOVELEFT(i, j)
In the following subsections, we investigate how to leverage the increased flexibility of the move framework, beginning with the hypothesis formed experimentally from section 1.3 that vineyards exhibits extraneous overhead maintaining the decomposition. In theory, decreasing the number of times the decomposition is restored to a valid state ought to reduce this overhead, motivating the question: can one simultaneously minimize the number of times the decomposition is restored to a valid state while retaining an efficient update scheme?
Continuous setting
In the original vineyards setting, a given homotopy F :
K ×[0, 1] → R continuously interpolating between (K, f ) and (K, f ′ )
is discretized into a set of critical events that alter the filtration order. As F determines the number of distinct filtrations encountered during the deformation from f to f ′ , a natural question is whether such an interpolation can be modified so as to minimize the number of times the decomposition is restored to a valid state. Towards explaining the phenomenon exhibited in Figure 2, in what follows we analyze a particular class of interpolation schemes in order to establish an upper bound on this quantity.
Let F : K × [0, 1] → R be a homotopy of x-monotone curves 7 between the filtrations f, f ′ : K → [m] whose function t → F (σ, t) is continuous and satisfies f (σ) = F (σ, 0) and f ′ (σ) = F (σ, 1) for every σ ∈ K. Note this family includes the straight-line homotopy F (σ, t) = (1−t)f (σ)+tf ′ (σ), studied in the original vineyards paper [22]. If we assume that each pair of curves t, F (·, t) ⊂ [0, 1] × R intersect in at most one point-at which they cross-the continuity and genericity assumptions on F imply that for σ, µ ∈ K distinct, the curves t → F (σ, t) and
t → F (µ, t) intersect if and only if f (σ) > f (µ) and f ′ (σ) < f ′ (µ), or f (σ) < f (µ) and f ′ (σ) > f ′ (µ).
In other words, the number of crossings in F is exactly the Kendall-τ distance [25] between f and f ′ :
K τ (f, f ′ ) = 1 2 (σ, µ) | sign f (σ) − f (µ) ̸ = sign f ′ (σ) − f ′ (µ)(6)
After slightly perturbing F if necessary, we can further assume that its crossings occur at
k = K τ (f, f ′ ) distinct time points 0 < t 1 < · · · < t k < 1. Let t 0 = 0, t k+1 = 1 and fix a i ∈ (t i , t i+1 ) for i = 0, . . . , k. Then, the order in K induced by σ → F (σ, a i ) defines a filtration f i : K → [m] so that f 0 = f , f k = f ′ and F = (f 0 , f 1 , . . . , f k ) is the ordered sequence of all distinct filtrations in the interpolation from f to f ′ via F .
The continuity of the curves t → F (·, t) and the fact that t i is the sole crossing time in the interval
(t i−1 , t i+1 ), imply that the permutation ρ i transforming f i−1 into f i , i.e. so that f i = ρ i • f i−1 , must be (in cycle notation) of the form ρ i = (ℓ i ℓ i + 1) for 1 ≤ ℓ i < m.
In other words, ρ i is an adjacent transposition for each i = 1, . . . , k. The ordered sequence of adjacent transpositions S F = (ρ 1 , ρ 2 , . . . , ρ k ) can thus be thought of as a schedule of permutations to be applied to the initial decomposition R = DV of (K, f ), in order to interpolate from f to f ′ via F . Going beyond transpositions, the idea of a schedule can be generalized as follows:
Definition 1 (Schedule)
. Given a pair of filtrations (K, f ), (K, f ′ ) and R = DV the initial decomposition of (K, f ), a schedule S = (s 1 , s 2 , . . . , s d ) is a sequence of permutations satisfying:
R = D 0 V 0 s1 → D 1 V 1 s2 → . . . s d → D d V d = R ′ (7)
where, for each i ∈ [d], R i = D i V i is a valid decomposition respecting invariants 2.1, and R ′ is a valid decomposition for (K, f ′ ).
Observe the size of the schedule S F defined from the homotopy F above is exactly K τ (f, f ′ ). On the positive side, the reduction of schedule planning to crossing detection implies the former can be that can be solved optimally in output-sensitive O(m log m + k) time by several algorithms [13], where k is the output-sensitive term and m is the number of simplices in the filtration(s). On the negative side, k = K τ (f, f ′ ) scales in size to ∼ O(m 2 ) in the worst case, achieved when f ′ = −f . As mentioned in 2.2, this quadratic scaling induces a number of issues in the practical implementations of the vineyards algorithm Remark 1. The grayscale image data example from section 1.3 exhibits this quadratic scaling. Indeed, the Freudenthal triangulation of the 9 × 9 grid contains (81, 208, 128) simplices of dimensions (0, 1, 2), respectively. Therefore, m = 417 and |S F | ≤ 1 2 m(m − 1) = 86, 736. As the homotopy given by the video is simulated, ≈ 70,000 transpositions are generated, approaching the worst case upper bound due to the fact that f ′ is nearly the reverse of f .
If our goal is to decrease |S F |, one option is to coarsen S F to a new schedule S F by e.g. collapsing contiguous sequences of adjacent transpositions to moves, via the map:
(i, i + 1)(i + 1, i + 2) · · · (j − 1, j) → (j, i + 1, · · · , j − 1, i) if i < j(8)
Clearly | S F | ≤ |S F | and the associated coarsened S F requires just O(m) time to compute. However, the coarsening depends entirely on the initial choice of F and the quadratic upper bound remains-it is always possible that there are no contiguous subsequences to collapse. Suggesting one must either abandon the continuous setting or make stronger assumptions on F to have any hope of keeping |S F | ∼ O(m) in size.
Discrete setting
Contrasting the continuous-time setting, if we discard the use of a homotopy interpolation and allow move operations in any order, we obtain a trivial upper bound of O(m) on the schedule size: simply move each simplex in K from its position in the filtration given by f to the position given by f ′ -which we call the naive strategy. However, in losing the interpolation interpretation it is no longer clear the O(m) bound is tight, and the "intermediate" filtrations need no longer respect the face poset of the underlying complex K. In this section, we investigate these issues from a combinatorial perspective.
Let S m denote the symmetric group. Given two fixed permutations p, q ∈ S m and a set allowable permutations Σ ⊆ S m , a common problem is to find a sequence of permutations s 1 , s 2 , . . . , s d ∈ Σ whose composition satisfies:
s d • · · · • s 2 • s 1 • p = q(9)
Common variations of this problem include finding such a sequence of minimal length (d) and bounding the length d as a function of m. In the latter case, the largest lower bound on d is referred to as the distance between p and q with respect to Σ. A sequence S = (s 1 , s 2 , . . . , s d ) of operations s ∈ Σ ⊆ S m mapping p → q is sometimes called a sorting of p. When p, q are interpreted as strings, these operations s ∈ Σ are called edit operations. The minimal number of edit operations d Σ (p, q) needed to sort p → q with respect to Σ is referred to as the edit distance [7] between p and q. We denote the space of sequences transforming p → q using d permutations in Σ ⊆ S m with Φ Σ (p, q, d).
Note the choice of Σ defines the type of distance being measured-otherwise if Σ = S m , then d Σ (p, q) = 1 trivially for any p ̸ = q ∈ S m .
Perhaps surprisingly, small changes to set of allowable edit operations Σ dramatically affects both the size of d Σ (p, q) and the difficulty of obtaining a minimal sorting. For example, while sorting by transpositions and reversals is NPhard and sorting by prefix transpositions is unknown, there are polynomial time algorithms for sorting by block interchanges, exchanges, and prefix exchanges [32]. Sorting by adjacent transpositions can be achieved in many ways: any sorting algorithm that exchanges two adjacent elements during its execution (e.g. bubble sort, insertion sort) yields a sorting of size K τ (p, q).
Here we consider sorting by moves. Using permutations, a move operation m ij that moves i to j in [m], for i < j, corresponds to the circular rotation:
1 · · · i − 1 i i + 1 · · · j − 1 j j + 1 · · · m 1 · · · i − 1 i + 1 · · · j − 1 j i j + 1 · · · m (10)
In cycle notation, this corresponds to the cyclic permutation:
( i j j-1 . . . i+2 i+1 )(11)
Similarly, in the context of edit operations, observe that a move operation can be interpreted as a paired delete-andinsert operation, i.e. m ij = (ins j • del i ), where del i denotes the operation that deletes the character at position i and ins j the operation that inserts the same character at position j. Thus, sorting by move operations can be interpreted as finding a minimal sequence of edits where the only operations allowed are (paired) insertions and deletions-this is exactly the well known Longest Common Subsequence (LCS) distance. Between strings p, q of sizes m and n, the LCS distance is given by [7]: d lcs (p, q) = m + n − 2|LCS(p, q)| (12) In general, one can compute the LCS itself in O(mn) with dynamic programming. One might hope computing the size d lcs (p, q) alone exhibits a lower complexity, however there is substantial evidence that for general string inputs the complexity cannot be much lower than quadratic [1]. However, if the pair of filtrations (K, f ), (K, f ′ ) come from the same underlying complex K, then their filter function f, f ′ may both be thought of a permutations in S m -the corresponding edit distance d then reduces to the permutation edit distance problem. With this insight in mind, we obtain the following bound on the minimum size of a sorting (i.e. schedule) using moves and the complexity of computing it. Proposition 3 (Schedule Size). Let (K, f ), (K, f ′ ) denote two filtrations of size |K| = m. Then, the smallest move schedule S * reindexing f → f ′ has size: Proof. Recall our definition of edit distance given above, depending on the choice Σ ⊆ S m of allowable edit operations, and that in order for any edit distance to be symmetric, if s ∈ Σ then s −1 ∈ Σ. This implies that d Σ (p, q) = d Σ (p −1 , q) for any choice of p, q ∈ S m . Moreover, edit distances are left-invariant, i.e. d Σ (p, q) = d Σ (r • p, r • q) for all p, q, r ∈ S m Conceptually, left-invariance implies that the edit distance between any pair of permutations p, q is invariant under an arbitrary relabeling of p, q-as long as the relabeling is consistent. Thus, the following identity always holds:
|S * | = d = m − |LCS(f, f ′ )| where we use LCS(f, f ′ ) tod Σ (p, q) = d Σ (ι, p −1 • q) = d Σ (q −1 • p, ι)
where ι = [m], the identity permutation. Suppose we are given two permutations p, q ∈ S n and we seek to compute LCS(p, q). Consider the permutation p ′ = q −1 • p. Since the LCS distance is a valid edit distance, if |LCS(p, q)| = k, then |LCS(p ′ , ι)| = k as well. Notice that ι is strictly increasing and that any common subsequence ι has with p ′ must also be strictly increasing. Thus, the problem of computing LCS(p, q) reduces to the problem of computing the longest increasing subsequence (LIS) of p ′ , which can done in O(m log log m) time using van Emde Boas trees [8]. The optimality of d follows from the optimality of the well-studied LCS problem [31].
Establishing a connection between the permutation edit distance and move scheduling allows use to exploit the combinatorial structure that comes from the developed theory on both LCS's and LIS's, which are both well-studied objects. We record a single corollary to demonstrate this fact.
L(p)
A large body of work dates back at least 50 years has focused on estimating this quantity, which is sometimes called the Ulam-Hammersley problem. Seminal work by Baik et al. [4] established that as m → ∞:
ℓ m = 2 √ m + cm 1/6 + o(m 1/6 )
where c = −1.77108.... Moreover, letting m → ∞, we have:
ℓ m √ m → 2 as m → ∞
Thus, if p ∈ S m denotes a uniformly random permutation in S m , then L(p)/ √ m → 2 in probability as m → ∞. Using the reduction from above to show that LCS(p, q) ⇔ LIS(p ′ ), the claimed bound follows.
Remark 2. Note the quantity from Corollary 1 captures the size of S * between pairs of uniformly sampled permutations, as opposed to uniformly sampled filtrations, which have more structure. However, Boissonnat [11] prove the number of distinct filtrations built from a k-dimensional simplicial complex K with m simplices and t distinct filtration values is at least ⌊ t+1 k+1 ⌋ m . Since this bound grows similarly to m! when t ∼ O(m) and k << m fixed, d ≈ n − √ n is not too pessimistic a bound between random filtrations.
In practice, when one has a time-varying filtration and the sampling points are relatively close [in time], the LCS between adjacent filtrations is expected to be much larger, shrinking d substantially. For example, for the complex from Section 1.3 with m = 417 simplices, the average size of the LCS across the 10 evenly spaced filtrations was 343, implying d ≈ 70 permutations needed on average to update the decomposition between adjacent time points.
We conclude this section with the main theorem of this effort: an output-sensitive bound on the simulation of persistence dynamically. Theorem 1. Given a pair of filtrations (K, f ), (K, f ′ ) and a decomposition R = DV of K, the size of a minimal sequence S = (s 1 , s 2 , . . . , s d ) of cyclic 'move' permutations s k = (i k , j k ) satisfying:
R = D 0 V 0 s1 → D 1 V 1 s2 → . . . s d → D d V d = R ′(13)
Constructing schedules
It is clear from Corollary 1 that one may compute the LCS between two permutations p, q ∈ S m via the LIS of a single permutation p * , and that computation may be carried out in O(m log log m) time. It is not immediately clear, however, how to obtain a sorting p → q from a given L = LIS(p ′ ) in an efficient way. We outline below a simple procedure which constructs such a sorting S = (s 1 , . . First, we require a few definitions. Recall that a sorting S with respect to two permutations p, q ∈ S m is an ordered sequence of permutations S = (s 1 , s 2 , . . . , s d ) satisfying q = s d • . . . s 1 • p. By definition, a subsequence of symbols in L common to both p and q satisfies:
p −1 (σ) < p −1 (τ ) =⇒ q −1 (σ) < q −1 (τ ) ∀σ, τ ∈ L(14)
where p −1 (σ) (resp. q −1 (σ)) denotes the position of σ in p (resp. q). Thus, obtaining a sorting p → q of size d = m − |L| reduces to applying a sequence of moves to symbols in the complement of L. Formally, we define a permutation s ∈ S m as a valid operation with respect to a fixed pair p, q ∈ S m if:
|LCS(s • p, q)| = |LCS(p, q)| + 1(15)
The problem of constructing a sorting S of size d thus reduces to the problem of choosing a sequence of d valid moves, which we call a valid sorting. To do this efficiently, let T denote a ordered set-like data structure that supports the following operations on elements σ ∈ M from the set M = {0, 1, . . . , m + 1}:
1 T ∪ σ-inserts σ into T , 2 T \ σ-removes σ from T , 3 T succ (σ)-obtain the successor of σ in T , if it exists, otherwise return m + 1 4 T pred (σ)-obtain the predecessor of σ in T , if it exists, otherwise return 0
Given such a T , an arbitrary valid sorting can be constructed by repeatedly querying and maintaining information about the LCS in T . To see this, suppose T contains all of the symbols in the current LCS between two permutations p and q. By definition of the LCS, we have:
p −1 (T pred (σ)) < p −1 (σ) < p −1 (T succ (σ))(16)
for every σ ∈ T . Now, suppose we choose a symbol σ / ∈ T . If p −1 (σ) < p −1 (T pred (σ)), then we must move σ to the right in p such that (16) holds. Similarly, if p −1 (T succ (σ)) < p −1 (σ), then we must move σ left in p to increase the size of the LCS. Assuming the structure T supports all of the above operations in O(log m) time, we easily deduce a O(dm log m) algorithm for obtaining a valid sorting, which for completeness is shown via Algorithm 5 in the appendix.
Minimizing schedule cost
The algorithm outlined in section 3.3 is a sufficient for generating move schedules of minimal cardinality: any schedule of moves S sorting f → f ′ above is guaranteed to have size |S| = m − |LCS(f, f ′ )|, and the reduction to the permutation edit distance problem ensures this size is optimal. However, as with the vineyards algorithm, certain pairs of simplices cost more to exchange depending on whether they are critical pairs in the sense described in [22], resulting in a large variability in the cost of randomly generated schedules. This variability is undesirable in practice: we would like to generate a schedule which not only small in size, but is also efficient in terms of its required column operations.
Greedy approach
Ideally, we would like to minimize the cost of a schedule S ∈ Φ Σ (p, q, d) directly, which recall is given by the number of non-zeros at certain entries in R and V :
cost(S) = d k=1 |I k | + |J k |(17)
where |I| + |J| are the quantities from Proposition 1. One advantages to the moves framework is that the cost of a single move on a given R = DV decomposition can be determined efficiently prior to any column operations-no more than O(m) time with row-oriented sparse matrices. It is natural to consider whether a greedy-type solution which chooses the minimal cost choice at every step yields an efficient strategy. We give a counter-example below demonstrating that a greedy procedure may lead to arbitrarily bad behavior. Table 1. Note the greedy strategy which always selects the cheapest move in succession would begin by moving x or z first, since these are the cheapest moves available, which implies one of S 1 , S 2 , S 5 , S 6 would be picked depending on the tie-breaker. While the cheapest schedule S 1 is in this candidate set, so is S 6 , the most expensive schedule. As a result, a greedy strategy which chooses the lowest-cost move may not yield an optimal schedule.
Counter-Example:
Proxy objective
Although the greedy approach from the last section can lead to arbitrarily high cost schedules, we find similar greedylike strategies can yield computationally efficient heuristics in practice, even if they are suboptimal. We seek a fast procedure for generating schedules that is not only substantially better than a random or simple schedule strategy in term of column reductions, but has a low enough time and storage complexity to be practical for larger data sets.
We seek a proxy objective that correlates with (17) and does not explicitly depend on the entries in the decomposition, as minimizing these terms directly is difficult due to the changing sparsity of the intermediate matrices
R k , V k , Given a pair of filtrations (K, f ), (K, f ′ ), consider a schedule S ∈ Φ(f, f ′ , d) of cyclic permutations (i 1 , j 1 ), (i 2 , j 2 ), . . . , (i d , j d ) minimizing: cost(S) = d k=1 2|i k − j k | ≥ d k=1 (|I k | + |J k |)(18)
In practice, this bound is very loose due to the sparsity of both R and V . Nonetheless, the complexities of the move operations discussed in Section 2.3 depend on |i−j|, and minimizing (18) has the intuitive interpretation as minimizing net displacement. A similar ℓ 1 -type distance for measuring the disarrangement between permutations is the Spearman distance, defined as:
F (p, q) = m i=1 |p(i) − q(i)| = m i=1 |i − (q −1 • p)(i)|(19)
The Spearman distance shares any similarities with K τ : it is a metric on S m that is invariant under consistent relabeling. Indeed, Diaconis [25] showed the Spearman distance approximates K τ within a factor of two:
K τ (p, q) ≤ F (p, q) ≤ 2K τ (p, q)(20)
Moreover, in contrast to K τ , the Spearman distance can be computed in O(m) time and is often used in the wellknown rank aggregation problem. Indeed, whereas obtaining a Kemeni optimal aggregation with respect to K τ is NP-hard, the optimal such aggregation with respect to F is obtainable in polynomial time [26]. To adapt F (·, ·) to sortings, we decompose the Spearman distance additively via the bound:
F S (p, ι) = d−1 i=1 F (ŝ i • p,ŝ i+1 • p) ≥ F (p, ι)(21)
whereŝ i = s i • · · · • s 2 • s 1 denote the composition of the first i permutations of a sorting S = (s 1 , . . . , s d ) that maps p → ι. The problem of minimizing the right hand side of (21) can be interpreted as a crossing minimization problem for a set of k-layered bipartite graphs. To see this, consider two permutations: p and m ij • p, where m ij is a move permutation. Drawing (p, m ij • p) as a bipartite graph, observe that F (p, m i,j • p) is twice the number of edge crossings in the graph, and that equality in (21) is achieved when the displacement of each symbol between its initial position in p to its value is non-increasing with every application of s i , which in general not guaranteed using the schedule construction method derived in section 3. Unfortunately, the k-layer crossing minimization problem is NP-hard for k sets of permutations, when k ≥ 4 [9].
Example: Let p = (1 2 3 4 5 6 7 8 9) and q = (9 4 2 7 1 8 6 3 5). An example of three possible schedules, S 1 , S 2 , and S 3 sorting p into q is given in the figure below. Each column represents the successive application of a move m ij in the schedule, and the edges track how each symbol has been moved. Black/red vertices correspond to symbols in and outside of LCS, respectively. All three schedules were generated from the same LCS(p, q) = (4 7 8) and each schedule transforms p → q in d = 6 moves. In this example, S 1 matches the minimal number of crossings amongst all possible schedules, since K τ (p, q) = 21.
In light of the discussion above, we propose a heuristic strategy to minimize (21) which we observed is both efficient to compute and effective in practice. The heuristic is inspired by the simplicity of computing the Spearman distance between cyclic permutations at the schedule construction phase. Suppose we begin with an array A of size m which provides O(1) access and modification, initialized with the (signed) displacement of every element in p to its corresponding position in q. Since the Spearman distance is simply the sum of the absolute value of these displacements, at any point during during the execution of Algorithm 5 we may obtain F (·, q) simply by having access to the sum of every entry in A. At every step, there are only two degrees of freedom in Algorithm 5: the choice of σ ∈ D to move, and choice of the target index j to move σ to. If we fix a heuristic for the latter, then each σ ∈ D induces a set of possible valid moves satisfying (15), which we denote as S D (σ). Now, observe that each permutation s ∈ S D (σ) changes the displacement of every symbol in at most three different ways:
A(s • p) = A(σ) ± |i − j| p −1 (σ) = i A(σ) ± 1 i < p −1 (σ) ≤ j A(σ) otherwise
Thus, if we replace A with a data structure supporting O(log m) access time to aggregate information and O(log m) modification time ability on |i − j| entries simultaneously, we could greedily choose the next permutation s to minimize: s greedy = arg min
s∈S D (σ) F (s • p, ι)(22)
in O(d log m) time. The former problem of accessing aggregate information reduces to the problem of efficiently calculating prefix sums, which is easily solved. It is not immediately clear how to achieve the latter modification complexity in O(log m) time, since |i − j| ≤ m is potentially larger than O(log m)-however, because we are working with displacements, note the additive modification to each entry in |i − j| is a constant. It is known that one can apply constant-factor updates to multiple values in a implicit treap data structure in O(log m) time via range updates [10]. Since single element modifications, removals, and insertions can all be achieved in O(log m) expected time with such a data structure, we conclude that equation (22) may solved in just O(d 2 log m) time.
Applications and Experiments
Video data
A common application of persistence is characterizing topological structure in image data. Since a set of "snapshot" frames of a video can be equivalently thought of as discrete 1-parameter family, our framework provides a natural extension of the typical image analysis to video data. To demonstrate the benefit of using minimal size schedules and the scalability of the greedy approach proposed in section 3.4.2, we re-use the video data from 1.3 as a baseline benchmark. We perform two performance tests: one to test the impact of shrinking the number of permutations a given reindexing operation needs and one to test the asymptotic behavior of the greedy approach.
In the first test, we fix a grid size of 9 × 9 and record the cumulative number of column operations needed to simulate persistence dynamically across 25 evenly-spaced time points using a variety of scheduling strategies. The primary three strategies we test are the greedy approach from section 3.4.2, the "simple" approach which uses upwards of O(m) move permutations via selection sort, and a third strategy which interpolates between the two. To perform this interpolation, we use a parameter α ∈ [0, 1] to choose m − α · d random symbols to move using the same construction method outlined in section 3.3. When α = 0, the strategy reduces to using selection sort to reindex f → f ′ using a random ordering of simplices; otherwise, α = 1 reduces to using a minimal sized schedule (with a random ordering of simplices). The results are summarized in the left graph on Figure 3, wherein the mean schedule cost of the random strategies are depicted by solid lines. To capture the variation in performance for the random sampling approach, we run 10 independent iterations and shade the upper and lower bounds of the schedule costs. As one can see from the figure, while using less move operations (lower α) does progressively reduce column operations, constructing random schedules of minimal size is no more competitive than the simple selection sort strategy. This suggests that efficient schedule construction needs to account for the structure of performing several permutations in sequence, like the greedy heuristic we introduced, to yield an adequate performance boost.
In the second test, we aim to measure the asymptotics of our greedy LCS-based approach. To do this, we generated 8 video data sets again of the expanding annulus outlined in section 1.3, each of increasing grid sizes of 5 × 5, 6 × 6, . . . , 12 × 12. For each data set, we simulate persistence over the duration of the video, again testing five evenly spaced settings of α ∈ [0, 1]-the results are shown in the right plot of Figure 3. On the vertical axis, we plot the total number of column operations needed to simulate the video again across 25 evenly-spaced time points as a ratio of the data set size (m); we also show the regression curves one obtains for each setting of α. As one can see from the Figure, the cost of using the greedy heuristic tends to increase sub-linearly as a function of the data set size, suggesting the move scheduling approach is indeed quite scalable. Moreover, schedules with minimal size tended to be cheaper than otherwise, confirming our initial hypothesis that repairing the decomposition less can lead to substantial reductions at runtime.
Crocker stacks
There are many challenges to characterizing topological behavior in dynamic settings. One approach is to trace out the curves constituting a continuous family of persistence diagrams in R 3 -the vineyards approach-however this visualization can be cumbersome to work with as there are potentially many such vines tangled together, making topological critical events with low persistence difficult to detect. Moreover, the vineyards visualization does not admit a natural simplification utilizing the stability properties of persistence, as individual vines are not stable: if two vines move near each other and then pull apart without touching, then a pairing in their corresponding persistence diagrams may cross under a small perturbation, signaling the presence of an erroneous topological critical event [45,47]. Acknowledging this, Topaz et al. [45] proposed the use of a 2-dimensional summary visualization, called a crocker 8 plot. In brief, a crocker plot is a contour plot of a family of Betti curves. Formally, given a filtration K = K 0 ⊆ K 1 ⊆ · · · ⊆ K m , a p-dimensional Betti curve β • p is defined as the ordered sequence of p-th dimensional Betti numbers:
β • p = { rank(H p (K 0 )), rank(H p (K 1 )), . . . , rank(H p (K m )) }
Given a time-varying filtration K(τ ), a crocker plot displays changes to β • p (τ ) as a function of τ . A example of a crocker plot generated from the simulation described below is given in Figure 4. Since only the Betti numbers at each simplex in the filtration are needed to generate these Betti curves, the persistence diagram is not directly needed to generate a crocker plot; it is sufficient to use e.g. any of the specialized methods discussed in 1.2. This dependence only on the Betti numbers makes crocker plots easier to compute than standard persistence, however what one gains in efficiency one loses in stability; it is known that Betti curves are inherently unstable with respect to small fluctuations about the diagonal of the persistence diagram.
Xian et al. [47] showed that crocker plots may be smoothed to inherit the stability property of persistence diagrams and reduce noise in the visualization. That is, when applied to a time-varying persistence module M = {M t } t∈[0,T ] an α-smoothed crocker plot for α ≥ 0 is the rank of the map M t (ϵ − α) → M t (ϵ + α) at time t and scale ϵ. For example, the standard crock plot is a 0-smoothed crocker plot. Allowing all three parameters (t, ϵ, α) to vary continuously leads to 3D visualization called an α-smoothed crocker stack.
f M (t, ϵ, α) = rank(M t (ϵ − α) → M t (ϵ + α)) and f M satisfies f M (t, ϵ, α ′ ) ≤ f M (t, ϵ, α) for all 0 ≤ α ≤ α ′ .
Note that, unlike crocker plots, applying this α smoothing efficiently requires the persistence pairing. Indeed, it has been shown that crocker stacks and stacked persistence diagrams (i.e. vineyards) are equivalent to each other in the sense that either one contains the information needed to reconstruct the other [47]. Thus, computing crocker stacks reduces to computing the persistence of a (time-varying) family of filtrations.
We test the efficiency of computing the necessary information to generate these crocker stacks using a spatiotemporal data set to illustrate the applicability of our method. Specifically, we ran a flocking simulation similar to the simulation run in [45] with m = 20 vertices moving around on the unit square equipped with periodic boundary conditions (i.e. S 1 × S 1 ). We simulated movement by equipping the vertices with a simple set of rules which control how the individual vertices position change over time. Such simulations are also called boid simulations, and they have been extensively used as models to describe how the evolution of collective behavior over time can be described by simple sets of rules. The simulation is initialized with every vertex positioned randomly in the space; the positions of vertices over time is updated according to a set of rules related to the vertices acceleration, distance to other vertices, etc. To get a sense of the time domain, we ran the simulation until a vertex made at least 5 rotations around the torus.
Given this time-evolving data set, we computed the persistence diagram of the Rips filtration up to ϵ = 0.30 at 60 evenly spaced time points using three approaches: the standard algorithm pHcol applied naïvely at each of the 60 time steps, the vineyards algorithm applied to (linear) homotopy connecting filtrations adjacent in time, and our approach using moves. The cumulative number of O(m) column operations executed by three different approaches. Note again that vineyards requires generating many decompositions by design (in this case, ≈ 1.8M ). The standard algorithm pHcol and our move strategy were computed at 60 evenly spaced time points of the simulations. As depicted in Figure 5, our move strategy is far more efficient than both vineyards and the naive pHcol strategies. Figure 5: One the left, the cumulative number of column operations (log-scale) of the three baseline approaches tested. On the right, the normalized K τ between adjacent filtrations depicts the coarseness of the discretization-about 5% of the ≈ O(m 2 ) simplex pairs between adjacent filtrations are discordant.
Multiparameter persistence
Given a procedure to filter a space in multiple dimensions simultaneously, a multifiltration, the goal of multi-parameter persistence is to identify persistent features by examining the entire multifiltration. Such a generalization has appeared naturally in many application contexts, showing potential as a tool for exploratory data analysis [37]. Indeed, one of the drawbacks of persistence is its instability with respect to strong outliers, which can obscure the detection of significant topological structures [14]. One exemplary use case of multi-parameter persistence is to detect these strong outliers by filtering the data with respect to both density and the associated metric. In this section, we show the utility of scheduling with a real-world use case: detecting the presence of a low-dimensional topological space which well-approximates the distribution of natural images. As a quick outline, in what follows we briefly recall the fibered barcode invariant 4.3.1, summarize its potential application to a particular data set with known topological structure 4.3.2, and conclude with experiments of demonstrating how scheduling enables such applications 4.3.3.
Fibered barcode
Unfortunately, unlike the one-parameter case, there is no complete discrete invariant for multi-parameter persistence. Circumventing this, Lesnick et al [35] associate a variety of incomplete invariants to 2-parameter persistence modules; we focus here on the fibered barcode invariant, defined as follows: Although an intuitive invariant, it is not clear how one might go about computing B(M ) efficiently. One obvious choice is fix L via a linear combination of two filter functions, restrict M to L, and compute the associated 1-parameter barcode. However, this is an O(m 3 ) time computation, which is prohibitive for interactive data analysis purposes. [44] depends on. Despite this significant complexity barrier, there is room for optimism: in practice, the external stability result from [33] justifies the use of a coarsening procedure which approximates the module M with a smaller module M ′ via a grid-like reduction, enabling practitioners to restrict the size of κ to a relatively small constant. This in-turn dramatically reduces the size of A(M ) and thus the number of barcode templates to compute. Moreover, the ordering of barcode templates given by the dual graph traversal implies that adjacent template points should be relatively close-so long as κ is not too small-suggesting adjacent templates may productively share computations due to the high similarity of their associated filtrations. Indeed, as algorithm 1 was designed for precisely such a computation, 2-parameter persistence is prototypical of the class of methods that stand to benefit from move scheduling.
Natural images dataset
A common hypothesis is that high dimensional data tend to lie in the vicinity of an embedded, low dimensional manifold or topological space. An exemplary demonstration of this is given in the analysis by Lee et al. [34], who explored the space of high-contrast patches extracted from Hans van Hateren's [28] still image collection 9 , which consists of ≈ 4, 000 monochrome images depicting various areas outside Groningen (Holland). In particular, [34] were interested in exploring how high-contrast 3 × 3 image patches were distributed, in pixel-space, with respect to predicted spaces and manifolds. Formally, they measured contrast using a discrete version of the scale-invariant Dirichlet semi-norm:
∥x∥ D = i∼j (x i − x j ) 2 = √ x T Dx
where D is a fixed matrix which upon application x T Dx to an image x ∈ R 9 yields a value proportional to the sum of the differences between each pixels 4 connected neighbors (given above by the relation i ∼ j). Their research was primarily motivated by discerning whether there existed clear qualitative differences in the distributions of patches extracted from images of different modalities, such optical and range images. By mean-centering, contrast normalizing, and"whitening" the data via the Discrete Cosine Transform (DCT) basis, they a convenient basis for D may be obtained via an expansion of 8 certain non-constant eigenvectors, shown below:
Since these images are scale-invariant, the expansion of these basis vectors spans the 7-sphere, S 7 ⊂ R 8 . Using a voronoi cell decomposition of the data, their distribution analysis suggested that the majority of data points concentrated in a few high-density regions.
After Lee et al published their work, Carlsson et al. [16] subsequently performed extensive experiments using persistent homology, wherein he found that the distribution of high-contrast 3 × 3 patches is actually well-approximated by a Klein bottle M-around 60% of the high-contrast patches from the still image data set lie within a small neighborhood around M which accounts for only 21% of the 7-spheres volume. Along a similar vein in the sparse coding context, Perea [42] introduced a dictionary learning framework for estimating the distribution of patches from texture images.
If one was not aware of the analysis done by [34,28,16,42], it is not immediately clear a priori that the Klein bottle model is a good candidate for capturing the non-linearity of image patches. Indeed, armed with a refined topological intuition, Carlsson still needed to perform extensive sampling, preprocessing, and model fitting techniques in order reveal the underlying the topological space with persistent homology [16]. One reason such preprocessing is needed is due to persistent homology's aforementioned instability with respect to strong outliers. In the ideal setting, a multiparameter approach that accounts for the local density of points should require far less experimentation.
Consider the (coarsened) fibered barcode computed from a standard Rips / codensity bifiltration on a representative sample of the image data from [28], shown in Figure 7. From the bigraded Betti number and the dimension function, one finds that a large area of dimension function is constant (highlighted as the blue portion in the middle of Figure 7), wherein the first Betti number is 5. Further inspection suggests one plausible candidate is the three-circle model C 3 , which consists of three circles, two of which (say, S v and S h ) intersect the third (say, S lin ) in exactly two points, but themselves do not intersect. Projecting the image data onto the first two basis vectors of leads to the projection shown in the top left of Figure 7, of which 15 landmark points are also shown. Observe the data are distributed well around three "circles"-the outside circle capturing the rotation gradient of the image patches (S lin ), and the other two capturing the vertical and horizontal gradients (S v and S h , respectively). Since the three circle model is the 1-skeleton of the Klein bottle, one may concur with Carlssons analysis [16] that the Klein bottle may be a reasonable candidate upon which the image data are distributed. The degree to which multi-parameter persistence simplifies this exploratory phase cannot be understated: we believe multi-parameter persistence has a larger role to play in manifold learning. Unfortunately, as mentioned prior, the compute barriers effectively bar its use in practice.
Accelerating 2D persistence
As we have outlined the computational theory of 2-parameter persistence and elucidated its relevance to our proposed move scheduling approach, we now demonstrate the efficiency of scheduling using the same high-contrast patch data set studied in [34] by evaluating the performance of various methods at computing the fibered barcode invariant via the parameterization from A.2.
Due to the aforementioned high complexity of the fibered barcode computation, we begin by working with a subset of the image patch data X . In particular, we combine the use of furthest-point sampling and proportionate allocation (stratified) sampling to sample landmarks X ⊂ X distributed within n = 25 strata. Each strata consists of the (1/n)thick level set given the k-nearest neighbor density estimator ρ 15 with k = 15. The use of furthest-point sampling gives us certain coverage guarantees that the geometry is approximately preserved within each level set, whereas the stratification ensures the original density of is approximated preserved as well. From this data set, we construct a Rips-(co)density bifiltration using ρ 15 equipped with the geodesic metric computed over the same k-nearest neighbor graph on X. Finally, we record the number of column reductions needed to compute the fibered barcode at a variety of levels of coarsening using pHcol, vineyards, and our moves approach. The results are summarized in Table 2. We also record the number of 2-cells in A(M ) and the number of permutations applied throughout the encountered along the traversal of the dual graph for both vineyards and moves, denoted in the table as d K and d LCS , respectively. As shown on the table, when the coarsening κ is small enough, we're able to achieve a significant reduction in the number of total column operations needed to compute T compared to both vineyards and pHcol. This is further reinforced by the observation that, when there is a high degree of coarsening, vineyards is particularly inefficient and moves requires only about 3x less column operations that naively computing T independently. As the coarsening becomes more refined and more 2-cells are added to A(M ), however, vineyards quickly becomes a much more viable option compared to pHcol-as predicted-though even at the highest coarsening we tested the gain in efficiency is relatively small. In contrast, our proposed moves approach scales quite well with this refinement, requiring about 12% and ≈ 5% of the number of column operations as vineyards and pHcol, respectively.
Conclusion and Future Work
In conclusion, we presented a scheduling algorithm for efficiently updating a decomposition in coarse dynamic settings. Our approach is simple, relatively easy to implement, and fully general: it does not depend on the geometry of underlying space, the choice of triangulation, or the choice of homology dimension. Moreover, we supplied efficient algorithms for our scheduling strategy, provided tight bounds where applicable, and demonstrated our algorithms performance with several real world use cases.
There are many possible applications of our work beyond the ones discussed in section 4. As mentioned in section 1. Indeed, we see our approach as potentially useful to any situation where the structure of interest is a parameterized family of persistence diagrams. Areas of particular interest include time-series analysis and dynamic metric spaces [30].
The simple and combinatorial nature of our approach does pose some limitations to its applicability. For example, better bounds or algorithms may be obtainable if stronger assumptions can be made on how the filtration is changing with time. Moreover, if the filtration (K, f ) shares little similarity to the "target" filtration (L, f ′ ), then the overhead of reducing the simplices from L \ K appended to the decomposition derived from K may be large enough to motivate simply computing the decomposition at L independently, especially if parallel processors are available. Our approach is primarily useful if the filtrations in the parameterized family is "nearby" in the combinatorial sense.
From an implementation perspective, one non-trivial complication of our approach is its heavy dependence on a particular sparse matrix data structure which permits permuting both the row and columns of a given matrix in at most O(m) time. As shown with the natural images example in section 4, there are often more permutation operations being applied than there are column reductions. In the more standard compressed sparse matrix representations 10 , permuting both the rows and columns generally takes at most O(Z) time, where Z is the number of non-zero entries, which can be quite expensive if the particular filtration has many cycles. As a result, the more complex sparse matrix representation from [22] is necessary to be efficient in practice.
Moving forward, our results suggest there are many aspects of computing persistence in dynamic settings yet to be explored. For example, it's not immediately clear whether one could adopt, for example, the twist optimization [18] used in the reduction algorithm to the dynamic setting. Another direction to explore would be the analysis of our approach under the cohomology computation [23], or the specialization of the move operations to specific types of filtrations such as Rips filtrations. Such adaptations may result in even larger reductions in the number of column operations, as have been observed in practice for the standard reduction algorithm [5]. Moreover, though we have carefully constructed an efficient greedy heuristic in section 3.4.2 and illustrated a different perspective with which to view our heuristic (via crossing minimization), it is an open question whether there exists a more structured reduction of (17) or (21) to a better-known problem.
Declarations
Ethical Approval: Not applicable.
Competing interests: The authors declare that they have no competing interests that could influence the interpretation or presentation of the research findings. There are no financial or personal relationships with individuals or organizations that could bias the outcomes of this work.
Authors' contributions: The contributions of each author to this article were as follows:
A Appendix
A.1 Algorithms
A.1.1 Reduction Algorithm
The reduction algorithm, also called the "standard algorithm," is the most often used modality for computing persistence. While there exists other algorithms for computing persistence, they are typically not competitive with the reduction algorithm in practice. We outline the reduction algorithm below in Algorithm 2. The algorithm begins by for j = 1 to m do 4:
while ∃ i < j such that low R (i) = low R (j) do 5: λ ← pivot R (j)/pivot R (i) 6: (col R (j), col V (j)) −= (λ · col R (i), λ · col V (i)) 7: return (R, V )
copying D to a new matrix R, to be subsequently modified in-place. After setting V is set to the identity, the algorithm proceeds with column operations on both R and V , left to right, until the decomposition invariants are satisfied. [39], though this seems to only be true on pathological inputs. Indeed, a more refined analysis by Edelsbrunner et al. [27] shows the reduction algorithm scales by the sum of squares of the cycle persistences, which is an output-sensitive bound.
Move Algorithms
As we've covered the moves algorithm extensively in section 2.3, we now record the algorithmic components of both MoveRight and MoveLeft. Though conceptually similar, note that there is an asymmetry between MoveRight and MoveLeft: moving a simplex upwards in the filtration requires removing non-zero entries along several columns of a particular row in V so that the corresponding permutation does not render V non-upper triangular. The key insight of the algorithm presented in [15] is that R can actually be maintained in all but one column during this procedure (by employing the donor column). In contrast, moving a simplex to an earlier time in the filtration requires removing non-zero entries along several rows of a particular column of V . As before, though R stays reduced during this cancellation procedure in all but one column, the subsequent permutation to R requires reducing a pair of columns which may cascade into a larger chain of column operations to keep R reduced. This is due to the fact that higher entries in columns in R (above the pivot entry) may very well introduce additional non-reduced columns after R is permuted. Since these operations always occur in a left-to-right fashion, its not immediately clear how to apply a donor column kind of concept. Fortunately, like move right, we can still separate the algorithm into a reduction and restoration phase-see Algorithm 4. Moreover, since R is reduced in all but one column by line 6 in Algorithm 4, we can still guarantee the number of column operations in R will scale with |i − j|. For a supplementary description of the move algorithm, see [15].
We recall an important claim given in [15] on the effect that move operations have on the status of simplices in the pairing. Recall from section 2 that simplices which create new homology classes are called creators and simplices that destroy homology classes are called destroyers. The effect of the movement on intermediate simplices depends on the direction of the movement. If i < j (respectively, j < i), all simplices at positions k ∈ [i + 1 : j] are shifted down (respectively, up) by 1.
A.1.2 LCS-Sort
Here we record explicitly the simple schedule construction algorithm outlined in section 3.3. The algorithm is simple enough to derive using the rules discussed in section 3.3 (namely, equation (15), but nonetheless for posterity sake we record it here for the curious reader; it is given in Algorithm 5. The high level idea of the algorithm is to first construct while D is not empty do 6: σ ← arbitrary element in D 7:
( i, i p , i n ) ← (p −1 (σ),p −1 (T pred (σ)),p −1 (T succ (σ)) ) ▷ O(log log m) 8: if i < i p then 9: j ← arbitrary element in [ i p , i n ) 10: else i n < i 11: j ← arbitrary element in ( i p , i n ] return S
A.2 2-parameter persistence
We now describe the reparameterization between the bigraded Betti numbers and the set of "critical lines" Lesnick and Wright [37] used to create their interactive 2d persistence algorithm, beginning with point-line duality. Let L denote the collection of all lines in R 2 with non-negative slope, L ⊂ L the collection of all lines with non-negative finite slope, and L • the collection of all affine lines with positive finite slope. Define the line and point dual transforms D ℓ and D p , respectively, as follows:
D ℓ : L → [0, ∞) × R D p : [0, ∞) × R → L y = ax + b → (a, −b) (c, d) → y = cx − d(23)
The transforms D ℓ and D p are dual to each other in the sense that for any point a ∈ [0, ∞) × R and any line L ∈ L, a ∈ L if and only if D ℓ (L) ∈ D p (a). Now, for some fixed line L, define the push map push L (a) : R 2 → L ∪ ∞ as:
push L (a) → min{v ∈ L | a ≤ v}(24)
The push map satisfies a number of useful properties. Namely:
1 For r < s ∈ R 2 , push L (r) ≤ push L (s) 2 For each a ∈ R 2 , push L (a) is continuous on L • 3 For L ∈ L • and S ⊂ R 2 , push L induces an ordered partition S L on S Property (1) elucidates how the standard partial order on R 2 restricts to a total order on L for any L ∈ L, whereas Properties (2) and (3) qualify the following definition: Definition 4 (Critical Lines). For some fixed S ⊂ R 2 , a line L ∈ L • is defined to be regular if there is an open ball B ∈ L • containing L such that S L = S L ′ for all L ′ ∈ B. Otherwise, the line L is defined as critical.
The set of critical lines crit(M ) with respect to some fixed set S ⊂ R 2 fully characterizes a certain planar subdivision of the half plane [0, ∞) × R. This planar subdivision, denoted by A(M ), is thus entirely determined by S under point line duality. A corollary from [35] shows that if the duals of two lines L, L ′ ∈ L are contained in the same 2-cell in A(M ), then S L = S L ′ , i.e. the partitions induced by push L are equivalent. Indeed, the total order on S L is simply the pullback of the total order on L with respect to the push map. Since A(M ) partitions the entire half-plane, the dual to every line L ∈ L is contained within A(M )-the desired reparameterization.
To connect this construction back to persistence, one requires the definition of bigraded Betti numbers. For our purposes, the i th -graded Betti number of M is simply a function β i (M ) : R 2 → N whose values indicate the the number of elements at each degree in a basis of the i th module in a free resolution for M -the interested reader is referred to [35,17] for a more precise algebraic definition. Let There are several approaches one can use to compute T , the simplest being to run Algorithm 2 independently on the 1-D filtration induced by the duals of some set of points (e.g. the barycenters) lying in the interior of the 2-cells of A(M ). The approach taken by [35] is to use the R = DV decomposition computed at some adjacent 2-cell e ∈ A(M ) to speed up the computation of an adjacent cell e ′ ∈ A(M ). More explicitly, define the dual graph of A(M ) to be the undirected graph G which has a vertex for every 2-cell e ∈ A(M ) and an edge for each adjacent pair of cells e, e ′ ∈ A(M ). Each vertex in G is associated with a barcode template T e , and the computation of T now reduces to computing a path Γ on G which visits each vertex at least once. To minimize the computation time, assume the n edges of G are endowed with non-negative weights W = w 1 , w 2 , . . . , w n whose values w i ∈ R + represent some notion of distance which is proportional to the computational disparity between adjacent template computations. The optimal path Γ * that minimizes the computation time is then the minimal length path with respect to W which visits every vertex of G at least once. There is a known 3 2 -approximation that can be computed efficiently which reduces the problem to the traveling salesman problem on a metric graph [20], and thus can be used so long as the distance function between templates is a valid metrics. [37] use the kendall distance between the push-map induced filtrations, but other options are available-for example, any of the combinatorial metrics we studied in Section 3.4.
Figure 1 :
1Top: A video of an expanding annulus. Bottom: Sublevel-set filtrations, via pixel intensity, of a Freudenthal triangulation of the plane.
Figure 2 :
2The cumulative column operations needed to compute persistence across the time-varying filtration of grayscale images. Observe 10 independent persistence computations evenly spaced in time (green line) captures the major topological changes and is the most computationally efficient approach shown.
and k = |i − j|. Moreover, let M denote the number of column operations to perform the same update R → R ′ with Move(i, j). Then the inequality M ≤ T holds.Proof. First, consider executing the vineyards algorithm with a given pair (i, j). As there are at most 2 column operations, any contiguous sequence of transpositions (i, i + 1), (i + 1, i + 2), . . .(j − 1, j) induces at most 2(|i − j|) column operations in both R and V , giving a total of 4(|i − j|) column operations. Now consider a single MoveRight(i,j) outlined in Algorithm 3. Here, the dominant cost again are the column operations (line 5). Though we need an extra O(m) storage allocation for the donor columns d * prior to the movement, notice that assignment to and from d * (lines (4), and (7) in RestoreRight of MoveRight, respectively) requires just O(1) time via a pointer swapping argument. That is, when d ′ low < d low , instead of copying col * (k) to d ′ * -which takes O(m) time-we instead swap their column pointers in O(1) prior to column operations. After the movement, d * contains the newly modified column and col * (k) contains the unmodified donor d ′ * , so the final donor swap also requires O(1) time. Since at most one O(m) column operation is required for each index in [i, j], moving a column from i to j where i < j requires at most 2(|i − j|) column operations for both R and V . The claimed inequality follows.
denote the LCS of the permutations of K induced by f and f ′ . Moreover, |S * | = d can be determined in O(m log log m) time.
Corollary 1 .
1If (K, f ), (K, f ′ ) are random filtrations of a common complex K of size m, then the expected size of longest common subsequence of simplices between f, f ′ is no larger than m − √ m, with probability 1 as m → ∞. Proof. The proof of this result reduces to showing the average length of the LIS for random permutations. Let L(p) ∈ [1, m] denote the maximal length of a increasing subsequence of p ∈ S m . The essential quantity to show the expected length of L(p) over all permutations: E L(p) = ℓ m = 1 m! p∈Sm
(
can be determined in O(m log log m) time and O(m) space, where R ′ = D d V d denotes a valid decomposition of (K, f ′ ). Moreover, if i k < j k for all k ∈ [d], then computing (13) requires O(κ) column operations, where: |I k | + |J k |) where the quantities |I k |, |J k | of the intermediate R k , V k are given in Proposition 1. Note that since κ depends on the sparsity of the intermediate entries V 1 , V 2 , . . . , V d and R 1 , R 2 , . . . , R d , the bound O(κ) is output-sensitive. Proof. Proposition 3 yields the necessary conditions for constructing S with optimal size d in O(m log log m) time and O(m). The definition of κ follows directly from Algorithm 3.
. , s d ) in O(dm log m) time and O(m) space, or O(m log m) time and O(m) space per update in the online setting.
A pair of filtrations is given below, each comprising the 1-skeleton of a 3-simplex. Relabeling (K, f ) to the index set f : K → [m] and modifying (K, f ′ ) accordingly yields the permutations given below:(K, f ) = {a b cThe subset of the filtration which corresponds to the simplices which lie in the LCS between these permutations is colored in red.For this example, the edit distance is d = m − |LCS(f, f ′ )|, implying exactly 3 moves are needed to map f → f ′ . There are six possible valid schedules of moves: S 1 = m xu , m yu , m zu S 3 = m yu , m xy , m zu S 5 = m zu , m xz , m yz S 2 = m xu , m zu , m yz S 4 = m yu , m zu , m xy S 6 = m zu , m yz , m xz where the notation m xy represents the move permutation that moves symbol x to the position of symbol y. The cost of each move operation and each schedule is recorded in
Figure 3 :
3Performance comparison between various scheduling strategies. On the left, the cumulative column operations required to simulate the 1-parameter family is shown for varying schedule sizes (d) and strategies. On the right, both the size of the schedule and the data set (m) are varied.
Figure 4 :
4A crocker plot (right) depicts the evolution of dimension p = 1 Betti curves over time. The green X marks correspond chronologically to the complexes (left), in row-major order. The large orange and purple areas depict 1-cycles persisting in both space (y-axis) and time (x-axis).
Definition 2
2(crocker stack). A crocker stack is a family of α-smoothed crock plots which summarizes the topological information of a time-varying persistence module M via the function f M : [0, T ] × [0, ∞) × [0, ∞) → N, where:
Definition 3 (
3Fibered barcode). The fibered barcode B(M ) of a 2D persistence module M is the map which sends each line L ⊂ R 2 with non-negative slope to the barcode B L (M ): B(M ) = { B L (M ) : L ∈ R × R + } Equivalently, B(M ) is the 2-parameter family of barcodes given by restricting M to the of set affine lines with nonnegative slope in R 2 .
Figure 6 :
6Bipersistence example on an 8 × 8 coarsened grid. On left, the input data, colored by density. In the middle, the bigraded Betti numbers β 0 (M ) and β 1 (M ) (green and red, respectively), the dimension function (gray), and a line L emphasizing the persistence of features with high density. On the right, the line arrangement A(M ) lying in the dual space derived from the β(M ).Utilizing the equivalence between the rank and fibered barcode invariants, Lesnick and Wright[35] developed an elegant way of computing B(M ) via a reparameterization using standard point-line duality. This clever technique effectively reduces the fibered barcode computation to a sequence of 1-D barcode computations at "template points" lying within the 2-cells of a particular planar subdivision A(M ) of the half-plane [0, ∞) × R. This particular subdivision is induced by the arrangement of "critical lines" derived by the bigraded Betti numbers β(M ) of M . As the barcode of one template point T e at the 2-cell e ∈ A(M ) may be computed efficiently by re-using information from an adjacent template point T e ′ ,[35] observed that computing the barcodes of all such template points (and thus, B(M )) may be reduced to ordering the 2-cells in A(M ) along a Eulerian path traversing the dual graph of A(M ). The full algorithm is out of scope for this effort; we include supplementary details for the curious reader in the appendix A.2.
Example 4 . 1 :
41Consider a small set of noisy points distributed around S 1 containing a few strong outliers, as shown on the left side ofFigure 6. Filtering this data set with respect to the Rips parameter and the complement of a kernel density estimate yields a bifiltration whose various invariants are shown in the middlefigure. The gray areas indicate homology with positive dimension-the lighter gray area dim 1 (M ) = 1 indicates a persistent loop was detected. On the right side, dual space is shown: the black lines are the critical lines that form A(M ), the blue dashedlines the edges of the dual graph of A(M ), the rainbow lines overlaying the dashed-lines form the Eulerian path, and the orange barycentric points along the 2-cells of A(M ) represent where the barcodes templates T e are parameterized. Despite its elegance, there are significant computational barriers prohibiting the 2-parameter persistence algorithm outlined from being practical. An analysis from [35] (using vineyards) shows the barcodes template computation requires on the order of O(m 3 κ + mκ 2 log κ) elementary operations and O(mκ 2 ) storage. Since the number of 2-cells in A(M ) is on the order O(κ 2 ), and κ itself is on the order of O(m 2 ) in the worst case, the worse-case complexity of the barcode templates computation O(m 5 )-this is both the highest complexity and most time-intensive sub-procedure the RIVET software
Figure 7 :
7Bipersistence example of natural images data set on an 12 × 16 coarsened grid. On the left, a projection the full data set is shown, along with the 15 landmark patches. (Middle) the bigraded Betti numbers and a fixed line L over parameter space. As before, the 0/1/2 dimension bigraded Betti numbers are shown in green/red/yellow, respectively, with the blue region highlighting where dim(M ) = 5. (Right) five persistent features representing B L (M ) are revealed from the middle, matching β 1 of the three-circle model.
homological critical points in time-varying filtrations
Algorithm 2
2Reduction Algorithm (pHcol) Require: D = (m × m) filtration boundary matrix Ensure: R is reduced, V is full rank upper triangular, and R = DV 1: function REDUCTION(
Since each column operation takes O(m) and there are potentially O(k) columns in D with identical low entries (line 4 in 2, observe the reduction algorithm below clearly takes O(m 2 k) time. Since there exists complexes where k ∼ O(m), one concludes the bound of O(m 3 ) is tight
(
S, D, T ) ← ( S ∪ (i, j), D \ σ, T ∪ σ ) ▷ O(log log m) 13:p −1 ←p −1 • m
S = supp β 0 (M ) ∪ supp β 1 (M ), where the functions β 0 (M ), β 1 (M ) are 0 th and 1 st bigraded Betti numbers of M , respectively. The main mathematical result from [35] is a characterization of the barcodes B L (M ), for any L ∈ L, in terms of a set of barcode templates T computed at every 2-cell in A(M ). More formally, for any line L ∈ L and e any 2-cell in A(M ) whose closure contains the dual of L under point-line duality, the 1-parameter restriction of the persistence module M induced by L is given by:B L (M ) = {[ push L (a), push L (b) ) | (a, b) ∈ T e , push L (a) < push L (b)}(25)Minor additional conditions are needed for handling completely horizontal and vertical lines. The importance of this theorem lies in the fact that the fibered barcodes are completely defined from the precomputed barcode templates Tonce every barcode template T e has been computed and augmented onto A(M ), B(M ) is completely characterized, and the barcodes B L (M ) associated to a 1-D filtration induced by any choice of L can be efficiently computed via a point-location query on A(M ) and a O(|B L (M )|) application of the push map.A.2.1 Invariant computationComputationally, the algorithm from[36] can be summarized into three steps:1Compute the bigraded Betti numbers β(M ) of M 2 Construct a line arrangement A(M ) induced by critical lines from (1) 3 Augment A(M ) with barcode templates T e at every 2-cell e ∈ A(M ) Computing (1) takes approximately ≈ O(m 3 ) using a matrix algorithm similar to Algorithm 2 [36]. Constructing and storing the line arrangement A(M ) with n lines and k vertices is related to the line segment intersection problem, which known algorithms in computational geometry can solve in (optimal) output-sensitive O((n+k) log n) time [12]. In terms of space complexity, the number of 2-cells in A(M ) is upper bounded by O(κ 2 ), where κ is a coarseness parameter associated with the computation of β(M ).
Table 1 :
1Move schedule costsCost of each permutation
1st
2nd
3rd
Total
S 1 2
3
1
6
S 2 2
2
4
8
S 3 4
2
2
8
S 4 4
3
3
10
S 5 2
2
4
8
S 6 2
5
3
10
Table 2 :
2Cost to computing T for various coarsening choices of β(M ).β(M )
A(M )
Col. Reductions / Permutations
Coarsening
# 2-cells
phCol
Vineyards / d K
Moves / d LCS
8 x 8
39
94.9K
245K / 1.53M
38.0K / 11.6K
12 x 12
127
318K
439K / 2.66M
81.9K / 33.0K
16 x 16
425
1.07M
825K / 4.75M
114K / 87.4K
20 x 20
926
2.32M
1.15M / 6.77M
148K / 154K
24 x 24
1.53K
3.92M
1.50M / 8.70M
184K / 232K
1 Matt Piekenbrock: conceptualization, methodology design, algorithm development, experimental design & analysis, software development, drafting of the article.
Dionysus 1 does have an implementation of vineyards, however the algorithm was never ported to version 2. Other major packages, such as GUDHI and PHAT, do not have vineyards implementations.6 For all accompanying software and materials, see: https://github.com/peekxc/move_schedules
This term has been used in reference to parameterized curves whose behavior with respect to a certain restricted set of geometric predicates is invariant, see[13].
crocker stands for "Contour Realization Of Computed k-dimensional hole Evolution in the Rips complex." Although the acronym includes Rips complexes in the name, in principle a crocker plot could just as easily be created using other types of triangulations (e.g.Čech filtrations).
See http://bethgelab.org/datasets/vanhateren/ for details on the image collection.
Jose Perea: Literature review, critical revision of the manuscript, final approval of the version to be published.10 By "standard," we mean any of the common sparse representations used in scientific computing packages, like SciPy's sparse module (https://docs.scipy.org/doc/scipy/reference/sparse.html)
All authors have reviewed and approved the final version of the manuscript and have agreed to be accountable for all aspects of the work.Funding: The research presented in this work is partially supported by the National Science Foundation through grants CCF-2006661 and CAREER award DMS-1943758. The funding source had no role in the study design, data collection and analysis, decision to publish, or preparation of the manuscript.Availability of data and materials: Links to the software and data used for the experiments can be found at: https://github.com/peekxc/move_schedules.( d low , d R , d V ) ← ( low R (I 1 ), col R (I 1 ), col V (I 1 ) )3:for k in I 2 , . . . , I s do 4:(col R (i), col V (i)) += (col R (k), col V (k)) 5:I ← I ∪ k + 1 the LCS between two permutations p, q ∈ S m . To do this efficiently, one re-labels q → ι to the identity permutation ι = [m] and applies a consistent re-labeling p →p. This relabeling preserves the LCS distance and has the additional advantage thatq = ι = [m] is a strictly increasing subsequence, and thus computing the LCS between p, q ∈ S m reduces to computing the LIS L ofp. By sortingp → p via operations which (strictly) increase the size of L, we ensure that the size of the set of corresponding permutations is exactly m − |L|.Suppose L has been computed fromp. Since L is strictly increasing, the only symbols left to permute are in L \p, which we denote with D. After choosing any symbol σ ∈ D, one then applies a cyclic permutation top that moves σ into any position that increases the size of L. To do this efficiently, we maintain a data structure T which enables us to query the successor and predecessor of any given symbol s ∈p in L. We also require a data structure to query the position of a given element σ ∈p, which for now we simply use the inverse permutationp (though a more efficient representation can be used based off of symbol displacements, see section 3.4.2. After the symbol is inserted into L, we updatep, its inverse permutationsp −1 , D and T prior to the next move. The final set of permutations which sort p → q (or equivalently,p → ι are stored in an array S, which is then returned for further use.
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| [
"https://github.com/peekxc/move_schedules",
"https://github.com/peekxc/move_schedules.("
]
|
[
"Interface Self-Referenced Dynamic Full-Field Optical Coherence Tomography",
"Interface Self-Referenced Dynamic Full-Field Optical Coherence Tomography"
]
| [
"Tual Monfort \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n\nCHNO des Quinze-Vingts\nINSERM-DGOS CIC 1423\n28 rue de CharentonF-75012ParisFrance\n",
"Salvatore Azzollini \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n",
"Tasnim Ben Yacoub \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n",
"Isabelle Audo \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n",
"Sacha Reichman \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n",
"Kate Grieve \nInstitut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance\n\nCHNO des Quinze-Vingts\nINSERM-DGOS CIC 1423\n28 rue de CharentonF-75012ParisFrance\n",
"Olivier Thouvenin *[email protected] \nInstitut Langevin\nESPCI Paris\nUniversité PSL\nCNRS\n75005ParisFrance\n"
]
| [
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"CHNO des Quinze-Vingts\nINSERM-DGOS CIC 1423\n28 rue de CharentonF-75012ParisFrance",
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"Institut de la Vision\nSorbonne Université\nINSERM\nCNRS\n17 rue MoreauF-75012ParisFrance",
"CHNO des Quinze-Vingts\nINSERM-DGOS CIC 1423\n28 rue de CharentonF-75012ParisFrance",
"Institut Langevin\nESPCI Paris\nUniversité PSL\nCNRS\n75005ParisFrance"
]
| []
| Dynamic full-field optical coherence tomography (D-FFOCT) has recently emerged as an invaluable live label-free and non-invasive imaging modality able to image subcellular biological structures and their metabolic activity within complex 3D samples. However, D-FFOCT suffers from fringe artefacts when imaging nearby reflective surfaces and is highly sensitive to vibrations. Here, we present interface Self-Referenced (iSR) D-FFOCT, an alternative configuration to D-FFOCT that takes advantage of the presence of the sample coverslip in between the sample and the objective by using it as a defocused reference arm, thus avoiding the aforementioned artefacts. We demonstrate the ability of iSR D-FFOCT to image 2D fibroblast cell cultures, which are among the flattest mammalian cells. | null | [
"https://export.arxiv.org/pdf/2302.08839v1.pdf"
]
| 257,019,856 | 2302.08839 | efe14f6446982216fb9022231065cf99f9db904c |
Interface Self-Referenced Dynamic Full-Field Optical Coherence Tomography
Tual Monfort
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
CHNO des Quinze-Vingts
INSERM-DGOS CIC 1423
28 rue de CharentonF-75012ParisFrance
Salvatore Azzollini
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
Tasnim Ben Yacoub
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
Isabelle Audo
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
Sacha Reichman
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
Kate Grieve
Institut de la Vision
Sorbonne Université
INSERM
CNRS
17 rue MoreauF-75012ParisFrance
CHNO des Quinze-Vingts
INSERM-DGOS CIC 1423
28 rue de CharentonF-75012ParisFrance
Olivier Thouvenin *[email protected]
Institut Langevin
ESPCI Paris
Université PSL
CNRS
75005ParisFrance
Interface Self-Referenced Dynamic Full-Field Optical Coherence Tomography
4 These authors contributed equally
Dynamic full-field optical coherence tomography (D-FFOCT) has recently emerged as an invaluable live label-free and non-invasive imaging modality able to image subcellular biological structures and their metabolic activity within complex 3D samples. However, D-FFOCT suffers from fringe artefacts when imaging nearby reflective surfaces and is highly sensitive to vibrations. Here, we present interface Self-Referenced (iSR) D-FFOCT, an alternative configuration to D-FFOCT that takes advantage of the presence of the sample coverslip in between the sample and the objective by using it as a defocused reference arm, thus avoiding the aforementioned artefacts. We demonstrate the ability of iSR D-FFOCT to image 2D fibroblast cell cultures, which are among the flattest mammalian cells.
Introduction
In recent years, full-field optical coherence tomography (FFOCT) has emerged as a versatile non-invasive label free optical imaging technique thanks to its high resolution, amplitude and phase contrasts [1][2][3], its sectioning ability, and its sensitivity and imaging speed [1][2][3][4]. Its use has been demonstrated in vivo on living rodent for single myelin resolution sheath disruption involved in neuropathies [5], as well as in vivo on human retina [6] and cornea [7] with cellularresolution capabilities.
One promising development of FFOCT is dynamic FFOCT (D-FFOCT), in which the temporal evolution of FFOCT signal is analysed in order to quantify the nanometric active displacements of subcellular organelles [8,9]. D-FFOCT provides a metabolic contrast [1,9,10] highly complementary to the structural contrast obtained from static FFOCT [11]. Static and dynamic FFOCT ((D)-FFOCT) have been combined for several in vitro and ex vivo studies, for example in retinal explants [10,12,13] and retinal organoids [9,13]. Taking advantage of morphology and specific dynamic contrast, (D)-FFOCT could resolve different cell types and subcellular structures [1,9,10,[13][14][15], and monitor different cell metabolic states, such as senescence and mitosis [10,13,15]. As a result, (D)-FFOCT is an appealing solution to drive biology research on unaltered systems at high resolution under live imaging conditions [9,13,15].
Despite its success, a few aspects still impede (D)-FFOCT. Indeed, based on a Linnik interferometer configuration, (D)-FFOCT relies on two symmetric but physically separated optical arms. This aspect leads to three main drawbacks; First, (D)-FFOCT is prone to fringe artefacts when imaging close to the reflective surface of sample holders for ex vivo and in vitro studies [10,[14][15][16][17] typically preventing imaging of the first micrometers of a sample, which especially impacts the imaging of thin samples. Yet, adherent (2D) cell culture is the main in vitro condition used in biology [18][19][20]. Groux et al. were recently able to partially suppress these fringe artefacts by using weakly scattering porous polycarbonate membranes to move the sample away from the sample holder in 2D retina pigmented epithelial (RPE) cell cultures [15]. Nonetheless, for thinner epithelial cells (<5µm thick), such as fibroblasts, the signal from the culture membranes was partially covering the cell bodies and decreased image quality. Also, these membranes are incompatible with the highest numerical aperture (NA) objectives due to their limited working distance, and are not adaptable to all cell culture conditions. Second, D-FFOCT is sensitive to subnanometric external vibrations which, even on specifically designed vibration-free optical tables, are hard to cancel completely. This reduces the image quality of D-FFOCT and makes it difficult to properly quantify the organelle motion and the metabolic contrast [10]. Finally, (D)-FFOCT is challenging to implement using objectives with very high numerical aperture (>1), or when the interference arms contain several optical elements [13]. Indeed, any dissymmetry, including spherical aberration due to small misalignments between interfering fields results in distortions, a loss of accessible field of view, and a loss of interference contrast [13]. This typically results in non-homogeneous and distorted (D)-FFOCT signal over the field of view, which makes efficient mosaicking complicated to implement [13].
In this work we present and characterize a new interferometric configuration which overcomes these (D)-FFOCT drawbacks in order to enable imaging of cells in the vicinity of the culture surface. This interface Self-Referenced (iSR) (D)-FFOCT method allowed imaging of 2D flat fibroblast samples, which are of interest to biologists in disease modelling applications of mitochondrial disease [21][22][23][24]. The technique could be used more widely for rapid and robust diagnosis directly from cell phenotypes.
Methods
FFOCT and iSR FFOCT microscopes
FFOCT measurements were performed using the setup described in [13], and shown in a simplified form in Fig. 1. A high-power LED S1 (either a M810L3 or M730L4, Thorlabs, Newport, NJ, USA, λ0= 810 nm Δλ=25 nm, coherence length Lc = 8.7 μm or λ0= 730 nm Δλ=40 nm, coherence length Lc = 4.4 μm), was used to illuminate a Linnik interferometer with high NA objectives (NA= 1.05, immersion medium n=1.4, silicon oil, UPLSAPO30XSIR, Olympus, Japan). The light reflected by the reference mirror, the light reflected by the glass coverslip supporting the sample, and the light backscattered by the sample are recombined by the nonpolarizing beamsplitter, focused by a tube lens L3 (AC254-300-B-ML, Thorlabs, Newport, NJ, USA) to overlap, and potentially interfere, on a high full well capacity (FWC) 2D CMOS sensor (Q-2HFW, Adimec, Netherland). The lateral and axial magnifications of the system are respectively: = 58, = 2400.
iSR FFOCT measurements were performed on the same setup by manually blocking the reference arm, hence allowing detection of the interference between the light reflected from the glass coverslip (standard coverslips, 170 μm thick, n=1.52, P24-1.5H-N, Cellvis, Canada) and the light backscattered by the different depths of the sample. The top surface of the glass coverslip here acts as the reference mirror and is slightly defocused compared to the focal plane of the objective. The reflectivity of the glass in water is about 0.4%, according to Fresnel coefficients and neglecting angular effects.
For both configurations, the Q-2HFW camera was configured to have a FWC of 1.6 Me -, with a signal to noise ratio (SNR) of 60.6 dB at saturation. Acquisitions were performed at 100 frames per second (FPS) when generating dynamic images. All samples were imaged inside a top-stage microincubator (H201-K-FRAME, H201-MW-HOLDER and OBJ-COLLAR-2532, Okolab, Italy) and were maintained at 37 C o and 5% of CO2 concentration during the acquisition. Data were acquired with a custom Matlab graphical user interface enabling continuous data logging while post-processing and saving the resulting metrics in parallel threads for maximal acquisition speed [13]. Fig.1a illustrates a classical D-FFOCT set-up with a spatially incoherent source illuminating a Linnik interferometer using a Köhler illumination. The incoming field, in yellow, is split by a non-polarizer beam splitter (NPBS, BS014, Thorlabs, Newport, NJ, USA) cube into a reference arm and a sample arm. In the reference arm, an objective focuses the light on a mirror, placed at the image focal plane (FPI ref) of the reference objectif (Obj.ref). The back reflected field is sketched in red. In the sample arm, an identical microscope objective (Obj. Sam) focuses the light onto a sample, laid on a coverslip (CoverSlip Sam) for the inverted microscope, at its image focal plane (FPI Sam). The backscattered light is illustrated in green. The objective (Obj. Sam) also collects the out-of-focus light (pictured in blue) reflected by the specular top surface of the coverslip (CoverSlip Sam). Both beams are recombined by the NPBS and focussed on a camera by intermediary of a tube lens L3. In iSR FFOCT, the reference arm is blocked so that only the two beams from the sample arm reach the camera and can interfere. Fig.1b illustrates the iSR D-FFOCT configuration showing the imperfect overlap of these two beams occurring on the camera.
Figure 1: Comparison between D-FFOCT setup versus iSR D-FFOCT.
Image acquisition protocol and dynamic FFOCT image generation
The dynamic images showcased were computed according to three metrics established by Scholler et al. in 2020 [9] using either 512 raw FFOCT or iSR FFOCT images and computing the averaged running standard deviation with a window of 50 images, the mean frequency of the power density spectrum, and the standard deviation frequency of the power density spectrum, and displayed together in a Hue Saturation and Brightness (HSB) base, respectively.
Fibroblast culture protocol
Human dermal fibroblasts from a healthy subject (male, 49 y.o, Caucasian) were obtained at the Quinze-Vingts hospital after a skin biopsy. Fibroblasts were cultured in T75 flasks in DMEM glutamax medium (Thermo Fisher Scientific, 61965026) supplemented with 10% Fetal Bovine Serum (FBS), 2% sodium pyruvate, 1% penicillin-streptomycin and 1% amphotericin B, and incubated at 37°C, in a 5% CO2 incubator. When fibroblasts reached 70% confluency, cells were dissociated using Enzyme Express (1X) TrypLE™ (Thermo Fisher Scientific, 12605010). Cells were plated in an uncoated 6-well glass bottom plate (Cellvis, P06-1.5H_N) at a density of 40 000 cells/well and the medium was changed every 2 days. Five days after seeding, fibroblasts were imaged.
Results and discussion
System characterization: Resolution, and sensitivity.
With iSR FFOCT, we successfully obtained static images of fibroblasts by recording and subtracting two successive planes so that a phase difference of is obtained (not shown). We also obtained dynamic images of fibroblasts (see Fig.2 -5). Interestingly, in iSR D-FFOCT fibroblasts show many small, distinguishable, highly contrasted features, such as filopodiacylindrical structures of around 20 to 200 nm diameter [26], below the diffraction limit-that can be used to characterize the optical system. For all characterizations in subsections 3.1 and 3.2, we only use and display the Brightness channel of the D-FFOCT or iSR D-FFOCT images. Evaluation of the system transverse and axial resolution were performed on such filipodiasee (1 − ( ( ⁄ )) ) = 751.9
We found that iSR D-FFOCT has a higher axial resolution than the expected theoretical resolution for FFOCT. However, we note that no theoretical model has been established for D-FFOCT in general and no PSF evaluation was previously measured in the literature, to the best of our knowledge. However, because D-FFOCT contrast relies on the non-linear postprocessing of multiple images, we expect that D-FFOCT resolution may depend on the SNR of the detection and on the stochastic diffusion of the scatterer. As a result, in favorable conditions, it can exceed the resolution of FF-OCT.
iSR FFOCT can be described as a standard FFOCT system, with a defocused reference arm. Interestingly, it was recently shown that FFOCT resolution was almost insensitive to defocus [27], so that the effect of the defocus mostly results in a loss of signal. Figure 2: Evaluation of iSR D-FFOCT spatial resolution using filipodia, which are sub-diffraction limit sized structures. Fig.2a shows an image including filipodia from which an intensity profile is displayed and fitted to a Gaussian in Fig.2b, with a half-width at the half-maximum (HWHM) of 378.4 nm. Fig.2d. shows an axial reslicing of a z stack including filipodia from which an intensity profile is displayed and fitted to a Gaussian in Fig.2c, with a half-width at the half-maximum (HWHM) of 415 nm.
In fibroblasts, we obtained a maximal sensitivity of 43.52 dB. Compared to standard D-FFOCT, the reference mirror in iSR D-FFOCT is made of glass of about 0.4% reflectivity, which can increase D-FFOCT sensitivity as long as incoherent reflections are lower than this value, and if the camera FWC can be saturated. Here, we used a NPBS to combine iSR D-FFOCT and a standard D-FFOCT system. However, incoherent reflections coming from the NPBS cube were very important when using a specular reflection glass/sample for creating a reference field, and still filled up the FWC of our camera: about 92% of the dynamic range of our camera is coming from incoherent reflection, of which 99% originates from the NPBS. These values were established by blocking out iteratively optics and/or sample contribution. We note further that the FFOCT configuration used in this paper picks up less incoherent reflection from the NPBS than more traditional setup [13]. Because iSR D-FFOCT is immune to mechanical vibrations, a pellicle beamsplitter could be used instead to increase the theoretical sensitivity by a factor of 12. However, more powerful light source would be required and its integration to a FF-OCT microscope would be more complex.
System advantages compared to D-FFOCT: fringe artefacts, vibration sensitivity and mosaicking
We first start by co-characterising iSR D-FFOCT and D-FFOCT responses in the vicinity of the coverslip, on the same areas and at the same distances from the interface coverslip/sample (ΔZ). Fig.3a-b show images at the same locations at ΔZ=0.8 μm, illustrating that iSR D-FFOCT contrast (Fig.3b), in the vicinity of a coverslip, is free of fringe-artefacts in comparison to D-FFOCT (Fig.3a) while displaying significant structural details, including nucleus, nucleoli, mitochondria, filipods and actin filament network (see subsection 3.3). We perform an additional Z-stacks using iSR D-FFOCT and D-FFOCT on a thicker sample in order to characterise iSR D-FFOCT and D-FFOCT axial signal response. A retinal organoid is used and iSR D-FFOCT and D-FFOCT signal is recorded on one plane every 100 nm. We found that iSR D-FFOCT (Fig.3d) maintains sufficient contrast up to ΔZ = 3.5 μm while D-FFOCT only becomes fringe-free from this distance (Fig.3c).
In the case of D-FFOCT (Fig.3c), we observe a strong signal (Brightness) in the vicinity of the glass coverslip due to the glass/water reflectivity being significantly higher than light scattered by the sample, before dropping to a level where the glass coverslip and sample contribution are equivalent. These two behaviours are highlighted in Fig.3c by an orange shaded area, corresponding to the depths where D-FFOCT is not able to image samples. In the case of iSR D-FFOCT, the signal is maximal at the glass coverslip interface and drops with the depths.
Figure 3: Illustration of differences between D-FFOCT and iSR D-FFOCT.
Two images at the same location and depth (ΔZ = 0.8 μm) are displayed using D-FFOCT (Fig.3a) and iSR D-FFOCT (Fig.3b). No fringes artefact is present when using iSR D-FFOCT (Fig.3b) unlike for D-FFOCT (Fig.3a). Signal strength (Brightness) is evaluated at different distances from the coverslip (ΔZ), on a 3D sample (retinal organoid), using D-FFOCT (Fig.3c) and iSR D-FFOCT (Fig.3d). Stability of D-FFOCT and iSR D-FFOCT is assessed using the mean frequency (Hue) at different depths in a 3D sample (retinal organoid) using the LED at 730 nm.
Furthermore, we qualitatively assess the relative immunity of iSR D-FFOCT to mechanical vibration compared to D-FFOCT (Fig.3e), by measuring how the averaged mean frequency of the power spectrum density (Hue channel) evolves during a Z-stack, at the equivalent depths and location. An improvement factor of 42.08 dB is found for iSR D-FFOCT over D-FFOCT in term of mean frequency stability (Hue). Indeed, the close distance ΔZ between the scatterer and the specular reflection, used to generate a reference field, seems to correlate mechanically both. As a result, phase shifts due to vibrations are stacking up identically to the reference and scattered field. This mechanical locking result in a virtually vibration-insensitive D-FFOCT imaging modality: we did not observe artefacts while stepping around the setup nor knocking on the optical table. As a result, iSR D-FFOCT is automatically quantitative and can function without the use of a mechanically damped optical table.
Finally, iSR D-FFOCT is able to produce very homogeneous and flat interferences compared to D-FFOCT [13], thanks to its insensitivity to setup-induced aberrations, and its low penetration depth. This aspect can clearly be seen when mosaicking, as shown in Fig.4 and Fig.5a, which shows a flawless mosaic over a wide field whilst using a basic stitching method [28]. Compared to previous work using 50% overlap between D-FFOCT tiles for mosaic reconstruction at the same magnification [13], and using high end stitching method, we could use a lower overlap of 20% in a flawless manner, and we forsee lower tile overlap possible. Furthermore, because field aberrations are particularly sensitive at high NAs, and are difficult to correct, this suggests that iSR D-FFOCT could potentially be performed at higher NA than standard D-FFOCT [13], with additional elements in overlaid in the sample arm.
Biological results on Fibroblasts
We imaged fibroblasts, which are flat epithelial cells, with iSR D-FFOCT to highlight cell structures and to characterise iSR D-FFOCT imaging responses. The shape of the fibroblast can be outlined, and using volumetric acquisition, intertwined fibroblasts can be individually identified using iSR D-FFOCT's sectioning ability. Morphologically, each fibroblast displays a nucleus, pointed out by a red arrows in Fig.5a,b, d,f, with an average frequency of 5 Hz and a slight desaturation, indicating a less chaotic oscillation of the nucleus compared to its surroundings. The boundary of the nucleus could be observed at specific depth displaying a further desaturated contrast, highlighted by a dash line in Fig.5d,f. Euchromatin structures, delineated with a drop in saturation, could be observed too (Fig.5d, f). Furthermore, we observe a drop in Saturation in the cytoplasm highlighting probably vesicles, as pointed out by pink arrows in Fig.5c-d.
Bright dots, especially present in Fig.5b-c,f, are mitochondria which have been specifically identified using immunochemistry labelling and cross correlated to D-FFOCT [15]. Stress fibers linked to sites of adhesion and composed by long actin-filament bundles crossing cell bodies can also be observed as bleu filaments at 3 Hz (Fig. 5b,c, grey arrows) [29]. Moreover, filipodia which are protrusions of cytoplasm from lamellipodia, observed during cell migration, were distinguished by their high saturation and high dynamic activity at 9-13 Hz (red bundles) and by their oriented bundle of dynamic actin filaments [30] (Fig. 5a,b,d,e, Yelow arrows). Their contrast has been specifically cross correlated with dynamic D-FFOCT contrast too [15].
Strikingly, Fig.5c [21]. In particular, membrane stress in Fig.5e can be observed on the edge of the fibroblast, with a ridge-like appearance (pointed out by a white arrow). Further comparison with this work, including comparison with specific fluorescence staining images, enables identification of actin fibres (Fig.5c, purple arrow). Lamellipodia, which guide cell movement, sensing the outer environment and extracellular stimuli, are clearly visualized in our images, with pronounced broad desaturation protrusions (Fig.5b,e, blue arrows).
Overall, these results show that iSR D-FFOCT is able to detect a wide variety of cytoplasm components, cell body shape and nuclear structures in cells, showing the potential of iSR D-FFOCT for label-free and non-invasive live assessment at high resolution. Furthermore, this demonstrates that iSR D-FFOCT could be used for thin 2D cell culture imaging. Fig.5a shows an iSR D-FFOCT HSB mosaic of fibroblasts composed of 5x5 tiles with 20% overlap. Hue scales from 3 to 13 Hz. Examples of nucleus observed are highlighted by red arrows in Fig.5a. Fig.5b highlights cases of lamellipodia structures (blue arrows). Fig.5c shows actin fibres, one example highlighted with a purple arrow. Euchromatin structures can be observed in Fig.5d within the nucleus, delimited by a white dashed line. Sharp membrane ridges and curved membrane pockets (white arrows) are shown in Fig.5e, as well as ridges in a nucleus in Fig.5f, delimited by a white dashed line. Examples of vesicles are highlighted by pink arrows in Fig.5c-d. An Example of filipodium is highlighted in Fig.5e with a yellow arrow. Scale bar is 90 μm for Fig.5a, 40 μm for Fig.5b, 20 μm for Fig.5c, 15 μm for Fig.5d, 30 μm for Fig.5e and 15 μm for Fig.5f.
Conclusion
In this work, we have created a new label free optical microscope based on a new selfreferenced configuration of static and dynamic FFOCT. This configuration is immune to fringe artefacts and mechanical vibrations. It is also less sensitive to interferometer arm misalignment and field curvature that can be critical with the use of very high NA objectives. It is simple to implement, especially on an existing D-FFOCT setup. Nonetheless, iSR D-FFOCT is restricted to an imaging depth within the coherence length of the source and is mostly interesting for thin 2D samples, as well as for recording structures at the surface of a 3D sample. These are situations where regular D-FFOCT typically struggles to obtain artefact-free images. As a result, by combining iSR D-FFOCT and D-FFOCT configurations, simply by blocking the reference arm close to the cell culture dish base, static and dynamic FFOCT images can now be obtained over a depth range from the sample surface to a few hundred microns.
Cytology plays an important role in diagnosing and managing human diseases. At a molecular level, the 'state' of a cell will depend on a large number of microscopic variables [31]. Moreover, most of the methods for measuring molecular state variables are destructive to cells, rendering it impossible to study temporal variation or correlate molecular states with downstream behaviour. In fact, organelle morphology can reflect neurological or metabolic diseases as well as cancers. For example, changes in mitochondrial morphology have been linked to several neurodegenerative disorders such as Alzheimer's [32], optic atrophy and Charcot-Marie-Tooth neuropathy [33]. Similar findings have been observed for the endoplasmic reticulum in pancreatic β-cells of type 2 diabetic patients [34]. At the nuclear level, changes in morphology have long been diagnostic of cancer with nuclear aberration correlated to the severity of prognosis [35]. A fast and easy analysis of subcellular structure with a noninvasive and live technique such as iSR D-FFOCT will contribute to enriching computational cytology and make a correct diagnosis. Moreover, this accessibility to image ultrastructure of live adherent cells will allow an efficient evaluation of new therapies targeting the pathological phenotype.
Thanks to its simplicity and its insensitivity to mechanical vibrations, iSR D-FFOCT could be deployed in much harsher environments than standard D-FFOCT, including operation rooms, classrooms, field measurements, or even within exploratory boats or zero-gravity environments, enabling the democratization of the use of D-FFOCT. In this direction, we could also imagine an adaptation of iSR D-FFOCT to cell phones, for example, based on the inexpensive extended source used for generating the flash and the camera of the mobile phone.
Fig. 2 .
2We obtained a very good gaussian fit (R>0.98) for both PSFs at 600 nm of depth, and measured PSFx,y (λ= 810 nm) = 378.4 nm and PSFz (λ= 810 nm) = 415 nm close to the theoretical values for the transverse resolution established for FFOCT[25], at high NA (NA>0.6), limited by diffraction[25]:
Figure 4 :
4iSR D-FFOCT (Brightness) mosaic of fibroblasts composed of 5x5 tiles with 20% overlap without numerical filtering. No overlap artefact could be observed nor image field distortions confirming iSR D-FFOCT signal uniformness and flatness.
Figure 5 :
5Fibroblasts structures analysis using iSR D-FFOCT.
FundingThe authors thank the following sources of funding: OREO [ANR-19-CE19-0023], IHU FOReSIGHT [ANR-18-IAHU-0001], OPTORETINA (European Research Council (ERC) (#101001841)).DisclosuresThe authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paperData Availability StatementData and codes are available upon reasonable request from the authors.ReferencesAcknowedgementWe thank Jeremy Brogard and Marilou Clémençon for providing some retinal organoid.Author ContributionsThe overall project was imagined and conceived by TM, but supervised by OT.The proofs of concept were made by TM.The acquisitions were carried out by TM, with assistance of SA.Optical design and construction were conceived and carried out by TM, supervised by OT. Acquisition protocols were designed by TM, TBY, on samples provided by TBY, SR and IA.Images and volumes were reconstructed by TM.OT, TM, KG, SR and SA discussed the results and wrote the article.
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| []
|
[
"Asymptotic Spectral Theory for Spatial Data",
"Asymptotic Spectral Theory for Spatial Data"
]
| [
"Wai Leong Ng \nThe Hang Seng University of Hong Kong\nThe Chinese University of Hong\nKong\n",
"Chun Yip Yau \nThe Hang Seng University of Hong Kong\nThe Chinese University of Hong\nKong\n"
]
| [
"The Hang Seng University of Hong Kong\nThe Chinese University of Hong\nKong",
"The Hang Seng University of Hong Kong\nThe Chinese University of Hong\nKong"
]
| []
| In this paper we study the asymptotic theory for spectral analysis of stationary random fields on Z 2 , including linear and nonlinear fields. Asymptotic properties of Fourier coefficients and periodograms, including limiting distributions of Fourier coefficients, and the uniform consistency of kernel spectral density estimators are obtained under various mild conditions on moments and dependence structures. The validity of the aforementioned asymptotic results for estimated spatial fields is also established.Mathematics Subject Classification: 62M15, 62E20, 62M30. | 10.1080/17442508.2022.2086806 | [
"https://arxiv.org/pdf/2005.13274v3.pdf"
]
| 218,900,574 | 2005.13274 | 4d18aa1ab82945670cdd312201c11eb828a786fc |
Asymptotic Spectral Theory for Spatial Data
27 Oct 2021 October 28, 2021
Wai Leong Ng
The Hang Seng University of Hong Kong
The Chinese University of Hong
Kong
Chun Yip Yau
The Hang Seng University of Hong Kong
The Chinese University of Hong
Kong
Asymptotic Spectral Theory for Spatial Data
27 Oct 2021 October 28, 2021and phrases: Fourier coefficientsgeometric moment contractionperiodogramsspectral analysisspectral density estimation
In this paper we study the asymptotic theory for spectral analysis of stationary random fields on Z 2 , including linear and nonlinear fields. Asymptotic properties of Fourier coefficients and periodograms, including limiting distributions of Fourier coefficients, and the uniform consistency of kernel spectral density estimators are obtained under various mild conditions on moments and dependence structures. The validity of the aforementioned asymptotic results for estimated spatial fields is also established.Mathematics Subject Classification: 62M15, 62E20, 62M30.
spectral density function and establishing asymptotic properties of Fourier coefficients and periodograms, which are relevant for many applications, e.g., frequency domain bootstrap methods, specification and testing of parametric models, detecting anisotropies, signal extraction, interpolation, prediction, and smoothing; see, e.g., [24,28]. There are two main types of estimators in nonparametric spectral density estimation for random fields, namely smoothed periodogram estimators and lag-windowed estimators, and they are closely related since the smoothed periodogram estimator can be viewed as a numerical integration approximation to the lag-windowed estimator; see [11,12,14,18,21,24].
Recently, there has been an increased interest in nonlinear random fields, see [7,15,17,10].
However, existing asymptotic spectral theories of random fields often require linearity, summability of joint cumulants functions, strong mixing property or Gaussianity. For example, [9] proposed a truncated autoregressive spectral density estimator and discussed its asymptotic properties under linearity; [1] established the uniform convergence of spectral density estimators on a homogeneous Gaussian random field; [24] constructed a lag-windowed estimator and a smoothed periodogram estimator under the linearity assumption, and proved the uniform convergence of the estimators.
In this paper, we focus on the spectral theory for stationary random fields on Z 2 , and study the asymptotic properties of Fourier coefficients and periodograms, and establish the uniform consistency property of kernel spectral density estimators of random fields under mild moment and dependence structure conditions, which are applicable to a variety of nonlinear and non-Gaussian random fields.
The rest of the paper is organized as follows. In Section 2, we describe the setting and background on random fields on Z 2 . Section 3 discusses the assumptions on the weak dependence structures of the random fields, and the kernel functions used in the spectral density estimation, which are required for establishing the main results. In Section 4, we establish the asymptotic normality of the Fourier coefficients and the asymptotic behaviors of the Fourier coefficients and periodograms. Section 5 considers the uniform consistency of the kernel spectral density estimators. Section 6 establishes the asymptotic results for estimated spatial fields. Proofs and technical details are given in Section 7.
Setting and Preliminary
In this section, we describe the basic settings and preliminary about the random fields of interest.
First, we introduce some notations. For any vector a = (a 1 , a 2 , . . . , a q ) ∈ R q , denote a = ∏ q i=1 a i ,
a = max i=1,...,q { a i } and a p = (∑ q i=1 a i p ) 1 p for p ≥ 1. Let T = {(t 1 , t 2 ) ∈ Z 2 , 1 ≤ t k ≤ d k , k = 1, 2}
be a spatial rectangular lattice, and denote d T = (d 1 , d 2 ) ∈ N 2 . For any set G, denote the cardinality of G by G . For any random variable X ∈ L p , denote the L p norm as X p = (E( X p )) 1 p . For any two real sequences {a n } and {b n }, denote by a n ≍ b n when a n = O(b n ) and b n = O(a n ), and by a n ∼ b n when a n b n → 1. For any x ∈ R, ⌊x⌋ is the greatest integer that is less than or equal to
x. All vectors are column vectors unless specified otherwise, hence for any a = (a 1 , a 2 , . . . , a q ) ∈ R q and b = (b 1 , b 2 , . . . , b q ) ∈ R q , the dot product between vectors a and b is defined as the vector
multiplication a ′ b = b ′ a = ∑ q i=1 a i b i .
Let V (t) ∶ t ∈ Z 2 be a stationary random field on a two-dimensional grid with mean µ = E(V (0)). Assume that we have observed {V (t) ∶ t ∈ T } on a rectangular lattice T = {(t 1 , t 2 ) ∈ Z 2 , 1 ≤ t k ≤ d k , k = 1, 2} with d T = (d 1 , d 2 ) ∈ N 2 and cardinality T = d 1 d 2 . Throughout the paper, T → ∞ denotes both d 1 , d 2 → ∞.
Fourier Coefficients for Spatial Data
Define the Fourier coefficients x(j) and y(j) as
x(j) = 1 T t∈T V (t)cos(−λ ′ j t) , y(j) = 1 T t∈T V (t)sin(−λ ′ j t) ,(2.1)
By using the symmetry property in (2.2), we now partition T as T = N ∪Ñ ∪ M such that the Fourier coefficients defined onÑ are determined by the Fourier coefficients defined on N . Also, the information about the covariance structure and mean of the random field are contained in N and M respectively. Hence, a spatial lattice can be reconstructed from the Fourier coefficients defined on N and M . In particular, when d 1 , d 2 are both odd, define
N = (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 , 1 ≤ t 2 ≤ d 2 − 1 2 ∪ (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 − 1 2 , t 2 = d 2 .
When d 1 is odd and d 2 is even (similar for d 1 is even and d 2 is odd), define
N = (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 , 1 ≤ t 2 ≤ d 2 2 − 1 ∪ (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 − 1 2 , t 2 = d 2 2 , d 2 .
When d 1 , d 2 are both even, define
N = (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 , 1 ≤ t 2 ≤ d 2 2 − 1 ∪ (t 1 , t 2 ) ∶ 1 ≤ t 1 ≤ d 1 2 − 1, t 2 = d 2 2 , d 2 .
Then, using the symmetry in (2.2), define the subsetÑ of T as
N = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ (t 1 , t 2 ) ∈ T ∶ t i = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ d i − s i , if s i < d i d i , if s i = d i , for i = 1, 2 , (s 1 , s 2 ) ∈ N ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ .
Note that the Fourier coefficients at j ∈Ñ can be completely determined by the Fourier coefficients at j ∈ N . From (2.1), for all c ∈ R, the Fourier coefficients of {V (t) − c ∶ t ∈ T } at j ∈ N are the same. In other words, the Fourier coefficients in N andÑ are invariant under additive constants and thus contain no information about the mean. In contrast, all of the information about the mean is contained in the Fourier coefficients
x(j) for j ∈ M , where M = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ {(d 1 , d 2 )} , when d 1 , d 2 are odd , {(d 1 , d 2 ), (d 1 2, d 2 )} , when only d 1 is even , {(d 1 , d 2 ), (d 1 , d 2 2)} , when only d 2 is even , {(d 1 , d 2 ), (d 1 2, d 2 ), (d 1 , d 2 2), (d 1 2, d 2 2)} , when d 1 , d 2 are even . (2.3)
Denote the sample mean asV T = T −1 ∑ t∈T V (t). Table 1 summarizes the values of the Fourier coefficients in M . Figure 1 illustrates the sets N and M in the rectangular lattices with different d 1 and d 2 . d1, d2 are both odd d1 is odd, d2 is even (vice versa) d1, d2 are both even
d1 = 7, d2 = 7 d1 = 7, d2 = 8 d1 = 8, d2 = 8 (1, 1) (1, 2) (1, 3)
. .
. (1,6) (1, 7)
(2, 1) (3, 1) . . . (6, 1) (7, 1)
(7, 7) ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ (1, 1) (1, 2) (1, 3) (1, 4)
.
.
(1, 7)
(1, 8)
(2, 1) (3, 1) . . . (6, 1) (7, 1) (8, 7) ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ (1, 1) (1, 2) (1, 3) (1, 4)
.
.
(1, 7)
(1, 8)
(2, 1) (3, 1) . . . (6, 1) (7, 1) (8, 1) (8, 8) ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀ ☀j (d 1 , d 2 ) (d 1 2, d 2 ) (d 1 , d 2 2) (d 1 2, d 2 2) x(j) T V T 1 T ∑ t∈T (−1) t 1 V (t) 1 T ∑ t∈T (−1) t 2 V (t) 1 T ∑ t∈T (−1) t 1 +t 2 V (t) y(j) 0 0 0 0
Kernel Spectral Density Estimators for Spatial Data
Kernel estimator is introduced by [20] and has been studied extensively in nonparametric estimation of probability density and spectral density functions. Similar to [24] and [19], we consider the following kernel spectral density estimator
f T (λ) = 1 4π 2 j∈Z 2 ⎛ ⎜ ⎝ K λ 1 −λ j1 h T 1 , λ 2 −λ j2 h T 2 ∑ k∈Z 2 K λ k1 h T 1 , λ k2 h T 2 ⎞ ⎟ ⎠ I(j) , for λ = (λ 1 , λ 2 ) ∈ [0, 2π] 2 , (2.4) where K(⋅) is the kernel function, λ j = (λ j1 , λ j2 ) = 2πj 1 d 1 , 2πj 2 d 2
for j = (j 1 , j 2 ) ∈ Z 2 , and similarly, for spectral density functions, see [11,18,19,21,24].
λ k = (λ k1 , λ k2 ) = 2πk 1 d 1 , 2πk 2 d 2 for k = (k 1 , k 2 ) ∈ Z 2 , h T = (h T 1 , h T 2 ) ∈ R 2 is the bandwidth satisfying h T k → 0, h T 1 ≍ h T 2 and h T k d k → ∞ for k = 1, 2, and I(j) = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ 0 , if j ∈ D = {(c 1 d 1 , c 2 d 2 ) ∶ (c 1 , c 2 ) ∈ Z 2 } , x 2 (j) + y 2 (j) , if j ∈ Z 2 D ,(2.
Assumptions
In this section, we impose some assumptions on the underlying random fields as well as kernel functions which are required for establishing the asymptotic results in Sections 4 to 6.
Assumptions on the underlying random fields
We impose the following assumptions about the lattice structure and underlying random fields to establish the asymptotic results.
Assumption P.1. For all sufficiently large T , there exist 0 < ξ ≤ 1 2 and c 1 , c 2 > 0 such that
d 1 > c 1 T ξ and d 2 > c 2 T ξ . Assumption P.2. The random field V (t) ∶ t ∈ Z 2 is stationary with absolutely summable auto- covariance function γ(⋅), i.e., ∑ j∈Z 2 γ(j) < ∞, where γ(j) = Cov(V (0), V (j)).
Assumption P.2 implies that the spectral density of the random field V (t) ∶ t ∈ Z 2 exists and can be expressed as Assumption P. 4(v). For some 0 < v ≤ 1 2 , the sample autocovariance function
f (λ) = 1 4π 2 j∈Z 2 e −iλ ′ j γ(j) ,(3.R V (r) = T −1 j,j+r∈T (V (j) − µ)(V (j + r) − µ) (3.2) satisfies that, uniformly on r ∈ Z 2 , R V (r) − E(R V (r)) = O p ( T −v ).
Assumption P.1 asserts that the asymptotic regime requires the sample to be increasing in both directions of space at a polynomial rate with respect to the total number of observations, see Assumption A.1 in [24] for similar setting. Assumptions P.2 and P.3 are common conditions on the autocovariance function and spectral density for the asymptotic theory of spectral analysis.
Assumption P.4(v) is fulfilled for a large class of linear and nonlinear random fields with mild moment and short-range dependence conditions. In particular, Assumption P.4(v) is fulfilled with v = 1 2 for linear random fields with existing fourth moments.
Assumption P.5(r). The random field V (t) ∶ t ∈ Z 2 is a real-valued linear random fields, i.e., V (j) − µ = ∑ s∈Z 2 a s ε j−s , where {ε i } i∈Z 2 is an i.i.d. random field with E(ε 0 ) = 0, E(ε 8 0 ) < ∞ and ∑ s∈Z 2 a s s r < ∞ for some r ≥ 0. Also, the innovation ε 0 satisfies Cramér's condition: there exist δ > 0 and u 0 > 0 such that, for all u ∈ R with u > u 0 , E exp(iuε 0 ) ≤ 1 − δ.
Note that Assumption P.5(r) with r ≥ 0 implies Assumptions P.2 and P. 4(v) with v = 1 2.
Assumption P.6(p). For j ∈ Z 2 , assume that V (j)−µ = G(ε j−s ∶ s ∈ Z 2 ), where G(⋅) is a measurable function and {ε i } i∈Z 2 is an i.i.d. random field. Let {ε i } i∈Z 2 be an i.i.d. copy of {ε i } i∈Z 2 . Define the coupled version of V (j) asṼ (j) − µ = G(ε * j−s ∶ s ∈ Z 2 ), where ε * j−s = ε j−s if j − s ≠ 0 , ε 0 if j − s = 0 .
Assume that there exists some p > 0 such that V (j) belongs to L p and
∆ p ∶= j∈Z 2 δ j,p ∶= j∈Z 2 V (j) −Ṽ (j) p < ∞ .
Assumption P.6(p) is the p-stable condition for random fields defined in [7] in which central limit theorems and invariance principles are established for a wide class of stationary nonlinear random fields. The next assumption is a geometric-moment contraction (GMC) condition:
Assumption P.7. Under the notation of Assumption P.6(p), define another coupled version of
V (j) asṼ † (j) − µ = G(ε † j−s ∶ s ∈ Z 2 ), where ε † j−s = ε j−s if s < j , ε j−s if s ≥ j .
Assume that there exist α > 0, C > 0 and 0 < ρ = ρ(α) < 1 such that for all j ∈ Z 2 ,
E( V (j) −Ṽ † (j) α ) ≤ Cρ j .
Assumption P.7 is the spatial extension of the geometric-moment contraction condition for time series, see [22]. This condition is fulfilled for short-range dependent linear random fields with finite variance, and a large class of nonlinear random fields such as nonlinearly transformed linear random fields, Volterra fields and nonlinear spatial autoregressive models, see [7] and [6].
Assumptions on the kernel function K
We impose the following mild regularity assumptions on the kernel function K(⋅).
Assumption K.1. The kernel K(⋅) is a real, positive, even function with ∫ R 2 K(λ)dλ = 1 and
4π 2 h T T j∈Z 2 K 2πj 1 h T 1 d 1 , 2πj 2 h T 2 d 2 = R 2 K(λ)dλ + o(1) = 1 + o(1) . Assumption K.2. Assume that sup λ∈[0,2π] 2 K h (λ) = O h T −1 = O (h T 1 h T 2 ) −1 , where K h (λ) = 1 h T j∈Z 2 K λ 1 + 2πj 1 h T 1 , λ 2 + 2πj 2 h T 2 . (3.3)
Assumption K.3. The kernel K(⋅) is absolutely integrable, i.e., ∫ R 2 K(λ) dλ < ∞. Furthermore, the inverse Fourier transform of K(⋅),
k(x) = R 2 K(λ) exp(ix ′ λ)dλ , (3.4) satisfies k(x) ≤k(x),
wherek(x) is monotonically decreasing with respect to x on [0, ∞) 2 , i.e., k(x) ≥k(y) if x ≤ y , and is an even function with
∫ R 2 +k (x)dx < ∞. Assumption K.4. The inverse Fourier transform k(x) in (3.4) is Lipschitz continuous with sup- port [−1, 1] 2 .
Assumption K.5. The quantity K h (λ) in (3.3) satisfies the following uniformly Lipschitz condition: for some constant L K > 0,
h T 3 2 K h (λ s ) − K h (λ t ) ≤ L K λ s − λ t , uniformly in λ s = 2πs 1 d 1 , 2πs 2 d 2 and λ t = 2πt 1 d 1 , 2πt 2 d 2 .
Remark 3.1. Assumptions K.1 and K.3 are general assumptions commonly used in kernel estimators, see [19] and [24]. Assumption K.2 is mild since it holds for any bounded kernel with compact support. The Lipschitz continuity in Assumption K.4 is also common in kernel spectral estimators; see, e.g., [24]. Moreover, under Assumptions K.3 or K.4, we have
K h (λ) = 1 4π 2 j∈Z 2 k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ) and K(λ) = 1 4π 2 k(x) exp(−ix ′ λ)dx ,
where k(⋅) is defined in (3.4). From the above representations it is clear that as soon as the sum in K h (λ) can be approximated by an integral for small enough h T = (h T 1 , h T 2 ), it holds for large T ,
K h (λ) ≅ 1 h T K λ 1 h T 1 , λ 2 h T 2 , (3.5)
which is of order O h T −1 for bounded K(⋅), hence Assumption K.2 holds. By (3.5), if the kernel K(⋅) is uniformly Lipschitz continuous with compact support, then Assumption K.5 holds for a small enough h T = (h T 1 , h T 2 ). For infinite support kernels, if K(⋅) is bounded and continuously differentiable, then Assumption K.5 also holds. In fact, Assumptions K.1 to K.5 hold for many commonly used kernels such as uniform kernels K(t 1 ,
t 2 ) = 1 4 1 { t 1 ≤1} 1 { t 2 ≤1} , multiplicative 2-d Epanechnikov kernels K(t 1 , t 2 ) = 9 16 (1 − t 2 1 )(1 − t 2 2 )1 { t 1 ≤1} 1 { t 2 ≤1} and Gaussian kernels K(t 1 , t 2 ) = 1 2π exp( −t 2 1 −t 2 2
2 ).
Asymptotic Properties of Fourier Coefficients and Periodograms
In this section, we establish some asymptotic results on the Fourier coefficients and the periodograms. The following theorem shows that the asymptotic normality of the Fourier coefficients holds in spatial lattice data. Specifically, linear combinations of Fourier coefficients are uniformly asymptotic normal.
Theorem 4.1. Let Assumptions P.1, P.2, P.3 and P.6(p) hold with some p ≥ 2. Denote, for j ∈ N ,s
T (j, k) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x(j) 2π 2 f (λ j ) , k = 1 , y(j) 2π 2 f (λ j ) , k = 2 , and F N ∶= {(j, k) ∶ j ∈ N, k = 1, 2}. Then, for each fixed q ∈ N, as T → ∞, sup a i ∈F N ,∀i=1,...,q, a i 1 ≠a i 2 ,∀i 1 ≠i 2 , c∈R q c 2 =1 , z∈R P (s T (a 1 ), ...,s T (a q )) ′ c ≤ z − Φ(z) = o(1) .
where Φ(⋅) is the standard normal cumulative distribution function.
{w j,N ∶ j ∈ N } such that ∑ j∈N w j,N = 1 and ∑ j∈N w 2 j,N → 0 as T → ∞, we have sup z∈R 1 2 j∈N w j,N 1 {x(j)≤z 2π 2 f (λ j )} + 1 {y(j)≤z 2π 2 f (λ j )} − Φ(z) p → 0 ,
where Φ(⋅) is the standard normal distribution function. If {{w j,N,s ∶ j ∈ N } ∶ s ∈ S} is a class of weights indexed by a countable index set S satisfying ∑ j∈N w j,N,s = 1 for all s ∈ S and sup s∈S ∑ j∈N w 2 j,N,s → 0, then the assertion remains true in the sense that, for any ε > 0,
sup s∈S P ⎛ ⎝ sup z∈R 1 2 j∈N w j,N,s 1 {x(j)≤z 2π 2 f (λ j )} + 1 {y(j)≤z 2π 2 f (λ j )} − Φ(z) ≥ ε ⎞ ⎠ → 0 .
The following example fulfills the required conditions for the underlying random fields in Theorem 4.1 and Corollary 4.2.
Example 4.1 (Linear Random Fields). Define the linear random field V (t) ∶ t ∈ Z 2 as V (j) = s∈Z 2 a s ε j−s , (4.1)
where {ε t } t∈Z 2 is an i.i.d. random field with E(ε 0 ) = 0. If we have E( ε 0 p ) < ∞ with some p ≥ 2 and ∑ j∈Z 2 a j < ∞, then Assumptions P.2, P.3 and P.6(p) with some p ≥ 2 hold since
∑ j∈Z 2 δ j,p = ∑ j∈Z 2 V (j) −Ṽ (j) p = ∑ j∈Z 2 a j ε 0 −ε 0 p < ∞.
Example 4.2 (Nonlinear Spatial Autoregressive Models). Let N ⊂ Z 2 be a finite set and 0 ∉ N . Define the nonlinear spatial autoregressive models V (t) ∶ t ∈ Z 2 as
V (t) = G({V (t − s)} s∈N ; ε t ) , (4.2)
where the function G satisfies the following condition: there exists nonnegative numbers u s , s ∈ N , with ∑ s∈N u s < 1 such that for all {v(−s)} s∈N and {v ′ (−s)} s∈N ,
G({v(−s)} s∈N ; ε t ) − G({v ′ (−s)} s∈N ; ε t ) ≤ s∈N u s v(−s) − v ′ (−s) .
If there exists {v(−s)} s∈N such that G({v(−s)} s∈N ; ε 0 ) ∈ L p for some p ≥ 2, then following the argument in Section 5 of [22] or Example 2 of [6], we have E ( V (0) p ) < ∞ and δ j,p = O(ρ j ) for some ρ ∈ (0, 1), and hence Assumptions P.2, P.3 and P.6(p) with some p ≥ 2 hold.
The following theorem establishes some asymptotic behaviors of the Fourier coefficients and periodograms relative to the spectral density function.
Theorem 4.3. Suppose that Assumptions P.1, P.2 and P.3 hold, and either Assumption P.5(r) with r > 1 2 holds, or Assumption P.7 with E V (0) 16 < ∞ holds, then with I(j) defined as in
(2.5), we have (a) 1 2 N j∈N x(j) + y(j) f (λ j ) p → 0 . (4.3) (b) If additionally the auto-covariance function γ(⋅) satisfies ∑ j∈Z 2 j u γ(j) < ∞ for some u > 0, then sup l,k∈N Cov(x(l), x(k)) − 2π 2 f (λ k )δ l,k = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ O( T −u ), 0 < u < 1 , O log T T , u = 1 , O( T −1 ), u > 1 . sup l,k∈N Cov(y(l), y(k)) − 2π 2 f (λ k )δ l,k = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ O( T −u ), 0 < u < 1 , O log T T , u = 1 , O( T −1 ), u > 1 . (4.4) (c) 1 4π 2 N j∈N I(j) f (λ j ) p → 1 . (4.5) (d) 1 N j∈N I 2 (j) f 2 (λ j ) = 2(4π 2 ) 2 + o p (1) . (4.6)
(e) There exists some q = 4 + ǫ with ǫ ∈ (0, 1) such that
1 N j∈N I q (j) f q (λ j ) < C 2 + o p (1) . (4.7)
Some statistical applications rely on the asymptotic behaviors of the weighted means of the Fourier coefficients and periodograms, for example, local bootstrap in frequency domain, see [16] and [13] in time series context. The next theorem describes the asymptotic behaviors of the weighted means of the Fourier coefficients and periodograms.
Theorem 4.4. Suppose that Assumptions P.1, P.2 and P.3 hold, and either Assumption P.5(r) with r > 1 2 holds, or Assumption P.7 with E V (0) 16 < ∞ holds. Also, assume that the bandwidth fulfills ( h T 4 T ) −1 = o(1), and the kernel K(⋅) fulfills Assumptions K.1 and K.5, and
1 T h T j∈Z 2 K 2 2πj 1 h T 1 d 1 , 2πj 2 h T 2 d 2 = O(1) . (4.8)
Define the weights p s,T as
p s,T = K 2πs 1 h T 1 d 1 , 2πs 2 h T 2 d 2 ∑ j∈Z 2 K 2πj 1 h T 1 d 1 , 2πj 2 h T 2 d 2 .
(4.9)
Then, with I(j) defined as in (2.5), the following results hold.
(a) If sup l,k∈N Cov(x(l), x(k)) − 2π 2 f (λ k )δ l,k = O 1 h T T , (4.10) sup l,k∈N Cov(y(l), y(k)) − 2π 2 f (λ k )δ l,k = O 1 h T T , (4.11)
are satisfied, then
sup j∈N s∈Z 2 p s,T (x(j + s) + y(j + s)) = o p (1) . (b) sup j∈N s∈Z 2 p s,T I(j + s) − 4π 2 f (λ j ) = o p (1). (c) sup j∈N s∈Z 2 p s,T I 2 (j + s) ≤ C 1 + o p (1) . (d) There exists some q = 4 + ǫ with ǫ ∈ (0, 1) such that sup j∈N s∈Z 2 p s,T I q (j + s) ≤ C 2 + o p (1) .
Note that the condition in (4.8) is satisfied when the kernel K(⋅) is square-integrable, i.e., if we have ∑ s∈Z 2 a s s r < ∞ for some r > 1 2, E(ε 8 0 ) < ∞ and the innovation ε 0 satisfies the Cramér's condition, for example, {ε t } t∈Z 2 are centered i.i.d. Gaussian white noises, then Assumption P.5(r) with r > 1 2 holds. Example 4.4 (Linear Random Fields). For the linear random fields defined in (4.1) in Example 4.1, if we have E( ε 0 p ) < ∞ with some p ≥ 16 and a j ≤ Cρ j for some ρ ∈ (0, 1) and C > 0, then we have E V (0) 16 < ∞ and δ j,p = O(ρ j ) which implies Assumption P.7, see Section 4 of [26].
∫ R 2 K 2 (λ)dλ < ∞,
Example 4.5 (Volterra Fields). Volterra Fields is a class of nonlinear random fields which plays an important role in the nonlinear system theory. Define the second order Volterra process
V (t) ∶ t ∈ Z 2 as V (j) = s 1 ,s 2 ∈Z 2 a s 1 ,s 2 ε j−s 1 ε j−s 2 , (4.12)
where {ε t } t∈Z 2 be an i.i.d. random field with E(ε 0 ) = 0 and {a s 1 ,s 2 } are real coefficients with a s 1 ,s 2 = 0 if s 1 = s 2 . Then, by Rosenthal inequality, there exists a constant C p > 0 such that
δ j,p = V (j) −Ṽ (j) p ≤ C p A 1 2 j ε 0 2 ε 0 p + B 1 p j ε 0 2 p , where A j = ∑ s 1 ,s 2 ∈Z 2 (a 2 s 1 ,j + a 2 j,s 2 ) and B j = ∑ s 1 ,s 2 ∈Z 2 ( a s 1 ,j p + a j,s 2 p ). If we have E( ε 0 p ) < ∞ for some p ≥ 32 and a s 1 ,s 2 = O(ρ max{ s 1 , s 2 } ) for some ρ ∈ (0, 1), then we have E V (0) 16 < ∞, and δ j,p = O(ρ j )
, and hence Assumption P.7 holds.
Example 4.6 (Nonlinear Spatial Autoregressive Models). For the nonlinear spatial autoregressive
models defined in (4.2) in Example 4.2, if there exists {v(−s)} s∈N such that G({v(−s)} s∈N ; ε 0 ) ∈ L p for p ≥ 16, then we have E V (0) 16 < ∞ and δ j,p = O(ρ j )
for some ρ ∈ (0, 1) which implies Assumption P.7.
Uniform Consistency for Kernel Spectral Density Estimators
In this section, we establish the uniform consistency of the kernel spectral density estimator for spatial lattice data. The following theorem establishes the uniform consistency off T (λ) in (2.4) under two different sets of regularity conditions.
h T + h T −1 T −v → 0, then max λ∈[0,2π] 2 f T (λ) − f (λ) p → 0 . (5.1) (b) If Assumptions K.4, P.3, P.6(p) and P.7 hold with some p ≥ 2, E( V (0) φ ) < ∞ for some 4 < φ ≤ 8, and the bandwidth satisfies h T → 0, ( h T T η ) −1 = O(1) for some 0 < η < (φ−4) φ , then max λ∈[0,2π] 2 f T (λ) − f (λ) p → 0 .
The following examples fulfill the required conditions for the underlying random fields in The-
.1, if we have E( ε 0 p ) < ∞ with some 4 < p ≤ 8 and a j ≤ Cρ j for some ρ ∈ (0, 1) and C > 0, then E( V (0) φ ) < ∞ for some 4 < φ ≤ 8, Assumption P.6(p) with some p ≥ 2 and Assumption P.7 hold since ∑ j∈Z 2 δ j,p = ∑ j∈Z 2 V (j) −Ṽ (j) p = ∑ j∈Z 2 a j ε 0 −ε 0 p < ∞ and δ j,p = O(ρ j ) respectively.4.5, if E( ε 0 p ) < ∞ for some 8 < p ≤ 16 and a s 1 ,s 2 = O(ρ max{ s 1 , s 2 } ) for some ρ ∈ (0, 1), then we have E( V (0) φ ) < ∞ for some 4 < φ ≤ 8
, and Assumptions P.6(p) with p ≥ 2 and P.7 hold since δ j,p = O(ρ j ).
Example 5.4 (Nonlinear Spatial Autoregressive Models). For the nonlinear spatial autoregressive models defined in (4
.2) in Example 4.2, if there exists {v(−s)} s∈N such that G({v(−s)} s∈N ; ε 0 ) ∈ L p for some 4 < p ≤ 8. Then, we have E( V (0) p ) < ∞ and δ j,p = O(ρ j )
for some ρ ∈ (0, 1) which implies Assumption P.6(p) with p ≥ 2 and Assumption P.7.
Asymptotic Results on Estimated Spatial Fields
In many applications, the Fourier coefficients and kernel spectral density estimators are not applied directly to a stationary spatial data set {V (t) ∶ t ∈ T }, but to an estimate {V (t) ∶ t ∈ T } computed from the spatial data {Y (t) ∶ t ∈ T }. For example, if the observed spatial data is nonstationary, then it is common to apply detrending, smoothing, filtering, or spatial regression to obtain a residual process, which is approximately stationary for statistical analysis. The residual process can be regarded as an estimated field of an unobserved stationary random field. The following theorem extends the asymptotic results in Theorems 5.1, 4.3 and 4.4 to estimated fields.
We use a subscriptV (resp. V ) on the notation to indicate the use ofV (resp. V ) in the calculations, e.g. xV (j), yV (j) (resp. x V (j), y V (j)) denote the Fourier coefficients based onV (⋅) (resp. V (⋅)). for q = 4 + ǫ with ǫ ∈ (0, 1) as defined in Theorem 4.3(e), then we have for q = 4 + ǫ with ǫ ∈ (0, 1) as defined in Theorem 4.4(d), then we have Consider F N 1 = {(j, 1) ∶ j ∈ N } and hences T (j, 1) corresponds to the real parts of x(j) + iy(j). For notation simplicity, denotes T (j) =s T (j, 1). Also, we assume here µ = E(V (0)) = 0. Denote
Theorem 6.1. Suppose that for the spatial data {Y (⋅)}, we have an estimatorV (⋅) of V (⋅) such that 1 T t∈T V (t) −V (t) 2 = o p α −1 T , as α T → ∞. Furthermore,(i) 1 N j∈N x V (j) − xV (j) + y V (j) − yV (j) f (λ j ) = o p (1) . (ii) 1 N j∈N I V (j) − IV (j) f (λ j ) = o p (1) . (iii) 1 N j∈N I 2 V (j) − I 2 V (j) f 2 (λ j ) = o p (1) . (iv) 1 N j∈N I q V (j) − I q V (j) f q (λ j ) = o p (1) .(i) sup k∈N j∈Z 2 p j,T [x V (k + j) − xV (k + j) + y V (k + j) − yV (k + j)] = o p (1) . (ii) sup k∈N j∈Z 2 p j,T (I V (k + j) − IV (k + j)) = o p (1) . (iii) sup k∈N j∈Z 2 p j,T (I 2 V (k + j) − I 2 V (k + j)) = o p (1) . (iv) sup k∈N j∈Z 2 p j,T (I q V (k + j) − I q V (k + j)) = o p (1) .A = (a 1 , ..., a q ) ⊂ F N 1 . Let H T = ∑ k∈T µ k V (k), where µ k = µ k (c, A) = q ∑ i=1 c i cos(k ′ λa i ) 2π 2 f (λa i ) . Since f * = min λ∈R 2 f (λ) > 0, there exists µ * such that µ k ≤ µ * for all c ∈ R q , c 2 = 1 and A ⊂ F N 1 . Let d T (h) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 T ∑ k∈T ∶ k−h∈T µ k µ k−h if {k ∈ T ∶ k − h ∈ T } ≠ ∅ , 0 if {k ∈ T ∶ k − h ∈ T } = ∅ , for h ∈ Z 2 ,
where ∅ is an empty set. Note that ∑
k∈T cos(k ′ λ au ) cos((k + h) ′ λ av ) = T 2 cos(h ′ λ au )1 {au=av} .
Thus, it is easily seen that there exists a constant K 0 > 0 such that for all h ∈ Z 2 ,
τ T (h) = sup a 1 ,...,aq∈F N1 sup c∈R q , c 2 =1 d T (h) − q i=1 c 2 i cos(h ′ λ a i ) 4π 2 f (λ a i ) ≤ K 0 h T . Clearly τ T (h) ≤ µ 2 * +(4π 2 f * ) −1 ∶= K 1 . By using the equality f (λ a i ) = 1 4π 2 ∑ h∈Z 2 γ(h) cos(h ′ λ a i ), hence ∑ h∈Z 2 γ(h) cos(h ′ λa i ) 4π 2 f (λa i )
= 1, we have uniformly over (a 1 , ..., a q ) ⊂ F N 1 and c that Set:
H T 2 2 T − 1 = 1 T E ⎛ ⎝ k 1 ∈T µ k 1 V (k 1 ) k 2 ∈T µ k 2 V (k 2 ) ⎞ ⎠ − 1 = 1 T k 1 ∈T k 2 ∈T µ k 1 µ k 2 γ(k 1 − k 2 ) − 1 = h∈Z 2 d T (h)γ(h) − 1 = h∈Z 2 d T (h)γ(h) − q i=1 c 2 i = h∈Z 2 d T (h)γ(h) − q i=1 ⎛ ⎝ c 2 i h∈Z 2 γ(h) cos(h ′ λ a i ) 4π 2 f (λ a i ) ⎞ ⎠ = h∈Z 2 γ(h) d T (h) − q i=1 c 2 i cos(h ′ λ a i ) 4π 2 f (λ a i ) ≤ h∈Z 2 τ T (h)γ(h) ≤ h∈Z 2 K 2 min h T , 1 γ(h) where K 2 = 2(K 1 + K 0 ) ≤ h∈H 1 K 2 γ(h) T 1 2 + h∈H 2 K 2 γ(h) → 0 , as T → ∞ , where H 1 = (h 1 , h 2 ) ∶ h 1 < d 1 2 1 and h 2 < d 1 2 2 and H 2 = (h 1 , h 2 ) ∶ h 1 ≥ d 1 2 1 or h 2 ≥ d 1 2 2 . LetĤ T = ∑ k∈T µ kV (k), whereV (k) = E(V (k) F k,<l T > ), F k,<l T > = {ε i ∶ i − k < l T }, where x = max i=1,2 x i for x = (x 1 , x 2 ) ∈ Z 2 .F k,<l T > F k,<l T > ⋂ F0,0 Locations of ε i included: k l T l T k l T l T F k,<l T > ⋂ F 0,0
Define projection operator P 0 for any F ∞,∞ -measurable random variable X(k) as
P 0 (X(k)) ∶= E[X(k) F 0,0 ] − E[X(k) F 0,−1 ] − E[X(k) F −1,0 ] + E[X(k) F −1,−1 ] .
If k < l T , then ε 0 ∈ F k,<l T > , and we have
P 0 (V (k)) = P 0 [E(V (k) F k,<l T > )] = i,j∈{−1,0} (−1) i+j E[E(V (k) F k,<l T > ) F i,j ] = i,j∈{−1,0} (−1) i+j E[V (k) F k,<l T > ⋂ F i,j ] = E[P 0 (V (k)) F k,<l T > ⋂ F 0,0 ] . If k ≥ l T , then ε 0 ∉ F k,<l T > , so P 0 (V (k)) = 0 . Clearly, P 0 (V (k) −V (k)) 2 ≤ 2δ l T for all k ∈ Z 2 .
Since by Lemma 1 of [7], we have P 0 V (j) 2 ≤ δ j,p for all j ∈ Z 2 , and hence by Assumption P.6(p) with p ≥ 2 and the Lebesgue dominated convergence theorem, it entails that
H T −Ĥ T 2 T = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 T j∈Z 2 P j (H T −Ĥ T ) 2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 T j∈Z 2 P j k∈T µ k (V (k) −V (k)) 2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 2 ≤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ µ 2 * T j∈Z 2 P j k∈T (V (k) −V (k)) 2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 2 ≤ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ µ 2 * T k∈T j∈Z 2 P j (V (k) −V (k)) 2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ µ 2 * j∈Z 2 P 0 (V (j) −V (j)) 2 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ 1 2 ≤ µ * j∈Z 2 P 0 (V (j) −V (j)) 2 ≤ µ * j∈Z 2 2 min( P 0 V (j) 2 , δ l T ) → 0 , as l T → ∞ . Let g T (r) = r 2 E(V (k) 2 1 { V (k) ≥ T r} ). Since E(V (k) 2 ) < ∞, lim T →∞ g T (r) = 0 for any fixed r > 0.
Note that g T is non-decreasing in r. Then, there exists a sequence r T ↑ ∞ such that g T (r T ) → 0. Figure 3 for a graphical illustration of the partition of the rectangular lattice T and the set L.
Let U (k) =V (k)1 { V (k) ≤ T r T } and H T,U = ∑ k∈T µ k U (k). Then U (k) −V (k) 2 = o (1 r T ). Since U (k) −V (k) are 2l T -dependent, H T,U −Ĥ T 2 ≤ a∈L µ b U (b) −V (b) 2 = o ⎛ ⎝ T r T ⎞ ⎠ , (7.1) where L = {(l 1 , l 2 ) ∶ 1 ≤ l 1 ≤ 2l T , 1 ≤ l 2 ≤ 2l T } with L = 4l 2 T , and the inner sum is over {b ∈ T ∶ b−a = (2ml T , 2nl T ), m, n ∈ Z + 0 = {0, 1, 2, ...}}. See
Let p T = ⌊r T 1 8 ⌋ and blocks
B m = {a = (a 1 , a 2 ) ∈ Z 2 ∶ 1 + (m k − 1)(p T + 2l T ) ≤ a k ≤ p T + (m k − 1)(p T + 2l T ) , for k = 1, 2} ,
where m = (m 1 , m 2 ) ∈ M B , and
M B = (m 1 , m 2 ) ∈ N 2 ∶ 1 ≤ m 1 ≤ m T 1 = ⌊1 + d 1 − p T p T + 2l T ⌋ , 1 ≤ m 2 ≤ m T 2 = ⌊1 + d 2 − p T p T + 2l T ⌋ .
See with similar argument in (7.1). Thus, by (7.1),
L 2l T 2l T 4l T 4l T 6l T 6l TT ∆ 2 ≤ E(S T ) + S T −Ĥ T 2 ≤ o ⎛ ⎝ T r T ⎞ ⎠ + S T − H T,U 2 + H T,U −Ĥ T 2 ≤ o ⎛ ⎝ T r T ⎞ ⎠ + O M B + o ⎛ ⎝ T r T ⎞ ⎠ = O M B . (7.2) Since T m 3 ≤ µ 3 * p 4 T ∑ k∈Bm U (k) 3 and E(U (k) 2 ) ≤ E(V (k) 2 ), we have E( T m 3 ) = O p 6 T T r T . By the Berry-Esseen theorem, we have sup x P(W ≤ x) − Φ x W 2 ≤ C m∈M B E( T m 3 ) S T − E(S T ) −3 2 = O ⎛ ⎝ M B p 6 T T r T ⎞ ⎠ × T − 3 2 = O(p −4 T ) . (7.3) Let δ = δ T = p − 1 2
T . By (7.2), (7.3) and
P(W ≤ w − δ) − P( ∆ ≥ δ) ≤ P(W + ∆ ≤ w) ≤ P(W ≤ w + δ) + P( ∆ ≥ δ) , we have sup x P Ĥ T ≤ T x − Φ T x Ĥ T 2 = O p −4 T + P( ∆ ≥ δ) + δ + δ 2 = O(δ) . (7.4) Note that sup x Φ x σ 1 − Φ x σ 2 ≤ C σ 1 σ 2 − 1 holds for some constant C. Let W 1 =Ĥ T T , ∆ 1 = H T −Ĥ T T and η = η l T ,T = H T −Ĥ T 2 T 1 2 . Applying (7.4) with w, ∆ replaced by W 1 , ∆ 1 , we have sup x P ⎛ ⎝ H T T ≤ x ⎞ ⎠ − Φ ⎛ ⎝ x H T 2 T ⎞ ⎠ = O P( ∆ 1 ≥ η) + δ + η + η 2 .
Thus, the conclusion follows by first letting T → ∞ and then l T → ∞. ∎
s T (j, k) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x(j) 2π 2 f (λ j ) , j ∈ N, k = 1 , y(j) 2π 2 f (λ j )
, j ∈ N, k = 2 . and F N ∶= {(j, k) ∶ j ∈ N, k = 1, 2} as in Theorem 4.1. The uniform asymptotic normality of vectors ofs T (⋅) is proven in Theorem 4.1. We will give the arguments only for vectors of length 2, but the same results hold true for any length p by induction. Precisely, we will prove that for each z = (z 1 , z 2 ), sup
a 1 ,a 2 ∈F N ,a 1 ≠a 2 P((s T (a 1 ) ≤ z 1 ,s T (a 2 ) ≤ z 2 )) − Φ(z 1 )Φ(z 2 ) = o(1) . (7.5)
Now, vectorizeS T,a 1 ,a 2 = (s T (a 1 ),s T (a 2 )), a 1 , a 2 ∈ F N , where N is the corresponding subset of T , to form a single sequence S t = (S t (1), S t (2)), t ∈ N in such a way that if S t 1 corresponds tõ S T 1 ,a 11 ,a 12 for some a 11 , a 12 ∈ F N 1 , and S t 2 corresponds toS T 2 ,a 21 ,a 22 for some a 21 , a 22 ∈ F N 2 , then T 1 ≤ T 2 implies that t 1 ≤ t 2 . By Levy's continuity theorem and Theorem 4.1, it holds for each
z = (z 1 , z 2 ) that φ S t (z) = E(e iz ′ St ) = E e i z z z ′ St = φ z ′ S t z ( z ) → φ G 1 ( z ) = E e i z G 1 = E e i z z z ′ (G 1 ,G 2 ) = φ (G 1 ,G 2 ) (z) ,
where φ X denote the characteristic function of X, and G 1 , G 2 are two independent standard normal random variables. Next, a second application of Levy's continuity theorem yields
P(S t (1) ≤ z 1 , S t (2) ≤ z 2 ) − Φ(z 1 )Φ(z 2 ) = o(1) ,
and by definition of S t we get (7.5). Consider F N 1 = {(j, 1) ∶ j ∈ N } and hences T (j, 1) corresponds to the real parts of x(j) + iy(j). For notation simplicity, denotes T (j) =s T (j, 1). The argument easily extends to general cases. Define P j (z) = P(s T (j) ≤ z) and P j 1 ,j 2 (z) = P(s T (j 1 ) ≤ z,s T (j 2 ) ≤ z). Then it holds by (7.5) and Theorem 4.1 that
E ⎛ ⎝ j∈N w j,N 1 {s T (j)≤z} ⎞ ⎠ − Φ(z) ≤ sup l∈F N1 P l (z) − Φ(z) j∈N w j,N = o(1) , and E ⎛ ⎝ j∈N w j,N 1 {s T (j)≤z} ⎞ ⎠ 2 − Φ 2 (z) ≤ sup j 1 ,j 2 ∈N j 1 ≠j 2 P j 1 ,j 2 (z) − Φ 2 (z) j 1 ,j 2 ∈N j 1 ≠j 2 w j 1 ,N w j 2 ,N + sup l∈F N1 P l (Z) − Φ(z) + Φ(z) − Φ 2 (z) j∈N w 2 j,N = o(1) ,
which remains true uniformly in s ∈ S if {{w j,N,s ∶ j ∈ N } ∶ s ∈ S} is a class of weights indexed by a countable index set S satisfying ∑ j∈N w j,N,s = 1 for all s ∈ S and sup s∈S ∑ j∈N w 2 j,N,s → 0. Since
E ⎛ ⎝ j∈N w j,N 1 {s T (j)≤z} − Φ(z) ⎞ ⎠ 2 = E ⎛ ⎝ j∈N w j,N 1 {s T (j)≤z} ⎞ ⎠ 2 − Φ 2 (z) − 2Φ(z) ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ E ⎛ ⎝ j∈N w j,N 1 {s T (j)≤z} ⎞ ⎠ − Φ(z) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = o(1) ,
we get both assertions by the Chebyshev inequality, and the uniformity in z follows from the continuity of Φ(z). ∎
Proof of Theorem 4.3 with lemmas
Before we prove Theorem 4.3, we need the following lemmas.
Cov(I(j), I(k)) = O 1 T , (7.6) max j∈N Cov(I(j), I(j)) − (4π 2 f (λ j )) 2 = O ⎛ ⎝ 1 T ⎞ ⎠ , (7.7) max j,k∈N,j≠k Cov(I 2 (j), I 2 (k)) = O 1 T . (7.8) max j∈N Cov(I 2 (j), I 2 (j)) − 4(4π 2 f (λ j )) 4 = O ⎛ ⎝ 1 T ⎞ ⎠ . (7.9)
Proof. An analogous proof of Theorem 10.3.2(ii) in [2] yields (7.6) and (7.7) for linear random fields, i.e., V (j) − µ = ∑ s∈Z 2 a s ε j−s , where {ε i } i∈Z 2 is an i.i.d. random field with E(ε 0 ) = 0, existence of 4-th moments E(ε 4 0 ) < ∞ and ∑ s∈Z 2 a s s 1 2 < ∞. Under the existence of 8-th moments E(ε 8 0 ) < ∞, one can also show that (7.8) and (7.9) hold. Hence, Assumption P.5(r) with r = 1 2 yields the results. ∎
Lemma 7.2. Suppose that Assumption P.7 with E V (0) 16 < ∞, then max j,k∈N Cov(I(j), I(k)) − (4π 2 f (λ j )) 2 δ j,k = O 1 T , max j,k∈N Cov(I 2 (j), I 2 (k)) − 4(4π 2 f (λ j )) 4 δ j,k = O 1 T .
Proof. Assumption P.7 with E V (0) 16
Cov(x(l), x(k)) − 2π 2 f (λ k )δ l,k → 0 , sup l,k∈N Cov(y(l), y(k)) − 2π 2 f (λ k )δ l,k → 0 . (7.10) Note that ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Cov(x(l), x(k)) + Cov(y(l), y(k)) = Re 1 T ∑ j,s∈T e −i(j ′ λ l −s ′ λ k ) Cov(V (j), V (s)) , Cov(x(l), x(k)) − Cov(y(l), y(k)) = Re 1 T ∑ j,s∈T e −i(j ′ λ l +s ′ λ k ) Cov(V (j), V (s)) . (7.11) Furthermore, it holds that ∑ j∈T e −ij ′ (λ l +λ k ) = d 1 ∑ j 1 =1 e −i 2π(l 1 +k 1 ) d 1 j 1 d 2 ∑ j 2 =1 e −i 2π(l 2 +k 2 ) d 2 j 2 = 0 . Hence 1 T j,s∈T e −i(j ′ λ l +s ′ λ k ) Cov(V (j), V (s)) = 1 T j∈T e −ij ′ (λ l +λ k ) s∈T e −i(s−j) ′ λ k γ(s − j) = 1 T j∈T e −ij ′ (λ l +λ k ) s∈T e −i(s−j) ′ λ k γ(s − j) − 4π 2 f (λ k ) = 1 T j∈T e −ij ′ (λ l +λ k ) ⎛ ⎝ s∈T e −i(s−j) ′ λ k γ(s − j) − s∈Z 2 e −i(s−j) ′ λ k γ(s − j) ⎞ ⎠ = 1 T j∈T e −ij ′ (λ l +λ k ) ⎛ ⎝ s∈Z 2 T −e −i(s−j) ′ λ k γ(s − j) ⎞ ⎠ ≤ 1 T j∈T s∈Z 2 T γ(s − j) = 1 T j∈T ⎛ ⎝ s∈Z 2 T * γ(s − j) + s∈T * T γ(s − j) ⎞ ⎠ ≤ s∈Z 2 T * * γ(s) + d 1 √ d 2 d 1 d 2 + d 2 √ d 1 d 1 d 2 + √ d 1 √ d 2 d 1 d 2 = o(1) , (7.12) where T * = {(t 1 , t 2 ) ∶ t k ∈ Z, 1 ≤ t k ≤ d k + √ d k , k = 1, 2} and T * * = {(t 1 , t 2 ) ∶ t k ∈ Z, 1 ≤ t k ≤ √ d k , k = 1, 2}
, uniformly in l, k by the absolute summability of the auto-covariance function. Analogously, we have, uniformly for l ≠ k, i.e. λ l ≠ λ k , that
1 T j,s∈T e −i(j ′ λ l −s ′ λ k ) Cov(V (j), V (s)) = o(1) . (7.13)
Finally, we have that, uniformly in k,
1 T j,s∈T e −i(j−s) ′ λ k Cov(V (j), V (s)) − 4π 2 f (λ k ) = 1 T h∶( h 1 , h 2 )∈T (d 1 − h 1 )(d 2 − h 2 )e −ih ′ λ k γ(h) − 4π 2 f (λ k ) = o(1) . (7.14)
Putting together (7.11)-(7.14) yields (7.10). Note that a refined version of (7.12)-(7.14) under the stronger assumption that ∑ j∈Z 2 j u γ(j) < ∞ holds for some u > 0, gives the uniform convergence rate
⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ O( T −u ), 0 < u < 1 , O log T T , u = 1 , O( T −1 ), u > 1 ,
which yields (4.4) in assertion (b). Note that E(x(k)) = E(y(k)) = 0 and ∑ j∈T e −ij ′ λ k = 0. We have
E(x(k) + iy(k)) = 1 T E(V (⋅)) j∈T e −ij ′ λ k = 0.
Thus, by (7.10), a simple application of Markov-inequality yields
1 2 N j∈N x(j) f (λ j ) = o p (1), 1 2 N j∈N y(j) f (λ j ) = o p (1) ,
hence assertion (a) follows. Since E(I(j)) = E(x 2 (j) + y 2 (j)) = Cov(x(j), x(j)) + Cov(y(j), y(j)) .
By (7.10), we then have sup
j∈N E(I(j)) − 4π 2 f (λ j ) = o(1) , (7.15)
and thus assertion (c) follows from an application of the Markov inequality, and Lemmas 7.1 and 7.2. Since E(I 2 (j)) = Var(I(j)) + E(I(j)) 2 = (4π 2 f (λ j )) 2 + (4π 2 f (λ j )) 2 = 2(4π 2 f (λ j )) 2 , it holds by Lemmas 7.1 and 7.2, and (7.15) that sup j∈N E(I 2 (j)) − 2(4π 2 f (λ j )) 2 = o(1). Hence by (7.8) and an application of the Markov inequality, assertion (d) follows. For assertion (e), if Assumption P.5(r) with r > 1 2 holds, by an analogous proof of Theorem 1 in [3] and Corollary 1 in [5], we have for any s < 5,
1 N j∈N I s (j) f s (λ j ) < C 2 + o p (11 2 N j∈N ( x(j) + y(j) ) = O p (1) . Let m T = (m T 1 , m T 2 ) = ⌊ d 1 1 2 h T 1 ⌋ , ⌊ d 2 1 2 h T 2 ⌋ . Define L M = {(l 1 , l 2 ) ∈ Z 2 ∶ l 1 ≤ m T 1 , l 2 ≤ m T 2 } and Q M = {(q, r) ∈ Z 2 × Z 2 ∶ q 1 − r 1 ≤ d 1 m T 1 + 1, q 2 − r 2 ≤ d 2 m T 2 + 1}.
Then, the supremum in (a) can be decomposed as
sup j∈N s∈Z 2 p s,T x(j + s) ≤ sup l∈L M s∈Z 2 p s,T x(s l + s) + sup (q,r)∈Q M s∈Z 2 p s,T (x(q + s) − x(r + s)) = O p h T −1 T − 1 4 , where s l = ⌊ l 1 d 1 m T 1 ⌋ , ⌊ l 2 d 2 m T 2 ⌋ for l = (l 1 , l 2 ) ∈ Z 2 .
The last line follows by the following two arguments. The first summation follows from the Chebyshev's inequality, the assumptions on K(⋅), f (⋅), and (4.10) and (4.11). Note that we have ∑
s∈Z 2 p 2 s,T = O( h T T ) O( h T 2 T 2 ) . Hence, we have P ⎛ ⎝ h T T 1 4 sup l∈L M s∈Z 2 p s,T x(s l + s) ≥ ε ⎞ ⎠ ≤ l∈L M h T 2 T 2 4 ε 2 Var ⎛ ⎝ s∈Z 2 p s,T x(s l + s) ⎞ ⎠ ≍ m T 1 m T 2 h T 2 T 2 4 ε 2 O(1) s∈Z 2 p 2 s,T = T 1 2 h T h T 2 T 2 4 ε 2 O(1) O ( h T T ) O ( h T 2 T 2 ) = O(1) .
For the second summand, using Assumptions K.1 and K.5, we have
sup (q,r)∈Q M s∈Z 2 p s,T (x(q + s) − x(r + s)) = sup (q,r)∈Q M s∈Z 2 (p s−q,T − p s−r,T )x(s) ≍ 1 T h T 3 2 m T 1 2 j∈T x(j) = O p h T −1 T − 1 4 .
Analogous arguments yield the assertion for y(⋅). Also, by using the results of Theorem 4.3, analogous arguments yield the results in (b), (c) and (d). ∎
Proof of Theorem 5.1(a)
Proof of Theorem 5.1 (a). Under the Assumptions of Theorem 5.1(a) and by using the union bound, we have for any ε > 0, η > 0, and T large enough, that P max
λ∈[0,2π] 2 f T (λ) − f (λ) > η = P max λ∈[0,2π] 2 f T (λ) − f (λ) > η, ε −1 ≤ h T −1 ≤ ε T v +P max λ∈[0,2π] 2 f T (λ) − f (λ) > η, ε −1 > h T −1 +P max λ∈[0,2π] 2 f T (λ) − f (λ) > η, h T −1 > ε T v ≤ P max λ∈[0,2π] 2 f T (λ) − f (λ) > η, ε −1 ≤ h T −1 ≤ ε T v +P(ε h T −1 < 1) + P( h T −1 > ε T v ) .
The last two probabilities converge to 0 under the conditions on h T . It remains to prove that
max ε −1 ≤ h T −1 ≤ε T v max λ∈[0,2π] 2 f T (λ) − f (λ) p → 0, as T → ∞. (7.17) Denotef T (λ) = 1 4π 2 ∑ j∈Z 2 R V (j)k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ), where R V is defined in (3.2). Recall that f T (λ) = 1 4π 2 ∑ j∈Z 2R V (j)k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ)+o(1) whereR V is defined in (3.2) with µ replaced byV T . Then, we havef T (λ) − f (λ) = a 1 + a 2 + a 3 ,
where
a 1 =f T (λ) −f T (λ), a 2 =f T (λ) − E(f T (λ)), and a 3 = E(f T (λ)) − f (λ).
For the third term a 3 , by a similar argument as in the proof of Theorem 5.1(b), see (7.27) below, it holds uniformly for
ε −1 ≤ h T −1 ≤ ε T v that max λ∈[0,2π] 2 ε −1 ≤ h T −1 ≤ε T v a 3 ≤ max λ∈[0,2π] 2 E(f T (λ)) − f (λ) = o(1) . (7.18)
For the second term a 2 , by Assumptions P.4(v) and K.3, we have
max λ∈[0,2π] 2 ε −1 ≤ h T −1 ≤ε T v a 2 ≤ max ε −1 ≤ h T −1 ≤ε T v 1 4π 2 j∈Z 2 k(j 1 h T 1 , j 2 h T 2 ) R V (r) − E(R V (r)) = O p ⎛ ⎝ T −v j∈Z 2k j 1 √ εd v 1 , j 2 √ εd v 2 ⎞ ⎠ = O p ε R 2 +k (x)dx = O p (ε) . (7.19)
Since ε is arbitrary, we have max λ∈[0,2π] 2
ε −1 ≤ h T −1 ≤ε T v a 2 p → 0.
Finally, for the first term a 1 , by Assumption
K.3, we have max λ∈[0,2π] 2 ε −1 ≤ h T −1 ≤ε T v a 1 ≤ max ε −1 ≤ h T −1 ≤ε T v ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 4π 2 T j∈Z 2 k (j 1 h T 1 , j 2 h T 2 ) × ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ l,l+j∈T (V (l) − µ)(V T − µ) + l,l+j∈T (V (l + j) − µ) V T − µ + V T − µ 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ = O p ⎛ ⎝ 1 T j∈Z 2k j 1 √ εd v 1 , j 2 √ εd v 2 ⎞ ⎠ p → 0 ,(7.
20) where boundedness of the spectral density f is used. As a result, by (7.18), (7.19) and (7.20), we have (7.17) and the proof is completed. ∎
Proof of Theorem 5.1(b) with lemmas
Before we prove Theorem 5.1(b), we need the following subsection about summability of joint cumulants implied by the geometric-moment contraction condition GMC(α) in Assumption P.7.
Summability of Cumulants under GMC(α) for spatial process
Assume that V (j) = G(ε j−s ∶ s ∈ Z 2 ), j ∈ Z 2 , where G(⋅) is a measurable function and {ε i } i∈Z 2 is an i.i.d. random field. Assume the geometric-moment contraction condition GMC(α) in Assumption P.7 holds. Let (U 1 , ..., U k ) be a random vector. Then the joint cumulant is defined
as cum(U 1 , ...U k ) = (−1) p (p − 1)!E ⎛ ⎝ j∈V 1 U j ⎞ ⎠ . . . E ⎛ ⎝ j∈Vp U j ⎞ ⎠ ,
where V 1 , ..., V p is a partition of the set {1, 2, ..., k} and the sum is taken over all such partition.
Lemma 7.3. Assume that there exists C 1 > 0, ρ ∈ (0, 1) and k ∈ N, k ≥ 2 such that E{ V (j) k } < ∞ and E{ V (j) −Ṽ (j) k } ≤ C 1 ρ j for j ∈ N 2 . Then, whenever 0 ≤ m 1 ≤ m 2 ≤ ... ≤ m k−1 where m 1 , ..., m k−1 ∈ N 2 , m k = (m k1 , m k2 ) and m k = max{ m k1 , m k2 }, cum(V (0), V (m 1 ), V (m 2 ), ..., V (m k−1 )) ≤ Cρ m k−1 2k(k−1) ,
where the constant C > 0 is independent of m 1 , ..., m k−1 .
Proof of Lemma 7.3. Let C > 0 be a generic constant which is independent of m 1 , . . . , m k−1 . In the proof, C may vary from line to line and it only depends on C 1 , ρ and the moments
E( V (j) i ), 1 ≤ i ≤ k. Let J = cum(V (0), V (m 1 ), ..., V (m k−1 )), where m 0 = 0, l ∈ N, 1 ≤ l ≤ k − 1. Define coupled versionṼ l (m i ) asṼ l (m i ) = G ε * m i −s , s ∈ Z 2 ∀m i ∈ Z 2 , l ∈ N , where ε * m i −s = ε m i −s , s < l , ε m i −s , s ≥ l .
Also, denoteṼ
l (m i ) = G ε * * m i −s , s ∈ Z 2 ∀m i ∈ Z 2 , where ε * * m i −s = ε m i −s , s < l , ε m i −s , s ≥ l .
Note that {ε i } i∈Z 2 , {ε i } i∈Z 2 and {ε i } i∈Z 2 are i.i.d. random fields. By the GMC property, we have
E V (m i ) −Ṽ l (m i ) α ≤ Cρ l and E V (m i ) −Ṽ l (m i ) α ≤ Cρ l .
Define n l = m l − m l−1 , 1 ≤ l ≤ k − 1 and m 0 = 0. We have
J = cum V (0), V (m 1 ), . . . , V (m l−1 ), . . . , V (m k−1 ) = cum V (m 0 − m l−1 ), V (m 1 − m l−1 ), . . . , V (0), . . . , V (m k−1 − m l−1 ) = cum V (m 0 − m l−1 ), . . . , V (0), V (m l − m l−1 ) −Ṽ n l 2 (m l − m l−1 ), V (m l+1 − m l−1 ), . . . , V (m k−1 − m l−1 ) + k−l−1 j=1 cum V (m 0 − m l−1 ), . . . , V (0),Ṽ n l 2 (m l − m l−1 ), . . . ,Ṽ n l 2 (m l+j−1 − m l−1 ), V (m l+j − m l−1 ) −Ṽ n l 2 (m l+j − m l−1 ), V (m l+j+1 − m l−1 ), . . . , V (m k−1 − m l−1 ) + cum V (m 0 − m l−1 ), . . . , V (0),Ṽ n l 2 (m l − m l−1 ), . . . ,Ṽ n l 2 (m k−1 − m l−1 ) =Ã 0 + k−l−1 j=1Ã j + cum V (m 0 − m l−1 ), . . . , V (0),Ṽ n l 2 (m l − m l−1 ), . . . ,Ṽ n l 2 (m k−1 − m l−1 ) =Ã 0 + k−l−1 j=1Ã j +B 0 + l−1 i=1B i + C 0 , wherẽ A 0 = cum V (m 0 − m l−1 ), . . . , V (0), V (m l − m l−1 ) −Ṽ n l 2 (m l − m l−1 ), V (m l+1 − m l−1 ), . . . , V (m k−1 − m l−1 ) , A j = cum V (m 0 − m l−1 ), . . . , V (0),Ṽ n l 2 (m l − m l−1 ), . . . ,Ṽ n l 2 (m l+j−1 − m l−1 ), V (m l+j − m l−1 ) −Ṽ n l 2 (m l+j − m l−1 ), V (m l+j+1 − m l−1 ), . . . , V (m k−1 − m l−1 ) , B 0 = cum V (E ⎛ ⎝ j∈F U j ⎞ ⎠ ≤ E V (0) F , and E ⎛ ⎝ U l j∈F U j ⎞ ⎠ ≤ U l 1+ F E ⎛ ⎝ j∈F U j F +1 F ⎞ ⎠ F 1+ F ≤ U l k E V (0) F +1 F 1+ F ≤ C 1 ρ n l 2k C * , where C * = k−1 ∑ i=0 E V (0) i+1m k−1 = k−1 l=1 ( m l − m l−1 ) = k−1 l=1 n l ≤ (k − 1) max 1≤l≤k−1 n l , we have m k−1 k−1 ≤ max 1≤l≤k−1 n l . Finally, we have J ≤ Cρ max 1≤l≤k−1 n l 2k ≤ Cρ m k−1 2k(k−1) . ∎ Lemma 7.4. Let the sequence of sets indexed by T be S T ⊂ N 2 satisfying S T = {(t 1 , t 2 ), 1 ≤ t k ≤ S T k , k = 1, 2} ⊂ T , where T = {(t 1 , t 2 ), 1 ≤ t k ≤ d k , k = 1, 2}, S T 1 ≤ d 1 and S T 2 ≤ d 2 , and define another sequence of sets indexed by T be B T ⊂ N 2 , B T = {(l 1 , l 2 ), 1 ≤ l k ≤ B T k , k = 1, 2} with (B T 1 , B T 2 ) = 1 h T 1 , 1 h T 2 and B T = o( S T ), and U j = U j (λ) = (4π 2 ) −1 l∈L B V (j)V (j + l)k(l 1 h T 1 , l 2 h T 2 ) cos(l ′ λ) , where L B = {(l 1 , l 2 ) ∈ Z 2 ∶ ( l 1 , l 2 ) ∈ B T }. Then, under GMC(4), we have ∑ j∈S T U j − E(U j ) 2 2 ∼ S T B T σ 2 . Proof of Lemma 7.4. Let L(s) = {(m 1 , m 2 , m 3 ) ∈ Z 2 ×Z 2 ×Z 2 ∶ max 1≤i≤3 m i = s} and C(m 1 , m 2 , m 3 ) = cum(X 0 , X m 1 , X m 2 , X m 3 )
. Thus, L(s) ≤ 24(2s + 1) 3 , By Lemma 7.3, we have
m 1 ,m 2 ,m 3 ∈Z 2 C(m 1 , m 2 , m 3 ) ≤ C * * ∞ s=0 (m 1 ,m 2 ,m 3 )∈L(s) C(m 1 , m 2 , m 3 ) ≤ C * * * ∞ s=0 s 3 ρ s 2(4)(4−1) < ∞ .
for some constants C * * > 0 and C * * * > 0. Then, the lemma follows from similar arguments in (3.9)-(3.12) of [21], page 1174. ∎
Uniform consistency of the smoothed periodogram spectral density estimators
Define the smoothed periodogram spectral density estimator as
f T (λ) = 1 4π 2 j∈Z 2 R V (j)k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ) , where R V (j) = 1 T ∑ l,l+j∈T V (l)− µ V (l+ j)− µ ,Theorem 7.5. Define B T = {(l 1 , l 2 ) ∈ Z 2 ∶ 1 ≤ l k ≤ B T k , k = 1, 2}, where (B T 1 , B T 2 ) = 1 h T 1 , 1 h T 2 → ∞. Assume GM C(α), α > 0, V (j) ∈ L 4+δ for some δ ∈ (0, 4], and B T = O( T η ), i.e., ( h T T η ) −1 = O(1) for some 0 < η < δ 4+δ and f * = min λ∈R 2 f (λ) > 0. Then, max λ∈[0,2π] 2 T h T f T (λ) − E(f T (λ)) = O p (log T ) 1 2 .
Remark 7.1. If GM C(α), α > 0 and V (j) ∈ L 4+δ , then V (j) satisfies GMC(4), see Lemma 2 of [27].
Proof of Theorem 7.5. Let
U j = U j (λ) = 1 4π 2 l∈L B V (j)V (j + l)k(l 1 h T 1 , l 2 h T 2 ) cos(l ′ λ) = 1 4π 2 l∈L B V (j)V (j + l)α l , where L B = {(l 1 , l 2 ) ∈ Z 2 ∶ ( l 1 , l 2 ) ∈ B T } and α l = k(l 1 h T 1 , l 2 h T 2 ) cos(l ′ λ). Note that f T (λ) = 1 4π 2 l∈L B ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 1 T j,j+l∈T (V (j)V (j + l)α l ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 1 T j∈T U j − 1 T T B T q T (λ) , where q T (λ) = 1 4π 2 T B T ⎛ ⎝ l∈L B j∶j∈T,j+l∉T V (j)V (j + l)α l ⎞ ⎠ .
By the summability of cumulants of order 2 and 4, we have
q T (λ) = ( T B T ) − 1 2 O( B T ) = O ⎛ ⎜ ⎝ B T T ⎞ ⎟ ⎠ = O ⎛ ⎜ ⎝ T η T ⎞ ⎟ ⎠ = o(1). Denote g T = g T (λ) = ∑ j∈T U j . We have T h T f T (λ) − E(f T (λ)) = g T − E(g T ) T B T − q T (λ) + E(q T (λ)
) .
.i.d. if i− j ≥ 2m andÛ i andÛ j are i.i.d. if i − j ≥ 2 max{B T 1 , B T 2 } + 2m. Defineĝ T =ĝ T (λ) = ∑ j∈TÛ j (λ). Then, g T −ĝ T 2 = o(1) since U j −Û j 2 ≤ 1 4π 2 l∈L B V (j)V (j + l) −V (j)V (j + l) 2 α l = O( B T )O V (j)V (j + l) −V (j)V (j + l) 2 = O B T ρ m 4 . (7.22)
Therefore, we have
g T −ĝ T 2 ≤ j∈T U j −Û j 2 = O T B T ρ m 4 = O T B T 1 T 2 = o(1) .
We now define four sets of blocks such that the blocks within the same set are i.i.d.. Let
p T = ⌊ T 1− 4η δ (log T ) − 8 δ −4 ⌋. We have p T → ∞, B T = o(p T ), and k T = (n 1 , n 2 ) ∶ 1 ≤ n 1 ≤ ⌊ d 1 p T ⌋ , 1 ≤ n 2 ≤ ⌊ d 2 p T ⌋ . Define k T A = {(n 1 , n 2 ) ∈ k T ∶ n 1 = 2z 1 + 1, n 2 = 2z 2 + 1 ∃ z 1 , z 2 ∈ N} , k T B = {(n 1 , n 2 ) ∈ k T ∶ n 1 = 2z 1 , n 2 = 2z 2 + 1 ∃ z 1 , z 2 ∈ N} , k T C = {(n 1 , n 2 ) ∈ k T ∶ n 1 = 2z 1 + 1, n 2 = 2z 2 ∃ z 1 , z 2 ∈ N} , k T D = {(n 1 , n 2 ) ∈ k T ∶ n 1 = 2z 1 , n 2 = 2z 2 ∃ z 1 , z 2 ∈ N} .
Then, define the blocks
A r = {(j 1 , j 2 ) ∈ N 2 ∶ (r 1 − 1)p T + 1 ≤ j 1 ≤ r 1 p T , (r 2 − 1)p T + 1 ≤ j 2 ≤ r 2 p T } ,
for r = (r 1 , r 2 ) ∈ k T A and B r , C r and D r are defined similarly. Let
A r = j∈ArÛ (j), B r = j∈BrÛ (j), C r = j∈CrÛ (j), D r = j∈DrÛ (j) . Observe that {A r } r∈k T A are i.i.d. Similarly, {B r } r∈k T B , {C r } r∈k T C and {D r } r∈k T D are i.i.d. Also, define the remaining part R T ∶= T (k T A ⋃ k T B ⋃ k T C ⋃ k T D ) and R = R(λ) = ∑ j∈R TÛ j (λ).
The following two lemmas state the order of the asymptotic behavior of A r , B r , C r , D r and R respectively.
Lemma 7.6. Let τ = 2 + δ 2. We have max λ∈[0,2π] 2 A 1 (λ) τ = O m 2 B T p T .
Proof of Lemma 7.6. By the definition of A 1 (λ),
A 1 (λ) τ = j∈A 1Û (j) τ = 1 4π 2 j∈A 1 l∈L BV (j)V (j + l)α l τ ≤ 1 4π 2 j∈A 1 l∈L fV (j)V (j + l)α l τ + 1 4π 2 j∈A 1 l∈LsV (j)V (j + l)α l τ ,
where L f = {(l 1 , l 2 ) ∈ Z 2 ∶ ( l 1 , l 2 ) ∈ B T , l 1 > 2m or l 2 > 2m}, L s = {(l 1 , l 2 ) ∈ Z 2 ∶ ( l 1 , l 2 ) ∈ B T , l 1 ≤ 2m, l 2 ≤ 2m}. For the first summation, we have
j∈A 1 l∈L fV (j)V (j + l)α l τ ≤ h∈H f j∈J f l∈L fV (h + 2mj)V (h + 2mj + l)α l τ ≤ O m 2 p T m l∈L fV (l)α l τ ≤ O(p T m) h∈H f j∈J B TV (h + 2mj)α h+2mj τ = O(p T m)O ⎛ ⎝ m 2 B T m 2 ⎞ ⎠ = O p T B T m 2 , where H f = {(h 1 , h 2 ) ∈ Z 2 ∶ 1 ≤ h 1 ≤ 2m, 1 ≤ h 2 ≤ 2m}, J f = {(j 1 , j 2 ) ∈ Z 2 ∶ j 1 ≤ ⌊ p T −h 1 2m ⌋ , j 2 ≤ ⌊ p T −h 2 2m ⌋}, and J B T = {(j 1 , j 2 ) ∈ Z 2 ∶ j 1 ≤ ⌊ B T 1 −h 1 2m ⌋ , j 2 ≤ ⌊ B T 2 −h 2 2m ⌋ , ( j 1 , j 2 ) ≠ (0, 0), (1, 1)}.
On the other hand, for the second summation, we have
j∈A 1 l∈LsV (j)V (j + l)α l τ ≤ l∈Ls h∈Hs j∈JsV h + j ′ (6m, 6m) + l V h + j ′ (6m, 6m) α l τ = O ⎛ ⎝ m 4 p T 2 m 2 ⎞ ⎠ = O m 3 p T , where H s = {(h 1 , h 2 ) ∈ Z 2 ∶ h 1 ≤ 6m, h 2 ≤ 6m}, J s = {(j 1 , j 2 ) ∈ Z 2 ∶ j 1 ≤ ⌊ p T −h 1 6m ⌋ , j 2 ≤ ⌊ p T −h 2 6m ⌋}. Then, we have A 1 (λ) τ = O p T B T m 2 + O m 3 p T = O p T B T m 2 . Thus, it holds uniformly over λ ∈ [0, 2π] 2 that max λ∈[0,2π] 2 A 1 (λ) τ = O m 2 B T p T . ∎ Next, define c T = T B T (log T ) − 1 2 and the truncated version A * r (λ) = A r (λ)1 { Ar(λ) ≤c T } . Also, we define B * r (λ), C * r (λ), D * r (λ)R(λ) = O p T (d 1 + d 2 − p T )m B T , (7.23) E max λ∈[0,2π] 2 q T (λ) = o(1) , (7.24) max λ∈[0,2π] 2 Var(A 1 (λ)) = O p 2 T B T , (7.25) Var(A * 1 (λ)) = Var(A 1 (λ))[1 + o(1)] ,(7.R(λ) ≤ C l∈L B E j∈R TV (j)V (j + l) .
For l 1 ≤ 2m and l 2 ≤ 2m, sinceV (j)V (j + l) is 4m-dependent, we have
j∈R TV (j)V (j + l) 2 = O R T m .
For l 1 > 2m or l 2 > 2m,
j∈R TV (j)V (j + l) 2 2 = j 1 ∈R T j 2 ∈R T E V (j 1 )V (j 1 + l)V (j 2 )V (j 2 + l) = O R T m 2 ,
since the sum vanishes if j 1 − j 2 ∞ > 2m, E(V (j)) = 0. As a result,
E max λ∈[0,2π] 2 R(λ) = O R T m B T = O p T (d 1 + d 2 − p T )m B T .
Letq T (λ) be the corresponding sum of q T (λ) with V (j)V (j + l) replaced byV (j)V (j + l). As in
(7.22), we have E max λ∈[0,2π] 2 q T (λ) −q T (λ) = o(1) .
To show (7.24), it suffices to show E max λ∈[0,2π] 2 q T (λ) = o(1), which follows from a similar arguments as in the proof of (7.23). Regarding (7.25) and let l = (l 1 , l 2 ), l * = (l * 1 , l * 2 ), we have
Var(A 1 (λ)) = j∈A 1 l∈L B [V (j)V (j + l) − γ(l)]α l 2 = j,j * ∈A 1 l,l * ∈L B γ(j − j * )γ(j − j * + l − l * ) + γ(j * − j + l * )γ(j * − j − l) + cum(V (0), V (l), V (j * − j), V (j * − j + l * )) α l α l * = I 1 + I 2 + I 3 ,
where
I 1 = j,j * ∈A 1 l,l * ∈L B γ(j − j * )γ(j − j * + l − l * )α l α l * , I 2 = j,j * ∈A 1 l,l * ∈L B γ(j * − j + l * )γ(j * − j − l)α l α l * , I 3 = j,j * ∈A 1 l,l * ∈L B cum(V (0), V (l), V (j * − j), V (j * − j + l * ))α l α l * .
Note that I 1 is bounded by
C j∈A 1 (p T − j 1 )(p T − j 2 ) γ(j) l∈L 2B (2B T 1 + 1 − l 1 )(2B T 2 + 1 − l 2 ) γ(j + l) , where L 2B = {(l 1 , l 2 ) ∈ Z 2 ∶ ( l 1 , l 2 ) ∈ 2B T } with 2B T = {(l 1 , l 2 ) ∈ Z 2 ∶ 1 ≤ l k ≤ 2B T k , k = 1, 2}, where (2B T 1 , 2B T 2 ) = 2 h T 1 , 2 h T 2
, which is less than
Cp 2 T (4 B T + o( B T )) ⎛ ⎝ l∈Z 2 γ(l) ⎞ ⎠ 2 = O p 2 T B T .
Similarly, smaller bounds can be obtained for I 2 and I 3 due to the summability of the second and fourth cumulants. Thus, we have max λ∈[0,2π] 2 Var(A 1 (λ)) = O p 2 T B T . For (7.26), let ν = Var(A 1 (λ) − A * 1 (λ)) and c = E(A * 1 (λ))E(A 1 (λ) − A * 1 (λ)). Then,
Var(A * 1 (λ)) = Var(A 1 (λ)) − ν + 2c .
By Markov's inequality, Holder's inequality and Lemma 7.6,
ν = Var(A 1 (λ) − A * 1 (λ)) = Var A 1 (λ)1 { A 1 (λ) >c T } ≤ E A 2 1 (λ) τ 2 2 τ E 1 { A 1 (λ) >c T } τ −2 τ ≤ E( A 1 (λ) τ ) c T τ −2 = O m 2 B T p T τ c T τ −2 = o p 2 T B T ,
and similarly, c ≤ A 1 (λ) τ +1 Let λ j = πj 1 t T 1 , πj 2 t T 2 where j 1 = 0, . . . , t T 1 ∶= ⌊B T 1 log(B T 1 )⌋, j 2 = 0, . . . , t T 2 ∶= ⌊B T 2 log(B T 2 )⌋. Let
C T i = 1 1 − 3π log B T i → 1 ,
for i = 1, 2. By the argument similar to Corollary 2.1 in [25], we have max λ∈[0,2π] 2 G T (λ) ≤ C T 1 C T 2 max j 1 ≤t T 1 j 2 ≤t T 2 G T (λ j ) . By (7.25) and (7.26), there exists a constant C 1 > 1 such that max λ∈[0,2π] 2 Var(A * 1 (λ)) ≤ C 1 p 2 T B T . Let β T = (C 1 T B T log T ) 1 2 , By Bernstein's inequality, we have
P ⎛ ⎜ ⎝ max 0≤j 1 ≤t T 1 0≤j 2 ≤t T 2 G * T (λ j ) ≥ 4β T ⎞ ⎟ ⎠ ≤ t T 1 j 1 =0 t T 2 j 2 =0 P( G * T (λ j ) ≥ 4β T ) = O(t T 1 t T 2 ) exp −16β 2 T 2 k T A C 1 p 2 T B T + 16c T β T = o(1) .
Let A * * r (λ) = A r (λ) − A * r (λ) and G * * T (λ) = G T (λ) − G * T (λ). Then, by Markov's inequality and Lemma 7.6,
P ⎛ ⎜ ⎝ max 0≤j 1 ≤t T 1 0≤j 2 ≤t T 2 G * * T (λ j ) ≥ 4β T ⎞ ⎟ ⎠ ≤ t T 1 j 1 =0 t T 2 j 2 =0 P( G * * T (λ j ) ≥ 4β T ) ≤ t T 1 j 1 =0 t T 2 j 2 =0
Var(A * * r (λ j )) k T A 16β 2 Proof of Lemma 7.8. For any λ = (λ 1 , λ 2 ) ∈ [0, 2π] 2 , we havê
T = O t T 1 t T 2 k T A p τ T B Tf T (λ) = ∑ j∈Z 2 K λ 1 −λ j1 h T 1 , λ 2 −λ j2 h T 2 I(j) 4π 2 ∑ j∈Z 2 K λ j1 h T 1 , λ j2 h T 2 = 1 h T T j∈Z 2 K λ 1 − λ j1 h T 1 , λ 2 − λ j2 h T 2 I(j) + o(1) = 1 h T T t∈T I(t) j∈Z 2 K λ 1 − λ t1 + 2πj 1 h T 1 , λ 2 − λ t2 + 2πj 2 h T 2 + o(1) = 1 T t∈T I(t)K h (λ − λ t ) + o(1) = 1 4π 2 T t∈T I(t) j∈Z 2 k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ (λ − λ t )) + o(1) = 1 4π 2 j∈Z 2 1 T t∈T I(t) exp(ij ′ λ t ) k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ) + o(1) = 1 4π 2 j∈Z 2R V (j)k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ) + o(1) =f T (λ) + o(1) .
For the last two equalities, note that by Assumption P.6(p) with p > 2, [V T − E(V (0))] 2 is asymptotically negligible.
E(f T (λ)) − f (λ) = 1 4π 2 j∈L T (d 1 − j 1 )(d 2 − j 2 ) T γ(j)k(j 1 h T 1 , j 2 h T 2 ) exp(−ij ′ λ) − 1 4π 2 j∈Z 2 γ(j) exp(ij ′ λ) ≍ j∶ j ≥ 1 h T γ(j) + j∶ j ≤ 1 h T (d 1 − j 1 )(d 2 − j 2 ) T k(j 1 h T 1 , j 2 h T 2 ) − 1 γ(j) = o(1) ,(7.27)
where L T = {j ∈ Z 2 ∶ ( j 1 , j 2 ) ∈ T }. The above arguments used the fact that for j ≥ 1 h T , we have
(d 1 − j 1 )(d 2 − j 2 ) T k(j 1 h T 1 , j 2 h T 2 ) − 1 ≤ (d 1 − j 1 )(d 2 − j 2 )
T C * + 1 ≤ C * + 1 , Proof of Theorem 6.1. First, note that we have j∈T cos(λ ′ j t 1 ) cos(λ ′ j t 2 ) + sin(λ ′ j t 1 ) sin(λ ′ j t 2 ) = T , t 1 = t 2 , 0, t 1 ≠ t 2 .
and for j ≤ 1 h T , we have (d 1 − j 1 )(d 2 − j 2 ) T k(j 1 h T 1 , j 2 h T 2 ) − k(0) = 1 − j 1 d 1 1 − j 2 d 2 [k(j 1 h T 1 , j 2 h T 2 ) − k(0)] + j 1 d 1 + j 2 d 2 − j 1 j 2 d 1 d 2 ≍ sup x ≤ h T k(x 1 , x 2 ) − k(0) + 1 h T d 1 + 1 h T d 2 = o(1) + o(1) + o(1) = o(1) .
For t 1 = t 2 , the equation is trivial by considering cos 2 x + sin 2 x = 1. For t 1 ≠ t 2 , we have j∈T cos(λ ′ j t 1 ) cos(λ ′ j t 2 ) + sin(λ ′ j t 1 ) sin(λ ′ j t 2 ) = j∈T cos(λ ′ j (t 1 − t 2 )) = j∈T cos(λ ′ j c) (By letting t 1 − t 2 = c ∈ Z 2 ) = j∈T [cos(λ j1 c 1 ) cos(λ j2 c 2 ) − sin(λ j1 c 1 ) sin(λ j2 c 2 )]
= d 1 j 1 =1 d 2 j 2 =1 cos 2πj 1 d 1 c 1 cos 2πj 2 d 2 c 2 − sin 2πj 1 d 1 c 1 sin 2πj 2 d 2 c 2 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ d 1 j 1 =1 cos 2πj 1 d 1 c 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ d 2 j 2 =1 cos 2πj 2 d 2 c 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ − ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ d 1 j 1 =1 sin 2πj 1 d 1 c 1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ d 2 j 2 =1 sin 2πj 2 d 2 c 2 ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ = 0 . Furthermore, denote F T (j) = t∈T (V (t) −V (t)) cos(λ ′ j t), G T (j) = t∈T (V (t) −V (t)) sin(λ ′ j t), j ∈ T .
By the previous equality and the definition of the kernel spectral density estimator in (2.4), applying the Cauchy-Schwarz inequality yields
F T (j) ≤ t∈T V (t) −V (t) ≤ t∈T (V (t) −V (t)) 2 t∈T 1 2 = 1 T t∈T (V (t) −V (t)) 2 T 2 = o p T α − 1 2 T .
For the proof of (b)(i), note that which yields the assertion in part (c)(iii). Finally, we have sup k∈N j∈Z 2 p j,T (I q V (k + j) − I q V (k + j)) ≍ sup k∈N j∈Z 2
p j,T (I V (k + j) − IV (k + j)) q ≍ sup k∈N ( h T T ) q−1 j∈Z 2 p j,T (I V (k + j) − IV (k + j)) q = o p T q−1 h T α q T = o p (1) ,
which yields the assertion in part (c)(iv). ∎
Figure 1 :
1Illustration for extracting Fourier coefficients at j ∈ N . In the grid, ⋆ are coefficients in the set N and ○ are coefficients containing information about the mean in the set M .
Theorem 4.1 is a nontrivial generalization of[22] from time series to spatial lattice data. The uniformity in Theorem 4.1 is helpful to show the convergence of the empirical distribution function of the Fourier coefficients in the following corollary.
Corollary 4. 2 .
2Let Assumptions P.1, P.2, P.3 and P.6(p) hold with some p ≥ 2. For any weights
which is satisfied by the commonly used kernels mentioned in Remark 3.1. Also, the conditions (4.10) and (4.11) are satisfied under the additional condition in Theorem 4.3(b) for h T = O( T u−1 ) when u < 1 or h T = o(1) when u ≥ 1. The following examples fulfill the required conditions for the underlying random fields in Theorems 4.3 and 4.4, .
Theorem 5. 1 .
1Suppose that Assumptions P.1, P.2 and K.1 hold. (a) If Assumptions K.3 and P.4(v) with some 0 < v ≤ 1 2 hold, and the bandwidth satisfies
Example 5 . 3 (
53Volterra Fields). For the second order Volterra process defined in(4.12) in Example
By
Assumptions K.1, K.5, and the continuity of the spectral density function f (λ), Theorem 4.4(b) implies the uniform consistency off T (λ) shown in Theorem 5.1. Note that Theorem 4.4(b) requires stronger conditions and thus achieves a stronger result than Theorem 5.1. To compare the results of Theorem 4.4(b) and Theorem 5.1, we compare the required moment and linearity conditions of the two theorems. First, the moment condition in Theorem 5.1 is generally weaker than that of Theorem 4.4(b). For Theorem 5.1(a), Assumption P.4(v) with v = 1 2 is satisfied under the existence of (4 + δ)-th moment and short-range weak dependence condition of the underlying fields, for example, mixing conditions, by which some forms of invariance principle or central limit theorem can be established. The moment condition required for Theorem 5.1(b) is similar to that of Theorem 5.1(a), whereas Theorem 4.4(b) requires the existence of 8-th moment. Note that Assumption P.5(r) with r > 1 2 which is required in Theorem 4.4 implies Assumption P.4(v) with v = 1 2 which is required in Theorem 5.1(a). Second, Theorem 5.1 allows non-linearity of the underlying fields, whereas Theorem 4.4(b) requires linearity of the underlying fields. As a result, the required conditions for Theorem 4.4(b) is in general stronger than that of Theorem 5.1.
assume that the kernel function of the spectral density estimator (2.4) satisfies Assumptions K.1 and K.2; and the kernel function used in defining the weights p s,T in (4.9) satisfies Assumptions K.1 and K.2. Then given {Y (⋅)}, we have the followings hold: (a) Under the setting of Theorem 5.1 and α T = O h T −1 , then we have sup j∈T f V (λ j ) −fV (λ j ) = o p (1) . (b) Under the setting of Theorem 4.3 and α T = O T
. 1 .
1The proof of Theorem 4.1 goes as follows. For presentational clarity we focus on the real parts of x(j) + iy(j). The argument easily extends to general cases which include the complex parts {y(j)}. We first express the linear combination of Fourier coefficients {x(j)} as linear combinations of {V (k)}, and then we construct a 2l T -dependent field {V (k)}, where l T is defined below, such that the central limit theorem can be applied to the linear combination of its truncated counterpart {U (k)}. The error of the above approximations are then examined.
Then we have that {V (k)} are 2l T -dependent and δ l T = V (0) −V (0) 2 → 0 as l T → ∞, andĤ T can be regarded as an approximation of H T . Denote F m,n = {ε i ∶ i = (i 1 , i 2 ) ∈ Z 2 with i 1 < m, i 2 < n}.
Figure 2 :
2Illustration of the locations of ε i included.
Figure 4 Figure 3 :
43for a graphical illustration of the locations of blocks in set B m . Define T m = ∑ k∈Bm µ k U (k), S T = ∑ m∈M B T m , R T = H T,U − S T , W = S T −E(S T ) T , and ∆ =Ĥ T T − W . Then, T m are independent and R T 2 = O M B since U (k) are 2l T -dependent. Note that The partition of the rectangular lattice T
Figure 4 :
4The
m 0 −
0m l−1 ), . . . , V (0) l − m l−1 ), . . . ,Ṽ n l 2 (m k−1 − m l−1 ) ,B i = cum V (m 0 − m l−1 ), . . . , V (m l−i−2 − m l−1 ), V (m l−i−1 − m l−1 ) −Ṽ n l 2 (m l−i−1 − m l−1 ), . . l − m l−1 ), . . . ,Ṽ n l 2 (m k−1 − m l−1 ) are independent,we have C 0 = 0. We shall now show that à 0 ≤ Cρn l 2k . To this end, let U j = V (m j − m l−1 ) for j = 0, 1, . . . , k − 1, j ≠ l and U l = V (m l − m l−1 ) −Ṽ n l 2 (m l − m l−1 ). Let F be the cardinality of the set F . For any subset F ⊂ {0, 1, . . . , k − 1} such that l ∉ F, by Hölder's and Jensen's inequity, we have
1
1i 1+i . By definition of joint cumulant, à 0 ≤ Cρ n l 2k for some constant C. Similarly, for j = 1, . . . , k − l − 1, we have à j ≤ Cρ n l 2k , and B i ≤ Cρ n l 2k for i = 0, . . . , l −
and k(⋅) is defined as in (3.4) satisfying Assumption K.4. The following theorem prove the uniform consistency of the smoothed periodogram spectral density estimatorf T (λ) using the results of Lemma 7.4. Without loss of generality, we assume that µ = E(V (0)) = 0 in the following.
= ρ(4) < 1 as in GMC(4). For l ∈ Z 2 , defineV (l) = E(V (l) F l,<m> ), where F l,<m> is defined the same as in the proof of Theorem 4.1 with m = m T = ⌊− 8 log T log ρ ⌋ ∈ N. LetÛ j =Û j (λ) be the corresponding sum with V (l) replaced byV (l). Observe thatV (i) andV (j) are i
1 = o p 2 T B T . ByLemma 7.4 and the fact that f is every-where positive, (7.26) follows.
G
T (λ) = O p (β T ). Using similar arguments, the same bound also holds for the sums ∑r∈k T B [B r (λ) − E(B r (λ))], ∑ r∈k T C [C r (λ) − E(C r (λ))], and ∑ r∈k T D [D r (λ) − E(D r (λ))]. By (7.22), we have E max λ∈[0,2π] 2 ĝ T (λ) − g T (λ) = o(1). As a result, by (7.23), (7.24) and (7.21), the proof is completed. ∎The following lemma states thatf T (λ) in (2.4) andf T (λ) are asymptotically equivalent.Lemma 7.8. Under Assumptions K.1 and P.6(p) with p > 2, we have max λ∈[0,2π] 2 f T (λ) −f T (λ) = o(1).
f
T (λ) − E(f T (λ)) = o(1) .It remains to show that max λ∈[0,2π] 2 E(f T (λ)) − f (λ) = o(1). This holds since by Assumption K.4, k(⋅) is bounded, continuous and with compact support [−1, 1] 2 . Also, k(0) = 1 as T → ∞. Thus, for any λ ∈ [0, 2π] 2 , we have
Table 1 :
1Fourier coefficients in M contain information about the mean.
Example 5.1 (Linear Random Fields). For the linear random fields defined in (4.1) in Example 4.1, if we have E( ε 0 p ) < ∞ with some p ≥ 4 and ∑ s∈Z 2 a s < ∞, then Assumptions P.4(v) with v = 1 2 holds. Example 5.2 (Linear Random Fields). For the linear random fields defined in (4.1) in Example 4orem 5.1.
Proof of Corollary 4.2. The results of Corollary 4.2 directly follow by the results of Theorem 4.1 and applications of Levy's continuity theorem as follows. Denotẽ7.2 Proof of Corollary 4.2
similarly. The following lemma investigates the asymptotic behavior of each term. Lemma 7.7. Under the Assumptions in Theorem 7.5, we have E max λ∈[0,2π] 2
26 )
26uniformly on λ ∈ [0, 2π] 2 .Proof of Lemma 7.7. Define τ = 2 + δ
2 . We have
E max
λ∈[0,2π] 2
With this definition, we getNow, we can prove the assertion in Theorem 6.1. For the proof of (a), recall that by the assumption K(x) ≥ 0, we have supBy the Cauchy-Schwarz inequality,which yields the assertion in part (a).It yields the assertion in part (b)(ii). For the proof of (b)(iii), similarly,From this we get, bywhich yields the assertion in part (b)(iii). Next, we have that the following holds:which yields the assertion in part (b)(iv). For the proof of (c)(i), we havewhich yields the assertion in part (c)(i). By the same arguments as for part (a), we get sup k∈N j∈Z 2which yields the assertion in part (c)(ii). Similar to the proof for part (b)(iii), we get sup k∈N j∈Z 2which yields sup k∈N j∈Z 2 p j,T (I 2 V (k + j) − I 2 V (k + j)) = sup k∈N j∈Z 2 p j,T [−(I V (k + j) − IV (k + j)) 2 + 2I V (k + j)(I V (k + j) − IV (k + j))]
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| []
|
[
"v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK",
"v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK"
]
| [
"Johannes Anschütz ",
"Ben Heuer ",
"Arthur-César Le Bras "
]
| []
| []
| We describe the category of continuous semilinear representations and their cohomology for the Galois group of a p-adic field K with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring B Sen in a geometric way.Notations. We will use the following notations. | null | [
"https://export.arxiv.org/pdf/2211.08470v2.pdf"
]
| 253,553,554 | 2211.08470 | 8448708c71c8761496d66a6b8237081aec7579a0 |
v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK
22 Dec 2022
Johannes Anschütz
Ben Heuer
Arthur-César Le Bras
v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK
22 Dec 2022
We describe the category of continuous semilinear representations and their cohomology for the Galois group of a p-adic field K with coefficients in a completed algebraic closure via vector bundles on the Hodge-Tate locus of the Cartier-Witt stack. This also gives a new perspective on classical Sen theory; for example it explains the appearance of an analogue of Colmez' period ring B Sen in a geometric way.Notations. We will use the following notations.
Introduction
Let K be a p-adic field, i.e., a mixed characteristic (0, p) complete discretely valued field with perfect residue field. Let C = K be the p-adic completion of an algebraic closure of K. The field C comes equipped with a continuous action of the Galois group G K := Gal(K/K) of K. Let Rep C (G K ) be the category of semilinear continuous G K -representations on finite dimensional C-vector spaces. The goal of this article is to give a new answer to the following classical question: 1.1. The perspective of Sen theory. In order to motivate Theorem 1.2 and explain its relevance, let us first recall that the first approach to answer Question 1.1 is provided by Sen theory: Let K ∞ be the (uncompleted) cyclotomic Z p -extension of K. Sen ( [Sen80]) constructs a functor S : Rep C (G K ) → finite dim. K ∞ -vector spaces W with an endomorphism θ : W → W .
This functor is not full, but on a certain full sub-category of "small" C-representations, it factors through a fully faithful functor S 0 into Sen modules over K, which is the category of pairs (M, θ), where M is a finite dimensional K-vector space and θ : M → M an endomorphism. As any object in Rep C (G K ) is in the domain of this functor after passing to a finite subextension K ⊆ K n ⊆ K ∞ , this defines the functor S in the colimit over n. One might try to descend this construction back from K n to K, but it is not true that the functor preserves Galois descent data: the dependence on the embedding K ⊆ C causes a subtle issue with functoriality of the construction.
Regarding essential surjectivity, if one tries to go into the other direction, from Sen modules to Galois representations, one can send a pair (M, θ) to the Galois representation on V = M ⊗ K C with σ ∈ G K acting via σ(m ⊗ c) = exp(log(χ(σ)) · θ) · m ⊗ σ(c) (χ is the cyclotomic character) assuming that the series defining exp(log(χ(σ)) · θ) converges for all σ ∈ G K . This defines a fully faithful functor from Sen modules satisfying this convergence condition to Rep C (G K ), which is a partial inverse to S 0 . But this does not capture all Sen modules.
When the residue field k of K is algebraically closed, Sen shows that it is still possible to extend S 0 to an equivalence
Rep C (G K ) ∼ − → {Sen modules}.
However, this equivalence is highly non-canonical as it depends on additional choices. For general K (i.e. when k is not algebraically closed, e.g. finite) the situation is even more subtle: one cannot expect even to have a fully faithful functor from Rep C (G K ) to Sen modules, as there are semilinear continuous C-representations V of G K for which End(V ) is a non-commutative division algebra (see [Sen80,Remark after Theorem 10] or [Fon04, Remarque 2 after Théorème 2.14]), but the endomorphism algebra End(M, θ) of a Sen module can never be of this form. But even in special cases where such a functor exists, one cannot expect to describe the essential image in a simple linear-algebraic way: for example, such a functor exists in the much simpler case of rank one, but we will show in Theorem 5.2 that it identifies the group of isomorphism classes of continuous semi-linear characters G K → C × with the kernel of a certain canonical surjection K → Br(K)[p ∞ ].
In summary, S allows us to characterise C-representations up to passage to a finite extension of K, or subject to a convergence condition, but in general does not describe the category Rep C (G K ). It therefore stops short of giving a complete answer to Question 1.1. Indeed, the relation between C-linear representations and Sen modules is not as simple as it may seem at first sight.
1.2. The perspective of the p-adic Simpson correspondence. To put the problem in perspective and analyze this issue, it is useful to recast the above setup in the much more general framework of p-adic non-abelian Hodge theory. Let X be an adic space over Spa(Q p ). The v-site of X is the category of all perfectoid spaces S over X endowed with Scholze's v-topology (generated by surjective morphisms between affinoid perfectoid spaces over X). It has a structure sheaf O, sending S to O S (S). The category of v-vector bundles Vec(X v , O) is defined as the category of finite locally free sheaves of O-modules on X v . If X = Spa(K), then the choice of K ⊆ C induces an equivalence Vec(X v , O) = Rep C (G K ) to the category studied by Sen theory. In particular, Question 1.1 can be recast as asking about the category of v-vector bundles on Spa(K).
If instead X is smooth over a complete algebraically closed extension L of Q p , then Vec(X v , O) is canonically equivalent to the category of generalized representations on X introduced by Faltings. It is a much richer category than the category of analytic vector bundles on X (which, in the example X = Spa(K) is nothing but the category of finite dimensional K-vector spaces). In general, p-adic non-abelian Hodge theory seeks to describe the category Vec(X v , O) in terms of Higgs bundles: let X be a smooth rigid space over a complete algebraically closed non-archimedean extension L of Q p . Then a Higgs bundle is a pair (E, θ), where E is an analytic vector bundle on X and θ : E → E ⊗ OX Ω 1 X (−1) is an O X -linear map such that θ ∧ θ = 0. In this setting, an instance of such a non-abelian Hodge correspondence is the "local p-adic Simpson correspondence", due to Faltings [Fal05] and studied further by Abbes-Gros-Tsuji [AGT16] and Wang [Wan21] among others: suppose that X admits an étale map (a toric chart )
c : X → T n L = Spa(L T ±1 1 , . . . , T ±1 n )
to the n-dimensional rigid torus over L for some n ≥ 1. Then any such chart induces an equivalence Here smallness is a technical condition which means that the considered v-vector bundle or Higgs bundles are p-adically close to the unit object. Towards the goal of describing v-vector bundles on X, the local p-adic Simpson correspondence has two serious drawbacks. First, it only relates small objects on both sides. As any smooth rigid space over L admits a toric chart étale-locally, and any v-vector bundle or Higgs bundle becomes small étale-locally, one might try to get rid of this smallness assumption (and of the condition on X) by an appropriate gluing procedure. However, and this is the second issue, the equivalence of categories is not functorial in X, but only in the pair (X, c) formed by X and the choice of a toric chart c. In fact, it is not true that one always has an equivalence between all v-vector bundles and all Higgs bundles on any smooth rigid space X over L (this fails already for the 2-dimensional unit ball, cf. [Heu22,§6]).
Formulated in this way, this approach to describing v-vector bundles on smooth X over L is entirely parallel to the arithmetic situation of X = Spa(K) in terms of Sen theory, with Higgs bundles (resp. small Higgs bundles) playing the role of Sen modules (resp. Sen modules satisfying the above convergence condition) 1 . In both cases, the statement in the "small" case is not functorial enough (it is only so after fixing a toric chart, or an embedding K → C) to be globalized for the étale topology, and thus a complete description of v-vector bundles on X via étale data is missing.
1.3.
A new perspective via the Hodge-Tate stack. The starting point of this article is the idea that the recent introduction of ring stacks in p-adic Hodge theory by Drinfeld [Dri20] and Bhatt-Lurie [BL22a], [BL22b] will help finding a more canonical formulation of such results and thereby shed new light on the (as yet still mostly conjectural) p-adic Simpson correspondence. The goal of this paper is to realize this concretely in the case X = Spa(K). Although this is quite a simple situation, it is already very much non-trivial as evidenced by the various subtleties of Sen theory, and we find it interesting that many features are already visible in this case. We regard this as a proof-of-concept for the general case.
Let us make this strategy precise in the case of interest. We will only need a small part of the work of Drinfeld and Bhatt-Lurie; we use the notations of [BL22a]. To any p-complete ring R, Bhatt-Lurie attach an fpqc stack Spf(R) HT , the Hodge-Tate stack of Spf(R), on the category of (commutative) rings in which p is nilpotent, with the property that, for R quasi-syntomic, the category D(Spf(R) HT ) of quasicoherent complexes on Spf(R) HT is naturally equivalent to the derived category of crystals of O ∆ -modules on the (absolute) prismatic site of Spf(R) 2 . If R is perfectoid, Spf(R) HT is canonically isomorphic to Spf(R). This applies for example to R = O C . As a consequence of this isomorphism, the natural morphism Spf(O C ) → Spf(O K ) lifts to a Galoisequivariant morphism
Spf(O C ) → Spf(O K ) HT .
Therefore, we obtain by pullback a canonical functor
α K : Perf (Spf(O K ) HT )[ 1 p ] → Perf (Spa(K) v )
from the isogeny category of the category of perfect complexes on the Hodge-Tate stack of Spf(O K ) towards v-perfect complexes on Spa(K).
On the other hand, any choice of a uniformizer π of K induces a Breuil-Kisin prism which induces a map
ρ π : Spf(O K ) → Spf(O K ) HT
that Bhatt-Lurie show is a faithfully flat torsor under a certain group G π . Considering the tangent action induces on the pullback of any module along ρ π a natural endomorphism Θ π . After inverting π, this defines a functor
β π : Perf (Spf(O K ) HT )[ 1 p ] → Perf (K[Θ π ]).
1 To make the analogy even more apparent, note that if ν * : Xv → Xet is the natural morphism of topoi, then for X smooth over L, Scholze proved in [Sch13] that there is an isomorphism R 1 ν * O = Ω 1 X (−1) , while for X = Spa(K) Tate's results on continuous cohomology [Tat67] imply that R 1 ν * O = O on Spa(K). So when Higgs bundles are defined in terms of R 1 ν * O, then Sen modules are Higgs bundles on Spa(K).
2 Hence, (a relative variant of) the Hodge-Tate stack can be thought of as an analogue for Hodge-Tate cohomology of p-adic formal schemes of Simpson's de Rham stack for de Rham cohomology of complex varieties.
The basic idea of our approach to non-abelian Hodge theory via the Cartier-Witt stack is now to study the diagram
Perf (Spf(O K ) HT )[ 1 p ] Perf (Spa(K) v ) Perf (K[Θ π ])
αK βπ Sen in which we note that the left map is completely canonical, whereas the right map depends on π even though neither the target nor source category do. Roughly, the idea is now that taking a preimage under α and mapping it down under β π should give an analogue of Sen's functor 3 . However, as is to be expected from the above discussion of Sen theory, the morphism α K is not essentially surjective, reflecting the fact that Sen is only partially defined. Instead, we have the following description of the above diagram, which is our second main result:
Theorem 1.3 (Theorem 4.2, Lemma 4.6, Corollary 2.15). Let E be the minimal polynomial of π over K 0 , the maximal unramified subfield of K. Let δ OK/Zp = (E ′ (π)) be the different of O K |Z p .
(1) The functor α K is fully faithful. Its essential image is the bounded derived category of nearly Hodge-Tate representations, i.e. semilinear continuous C-representations of G K whose Sen operator has all its eigenvalues in Z + δ −1 OK /Zp · m K . (2) The functor β π is fully faithful. Its essential image consists of those perfect complexes M of K[Θ π ]-modules for which H * (M ) is finite dimensional over K and Θ p π − E ′ (π) p−1 Θ π acts topologically nilpotently on H * (M ).
Equivalently, and in close parallel to (1), condition (2) means that on each cohomology group,
E ′ (π) −1 Θ π has all generalised eigenvalues in Z + δ −1 OK/Zp · m K . Moreover, in Theorem 2.5 we describe D(Spf(O K ) HT ) in terms of complexes of O K [Θ π ]-modules.
Remark 1.4. Related results were proved recently by Min-Wang [MW21][MW22] and Gao [Gao22] in terms of Hodge-Tate prismatic crystals, which are equivalent to vector bundles on Spf(O K ) HT by [BL22b,Proposition 8.15]. Their results only hold at the abelian level (i.e. for vector bundles instead of perfect complexes), but work more generally for smooth rigid spaces of good reduction over a p-adic field (instead of just X = Spa(K)), also using the prismatic formalism. While they use the prismatic site, we adopt the perspective furnished by Drinfeld and Bhatt-Lurie's stacks. For (2) we basically follow Bhatt-Lurie's arguments for K = Q p .
It follows that any object is in the essential image of β π after passing to a finite extension L|K. For Question 1.1, the crucial point is now that since α K is completely canonical and functorial, the subtleties that keep us from descending Sen's functor disappear for α K . We deduce:
Theorem 1.5. If L|K is a finite Galois extension, the functor α L induces a fully faithful functor
α L/K : Perf ([Spf(O L ) HT /Gal(L/K)])[ 1 p ] → Perf (Spa(K) v )
and any object in Perf (Spa(K) v ) lies in the image of α L/K for some L.
This allows us to compute Galois cohomology of semilinear G K -representations, i.e. cohomology of vector bundles on Spa(K) v : Let E ∈ Perf (Spf(O K ) HT ) and let V :
= α K (E), (M, θ M ) := β π (E). Then RΓ(X v , V ) = RΓ(Spf(O K ) HT , E)[ 1 p ] = fib(M θM − − → M )
. Theorem 1.5 is an easy consequence of Theorem 1.3.(1). The proof of Theorem 1.3 relies crucially on the geometry of Spf(O K ) HT . Namely, Bhatt-Lurie show that the map ρ π : Spf(O K ) → Spf(O K ) HT realizes Spf(O K ) HT as the (relative) classifying stack of an explicit group G π (which is either isomorphic to G ♯ m or to G ♯ a ). This gives a concrete description of perfect complexes on this stack, leading to Theorem 1.3.(2); in particular, let us point out that the (seemingly strange) nearly Hodge-Tate condition appears very naturally through the description of the Cartier dual of G π . Using this presentation, one can also unravel what the functor α K is doing in terms of the description provided by (2): let Z π → Spf(O C ) be the base change of ρ π along the map Spf(O C ) → Spf(O K ) HT mentioned above. Let A en = O(Z π ) be the ring of functions on Z π and let B en = A en [1/p]. The ring B en is endowed with an endomorphism Θ π and a commuting continuous C-semilinear G K -action. Then α K sends a complex E on Spf(O K ) HT with β π (E) = (M, θ M ) to
fib M ⊗ K B en 1⊗Θπ +θM ⊗1 − −−−−−−−− → M ⊗ K B en .
3 More precisely, it is an analogue for Sen's functor defined using the Kummer tower of π rather than the cyclotomic tower. These two constructions are equivalent up to a non-canonical transformation.
In other words, the ring B en functions as a period ring in this context. This ring is closely related to Colmez' ring B Sen (introduced by him in [Col94] to reformulate the construction of the Sen functor in the style of Fontaine) but is different and has not been considered earlier in the literature as far as we know. (Interestingly, it seems to be more easily related to a variant of Sen theory using the Kummer tower, rather than the cyclotomic tower as is usually done. Compare [Gao22].)
Let us remark at this point that our proof of the essential surjectivity aspect in Theorem 1.5.(1) uses Sen theory: we have nothing new to say regarding the (key) decompletion process in the theory. But we stress that α K is a canonical, geometrically defined, functor (which after a choice becomes identified with an explicit Fontaine-type functor). This is the main point of the Hodge-Tate stack approach and we believe that this point of view will give a fruitful new perspective on p-adic non-abelian Hodge theory in general. In work in progress, we investigate the case of smooth varieties over algebraically closed non-archimedean fields and p-adic fields by the same methods.
One might ask if one can use this description to obtain a more explicit description of the category Rep C (G K ) in terms of linear algebra data. In order to explore what such a description could look like, we investigate in the last section the rank one case. In dimension 1, Sen's division algebra counterexample discussed at the beginning of the introduction does not apply and one might hope to extend the fully faithful functor of point (1) of Theorem 1.3 to an equivalence between continuous semilinear C-representations of G K and rank one Sen modules. We prove that this is not possible, by showing that there are more objects on the side of Sen modules than on the Galois side. Rather, there is in general a canonical short exact sequence
0 → Pic v (K) → K → Br(K)[p ∞ ] → 0 where Br(K) = H 2 et (K, G m )
is the Brauer group. This means that already in the very simple case of rank 1, the category of representations of G K sees subtle arithmetic information of the p-adic field K like the p-primary part of the Brauer group (which we recall is related to local class field theory), which are therefore captured by the Hodge-Tate stacks in Theorem 1.2.
1.4. Plan of the paper. After some brief recollection on the Hodge-Tate stack (the Hodge-Tate locus in Bhatt-Lurie's Cartier-Witt stack), in Section 2.1, we prove (a more general version of) point (2) of Theorem 1.3 in Section 2.2. Section 3 contains some explicit computations describing the Galois action on B en (in particular a short discussion of its relation to Colmez' B Sen ) and its cohomology, which allow us to complete the proof of Theorem 1.3 in Section 4. Finally, in section Section 5 we compute the v-Picard group of p-adic fields. (1) p is a prime,
(2) K is a p-adic field (complete discretely valued with perfect residue field) with ring of integers O K and residue field k.
(3) Nilp p is the category of (classical, commutative) rings R such that p is nilpotent in R, (4) If X is a p-adic formal scheme, we denote by X ∆ (resp. X HT ) the Cartier-Witt stack of X (resp. the Hodge-Tate locus in the Cartier-Witt stack of X, or Hodge-Tate stack of X), which is denoted WCart X (resp. WCart HT X ) in [BL22a], [BL22b].
α → W (R) δ → W (R)
generates the unit ideal. Since p is nilpotent in R, Zariski-locally on Spec(R), I is principal generated by some element d and the above two conditions can be formulated concretely by saying that
α(d) = n V n [r n ],
with r 0 nilpotent and r 1 ∈ R × . The functor WCart in fact defines an fpqc stack on Nilp p . The
Hodge-Tate locus
WCart HT in the Cartier-Witt stack is defined to be the closed substack defined by the condition that, for R ∈ Nilp p , a Cartier-Witt divisor α : I → W (R) belongs to the full subcategory WCart HT (R) of WCart(R) if the composite I α → W (R) → R is zero (equivalently, in terms of the above explicit formula, if r 0 = 0).
If X is a bounded p-adic formal scheme, we define the Cartier-Witt stack of X, or prismatization of X, to be the groupoid-valued functor X ∆ on Nilp p sending R to the groupoid of pairs (α : I → W (R), η : Spec(W (R) := cone(α)) → X), where (α : I → W (R)) ∈ WCart(R) is a Cartier-Witt divisor and η is a map of derived formal schemes. The Hodge-Tate stack of X, which will be the main player in this text, is defined as the fiber product
X HT / / X ∆ WCart HT / / WCart. For R ∈ Nilp p , if (α : I → W (R)) ∈ WCart HT (R)
, the map α factors through V W (R) and hence one gets a map W (R) → W (R)/V W (R) = R. Therefore any point (α : I → W (R), η : Spec(W (R)) → X) ∈ X HT (R) gives rise to a map Spec(R) → Spec(W (R)) η → X. This defines a morphism X HT → X, called the Hodge-Tate structure morphism.
Remark 2.1. Note that we have WCart = Spf(Z p ) ∆ and WCart HT = Spf(Z p ) HT , as Spf(Z p ) is the terminal object in the category of p-adic formal schemes.
A fundamental feature of these stacks is their relation to prisms, which is given by the following simple construction ([BL22b, Construction 3.10]). Let X be a bounded p-adic formal scheme and (A, I) ∈ (X) ∆ an object of the absolute prismatic site of X. If R is an A-algebra in Nilp p , the A-algebra structure on R uniquely lifts to a δ-A-algebra structure on W (R). Base changing the inclusion I ⊂ A along A → W (R) gives a Cartier-Witt divisor α : I ⊗ A W (R) → W (R) together with a map η : Spf(W (R)) → Spf(A := A/I) → X (where the last map comes from the assumption that (A, I) ∈ (X) ∆ ), hence a point in X ∆ (R). Therefore, we get a map ρ X,A : Spf(A) → X ∆ (A is endowed with the (p, I)-adic topology). It induces a map
ρ X,A : Spf(A) → X HT .
Let us make this more explicit when X = Spf(O K ). Let π ∈ O K be a uniformizer. This choice yields the Breuil-Kisin prism
(A π , I π ) := (W (k)[[u]], (E(u))) with A π := A π /I π ∼ = O K , u → π, where E(u) ∈ W (k)[u] is the minimal polynomial of π ∈ K over K 0 := W (k)[1/p], and the δ-algebra structure is defined by δ(u) = 0, meaning ϕ(u) = u p . Set 4 e := E ′ (π) ∈ O K . We let ρ π = ρ Spf(OK ),Aπ : Spf(O K ) → Spf(O K ) HT be the map induced by (A π , I π ) introduced above. Concretely, an O K -algebra S is mapped to the image along Spf(O K ) HT (O K ) → Spf(O K ) HT (S) of the point (I π ⊗ Aπ W (O K ) → W (O K ), O K ∼ = A π /I π → W (O K ))
. 4 We note that usually e denotes the absolute ramification index of K, i.e., the degree of E(u). As this will not be relevant to us, we use the symbol e for something else.
Definition 2.2. Let G π be the group sheaf of automorphisms of the point ρ π , i.e. G π is the functor
S → Aut Spf(OK ) HT (S) (Spf(S) → Spf(O K ) ρπ −→ Spf(O K ) HT ) on p-complete O K -algebras.
By [BL22b,Example 9.6] the group G π identifies concretely with the subgroup
G π = {(t, a) ∈ G ♯ m ⋉ G ♯ a | t − 1 = e · a} (see [BL22a, §3.4] for the definition of G ♯ m and G ♯ a ). The action of G ♯ m (S) ∼ = W × [F ](S) on ρ π is via the natural W (S)-multiplication on I π ⊗ Aπ W (S) while the action of G ♯ a (S) ∼ = W [F ](S) is via homotopies on the morphism O K → W (S) of animated rings, cf. [BL22b, Construction 9.4].
It is useful to note that if S is p-torsion free, then the action of an element (t, a) ∈ G π (S) is uniquely determined by its action via the projection
G π → G ♯ m ∼ = W × [F ] on I π ⊗ Aπ W (S).
Moreover, under this torsion freeness assumption, an element (a 0 , a 1 , . . .) ∈ W × [F ](S) ⊆ W (S) is uniquely determined by a 0 . We now have:
Proposition 2.3 ([BL22b, Proposition 9.5]). The action of G π makes ρ π : Spf(O K ) → Spf(O K )
HT into a G π -torsor for the fpqc-topology. In particular, ρ π is affine and faithfully flat and induces an isomorphism
Spf(O K ) HT = BG π .
To describe G π even more concretely, note that the projection (t, a) → a yields an isomorphism
G π ∼ = G ♯ a = Spf(O K [ a n n! | n ≥ 0] ∧ p ) of formal schemes, such that the comultiplication on O Gπ identifies with the formal group law a + b + e · ab.
If e ∈ O × K , then G π ∼ = G ♯ m via the first projection, and if e ∈ m K := π · O K , then G π ∼ = G ♯ a given by the maps
G π → G ♯ a , a → log(ea + 1) e = n≥1
(−e) n−1 a n n and G ♯ a → G π , a → exp(ea) − 1 e = n≥1 e n−1 a n n! .
2.2.
From complexes on the Hodge-Tate stack to Sen modules. We keep the notations of the previous section. In this section, following very closely an argument of [BL22a], we want to give an explicit description of the ∞-category of quasicoherent complexes
D(Spf(O K ) HT ) on the Hodge-Tate stack Spf(O K ) HT of O K (cf. [BL22a][Definition 3.5.1])
. We stress that this description is not canonical but involves the choice of a uniformizer π ∈ O K . The choice of such a π ∈ O K yields a faithfully flat morphism
ρ π : Spf(O K ) → Spf(O K ) HT ,
whose automorphism group is G π as in Section 2.1. As in [BL22a, Construction 3.5.4] we can construct an automorphism of the projection
Spf(O K ) HT × Spf(OK ) Spf(O K [ε]/(ε 2 )) → Spf(O K ) HT yielding a Sen operator for O K . Namely, for E ∈ D(Spf(O K ) HT ) multiplication by the element 5 (1 + eε, ε) ∈ G π (R[ε]/(ε 2 )) yields an automorphism of E ⊗ OK O K [ε]/(ε 2 ),
which can be interpreted as a morphism Id+εΘ π,E :
E → E ⊗ OK O K [ε]/(ε 2 ) for some endomorphism Θ π,E : E → E in D(Spf(O K ) HT ).
The endomorphism Θ π,E is called the Sen operator of E. By a slight abuse of notation, we denote by Θ π,E also the pullback of Θ π,E along the map ρ π . The construction of Θ π,E is clearly functorial in E, thus pullback along ρ π defines a natural functor
β + π : D(Spf(O K ) HT ) → D(O K [Θ π ]), E → (ρ * π E, Θ π,E ). Example 2.4. The natural map h : Spf(O K ) HT → Spf(Z p ) HT ∼ = BG ♯ m is induced by the morphism of groups G π → G ♯ m , (t, a) → t.
From the constructions of the Sen operators for O K and Z p , respectively, we can conclude that if E ∈ D(Spf(Z p ) HT ) with Sen operator Θ E : E → E, then h * E has Sen operator Θ π = e · h * Θ. HT {n}, whose associated Sen operator is given by e · n, cf. [BL22a, Example 3.5.6]. We stress that O Spf(OK ) HT {n} is canonically defined, but that its associated Sen operator depends on the choice of π because e depends on π.
O Spf(OK ) HT {n} := h * O Spf(Zp)
As in [BL22a, Example 3.5.5] the Sen operator satisfies the Leibniz rule for tensor products in
D(Spf(O K ) HT ).
The main aim of this section is to prove the following analog of [BL22a, Theorem 3.5.8].
Theorem 2.5. The functor
β + π : D(Spf(O K ) HT ) → D(O K [Θ π ]), E → (ρ * π E, Θ π,E )
is fully faithful and its essential image consists of complexes M ∈ D(O K [Θ π ]) which are (derived) π-adically complete and such that the action of Θ p π − e p−1 Θ π on the cohomology H * (k ⊗ L OK M ) is locally nilpotent.
We note that the functor depends on the choice of the uniformizer
π ∈ O K . If e ∈ O K is not a unit, then Θ p π − e p−1 Θ π = Θ p π on H * (k ⊗ L OK M )
, which is in accordance with the description of representations of G ♯ a as finite free O K -modules with a topologically nilpotent endomorphism. For the proof of Theorem 2.5 we will follow the arguments of [BL22a, Theorem 3.5.8].
Lemma 2.6. Set E := ρ π, * O Spf(OK) . Then (1) ρ * π E ∼ = O Gπ = n≥0 O K · a n n! (where on the right we write for simplicity O K for what should be denoted O Spf(OK ) ), (2) Θ π,E = (1 + ea) ∂ ∂a , (3) the sequence 0 → O K → ρ * π E Θπ,E − −− → ρ * π E → 0 is exact. Proof. The isomorphism ρ * π E ∼ = O Gπ follows from the projection formula. The description of O Gπ in terms of n≥0 O K · a n
n! follows if we identify G π ∼ = G ♯ a (as formal schemes) via the map (t, a) → a. Under the isomorphism G π ∼ = G ♯ a , which transfers the comultiplication to the map
c : O G ♯ a → O G ♯ a ⊗ OK O G ♯ a , a → a + b + e · ab,
the Sen operator Θ π,E is constructed using the action of the element ε ∈ G ♯ a (O K [ε]/(ε 2 )). In other words, we have to look at the composition
O G ♯ a c − → O G ♯ a ⊗ OK O G ♯ a Id⊗(a →ε) −−−−−−→ O G ♯ a ⊗ OK O K [ε]/(ε 2 ), which sends some f (a) ∈ O G ♯ a to the element f (a + (1 + ea)ε) = f (a) + ε(1 + ea) ∂f ∂a (a) as desired. Let f (a) = n≥0 c n a n n! ∈ O G ♯ a . Then Θ π,E (f ) = n≥1 c n (en a n n! + a n−1 (n − 1)! ) = n≥0 (c n+1 + enc n ) a n n! .
Thus, if Θ π,E (f ) = 0, then c n+1 = −enc n for all n ≥ 0, which implies c n+1 = 0 for all n ≥ 0, i.e., f ∈ O K . Given an element g(a) = n≥0 b n a n n! , then we can inductively solve the system b n = c n+1 + enc n by starting with
c 0 := 0 c 1 := b 0 c 2 := b 1 − e · 1 · c 1 = b 1 − eb 0 . . . c n+1 := b n − enc n = . . . = b n − . . . − (−e) n−1 n!b 0 .
We see that if for the p-adic topology b n → 0, n → ∞, then also c n → 0, n → ∞. This finishes the proof of the lemma.
Similarly to [BL22a, Proposition 3.5.11] we can draw the following consequence.
Proposition 2.7. For any E ∈ D(Spf(O K ) HT ) there exists a canonical fiber sequence
RΓ(Spf(O K ) HT , E) → β + π (E) Θπ,E − −− → β + π (EO Spf(OK ) HT → ρ π, * ρ * π O Spf(OK ) HT = ρ π, * O Spf(OK ) induces a natural map O Spf(OK ) HT → fib(ρ π, * (O Spf(OK ) ) Θπ,ρ π, * O Spf(O K ) − −−−−−−−−−− → ρ π, * (O Spf(OK ) )),
which by faithfully flat descent along ρ π and Lemma 2.6 is an isomorphism. Tensoring the resulting fiber sequence with E yields a fiber sequence
E → ρ π, * (ρ * π (E)) α − → ρ π, * (ρ * π (E))
. From here we apply the functor RΓ(Spf(O K ) HT , −) to get a fiber sequence
RΓ(Spf(O K ) HT , E) → β + π (E) α − → β + π (E) and one checks that α = Θ π,E .
Remark 2.8. Given Theorem 2.5 the same argument as in [BL22a, Corollary 3.5.14] characterizes
objects isomorphic to O Spf(OK ) HT {n} ∈ D(Spf(O K ) HT ) as those E ∈ D(Spf(O K ) HT ) such that β + π (E) ∼ = O K and Θ π,E = e · n.O Gπ = i≥0 O K · a i i! given by F ≤n := n i=0 O K · a i i! , the formula
(1 + ea) ∂ ∂a a n n! ≡ en a n n! mod F ≤n−1 We can now prove Theorem 2.5.
Proof of Theorem 2.5. Given Proposition 2.9, well-definedness and fully faithfulness follow as in [BL22a, Theorem 3.5.8]. The condition that Θ p e − e p−1 Θ e is locally nilpotent on the cohomology modulo π appears as (en) p − e p−1 (en) = e p (n p − n) is zero modulo p for any n ∈ Z. To show essential surjectivity let M ∈ D(O K [Θ π ]) be an object satisfying the two conditions in Theorem 2.5.
Let O K {n} ∈ D(O K [Θ π ]) be the image of O Spf(OK ) HT {n}. If M is non-zero, we have to cook up a non-zero morphism O K {n}[m] → M for some n, m ∈ Z, n ≥ 0. As RΓ(Spf(O K ) HT , (O K {−n}[−m] ⊗ OK M ) ⊗ L OK k) ∼ = RΓ(Spf(O K ) HT , O K {−n}[−m] ⊗ OK M ) ⊗ L OK k and RΓ(Spf(O K ) HT , O K {−n}[−m] ⊗ M ) is p-complete (e.g., by Proposition 2.7) it suffices to replace M by M ⊗ L OK k (this uses p-completeness of M to ensure that M ⊗ L OK k = 0 if M = 0). Then Θ p π − e p−1 Θ π = p−1 i=0 (Θ π − e · i)Spf(lim ← − n O K [u]/ n i=0 (u − e · i)).
Remark 2.12. Theorem 2.5 could also be deduced from Remark 2.11 and Cartier duality (switching to modules over the Cartier dual of G π ) and the observation we already made that
u p − e p−1 u = p−1 i=0 (u − e · i) in k[u].
For perfect complexes we get the following version of Theorem 2.5.
Lemma 2.13. The functor β + π from Theorem 2.5 restricts to a fully faithful functor
β + π : Perf (Spf(O K ) HT ) → Perf (O K [Θ π ]) whose essential image consists of π-adically complete perfect complexes M over O K [Θ π ] for which Θ p π − e p−1 Θ π is nilpotent on H * (k ⊗ L OK M ). The perfect O K [Θ π ]-module M = O K [Θ π ]/(πΘ π − 1)-module is zero modulo π.
Hence, the π-completeness assumption on M is necessary.
Proof. By regularity of O K [Θ π ], a complex of O K [Θ π ]-modules is perfect if its underlying complex of O K -modules is perfect. This shows that β + π sends Perf (Spf(O K ) HT ) to Perf (O K [Θ π ])
. Then fully faithfulness follows from Theorem 2.5.
Conversely, let M ∈ Perf (O K [Θ π ]) be π-adically complete perfect complex such that Θ p π − e p−1 Θ π is nilpotent on H * (k ⊗ L OK M )
. We need to see that M is perfect as a complex of O Kmodules. It suffices to check this for M ⊗ L OK O K /π n for all n ≥ 0. Indeed, as O K is π-adically complete this implies that M is perfect as a complex of O K -modules. As canonical truncations of M are again perfect O K [Θ π ]-modules, this reduces to the case that M is concentrated in a single degree. Using dévissage, we reduce to the case that πM = 0. But then M is a finitely generated
O K [Θ π ]/(Θ p π − e p−1 Θ π ) i -module for some i ≥ 0 and hence perfect as an O K -module. Let S be a p-complete O K -algebra and g = (s, b) ∈ G π (S) an S-valued point of G π . As G π (S)
is commutative, the multiplication by g will induce an automorphism
γ g,M : M ⊗ OK S → M ⊗ OK S on any pair (M, Θ π ) ∈ D(BG π )
. The next lemma makes this action explicit, and also sheds more light on the condition on Θ π in Theorem 2.5. The formula also occurs in [MW21, Theorem 1.3].
Lemma 2.14. In the above notation, we have the equality
γ g,M = (1 + e · b) Θπ/e := ∞ n=0 b n n! n−1 i=0 (Θ π − e · i)
as endomorphisms of each cohomology module of M ⊗ OK S.
We recall that there exist the formal equalities
exp(x · log(1 + y)) =(1 + y) x = n≥0 x n y n = n≥0 y n n! n−1 i=0 (x − i) of power series in Q[[x, y]].
Proof. For simplicity of notation, we assume O K = S. The general case is similar. Using a reduction to M concentrated in degree 0 and the natural morphism M → ρ π, * (ρ * π M ), it suffices to check the statement in the case that
M = O Gπ with Θ π = (1+e·a) ∂ ∂a . Let f (a) ∈ O Gπ = n≥0 O K a n n! . Then γ g,OG π (f )(a) = f (a + b + e · ab)
. We write t := 1 + e · a and h(t) = f ( t−1 e ). Then γ g,OG π (f )(a) = h(s · t) with s = 1 + e · b. From the proof of [BL22a, Proposition 3.7.1] we get
h(s · t) = n≥0 log(s) n n! (t ∂ ∂t ) n h(t) = exp( Θ π e log(s))h(t)
from which we deduce the formal equality of power series in a
(1) f (a + b + e · ab) = (1 + e · b) Θπ/e f (a).
From the formal equality
(2) (1 + y) x = n≥0 y n n! n−1 i=0 (x − i)
mentioned before we conclude that (Equation (1)) converges in O Gπ by the convergence condition on Θ π . Namely, setting y = e · b and x = Θ π /e we get
(1 + e · b) Θπ/e = n≥0 e n b n n! n−1 i=0 (Θ π /e − i) = n≥0 b n n! n−1 i=0 (Θ π − i · e)
and n−1 i=0 (Θ π − i · e) converges to 0 if n → ∞.
2.
3. An analytic variant. Finally, when we restrict attention to perfect complexes, we deduce an "analytic" variant of the functor β + π , which will be more closely related to v-vector bundles: Namely, by formally inverting p on the source, β + π defines a functor β π :
Perf (Spf(O K ) HT )[ 1 p ] → Perf (K[Θ π ]).
Corollary 2.15. The functor β π is fully faithful. Its essential image consists of complexes M ∈ Perf (K[Θ π ]) such that H * (M ) is finite dimensional over K and the action of Θ p π − e p−1 Θ π on H * (M ) is topologically nilpotent.
Here, the cohomology H * (M ) has its canonical topology as a finite dimensional K-vector space.
Proof. By Lemma 2.13 we can identify Perf (Spf(O K ) HT ) with the full subcategory of Perf (O K [Θ π ]) given by π-complete objects M such that Θ p π − e p−1 Θ π is nilpotent on H
* (k ⊗ L OK M ). The natural functor Perf (O K [Θ π ])[ 1 p ] → Perf (K[Θ π ])
is fully faithful because perfect complexes are compact objects. This establishes fully faithfulness of β π .
To see the description of the essential image, by induction on the amplitude and via considering cones, we can reduce to the case that M is concentrated in degree 0. Then Θ π : M → M is an endomorphism of a finite dimensional K-vector space whose characteristic polynomial has coefficients in O K . This implies that there exists a Θ π -stable O K -lattice M 0 in M . Indeed, if L/K is a finite extension and N 0 ⊆ M ⊗ K L a Θ π -stable O L -lattice, then M 0 := N 0 ∩ M is a Θ π -stable O K -lattice in M . Hence, for the existence of M 0 we may enlarge K. Then Θ π can be assumed to have Jordan normal form, in which case the existence of M 0 is clear as all eigenvalues lie in O K by integrality of the characteristic polynomial. Now any Θ π -stable O K -lattice M 0 in M is π-adically complete and the Θ p π − e p−1 Θ π -action on M 0 /π is nilpotent by the assumed topological nilpotence. By Lemma 2.13 this implies that M lies in the essential image as desired.
Galois actions on Hodge-Tate stacks
Let C = K be the completion of an algebraic closure of K. Let In this section, we would like to analyze explicitly the action of Gal(K/K) on the ring of functions O(Z π ) under some additional choices. We do so in Section 3.1 in a slightly more general context. Using this description, we will relate in Section 3.2 the ring O(Z π )[1/p] to other more familiar period rings from p-adic Hodge theory (most notably the ring B Sen previously introduced by Colmez) and compute its Galois cohomology in Section 3.3.
3.1. Explicit functoriality. Let K, K ′ be p-adic fields with rings of integers R := O K , R ′ := O K ′ and (perfect) residue fields k, k ′ . We assume that K → C, K ′ → C ′ are two extensions of nonarchimedean fields with C, C ′ algebraically closed. Moreover, we let σ : C → C ′ be a continuous homomorphism, which we require to induce a homomorphism τ :
K → K ′ . Let f : Spf(O C ) → Spf(R), f ′ : Spf(O C ′ ) → Spf(R ′ )
denote the natural morphisms. As recalled above in the particular case C = K, the natural map Example 3.12]) and thus the map f lifts naturally to a mapf :
Spf(O C ) HT → Spf(O C ) is an isomorphism ([BL22b,Spf(O C ) → Spf(R) HT over Spf(R). Similarly, f ′ lifts naturally to a mapf ′ : Spf(O C ′ ) → Spf(R ′ ) HT . Explicitly, if S is a p-adically complete R-algebra, (A inf , J) the perfect prism associated to O C , thenf maps a morphism g : O C → S to the image of the O C -point J ⊗ A inf W (O C ) → W (O C ), R → O C ∼ = A inf /J → W (O C ) ∈ Spf(R) HT (O C ) along the map Spf(R) HT (O C ) → Spf(R) HT (S) induced by g.
Similarly the mapf ′ can be described using the perfect prism (A ′ inf , J ′ ) associated with O C ′ . From the naturality of the Hodge-Tate stack we deduce that there exists a natural 2-commutative diagram
Spf(O C ′ ) σ / / f ′ Spf(O C ) f Spf(R ′ ) HT τ / / Spf(R) HT , or more precisely a natural isomorphism ι σ :f • σ ∼ = τ •f ′ between two points in the groupoid Spf(R) HT (O C ′ ) ∼ = Mor R (Spf(O C ′ ), Spf(R) HT ).
Setπ := τ (π) ∈ R ′ and assume that this element is a uniformizer in R ′ . We get two prisms
(A π , I π ) = (W (k)[[u]], (E π (u))) and (A ′π , I ′ π ) = (W (k ′ )[[u ′ ]]
, (Eπ(u ′ ))) lifting R and R ′ respectively via the morphisms
A π = W (k)[[u]] → R, u → π, A ′π = W (k ′ )[[u ′ ]] → R ′ , u ′ →π.
As explained in Section 2.1, these choices give rise to two maps ρ π : Spf(R) → Spf(R) HT , ρ ′π : Spf(R ′ ) → Spf(R ′ ) HT with group sheaves of automorphisms
G π = {(t, a) ∈ G ♯ m ⋉ G ♯ a | t = 1 + e · a}, Gπ = {(t, a) ∈ G ♯ m ⋉ G ♯ a | t = 1 +ẽ · a}, where e := E ′ π (π),ẽ := E ′ π (π)
. Let τ 0 : W (k) → W (k ′ ) be the homomorphism induced by τ . Then τ 0 extends to a homomorphism τ A : A π → A ′π by sending u to u ′ . This yields a homomorphism (A π , I π ) → (A ′π , I ′ π ) of prisms, which reduces to the homomorphism τ : R ∼ = A π /I π → A ′π /I ′ π ∼ = R ′ . By naturality of [BL22b, Construction 3.10] we obtain a 2-commutative diagram
Spf(R ′ ) τ / / ρ ′ π Spf(R) ρπ Spf(R ′ ) HT τ / / Spf(R) HT .
To describe the implicit isomorphism in Spf(R) HT (R ′ ), note that the diagram
A π / / τA W (R) W (τ ) A ′π / / W (R ′ )
commutes and that the natural map I π ⊗ Aπ A ′π → I ′ π is an isomorphism. From here one sees that the implicit isomorphism is induced by the isomorphism
I π ⊗ Aπ W (R) ⊗ W (R) W (R ′ ) → Iπ ⊗ Aπ W (R ′ ), i ⊗ x ⊗ y → τ A (i) ⊗ W (τ )(x)y.
As in the beginning of Section 3, we define Z π → Spf(O C ) by the fiber product diagram
Z π / / Spf(O C ) f Spf(R) ρπ / / Spf(R) HT .
By faithfully flat descent for G π -torsors
Z π ∼ = Spf(A en ) for some O C -algebra A en with G π -action. Similarly, we can define Z ′ π = Spf(A ′ en ). From the 2-commutative diagram Spf(R ′ ) Spf(R ′ ) HT Spf(O C ′ ) Spf(R) Spf(R) HT Spf(O C ) ρ ′ π τf ′ σ ρπf
we deduce from the universal property of the fiber product a natural map σ Z : Z ′ π → Z π Our aim is to make σ Z more explicit. For this we first have to construct a suitable trivialization of the G π -torsor Z π → Spf(O C ) and the Gπ-torsor Z ′ π → Spf(O C ′ ). If π ♭ = (π, π 1/p , . . .) ∈ O ♭ C is a compatible system of p-power roots of π, then the embedding ι π ♭ : A π → A inf , u → [π ♭ ] extends to a morphism of prisms (A π , I π ) → (A inf , J) and yields an isomorphism γ π ♭ in Spf(R) HT (O C ) between the points corresponding tof and the composition Spf
(O C ) → Spf(R) ρπ −→ Spf(R) HT . Concretely, γ π ♭ is the isomorphism of objects in Spf(R) HT (O C ) I π ⊗ Aπ W (O C ) W (O C ) , R A π /I π W (O C ) J ⊗ A inf W (O C ) W (O C ) , R O C ∼ = A inf /J W (O C ) induced by the W (O C )-linear isomorphism I π ⊗ Aπ W (O C ) ∼ = J ⊗ A inf W (O C ), i ⊗ x → ι π ♭ (i) ⊗ x.
Similarly, a choiceπ ♭ = (π,π 1/p , . . .) ∈ O C ′♭ of p-power compatible p-power roots ofπ in O C ′ yields an isomorphism betweenf ′ and the composition
Spf(O C ′ ) → Spf(R ′ ) ρ ′ π −→ Spf(R ′ ) HT .
We note that a possible such choice is given by σ(π ♭ ), but we explicitly want to allow greater freedom for this choice to later compute Galois actions.
From here we see that γ π ♭ , γπ♭ induce isomorphisms
γ π ♭ : G π,OC := G π × Spf(R) Spf(O C ) ≃ − → Z π and γπ♭ : Gπ ,O C ′ := Gπ × Spf(R ′ ) Spf(O C ′ ) ≃ − → Z ′ π .
We can conjugate the map σ Z : Z ′ π → Z π to get the composition
Gπ ,O C ′ γπ ♭ − − → ∼ Z ′ π σZ − − → Z π γ −1 π ♭ −−→ ∼ G π,OC ,
which lies over the morphism σ : Spf(O C ′ ) → Spf(O C ). Base changing along σ yields a map
σ G : Gπ ,O C ′ → σ * G π,OC := G π × Spf(R) Spf(O C ′ ) (we write σ π ♭ ,π ♭ G
if we want to stress its dependence on the choices of π ♭ andπ ♭ ). We now choose a compatible system
ε = (1, ζ p , . . .) ∈ O ♭ C ′
of p-power primitive roots of unity. Then we can write
σ([π ♭ ]) = [ε] c(σ) [π ♭ ] ∈ A ′ inf
for a unique element c(σ) ∈ Z p . Indeed, this follows as σ(π) = τ (π) =π. Let us note that τ A (E π (u)) = Eπ(u ′ ).
We furthermore introduce the elements
µ := [ε] − 1 ∈ A ′ inf and z :=π · θ ′ ( µ Eπ([π ♭ ]) ) ∈ O C ′ (here θ ′ : A ′ inf → O C ′ is Fontaine's map). Let us note that if ξ := µ ϕ −1 (µ) = 1 + [ε 1 p ] + . . . + [ε p−1 p ], then ξ Eπ([π ♭ ]) ∈ A ′
inf is a unit and hence also θ ′ ( ξ Eπ([π ♭ ]) ) ∈ O C ′ is a unit. Therefore
z =π · θ ′ ( ξ Eπ([π ♭ ]) )θ ′ (ϕ −1 (µ)) ∈π · (ζ p − 1)O C .
In particular, z ∈ G ♯ a (O C ′ ). The main result of this subsection is the following statement.
Proposition 3.1. The map σ G : Gπ ,O C ′ → σ * G π,OC
is given by the map (t, a) → ((1 +ẽ · z · c(σ)) −1 · t, 1 1 +ẽzc(σ)ẽ e (a − z · c(σ)).
Proof. By flatness of both formal schemes over
Spf(O C ′ ) it suffices to identify σ G on O C ′ -algebras S which are p-torsion free. In particular, a point (s, b) ∈ σ * G π,OC (S) ⊆ G ♯ m (S) ⋉ G ♯ a (S)
is already determined by s as the equation s = 1 + e · b ∈ S holds. Now, assume that (t, a) ∈ Gπ ,OC (S) is a given point, and let g ∈ W × [F ](S) ∼ = G ♯ m (S) be the element corresponding to t (in particular, g = (t, . . . , ) in Witt coordinates). Now σ G (t, a) is by definition the element in σ * G π,OC determined by the isomorphism of the two outer compositions in the 2-commutative diagram
Spf(S) Spf(R ′ ) Spf(R ′ ) HT Spf(R) Spf(O C ′ ) Spf(R ′ ) Spf(R ′ ) HT Spf(O C ′ ) Spf(R) Spf(R) HT Spf(O C ) Spf(R) Spf(R) HT Spf(R) Spf(O C ) ρπ ρπ ρπf σ τ τ ρ ′ πf ′ ρ ′ π ρ ′ π
where the isomorphism of the top roof is defined by g, the isomorphism for the upper right square is given by γπ♭ and the one for the lower right by γ π ♭ . For the other squares the implicit isomorphisms are obtained by functoriality of the Hodge- Tate
I π ⊗ Aπ W (O C ) ⊗ W (OC ) W (O C ′ ) τA⊗Id W (O C ′ ) − −−−−−−−− → I ′ π ⊗ A ′ π W (O C ′ ) ·g − → I ′ π ⊗ A ′ π W (O C ′ ) γπ ♭ − − → J ′ ⊗ A ′ inf W (O C ′ ) and I π ⊗ Aπ W (O C ) ⊗ W (OC ) W (O C ′ ) ·h − → I π ⊗ Aπ W (O C ) ⊗ W (OC ) W (O C ′ ) γ π ♭ ⊗Id W (O C ′ ) −−−−−−−−−→ J ⊗ A inf W (O C ) ⊗ W (OC ) W (O C ′ ) σ − → J ′ ⊗ A ′ inf W (O C ′ ) agree. Evaluating both morphisms on the element E π (u) ⊗ 1 ⊗ 1 yields the equality (in J ′ ⊗ A ′ inf W (O C ′ )) Eπ([π ♭ ]) ⊗ g = σ(E π ([π ♭ ])) ⊗ h. Now, σ(E π ([π ♭ ])) = Eπ(σ([π ♭ ])), which implies that h = Eπ([π ♭ ]) Eπ(σ([π ♭ ])) · g, where Eπ([π ♭ ]) Eπ(σ([π ♭ ])) ∈ A ′
inf is a unit and the multiplication with it refers to the A ′ inf -algebra structure of W (S). By Lemma 3.2 below, we can conclude that
(1 +ẽ · z · c(σ)) −1 · t = s. From this we can conclude by a small calculation that b = 1 1 +ẽzc(σ)ẽ e (a − z · c(σ)) as desired.
Lemma 3.2. Under the map θ ′ : A ′ inf → O C ′ the element Eπ(σ([π ♭ ])) Eπ([π ♭ ]) ∈ A ′ inf is mapped to the element 1 +ẽ · z · c(σ) ∈ O C . Proof. We write σ([π ♭ ]) = [ε] c(σ) [π ♭ ] and µ := [ε] − 1. Recall that ker(A ′ inf → W (O C ′ )) = µ · A ′ inf , cf. [BMS18, Lemma 3.23]. Now Eπ(σ([π ♭ ])) = Eπ([π ♭ ](1 + µ) c(σ) ) = Eπ([π ♭ ](1 + c(σ)µ)) + x · µ 2 = Eπ([π ♭ ]) + E ′ π ([π ♭ ])[π ♭
]c(σ)µ + y · µ 2 for some x, y ∈ A ′ inf . As µ is divisible by Eπ([π ♭ ]), we can divide this equation by Eπ([π ♭ ]) and then apply θ ′ . We conclude that
θ ′ ( Eπ(σ([π ♭ ])) Eπ([π ♭ ]) ) = 1 + E ′ π (π)θ ′ ( µ Eπ([π ♭ ])
)πc(σ)
because θ ′ (µ) = 0. By definition of z we can conclude the lemma.
We now specialize to the case that K = K ′ , C = C ′ = K and τ = Id K . In this case, let G K := Gal(K/K) be the Galois group of K, which acts on C by continuous automorphisms. As the morphism f : Spf(O C ) → Spf(O K ) HT is G K -equivariant (by naturality of the Hodge-Tate stack), the G π -torsor Z π → Spf(O C ) aquires an action of G K . From Proposition 3.1 we can deduce an explicit description of this action. To formulate it, let us first make a definition.
Definition 3.3. We set
χ π ♭ : G K → G ♯ m (O C ) = 1 + p α · O C , σ → 1 + e · z · c(σ) = θ σ(E π ([π ♭ ])) E π ([π ♭ ]) ,
(here α = 1 p−1 if p = 2 and α = 1/2 if p = 2 defines the convergence radius for the exponential) where the last equality follows from Lemma 3.2.
We recall that the choice π ♭ of p-power compatible roots of π yields an isomorphism γ π ♭ : G π,OC ∼ = Z π . As a corollary of the previous proposition, we obtain:
Corollary 3.4. The isomorphism γ π ♭ : G π,OC → Z π is equivariant for the G K -action if σ ∈ G K acts on G π,OC by
G π,OC ·dσ − − → G π,OC IdG π ×σ − −−−− → G π,OC , where d σ := (χ π ♭ (σ), z · c(σ)) −1 .
The map d : G K → G π (O C ), σ → d σ is a 1-cocycle, and we can formulate Corollary 3.4 by saying that the G K -action on G π,OC induced by the natural action on Z π via transport of structure through γ π ♭ : G π,OC → Z π is given by twisting the usual action on coefficients by d.
Proof. This follows by settingπ ♭ = π ♭ in Proposition 3.1, as in this caseẽ = e.
Remark 3.5. The object Z π is also implicit in the work of Min-Wang when they consider the coproduct of the prisms (A π , I π ) and A inf (O C ) in the prismatic site of O K , see [MW22, Lemma 2.11 and §5]. But it plays a different role in their work, as they use it to study τ -connections. For us, O(Z π ) will play the role of a period ring in the style of Fontaine, as we explain in the following.
3.2. Comparison to Colmez' ring B Sen . We will continue with the notation from Section 3.1 in the situation that K = K ′ , C = C ′ = K, τ = Id K ,π ♭ = π ♭ . Let A en := O(G π,OC ) and set B en := A en [ 1 p ]. The map G π,OC → G ♯ a , (t, a) → a is an isomorphism of formal schemes. In particular, we have an identification
A en ∼ = n≥0 O C · a n n! .
As shown in Corollary 3.4, on the element t := 1 + e · a ∈ A en the G K -action is given by
σ(t) = χ π ♭ (σ) −1 · t,
with χ π ♭ (σ) ∈ 1 + p α · O C as defined in Definition 3.3. Hence the 1-cocycle χ π ♭ plays a similar role in our setting as the cyclotomic character χ : G K → Z × p plays in p-adic Hodge theory via the cyclotomic tower (in the sense that on the usual element t = [ε] − 1, the action of G K is via χ).
To make this analogy more precise, let us first introduce a "cyclotomic" version of B en by setting
B cycl en = B cycl en,K := n≥0 O C · 1 n! t ′ − 1 e n [ 1 p ].
We note that this Banach algebra over C depends on K, but not on the choice of π. Indeed, the implicit orthonormal basis ( 1 n! t ′ −1 e n ) n∈N depends on e = E ′ π (π), but the ideal e · O K is the different of K over W (k)[1/p] and thus independent of π.
The C-Banach algebra B cycl en admits a (continuous) Galois action by G K via σ(t ′ ) := χ(σ)t ′ , where χ denotes the cyclotomic character.
We now wish to compare B en and B cycl en , so we need to find an appropriate coordinate transformation from t to t ′ that transforms the 1-cocycle χ π ♭ to the cyclotomic character. For this we use:
Lemma 3.6. For any σ ∈ G K , σ(z) z = χ(σ)χ π ♭ (σ) −1 .
Here, the element z = πθ( µ Eπ([π ♭ ]) ) was introduced before Proposition 3.1. Proof. Since π is fixed by the action of σ, we have σ(z)/z = σ(z 0 )/z 0 for z 0 := θ( µ Eπ([π ♭ ]) ). Observe now that
σ(z) z · χ π ♭ (σ) = θ σ( µ Eπ([π ♭ ]) ) µ Eπ([π ♭ ]) · θ σ(E π ([π ♭ ])) E π ([π ♭ ]) = θ σ(µ) µ = χ(σ)
as desired.
Focusing just on the Galois action, it therefore looks promising to send
t = 1 + e · a → z t ′ since σ( z t ′ ) = zχ(σ)χ π ♭ (σ) −1 χ(σ)t ′ = χ π ♭ (σ) −z t ′ ∈ 1 + e · p α O C ⊆ 1 + p α O C for all t ′ ∈ G ♯ m (O C ) = 1 + p α O C
where as before, α is the convergence radius of the exponential (α = 1 p−1 or α = 1 2 if p = 2). This is equivalent to requiring
z ∈ 1 + p α O C ,
which is not necessarily true: indeed, specializing z along the map O C → k, we can see that z need not lie in 1 + m C if the constant term of E π is not p. This raises the question whether one can always modify z so that it is of the desired form: Suppose that we have a different z ′ ∈ C × for which σ(z ′ )/z ′ = χ(σ)χ π ♭ (σ) −1 as in Lemma 3.6, but also z ′ ∈ 1 + ep α O C . Then we have σ(z/z ′ ) = z/z ′ , so z ′ = bz for some b ∈ K × . This shows that z ′ is of the desired form if and only if
z ′ ∈ (1 + ep α O C ) ∩ (z · K × ).
Such an element might not exist, but due to the fact that K ⊆ C × is dense, we can always find such a z ′ after replacing K by a finite extension K ′ . Crucially, however, there is in this case not a unique element z ′ , but rather there is an ambiguity by a factor 1 + ep α O C ∩ K ′ . In summary:
Proposition 3.7. There exists z ′ ∈ C × such that there is an O C -linear continuous isomorphism
B en ∼ − → B cycl en , t = 1 + e · a → z ′ t ′
which is G K ′ -equivariant for some finite extension K ′ |K. The element z ′ is in general noncanonical and only well-defined up to a factor in 1 + e · p α O K ′ .
Proof. This follows from our above discussion.
Remark 3.8. There is a slightly different perspective on this which also allows us to make K ′ slightly more precise: Consider χ · χ −1 π ♭ : G K → C × as a continuous 1-cocycle which defines a trivial class in H 1 cts (G K , C × ) by Lemma 3.6. For any k ∈ N, there is m ∈ N (by continuity) such that for K ′ := K(π 1/p m , ζ p m ), this cocycle restricts on G K ′ to a cocycle with image in 1 + p k O C . Consider the commutative diagram
H 1 cts (G K ′ , 1 + p k+1 O C ) H 1 cts (G K ′ , 1 + p k O C ) H 1 cts (G K ′ , C × ) H 1 cts (G K ′ , O C ) H 1 cts (G K ′ , O C ) H 1 cts (G K ′ , C) 1 p k+1 log ∼ 1 p k log ∼ log ·p ·p k
as we will recall in Lemma 3.13 below that the torsion in H 1 cts (G K ′ , O C ) is uniformly bounded among all finite extensions K ′ of K. Since χ · χ −1 π ♭ defines an element in the top left that vanishes in the top right, it follows that for k ≫ 0, the class of χ · χ −1 π ♭ in H 1 cts (G K , 1 + p k O C ) vanishes. Thus there exists z ′ ∈ K ′ whose coboundary equals χ · χ −1 π ♭ . This shows that we can in fact choose K ′ to be K(π 1/p m , ζ p m ) for some m ≫ 0. From this perspective, the reason that z ′ is non-canonical is that we can only prove the vanishing of a class in group cohomology, but we did not produce a canonical coboundary representing it.
To reformulate Sen theory in the style of Fontaine, Colmez, [Col94], has introduced the period ring B Sen . As a ring, B Sen identifies with the ring C{{u}} of power series in u with coefficients in C having a positive radius of convergence 6 . It is filtered by the subrings B n Sen , n ≥ 0, defined as the power series with radius of convergence ≥ p − max(1,n) for p odd and p − max(2,n) for p = 2. If for σ ∈ G K we set σ(u) = u + log(χ(σ)), one checks that this defines an action of G Kn on B n Sen , where K n = K(µ p n ). Note that there is an embedding
(3) B cycl en ֒→ B Sen , t ′ → exp(u). which factors through B 1
Sen and is Galois equivariant with respect to the action on B 1 Sen . Therefore, as a consequence of Proposition 3.7, we deduce: Corollary 3.9. There exists z ′ ∈ C × such that there is an O C -linear continuous embedding
B en ֒→ B Sen , t = 1 + e · a → z ′ exp(u)
that factors through B 1 Sen and is G K ′ -equivariant for some finite extension K ′ |K. The element z ′ is in general non-canonical and only well-defined up to a factor in 1 + e · p α O K ′ .
Remark 3.10. The subscript "en" in B en stands for "enhanced", the motivation of which is twofold: First, the notation is to suggest that B en is closely related to B Sen , but has a finer convergence condition. As we will explain in Remark 3.15 below, this makes B en better behaved with respect to Galois cohomology. Second, the ring B en has a natural generalisation for X HT when X is a smooth formal scheme over O K , and in this case it encodes a close relation with the functor from v-vector bundles to the "enhanced Higgs modules" of Min-Wang [MW22], which are roughly a hybrid of Sen modules and Higgs modules.
We end this subsection with a slightly different viewpoint on the ring B en . As we saw above, the G K -action on t ′ := z t is via the cyclotomic character. In particular, Fontaine's period ring
B HT := C[(t ′ ) ±1 ] ∼ = n∈Z C(n)
embeds as a subring into B en . In fact, B en is the completion of B HT for the Banach norm induced by the O C -lattice A en ∩ B HT , and this sublattice can be made rather explicit.
Let us give an explicit consequence of this completion.
Example 3.11. The class of continuous semilinear C-representations V of G K , which satisfy
dim K (B en ⊗ C V ) GK = dim C V
is different from the class of Hodge-Tate representations. As a concrete example, let M be a free rank 2 vector space over K and equip M with the Sen operator Θ π,M := 0 −1 0 0 for some choice of basis. As we explain in detail in §4, the G K -representation associated to (M, Θ π ) via α K • β −1 π is explicitly given by V := (M ⊗ OK B en ) Θπ +Θπ,M =0 . Firstly, the constants C lie in V . To get the whole of V we have to solve the differential equation
(1 + ea) ∂ ∂a (f (a)) = 1,
which yields that f (a) = 1 e log(1 + ea) + c for some constant c ∈ C, and thus V = 1, log(1 + ea) C . In particular, V is 2-dimensional and verifies dim K (B en ⊗ C V ) GK = 2. But V is not Hodge-Tate as the Sen operator is not semi-simple.
3.3. Calculation of Galois cohomology of B en . We continue to use the notation of Section 3.2. The following theorem is the crucial input in our proof of the full faithfulness part of Theorem 1.3.2 in Theorem 4.2 below.
Theorem 3.12. The natural map
K → RΓ(G K , B en )
is an isomorphism.
Here, the right hand side is continuous group cohomology for the Banach space topology on B en . Before starting the proof, let us recall Tate's celebrated theorem ([Tat67, Proposition 8(a)]) which will be used in the proof:
RΓ(G K , C) = K ⊕ K[−1].
We will also need the following result, which is a more precise version of the result that H 1 (G K , C(n)) = 0 for n = 0:
Lemma 3.13 ([Tat67, Proposition 7(c) and its proof]). For any 0 = n ∈ Z, the K-linear operator gχ(g) n − 1 : C → C has a bounded inverse ρ n . More precisely, there is δ > 0 such that |ρ n (x)| ≤ p δ |x|. This δ only depends on K, and in fact we can choose δ such that it works uniformly for all finite extensions K(µ p m )|K for m ∈ N.
Proof. Tate proves this for g − λ where λ ∈ 1 + pZ p . We can take λ = χ(g) −n and then multiply g − λ by χ(g) n which does not change the estimates. The last statement on independence of K is [Tat67, Remark on page 172].
Proof of Theorem 3.12. First note that it suffices to prove
K ′ ∼ = RΓ(G K ′ , B cycl en,K ) for some finite extension K ′ of K where B cycl
en,K is the Banach C-algebra from Section 3.2 and with respect to the field K.
Indeed, given this claim we can use Proposition 3.7 to deduce that
K ′ ∼ = RΓ(G K ′ , B en )
for some finite Galois extension K ′ /K with Galois group H. Applying RΓ(H, −), then yields the theorem by Shapiro's lemma as K ′ is an induced H-representation (by the normal basis theorem) with invariants K.
To prove the claim we first set up some notation. We set B := n≥0 C a n n! and A := n≥0 O C a n n! . We fix some e ∈ O K \{0} and set t := 1 + e · a. We equip B with the unique continuous Galois action such that σ(t) = χ(σ)t for σ ∈ G K and χ : G K → Z × p the cyclotomic character. (Thus, in particular, if e = E ′ (π) up to O × K , then B ∼ = B cycl en from Section 3.2.) We can conclude (4) σ(a) = χ(σ) · a + χ(σ) − 1 e for σ ∈ G K as t = 1 + e · a.
Step 1: Calculation of H 0 (G K , B). We start by proving that
B GK = K.
For this we can use an analog of the embedding Equation (3) and use [Col94, Théorème 2.(i)]. Alternatively, we can argue directly and follow Colmez' argument for B Sen , as we shall now demonstrate. In the following we get slightly different series expansions due to the difference in Galois actions on u and a.
Let K ∞ ⊆ C be the completed p-power cyclotomic extension of K and let Γ := Gal(K ∞ |K). Since the action of G K∞ := Gal cts (C/K ∞ ) on a is trivial, we see from the definition of B that n≥0 C a n n! GK = n≥0 K ∞ a n n! Γ .
Via the cyclotomic character χ : Γ → Z × p → O × K , we can identify Γ with a closed subgroup of O × K . Since it suffices to prove that there is a finite extension K ′ |K for which B G K ′ = K ′ , we may for any k ∈ N assume after passing to K ′ large enough that χ has image in 1 + p k Z p . Hence we may assume that
y := 1 e (χ(g) − 1) ∈ pO K
Let now g ∈ Γ and b := b n a n n! ∈ B. Then since G K acts trivially on y, we have
g( n≥0 b n a n n! ) = n≥0 g(b n ) (χ(g)a + y) n n! = n≥0 g(b n ) k+m=n χ(g) m a m m! y k k! = ∞ m=0 χ(g) m ∞ k=0 g(b m+k ) y k k! a m m! . (5)
Here we note that the sum in brackets converges in C because y ∈ pO K . Assume now that g(b) = b for all g, then by comparing coefficients of a n n! , we see that this is equivalent to
b n = χ(g) n ∞ k=0 g(b n+k ) y k k!
for all n ≥ 0. Following Colmez' argument, we now multiply both sides by g −1 and apply for any j > 0 the normalised trace tr j : K ∞ → K j to the j-th cyclotomic subextension K j |K. Then since tr j is Γ-equivariant, we have
g −1 tr j (b n ) = (e · y + 1) n ∞ k=0 tr j (b n+k ) y k k! .
We can consider this expression as a function Γ → K j of the variable g ∈ Γ ⊆ Z × p (recall that y = 1 e (χ(g) − 1)). Then the left hand side shows that this function is locally constant, whereas the right hand side shows that this function is analytic. It follows that it is constant, hence tr j (b n ) ∈ K for all j, hence b n ∈ K. Consequently, for all n ≥ 0 we have
(6) b n = (e · y + 1) n ∞ k=0 b n+k y k k! .
For n = 0, this means that b 0 = ∞ k=0 b k y k k! , and comparing these inside C[[y]] we deduce that b k = 0 for k ≥ 1. Hence b = b 0 ∈ K as desired.
Step 2: Calculation of H 1 (G K , B). In order to compute H 1 , we first wish to see that the natural map
(7)
H 1 cts (G K , C) → H 1 cts (G K , B) vanishes. To see this, we consider the short exact sequence of C-linear G K -modules
0 → C → B → B/C → 0.
From Tate's theorem we know that
RΓ(G K , C) ∼ = K ⊕ K[−1].
We claim that also (B/C) GK ∼ = K as a K-vector space. Once this is known, we see by comparing K-vector space dimensions in the long exact sequence
0 → H 0 (G K , C) =K → H 0 (G K , B) =K, by Step 1 → H 0 (G K , B/C) → H 1 (G K , C) (7) − − → H 1 (G K , B) → H 1 (G K , B/C) that (7) is zero.
In fact, it suffices to see that (B/C) GK = 0 as it must then be of K-dimension 1. The nonvanishing is witnessed by the element b := log(t) = ∞ n=1 (−1) n−1 (e · a) n n ∈ B
for which g(log(t)) = log(t) + log(χ(g)) ≡ log(t) ∈ B/C. Before going on, we note that one can also prove the claim (B/C) GK ∼ = K by arguing exactly as in the first part: Observe that as a C-module there is a description
B/C = n≥1
C · a n n! .
Like for B, we deduce that an element b n a n n! in the right hand side satisfies g(b n ) = b n if and only if (6) holds, but this time only for n ≥ 1 instead of n = 0. From this point on, the arguments diverge: We cannot conclude anymore by considering n = 0, instead we need to expand the right hand side for n = 1, where we find
b 1 = b 1 + ∞ k=1 (b n+k + ekb n+k−1 ) y k k! ,
and similarly for n ≥ 2, where the expression is independent of b 1 . This means that we can choose b 1 ∈ K freely, and then the vanishing of the coefficients in front of y k for k ≥ 1 impose linear relations on the b k for k ≥ 2 which determine b uniquely. Hence dim K (B/C) GK = 1.
To finish the proof of the Theorem, it remains by the above long exact sequence to show that
H i (G K , B/C) = 0
for i ≥ 1. For this we again first reduce to the extension K ∞ |K: First note that H i (G K∞ , B/C) = n≥1 H i (G K∞ , C) = 0 for i ≥ 1 because G K∞ acts trivially on a and K ∞ is perfectoid. Thus it suffices to compute H 1 (Γ, (B/C) GK ∞ ) as this is the only group having a potential non-zero contribution to RΓ(G K , B). We first note that (B/C) GK ∞ =⊕ n≥1 K ∞ a n n! =: D ⊆ B/C. We may by enlarging K assume that Γ is pro-cyclic. Then we can choose a topological generator g ∈ Γ and compute H 1 (Γ, D) as the cokernel of the map g − 1 : D → D. It remains to prove that this morphism is surjective. To see this, it suffices to prove the statement for g m − 1 (for example because g − 1 divides g m − 1, or by inflation-restriction for the closed subgroup generated by g m ). So we are later free to replace g by g m .
Let now b = ∞ n=1 b n a n n! ∈ D, then using (5), we have
(g − 1)b = ∞ n=0 χ(g) n ∞ k=0 g(b n+k ) y k k! − b n a n n! .
Let furthermore c = ∞ n=1 c n a n n! ∈ D, then to show that c lies in the image we need to solve the equation (g − 1)b = c, which is equivalent to asking that for all n ≥ 1, we have
c n = (g(b n )χ(g) n − b n ) + ∞ k=1 χ(g) n g(b n+k ) y k k! .
In other words, if we consider (g − 1) as a continuous endomorphism of the Banach K ∞ -module D, then in terms of the orthonormal basis ( a n n! ) n≥1 , this is represented by an infinite upper triangular block matrix
g − 1 = gχ(g) − 1 * * . . . gχ 2 (g) − 1 * . . . gχ 3 (g) − 1 . . . . . .
(each block represents a semilinear endomorphism of a 1-dimensional K ∞ -vector space, and the whole matrix a K-linear endomorphism). Moreover, we observe that the entries above the diagonal are all p-adically small: Namely, let M be the matrix given by the upper triangular entries, then we can assume that |M | ≤ |y|.
Indeed, after enlarging K we may assume that y ∈ pO K , and then | y k k! | ≤ |y| for all k ≥ 1. Lemma 3.13 shows that the K-linear operator defined by the block matrix ρ := diag(ρ 1 , ρ 2 , . . . ) is bounded and continuous. We now consider
ρ • (g − 1) = 1 + ρ • M,
then |ρM | ≤ p δ |y|. Passing to an extension of K and replacing g by g m , we can make y = 1 e (χ(g)−1) arbitrarily small. As passing to an extension of K does not change δ, we can therefore assume without loss of generality that |ρM | < 1. Thus the matrix M is "topologically nilpotent". In particular, the matrix 1+ρM is indeed invertible with inverse the bounded operator ∞ k=0 (−ρM ) k . Thus g − 1 is invertible, showing that H 1 (Γ, D) = 0 as desired.
This finishes the proof of Theorem 3.12.
Remark 3.14. More axiomatically, in the spirit of the Sen axioms of Berger-Colmez [BC08], the above proof works for any field K and with the coefficient field C of B en replaced by the completed cyclotomic extension K ∞ |K with Galois group Γ whenever the following conditions are satisfied:
(1) We have Tate's Galois equivariant normalised traces K ∞ → K, and (2) The analogue of Lemma 3.13 holds, i.e. for Γ = Gal(K ∞ |K) we have H 1 (Γ, K ∞ ) = K, and for any generator g ∈ Γ and n ≥ 1, the map gχ(g) n − 1 : K ∞ → K ∞ has a continuous inverse that is bounded, uniformly in n and for all finite subextensions of K ∞ |K. Incidentally, we point out that the element α introduced in [BC08, §4.5] to describe the locally analytic vectors for the Galois action in the Kummer tower can be taken to be our element (ez) −1 .
Remark 3.15. It is natural to ask if the analogue of the Theorem also holds for Colmez' ring B m Sen with its natural Galois action by G K∞ 7 , but this is not the case. Instead of filtering B Sen by the subspaces B m Sen , let us filter it by the subspaces
Fil m B Sen = n≥0 b n u n n! , |b n |p − sup(1,m)·n → 0 .
One has Fil m B Sen ⊂ B m−1 Sen for m ≥ 1, and G Km−1 acts on Fil m B Sen . Take an element in Fil m+1 B Sen , m ≥ 0. Following the same line of argument with a replaced by u, we see that since the action on u is given by g(u) = u + log χ(g) instead of (4), we need to replace (5) by
g n≥0 b n u n n! = ∞ l=0 ∞ k=0 g(b l+k ) y k k! u l l!
where this time y := log(χ(g)). The essential difference is that there is no longer a factor of χ(g) l before the coefficient of u l l! . Consequently, following the same line of argument, the action of g − 1 with respect to ( u n n! ) n≥1 is now given by a block matrix of the form
g − 1 = g − 1 * * g − 1 * . . . .
But this has large cokernel as the cokernel of g − 1 on C is H 1 (G Km , C) = K m . The conclusion is that H 1 (G Km , Fil m B Sen ) is also large. This is one way in which the ring B en is "enhanced".
C-semilinear Galois representations via Hodge-Tate stacks
In this section we prove our main result, Theorem 4.9, which gives a description of v-perfect complexes on Spa(K) in terms of Hodge-Tate stacks. Let where the limit is taken over the category of all (discrete) rings S with a morphism Spec(S) → Spf(O K ) HT . Using the maps f : Spf(R + ) = lim − → n Spec(R + /p n ) → Spf(O K ) HT , the construction of α + K can now be stated as
(E S ) Spec(S)→Spf(OK ) HT → ((R lim ← − n E Spec(R + /p n )→Spf(OK ) HT )[1/p]) Spa(R,R + )→Spa(K,OK ) .
By construction, the functor α + K induces a functor
α K : Perf (Spf(O K ) HT )[1/p] → Perf (Spa(K) v )
by passing to the isogeny category on the source. By v-descent of perfect complexes on perfectoid spaces, cf. [AB21, Theorem 2.1], the category Perf (Spa(K) v ) identifies with the category of "continuous, semilinear representations of G K on perfect complexes of C-vector spaces" (here C = K as before). 9 Let us note that the usual truncations equip Perf (Spa(K) v ) with a t-structure whose heart is the usual category of continuous semilinear representations of G K on finite dimensional C-vector spaces.
Our next aim is to prove the following result. However, this strategy does not work: the category Perf (Spf(O K ) HT )[1/p] is not generated under colimits by Breuil-Kisin twists. 10 To remedy this one could hope to prove a more general statement involving a certain rationalized version of D(Spf(O K ) HT ). However, one then also needs to enlarge Perf (Spa(K) v ) accordingly in such a way that α K extends to a functor, which still commutes with arbitrary colimits in D(Spf(O K ) HT ). We do not know if this possible. This is why we pursue a different strategy and use Theorem 3.12.
We will start with some preparations. Fix a uniformizer π ∈ O K and consider the G π -torsor Z π := Spf(A en ) → Spf(O C ) as in Section 3.1. The O C -algebra A en is equipped with a continuous G K -action and a commuting endomorphism Θ = Θ Aen . Let (A inf , J) be the perfect prism associated to O C . After fixing a choice π ♭ := (π, π 1/p , . . .) ∈ O ♭ C of compatible p-power roots of π, the Galois action and the operator Θ have been made explicit in Section 3.1 by mapping u ∈ A π to [π ♭ ]. Note that under the equivalence from Theorem 2.5 the pullback along the map
[Spf(O C )/G K ] → Spf(O K ) HT 9
To make the continuity precise, one could use the solid formalism, or just use the ∞-limit For the right hand side we use Theorem 3.12. Namely,
lim ← − ∆ Perf (Spa(C, O C ) •+1 ) ∼ = lim ← − ∆ Perf (Spa(C, O C ) × G K • ).RΓ(G K , V ) ∼ = RΓ(G K , fib(M ⊗ OK B en 1⊗Θ+Θπ,M ⊗1 −−−−−−−−−→ M ⊗ OK B en )) ∼ = fib(M ⊗ OK RΓ(G K , B en ) 1⊗Θ+Θπ,M ⊗1 −−−−−−−−−→ M ⊗ OK RΓ(G K , B en )) ∼ = fib(M Θπ,M −−−→ M ), where we used RΓ(G K , B en ) ∼ = K (with Θ = 0) in the last step.
We now turn to the description of the essential image of the functor
Perf (Spf(O K ) HT )[1/p] → Perf (SpaK v , O).
As the target admits a t-structure with heart the abelian category Rep C (G K ) the claim reduces to the question when a given V ∈ Rep C (G K ) lies in the image. Following [MW21], [Gao22] the essential image should be given by the nearly Hodge-Tate representations.
Definition 4.4 ([Gao22, Definition 1.1.3]). A representation V ∈ Rep C (G K ) is nearly Hodge-Tate if its Sen operator Θ has eigenvalues in Z + δ −1 OK/Zp m K ⊆ K where δ OK /Zp = (E ′ π (π)) is the inverse different ideal and Θ : V → V denotes the classical Sen operator as constructed in [Sen80].
To see this, we start with the following lemma:
M ⊗ OK O C Id⊗σ − −− → M ⊗ OK O C γ −1 d(σ),M − −−−− → M ⊗ OK O C
with d(σ) ∈ G π (O C ) defined in Corollary 3.4 and γ d(σ),M is as described in Lemma 2.14. Thus, explicitly σ(m ⊗ c) = χ π ♭ (σ) Θπ,M /e (m ⊗ 1)σ(c).
Proof. First of all note that by Corollary 3.4 the Galois action of σ ∈ G K on A en ∼ = O Gπ ⊗ OK O C is given by first applying Id OG π ⊗ σ, and then γ d(σ),OG π ⊗ O K OC . Now, we can apply the following general observation: If H is any group (in any topos) and V any H-representation with underlying abelian group |V |, then the map
H × V → H × |V |, (h, v) → (h, h −1 v)
is an isomorphism, which converts the diagonal action on the left hand side to the left multiplication on H on the right hand side, and the right multiplication on H in H ×V to the H-action on H ×|V | given by g·(h, v) = (hg, h −1 v). In particular, the invariants of the diagonal action on H ×V identify with |V | and the remaining right action of H with the inverse of the given action on V . Now we can apply this to H = G π , V = M and to the right multiplication on H. Formulated at the level of comodules, this implies that
M ⊗ OK O C ∼ = fib(M ⊗ OK A en 1⊗ΘA en +Θe,M ⊗1 − −−−−−−−−−−− → M ⊗ OK A en )
and that the right G π -action on A en ∼ = O Gπ ⊗ OK O C transforms to the inverse of the G π -action on M . This implies the claim.
Fix compatible systems π ♭ ,π ♭ , which yield identifications between the three morphisms
f : Spf(O C ) → Spf(O K ) HT , Spf(O C ) → Spf(O K ) ρπ −→ Spf(O K ) HT , Spf(O C ) → Spf(O K ) ρπ −→ Spf(O K ) HT .
Let E ∈ Vec(Spf(O K ) HT )[1/p] and denote by (M π , Θ π ) ∈ Vec(BG π )[1/p], (Mπ, Θπ) ∈ Vec(BGπ)[1/p] the corresponding objects. Set V := f * E ∈ Rep C (G K ). Then γ π ♭ induces an isomorphism V ∼ = M π ⊗ K C with action via χ π ♭ , cf. Lemma 4.5. Now Theorem 4.2 and the proof of Lemma 4.6 imply that M π is the unique G K -stable K-subspace of V such that G K acts on M π via χ π ♭ (−) Θ , where Θ : V → V is the Sen operator. Similarly, Mπ identifies with the unique G K -stable K-subspace of V on which G K -acts via χπ♭(−) Θ . This gives, in principle, a way of constructing Mπ out of M π or vice versa, or in other words describing instances of β * π,π : Set z := θ( Eπ ([π ♭ ]) Eπ([π ♭ ]) ). Then σ(z) z = χ π ♭ (σ)χπ♭ (σ) −1 for σ ∈ G K . If z Θ is well-defined, then M π = z Θ · Mπ as subspaces of V . Thus, in this case the relation between M π and Mπ is more direct.
We finally arrive out our desired description of the whole category Perf (Spa(K) v ). where L runs over finite Galois extensions of K contained in K.
Proof. Full faithfulness of α L is just Theorem 4.2 applied to K = L. By naturality of the Hodge-Tate stacks the functor α L passes to Gal(L/K)-equivariant objects, and the resulting functor α L/K is again fully faithful by finite étale descent of perfect complexes and full faithfulness of α L . If V ∈ Rep C (G K ), then its restriction W := V |GL to some G L for L/K a large enough finite Galois extension will become nearly Hodge-Tate (with respect to L), and hence lies in the essential image of α L . As W ∈ Perf (Spa(L) v ) is Gal(L/K)-equivariant and α L fully faithful, the final claim follows (note that each E ∈ Perf (Spa(K) v ) can be represented as a complex of continuous G K -representations). Perf (Spa(K) HT ) ∼ = Perf (Spa(K) v ). This would yield a description of v-bundles on Spa(K) in terms of the étale/analytic topology on a certain analytic stack over Spa(K). We plan to come back to this question in future work.
5.
The v-Picard group of p-adic fields As pointed out in the introduction, Theorem 4.2 (together with Lemma 4.6) can be regarded as an analogue of the local Simpson correspondence, in that it relates "small" semilinear Crepresentations of G K to "small" Sen modules.
Theorem 4.9 provides (formally) a description of all G K -representations. One might still ask whether there exists a simpler, more classical description. As already mentioned in the introduction, one usually cannot hope for an equivalence of categories between v-bundles and Sen modules, but is it still possible to give a more precise relation between the whole two categories?
As a first partial answer in this direction, we note that it is indeed possible in general to extend the local correspondence between small objects, by the following immediate consequence of Lemma 4.6 and Theorem 4.2:
Corollary 5.1. There is a natural fully faithful functor 11 nearly Hodge-Tate v-vector bundles on Spa(K) ֒→ Sen modules over K .
One might hope optimistically that there is an equivalence between all v-bundles and all Higgs bundles, like in the geometric situation. However, this is not the case. Indeed, the goal of this section is to prove the following result, which evidences that over a local field, there are always "fewer v-vector bundles than Higgs bundles".
Theorem 5.2. For any complete discretely valued field K|Q p , there is a short exact sequence
0 → Pic v (K) HT log − −−− → K → Br(K)[p ∞ ] → 0
that is functorial in K. If K|Q p is finite, the last map can more explicitly be identified with 1 p Tr K|Qp : K → Q p /Z p . The map HT log will be defined in the proof. It has already been described by Sen [Sen80, Theorem 7'], as well as by Serre [Ser98, §III, A2, Proposition 2], in terms of the continuous cohomology Pic v (K) = H 1 (G K , C × ). Sen also shows that if the residue field is algebraically closed, then HT log is an isomorphism [Sen80, Theorem 9']. From the perspective of Theorem 5.2, this is because Br(K) = 1 in this case. Beyond this case, we are not aware of any previous description of the cokernel of HT log.
Proof. Consider the morphism of sites ν : Spa(K) v → Spa(K)é t . By Tate's cohomological result, we have R 1 ν * O = O. We instead consider the sheaf G m on Spa(K) v defined by T → O(T ) × . Following [Ser98, §III, A2] or alternatively arguing as in [Heu22,§2], we see:
Lemma 5.3. The exponential induces a natural isomorphism R 1 ν * G m = O.
Proof. It suffices to construct natural isomorphisms R 1 ν * (1 + mO + ) ∼ − → R 1 ν * G m exp : R 1 ν * (1 + mO + ) ∼ − → R 1 ν * O. Namely let K ∞ |K be any totally ramified perfectoid Galois cover with group ∆ ∼ = Z p . Then the first isomorphism is obtained by applying H 1 (∆, −) to the short exact sequences
0 → O × K∞ → K × ∞ → Z[ 1 p ] → 0 0 → 1 + m K∞ → O ×
K∞ → k × → 0 and using that H 1 (∆, k × ) = Hom cts (∆, k × ) = 1 and H 1 (∆, Z[ 1 p ]) = 1, where k is the residue field. The second isomorphism then follows from the exponential sequence over C = K .
Since Picé t (K) = 1, it follows that the Leray sequence of G m for ν is of the form
1 → Pic v (K) HT log − −−− → K ∂K − − → H 2 et (K, G m ) → H 2 v (K, G m )
, functorially in K. Let Q =: coker HT log = ker ∂ n . It follows from the construction that HT log admits a canonical splitting over pO K ⊆ K defined by the p-adic exponential function, hence Q is a p-power torsion group. This shows that Q ⊆ Br(K)[p ∞ ].
To see the other containment, it now suffices to prove:
Lemma 5.4. H 2 v (K, G m ) is p-torsionfree.
Question 1 . 1 .
11How can we describe the category Rep C (G K ) in terms of linear algebraic data? Our first main result is a new description of this category in terms of the Hodge-Tate locus in the Cartier-Witt stack of Bhatt-Lurie (notations to be explained below): Theorem 1.2. There is a natural equivalence of categories Rep C (G K ) → 2-lim − → L/K Vec([Spf(O L ) HT /Gal(L/K)])[ 1 p ] where L|K ranges through finite Galois extensions of K in C and Vec is the category of vector bundles on the stack quotient [Spf(O L ) HT /Gal(L/K)].
2 .
2Complexes on the Hodge-Tate stack 2.1. Recollections on the Hodge-Tate stack for O K . The Cartier-Witt stack WCart is the groupoid-valued functor on Nilp p sending R to the groupoid of Cartier-Witt divisors, i.e. W (R)-linear maps α : I → W (R) from an invertible W (R)-module I satisfying the conditions that the image of the composite I α → W (R) → R is a nilpotent ideal of R and the image of the composite I
Now we prove the analog of [BL22a, Proposition 3.5.15]. Proposition 2.9. The category D(Spf(O K ) HT ) is generated under colimits by the Breuil-Kisin twists O Spf(OK) HT {n}, n ≥ 0. Proof. Following the argument of [BL22a, Proposition 3.5.15] the claim reduces to showing that the regular representation O Gπ of G π lies in the category spanned by the (objects corresponding to the) O Spf(OK ) HT {n}, n ≥ 0. For this we can use the filtration of
f : Spf(O C ) → Spf(O K ) denote the natural morphism. As the natural map Spf(O C ) HT → Spf(O C ) is an isomorphism ([BL22b, Example 3.12]), the map f lifts naturally to a map f : Spf(O C ) → Spf(O K ) HT over Spf(O K ). Let Z π be the base change alongf of the G π -torsor ρ π : Spf(O K ) → Spf(O K ) HT .
stack or of [BL22b, Construction 3.10]. Write σ G (t, a) = (s, b) and let h ∈ W × [F ](S) ∼ = G ♯ m (S) be the point determined by s. Let us now for simplicity set S = O C ′ . Then the commutativity of the outer diagram implies that the two compositions
Spa(K) := Spa(K, O K )be the diamond associated to K, cf.[Sch17, Definition 15.5], and its v-siteSpa(K) v , cf. [Sch17, Definition 14.1]. If Spa(R, R + ) ∈ Spa(K) v is an affinoid perfectoid space over K 8 , then the natural map π HT R + : Spf(R + ) HT → Spf(R + ) is an isomorphism, cf. [BL22b, Example 3.12]. This implies that for any Spa(R, R + ) ∈ Spa(K) v the morphism f : Spf(R + ) → Spf(O K ) lifts naturally to a map f : Spf(R + ) → Spf(O K ) HT over Spf(O K ). From here we get a symmetric monoidal, exact functorα + K : Perf (Spf(O K ) HT ) → Perf (Spa(K) v ),where the latter category means perfect complexes on Spa(K) v for the "completed structure sheaf" (Spa(R, R + ) → Spa(K, O K )) → R.7 In the terminology of Colmez: i.e., for each m ≥ 0, B m Sen has an action of G Km . 8 Here and in the following we identify the v-site of K, which consists of perfectoid spaces S in characteristic p and an untilt S ♯ over Spa(K, O K ), with the site of perfectoid spaces over Spa(K, O K ).
Remark 4 . 1 .
41The generic fiber Spf(O K ) HT η of the Hodge-Tate stack Spf(O K ) HT can be defined as the (analytic sheafification of the) functor (B, B + ) → lim − → B0⊆B + ring of definition
Theorem 4. 2 .
2The functor α K : Perf (Spf(O K ) HT )[1/p] → Perf (Spa(K) v ) is fully faithful.
Remark 4. 3 .
3Given Proposition 2.9 and Tate's calculation of the continuous Galois cohomology H * (G K , C(n)), n ∈ Z, Theorem 4.2 might first seem trivial to prove: first, reduce checking fully faithfulness to the case of morphisms between Breuil-Kisin twists O Spf(OK) HT {n}, and then apply Tate's theorem.
M ⊗ OK A en ), where G K -acts via A en on the fiber. As in Section 3.1 we set B en := A en [1/p]. Given Theorem 3.12 we can conclude the proof of Theorem 4.2.Proof of Theorem 4.2. Let E ∈ Perf (Spf(O K ) HT ) and let (M, Θ π,M ) = β + π (E). Then α K (E) is the perfect complex of C-vector spaces with G K -actionV := fib(M ⊗ OK B en 1⊗Θ+Θπ,M ⊗1 −−−−−−−−−→ M ⊗ OK B en ).It suffices to show that the natural mapRΓ(Spf(O K ) HT , E)[ 1 p ] → RΓ(G K , V )is an isomorphism. Here, the left hand side identifies (by Proposition 2.
Lemma 4 . 5 .
45Let (M, Θ π,M ) ∈ D(BG π ) ∼ = D(Spf(O K ) HT ). Then its pullback to Spf(O C ) is isomorphic to the complex M ⊗ OK O C and if σ ∈ G K , then the resulting action of σ on M ⊗ OK O C is given by the composition
Theorem 4 . 9 .
49For any finite Galois extension L/K the functorα L : Perf (Spf(O L ) HT )[1/p] → Perf (Spa(L) v )is fully faithful and induces a fully faithful functorα L/K : Perf ([Spf(O L ) HT /Gal(L/K)])[1/p] → Perf (Spa(K) v )on Gal(L/K)-equivariant objects. Each E ∈ Perf (Spa(K) v ) lies in the essential image of some α L/K . Consequently, we get an equivalencePerf ([Spf(O L ) HT /Gal(L/K)])[1/p] ∼ = Perf (Spa(K) v ),
Remark 4. 10 .
10More geometrically, one could consider the pro-systemY := " lim ← − "[Spf(O L ) HT /Gal(L/K)]over finite Galois extensions L/K contained in K and consider the morphismf pro : [Spf(O C )/G K ] → Y,obtained by the natural lifts Spf(O C ) → Spf(O L ) HT . Probably, it makes sense to pass to a "generic fiber Spa(K) HT " of Y and define a reasonable notion of perfect complexes on Spa(K) HT (for the analytic topology) in such a way that Perf (Spa(K) HT ) ∼ = 2-lim − → L/K Perf ([Spf(O L ) HT /Gal(L/K)])[1/p], cf. Remark 4.1. Then Theorem 4.9 could more cleanly be stated as an equivalence.
In particular, for the Breuil-Kisin line bundles O Spf(OK ) HT {n} from [BL22a, Example 3.3.8], this means that O Spf(Zp) HT {n} pulls back to
). In particular, the functor RΓ(Spf(O K ) HT , −) commutes with all colimits/limits.Proof. Note that the Sen operator for O Spf(OK) HT is 0, e.g., by Example 2.4. This implies (by naturality of the Sen operator) that the natural map
and Remark 2.8 to conclude the proposition as in [BL22a, Proposition 3.5.15].Remark 2.10. In fact, when K is ramified, one only needs the structure sheaf O Spf(OK ) HT to generate the category under colimits: see [BL22b, Remark 9.7].
acts locally nilpotently on H * (M ). Now we can finish the argument as in the proof of [BL22a, Theorem 3.5.8.]. Remark 2.11. The Cartier duals of G ♯ m and G ♯ a are known, cf. [BL22a, Remark 3.5.17], [Dri21, Appendix B]. For G π one can uniformly describe the Cartier dual as the formal group scheme
1 z t ′ transforms as desired to get Galois equivariance of the transformation. However, this is problematic due to convergence issues: in order for this rule to extend to a morphismB en → B cycl
en ,
the sum
1
n!
z
t ′ − 1
e
n
would have to converge in B cycl
en , or in other words we would have to have
Spf(O K ) HT (B 0 ) on complete adic Huber pairs over (K, O K ), cf. [SW13, Proposition 2.2.2.]. Ideally, the category Perf (Spf(O K ) HT )[1/p] should be defined as a category of perfect complexes on Spf(O K ) HT η . By [Mat22, Theorem 7.8] one can by descent define such a category of perfect complexes, and using Cartier-duality over O K for the Hopf algebra O Gπ (and Theorem 2.5) see that each perfect complex extends to Spf(O K ) HT so that the resulting category of perfect complexes should be equivalent to our more concrete version Perf (Spf(O K ) HT )[1/p].
Here all necessary higher divided powers of ε, 1 + eε are defined to be 0.
Note that u ∈ B Sen is not related to the element u ∈ Aπ from Section 3.1.v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK
This is not a contradiction as the generation in D(Spf(O K ) HT ) involves infinite colimits.
v-VECTOR BUNDLES ON p-ADIC FIELDS AND SEN THEORY VIA THE HODGE-TATE STACK
We note that the notion of "nearly Hodge-Tate" for a representation is independent of any choice and hence passes to an intrinsic notion for v-bundles on Spa(K).
Acknowledgments. The authors would like to thank Bhargav Bhatt, Yu Min, Juan Esteban Rodríguez Camargo and Lue Pan for helpful conversations on Sen theory and feedback on a preliminary version of this paper, and Peter Scholze and Matti Würthen for useful discussions. One author (J.A.) wants to thank the organizers of the conference in Darmstadt in October 2022, where he could present the results of this paper.The second author was funded by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -EXC-2047/1 -390685813, as well as by DFG Project-ID 444845124 -TRR 326, and was moreover supported by DFG via the Leibniz-Preis of Peter Scholze.For example, if (M, Θ π,M ) = (O K , e · n), then with z ∈ C × as in Lemma 3.6 σ(1 ⊗ z n ) = χ π ♭ (σ) n · 1 ⊗ χ(σ) n χ π ♭ (σ) −n z n = χ(σ) n 1 ⊗ z n as expected.Lemma 4.6. An object V ∈ Rep C (G K ) lies in the essential image of α K if and only if it is nearly Hodge-Tate.Phrased in terms of prismatic crystals this result was conjectured in[MW21]and proved in [Gao22, Theorem 1.1.5]. Note that the A 1 appearing in[MW21],[Gao22]is (probably) equal to −Θ π by unraveling all identifications.Proof. Assume (M, Θ π,M ) ∈ Vec(Spf(O K ) HT )[1/p] maps to V , i.e., V = M ⊗ OK C with the action as described in Lemma 4.5. Set z ∈ C × as in Lemma 3.6. ThenPassing to a finite field extension K ′ of K as in Proposition 3.7 yields z ′ ∈ C × such thatThis implies that eΘ = Θ π,M by the unique characterisation of the classical Sen operator Θ for V , cf.[Sen80,Theorem 4]. In particular, Θ has eigenvalues in Z + E ′ π (π)m K because Θ π,M has eigenvalues in e · Z + m K by Theorem 2.5. Conversely, assume that V is nearly Hodge-Tate. Then for any σ ∈ G K the sum ψ σ := χ π ♭ (σ) −Θ : V → V converges (see the proof of Lemma 2.14). Now define a new G K -action on V by settingAgain by classical Sen theory, we can find v 1 , . . . , v n ∈ V such that for a suitable open subgroup U ⊆ G K , any σ ∈ U acts viaUsing multiplication by w Θ on the K-span of v 1 , . . . , v n we see that V U, * , the space of invariants of U for the action via * , has dimension n. In particular, V for the action via * has trivial Sen operator. But this implies that it is generated by its G K -invariants M (more precisely, M ⊗ K C → V is an isomorphism for M := V GK , * ), cf. [Sen80, Theorem 6]. By construction, the given action on V equips M with an action via χ π ♭ (σ) Θ , i.e., V is associated with (M, eΘ) ∈ Perf (BG π ). This finishes the proof.Remark 4.7. Alternatively, one could use Corollary 3.9 ro relate Θ π,M and Θ.Remark 4.8. The construction of α K is independent of any choice, but β depends on the choice of a uniformizer π. Conversely, we can use these statements to clarify the way this description of perfect complexes depends on these choices. Namely, letπ ∈ O K be a second uniformizer. Then there exists a unique isomorphism δ π,π : Gπ → G π , which is compatible with the two respective projections to G ♯ m . Explicitly,whereẽ := E ′ π (π). The functor δ * π,π : D(BG π ) → D(BGπ) sends a complex (M, Θ π,M ) to the pair (M, ẽ e Θ π,M ). But there is also a different equivalence, more relevant to us. Namely, letHTbe the two morphisms associated with π,π. Then we obtain an equivalence β π,π : BG π ∼ = Spf(O K ) HT ∼ = BGπ, which does not agree with δ π,π . For example, β π,π does not map the trivial G π -torsor to the trivial Gπ-torsor.Proof. The Cartan-Leray sequence of the ∆-torsor Spa(K ∞ ) → Spa(K) is of the formUsing the above short exact sequences, we moreover see that for any n ≥ 2,p n } as a colimit of p-complete Z p -algebras, for which H n (∆, −) vanishes for n ≥ 2. Thus the first column E •0 2 vanishes in degree n ≥ 2. It follows that the natural mapWe thus get the first exact sequence, and it remains to identify the transition morphism in the case of local fields: recall from local class field theory that there is a natural isomorphism inv K : Br(K) → Q/Z such that for any extension L|K, we have inv L (x) = [L : K] · inv K (x) for any x ∈ Br(K). Via this identification, we obtain a morphism, functorial in K,It follows formally that this morphism must factor through the trace Tr K|Qp : K → Q p : namely, assume without loss of generality that K|Q p is Galois, then the functoriality in K means that ∂ K is G := Gal(K|Q p )-equivariant. Let n := |G| = [L : K], then for any x ∈ K, we thus haveSince 1 n Tr K|Qp (x) ∈ Q p , this now equalsWe can thus reduce to computing the homomorphism ∂ Qp : Q p → Q p /Z p which is necessarily given by multiplication by some uniquely determined a ∈ Q p , followed by the quotient map. It therefore suffices in a second step to reduce to K = Q p (µ p ), where some computations become easier. In order to determine the element a, we consider the Galois cohomology of the short exact sequenceLet G K := Gal(C|K), then by tracing through the identification R 1 ν * O × = O, we easily verify that this induces an exact sequencein which the first vertical map is surjective, the second is an isomorphism, and the third is injective with image the p-primary part of the Brauer group. It thus suffices to identify the morphism ∂ ′ . This is a boundary morphism for Galois cohomology, which we can make explicit: Let Q cyc p |Q p (µ p ) = K be the completed cyclotomic extension with Galois group Γ := Z p . Then any element of H 1 (G K , C) is represented by a continuous homomorphism Γ → K given by γ → γ · b for some element b ∈ K. We note that since log is split by exp over pO K , it is clear that ∂ ′ (b) = 0 for b ∈ pO K . On the other hand, let b ∈ K be any element and let y ∈ 1 + m C be a preimage of b under log. Then ∂ ′ sends y to the 2-cocycle Γ → µ p ∞We see from this explicit description that this 2-cocycle can be captured as follows: consider the extensionwith Galois groups as indicated. Due to the functoriality properties of inv, we know that any p n torsion class in Br(K) is killed by an extension of K of degree p n . It follows that the kernel of the map. By a Hochschild-Serre sequence, and due to vanishing of H i (Γ, −) and H i (∆, −) for any i ≥ 2, we get a natural identificationwhere the last map comes from the identification ∆ = Z p (1).It is now clear from construction that the explicit cocycle ((8)) defines an element in the middle term. To see when this vanishes in Br(K), it thus suffices to see when the 1-cocycle g → g * y/y is non-trivial. Via the identification from the Kummer sequencewe see that this happens if and only if y is already contained in Q cycl p . We claim that this is never the case for b ∈ pZ p , for which it suffices to prove that no element b ∈ K with |b| = |p| is such that exp(b) has a p-th root in Q cycl p . For this we use that the natural map Q × p /Q ×p p ֒→ Q cycl× p /Q cycl×p p is injective: Its kernel can be described via the Kummer sequence and the Cartan-Leray sequence as the cohomology group H 1 (Z × p , µ p ) for the action of Z × p on µ p via aζ p = ζ a p , which vanishes. Thus exp(b) has a p-th root in Q cycl× p if and only if it has as in Q p . Such a root cannot exist due to the assumption that |b| = |p|, which would force 0 < |1 − exp(b 1/p )| < 1, a contradiction to b 1/p ∈ Q p .
) the canonical map inv : Br(K) → Q/Z from local class field theory? This seems plausible, but a bit cumbersome to prove since this morphism is constructed using the maximal unramified extension rather than the highly ramified one we consider above. In any case, we do not need it for the description of the image of the map HT log : Pic v (K) → K. Since any morphism between v-line bundles on K is either 0 or an isomorphism, and the same is true for Sen modules, we deduce from Theorem 5.2 that in the case of rank one, Corollary 5.1 yields: Corollary 5.6. There is a fully faithful functor, which is however not essentially surjective: v-line bundles on Spa(K) ֒→ v-vector bundles and all Sen modules over Spa(K). Remark 5.5. Does the natural map (9) agree with (the p-primary map of. that is compatible with both the functor from Corollary 5.1 as well as passage to finite extensions. In particular, this means that theRemark 5.5. Does the natural map (9) agree with (the p-primary map of) the canonical map inv : Br(K) → Q/Z from local class field theory? This seems plausible, but a bit cumbersome to prove since this morphism is constructed using the maximal unramified extension rather than the highly ramified one we consider above. In any case, we do not need it for the description of the image of the map HT log : Pic v (K) → K. Since any morphism between v-line bundles on K is either 0 or an isomorphism, and the same is true for Sen modules, we deduce from Theorem 5.2 that in the case of rank one, Corollary 5.1 yields: Corollary 5.6. There is a fully faithful functor, which is however not essentially surjective: v-line bundles on Spa(K) ֒→ v-vector bundles and all Sen modules over Spa(K) that is compatible with both the functor from Corollary 5.1 as well as passage to finite extensions. In particular, this means that the
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"Jun Gao \nTencent YouTu Lab\nShenzhenChina\n",
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| Recently, density map regression-based methods have dominated in crowd counting owing to their excellent fitting ability on density distribution. However, further improvement tends to saturate mainly because of the confusing background noise and the large density variation. In this paper, we propose a Hierarchically Decoupled Network (HDNet) to solve the above two problems within a unified framework. Specifically, a background classification sub-task is decomposed from the density map prediction task, which is then assigned to a Density Decoupling Module (DDM) to exploit its highly discriminative ability. For the remaining foreground prediction sub-task, it is further hierarchically decomposed to several density-specific sub-tasks by the DDM, which are then solved by the regression-based experts in a Foreground Density Estimation Module (FDEM). Although the proposed strategy effectively reduces the hypothesis space so as to relieve the optimization for those task-specific experts, the high correlation of these sub-tasks are ignored. Therefore, we introduce three types of interaction strategies to unify the whole framework, which are Feature Interaction, Gradient Interaction, and Scale Interaction. Integrated with the above spirits, HDNet achieves state-of-the-art performance on several popular counting benchmarks. | 10.1109/icme52920.2022.9859688 | [
"https://export.arxiv.org/pdf/2212.05722v1.pdf"
]
| 251,847,419 | 2212.05722 | d02a6f5bf4941fd17b194c3f1a5ae0cdf9cf5516 |
HDNET: A HIERARCHICALLY DECOUPLED NETWORK FOR CROWD COUNTING
Chenliang Gu [email protected]
Tencent YouTu Lab
ShenzhenChina
Bin-BinChangan Wang [email protected]
Tencent YouTu Lab
ShenzhenChina
Jun Gao
Tencent YouTu Lab
ShenzhenChina
Tianliang Liu [email protected]
Tencent YouTu Lab
ShenzhenChina
Zhang
Tencent YouTu Lab
ShenzhenChina
HDNET: A HIERARCHICALLY DECOUPLED NETWORK FOR CROWD COUNTING
Index Terms-Crowd CountingDensity DecouplingForeground Density EstimationInteraction
Recently, density map regression-based methods have dominated in crowd counting owing to their excellent fitting ability on density distribution. However, further improvement tends to saturate mainly because of the confusing background noise and the large density variation. In this paper, we propose a Hierarchically Decoupled Network (HDNet) to solve the above two problems within a unified framework. Specifically, a background classification sub-task is decomposed from the density map prediction task, which is then assigned to a Density Decoupling Module (DDM) to exploit its highly discriminative ability. For the remaining foreground prediction sub-task, it is further hierarchically decomposed to several density-specific sub-tasks by the DDM, which are then solved by the regression-based experts in a Foreground Density Estimation Module (FDEM). Although the proposed strategy effectively reduces the hypothesis space so as to relieve the optimization for those task-specific experts, the high correlation of these sub-tasks are ignored. Therefore, we introduce three types of interaction strategies to unify the whole framework, which are Feature Interaction, Gradient Interaction, and Scale Interaction. Integrated with the above spirits, HDNet achieves state-of-the-art performance on several popular counting benchmarks.
INTRODUCTION
Crowd counting aims to estimate the number of persons in a still image or video frame, which recently draws increasing attention in the application of public safety.
Despite the progressive advancement in crowd counting, there are still two tricky problems remaining to be solved: the cluttered background noise [1] and the large density variation [2]. For the former, some methods [3] focus on learning more robust features, but ignore that it is intrinsically hard to directly regress an exact zero value for various background regions. Instead, some other methods [1] introduce a segmentation branch to exploit the highly discriminative ability of classification network. But they treat foreground with various densities as a single class, which ignores the large intra-class variation within foreground regions and leads to inferior background modeling accuracy. For the latter problem, [2] adopts a multi-column based feature fusion strategy without taking the corresponding relations between receptive field and density distribution into consideration, thus leading to significant feature redundancy. Differently, we bypass these defects by simultaneously solving the two problems with a novel hierarchically decoupled strategy under a unified framework, being more simple, effective and consistent.
We firstly decompose a background classification subtask from the whole density prediction task, inspired by its highly discriminative ability. Then, according to the spatial density distribution, the remaining foreground prediction sub-task is further hierarchically decomposed to several density-specific sub-tasks. The foreground related sub-tasks are solved by several regression based experts in a Foreground Density Estimation Module (FDEM). To guide the above decoupling process, we propose a Density Decoupling Module (DDM) which is supervised by a fine-grained classification loss. The proposed hierarchically decoupled strategy helps each task-specific expert to focus on their most skilled subtask, collaboratively contributing to the final prediction. In addition, three types of interaction strategies, including Feature Interaction, Gradient Interaction and Scale Interaction, are introduced for the use of the intrinsic relations among these sub-tasks. Combining with the above spirits, we propose an effective and compact counting model termed as Hierarchically Decoupled Network (HDNet), achieving state-ofthe-art performance on several popular benchmarks.
Contributions of our paper are summarized as follows:
• We propose a novel hierarchically decoupled strategy for crowd counting to simultaneously solve its two tricky problems, i.e., the cluttered background noise and the large density variation.
• We propose three types of interaction strategies to collaboratively integrate the decoupled components, yielding a compact and unified counting model. The impact from cluttered background noise attracts increasing attention in the literature but remains a tricky problem. Specifically, [4] introduces an explicit foregroundbackground segmentation to its multi-task architecture. Then the predicted foreground masks are used to form a better learning target. [5] trains a semantic prior network on the ADE20K dataset to reweight the feature maps for crowd counting, and the semantic prior helps to eliminate the side effect of noisy false alarms in background region. For the first time, [1] shows the importance of tackling with the noisy background problem by quantifying the counting errors from background region. Then they introduce a simple foreground segmentation branch to suppress background mistakes. However, previous works ignore the large intra-class variance within foreground regions and result in inferior background modeling accuracy. In this paper, we propose a simple and unified framework to deal with this problem, in which a finegrained supervision for foreground helps the accurate modeling for background region.
2.2. Density-aware Crowd Counting.
The major density variation problem is a long-standing challenge in crowd counting. To remedy this issue, some methods [6,7] try to fuse multiple columns convolutions with different kernel sizes or receptive fields to obtain density-invariant features. Despite their effectiveness, they ignore the corresponding relations between receptive fields and density distributions, thus leading to significant feature redundancy [2]. Besides, attention mechanism is also exploited to tackle with the density variation problem. Specially, DADNet [8] uses different dilated rates in each parallel column to obtain attention maps and multi-scale features. ADCrowdNet [9] designs an attention map generator to indicate the locations of congested regions for latter density map estimator. ASNet [10] learns attention scaling factors and automatically adjusts the density regions by multiplying density attention masks on them. These sub-tasks are optimized separately, which ignores the correlation between tasks. Differently, with the proposed hierarchically decoupled strategy, we distribute regions with various densities to multiple density-specific experts to collaboratively contribute to the final prediction, accounting for the internal relations between receptive field and density distribution whilst being more simpler.
THE PROPOSED APPROACH
In this section, we first describe the Hierarchically Decoupled Network, which is combined with the Density Decoupling Module and the Foreground Density Estimation Module. Then, we introduce the three interaction strategies including Feature Interaction, Scale Interaction, and Gradient Interaction.
Density Decoupling Module
As shown in Fig. 1, considering that feature maps on different levels have various resolutions, we firstly construct the feature X dec in decoupling branch with multi-level feature maps {X i } n i=1 from the backbone via upsampling. The decoupling head of our DDM is simply equipped with a 3×3 convolution block followed by a 1×1 convolution block. The convolution block consists of a convolution layer, a batch normalization layer and ReLU layer. Each decoupling head converts
X dec into M ∈ R H 4 × W 4 ×(n+1) ,
where n is the number of density levels, plus 1 for background. Note that, DDM only decouples foreground and background if we set n to 1. When n > 1, DDM is responsible not only for decoupling foreground and background but also for decoupling foreground into regions with multiple densities.
Accounting for the continuous spatial density variation across an image, we limit the range of activation values in M to [0, 1]. To this end, M is fed into a softmax function to get its soft outputM
= {M i } n i=0 , that iŝ M j,k i = e M j,k i n i=0 e M j,k i ,(1)
the value ofM j,k i represents a probability that it belongs to the i-th density level at spatial location (j, k). The soft output M is supervised by the ground-truth M gt as following:
L dec = L(M, M gt ),(2)
where L(·) denotes the cross-entropy loss. The ground-truth M gt is generated in the similar way as [11].
Foreground Density Estimation Module
We propose a Scale Adaptive Feature Fusion (SAFF) block to get multi-scale features{X i } n i=1 , which will be described in detail in later section.
The Foreground Density Estimation Module (FDEM) contains n density heads, and each density head is responsible for predicting image regions within a specific range of density. Similar to the architecture of the decoupling head, our each density head is implemented by a 3×3 convolution block followed by a 1×1 convolution block. The i-th density head takesX i as input and predicts the corresponding singlechannel density map to produce {D i } n i=1 . Then, in order to weaken the learning of the density prediction of the background region and focus on the different density regions of the foreground, we take the n foreground soft masks {M i } n i=1 as the attention maps to multiply the corresponding foreground density map D i . Then the final density map D is the sum of the prediction results of all density heads, and it is calculated as :
D = n i=1D i = n i=1 D i M i ,(3)
where denotes the Hadamard operation. The overall loss of the hierarchically decoupled framework is defined as
L = L reg + λL dec ,(4)
where L reg is L 2 loss between predicted density map and ground-truth density map, and λ is a weight which balances the importance between L reg and L dec .
Interaction in the HDNet
Density estimation and density decoupling are two highly correlated tasks in nature. Therefore, three types of interaction strategies are proposed to integrate these sub-tasks in consideration of the intrinsic relations among them, thus unifying the whole framework. Feature Interaction. We combine the density estimation and density decoupling into an unified framework with a shared backbone in an end-to-end manner. Through a joint optimization, this encourages the co-evolution of backbone features by using sharing weights for different tasks and reduces the parameters of the network. Scale Interaction. It is widely acknowledged that high resolution features with more detailed textures are useful to detect tiny objects while low resolution features with more contextual information are useful to suppress false alarms. However, for crowd counting, features with rich contextual information are also demonstrated useful for the estimation in highly congested regions [8]. Therefore, in order to make full use of multi-scale features {X i } n i=1 adaptively, we introduce the Scale Adaptive Feature Fusion (SAFF) block. Inspired by SENet [12], we use a learnable channel-wise parameter, w, to multiply with each transformed feature, through which it can learn to selectively emphasise informative features and suppress less useful ones from other layers. The output featureX is formulated aŝ
X i = X i + n k=1 w i,k F i,k (X k )1 [k =i] ,(5)
where w i,k ∈ R C is a channel-wise learnable parameter, C is the number of channel of X k , and F is an up-sampling or down-sampling operation according to the sizes of X i and X k . The up-sampling operation uses a set of 1×1 convolutional blocks and bilinear up-sampling layers, and the downsampling operation uses a set of 3×3 convolutional blocks with a stride 2. 1 [k =i] ∈ {0, 1} is an indicator function evaluating to 1 iff k = i. The transformed feature pyramid is in the same size with input features but much richer contexts than the original. Gradient Interaction. For the decoupling branch, it would be helpful to its learning if the density intensity in certain regions is aware of. While for the regression branch, it could pay less attention to regions which are classified as background in the decoupling branch. So we construct gradient interaction by multiplying soft masksM into intermediate density maps D, resulting in FDEM and DDM optimized jointly by the two losses simultaneously. This helps to learn density value regression and densityspecific classification jointly so that the density-aware knowledge contained in the regression task can be leveraged by the classification task. At the same time, by a joint supervision from the regression loss in FDEM, the density-aware knowledge could also be transferred to classification tasks.
EXPERIMENTS
In this section, we firstly describe the experimental settings about datasets, evaluation metrics, and implementation details. Then the effectiveness of the proposed DDM and FDEM is evaluated on the benchmark datasets. Finally, the performance of HDNet and the comparisons with state-of-the-art crowd counting estimators are presented.
Experimental setups
Datasets. We evaluate the HDNet on four most challenging datasets: ShanghaiTech PartA dataset [6], UCF-QNRF [13], JHU-CROWD++ [14], and NWPU-Crowd [15]. Evaluation metrics. As in previous works [6], we adopt the Mean Absolute Error (MAE) and the Mean Squared Error (MSE) to evaluate our method. The MAE and MSE are defined by
M AE = 1 N N i=1 D gt i − D i ,(6)M SE = 1 N N i=1 D gt i − D i 2 ,(7)
where N is the number of the test images, D gt i and D i are the ground-truth and estimated counts of the i th image, respectively. Training. We adopt HRNet [16] as the backbone and set λ = 1 to balances two losses in Eq. 2. SGD is used to optimize the model with the learning rate of 0.001, and the weight decay is set as 0.0005. The training batch size is 6. We resize the images to ensure that the longer side is 2048 for all datasets. The number of density levels is set to 3 in the experiments. Ground Truth Generation. Ground truth annotations for crowd counting typically consist of a set of coordinates that indicate the center point of the human head. We follow the standard procedure of the generation of ground truth [15], which converts the points to crowd density map using a Gaussian kernel with standard deviation of 15 pixels.
Ablation Study
In this section, we perform ablation studies on ShanghaiTech PartA dataset to analyze effectiveness of proposed modules. Foreground and Background Decoupling. To solve the cluttered background noise, we adopt the decoupling foreground and background method. We define a base estimator only with one density head as the baseline. As shown in first Table 2. Ablation study of feature interaction.
Foreground Density Decoupling. To demonstrate the effectiveness of decoupling foreground density, the foreground density decoupling further divides the foreground into multiple density levels according to its density. We define the baseline with Foreground and Background (FB) Decoupling and Foreground Density (FD) Decoupling as HDNet (n = 3, w/o SAFF). As shown in Table 1, it obtains an MAE of 54.61. Compared HDNet (n = 1, w/o SAFF), the MAE decreases by 4.2%, which proves the necessity of decoupling foreground density decoupling. FB Decoupling ignores the large intraclass variance within foreground regions, while FD Decoupling provides a fine-grained supervision. It helps the accurate modeling for background region owing to the decreasing intra-class variance. At the same time, FD Decoupling allows each task-specific expert to focus on their most skilled subtask, thus reducing the risk of overfitting. Fig. 2 shows examples of initial density maps D, soft masksM, intermediate density mapsD and final density map D. Density Decoupling and Interaction suppress noise in red rectangles successfully, which also make each expert of density estimation focus on the density-specific region. Feature Interaction. In order to verify the effectiveness of Table 4. Ablation study of gradient interaction.
Feature Interaction, we split the HDNet (n = 3, w/o SAFF) into two independent networks (i.e., FDEM and DDM). Here, each network has its own backbone. Compared to the HDNet with a sharing backbone, the two independent networks without Feature Interaction increase an MAE from 54.61 to 55.77, as shown in Table 2. This proves that the feature sharing of the two tasks can encourage the co-evolution of the backbone feature and achieve a better performance. Scale Interaction. Follow the setting in the previous paragraph, the HDNet (n = 3) without SAFF block obtains an MAE of 54.61, as shown in first row of
Comparisons with State-of-the-Arts
To demonstrate the effectiveness of our proposed approach, we compare our approach with state-of-the-art methods on four challenging datasets with various densities. The results are illustrated in Table 5. As we can see that our proposed approach is the state-of-the-art or close to state-of-the-art on the four challenging datasets. Fig. 3 Table 5. Comparison with state-of-the-art methods on four challenging datasets. Smaller number indicates better performance. In each column, the best result is blod, and the second best is underlined.
CANNet HDNet (Ours) Image GT ASNet from left to right columns on the ShanghaiTech PartA dataset. Compared with CANNet [18] (second column) and ASNet [10] (third column), the proposed HDNet significantly eliminates noises caused by cluttered backgrounds. For the second row, the dense crowd at the center, our method obtains a more discrete density prediction due to the use of the scale adaptive features with high-resolution and more context. For the last row, HDNet reserves more consistency with the real crowd distributions and is robust to occlusions. It shows that HD-Net is very powerful and achieves much more accurate count estimations.
CONCLUSION
In this work, we firstly propose a novel hierarchically decoupled strategy to simultaneously solve two long-standing problems: the cluttered background noise and the large density variation. Specifically, a Density Decoupling Module is proposed to guide this key decoupling process, which is supervised by a fine-grained density-aware learning target. Then the decoupled components are distributed to several task-specific experts according to their most skilled sub-task. As a complement to this effective decoupling strategy, three kinds of interaction strategies are proposed to collaboratively integrate those decoupled components. By combining these spirits together, we propose a compact, effective and unified counting model named as HDNet. The effectiveness of our contributions is demonstrated by the state-of-the-art performance on several dominant counting benchmarks.
Fig. 1 .
1The overall architecture of the proposed Hierarchically Decoupled Network mainly consists of two components: Density Decoupling Module and Foreground Density Estimation Module. Scale Adaptive Feature Fusion (SAFF) block is introduced to fuse high-resolution and rich context information.
Fig. 2 .
2Examples of initial density maps D, soft masksM, intermediate density mapsD and final density map D.
FB
Fig. 3 .
3The demonstration of different methods on the Shang-haiTech PartA dataset. GT means the ground truth.
Table 3 .
3Ablation study of SAFF.Gradient Interaction MAE
MSE
×
56.60 95.71
√
53.39 89.87
Table 3 .
3The SAFF without the channel-wise parameter, w, brings a slight performance improvement on MAE but a larger MSE of 93.63. The reason could be that the complete fusion of different resolution features results in feature redundancy. In the last row, a channel-wise parameter w is introduced in SAFF. The w se-FDEM and DDM in the training stage. As shown inTable 4, this change obtains an MAE of 56.60, while using soft masks with back propagation ability can get an MAE of 53.39. It shows that Gradient Interaction ensures a reciprocal optimization between FDEM and DDM by propagating gradient signals to each other. This leads to a better knowledge sharing among tasks and improve each other.lectively emphasises informative features and suppresses less
useful ones from other layers, achieving an MAE of 53.39
and an MSE of 89.87. This improved performance shows that
Scale Interaction can enable the network to adaptively fuse
complementary features, resulting in increasing the density-
aware ability of different density regions.
Gradient Interaction. To illustrate the improvement of the
gradient interaction, we change soft masksM in the DDM to
truncated binary masks, which cuts off the back propagation
between
compares the density maps of different methods MSE MAE MSE MAE MSE MAE MSEMethods
Venue
NWPU
UCF-QNRF
JHU++
ShTechA
MAE CSRNet [7]
CVPR 121.3 387.8 110.6 190.1 85.9 309.2 68.2 115.0
SANet [17]
ECCV 190.6 491.4
-
-91.1 320.4 67.0 104.5
DSSINet [18]
ICCV
-
-99.1 159.2 133.5 416.5 60.6 96.0
MBTTBF [19] ICCV
-
-97.5 165.2 81.8 299.1 60.2 94.1
BL [20]
ICCV 105.4 454.2 88.7 154.8 75.0 299.9 62.8 101.8
LSCCNN [21] TPAMI
-
-120.5 218.2 112.7 454.4 66.5 101.8
KDMG [22]
TPAMI 100.5 415.5 99.5 173.0 69.7 268.3 63.8 99.2
ASNet [10]
CVPR
-
-91.6 159.7
-
-57.8 90.1
LibraNet [23]
ECCV
-
-88.1 143.7
-
-55.9 97.1
DM-count [24] NIPS
88.4 357.6 85.6 148.3 68.4 283.3 59.7 95.7
NoiseCC [25]
NIPS
96.9 534.2 85.8 150.6 67.7 258.5 61.9 99.6
LA-Batch [26] TPAMI
-
-113.0 210.0
-
-65.8 103.6
UOT [27]
AAAI
87.8 387.5 83.3 142.3 60.5 252.7 58.1 95.9
SASNet [2]
AAAI
-
-85.2 147.3
-
-53.6 88.4
GLoss [28]
CVPR
79.3 346.1 84.3 147.5 59.9 259.5 61.3 95.4
HDNet (Ours)
-
76.1 322.3 83.2 148.3 62.9 276.2 53.4 89.9
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CSR-Net: Dilated Convolutional Neural Networks for Understanding the Highly Congested Scenes. Yuhong Li, Xiaofan Zhang, Deming Chen, CVPR. Yuhong Li, Xiaofan Zhang, and Deming Chen, "CSR- Net: Dilated Convolutional Neural Networks for Under- standing the Highly Congested Scenes," in CVPR, 2018.
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| []
|
[
"NAS-Navigator: Visual Steering for Explainable One-Shot Deep Neural Network Synthesis A Reduction Cell Candidate Information",
"NAS-Navigator: Visual Steering for Explainable One-Shot Deep Neural Network Synthesis A Reduction Cell Candidate Information"
]
| [
"Anjul Tyagi [email protected]. \nImaging Lab at Computer Science Department\nStony Brook University\nNew York\n",
"Cong Xie [email protected]. \nImaging Lab at Computer Science Department\nStony Brook University\nNew York\n",
"Klaus Mueller [email protected]. \nImaging Lab at Computer Science Department\nStony Brook University\nNew York\n"
]
| [
"Imaging Lab at Computer Science Department\nStony Brook University\nNew York",
"Imaging Lab at Computer Science Department\nStony Brook University\nNew York",
"Imaging Lab at Computer Science Department\nStony Brook University\nNew York"
]
| []
| Fig. 1. NAS-Navigator is visual analytics (VA) framework for explainable and human-in-the-loop neural network architecture search (NAS). NAS-Navigator implements a one-shot NAS, using an iterative evolutionary search algorithm. The interface supports the visualization of NAS with the human-in-the-loop search control. Analysts start by designing a large template network, through a lego view (A); capable of emulating the search space of candidate neural networks. This network is then trained for a few epochs to initialize meaningful weights, useful for candidate NN search, via a loss chart view (B). Following this, our evolutionary search algorithm evaluates possible candidate NNs iteratively, with iteration counter (C); sampled from the large NN, and these accuracy evaluation results are then presented in the form of a candidate NN projection on a scatterplot, via a search space view (D). Analysts can further pause/stop the search and edit the template NN based on the fitness scores generated by our search algorithm, on the candidate information view (E); to generate the final NN architecture, or to reduce the search space size. The fitness scores are calculated for each node of the candidate neural networks which are sampled from the large template network during the search.Abstract-The success of DL can be attributed to hours of parameter and architecture tuning by human experts. Neural Architecture Search (NAS) techniques aim to solve this problem by automating the search procedure for DNN architectures making it possible for non-experts to work with DNNs. Specifically, One-shot NAS techniques have recently gained popularity as they are known to reduce the search time for NAS techniques. One-Shot NAS works by training a large template network through parameter sharing which includes all the candidate NNs. This is followed by applying a procedure to rank its components through evaluating the possible candidate architectures chosen randomly. However, as these search models become increasingly powerful and diverse, they become harder to understand. Consequently, even though the search results work well, it is hard to identify search biases and control the search progression, hence a need for explainability and human-in-the-loop (HIL) One-Shot NAS. To alleviate these problems, we present NAS-Navigator, a visual analytics (VA) system aiming to solve three problems with One-Shot NAS; explainability, HIL design, and performance improvements compared to existing state-of-the-art (SOTA) techniques. NAS-Navigator gives full control of NAS back in the hands of the users while still keeping the perks of automated search, thus assisting non-expert users. Analysts can use their domain knowledge aided by cues from the interface to guide the search. Evaluation results confirm the performance of our improved One-Shot NAS algorithm is comparable to other SOTA techniques. While adding Visual Analytics (VA) using NAS-Navigator shows further improvements in search time and performance. We designed our interface in collaboration with several deep learning researchers and evaluated NAS-Navigator through a control experiment and expert interviews. | 10.1109/tvcg.2022.3209361 | [
"https://export.arxiv.org/pdf/2009.13008v3.pdf"
]
| 250,264,716 | 2009.13008 | 7cb4b322ff3953a521af72e10e9d8b20e7105949 |
NAS-Navigator: Visual Steering for Explainable One-Shot Deep Neural Network Synthesis A Reduction Cell Candidate Information
Anjul Tyagi [email protected].
Imaging Lab at Computer Science Department
Stony Brook University
New York
Cong Xie [email protected].
Imaging Lab at Computer Science Department
Stony Brook University
New York
Klaus Mueller [email protected].
Imaging Lab at Computer Science Department
Stony Brook University
New York
NAS-Navigator: Visual Steering for Explainable One-Shot Deep Neural Network Synthesis A Reduction Cell Candidate Information
F G H D E B C Normal Cell Search Space View Low High Not evaluated Node Information View Loss ChartIndex Terms-Deep LearningNeural Network Architecture SearchVisual AnalyticsExplainability • Anjul TyagiCong Xie and Klaus Mueller are with the Visual Analytics and
Fig. 1. NAS-Navigator is visual analytics (VA) framework for explainable and human-in-the-loop neural network architecture search (NAS). NAS-Navigator implements a one-shot NAS, using an iterative evolutionary search algorithm. The interface supports the visualization of NAS with the human-in-the-loop search control. Analysts start by designing a large template network, through a lego view (A); capable of emulating the search space of candidate neural networks. This network is then trained for a few epochs to initialize meaningful weights, useful for candidate NN search, via a loss chart view (B). Following this, our evolutionary search algorithm evaluates possible candidate NNs iteratively, with iteration counter (C); sampled from the large NN, and these accuracy evaluation results are then presented in the form of a candidate NN projection on a scatterplot, via a search space view (D). Analysts can further pause/stop the search and edit the template NN based on the fitness scores generated by our search algorithm, on the candidate information view (E); to generate the final NN architecture, or to reduce the search space size. The fitness scores are calculated for each node of the candidate neural networks which are sampled from the large template network during the search.Abstract-The success of DL can be attributed to hours of parameter and architecture tuning by human experts. Neural Architecture Search (NAS) techniques aim to solve this problem by automating the search procedure for DNN architectures making it possible for non-experts to work with DNNs. Specifically, One-shot NAS techniques have recently gained popularity as they are known to reduce the search time for NAS techniques. One-Shot NAS works by training a large template network through parameter sharing which includes all the candidate NNs. This is followed by applying a procedure to rank its components through evaluating the possible candidate architectures chosen randomly. However, as these search models become increasingly powerful and diverse, they become harder to understand. Consequently, even though the search results work well, it is hard to identify search biases and control the search progression, hence a need for explainability and human-in-the-loop (HIL) One-Shot NAS. To alleviate these problems, we present NAS-Navigator, a visual analytics (VA) system aiming to solve three problems with One-Shot NAS; explainability, HIL design, and performance improvements compared to existing state-of-the-art (SOTA) techniques. NAS-Navigator gives full control of NAS back in the hands of the users while still keeping the perks of automated search, thus assisting non-expert users. Analysts can use their domain knowledge aided by cues from the interface to guide the search. Evaluation results confirm the performance of our improved One-Shot NAS algorithm is comparable to other SOTA techniques. While adding Visual Analytics (VA) using NAS-Navigator shows further improvements in search time and performance. We designed our interface in collaboration with several deep learning researchers and evaluated NAS-Navigator through a control experiment and expert interviews.
INTRODUCTION
With the recent advances in computing power, deep learning (DL) has made it possible to automate the problem of feature engineering through neural networks (NNs). Highly complex features can be automatically learned from the data. However, this requires carefully designed DNN architectures, which transform the problem of feature engineering to architecture engineering [20]. Some well-studied DNNs like AlexNet [29] and ResNet [24] have been the results of extensive architecture search studies and required many hours of manual parameter tuning by experts. Most of the current automated approaches find the optimal solution of a NN architecture based on adaptive experiments [42], and most of them rely on strong computing power. As a result, these networks are hard to generalize because of the very high hardware equipment demands and associated costs. Network Architecture Search (NAS) techniques aim at alleviating these problems for deep learning researchers by automatically finding the best candidate NN architectures based on validation accuracy. NAS designed methods have outperformed manually curated networks as shown by Zoph et al. [42], Real et. al [44] and the SMASH model [7].
Typical NAS algorithms apply techniques like Reinforcement Learning (RL) [58,62,63] or Evolutionary Search (EA) [33,44] to search for candidate NNs directly. However, these approaches have been shown to be computationally very expensive. One-Shot NAS techniques aim at reducing the search time for NAS through training candidate neural networks via weight sharing. The idea is that instead of training each candidate NN separately, one trains a large template NN which is a super-set network of all candidates, and then uses the same weights to randomly sample candidate NN from this main network.
Most NAS algorithms use Recurrent NNs (RNNs) or DNNs as the backbone to run the search. These types of algorithms typically have a low capacity for explaining their actions and strategies [2,20]. Explainability, however, is critical when deploying real-world systems which have a high need for process auditing and often also have high legal liability [52]. In this work, we talk about explainability from the context of NAS, and NAS-Navigator focuses on adding explainability to the one-shot NAS process. Possible benefits of this procedure include the reasoning behind why a particular NN architecture was chosen over the others. Users can also compare how different candidate NNs behave on the data. Human-in-the-loop (HIL) assistance in NAS approaches can influence the choice of search in high stake decision making [22] and so assist human users in building trust into the process. The need for explainability and HIL is crucial for many situations and has been the subject of a long debate in the HCI community, commonly referred to as the control and automation trade-off [4,8,25,26,48,55,56].
Evolving from a formative study done with deep learning and NAS researchers, we developed NAS-Navigator to solve the issues revolving around these problems. NAS-Navigator is a visual analytics (VA) system that reconciles both automation and user control for NAS, where expert knowledge and automated intelligent services can be combined effectively. We present a One-Shot NAS algorithm developed using evolutionary search (EA) to support our explainability and HIL use cases. Evaluation results show that our EA algorithm is fast and more effective than typical One-Shot NAS algorithms. It learns to sample better candidates given the history of selected candidates and their validation accuracy data. We find that our scheme performs better than the random sampling strategy used in the existing One-Shot NAS techniques. Overall, our contributions include solutions to three problems with existing NAS techniques:
• One-Shot NAS search speed: Our One-Shot EA NAS algorithm provides faster search results • Explainability: Our VA interface NAS-Navigator supports search tracking and progress, search space visualization, candidate ranking, and score visualizations to provide cues to the users • Human-in-the-loop control: NAS-Navigator provides a usercontrollable HIL NAS paradigm, where users can improve the Our evaluation through a user study and expert interviews show that NAS-Navigator is effective in adding explainability and HIL to NAS. We separately evaluate our EA One-Shot NAS algorithm for its efficiency against SOTA methods. The results we obtained show better search convergence at a similar accuracy to the final candidate model compared to existing fully automated NAS techniques.
Our paper is organized as follows. Section 2 presents related work. Section 3 describes our formative inquiry with deep learning researchers and practitioners. Section 4 presents our Explainable One-Shot NAS method. Section 5 describes our visual interface. Section 6 presents our user study and its findings. Section 7 ends with conclusions.
RELATED WORK
We summarize several NAS research works by comparing them based on five factors as shown in Table 1. Overall, these methods can be divided into 3 categories; automated NAS, One-Shot NAS, and HIL NAS.
Automated NAS. Automated NAS has a long history [37,46]. NAS designed methods have outperformed manually curated networks as shown by Zoph et al. [42], Real et al. [44] and the SMASH model [7]. They use several trained networks to provide the final architecture design after evaluating each network on a validation set. DARTS [34] is another popular NAS algorithm that searches for good candidate NNs through gradient descent. However, training networks with NAS is expensive since many different networks have to be trained before evaluation. Also, these methods lack the human-in-the-loop control and visual analytics, supporting explainability, as shown in Table 1. To overcome this, another technique called the MorphNet [23] uses a different approach where the final architecture design is decided directly as a subset of a single hypernetwork where the candidate NNs share the same parameters. Liu et al. [33] proposed an evolutionary search algorithm for automated NAS without the weight sharing network search method. Evolutionary search is used to generate model architectures by manipulating operations and editing edges in the network.
One-Shot NAS. Following the work on MorphNets, a slightly different approach known as One-Shot Architecture Search [6] has evolved, which involves searching for the best neural network architecture as a subset of a largely trained hypernetwork. The hypernetwork in One-Shot NAS has a lesser number of parameters than training several different architectures independently [6]. Our EA algorithm used in NAS-Navigator is similar to previous one-shot approaches [7,17,60] where we train a hypernetwork to generate representative weights for every network in the search space (shared parameters). Although these algorithms are less resource-intensive than typical NAS algorithms, they still lack in the explainability and HIL aspects. Our EA algorithm is developed to sample optimal NN candidates along with explainability and HIL support to the typical one-shot NAS pipeline. Real et al. [44] proposed a one-shot algorithm specifically developed to generate a model AmoebaNet-A for hand sketches. The model performance is evaluated after generation by separate training. This is different in NAS-Navigator where we refer to that information from the template network and hence it is faster. The child models are generated by mutating the NNs from the highest accuracy models, whereas in our algorithm, children are generated from blocks with the highest fitness scores. Fitness scores include the history of that block and how it performed in all the previous models.
Interactive NAS. There has been significant research in the visualization community to make NN model selection and search more effective. Techniques exist to support NAS where the model parameters are known, and the model has to be evaluated only with a single dataset [9,39,47,61]. VA frameworks have also been proposed for HIL ML applications [30,54]. BEAMES [15] helps the users to find the best regression models for a given dataset iteratively. TreePOD [38] provides an interface to manage the trade-off between accuracy and interpretability of different existing ML models. REMAP [10] allows interactive CNN NAS starting with a few pre-trained models. Besides designing NNs, other tools allow interactive design and filtering of clustering techniques [11,31,40,45] and dimension reduction [5,14,27,35,41]. However, it is considerably harder to interactively optimize a DNN compared to optimizing a regression, clustering, or dimension reduction model; NAS-Navigator contributes by adding a HIL VA interface to one-shot NAS.
Out of all these existing techniques, the work by Cashman et al. [10], REMAP is the most closely related to NAS-Navigator. However, REMAP does a global search by evaluating some set of models in a given search space where the accuracies of each model on a given dataset are already known. Getting this initial data where accuracy information is known is resource-intensive, and is not easily available for different applications. Datasets like NAS benchmarks [19,49,59] are not available for most applications where deep learning is applied. Hence, in NAS-Navigator, using the one-shot technique, there is no requirement for an initial dataset where every candidate is pre-evaluated.
Another variation of NAS-Navigator from REMAP is in the ablation and variation phases. REMAP provides the user with options to drop some layers in the original NN for evaluation of different connections in the NN search space. In NAS-Navigator, we implement this operation automatically using the dropout operation during the search procedure. Also, users can still edit layers manually using the lego-view. Similarly, in the variation step of REMAP, users generate variations of different components of a candidate NN. NAS-Navigator handles that automatically through repeated components in the template network. Each layer in the template network consists of differently parameterized components. And the search procedure searches through all these combinations automatically.
Besides, there are other features in NAS-Navigator which are not available in existing works. The fitness scores of each block visualization allow the users to see which regions in the search space are impactful. For networks with skip connections, the lego view provides better visualization of the architecture. Also, users can easily set parameters using the parameter visualization sidebar, which can be easily viewed for each block. The search space view provides a single view to compare the accuracy and architectural similarities of the full search space.
FORMATIVE STUDY
To systematically evolve our idea of an explainable and HIL NAS framework, we first conducted a formative study to collect requirements from deep learning researchers, their views of explainable NAS, and general workflows. This approach helped concertize our framework and tool design with a user-centered evaluation at an earlier development stage.
The formative study participants were carefully chosen to be data analysts and researchers with different experiences, working in deep learning applications with a basic understanding of NAS techniques. Out of ten participants, two were deep learning professionals working in the industry, two were professors working in computer vision and NLP, and three were Ph.D. students working in computer vision and NLP, categorized as experts for this study (E1-E7). Three were graduate students in Computer Science studying deep learning with a basic understanding of NAS, categorized as non-experts for this study (NE1-NE3). Each participant was interviewed for about 45 minutes discussing their experience in DNNs and NAS. We covered the following topics during the interview, categorized as who proposed the design ideas.
• Their experience with the general NAS and One-Shot NAS workflows. • Principles, practices, and difficulties of NAS.
• Benefits and frustrations of existing VA machine learning tools.
Key Findings -Design Components
The purpose of the formative study was to gather a list of requirements from domain experts and potential users, which are expected to be met by our framework. Our many discussions culminated in the following list of requirements.
OUR EXPLAINABLE ONE-SHOT NAS METHODOLOGY
In this section, we discuss the technical details of designing a One-Shot NAS technique. NAS problems are often confined to predicting the structure of different subsets of a large template NN (known as cells) instead of the complete architecture designs. This strategy is shown to be more effective than finding complete architectures of DNNs [32,34,51,60]. These cells can be combined via an evaluation strategy to form a complete DNN structure, where NAS aims to find the structure of each of these cells in a DNN. A cell (in the context of CNNs) is a fully convolutional structure that maps an input tensor to an output tensor. Following the previous NAS works [33,50], two types of cell structures have been found useful in the context of designing CNNs, i.e. Normal Cells and Reduction Cells.
A normal cell has CNN components with a stride of 1, which maps the input size feature map of a given height, width, and the number of feature maps (H, W, F) to the same size output feature maps (H', W', F'). A reduction cell is used to reduce the height and width of the input feature map by a factor of 2, hence (H', W', F') = (H/2, W/2, 2F).
Each cell contains B number of nodes, with a default value of 4, they can be changed through the interface. Each node is connected to every other node through six operations (O's). These values are kept similar to past works in NAS [17,33] and are: 3x3 Max Pooling, 3x3 Average Pooling, Skip connection, 3x3 Separable Conv, 5x5 Separable Conv, and 1x3 then 3x1 Conv. Each node inside a cell takes two inputs (I 1 , I 2 ) and returns a transformed tensor T o = t 1 (I 1 ) + t 2 (I 2 ) where t refers to the transformed input tensor through an operation O. The task is to find which of these O's work best for every pair of connected nodes for every cell.
Once the cells are identified, the overall structure of the CNNs is created using the structures identified in [63]. In a typical NAS algorithm, the cell structures are fixed and users cannot change or skip the search of particular cells/operations based on evaluated candidates. However, NAS-Navigator through a VA interface, allows users to visualize these structures and make changes at any stage of the search. Users can edit the number of normal and reduction cells, and edit nodes (fix, remove and add) inside cells in real-time based on search progression results being displayed on the interface.
This idea of cell-based construction has been extended to transformer models related to non computer vision tasks. Different templates for transformer models have been suggested [21], which can be directly imported into NAS-Navigator to perform the search as the other computer vision model counterpart. Figure 2 shows our methodology for implementing an explainable HIL One-Shot NAS framework with NAS-Navigator. We explain all the five stages of the process in the following text. S1: Dataset. Users can choose a dataset using the menu bar on NAS-Navigator which controls the type of template networks the user can choose through the interface. While designing a template network, the structure of the network depends on the dataset and the global structure of the DNN. For example, CNNs for CIFAR10 and ImageNet have different template structures [63]. With the help of a formative study discussed in Section 3, we have designed template networks for 10 common datasets.
Methodology
S2: Design a template NN (T2). Through NAS-Navigator, users can edit the number of normal and reduction cells in the default template network; choose the number of nodes in a cell, and change the number of cells to control the depth of the template NN. To support the tasks where the default template network is not available, users can create their template network by combining components available in the sidebar (See F in Figure 1). The sidebar provides template structures for CNN and LSTM components which can be combined to create template networks. This satisfies the design component requirement (T2) discussed in Section 3.1.
S3: Train the template NN (T2). The next step after selecting the template network structure is to train the template NN for a few epochs. This is important to assign meaningful weights to each node of the network. Users can choose when to stop the training, based on time and resource usage demands. The template network train accuracy affects the search results, hence more training will result in better search results, but also higher training resource consumption. Also, using the cues from the search iterations, users can edit the template network to add or remove nodes or cells. This satisfies the design requirement (T2) from Section 3.1.
S4: Search Algorithm (T1, T3, T4). Training the template network assigns meaningful weights to each path (node and their respective operations inside a cell) in the network. After training, the purpose of the search algorithm is to evaluate these paths in the template NN cells and choose the operations with the highest fitness scores. This can be presented as a graph search algorithm where each candidate NN is a path in the super-graph connecting the start and end nodes. In every search iteration, a few of these subgraphs (candidate NNs) are evaluated for validation accuracy and corresponding nodes and operations contained in that candidate are ranked. This procedure ranks and helps finalize the best performing cell structure, and hence the candidate NN. Our EA-based search algorithm evaluates the candidate NNs in each iteration and learns from their validation accuracies, thus using this information to choose better candidates in the next iteration. The algorithm updates the candidate information view (E) and the search space view (D) as shown in Figure 1. More details on our search algorithm are discussed in Section 4.2. This satisfied the design requirement (T1, T3, and T4) formulated in Section 3.1.
S5: Final Architecture Evaluation. The last stage of this complete procedure is to evaluate the best candidates found with the help of the search algorithm. Users can choose to stop the search at any iteration based on the intermediate results and the amount of search space explored by the algorithm. The final model consists of one of the final candidates with the highest validation accuracy.
Evolutionary Search Algorithm.
We devised an evolutionary search algorithm to search for the candidate NN architectures with HIL and add explainability to the process. The stages of our EA search algorithm are summarized in Figure 3 and are discussed in detail below.
EA1: Selecting the candidate models. To select each candidate NN, we generate a bitmask vector with each bit corresponding to a path inside each cell, an example shown in Figure 3 for candidate NNs A and B. Path refers to the connection between blocks in the network. This mask represents a subgraph from the template NN which includes the paths corresponding to set bits. We generate a candidate NN from the mask by zeroing out the weights of the paths excluded from the template network. This way, only the paths with a corresponding set bit in the mask are activated. The number of candidate NNs in the population is heuristically set to 1.5 times the number of cells in the template neural network based on the studies shown in [12], that relate the dataset dimensionality with the population size in EA algorithms.
EA2: Calculating the fitness scores. To calculate the fitness of a candidate NN (subgraph), we calculate its validation accuracy and use Equation 1 to alter the fitness scores of the paths existing in the candidate NN. α is an accuracy threshold that we fix for our experiments, which serves as a tradeoff on how fast the search algorithm learns from the current accuracy scores. Following this, all the fitness scores for a particular cell are normalized by dividing the sum of all fitness scores from individual node fitness values.
individualPathFitness = ValidationAccuracy − α(1)
EA3: Choose Parents. The choose parents procedure returns the father and the mother candidate NNs from the fitness probability distributions of the population. This means that there is a higher probability of choosing the models with better fitness scores, as shown in the second block in Figure 3.
EA4: Cross-Over. To generate the child architectures from the father and mother NNs, a cross-over procedure is used, shown in Algorithm 1 (Cross-Over). Firstly, a mask is generated with the procedure described in the state EA1 above. This mask is compared to the masks of the father and mother models to generate a child mask. Both the father and mother models are chosen from the best performing candidates in the population. Also, the mask is generated from the probability density of the fitness scores, i.e. more probability of bits being set at indices pointing to well-performing paths.
EA5: Mutation. The child mask is mutated as shown in Algorithm 1 (Mutation), to generate a final child candidate. We chose the mutation rate to be 0.05 based on the work by Suganuma et al. [53].
EA6: Get Mask Probabilities. At each search iteration, when the new population is updated, each path in the template network is given a fitness score between 0 and 1, see part EA2 above. These values combined, form the probability distribution of the paths from which paths for the next candidate NNs are sampled.
Complete Evolutionary Search Algorithm. The complete search algorithm combines all these different stages in each iteration with a goal to find better candidates in each iteration, learning from previous candidates' performances. Algorithm 1 explains our complete EA search, also shown in Figure 3. (2). This template network consists of multiple repeated cells (normal and reduction cells, as described in Section 4) which are initially randomly initialized and are assigned meaningful weights after training the template network (3). Then one-shot search generates candidate NNs for evaluation from the template network (4), from which a final candidate is sampled iteratively; with the search process. During the search, users can edit the template NN using the search feedback (5); to make changes to the template NN architecture, which in-turn impacts the search progress and generation of candidate NNs (6).
THE INTERFACE
To implement our idea of explainable HIL NAS, we developed NAS-Navigator with the help of principles discovered during the formative study (Section 3). NAS-Navigator allows users to interactively control the search algorithm of our framework. As shown in Figure 1, our interface consists of six views, which we discuss in the following text.
The T# and S# next to each view show which of the formative study requirement and the One-Shot NAS stage that view satisfies.
Lego View (T1, T2, T3, S2, S4)
Shown in Figure 1(A)
Loss Chart (T1, T2, S3, S5)
As shown in Figure 1 (B), the loss chart allows users to visualize and control the training of template NN before running our EA search algorithm. As discussed in Section 4.1 (S3), training of the template NN is crucial to assign meaningful weights to the nodes. Users can make choices on how much to train the template NN based on this loss chart and depending on the resource availability for training. We can also view the final evaluation of the selected candidate NN in this view.
Search Space View (T1, T4, S4)
As shown in Figure 1 (D), the search space view is a scatterplot projection of candidate NN space. This is one of the key components of NAS-Navigator since it allows users to visualize and interact directly After the validation accuracies are available for these two candidates, the fitness scores associated with each path of the template NN cell are updated. In this example, all the paths had equal fitness scores before the search iteration, which are updated after the validation of candidate NNs. Based on these new fitness scores, a father and mother candidate NNs are a sample from the distribution of fitness scores, and a child mask is generated using the cross-over algorithm. This child mask is the candidate NN for the next search iteration. This way, our EA algorithm generates more child candidates coming from higher fitness score paths.
Algorithm 1 One-Shot Evolutionary Search Algorithm 1: procedure CROSSOVER(father, mother) 2: mask ← Randomly generated mask 3: childMask ← (mask == 0) * ( f ather.mask) + (mask == 1) * (mother.mask) 4: return childMask mask ← Uniformly generated numbers from 0 to 1 8: mutationRate ← 0.05 9: childMask ← newModelMask AND (mask > mutationRate) OR (1 − childMask) AND (mask <= mutationRate) 10: childModel ← getModel(childMask) 11: return childModel 12:
13: procedure ARCHITECTURE SEARCH 14: population ← Set of candidate models 15: loss ← Set of loss values 16: for each iteration do 17: population, loss, maskProb ← EVOLV E(population) 18: loss.append([max(loss), mean(loss), min(loss)]) 19: 20: procedure EVOLVE(population) 21: f itness ← fitness scores for each NN in population 22: newPop ← Top k NN from population with highest fitness 23: k <length(population) 24: for i=0 to length(population)-k step 1 do 25:
f ather, mother ← chooseParents(population, f itness) 26: newModel ← crossOver( f ather, mother) 27: newModelMask ← mutate(newModel) 28: newPop.append(newModelMask) 29: loss ← getLoss(newPop) 30: maskProb ← getMaskProb(newPop) 31: return newPop, loss, maskProb with the candidate NN space. These projections are obtained using t-SNE [36] and graph edit distance [1] on a randomly sampled set of candidate NNs. The sampled candidate NNs are subgraphs of the template NN with nodes labeled by the component or operation type, e.g. C for convolution and R for Relu. Using these labeled directed subgraphs, the distances between each of the sampled candidate architectures are calculated using the graph edit distance which is stored in a distance matrix. A t-SNE projection is generated from this distance matrix in 2-D shown as the search space view. This clusters the search space based on the architectural similarity of the candidate models. As the search algorithm progresses, these candidates are colored based on their evaluation accuracy. Hence, the search space view acts as a dual clustered space for architecture and accuracy similarity.
There are several user interactions supported in the search space view. Hovering over each dot highlights the candidate NN in the lego view. As the search iterations progress, the evaluated candidate NNs are colored based on their evaluation scores, see Figure 1 (D). This is useful to cluster the candidate search space, as more search iterations (shown as C in Figure 1) are elapsed, the search space view will be clustered based on the candidate performances. This can help the users to separate the search space into high-scoring and low-scoring candidate regions. Using this information, users can also select a region in the search space, which then limits the search algorithm to select candidates from the selected region for further iterations.
Another useful piece of information presented with the search space view is the set operations on nodes and operations. Users can drag areas on the scatterplot and find the Union, Intersection or Complement of the nodes and operations in that region. This is a helpful operation to find the most useful components of the search space which allows users to edit the template NN, thus impacting the search algorithm directly. For example, users can remove the most common cells (intersection) from a low-scoring search space region from the template NN, which reduces the search space, thus allowing for faster convergence of the EA algorithm.
Candidate Information View (T1, T3, S4)
As shown in Figure 1 (E), the candidate information view is a scatterplot showing the relationship between frequency and fitness scores of paths in the template NN. The frequency shows how many candidate NNs contain a particular path, hence more frequency score means higher occurrence. The fitness scores are assigned during each search iteration.
The idea behind the candidate information view is to show confidence in the fitness scores of the paths. For example, removing low-scoring, high-frequency paths from the template NN can prune the search space and help converge the EA algorithm faster. Also, this view helps analyze each operation path in the template NN and their respective fitness scores which have been accumulated over the search progression. Hovering over an operation in the lego view highlights the corresponding dot on the candidate information view. Similarly, dragging an area on the scatterplot highlights all the paths contained in that area on the lego view.
Menu Bar, Properties Sidebar (T2, S1, S2)
Shown in Figure 1 (F,G), the menu bar provides buttons for selecting the model, optimizers, loss functions, datasets, saving a model, and set operations (discussed in Section 5.3). The properties sidebar is linked with the lego view and displays the properties of each node and operation in the template NN. Users can change the parameters for each node through this sidebar.
EVALUATION
In this section, we evaluate NAS-Navigator for its effectiveness and design efficiency through a fully automated One-Shot NAS comparison with the state-of-the-art (SOTA), followed by case studies to show how HIL and VA can support better NAS. Finally, we evaluate the design experience of using NAS-Navigator through a user study and expert user interviews.
Comparison of SOTA and our EA One-Shot NAS algorithms
In this study, we evaluate the performance of our EA algorithm against the existing NAS techniques on ImageNet [16]. ImageNet is a largescale image classification dataset that has been extensively used for evaluating computer vision object detection research. The dataset contains 1.28 million training images with 50k validation images. For this study, we ran our EA algorithm for 1k iterations and experimented with different fitness-scores threshold values to create our final NN. Based on our experiments, the fitness-score threshold of 0.68 was used to obtain maximum accuracy on ImageNet with our model. We train our template network with a batch size of 256 for 400 epochs using an SGD optimizer. We use the setting similar to previous NAS training methods [17] for setting the learning rate to 0.025 and decay to zero using the cosine scheduler. The probability of dropout for our EA algorithm was set to 0.1. Table 2 shows the comparison of our algorithm against the existing techniques. We separate the existing techniques based on their efficiency of NAS along with human-derived NNs, shown under Task Category. As shown in the results, Table 2, our model obtained the best Top-5 Acc, the same as the previous best performing model GDAS [18] (best of 5 experiment runs). Other parameters for our model are comparable to the existing NAS techniques. Our search algorithm took about 1.7 GPU days on a Tesla V100 GPU to run for 200 iterations.
Case Studies
Compared to a fully automated study, we separately did case studies with real users to see the impact of HIL and VA on One-Shot NAS. The goal of this study was to compare the efficiency of our framework for search time, resource usage, and usability. We set a baseline accuracy range for our experiments which was devised based on the best performing existing NAS techniques and our automated EA algorithm. For each experiment, we noted the amount of GPU days it requires for our users to achieve that accuracy through our system. For this study, we used the CIFAR10 and CIFAR100 datasets [28] containing 60k images categorized into 10 and 100 categories for object detection.
Participants. We worked with six participants for this study, who were chosen based on their experience levels with Deep Learning on Computer Vision tasks with NAS techniques. 3 were categorized as experts and 3 as Non-experts according to their experience, with experts having experience working with NAS and DL for more than 3 years. The experts were Ph.D. students in Computer Vision and non-experts were graduate students working in Computer Science with basic knowledge of Deep Learning and Computer Vision. Out of the six participants, 4 were males and 2 were females. 2 experts and 2 non-experts were the users from the formative study, who proposed the design of the original system (See Sec 3). 3 experts used in the further evaluation are listed as E1-E3.
Task Description. We initially informed the participants of the concepts of NAS and related terminologies for our framework. Next, we showed them a few examples from the template networks and the filtering steps possible with our interface. The participants were then allowed to experiment with the framework and ask clarifying questions regarding the tasks. The task for the participants in this study was to achieve a baseline accuracy range on both the CIFAR datasets.
The accuracy values were decided based on existing NAS techniques and CNN models evaluated on these datasets, as shown in Table 3. To get the accuracy baselines for the case study tasks, we used the previous works from Table 1 and Table 2 which have published results on CIFAR10 and CIFAR100 datasets, separated into three categories of Human Experts, Older NAS techniques taking > 100 GPU days and newer one-shot techniques taking < 5 GPU days. The task for the users was to use NAS-Navigator and based on their knowledge and cues provided in the interface, get the accuracy results within the baseline range. The time taken for our expert and non-expert users to achieve this task is reported in Table 3.
Results. Table 3 shows the comparison of time and accuracies achieved by NAS-Navigator with Experts and non-expert users. We can see that using NAS-Navigator both the expert and non-expert users achieved the desired accuracies in 0.6 to 1.5 GPU days. For expert users, using domain knowledge and search pruning helped achieve the accuracies in 0.6-0.9 GPU days, which is considerably less than all the existing NAS techniques. This evaluates the efficacy of VA and HIL in NAS and the impact of domain knowledge in reducing the search times. Even for non-expert users, the time taken is less than our fully automated EA algorithm, achieving similar accuracies.
NAS-Navigator architecture search on Imagenet dataset
In this section, we discuss the architecture search process followed by one of the experts in the study (E1) for architecture search on Imagenet [29]. E1 was first given a short demonstration of using our interface followed by an explanation of the task to be performed. All search steps performed by E1 were logged along with the time taken for each search step. E1 started by loading a customized AlexNet into the interface, which has multiple options of blocks to choose from and follows the basic architecture model of AlexNet. This customized version of AlexNet contains the same number of layers as the original network but each layer has multiple blocks. For example, layer 1 has multiple convolutional blocks, i.e. Conv 3x3, Conv 5x5, and Conv7x7, and similarly for other layers. E1 explained that the reason for choosing the AlexNet template as a template network was his experience in using AlexNet for image classification tasks. After a careful understanding of the template AlexNet, E1 started by analyzing the results of the first 20 iterations of the evolutionary search algorithm, which gives a fitness value for each block of the template network. After the search results from the first four iterations, E1 decided to further analyze the 7x7 Conv blocks from Layers 1 and 2 because the fitness values of these blocks dropped to zero. Focusing on the search space view, E1 was able to find a subspace where the most common block was a7x7 Conv block in Layer1. E1 then dragged this region on the search space view which forced the search algorithm to sample candidate architectures from this search space. After 5 more search iterations, it was confirmed that the presence of this block resulted in a below-par performance of the neural networks; E1 decided to remove this block from the search space using the lego view and then continued the search further. Another 5 iterations of the evolutionary search suggested the removal of 3x3 convolution from Layer 1 and 5x5 convolution from Layer 2; these blocks were removed by E1. Additionally, the search results also suggested that 7x7 convolution was the best at Layer 3 on the evaluation dataset but E1 wasn't convinced about this result because of where the most common block was the 7x7 Conv block at Layer 3 to evaluate more candidate neural networks from this subspace. It was confirmed after a single search iteration that most neural networks from this search space had high accuracy, hence, giving further evidence that 7x7 convolution was the best option among the other blocks at layer 3. The search also suggested that the linear block with 2,304 input and 4,096 output parameters worked the best at layer 6. This yielded the final architecture of the suggested neural network. Results: E1 compared the results of the suggested network with a baseline of AlexNet performance on the Imagenet dataset after training for 10 epochs. While the baseline AlexNet has an accuracy of 72.70% on the test data, the network derived from our interface had an accuracy of 74.72%. This accuracy was further improved to 76.34% after E1 used his expertise and added batch normalization layer after every convolutional layer in the network. E1 was satisfied with the final network since it resulted in better accuracy than the baseline AlexNet model. This study confirmed that our tool can help computer vision researchers effectively search for and identify high-performing convolutional neural network architectures.
Interface Evaluation Through User Study
In this study, we evaluated NAS-Navigator for its support for multiple factors of usability and creativity. Considering that the main goal of creating NAS-Navigator was to add VA and HIL in NAS, we carefully evaluated our interface for its ability to support user thought processes and creativity. Since there are no existing frameworks publicly available that can be used as a suitable baseline for this task (see Table 1 user study helped us to explore the strengths and weaknesses of our interface through user feedback. Participants. The same participants, as described in Section 6.2 performed this user study. A total of six people participated, out of which three were considered Experts and three were non-experts.
Task and Procedure. The task was to answer questions regarding the procedure they applied in the Case Study discussed in Section 6.2 and rate their experience on a five-point Likert Scale from 1 (Strongly Disagree) to 5 (Strongly Agree).
The questionnaire was based on 7 factors, 5 of which are taken from the work by Cherry et al. [13] for quantifying the creativity support for design tools (Q1 to Q5 in Figure 4). We added two additional factors in the questionnaire to rate the domain-specific questions (Q6 and Q7 in Figure 4). The whole study took about 45 minutes for each participant.
Questionnaire Results
As shown in Figure 4, out of the total 42 ratings on 7 questions, only 4 ratings had a low score of 1 or 2. Overall, 81% of the votes rated the questions with a 4 or 5, with Q4 and Q5 being the highest-rated questions with the most Agree votes. Overall, positive feedback and high ratings for design and usability questions show the efficacy of our interface in supporting user creativity in NAS. However, detailed feedback was collected from the experts about the low-scoring questions and some directions of possible improvement in NAS-Navigator, discussed in the following text.
Expert Interview Results
Besides the general results, we separately collected detailed feedback from two of the experts (E2 and E3) about their experience in working with NAS-Navigator. This interview helped compare NAS-Navigator with the existing works from an expert's perspective and helped us gather deeper insights into the VA aspect of our framework. We discuss our results organized by the themes of the questionnaire in the following text.
Enjoyment (Q1). Both the participants found NAS-Navigator to be useful in their tasks. E2 explained "Drag and drop on the template NN was a great way to edit the search and get desired results". E3 added "I like that there are template NN provided for major DL tasks, which we can directly load on the interface and start playing with them." Exploration (Q2). Both the participants liked the search space view to exploring the candidate NN search space. E2 commented "Search Space view is a great idea and the fact that we can see clusters in the candidate NN search space shows how the structures of candidate NN can change the performance of the NNs." E3 suggested an improvement to NAS-Navigator commenting "A useful feature can be to suggest changes in the NN model based on current fitness scores for some nonexperts or in case the user has no prior idea on what NN will perform better on a given dataset."
Expressiveness (Q3). Both the experts suggested improvements in the expressiveness aspects of NAS-Navigator. E2 suggested "the user has great power to do pretty much any change with the interface, which can be great if they know what they are doing. However, in many cases, this can be a disaster if the user mistakenly updates something which later turned out to be useful." Adding on to this, E3 suggested, "it's a good idea for future will be to see the impact of user changes on the search results compared to the fully automated EA algorithm. This will allow the users to know the impact of their decision and will make the process more transparent."
Immersion (Q4). The participants were positive about the immersion part, suggesting a few improvements on top of the existing interactions. E2 commented "NAS processes are slower than the common design tasks, which are more commonly done through dashboards. Hence, the users cannot see immediate results of their actions in this case." E3 added to this comment and mentioned "While the users are waiting for the search iteration to complete, a notification will be useful to see if a particular search iteration has found something useful which can have some impact on the NN performance. Every search result, if it can be linked with user action and its impact, will be useful information to have on the interface."
Results worth effort (Q5). Both the participants agreed that even small interactions if done correctly, can have a great impact on the search convergence time and the final NN performance.
Interactions (Q6). The participants had mixed reactions to the interaction effects of NAS-Navigator. E2 commented "It would be great if we could track and visualize how the scores have changed over the search iterations. It'll further add to the decision-making as we will be able to see the history of changing scores and not just the last timestamp." E3 was satisfied with the interactions in NAS-Navigator and commented "The search space interactions are useful in controlling the areas to search from. This is a great idea and the fact that I can see all the candidate NNs in a single space is very useful."
Results Quality (Q7). Both the participants were satisfied with the quality of the results. An improvement was suggested by E2 who said "Sometimes the clusters in the search space view scatterplot are not very clear, maybe adding supporting plots to show further details of each network architecture would be useful."
DISCUSSION
Several important lessons were learned while designing this framework. Our initial discussion with domain experts was decisive in pinning down the main interface design. After all, tasks were formulated within comprehensive discussions with the experts, it was easier to design the visual interface and its components. Also, we realized that adding strong user interaction facilities was important, as a means to allow users to infuse their domain knowledge into the search process to accelerate convergence to the final solution. This design allows analysts to use their domain knowledge and the one-shot search results to quickly converge to the best performing neural network architecture for a given task. Analysts also have the freedom to apply certain soft constraints at their discretion, for example trading off between neural network size and accuracy, for example, different Resnet [24] sizes.
CONCLUSION
This paper presents a visual analytics framework to assist in deep neural network architecture search. Our interface combines the automated one-shot neural network architecture search approach with a humanin-the-loop design. Our interface is also less resource-intensive than conventional automatic neural network architecture search algorithms. Analysts can quickly load a template neural network along with their dataset and explore different subset neural network architectures to find the best one. Our evolutionary search algorithm allows for quick sampling of well-performing candidate architectures which can then be further evaluated for their performance. A design study was conducted in collaboration with several researchers working in the deep learning domain to lay down the tasks to be performed by our interface. We evaluated our framework for its ability to better search for the best performing neural network architecture with the help of a user study, case studies, and expert interviews. However, besides the effectiveness of our present interface, there remains some scope for improvement, which will be taken on in future work for this project. First, we would like to run comparison experiments to compare the performance of our EA algorithm against other solutions including reinforcement learning and bayesian optimization. Taking some points from our interview evaluation, we will add supporting visualization to the search space view to better present the clusters and candidate NN architectures inside these clusters. We will work on identifying user actions for each search iteration result, that will predict and suggest these actions to the non-expert users. Also, tracking of search iteration results and changes in the fitness scores will be added. We would also like to evaluate NAS-Navigator on language models to extend its applicability. Currently, all our evaluation is based on computer vision networks, but as more NAS algorithms start to come from the language domain, it will be interesting to see how NAS-Navigator compares against the state-of-the-art transformer models. These features are not yet supported and we will continue to work on our interface to incorporate them in the future. We plan to deploy NAS-Navigator for real users as a long-term study to collect in-depth feedback and usage scenarios.
Fig. 3 .
3Our EA One-Shot NAS search algorithm overview. Assuming two candidate NNs A and B sampled from a template NN cell, are evaluated in the current search iteration.
Table 1 .
1Comparison of NAS-Navigator with different NAS algorithms based on five aspects. VA stands for Visual Analytics, showing whether the technique supports HIL interaction with NAS. Eff shows if the search algorithm is efficient, i.e. the search time is less than 5 GPU days for a CNN. Sh stands for shared parameters, meaning whether the algorithm supports search through a shared parameter template network. Exp means explainable, comparing algorithms that support explainability in NAS. No PT means no pre-training of the candidate networks is required for the search algorithm. The table is divided into 4 sections of rows, separating the manually designed DNNs, automated NAS techniques, VA techniques for NAS, and our work (NAS-Navigator).Method VA Eff Sh Exp No PT
ResNet [24]
-
NASNet [63]
EAS [42]
PNAS [32]
SMASH [7]
DARTS [34]
SETN [17]
BEAMES [15]
-
-
REMAP [10]
TREEPOD [38]
-
-
NAS-Navigator
search through VA. Users can control the final NN architecture
depending on the resource-accuracy trade-off
• T1 :
T1HIL One-Shot NAS search. Develop a one-shot NAS algorithm that can be controlled through user feedback. Complete transparency on how the algorithm is searching through different NN candidates is important for explainability. Users should be given control of the search process via a VA interface. E1, E3, NE1, NE2, NE3. • T2: Template models VA. The design of hypernetworks (tem-T3: Candidate models VA. Through NAS-Navigator, users should be able to see which parts of the template NN combine to form a candidate NN for search. The interface should allow for comparing and contrasting different candidate NNs. E3, E4, E6, E7, NE1. • T4: Search Space VA. Users should be able to see a visualization of the candidate NNs search space. This is useful to compare how different candidates are related to each other. Some cues to validate candidate performance during search evaluation would be useful. This can be used to cluster the search into different regions based on validation accuracy. E4, E7, NE3.plate NNs) plays a crucial role in the results of One-Shot NAS
techniques. Through NAS-Navigator, users should be able to
design and edit template models, make choices based on their
experience and search progression feedback. E2, E4, E5, NE2.
•
Fig. 2. Stages in a one-shot NAS search process are mainly divided into two parts, template network training, and the search algorithm to find possible candidates from the trained template NN. After loading the dataset (1), a template network is trained (shown as the transition from dashed lines to solid lines in the figure) with shared parametersTemplate network training
Search NN candidates
1-Load Dataset
2 -Design a template NN
3 -Train
template NN
4 -Search and
evaluate candidate
NN
5 -Edit template NN
6 -Generate Final NN
, the goal of the Lego view is to allow editing of the template NN. Users can control the network depth, edit different components and visualize how the components are placed in this view. Controlling the template NN gives the power to the users to control the search. Also, users can visualize candidate NNs, which are a subgraph of the template NN in this view, where the cells and the nodes contained inside a candidate NN can be highlighted over the template NN.
Table 2 .
2Comparing our EA algorithm to existing State-of-the-art NAS techniques and human designed CNNs on ImageNet. Task Category groups
the models based on their NAS effectiveness, separately placing the human designed CNNs. We run our EA algorithm on Testla V100 GPUs for 200
iterations, achieving best Top-5 Acc with our experiments, similar to GDAS. The experiments took approximately 5 hrs on NAS-Navigator with 8
GPUs.
Task Category
Method
GPU
days
Parameters (MB)
Top-1 Acc
Top-5 Acc
Human Experts
ResNet [24]
-
11.7
69.8
89.1
Inception-v1 [3]
-
6.6
69.8
89.9
NAS with more than 100 GPU days
Progressive NAS [32]
150
5.1
74.2
91.9
NASNet [43]
2000
5.3
72.8
91.3
NAS with less than 5 GPU days
DARTS [34]
4
4.9
73.1
91
SNAS [57]
1.4
4.3
72.7
90.8
GDAS [18]
0.85
5.3
74
91.5
SETN [17]
1.8
5.3
74.1
91.4
NAS-Navigator (T=200)
1.7
5.3
73.9
91.5
Table 3. Evaluating our HIL One-Shot NAS framework with fully automated NAS techniques. We categorize existing NAS techniques and CNNs
into three categories shown as first three rows in Methods. The time taken by users through NAS-Navigator to achieve comparable accuracies
on CIFAR10 and CIFAR100 datasets is less than existing fully automated techniques. This shows the importance of HIL and VA in reducing the
resource footprint of NAS. The experiments took approximately 3.5 hrs on NAS-Navigator with 8 Tesla V100 GPUs.
Method
GPU days range Params (MB)
range
Error on CIFAR 10
Error on CIFAR 100
Human Experts
-
24-30
3.38 ± 0.8
21.76 ± 6.14
NAS > 100 GPU days
150-2000
3.3-10.6
3.52 ± 0.34
21.62 ± 6.34
NAS < 5 GPU days
0.84-50
3.3-5.7
3.25 ± 0.8
18.5 ± 2.7
NAS-Navigator (No VA)
(T=1K)
1.4-2.0
3.3
2.6 ± 0.1
17.8 ± 0.3
NAS-Navigator (Experts)
0.6-0.9
3.1 -3.7
2.83 ± 0.2
17.82 ± 0.7
NAS-Navigator
(Non-
Experts)
0.8-1.5
2.9-4
3.2 ± 0.3
18.6 ± 0.5
his experience. Hence, E1 selected a region in the search space view
), thisFig. 4. The user study results to evaluate our framework with the Creative Support Index[13], on a five-point Likert scale. We asked 7 questions to the participants, covering different aspects of usability.Q4: Immersion
Q5: Results worth effort
Q2: Exploration
Median
IQR
Q1: Enjoyment
Q3: Expressiveness
Q6: Interactions
User Study Results
© 2022 IEEE. This is the author's version of the article that has been published in IEEE Transactions on Visualization and Computer Graphics. The final version of this record is available at: xx.xxxx/TVCG.201x.xxxxxxx/
ACKNOWLEDGEMENTSWe would like to thank the anonymous VIS 2022 reviewers for their valuable comments. This work was partially funded by NSF grants CNS 1900706, IIS 1527200 and 1941613, and NSF SBIR contract 1926949.
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| []
|
[
"Weak notions of nondegeneracy in nonlinear semidefinite programming",
"Weak notions of nondegeneracy in nonlinear semidefinite programming"
]
| [
"Roberto Andreani ",
"Gabriel Haeser ",
"Leonardo M Mito ",
"Héctor Ramírez "
]
| []
| []
| The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, ℓ-dimensional kernel of the constraint matrix, by the linear independence of a set of ℓ(ℓ + 1)/2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of ℓ derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. We use some of these ideas to revisit an approach of Forsgren [Math. Prog. 88, 105-128, 2000] for exploiting the sparsity structure of a transformation of the constraints to define a constraint qualification, which led us to develop another relaxed notion of nondegeneracy using a simpler transformation. If the zeros of the derivatives of the constraint function at a given point are considered, instead of the zeros of the function itself in a neighborhood of that point, we obtain an even weaker constraint qualification that connects Forsgren's condition and ours. | 10.1007/s10107-023-01970-4 | [
"https://arxiv.org/pdf/2012.14810v2.pdf"
]
| 229,923,513 | 2012.14810 | ff0d620bd73e56dfc918b8657fdfce8f9c773244 |
Weak notions of nondegeneracy in nonlinear semidefinite programming
15 Mar 2022 March 16, 2022
Roberto Andreani
Gabriel Haeser
Leonardo M Mito
Héctor Ramírez
Weak notions of nondegeneracy in nonlinear semidefinite programming
15 Mar 2022 March 16, 2022arXiv:2012.14810v2 [math.OC]Semidefinite programmingConstraint qualificationsConstraint nondegeneracy
The constraint nondegeneracy condition is one of the most relevant and useful constraint qualifications in nonlinear semidefinite programming. It can be characterized in terms of any fixed orthonormal basis of the, let us say, ℓ-dimensional kernel of the constraint matrix, by the linear independence of a set of ℓ(ℓ + 1)/2 derivative vectors. We show that this linear independence requirement can be equivalently formulated in a smaller set, of ℓ derivative vectors, by considering all orthonormal bases of the kernel instead. This allows us to identify that not all bases are relevant for a constraint qualification to be defined, giving rise to a strictly weaker variant of nondegeneracy related to the global convergence of an external penalty method. We use some of these ideas to revisit an approach of Forsgren [Math. Prog. 88, 105-128, 2000] for exploiting the sparsity structure of a transformation of the constraints to define a constraint qualification, which led us to develop another relaxed notion of nondegeneracy using a simpler transformation. If the zeros of the derivatives of the constraint function at a given point are considered, instead of the zeros of the function itself in a neighborhood of that point, we obtain an even weaker constraint qualification that connects Forsgren's condition and ours.
Introduction
The study of linear and nonlinear semidefinite programming (for short, SDP and NSDP, respectively) problems has been consistently growing over the last decades. There are several models for real world problems that can be reformulated as SDPs or NSDPs (we refer to the handbooks [9,Part 4] and [29, Part 3] for a vast collection of applications), which motivate and are motivated by the development of theoretical results regarding optimality conditions and constraint qualifications (CQs) for (N)SDPs. Loosely speaking, CQs are assumptions over the feasible set of an optimization problem that ensure that it can be locally described in terms of its first-order approximation. This leads to the possibility of characterizing all solutions of an (N)SDP problem in terms of the derivatives of the functions that describe it, which gives CQs a pivotal role in building convergence theories for practical algorithms. The standard way to do this is to prove that every feasible limit point of the output sequence of the algorithm satisfies the Karush-Kuhn-Tucker (KKT) conditions under a given CQ. Thus, employing a weaker CQ leads to a more robust convergence theory.
One of the most relevant CQs in the literature of (N)SDP is the so-called nondegeneracy (or transversality) condition, introduced by Shapiro and Fan in [26,Sec. 2] in the context of eigenvalue optimization, and later reformulated by Shapiro [24,Def. 4] for general NSDPs. This condition has been widely used for characterizing sensitivity results (see, for instance, [13,16,18,19,20,27]), and also for proving global convergence and the rate of convergence of numerical algorithms (we refer to Yamashita and Yabe [30,Secs. 3,4,and 5] for a survey on this topic). However, it is known that even in the linear case, the solutions of large scale SDP problems tend to be degenerate, even though nondegeneracy is expected to hold in a generic sense. Besides, when the constraint of an NSDP problem has some sparsity structure near one of its solutions -for instance, a diagonal structure -then nondegeneracy is not satisfied at that solution [24]. This means that the convergence theory of an algorithm supported by nondegeneracy does not cover such points.
The explanation for such kind of issue, in our opinion, is the degree of generality of the nondegeneracy condition. That is, although it was born in NSDP, nondegeneracy does not capture any particularity of the constraints, being straigthforwardly extended for any general conic optimization problem, as long as the cone is closed and convex. However, embedding specific traits of matrix-valued functions into nondegeneracy may be more or less direct, depending on how it is characterized. For example, it is well known that (block-)diagonal problems can be remodelled as multiple potentially dense constraints, such that the nondegeneracy condition, when applied to this remodelled problem, may hold. But what about other types of sparsity? While this question has once been addressed by Forsgren [12], his approach is somewhat intricate and it was not the main topic of his paper, leaving room for a more dedicated analysis. In this paper, instead of defining nondegeneracy as the transversality of two particular subspaces -which is the most usual definition -we exploit an equivalent characterization by Shapiro [24,Prop. 6], which is phrased in terms of the gradients of the entries of an isolated "active block" of the constraints. One particularly interesting detail about this characterization is that it treats all representations of such an "active block" equally, but we show that some of them are more meaningful than others.
The contributions of this paper revolve around the following results:
• We provide a new characterization of nondegeneracy that induces a weaker variant of it, here called weak-nondegeneracy, which uses information of the eigenvectors of the constraints evaluated at nearby points;
• We incorporate a sparsity treatment in [24,Prop. 6], which leads to another weak variant of nondegeneracy, called sparse-nondegeneracy.
• We connect sparse-nondegeneracy with Forsgren's CQ by means of replacing, in both conditions, the strucutural zeros of the constraint function in a neighborhood of a point, with the zeros of the gradients of its entries at such point. This new condition happens to be a constraint qualification also, which we call gradient sparse-nondegeneracy.
These conditions are designed with the sole goal of assisting in proving global convergence of algorithms by means of sequential optimality conditions [3,8]; however, we envision that they may be further employed in sensitivity analysis, second-order analysis, among other applications. All variants of nondegeneracy we present are proved to be constraint qualifications strictly weaker than nondegeneracy. We also show that when our conditions are applied to diagonal matrices, they are reduced to the linear independence constraint qualification (LICQ) from nonlinear programming (NLP). More generally, the conditions are invariant to block representations of (N)SDP problems as a single semidefinite block diagonal matrix or as multiple semidefinite constraints. Then, we compare our definitions with other CQs from the literature. This paper is structured as follows: In Section 2, we introduce our notation; in Section 3 we recall the nondegeneracy condition and we prove a new characterization of it, which is where the definition of weaknondegeneracy comes from. In Section 4, we present our definition of sparse-nondegeneracy and a relaxation of it with distinct properties. Finally, in Section 5, we discuss some possibilities for prospective work.
Preliminaries
Let f : R n → R and G : R n → S m be continuously differentiable functions, where S m is the linear space of all m×m symmetric matrices, and let S m + be the closed convex pointed cone of all m×m positive semidefinite matrices. The problem of interest in this paper is the following:
Minimize x∈R n f (x), subject to G(x) 0,(NSDP)
where is the partial order induced by S m + , characterized by the relation:
M N ⇔ M − N ∈ S m + , for all M, N ∈ S m .
It is worth pointing out that all results in this paper can be straightforwardly extended to NSDP problems with separate equality constraints, but we omit them for simplicity. The feasible set of (NSDP) will be denoted by F . = G −1 (S m + ). It is well-known that S m is an Euclidean space when equipped with the (Frobenius) inner product M, N .
= trace(M N ) . = m i,j=1
MijNij . The derivative of G at a point x ∈ R n is the linear mapping DG(x) : R n → S m that can be described (in the canonical basis of R n ) by the action
d → DG(x)[d] . = n i=1 Dx i G(x)di
for all d = (d1, . . . , dn) ∈ R n , where Dx i G(x) ∈ S m is the partial derivative of G with respect to the variable xi at x = (x1, . . . , xn) ∈ R n . Also, for each fixed x, the adjoint of DG(x) is the unique linear mapping
DG(x) * : S m → R n that satisfies DG(x)[d], M = d, DG(x) * [M ]
, for all (d, M ) ∈ R n × S m . Hence,
DG(x) * [M ] = Dx 1 G(x), M . . . Dx n G(x), M = m i,j=1
Mij ∇Gij (x) for all M ∈ S m , where ∇Gij(x) denotes the gradient of the (i, j)-th entry of G as a function of x. Similarly, we shall denote the gradient of any real-valued function F : R n → R at a point x ∈ R n by ∇F (x). For any given M ∈ S m , we consider its spectral decomposition in the form
M = m i=1 λi(M )ui(M )ui(M ) ⊤ ,
where λi(M ) ∈ R denotes the i-th eigenvalue of M arranged in non-increasing order (that is, λ1(M ) λ2(M ) . . . λm(M )), and ui(M ) ∈ R m corresponds to any associated eigenvector such that the set {ui(M ) : i ∈ {1, . . . , m}} is an orthonormal basis of R m (that is, ui(M ) T ui(M ) = 1 and ui(M ) T uj(M ) = 0 when i = j, for all i, j ∈ {1, . . . , m}).
A useful fact for our analyses is that the orthogonal projection of M onto S m + with respect to the induced (Frobenius) norm, denoted by Π S m + (M ), can be characterized in terms of its spectral decomposition as follows:
Π S m + (M ) = m i=1 [λi(M )]+ui(M )ui(M ) ⊤ , where [λ]+ . = max{0, λ} for all λ ∈ R.
Given any x ∈ F and any orthogonal matrix U ∈ R m×m whose columns are eigenvectors of G(x), we partition U = [P , E] such that the columns of P ∈ R m×r correspond to the eigenvectors associated with the positive eigenvalues of G(x) and the columns of E ∈ R m×m−r correspond to the eigenvectors associated with the null eigenvalues of G(x), where r = rank(G(x)). To abbreviate, as an abuse of notation and language, we will say that E spans Ker G(x) in this context. That is, E spans Ker G(x) if, and only if, E ⊤ E = Im−r and G(x)E = 0, where Im−r denotes an (m − r)-dimensional identity matrix.
There are multiple ways of describing optimality in NSDP problems, but in this paper we direct our attention to necessary optimality conditions that are based on the classical Karush-Kuhn-Tucker (KKT) conditions:
Definition 2.1 (KKT). We say that a point x ∈ F satisfies the KKT conditions when there exists some Y 0 such that ∇xL(x, Y ) . = ∇f (x) − DG(x) * [Y ] = 0, G(x), Y = 0,(KKT)
where L : R n × S m → R is the Lagrangian function of (NSDP), given by
L(x, Y ) . = f (x) − G(x), Y .
As usual, the matrix Y is called a Lagrange multiplier associated with x and we denote the set of all Lagrange multipliers associated with x by Λ(x). When Λ(x) = ∅, x is called a KKT point of (NSDP). Let r be the rank of G(x) and let E ∈ R m×m−r be a matrix that spans Ker G(x); then, for any Y ∈ Λ(x), since both Y and G(x) are positive semidefinite, the complementarity relation G(x), Y = 0 is equivalent to G(x)Y = 0, which is in turn equivalent to saying that Im Y ⊆ (Im G(x)) ⊥ = Ker G(x), where (Im G(x)) ⊥ denotes the orthogonal complement of Im G(x). Therefore, Y is complementary to G(x) if, and only if, it has the form
Y = EỸ E ⊤ ,(1)
whereỸ ∈ S m−r + is not necessarily a diagonal matrix. Moreover, note thatỸ is not necessarily positive definite; that is, dim(Ker Y ) does not necessarily coincide with r. When they do coincide, x and Y are said to be strictly complementary [24].
It is known that the KKT conditions are not necessary for local optimality unless they are paired with a constraint qualification. For instance, one of the most studied constraint qualifications for (NSDP) is Robinson's CQ [23,Def. 3], which holds at a point x ∈ F if there exists d ∈ R n such that
G(x) + DG(x)[d] ∈ int S m + ,
where int S m + denotes the topological interior of S m + , which in turn coincides with the set of m × m symmetric positive definite matrices. Alternatively, following Bonnans and Shapiro [10,Prop. 2.97], it is possible to say that (the dual form of) Robinson's CQ holds at x ∈ F if, and only if,
DG(x) * [Y ] = 0 G(x), Y = 0 Y 0 ⇒ Y = 0.(2)
Another well-known fact is that, for every local minimizer x ∈ F, the set Λ(
The nondegeneracy condition for NSDP
In this section, we discuss the well-known nondegeneracy condition introduced by Shapiro and Fan [26,Sec. 2]. We derive a different characterization for it that suggests a way of obtaining a weaker constraint qualification with potentially interesting properties. But firstly, we briefly recall some elements of convex analysis.
The (Bouligand ) tangent cone to a set C at a point y ∈ C is defined as
TC(y) . = d : ∃{d k } k∈N → d, ∃{t k } k∈N → 0, t k > 0, ∀k ∈ N, y + t k d k ∈ C .
In particular, when C = S m + , at a given M ∈ S m + , it can be characterized as follows
T S m + (M ) = N ∈ S m : d ⊤ N d 0, ∀d ∈ Ker M .
Therefore, for every feasible x we have
T S m + (G(x)) = N ∈ S m : E ⊤ N E 0 ,(3)
whenever E spans Ker G(x). It is clear from (3) that the largest subspace contained in T S m + (G(x)), that is, its lineality space, can be characterized as follows:
lin(T S m + (G(x))) = N ∈ S m : E ⊤ N E = 0 .(4)
The nondegeneracy condition of Shapiro and Fan is verified at x when the linear subspaces Im DG(x) and lin(T S m + (G(x))) of S m meet transversally, which is why it was originally called transversality in [26]. In mathematical language: Definition 3.1 (Def. 4 from [24]). A point x ∈ F is said to satisfy the nondegeneracy condition when the following relation is satisfied:
Im DG(x) + lin(T S m + (G(x))) = S m .(5)
If x is a local solution of (NSDP), then nondegeneracy implies that Λ(x) is a singleton; and the converse is also true in the presence of strict complementarity (see [25,Thm. 2.2 and Sect. 3]). Hence, Definition 3.1 is generally seen as an analogue of LICQ, from NLP, in NSDP. However, this analogy is tied to how the link between NLP and NSDP is made [24]. For example, when an NLP problem with constraints g1(x) 0, . . . , gm(x) 0 is modelled as an NSDP with a single structurally diagonal conic constraint; that is, with G in the form
G(x) . = g1(x) . . . gm(x) 0;(6)
then Definition 3.1 fails whenever there is some Y ∈ Λ(x) and some nonzero H ∈ S m with only zeros in its diagonal, such that H −Y , regardless of the linear independence of the set {∇g1(x), . . . , ∇gm(x)}. In fact, structurally diagonal NSDP problems are in general expected to lack uniqueness of the Lagrange multiplier. On the other hand, it is well-known (cf. [10, Sect. 4.6.1]) that a feasible point x satisfies the nondegeneracy condition if, and only if, either Ker G(x) = {0} or the linear mapping ψx : R n → S m−r , defined by ψx(d)
. = E ⊤ DG(x)[d]E,(7)
is surjective for any E that spans Ker G(x). As a direct consequence of the equivalence above, it is possible to characterize Definition 3.1 as follows:
vij (x, E) . = e ⊤ i Dx 1 G(x)ej, . . . , e ⊤ i Dx n G(x)ej ⊤ = DG(x) * eie ⊤ j + eje ⊤ i 2 , 1 ≤ i ≤ j ≤ m − r(8)
are linearly independent, where E ∈ R m×m−r is an arbitrary fixed matrix that spans Ker G(x), and ei denotes the i-th column of E, for all i ∈ {1, . . . , m − r}.
Now, inspired by Proposition 3.1, we present a similar characterization of nondegeneracy that evaluates the linear independence of a narrower set of vectors at the cost of looking at all possible choices of E instead of a fixed one. Since our reasoning can be extended to Robinson's CQ, we also characterize it in a similar manner. is linearly independent for every matrix E ∈ R m×m−r that spans Ker G(x).
2.
Robinson's CQ if, and only if, either r = m or (9) is positive linearly independent for every matrix E ∈ R m×m−r that spans Ker G(x).
Proof. Let us assume that r < m since the result follows trivially otherwise. Then, for any fixed E ∈ R m×m−r such that G(x)E = 0 and E ⊤ E = Im−r, note that (9) is (positive) linearly independent if, and only if, the following holds: if the scalars α1, . . . , αm−r ∈ R (with α1 0, . . . , αm−r 0, respectively) satisfy
m−r i=1 αiDG(x) * [eie ⊤ i ] = m−r i=1 αivii(x, E) = 0,(10)
then one must have α1 = . . . = αm−r = 0. That is, (9) is (positive) linearly independent if, and only if, for every matrix Y of the form
Y . = m−r i=1 αieie ⊤ i = E α1 . . . αm−r E ⊤(11)
where α1, . . . , αm−r ∈ R (with α1 0, . . . , αm−r 0, respectively), we have that
DG(x) * [Y ] = 0 ⇒ Y = 0.(12)
With this in mind, we recall that:
• For any fixed choice of E spanning Ker G(x), nondegeneracy holds at x if, and only if, (12) holds for every Y in the form Y = EZE ⊤ with Z ∈ S m−r (Proposition 3.1);
• Robinson's CQ holds at x if, and only if, (12) holds for every Y 0 such that G(x), Y = 0 (see (2));
and it becomes clear that nondegeneracy (respectively, Robinson's CQ) implies that (9) is (positive) linearly independent, for every E as described above, because every Y as in (11)
satisfies G(x), Y = 0.
To prove the converse of item 1, assume that (9) is linearly independent for all E that spans Ker G(x). Let Y = EZE ⊤ be such that Z ∈ S m−r and let C ∈ R m−r×m−r be an orthogonal matrix such that C ⊤ ZC = Diag(z1, . . . , zm−r), where Diag(z1, . . . , zm−r) ∈ S m−r is a diagonal matrix whose i-th diagonal entry is zi, with i ∈ {1, . . . , m − r}. Then, note that EC ⊤ also spans Ker G(x), which puts (11); by our previous assumption (12) holds for Y and we conclude that nondegeneracy holds at x. Now, to prove the converse of item 2, assume that (9) is positive linearly independent for all E that spans Ker G(x), and let Y be such that DG(x) * [Y ] = 0, G(x), Y = 0 and Y 0. It is elementary to see that there exists some matrix E that spans Ker G(x), such that Y has the form (11). It follows from our hypothesis that Y = 0 and because Y is arbitrary, Robinson's CQ holds at x.
Y = EC ⊤ Diag(z1, . . . , zm−r)(EC ⊤ ) ⊤ in format
The characterizations of nondegeneracy and Robinson's CQ from Proposition 3.2 may seem less practical than the one from Proposition 3.1, but it reveals a clear path for defining weaker CQs by ruling out some particular choices of E, which is the main result of the next subsection.
We recall that Wachsmuth [28] proved for NLPs that LICQ is equivalent to the uniqueness of the Lagrange multiplier for any objective function f (the unique multiplier may vary with f ). Thanks to Proposition 3.2 this characterization can be straightforwardly extended to NSDP replacing LICQ by nondegeneracy, which we omit.
Sequences of eigenvectors and weak-nondegeneracy
In [8], Andreani et al. introduce a constructive technique for proving the existence of Lagrange multipliers for (NSDP), which is based on the so-called sequential optimality conditions from NLP [3]. The core idea of their proof is to apply an external penalty algorithm to (NSDP) after regularizing it around a given local minimizer, to obtain a sequence of approximate KKT points converging to it, as follows:
Theorem 3.1 (Thm. 3.2 from [8])
. Let x be a local minimizer of (NSDP). Then, for any sequence {ρ k } k∈N → +∞, there exists some {x k } k∈N → x, such that for every k ∈ N, x k is a local minimizer of the regularized penalty function
f (x) + 1 2 x − x 2 2 + ρ k 2 Π S m + (−G(x)) 2 . In particular, computing derivatives we obtain ∇xL(x k , Y k ) → 0, where Y k . = ρ k Π S m + (−G(x k )).
With this result at hand, the authors prove that the sequence {Y k } k∈N must be bounded in the presence of Robinson's CQ, and that all of its limit points are Lagrange multipliers associated with x [8, Thm. 6.1]. Furthermore, the proof of this fact under nondegeneracy follows easily by contradiction: suppose that {Y k } k∈N is unbounded, and take any limit point Y of the sequence Y k / Y k k∈N ; then:
1. It follows from ∇xL(x k , Y k ) → 0 that DG(x) * [Y ] = 0, which means Y ∈ Ker DG(x) * = Im DG(x) ⊥ ; 2. By the definition of Y k , we have 0 = Y 0 and G(x), Y = 0, so Y ∈ lin(T S m + (G(x))) ⊥ ; Hence, Y ∈ Im DG(x) ⊥ ∩ lin(T S m + (G(x)
)) ⊥ , which contradicts nondegeneracy. With a single extra step, which is to take a spectral decomposition of Y k for each k, the reasoning of the previous paragraph can be put in the same terms as Proposition 3.2. Indeed, observe that λi(Y k ) = [ρ k λi(−G(x k ))]+ = 0 for all i ∈ {m − r + 1, . . . , m} and all k large enough, because
λi(−G(x k )) = −λm−i+1(G(x k )). So ∇xL(x k , Y k ) = ∇f (x k ) − m−r i=1 [ρ k λi(−G(x k ))]+vii(x k , E k ) → 0, where E k ∈ R m×m−r is a matrix whose i-th column is um−i+1(G(x k ))
. Then, note that if E k can be chosen such that at least one of its limit points E ensures linear independence of vii(x, E) : i ∈ {1, . . . , m − r} , then {Y k } k∈N must be bounded. Although the first clause of the previous sentence resembles nondegeneracy (as in Proposition 3.2), note that asking for the linear independence of the set vii(x, E) : i ∈ {1, . . . , m − r} when E is not a limit point of some sequence {E k } k∈N of eigenvectors of G(x k ) seems unnecessary for defining a constraint qualification. This motivates us to propose a weaker variant of nondegeneracy in a way that can also be extended to Robinson's CQ, which goes as follows:
Definition 3.2 (Weak-nondegeneracy and weak-Robinson's CQ). Let x ∈ F and let r be the rank of G(x). We say that weak-nondegeneracy (respectively, weak-Robinson's CQ) holds at x if either Ker G(x) = {0} or: for every sequence {x k } k∈N → x, there exists some sequence of matrices with orthonormal columns {E k } k∈N ⊆ R m×m−r such that:
1. The columns of E k are eigenvectors associated with the m − r smallest eigenvalues of G(x k ), for each k ∈ N; 2. There exists a limit point E of {E k } k∈N such that the set vii(x, E) : i ∈ {1, . . . , m − r} , as defined in (8), is (positive) linearly independent.
There are a couple of nuances about Definition 3.2 that should be properly addressed (see also the discussion after Remark 3.2). First, we recall that the eigenvector functions ui(G(x)), i ∈ {m − r + 1, . . . , m} are not necessarily continuous at a given point x; so weak-nondegeneracy (and weak-Robinson's CQ) relies on the "sequential continuity of eigenvectors" along a given path. Second, for any fixed x ∈ F and any {x k } k∈N → x, the sequence {E k } k∈N described in Definition 3.2 is well-defined for k sufficiently large, since the r largest eigenvalues of G(x k ) are necessarily bounded away from zero. Remark 3.1. Based on the previous discussion, it is worth mentioning that weak-nondegeneracy (and weak-Robinson's CQ) can be equivalently stated in terms of a certain notion of continuity of the eigenvectors of G(x). To see why, consider a feasible point x ∈ F and let r be the rank of G(x). Because r < m, it follows that λr(G(x)) > λr+1(G(x)) for every x close enough to x, so the following set is well-defined:
B(x) . = E ∈ R m×(m−r) : G(x)ei = λm−i+1(G(x))ei, ∀i ∈ {1, . . . , m − r} E ⊤ E = Im−r(13)
where E . = [e1, . . . , em−r]. The set above consists of all matrices whose columns are orthonormal eigenvectors associated with the m − r smallest eigenvalues of G(x). Moreover, for any sequence
{x k } k∈N → x recall the Painlevé-Kuratowski upper limit [10, Def. 2.52]) of the sequence of images {B(x k )} k∈N , defined as lim sup k→∞ B(x k ) . = z : ∃I ⊆ N infinite, ∃{z k } k∈I → z, ∀k ∈ I, z k ∈ B(x k ) .
In these terms, it is easy to see that weak-nondegeneracy (respectively, weak-Robinson's CQ) holds at x if, and only if, either Ker
G(x) = {0} or, for every sequence {x k } k∈N → x, there exists some E ∈ lim sup k→∞ B(x k ) such that vii(x, E) : i ∈ {1, . . . , m − r} is (positive) linearly independent.
Although the characterization of Remark 3.1 may shorten notation, in order to check whether weaknondegenearcy holds or not at a given point x requires the computation of the set B(x), which may be complicated in practice. Therefore, it is important to emphasize that B(x) is not meant to be explicitly computed because weak-nondegeneracy is not meant to be manually checked at any point, except for very specific cases with a convenient eigenvector structure (see Examples 3.1 and 3.2); instead, the main purpose of weak-nondegeneracy (and weak-Robinson's CQ) is to serve as a theoretical tool for building the convergence theory of iterative algorithms, as it was presented in the proof of Theorem 2 for the external penalty method. In this context, knowledge of the problem solution is usually limited to an approximation obtained by truncating the method's output sequence, which ends up taking away some of the meaning of checking constraint qualifications in practice.
The discussion that motivated Definition 3.2 already suggests that it indeed describes a genuine constraint qualification, and it also provides an outline of how to prove it. Nevertheless, we state and prove this fact with appropriate mathematical rigor below. Although we prove the next result for weak-Robinson's CQ, observe that the analogous statement for weak-nondegeneracy follows as a corollary.
Theorem 3.2. Every local minimizer x ∈ F of (NSDP) that satisfies weak-Robinson's CQ also satisfies the KKT conditions. By extension, the same holds for weak-nondegeneracy.
Proof. Let x be a local minimizer of (NSDP) that satisfies weak-Robinson's CQ and let {x k } k∈N → x and {Y k } k∈N be the sequences described in Theorem 3.1, for an arbitrary sequence {ρ k } k∈N → ∞. If r = m, set Y = 0 as a Lagrange multiplier associated with x and we are done; so let us assume that r < m from now on. From the local optimality of x k , for each k ∈ N, we obtain
∇f (x k ) + (x k − x) + DG(x k ) * [Y k ] = 0.(14)
Recall that we assume, without loss of generality, that λ1(−G(x k )) . . . λm(−G(x k )), for every k; and note that when k is large enough, say greater than some k0 ∈ N, we necessarily have λi(−G(x k )) < 0 for all i ∈ {m − r + 1, . . . , m} since G(x k ) → G(x) and eigenvalues λi(·) are continuous mappings. Then, for each k > k0, we have
Y k = m−r i=1 α k i e k i (e k i ) ⊤ , where α k i . = [ρ k λi(−G(x k ))]+ 0 and e k i . = um−i+1(G(x k )) is an arbitrary unitary eigenvector associated with λm−i+1(G(x k )), for each i ∈ {1, . . . , m − r}. Set E k . = [e k 1 , . . . , e k m−r ]. Since {E k } k∈N is bounded,
we may pick any of its limit points E = [e1, . . . , em−r] and assume, taking a subsequence if necessary, that it converges to E, which spans Ker G(x). Then, observe that (14) implies (8)), we can rewrite it as
∇f (x k ) − m−r i=1 α k i DG(x k ) * [e k i (e k i ) ⊤ ] → 0, but since DG(x k ) * [e k i (e k i ) ⊤ ] = vii(x k , E k ) (see∇f (x k ) − m−r i=1 α k i vii(x k , E k ) → 0.(15)
If (α k i , . . . , α k m−r ) k∈N has any convergent subsequence, denote its limit point by α . = (αi, . . . , αm−r), and note that α generates a Lagrange multiplier for x, which is
Y . = m−r i=1 αieie ⊤ i .(16)
Hence, it suffices to prove that {α k i } k∈N , i ∈ {1, . . . , m − r}, must be bounded under weak-Robinson's CQ. Let us assume for a moment that the sequences {α k i } k∈N are unbounded, which means
m k . = max α k i : i ∈ {1, . . . , m − r} → ∞.
Note that (α k 1 , . . . , α k m−r )/m k k∈N must be bounded and it must also have a nonzero limit point, which we will denote by (α1, . . . ,αm−r) 0. We assume without loss of generality, that (α k 1 , . . . , α k m−r )/m k k∈N → (α1, . . . ,αm−r). After dividing (15) by m k for each k and taking limit k → +∞, we obtain Let us briefly analyse a direct application of weak-Robinson's CQ: As an intermediary step of the proof of Theorem 3.2, we proved that every feasible limit point of a sequence described in Theorem 3.1 must satisfy the KKT conditions under weak-Robinson's CQ. However, the sequences {x k } k∈N and {Y k } k∈N described in Theorem 3.2 are precisely the ones that are generated by a standard external penalty method (that is, [8,Algorithm 1] with the parameter Ω max fixed at zero). Thus, every feasible limit point of the external penalty method that satisfies weak-Robinson's CQ must also satisfy the KKT conditions. By extension this also holds for weak-nondegeneracy.
Another interesting property of the weak variants of nondegeneracy and Robinson's CQ is that they are equivalent to LICQ and MFCQ, respectively, when G is a structurally diagonal matrix function (as in (6)) that models an NLP problem, which in some sense resolves the inconsistency between nondegeneracy and LICQ noted by Shapiro [24, Page 309].
Remark 3.2. When G is structurally diagonal, as in (6), then x ∈ F satisfies weak-nondegeneracy if, and only if, the set {∇gi(x) : gi(x) = 0} is linearly independent. Indeed, if r = m the result follows trivially, so let us assume that r < m. Also, suppose that {i ∈ {1, . . . , m} : gi(x) = 0} = {r + 1, . . . , m}, where r is the rank of G(x). Clearly, if {∇gr+1(x), . . . , ∇gm(x)} is linearly independent, then we may take
E k . = 0 Im−r ∈ R m×m−E = 0 Q , where Q ∈ R m−r×m−r is orthonormal,
due to the diagonal structure of G and the fact that gi(x) = 0 for all i ∈ {1, . . . , r}. Hence,
vii(x, E) = m j=r+1 ∇gj(x)Q 2 i,j−r = Dg(x) ⊤ (Qi ⊙ Qi),(17)
where Dg(x) is the Jacobian matrix of g(x) . = (gr+1(x), . . . , gm(x)) at x, the operator ⊙ is the (Hadamard) entry-wise vector product, and Qi is the i-th column of Q, with i ∈ {1, . . . , m − r}. Then, It is clear from Proposition 3.2 that weak-nondegeneracy is implied by nondegeneracy; and we see in the example below that the converse is not true.
span vii(x, E) : i ∈ {1, . . . , m − r} ⊆ Im Dg(x) ⊤ and, consequently, m − r = dim(span vii(x, E) : i ∈ {1, . . . , m − r} ) dim(Im Dg(x) ⊤ ) = rank(Dg(x) ⊤ ) m − r Hence, rank(Dg(x) ⊤ ) = m − r,
Example 3.1. Consider the following constraint:
G(x) . = x1 x2 x2 x1
at the point x . = (0, 0), which clearly does not satisfy nondegeneracy. Weak-nondegeneracy, on the other hand, holds at x as
B(x) = ±1 √ 2 −1 1 1 1 , ±1 √ 2 1 1 −1 1 , if x2 = 0 E ∈ R 2×2 : E ⊤ E = I2 , if x2 = 0
for every x ∈ R 2 , according to (13), so it suffices to take
E k . = 1 √ 2 −1 1 1 1 ∈ B(x k ) and E . = 1 √ 2 −1 1 1 1 ∈ lim sup k→∞ B(x k )
for all sequences {x k } k∈N → x to obtain that v11(x, E) = [1, −1] and v22(x, E) = [1, 1] are linearly independent. This simple example is also important to show that weak-nondegeneracy does not guarantee uniqueness of Lagrange multipliers. For instance, consider the constraint above with the objective function f (x) . = 2x1 which has x as its global minimizer; then every Y in the form
Y . = 1 − α 0 0 1 + α for α ∈ [−1, 1] \ {0}
is a Lagrange multiplier associated with x.
Another example that serves the same purpose, which can also be used to show how the sparsity structure of the eigenvectors of G is grasped by weak-nondegeneracy is the following: Example 3.2. Consider the constraint:
G(x) . = x11 0 x13 0 x22 0 x13 0 x33 0
and let x . = 0. Nondegeneracy fails at x, but weak-nondegeneracy holds. To see this, take any sequence {x k } k∈N → x, if x k 13 = 0 for all k (the other case is trivial, so we will omit it), and
E k . = −η k 1 √ (η k 1 ) 2 +1 0 −η k 2 √ (η k 2 ) 2 +1 0 1 0 1 √ (η k 1 ) 2 +1 0 1 √ (η k 2 ) 2 +1 , where η k 1 . = −x k 11 + x k 22 + (x k 11 ) 2 − 2x k 22 x k 33 + (x k 33 ) 2 + 4(x k 13 ) 2 2x k 13 and η k 2 . = −x k 11 + x k 22 − (x k 11 ) 2 − 2x k 22 x k 33 + (x k 33 ) 2 + 4(x k 13 ) 2 2x k 13 .
In this case, assuming that x k 13 > 0 for all k (which can be done without loss of generality since the other cases are analogous), we have
lim k→∞ η k 1 = lim k→∞ |x k 13 | x k 13 = 1 and lim k→∞ η k 2 = lim k→∞ − |x k 13 | x k 13 = −1, hence E k → E . = −1 √ 2 0 1 √ 2 0 1 0 1 √ 2 0 1 √ 2 ,
and computing the vectors of interest we arrive at
v11(x, E) = 1 2 1 0 1 −2 , v22(x, E) = 0 1 0 0 . v33(x, E) = 1 2 1 0 1 2 ,
which are linearly independent, so weak-nondegeneracy holds at x. Observe that, in this case, the matrix E k has the same sparsity structure as G.
Moreover, note that weak-nondegeneracy imposes a less demanding dimensionality constraint over (NSDP); in fact, in order to verify nondegeneracy, one must have n (m − r)(m − r + 1)/2, while weak-nondegeneracy may hold as long as n m − r (Remark 3.2). It is also clear from their definitions that weak-nondegeneracy implies weak-Robinson's CQ; and it is possible to show that the converse is not necessarily true. For instance, consider the constraint defined by:
G(x) . = x 0 0 x ,
and note that all orthogonal matrices E ∈ R 2×2 have in their columns eigenvectors of G(x), for every x.
Since v11(x, E) = v22(x, E) = 1 for every E, it follows that weak-nondegeneracy and weak-Robinson's CQ are equivalent to their strong counterparts in this case. Thus, from Proposition 3.2 we see that (weak-)nondegeneracy does not hold, while (weak-)Robinson's CQ does. It is also clear from Proposition 3.2 (item 2) that Robinson's CQ implies weak-Robinson's CQ; however, we were not capable of finding a counterexample for the converse. We conjecture that they are equivalent. = λi(G(x)) otherwise. So, clearly M k → G(x) and the only convergent sequence E k to E is E itself. Consequently, when we assume Definition 3.2 it necessarily follows that {vii(x, E) : i ∈ {1, . . . , m − r}} is linearly independent. Then, since E was chosen arbitrary, Proposition 3.2 implies that nondegeneracy holds true. where each "block" is defined by a continuously differentiable function G ℓ : R n → S m ℓ , with ℓ ∈ {1, . . . , q}, and m1 + . . . + mq = m. In fact, let x ∈ F and r ℓ . = rank(G ℓ (x)) for each ℓ; and, for simplicity, let us assume that r ℓ < m ℓ for all ℓ. Since Ker G(x) = Ker G1(x) × . . . × Ker Gq(x), then r = r1 + . . . + rq. Then, weak-nondegeneracy (respectively, weak-Robinson's CQ) holds at x if, and only if, for all sequences {x k } k∈N → x, there are sequences of matrices {E k ℓ } k∈N such that: • The columns of E k ℓ are unitary eigenvectors associated with the m ℓ − r ℓ smallest eigenvalues of G ℓ (x k ), for each k ∈ N and each ℓ ∈ {1, . . . , q};
• There are limit points E ℓ of {E k ℓ } k∈N , ℓ ∈ {1, . . . , q}, such that the set
q ℓ=1 v ℓ ii (x, E ℓ ) : i ∈ {1, . . . , m ℓ − r ℓ } is (positive) linearly independent, where v ℓ ij (x, E ℓ ) . = e ⊤ ℓ,i Dx 1 G ℓ (x)e ℓ,j , . . . , e ⊤ ℓ,i Dx n G ℓ (x)e ℓ,j ⊤ ,(18)
and e ℓ,1 , . . . , e ℓ,m ℓ −r ℓ denote the columns of E ℓ , for each ℓ ∈ {1, . . . , q}.
The proof of this fact is elementary with [14, Lem. 1.3.10] and (8) at hand. Moreover, note that this is precisely the way weak-nondegeneracy would be defined for an equivalent multifold NSDP with constraints G1(x) 0, . . . , Gq(x) 0. Thus, weak-nondegeneracy and weak-Robinson's CQ are invariant to block diagonal and multifold representations of (Block-NSDP). This is especially meaningful in problems that do not present an explicit block-diagonal representation, in which case it is not necessary to have prior knowledge of such a representation to talk about weak-nondegeneracy (or weak-Robinson's CQ), contrary to nondegeneracy.
Recall that the analysis we presented until this point showed, among other things, that some choices of E may be more meaningful than others. With this in mind, we are now led to revisit the work of Forsgren [12], who presented a very interesting way of talking about nondegeneracy in the presence of any sparsity structure that appears after applying a particular transformation to the problem. In the next section, we improve some of Forsgren's ideas by presenting a simplified and more general way of dealing with sparsity.
Dealing with structural sparsity
In this section, we take inspiration from a regularity condition introduced by Forsgren [12, Sect. 2.3], whose primary goal was to prove second-order optimality conditions for (NSDP). However, what makes Forsgren's condition specially interesting for us is the fact it can benefit from some sparsity structure of a certain Schur complement related to the constraint function. The main objective of this section is to present a more straightforward way of enjoying sparsity, based on Forsgren's results and Section 3. But before that, we present some of the notation used by Forsgren.
Given a point x and a matrix-valued function F : R n → S β , consider the set S(F, x) defined as follows:
S(F, x) . = M ∈ S β : Mij = 0 if Fij (x) is structurally zero near x = M ∈ S β : Mij = 0 if ∃ε > 0 such that Fij (x) = 0, ∀x ∈ B(x, ε) .
For example, if β = 3 and for all x close to x, we are able to identify non trivial mappings Fij such that
F (x) = F11(x) 0 F13(x) 0 F22(x) 0 F13(x) 0 F33(x) , then M ∈ S(F, x) ⇔ M = M11 0 M13 0 M22 0 M13 0 M33 ,
where M11, M13, M22, and M33 may or may not be zero. Also, we define
I(F, x) . = {(i, j) : ∀ε > 0, ∃x ∈ B(x, ε) such that Fij (x) = 0, 1 i j β}
as the set of indices that define the elements of S(F, x). Forsgren's results are obtained in terms of the functioñ
G(x) . = G(x) − G(x)P (P ⊤ G(x)P ) −1 P ⊤ G(x),
where U = [P , E] has columns that form an orthonormal eigenvector basis for G(x), such that E spans the kernel of G(x) and P
⊤ G(x)P ≻ 0. Note that E ⊤G (x)E is the Schur complement of P ⊤ G(x)P inside U ⊤ G(x)U = P ⊤ G(x)P P ⊤ G(x)E E ⊤ G(x)P E ⊤ G(x)E .
Moreover, following Forsgren [12, Lem. 1], we see thatG(x) 0 if, and only if G(x) 0, for all x sufficiently close to x, so the original NSDP problem can be locally reformulated as a minimization problem overG(x) 0, around x. In fact, since
P (P ⊤ G(x)P ) −1 P ⊤ = P λ+(G(x)) −1 P ⊤ = U λ+(G(x)) −1 0 0 0 U ⊤ = G(x) † ,
where G(x) † is the Moore-Penrose pseudoinverse of G(x), it follows thatG(x) = 0 [12, Lem. 2], soG can be considered a reduction to the kernel of G(x) near x. The regularity condition introduced by Forsgren is as follows: Forsgren's CQ). Let x ∈ F and let U . = [P , E] be an orthogonal matrix that diagonalizes G(x), such that the columns of E span Ker G(x). Then, Forsgren's CQ holds at x with respect to U when
Definition 4.1 (span E ⊤ Dx i G(x)E : i ∈ {1, . . . , n} = E ⊤ S(G, x)E (F1)
and
∃M ∈ E ⊤ S(G, x)E, such that M ≻ 0. (F2)
Forsgren's CQ is indeed a constraint qualification, for when (F1) holds, then (F2) is equivalent to Robinson's CQ [12,Lem. 5]. However, although Forsgren states that any choice of U leads to a valid CQ, there is no discussion on the effects of this choice over the condition proposed. Under a specific condition, Forsgren's CQ provides uniqueness of the Lagrange multiplier [12, Thm. 1], but this condition varies with U . Thus, different choices of U are likely to define different variants of Forsgren's CQ. This is not necessarily a negative point, but a comparison among those variants would be appropriate. For instance, from the practical point of view, one may be interested in knowing which choice of U defines the weakest CQ, or which one is easier to compute.
A result from Dorsch, Gómez, and Shikhman [11] shows that, ignoring the sparsity treatment, (F1) becomes equivalent to nondegeneracy. However, similarly to weak-nondegeneracy, Forsgren's CQ also reduces to LICQ from NLP when G is structurally diagonal (as in (6) Proof. Let us assume that r < m, since otherwise the proof is trivial. We employ [12,Lem. 2], which states that
E ⊤ Dx i G(x)E = E ⊤ Dx iG (x)E for all i ∈ {1, .
. . , n}, to ensure that the linear operator ψ :
R n → E ⊤ S(G, x)E, defined by the action ψ(d) . = E ⊤ DG(x)[d]E is well-defined.
With this in mind, note that
Im (ψ) = span E ⊤ Dx i G(x)E : i ∈ {1, . . . , n} = E ⊤ S(G, x)E
if, and only if,
Ker (ψ * ) = M ∈ E ⊤ S(G, x)E : E ⊤ Dx ℓ G(x)E, M = 0, ∀ℓ ∈ {1, . . . , n} = {0},(19)
whence the result follows since
E ⊤ Dx ℓ G(x)E = [(vij (x, E)) ℓ ] i,j∈{1,...,m−r} ,
where (vij (x, E)) ℓ is the ℓ-th entry of the vector vij (x, E). As far as we understand, the relation between Forsgren's CQ and nondegeneracy was not formally established in [12]. To clarify this important detail, note that it is clear from Propositions 4.1 and 3.1 that nondegeneracy implies Forsgren's CQ. Moreover this implication is clearly strict, as nondegeneracy does not recover LICQ in a diagonal example.
The above discussion leads us to deal with sparsity in a more straightforward way, namely without taking Schur complements, which induces another weak variant of nondegeneracy.
A sparse variant of nondegeneracy
For any matrix E that spans Ker G(x), consider the function
G E (x) . = E ⊤ G(x)E and note that ∇ G E ij (x) = vij (x, E) for all i, j ∈ {1, .
. . , m − r} with i j. We incorporate structural sparsity into nondegeneracy directly, but in a similar style of Forsgren's CQ (as characterized in Proposition 4.1), to introduce a new constraint qualification.
Definition 4.2 (Sparse-nondegeneracy). We say that sparse-nondegeneracy holds at x ∈ F when either Ker G(x) = {0} or there exists a matrix E ∈ R m×m−r that spans Ker G(x) and such that:
1. The set vij (x, E) : (i, j) ∈ I( G E , x), 1 i j m − r is linearly independent; 2. (i, i) ∈ I( G E , x) for all i ∈ {1, . . . , m − r}.
There are two natural questions about sparse-nondegeneracy that we shall answer in the following paragraphs. The first one consists of knowing whether the sparse-nondegeneracy condition is a genuine constraint qualification; and the second one concerns about the relation between Definition 4.2 and other constraint qualifications, such as nondegeneracy, Forsgren's CQ, and Robinson's CQ. To address these questions, we first prove an elementary characterization of sparse-nondegeneracy: Proof. The result follows directly by noticing that
(i,j)∈I( G E ,x) vij (x, E)Ỹij = DG(x) * [EỸ E ⊤ ](20)
for everyỸ ∈ S( G E , x).
Next, we prove that sparse-nondegeneracy implies Robinson's CQ, which also shows that it is indeed a constraint qualification. Proof. The result follows trivially when Ker G(x) = {0}, so let us assume that r = rank(G(x)) < m. Suppose that sparse-nondegeneracy holds at x ∈ F, and take any Z 0 such that Z, G(x) = 0 and DG(x) * [Z] = 0, then there exists some Y ∈ S m−r + such that Z = EY E ⊤ . Define the matrixỸ ∈ S( G E , x) whose (i, j)-th entry is given byỸ Since Z is arbitrary, Robinson's CQ holds.
ij . = Yij, if (i, j) ∈ I( G E , x) 0, otherwise, and note that DG(x) * [Z] = DG(x) * [EY E ⊤ ] = DG(x) * [EỸ E ⊤ ] = 0,(21)
We highlight that item 2 of Definition 4.2 is not superfluous, for removing it may cause us to lose the property of being a constraint qualification. Indeed, the following example illustrates that: Let us show that when item 2 fails, the problem can be reformulated such that it holds. Let x ∈ F and E be a matrix that spans Ker G(x). If item 2 of Definition 4.2 is not satisfied, then let J .
= {i ∈ {1, . . . , m − r} : (i, i) ∈ I( G E , x)} and note that there exists some ε > 0 such that
G(x) ∈ S m + if, and only if, G(x) ∈ S m + i∈J {eie ⊤ i } ⊥ ,
for every x ∈ B(x, ε), where ei denotes the i-th column of E. That is, the feasible set F coincides locally with the preimage of the face F .
= S m + i∈J {eie ⊤ i } ⊥ of S m + . Moreover, since F is a face of S m + , then there is an orthogonal matrix V . = [V1, V2] ∈ R m×m such that V ⊤ F V = M 0 0 0 : M ∈ S m−ω + ,
where ω is the cardinality of J [21, Eq. 2.3]. This means that it is possible to locally replace the original constraint of (NSDP) by the equality constraint V ⊤ 2 G(x) = 0 and a smaller semidefinite constraint G(x)
. = V ⊤ 1 G(x)V1 ∈ S m−ω + .
If F is minimal, then the new constraint G(x) ∈ S m−ω + satisfies item 2 of Definition 4.2 at x. Otherwise, this process can be repeated until the minimal face is reached. Thus, every problem can be equivalently reformulated (reducing dimension if necessary), such that item 2 always holds. In particular, when G is an affine function, then this procedure can be computed via a popular preprocessing technique called facial reduction (we refer to Pataki [21] and references therein for more details about it). When G(x) = 0 and E = Im, this procedure can be done by simply removing the i-th row and the i-th column of G, for every i such that (i, i) ∈ I(G, x), and including the correspondent equality constraints into the problem. We recall that all of our results can be easily extended to NSDP problems with separate equality constraints.
Let us illustrate this procedure using Example 4.1. In this case we have e ⊤ 2 G(x)e2 = 0 for every x; then x ∈ F if, and only if,
G(x) ∈ F , where F . = S 2 + 0 0 0 1 ⊥ = α 0 0 0 : α 0,
which means that the constraint of the problem can be equivalently written as x2 = 0 and x1 0; for which x satisfies Definition 4.2 and the KKT conditions.
Remark 4.3.
If G is structurally diagonal as in (6), then x satisfies sparse-nondegeneracy if, and only if, the set {∇gi(x) : gi(x) = 0} is linearly independent. Moreover, this can be extended to block-diagonal constraints. In this case, assuming the same notation as Remark 3.4, sparse nondegeneracy holds at a feasible point x of (Block-NSDP) if, and only if, for each ℓ ∈ {1, . . . , q} there is some matrix E ℓ that spans Ker G ℓ (x), such that:
• For all i ∈ {1, . . . , m ℓ − r ℓ }, we have (i, i) ∈ I(G E ℓ ℓ , x); • The set q ℓ=1 v ℓ ij (x, E ℓ ) : (i, j) ∈ I(G E ℓ ℓ , x)
is linearly independent, where v ℓ ij (x, E ℓ ) is defined as in (18). Note that this is how sparse-nondegeneracy would be defined for a multifold equivalent representation of (Block-NSDP), with constraints G1(x) 0, . . . , Gq(x) 0.
In view of Remark 4.3, it is easy to build a diagonal counterexample for the converse of Proposition 4.2. For instance, take m = 2 and set x = 0; then, define the constraint
G(x) . = x 0 0 x ,(22)
and note that v11(x, E) = v22(x, E) = 1 for every matrix E that spans Ker G(x). Hence, sparse-nondegeneracy does not hold, although Robinson's CQ does. Furthermore, Remark 4.3 reveals a similarity among sparse-nondegeneracy, Forsgren's CQ, and weaknondegeneracy, which is the fact they all reduce to LICQ when considering a diagonal matrix constraint. Moreover, it follows directly from Propositions 3.1 and 3.2 that nondegeneracy also strictly implies sparsenondegeneracy. To make a rough comparison between Forsgren's CQ and sparse-nondegeneracy, note that both evaluate linear independence of the set vij (x, E) : 1 i j m − r , but while item 1 of Definition 4.2 takes coefficients structured as in S( G E , x), condition (F1) takes coefficients structured as in E ⊤ S(G, x)E. This suggests that they are different conditions. In fact, Example 3.1 can also be used to show that neither weak-nor sparse-nondegeneracy imply Forsgren's CQ. = (0, 0), which satisfies weak-nondegeneracy and violates nondegeneracy (Example 3.1). Also:
• Sparse-nondegeneracy holds at x: take the same E as above and we have
G E (x) . = x1 − x2 0 0 x1 + x2
and I( G E , x) = {(1, 1), (2, 2)};
• Forsgren's CQ does not hold at x: in this case Forsgren's CQ is equivalent to nondegeneracy, which does not hold because if E . = Im, then v11(x, E) = v22(x, E) = [1,0].
Thus, neither weak-nor sparse-nondegeneracy imply Forsgren's CQ.
Moreover, if G(x) = 0 thenG = G and in this case Forsgren's CQ implies sparse-nondegeneracy (see Proposition 4.1 and the discussion afterwards). Whether this still holds or not when G(x) = 0 is an open problem that we are currently unable to address, due to the intricate form ofG in the general case.
An elementary consequence of Lemma 4.2 is that sparse-nondegeneracy guarantees uniqueness of the Lagrange multiplier with respect to a fixed sparsity pattern, which is similar to a result proven for Forsgren's CQ [12].
Y k . = ρ k Π S m + (−G(x k ))
must belong to Λ(x). But clearly, for all k ∈ N large enough, we see that Another important property of sparse-nondegeneracy is that the number of structural zeros of G E , at points that satisfy it, remains the same regardless of E.
Y k ∈ ES( G E , x)E ⊤ and so does Y . Now let Y1, Y2 ∈ Λ(x) ES( G E , x)E ⊤ be Lagrange multipliers associated with x, define Y . = Y1 − Y2,
which can be rephrased in terms of the Kronecker product as unfold(D G W (x)) = unfold(D G E (x))Z ⊗ Z. But since Z is invertible, Z ⊗ Z is also invertible, which means that
span ∇ G W ij (x) : 1 i j m − r = span ∇ G E ij (x) : 1 i j m − r .
Then, since ∇ G E ij (x) = 0 for all (i, j) ∈ I( G E , x) (and the same holds for W ), it follows that
span ∇ G W ij (x) : (i, j) ∈ I( G W , x) = span ∇ G E ij (x) : (i, j) ∈ I( G E , x) , .
Finally, since item 1 of Definition 4.2 holds for both E and W , we conclude that #I( G E , x) = #I( G W , x).
Proposition 4.4 tells us that the strength of sparse-nondegeneracy is invariant with respect to E. That is, if there are multiple matrices E certifying sparse-nondegeneracy at a point x, then they all induce similar conditions. In our opinion, this is an advantage with respect to Forsgreen's CQ. As for weak-nondegeneracy, we were not able to find any counterexample nor prove any relation between them. In fact, finding this relation seems a challenging task since there is no clear relation between the eigenvectors of G(x) and its sparsity structure, in general.
One should also keep in mind that if sparse-nondegeneracy holds at some x, then it also holds in a neighborhood of x.
Theorem 4.1. Let x ∈ F satisfy sparse-nondegeneracy. Then, there exists a neighborhood V of x such that every x ∈ V satisfies sparse-nondegeneracy.
Proof. Suppose that the statement above is false. That is, suppose that there exists a feasible sequence {x k } k∈N → x such that sparse-nondegeneracy fails at each x k , but it holds at x. Our aim is to prove that this leads to an absurd. So let E be any matrix with orthonormal columns that span Ker G(x) and, for each k ∈ N let Π k be the projection matrix onto the space spanned by the m − r smallest eigenvectors of G(x k ), which is well defined when k is sufficiently large. DefineW k .
= Π k E, for all such k ∈ N. It is well-known (see, for instance, [10,Ex. 3.98]) that the columns ofW k are linearly independent, which allows us to apply the Gram-Schmidt orthonormalization process to them and arrange its output in the columns of a new matrix, which we will denote by W k . It is also known that W k → E as k → ∞.
Because sparse-nondegeneracy fails at x k , we know that the rank r k of G(x k ) is smaller than m, and by the pigeonhole principle we can even assume that r k is the same, sayr, for every k ∈ N. Also, note that m −r m − r and that, by construction, we can assume that the first m −r columns of each W k , which we will arrange in a matrix denoted by E k , span Ker G(x k ). Since sparse-nondegeneracy fails at x k it holds that {vij (x k , E k )} (i,j)∈I( G E k ,x k ) linearly dependent for each k ∈ N. Observe that since lim k→∞ E k is a submatrix of E we have that lim
k→∞ I( G E k , x k ) ⊆ I( G E , x) and lim k→∞ {vij (x k , E k )} (i,j)∈I( G E k ,x k ) ⊆ {vij (x, E)} (i,j)∈I( G E ,x) .
The left-hand side of the expression above is linearly dependent, which makes {vij (x, E)} (i,j)∈I( G E ,x) linearly dependent as well. Because E is arbitrary, it follows that sparse-nondegeneracy fails at x, which is a contradiction.
Remark 4.4. It is noteworthy that it is also possible to define another variant of Robinson's CQ that enjoys sparsity, by replacingỸ ∈ S(
G E , x) byỸ ∈ S( G E , x) ∩ S m−r + in Lemma 4.2.
This definition is strictly implied by sparse-nondegeneracy (see the example given in (22)). Moreover, it is clear that this variant of Robinson's CQ is implied by Robinson's CQ, but the converse is also an open question. The proof that this is a CQ follows similarly to the proof of Theorem 3.2. = S m + ∩ H which is still closed and convex, and in this setting sparse-nondegeneracy induces a second-order optimality condition, which is inherited from [10,Thm. 3.45]. Namely, for every d ∈ DG(x) −1 (T S m
+ ∩H (G(x)) ∩ {∇f (x)} ⊥ it holds that sup Y ∈Λ(x)∩H d ⊤ ∇ 2 L(x, Y )d − σ(Y, T 2 S m + ∩H (G(x), DG(x)[d])) 0,(23)
because sparse-nondegeneracy implies Robinson's CQ, which in turn is carried over to the reduced problem, but since Λ(x) ∩ H is a singleton, we have for Y ∈ Λ(x) ∩ H that
d ⊤ ∇ 2 L(x, Y )d − σ(Y , T 2 S m + ∩H (G(x), DG(x)[d])) 0.
Although this condition concerns the reduced problem, mostly, it can also bring some information about the original problem, for an inequality analogous to (23)
Zeros of the gradients and sparse-nondegeneracy
In this short ending section, we discuss how to improve sparse-nondegeneracy even further. This is mainly motivated by the realization that the idea of disregarding "structural zeros" in the study of regularity is actually too conservative. Since nondegeneracy is mainly concerned with the derivative of G at x instead of the value of G in a neighborhood of x, we can in fact ignore all entries of G whose gradients are zero at x, which is done by considering the following sets:
S∇(F, x) . = M ∈ S β : Mij = 0 if ∇Fij (x) = 0 and I∇(F, x) . = {(i, j) : ∇Fij (x) = 0, 1 i j β} .
For example, if n = 1 and β = 3, for all x close to x . = 0 we have, as an example,
if F (x) . = x 0 x 2 0 x 1 x 2 1 x then M ∈ S(F, x) ⇔ M = M11 0 0 0 M22 0 0 0 M33 ,(24)
where M11, M22, and M33 may or may not be zero. Then, we can define a condition similarly to Definition 4.2 but in terms of I∇:
Definition 4.3 (GS-nondegeneracy).
We say that the condition gradient-sparse-nondegeneracy (GS-nondegeneracy) holds at x ∈ F if either Ker G(x) = {0} or there exists a matrix E ∈ R m×m−r that spans Ker G(x) such that:
1. The set vij (x, E) : (i, j) ∈ I∇( G E , x), 1 i j m − r is linearly independent; 2. (i, i) ∈ I∇( G E , x) for all i ∈ {1, . . . , m − r}.
The interesting properties of GS-nondegeneracy that make it worth an extended comment are twofold. The first one is that sparse-nondegeneracy is strictly stronger than GS-nondegeneracy. Noticing that I∇( G E , x) ⊆ I( G E , x) is enough to see the implication and the next example shows that the converse is not necessarily true. are linearly independent, so GS-nondegeneracy holds at x.
We remark that Lemma 4.2 and Propositions 4.2, 4.3, and 4.4, can be also stated and proved in terms of GS-nondegeneracy. Moreover, if Forsgren's CQ was defined in terms of I∇(G, x) instead of I(G, x), we would obtain precisely Definition 4.3 (due to [12,Lem. 2]), which is quite unexpected. The second interesting aspect of GS-nondegeneracy is that, although an analogue of Theorem 4.1 may not be true, it presents at least a different notion of stability, in the sense of ignoring small perturbations. Formally: Theorem 4.2. Let x ∈ F and δ : R n → S m be any continuously differentiable function such that δ(x) = 0 and Dδ(x) = 0. Then, GS-nondegeneracy holds at x for the constraint G(x) 0 if, and only if, it holds for the constraint G δ (x) . = G(x) + δ(x) 0 at the same point.
Proof. Direct from the fact DG(x) = DG δ (x) and I∇( G E , x) = I∇( G E δ , x).
Despite the apparent triviality of Theorem 4.2, observe that it is essentially telling us that any noise of order two can be disregarded, as we could observe in Example 4.3.
Conclusions
In this paper, we studied the nondegeneracy condition of Shapiro and Fan [26] with the purpose of incorporating some matrix structure into it, such as spectral decompositions and structural sparsity. Our work was motivated by a well-known limitation of nondegeneracy, which is the fact it generally fails in the presence of structural sparsity in the constraint function. For example, we recall that a NSDP problem with multiple constraints may be equivalently reformulated as a single block diagonal constraint, but nondegeneracy is not expected to be preserved in the process. This limitation may have important consequences in practice, since many algorithms are theoretically supported by nondegeneracy and, on the other hand, structural sparsity is a very common trait of optimization models of real world problems.
To address this issue, we proposed three variants of nondegeneracy, here called weak-nondegeneracy, sparse-nondegeneracy, and GS-nondegeneracy. They were proven to be strictly weaker than the classical nondegeneracy. In particular, all new constraint qualifications only require the dimension constraint n m − r, which is considerably less demanding than the constraint n (m − r)(m − r + 1)/2 imposed by nondegeneracy. Also, they are invariant to multifold or block diagonal formulations of (NSDP) and, consequently, they recover the LICQ condition from NLP when the constraint function is structurally diagonal.
All our conditions are inspired by sequential optimality conditions [3,8] which provide simple proofs for the facts that the conditions we define are CQs (the proof for sparse-nondegeneracy and GS-nondegeneracy were not presented but they are left for the reader). Besides the simplicity of the approach, the convergence of an external penalty method to KKT points under these CQs is obtained automatically (see the discussion after Theorem 3.2), which is a direct application of the new CQs. Also, several other CQs for NLP have been recently (re)invented with sequential optimality conditions in mind. In particular, the so-called constant rank constraint qualification (CRCQ) by Janin [15], and the constant positive linear dependence (CPLD) of Qi and Wei [22], together with their weaker counterparts [6,7,17]. Previous attempts have been made to extend these CQs to the conic context, but they have turned out to be flawed [2] or incomplete [4], since the results in [4] are only relevant for multifold conic problems where at least one block of constraints is such that the zero eigenvalue is simple. The approach we present in this paper gives the proper tools for providing the extension of all mentioned CQs to the context of general NSDPs and, more generally, to optimization over symmetric cones, also extending the global convergence results to more practical algorithms. For instance, in NLP, it is known that the convergence theory of a safeguarded augmented Lagrangian method can be built around CPLD [1], which will also be the case for its NSDP variant [8]. A continuation of this paper will appear shortly with these results.
With this in mind, we believe that the concepts introduced in this paper are interesting enough to shed a new light to the classical theme of constraint nondegeneracy for conic programming, showing, in particular, how to redefine it in such a way that linear independence can be replaced by weaker notions. In this process, new and interesting challenging open questions have appeared which we believe should be addressed. In particular, new studies should be conducted to clarify the relationship between weak-nondegeneracy and sparse-nondegeneracy, together with the relationship between weak-Robinson's CQ and Robinson's CQ (see Figure 1).
Nondegeneracy
Robinson's CQ
GS-nondegeneracy
Forsgren's CQ Weak-nondegeneracy Sparse-nondegeneracy Weak-Robinson's CQ
Proposition 3. 1 (
1Prop. 6 from[24]). Let x ∈ F and let r denote the rank of G(x). Then, x satisfies the nondegeneracy condition if, and only if, either Ker G(x) = {0} or the vectors
Proposition 3 . 2 .
32Let x ∈ F and r = rank(G(x)). Then, x satisfies: 1. Nondegeneracy if, and only if, either r = m or vii(x, E) : i ∈ {1, . . . , m − r} (9)
vii(x, E) : i ∈ {1, . . . , m − r} is positive linearly dependent. However, since our analyses hold for any arbitrary choice of {E k } k∈N and any E, this contradicts weak-Robinson's CQ.
which means that {∇gr+1(x), . . . , ∇gm(x)} is linearly independent. Using similar arguments, thanks to(17) which states that the vectors vii(x, E) are nonnegative linear combinations of the columns of Dg(x) ⊤ , it is possible to prove that x ∈ F satisfies weak-Robinson's CQ if, and only if, {∇gi(x) : gi(x) = 0} is positive linearly independent, which is in turn equivalent to Robinson's CQ.
Remark 3 . 3 .
33If we replace the sequences {x k } k∈N → x by matrix sequences {M k } k∈N → G(x) in Definition 3.2, then we recover the nondegeneracy condition. Indeed, for any E ∈ R m×m−r that spans Ker G(x),consider M k . = U Λ k U ⊤ , with U . = [E,um−r+1(G(x)), . . . , um(G(x))], and Λ k . = Diag(y k ) such that y k i . = i/k for i ∈ {1, . . . , m−r}, and y k i .
Remark 3 . 4 .
34Remark 3.2 can be straightforwardly extended to structurally block diagonal matrix constraints, such as
Lemma 4 . 1 (
41Lem. 5 from[11]). Let x ∈ F and assume that S(G, x) = S m . Then, condition (F1) of Forsgren's CQ holds if, and only if, nondegeneracy holds at x.
), contrasting with nondegeneracy. To put Forsgren's CQ in the same terms as the previous sections, we present an elementary characterization of it using the vectors vij (x, E) defined in Proposition 3.1: Proposition 4.1. Let x ∈ F and let E ∈ R m×m−r span Ker G(x). Then, condition (F1) of Forsgren's CQ holds at x if, and only if, vij (x, E) = 0, M ∈ E ⊤ S(G, x)E ⇒ M = 0,where r = rank(G(x)).
As an abuse of language, (F1) consists of the "linear independence" of vij (x, E) : 1 i j m − r with respect to the set E ⊤ S(G, x)E. In particular, when G(x) = 0 and (F2) holds, take U = E = Im and note that Forsgren's CQ holds for this particular choice of U if, and only if, the set {∇Gij (x) : (i, j) ∈ I(G, x)} is linearly independent, with (i, i) ∈ I(G, x) for all i ∈ {1, . . . , m}.
Lemma 4 . 2 .
42Let x ∈ F be such that Ker G(x) = {0}, and let E be a matrix that spans Ker G(x). Then, item 1 of Definition 4.2 holds at x if, and only if, there is no nonzeroỸ ∈ S( G E , x) such that DG(x) * [EỸ E ⊤ ] = 0.
Proposition 4 . 2 .
42If x ∈ F satisfies sparse-nondegeneracy, then it also satisfies Robinson's CQ.
soỸ = 0
0due to Lemma 4.2. Moreover, from item 2 of Definition 4.2, (i, i) ∈ I( G E , x) for all i ∈ {1, . . . , m − r}, so the diagonal of Y must consist only of zeros, which implies that Y = 0 and, consequently, Z = 0.
x . = (0, 0) as one of its solutions. The point x satisfies Definition 4.2 after removing item 2, with E . = I2, because v11(x, E) = (1, 0) and v12(x, E) = (0, 1) are linearly independent; but x does not satisfy the KKT conditions since there is no Y 0 such that Y 11 = 0 and Y 12 = Y 21 = 1/2. Thus, Definition 4.2 is not a constraint qualification without item 2.
Example 4.2 (same as Example 3.1). Consider the constraint:G(x) . = x1 x2 x2 x1and the point x .
Proposition 4 . 3 .
43Let x be a KKT point of (NSDP) that satisfies item 1 of Definition 4.2 and let E be the matrix that certifies it, which spans Ker G(x). Then,Λ(x) ES( G E , x)E ⊤ is a singleton.Proof. Firstly, to see why Λ(x)ES( G E , x)E ⊤ = ∅ weresort to a result of [8, Thm. 7] which states that under Robinson's CQ any accumulation point Y of the sequence
and by definition there exists some Z ∈ S( G E , x) such that Y = EZE ⊤ and DG(x) * [EZE ⊤ ] = 0. By Lemma 4.2 we must have Z = 0 and, consequently, Y1 = Y2.
Remark 4. 5 .
5Regarding second-order optimality conditions, we call the reader's attention to the fact that for each x ∈ F and each E that spans Ker G(x), there exists a neighborhood V of x such thatG(V) = G(V) ∩ ES( G E , x)E⊤ . =H This means that near x we can consider a new space S H . = S m ∩ H, define a new cone S H + .
the constraint G(x) 0 at the point x . = (0, 0). In this case, Forsgren's CQ fails at x with E . (G, x) = S 2 .In fact, (F1) fails for every orthogonal matrix E. Furthermore, regardless of E the vectors v11(x, E), v22(x, E), and v12(x, E) ∈ R 2 , are linearly dependent and I( G E , x) = {(1, 1), (1, 2), (2, 2)}, hence sparse-nondegeneracy also fails to hold at x. On the other hand, note that for E = I2, we obtain I∇( G E , x) = {(1,
Figure 1 :
1Relationship among some CQs for NSDP. Classical CQs are in blue boxes, while new CQs are in green boxes. Arrows indicate strict implications, except for the dashed arrow where the reverse implication is unknown.
r for all sequences {x k } k∈N → x to conclude that x satisfies weak-nondegeneracy. Conversely, suppose that weak-nondegeneracy holds at x, take any sequence {x k } k∈N → x and any {E k } k∈N → E . = [ei, . . . , em−r] such that vii(x, E) : i ∈ {1, . . . , m − r} is linearly independent. Note that E must have the form
in terms of sup Y ∈Λ(x) is also true. Above, T 2 S m + ∩H (G(x), DG(x)[d]) denotes the second-order tangent set to S m + ∩ H at G(x) along DG(x)[d] (see [10, Def. 3.28]), and σ(Y, T 2 S m + ∩H (G(x), DG(x)[d])) denotes its support function.
Proposition 4.4. Let x ∈ F be such that Ker G(x) = {0}, and let E and W be matrices that span Ker G(x), such that item 1 of Definition 4.2 holds. Then, #I( G E , x) = #I( G W , x).Proof. Let Z . = [z ℓs ] ℓ,s∈{1,...,m−r} be an invertible matrix such that EZ = W and note that Gwhere zi denotes the i-th column of Z, andRephrasing,where unfold : R m−r×m−r×n → R n×(m−r) 2 is an unfolding operator for the tensor D G E (x) when it is seen as an m − r × m − r matrix with n-dimensional entries. Also, vec : R m−r×m−r → R (m−r) 2 is the usual vectorization operator, which transforms a matrix into a vector by stacking up its columns, from left to right. Consequently,
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"Long-tailed Distribution Adaptation",
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| [
"Zhiliang Peng \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Wei Huang \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Zonghao Guo \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Xiaosong Zhang [email protected] \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Jianbin Jiao \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Qixiang Ye [email protected] \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Zhiliang Peng \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Wei Huang \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Zonghao Guo \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Xiaosong Zhang \nUniversity of Chinese Academy of Sciences\nBeijingChina\n",
"Jianbin Jiao \nUniversity of Chinese Academy of Sciences\nBeijingChina\n"
]
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"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina",
"University of Chinese Academy of Sciences\nBeijingChina"
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"Proceedings of the 29th ACM International Conference on Multimedia (MM '21)"
]
| Recognizing images with long-tailed distributions remains a challenging problem while there lacks an interpretable mechanism to solve this problem. In this study, we formulate Long-tailed recognition as Domain Adaption (LDA), by modeling the long-tailed distribution as an unbalanced domain and the general distribution as a balanced domain. Within the balanced domain, we propose to slack the generalization error bound, which is defined upon the empirical risks of unbalanced and balanced domains and the divergence between them. We propose to jointly optimize empirical risks of the unbalanced and balanced domains and approximate their domain divergence by intra-class and inter-class distances, with the aim to adapt models trained on the long-tailed distribution to general distributions in an interpretable way. Experiments on benchmark datasets for image recognition, object detection, and instance segmentation validate that our LDA approach, beyond its interpretability, achieves state-of-the-art performance. Code is available at github.com/pengzhiliang/LDA. | 10.1145/3474085.3475479 | [
"https://arxiv.org/pdf/2110.02686v1.pdf"
]
| 238,407,752 | 2110.02686 | 7d8a2aa6e3a0568305a8ddaa727016cfc7653bb9 |
Long-tailed Distribution Adaptation
Virtual Event, China. ACMCopyright Virtual Event, China. ACM2021. October 20-24, 2021
Zhiliang Peng
University of Chinese Academy of Sciences
BeijingChina
Wei Huang
University of Chinese Academy of Sciences
BeijingChina
Zonghao Guo
University of Chinese Academy of Sciences
BeijingChina
Xiaosong Zhang [email protected]
University of Chinese Academy of Sciences
BeijingChina
Jianbin Jiao
University of Chinese Academy of Sciences
BeijingChina
Qixiang Ye [email protected]
University of Chinese Academy of Sciences
BeijingChina
Zhiliang Peng
University of Chinese Academy of Sciences
BeijingChina
Wei Huang
University of Chinese Academy of Sciences
BeijingChina
Zonghao Guo
University of Chinese Academy of Sciences
BeijingChina
Xiaosong Zhang
University of Chinese Academy of Sciences
BeijingChina
Jianbin Jiao
University of Chinese Academy of Sciences
BeijingChina
Long-tailed Distribution Adaptation
Proceedings of the 29th ACM International Conference on Multimedia (MM '21)
the 29th ACM International Conference on Multimedia (MM '21)New York, NY, USAVirtual Event, China. ACM102021. October 20-24, 202110.1145/3474085.3475479ACM Reference Format:CCS CONCEPTS • Computing methodologies → Cost-sensitive learningIm- age representationsObject detection KEYWORDS long-tail distribution, domain adaptation, classification, object de- tection, instance segmentation
Recognizing images with long-tailed distributions remains a challenging problem while there lacks an interpretable mechanism to solve this problem. In this study, we formulate Long-tailed recognition as Domain Adaption (LDA), by modeling the long-tailed distribution as an unbalanced domain and the general distribution as a balanced domain. Within the balanced domain, we propose to slack the generalization error bound, which is defined upon the empirical risks of unbalanced and balanced domains and the divergence between them. We propose to jointly optimize empirical risks of the unbalanced and balanced domains and approximate their domain divergence by intra-class and inter-class distances, with the aim to adapt models trained on the long-tailed distribution to general distributions in an interpretable way. Experiments on benchmark datasets for image recognition, object detection, and instance segmentation validate that our LDA approach, beyond its interpretability, achieves state-of-the-art performance. Code is available at github.com/pengzhiliang/LDA.
INTRODUCTION
The success of deep learning depends on large-scale high-quality datasets, e.g., ImageNet ILSVRC 2012 [24], MS COCO [19] and Places Database [34], where the data uniformly distributes with respect * Corresponding author.
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. to categories. The real-world data, however, typically follows longtailed distributions [7,9], i.e., a few classes (a.k.a. head classes) occupy most of the data while most classes (a.k.a. tail classes) have few samples.
To adapt the models trained upon long-tailed distributions to uniform distributions, re-sampling and re-weighting methods [3,6] have been explored to re-balance the data distributions. Recently, feature-classifier decoupling learning strategies [14,16,23,33] are proposed to learn representation and classifiers, step by step. Despite of the effectiveness, these approaches require complicated training strategies and/or work in empirical fashions, which hinder the interpretability and further progress of long-tailed recognition problems.
In this study, analogous to [14], we treat the unbalanced training set as an unbalanced domain and the balanced test set as a balanced domain, solving the Long-tailed recognition problem in a Domain Adaptation (LDA) fashion. Based on the upper bound of generalization error of domain adaptation [1,2], LDA firstly defines the relaxed upper bound of generalization error for the balanced domain. The training objective of LDA is to minimize the defined relaxed upper bound, which is composed of empirical risks of the unbalanced domain and balanced domain and the divergence between these two domains. By the way, the proposed relaxed upper bound has a great potential in explaining popular decoupling learning strategies [14,16,23,33].
During training, LDA minimizes the empirical risk of the unbalanced and balanced domains in an end-to-end manner. Considering that balanced and unbalanced domains follow the same distributions but have different sampling ratios, empirical risk of the balanced domain is estimated by using the sample re-weighting method [31]. However, it is challenging for a classifier ℎ to simultaneously minimize two homogeneous but different risks (balanced and unbalanced empirical risks). Therefore, an auxiliary classifier ℎ ′ , is introduced to collaborate with ℎ and minimize the unbalanced and the balanced empirical risks, respectively.
Meanwhile, a dynamic balance factor defined on the difference between the two empirical risks is used to regularize the two risks, which prevents the model from favoring either risk. Compared with recent works (including decoupling learning strategy [16] and BBN [33]) which focus on tail classes when the network tends to converge, our LDA approach consistently considers tail and head classes during the entire training procedure, which can prevent the network from falling into a local minimum.
LDA involves a plausible strategy to reduce the divergence between balanced and unbalanced domains. The primary difference between long-tail recognition with general domain adaptation is that the divergence between balanced and unbalanced domains is not observable. We therefore propose to minimize the divergence by constraining feature distribution of the unbalanced domain, Figure 2(c). Specially, we construct an ideal balanced domain, where features of each class have minimal overlap, as the learning objective, Figure 2(e). Furthermore, a self-regularization strategy that increases inter-class distance and decreases intra-class distance is proposed to reduce the divergence between the unbalanced domain and the ideal balanced domain. Consequently, the divergence, as defined in [17], has been calculated and shown the effectiveness of the self-regularization strategy.
We evaluate LDA on long-tailed tasks including image classification, object detection and instance segmentation, and validate its superiority over competing approaches. The last but not the least, LDA has a negligible computational cost overhead compared with the baseline method, which is trained with the cross entropy loss.
RELATED WORK 2.1 Long-tailed Recognition
Re-balancing Method. This line of methods can be categorized to re-sampling and re-weighting ones. The re-sampling methods are further categorized into two: 1) Over-sampling by simply repeating data for minority classes [3,4,9,25] and 2) under-sampling by abandoning data for dominant classes [3,10,15]. The re-weighting methods usually assign larger weights to samples from tail classes in the loss function [4,6,13] or drop some scores of tail classes in the Softmax function to balance the positive and negative gradients [27].
Most re-balancing strategies try to restore the balanced distribution from the unbalanced set, so as to directly minimize the balanced empirical risk. Nevertheless they unfortunately ignore the unbalanced empirical risk and distort the feature distribution to some extent. Decoupling Method. Considering that the balanced and unbalanced data share a feature space but different distributions, various decoupling strategies [5,7,14,16,21,23] separated the training process into two stages. In the first stage, they train the network upon the unbalanced data to construct the feature space. In the second stage, they utilize re-balancing strategies to fine-tune the classifiers with small learning rates to elaborate the classification boundaries. The representative BBN method [33] has a two-branch architecture, which equips with two samplers and shifts attention from head classes to tail classes during the training process.
Despite of the effectiveness, these methods either significantly increase the parameters or require a complicated training strategy. The proposed LDA overcomes these disadvantages by training a simpler yet more effective paradigm, as well as considering the divergence between two domains for distribution alignment.
Domain Adaptation
Domain adaptation methods aim to mitigate the divergence between the distribution of training data and test data so that the learned models can be well generalized to target domains [8,26,32]. There are some approaches that handle the imbalance problem during domain adaptation [12,35]. The test data in the domain adaptation setting can be used to explicitly calculate divergence, which is however invisible for long-tailed recognition during the training procedure. One recent method [14] estimated the conditional probability within balanced empirical risk by the meta-learning method and adopt the decoupling training strategy. However, the divergence between the unbalanced training set and the balanced test set remains overlooked and requires to be elaborated.
METHODOLOGY 3.1 Modeling
We formulate long-tailed recognition as a domain adaptation task, where the source domain corresponds to a unbalanced domain (the unbalanced training data distribution) and the target domain to a balanced domain (the balanced test data distribution). Denoting the two domains as D and D , their distributions in feature space are defined asD andD , respectively. Notably, we not only consider the long-tailed recognition problem as a domain adaptation task, analogous to [14], but also propose to define and slack the generalization error bound.
In domain adaptation, generalization error upper bound of the target domain has been extensively studied [1,2]. Let R be a representation function from the input sample space X to the feature space Z and H a hypothesis space with VC-dimension . If a random labeled sample set of size is generated by applying R to a D -i.i.d. labeled samples, then with probability 1 − at least for every ℎ ∈ H the expect risk of the balanced domain (ℎ) satisfies the following inequality:
(ℎ) ≤ˆ(ℎ) + √︄ 4 log 2 + log 4 + H (D ,D ) + ,(1)
whereˆ(ℎ) denotes empirical risk of the unbalanced domain and is a constant. H (D ,D ) specifies the divergence betweenD andD .
Based on the theory [29], expect risk (ℎ) is bound to its empirical estimationˆ(ℎ). Namely, if is an ′ -size i.i.d. sample, then with probability exceeding 1 − , (ℎ) satisfies the following inequality: Notably, images in an unbalanced training set and balanced validation/test set are sampled from the same distribution but with different sample ratios. It remains satisfying the hypothesis that D is contained by D [31]. Therefore,ˆ(ℎ) can be estimated upon (ℎ):ˆ(
(ℎ) ≤ˆ(ℎ) + √︄ 4 ′ log 2 ′ + log 4 .(2)ℎ) = E ( , )ˆ( ℎ( ), ) = E ( , ) ( , ) ( , )ˆ( ℎ( ), )(3)
where E is expectation function and ( , ) and ( , ) are joint probability in unbalanced and balanced domain, respectively. So far, (ℎ) is confined to two upper bounds defined in Equations 1 and 2. However, it is nontrivial to determine its strict minimal upper bound in theory and we propose to approximate an upper bound by
(ℎ) ≲ˆ(ℎ) +ˆ(ℎ) + H D ,D + * ,(4)
where * is the constant term. As shown in Equation 4, expect risk of the balanced domain (ℎ) is connected with empirical risks of the unbalanced domainˆ(ℎ) and the balanced domainˆ(ℎ) and the divergence H (D ,D ) between two domains. In what follows, we propose to optimize these items one by one while minimizing (ℎ).
Empirical Risk Minimization
With a single classifier ℎ (termed balanced classifier), it is difficult to minimize two homogeneous risk functions defined in Equation 4. We therefore introduce an auxiliary classifier ℎ ′ (termed unbalanced classifier) to minimizeˆ. In this way, we relax the contraint for the balanced classifier ℎ, by solely minimizingˆ, without reducinĝ explicitly.
Notably, the representation function R in [1,2] is assumed be fixed, but it's learnable in the CNN framework with gradient comes from the classifier. If we regard the whole network as a complex classifier (including R, ℎ and ℎ ′ ), should also subject to the slack upper bound defined by Equation 4. In this case, R can be optimized regardless of which classifier the empirical risk comes from. Therefore, introducing ℎ ′ is plausible, as it relaxes the constraint on ℎ but does not affect the optimization objective of .
Based on the above analysis, we conclude the risk/loss function for ℎ and ℎ ′ . For ℎ ′ , it requires to minimize empirical risk for the unbalanced domain, i.e.,ˆ(R,
ℎ ′ ) = 1 =1 L ( ( , R, ℎ ′ ), ), where
denotes the samples number in a batch and L is the vanilla cross entropy loss. For ℎ, we would like to have a closer look atˆ(R, ℎ), from the view of target shift case in domain adaptation [31]. In this case, the conditional probability is shared between balanced and unbalanced domains, i.e., ( | ) = ( | ), which is different from [14], while the marginal probability is different, i.e., ( ) ≠ ( ). Therefore, the estimatedˆis re-defined as:
(R, ℎ) = E ( , ) ( , ) ( , ) L ( ( , R, ℎ), ) = E ( , ) ( ) ( | ) ( ) ( | ) L ( ( , R, ℎ), ) = E ( , ) ( ) ( ) L ( ( , R, ℎ), ) = 1 ∑︁ =1 L ( ( , R, ℎ), ),(5)
Empirical Risk Balance
Two empirical risks jointly determine the optimization of the network while R receives supervised information to minimize the risks. If either of them dominates the training process, the optimization tends to get stuck to a local minimum. Therefore, how to balance two empirical risks remains critical.
We propose a self-adaptive adjustment method, based on the real-time performance difference between ℎ and ℎ ′ during training to balance the two empirical risks. For image , letˆℎ′ and ℎ denote the predict labels of classifier ℎ ′ and ℎ respectively, i.e.,ˆℎ′ = arg max ( ; R, ℎ ′ ),ˆℎ = arg max ( ; R, ℎ). Let parameter regularize the importance ofˆ(ℎ ′ ) andˆ(ℎ). The empirical risk is a trade-off betweenˆ(ℎ ′ ) andˆ(ℎ):ˆ(ℎ ′ ) +ˆ(ℎ). Let Δ denote the performance differences. is defined as:
= Δ = ( 1 1 ∑︁ =1 ∑︁ =1 (I[ˆℎ ≠ ] − I[ˆℎ′ ≠ ])) ,(6)
where is the sample number in a few previous mini-batches. It avoids Δ changing dramatically. I[ˆ≠ ] ∈ {0, 1} is an indicator function evaluating to 1 ifˆ≠ else 0. is an hyper-parameter which controls the degree of self-adaptive adjustment.
Different from the decoupling methods [5,16], which optimize the head classes at early training epochs and optimize tail classes after networks are adequately trained, our approach treats all classes equally during the whole training process. The self-adaptive adjustment strategy dynamically balances two empirical risks, Figure 6(b), as: a) When Δ increases increases, because is proportional to Δ ; b) When increases the network pays more attention tô (ℎ), driving Δ to decrease; c) when Δ decreases decreases for the proportional relation; d) when decreases, the network reduces its attention toˆ(ℎ), so that Δ increases. These processes iterate until the convergence of the training procedure.
Divergence Minimization
With respect to divergence, the differences between general domain adaptation and long-tailed recognition are twofold. On the one hand,D is invisible in long-tailed recognition because images in validation/test set are inaccessible during training. On the other hand, we can obtain an approximately balanced classifier when minimizing the balanced empirical risk, as shown in Figure 2 Considering the difficulty to estimate the hidden spaceD , we propose to approach it in an indirect fashion. Assuming there exists an ideal balanced domain spaceD * , as shown in Figure 2(e), which is the alignment objective of the unbalanced domain. Unfortunately, D * can't be defined explicitly. We thereby propose to take advantage of the nature ofD * , where the feature space of each class has minimal intersection. In specific, we propose a self-regularization strategy to approximate the divergence H D ,D by minimizing the intersection space among classes. This is implemented by increasing the inter-class distance and decreasing the intra-class distance, as shown in Figure 2(c).
Based on above analysis, we propose to separate features of different classes and pull together features of same classes to the largest extent. In specific, we add a projection head , i.e. a fully connect layer, to project the representation before the classifier to a latent space of higher dimension. To increase the distance between class in the latent space, the feature centers of different classes are required to stay away from each other.
Let F be the feature of in the latent space, the feature center of class , . ., = 1 | | ∈ { = } F , where | | denotes the feature number for class . The intra-class distance of class is defined as
= 1 | | ∈ { = } 1 − (F , ),
where is the funciton. Similarly, the inter-class distance between class and is defined as = 1 − ( , ). By defining as the number of class appears in a mini-batch, the intra-distance loss L , which aims to reduce , is defined as:
L = 1 ∑︁ =1 1 | | ∑︁ ∈ { = } 1 − (F , ).(7)
Minimizing the inter-distance loss L aims to maximize . It is a variant of hinge loss with margin Δ , which is 1 in our experiments unless otherwise noted. L is defined as:
L = ∑︁ =1 ∑︁ =1, ≠ max(0, Δ − (1 − ( , ))),(8)
where = + , and , which are defined in Equation 5, are respectively the weight for class and .
Long-tail Recognition
The implementation of long-tailed recognition is illustrated in Figure 3. In the training phase, we minimizeˆ(R, ℎ ′ ) andˆ(R, ℎ) to optimize the unbalanced classifier ℎ ′ and balanced classifier ℎ, respectively. We also minimize L (R, ) and L (R, ) to optimize projection head . The total loss function is concluded as:
L =ˆ(R, ℎ ′ ) +ˆ(R, ℎ) + (L (R, ) + L (R, )),(9)
where is defined in Equation 6 and is the weight of L and L penalty term, which is set to 1 in experiments unless otherwise noted.
In the test phase, we utilize the output from balanced classifier to evaluate performance because the test set is a balanced domain. As there is not additional parameters introduced, the proposed LDA method has negligible computational cost in the inference phase compared with the baseline method, which is trained with the cross entropy loss.
EXPERIMENT 4.1 Datasets and Setup
We evaluate the proposed approach on four datasets including the CIFAR-100-LT [6], ImageNet-LT [20], Places-LT [20], and the LVIS v1.0 dataset [9]. Following [6], we define the imbalance ratio (IR) of a dataset as the class size of the first head classes divided by the size of the last tail classes in the training set. The details of these datasets are present in the Supplemental. For classification, after training on the long-tailed training sets, we evaluate the models on the balanced validation/test sets and report top-1 recognition accuracy over all classes. To examine performance variations across classes with different examples numbers during training, we follow [20] to report accuracy on three splits of these classes: Many-shot (more than 100 images), Medium-shot (20-100 images) and Few-shot (less than 20 images). For object detection and instance segmentation, we train on the LVIS training set and report mean Average Precision (mAP) on the validation set for all classes as well as three split groups including rare (appears in ≤ 10 images), common (appears in 10−100 images) and frequent (appears in > 100 images)), respectively.
Implementation Detail
ImageNet-LT. The model is trained for 90 epochs in total from scratch. Following [16], ResNeXt-50-32x4d [30] is set as the backbone network. The SGD optimizer with momentum 0.9, batch size 256, cosine learning rate schedule decaying from 0.2 to 0 and image resolution 224×224 is used. The parameter in Equation 6 is set as 2.0. Unless otherwise specified, the warm-up strategy is used in first 5 epochs. Places-LT. Following [16], ResNet-152 pre-trained on ImageNet is set as the backbone network. During training, weights of the last block changes while other weights are frozen. and are respectively set as 0.6 and 0.5. CIFAR100-LT. Following [33], ResNet-32 is set as the backbone network. The network is trained for 200 epochs and the SGD optimizer has a momentum 0.9, batch size 128 and base learning rate 0.1. The learning rate is decayed at 160-th and 180-th epoch by 0.01. and are respectively set as 1.0 and 0.5.
LVIS.
We equip the Mask R-CNN with three classifiers (present in the Supplemental), as in Figure 3. For training, we apply repeat factor sampling [9], scale jitter (sampling image scale for the shorter side from 640, 672, 704, 736, 768, 800) and random flip at training time. For testing, images are resized to a shorter image edge of 800 pixels and no test-time augmentation is used. The SGD optimizer has a momentum 0.9, weight decay 0.0001, batch size 8 and base leaning rate 0.01. and are respectively set as 1.5 and 0.1. The learning rate is dropped by a factor of 10 at both 8-th(16-th, 27-th) and 11-th(22-th, 33-th) epoch when total epochs is 12(24, 36).
Main Result
Classification. We conduct experiments on ImageNet-LT, Place-LT and CIFAR100-LT, and report the results in Tables 1, 3 and 2, respectively. Additional results about backbone networks are presented in the Supplemental. One can see that LDA consistently outperforms the state-of-the-art methods [16,28,33] for all datasets and backbone networks. On ImageNet-LT, it achieves 53.4% accuracy, 1.6% higher than that of De-confound-TDE[28] and 3.5% higher than that of LWS [16]. On Place-LT, LDA achieves 39.1% accuracy which is slightly higher than the state-of-the-art (BALMS [23]). On CIFAR100-LT, with a pre-trained model, LDA respectively achieves 2.1%, 3.4% and 3.3% higher accuracy than De-confound-TDE [28] when IR is 10, 50 and 100. These are significant margins for the challenging task. Object Detection and Instance Segmentation. Table 4 shows experiment results on the LVIS dataset, and further exploration can be found in the Supplemental. In Table 4, when trained 12 epochs, LDA achieves 25.2 AP for object detection and 24.4AP for instance segmentation, which is 2.7 and 2.6 higher than the baseline method. When trained 24 epochs, LDA achieves 25.7 AP, which is 0.6 higher than the Balanced sigmoid method, which is implemented upon [16].
Visualization
Classifier Weight. The norm of the weight || || is correlated with the number of samples of class , which is consist with the observation in [16]. As shown in Figure 4(a), the balanced classifier has uniform weight values for all categories, which demonstrate the bias of classifiers is largely solved by the proposed method. In comparison, weight values of the baseline classifier are significantly different for different categories. As shown in Figure 1, when trained with the plain cross entropy loss, the classifier weight vectors tend to mix together, which implies that the decision boundaries in the high-dimensional space are not clear. In contrast, weight vectors of the balanced classifier are more separated, which suggests the classifier is more discriminative. Domain Divergence. To verify the effectiveness of the proposed self-regularization strategy in Subsection 3.4, after inference phase, we calculate the contrastive domain divergence, which can reflects the intra-class and inter-class domain discrepancy for unsupervised domain adaptation. Please refer to [17] for more details. Figure 4 divergence, which demonstrates the rationality and validity of the proposed self-regularization strategy. Feature Representation. To visualize the impact of LDA to feature representation, we measure the ratio of to on the test set, as shown in Figure 4
Ablation Study
Empirical Risk and Domain Divergence. As shown in Figure 5, by using empirical risks minimization (E.M), LDA achieves 5% improvement over the baseline method on ImageNet-LT, which demonstrates that combining risks from both the balanced and unbalanced domains is plausible. Moreover, with divergence minimization (D.M), LDA further improves the accuracy by 0.5%, which not only confirms that the domain divergence is an important factor to be considered in long-tailed recognition but also validates the proposed divergence minimization strategy. Joint Optimization. As shown in Table 5, we compare optimization strategies. It can be seen that solely optimizing the emperical risk of the unbalanced (ˆ) or balanced domain (ˆ) achieves poor performance (41.3%-47.9%). Optimizing the unbalanced (ˆ) and balanced domain (ˆ) step by step achieves much higher performance (52.1%). Such strategy has also been validated in [16] (CE→RS) and in [5](CE→RW), which can be regarded as specific cases of the step-by-step optimization. Our joint optimization strategy (ˆ&ˆ) achieves the best performance. Balance Strategy. To make a trad-off between the empirical risks of the unbalanced and balanced domains, we design the self-adaptive strategy based on the performance difference of unbalanced and balanced classifier. in Equation 6 is determined by the grid search method. In Figure 6(a), we compare grid search with our selfadaptive adjustment strategy. It can be seen that our strategy outperforms the grid search method. When is set as 2.0, LDA achieves the best performance. is the balance factor generated by grid search to make a trade-off between empirical risks. is a hyper-parameter defined in Equation 6, (b) Change trend of adaptive alpha with the real-time performance difference Δ .
CONCLUSION
In this paper, we re-visited the long-tailed problem from a domain adaptation perspective. In specific, after formulating Long-tailed recognition as Domain Adaption (LDA), we propose the slack the generalization error bound, which is defined upon the empirical risks of unbalanced and balanced domains the the divergence between them, which also has a great potential in explaining popular decoupling learning strategies. By jointly minimizing unbalanced and balanced empirical risks and minimizing the domain divergence, we adapted the feature representation and classifiers trained on the long-tailed distributions to balanced distributions. Experiments on four benchmark datasets for image recognition, object detection, and instance segmentation demonstrated effectiveness of our proposed approach, in striking constant with the state-of-thearts. In an explainable way, our research provides a fresh insight to the long-tail recognition problem in real-world scenarios.
A DATASET DETAILS
Long-tailed CIFAR-100: This dataset is the long-tailed version of CIFAR-100 manually created in [6]. The imbalance ratios used in experiments are 100, 50 and 10 following existing works [6,20,23,28,33].
ImageNet-LT: To construct an unbalanced real-world scene dataset, a long-tailed version of ImageNet [24] termed ImageNet-LT was created [20]. The details about how to sample data from ImageNet can be found in [20]. The ImageNet-LT dataset has 115.8K training images about 1,000 classes of images, with an imbalance ratio 1280/5. The validation/test set has 20/50 images per class, following a uniform distribution.
Places-LT: A long-tailed version of Places [34] is introduced in [20] with an imbalance ratio 4980/5, which is more challenging than the ImageNet-LT. The validation/test set has 20/100 images per class, following a uniform distribution.
LVIS: LVIS v1.0 (LVIS for simplicity) is a long-tailed dataset for object detection and instance segmentation [9]. The images are sampled from the MS COCO dataset [19]. Compared with COCO, LVIS contains approximately 160k images from 1203 classes with 2M instance annotations. It also expanded the validation set from 5k images to 20k images. Compared with the datasets for image classification, LVIS has a significantly larger imbalanced ratio 50552/1.
B OBJECT DETECTION AND INSTANCE SEGMENTATION
We choose Mask R-CNN [11] as the baseline for object detection and instance segmentation, following the settings in [20]. The head of Mask R-CNN has three branches, . ., a classifier, a regression module, and a segmentation module. In detector implementation, the a regression module and segmentation module are kept as they are. The classifier is updated to three classifiers: the first for Fg and Bg classification, the second and the third are unbalanced and balanced classifiers, as shown in Figure 7. Following the long-tailed classification, we further use a fully connection layer ( Figure 3 in the paper) for divergence minimization. In Figure 7, Fg&Bg is the foreground and background classifier, which is used to distinguish proposal is foreground or not. All the prediction results of the proposal are used to calculate the loss for Fg&Bg classifier. The unbalanced and balanced classifiers are respectively used to minimize the empirical risks of unbalance and
Where L & denotes the cross entropy loss for the Fg&Bg classifier, L denotes the loss defined in Equation 9 in the paper, L and L are kept same with [11]. During inference, we multiple the score of Fg&Bg with the score of the balanced classifier for object detection and instance segmentation.
C COMPARED METHODS
We compare our model with the following methods:
Cross-entropy loss. We choose the vanilla cross-entropy loss as our baseline.
Label-distribution-aware margin loss. Motivated by minimizing a margin-based generalization bound, It [5] assigns different margins between classes according to classes frequency.
Focal loss. The focal loss [18] is a kind of hard example mining method, which aims to emphasize on learning hard examples by down-weight easy examples. It is proposed to tackle the foregroundbackground class imbalance in object detection. Therefore, it treats tailed class as hard sample in long-tailed classification.
Decoupling representation and classifier. [16] decouples the learning pipeline into representation learning and classifier learning. Concretely, After learning representations with the random sampler, it adjusts the classifier by various methods(NCM, cRT, LWS or -norm) with the class-balanced sampler.
Equalization loss. [27] analyzes the long-tailed recognition problem from the perspective of gradient, and tackles the problem by ignoring discouraging gradients for tail classes.
Bilateral-Branch Network. [33] proposes a novel bilateralbranch architecture to simultaneously take care of the representation learning and classifier learning with a random sampler and a class-balanced sampler.
De-confound training and TDE. [28] establishes a causal inference framework to interpret what misleads the tail prediction biased towards head classes. It uses causal intervention in training, and counterfactual reasoning in inference.
D BACKBONE NETWORKS
To further validate the general applicability of the proposed LDA, we test it with different backbone networks for image classification, object detection and instance segmentation. The results are included in Tables 6 and 7.
Figure 1
1MM '21, October 20-24, 2021, Virtual Event, China © 2021 Association for Computing Machinery. ACM ISBN 978-1-4503-8651-7/21/10. . . $15.00 https://doi.org/10.1145/3474085.: t-SNE visualization of classifier weight vectors. (a) Classifier weight vectors trained with the cross entropy loss, and (b) with the proposed LDA approach. Larger distances among vectors imply higher discriminative capacity.
Figure 2 :
2Overview of the proposed LDA approach. (a) and (b) are the feature spaces of balanced and unbalanced domains generated by the baseline model, which is trained with the cross entropy loss. The model biases towards head classes as their loss dominates the unbalanced empirical risk, . ., the classifier and generated feature distribution (c) and (d) respectively denote feature spaces of the unbalanced and balanced domains generated by our LDA approach. The model has significantly smaller bias towards head classes. (e) is the feature space of the ideal balanced domain as the alignment goal for unbalanced domain. (Best viewed in color)
(d). Different from general domain adaptation, where transfers the features of the target domain to the source domain to guarantee the classifier trained within the source domain yields accurate results in the target domain, the proposed LDA requires to transfer the features of the unbalanced domain to the balanced domain on the contrary.
Figure 3 :
3Network architecture. ℎ ′ , ℎ, and are jointly optimized classifiers (fully-connect layers). During inference, ℎ predicts classification results while ℎ ′ and do not predict.
Figure 4 :Figure 5 :
45(b) shows the divergence on CIFAR-100-LT with different imbalance ratios. The results show that, under different imbalance ratios, empirical risk minimization (E.M) slightly increases the divergence while divergence minimization (D.M) significantly decreases the Visualization analysis. (a) Weight norm || || of classifiers. (b) Divergence between unbalanced and balanced domains. (c) Ratio between between the intra-class and inter-class distances ( / ). For short, the joint minimization of empirical risks and divergence minimization are denoted as E.M and D.M, respectively. Analysis of E.M and D.M in LDA. E.M and D.M respectively denote empirical risk minimization and divergence minimization.
(c). With LDA(E.M), the ratio almost keep the same as baseline. But with LDA (E.M+D.M), the ratio becomes significantly smaller, showing the higher representative power of features for the tail classes.
Figure 6 :
6Ablation and visualization for self-adaptive strategy on ImageNet-LT. (a) Comparison of balance strategies.
Figure 7 :
7The classifier of our LDA in object detection and instance segmentation.
balance as defined in Subsection 3.2. Only the positive proposals are fed to the balanced and unbalanced classifiers.During training, the loss function for Mask R-CNN is concluded by:
Table 1 :
1Classification accuracy on ImageNet-LT. † means copied from[28].Methods
Few Medium Many All
Focal † [18]
8.2
37.1
64.3
43.7
OLTR † [20]
20.8
40.8
51.0
41.9
NCM[16]
28.1
45.3
56.6
47.3
cRT[16]
27.4
46.2
61.8
49.6
-norm[16]
30.7
46.9
59.1
49.4
LWS[16]
30.3
47.2
60.2
49.9
De-confound[28]
14.7
42.7
67.9
48.6
De-confound-TDE[28] 31.6
48.8
62.7
51.8
Baseline
11.1
41.6
68.8 47.9
(Ours) LDA
31.5
50.9
64.5 53.4
where
= ( )/ ( ). Elaborate designs for
can be found
in the literature [6, 18] but we implement it as
= 1/ ( ) for
simplicity.
Table 2 :
2Classification accuracy on Long-tailed CIFAR-100.* means using pre-trained models[28].Methods
Imbalance Ratios
10
50
100
Focal[18]
55.8 44.3 38.4
CB-Focal[6]
58.0 45.3 39.6
CE-DRW[5]
58.1 45.3 41.5
CE-DRS[5]
58.1 45.5 41.6
LDAM-DRW[5]
58.7 46.6 42.0
BBN[33]
59.2 47.0 42.6
De-confound*[28]
59.5 48.9 43.9
De-confound-TDE*[28] 59.8 51.2 47.3
Baseline
55.7 43.9 38.4
(Ours) LDA
59.7 48.1 43.8
(Ours) LDA*
61.9 54.6 50.6
Table 3 :
3Classification accuracy on Places-LT. † denotes results from[20].Methods
Few Medium
Many
All
Focal † [18] 22.4
34.8
41.1
34.6
OLTR † [20] 25.3
37.0
44.7[16] 35.9
NCM[16]
27.3
37.1
40.4
36.4
cRT[16]
24.9
37.6
42.0
36.7
-norm[16] 31.8
40.7
37.8
37.9
LWS[16]
28.6
39.1
40.6
37.6
BALMS[23] 31.6
39.8
41.2
38.7
Baseline
7.9
25.9
43.7
28.8
(Ours) LDA 32.1
40.7
41.0
39.1
Table 4 :
4Performance comparison on the LVIS val set with ResNet-50. † denotes data from[22].Methods
Epochs AP
AP
AP
AP
AP
Mask R-CNN[11]
12
22.5
9.6
21.0 27.8
21.7
cRT † [16]
25+10
24.8
14.7 22.5 27.8
23.2
LWS † [16]
25+10
24.8
14.9 22.5 27.8
23.3
EQL † [27]
25+10
24.2
16.5 24.2 27.5
24.2
Balanced sigmoid[22] 25+10
26.7
18.3 24.0 29.3
25.1
Baseline
12
22.5
10.2 20.8 28.0
21.8
(Ours) LDA
12
25.2
18.2 23.7 27.9
24.4
(Ours) LDA
24
26.6
18.1 25.4 29.4
25.7
(Ours) LDA
36
26.7
18.4 25.3 29.8
25.9
Table 5 :
5Comparison of optimization strategies. Top-1 accu-
racy of ResNeXt-50 on ImagetNet-LT test set.ˆandˆde-
note optimizing unbalanced and balanced empirical risks,
respectively.ˆ→ˆmeans optimizingˆandˆstep-by-
step.ˆ&ˆmeans joint optimization ofˆandˆ. Ours-
u and Ours-b respectively denote the unbalanced and bal-
anced classifiers of the proposed LDA models.
Methodsˆˆˆ→ˆˆ&ˆaccuracy
CE
✓
47.9
RW
✓
41.3
CE→RW
✓
51.9
CE→RS
✓
52.1
Ours-u
✓
48.7
Ours-b
✓
52.9
Table 6 :
6Results on ImageNet-LT with different backbone networks.Methods
Backbones
Few Medium Many All
BALMS[23]
ResNeXt-10
25.3
39.5
50.3 41.8
cRT[16]
-
-
-
41.8
LWS[16]
-
-
-
41.4
LDA(Ours)
26.5
39.9
49.6 41.8
cRT[16]
ResNet-50
26.1
44.0
58.8
47.3
LWS[16]
29.3
45.2
57.1
47.7
[14]
-
-
-
48.0
LDA(Ours)
31.2
48.4
61.1 50.9
cRT[16]
ResNet-101
28.0
46.5
61.6
49.8
LWS[16]
31.2
47.6
60.1
50.2
LDA(Ours)
31.9
49.6
62.2 52.1
cRT[16]
ResNeXt-101
27.0
46.0
61.7
49.4
LWS[16]
31.2
47.2
60.5
50.1
De-con[28]
16.5
44.3
68.9 50.0
De-con-TDE[28]
33.0
50.0
64.7
53.3
LDA(Ours)
32.5
51.7
65.5 54.4
Table 7 :
7Results on LVIS v1.0 val set with different backbone networks.Methods
Backbone
AP
AP
AP
AP
AP
Mask R-CNN
ResNet-50
22.5
9.6 21.0 27.8
21.7
LDA
25.2
18.2 23.7 27.9
24.4
Mask R-CNN
ResNet-101
24.6
13.2 22.7 29.3
23.6
LDA
26.9
18.9 25.5 29.5
25.9
Mask R-CNN ResNeXt-101-32x4d
26.7
16.0 24.8 30.5
25.5
LDA
28.2
20.1 26.2 30.7
26.9
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| []
|
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"A Stronger Case for Superunification Post Higgs Boson Discovery",
"A Stronger Case for Superunification Post Higgs Boson Discovery",
"A Stronger Case for Superunification Post Higgs Boson Discovery",
"A Stronger Case for Superunification Post Higgs Boson Discovery"
]
| [
"Pran Nath [email protected] \nDepartment of Physics\nNortheastern University\n02115-5000BostonMAUSA\n",
"Raza M Syed \nDepartment of Physics\nNortheastern University\n02115-5000BostonMAUSA\n\nDepartment of Physics\nAmerican University of Sharjah\nP.O. Box 26666SharjahUAE\n",
"Pran Nath [email protected] \nDepartment of Physics\nNortheastern University\n02115-5000BostonMAUSA\n",
"Raza M Syed \nDepartment of Physics\nNortheastern University\n02115-5000BostonMAUSA\n\nDepartment of Physics\nAmerican University of Sharjah\nP.O. Box 26666SharjahUAE\n"
]
| [
"Department of Physics\nNortheastern University\n02115-5000BostonMAUSA",
"Department of Physics\nNortheastern University\n02115-5000BostonMAUSA",
"Department of Physics\nAmerican University of Sharjah\nP.O. Box 26666SharjahUAE",
"Department of Physics\nNortheastern University\n02115-5000BostonMAUSA",
"Department of Physics\nNortheastern University\n02115-5000BostonMAUSA",
"Department of Physics\nAmerican University of Sharjah\nP.O. Box 26666SharjahUAE"
]
| []
| Supersymmetry and more specifically supergravity grand unification allow one to extrapolate physics from the electroweak scale up to the grand unification scale consistent with electroweak data. Here we give a brief overview of their current status and show that the case for supersymmetry is stronger as a result of the Higgs boson discovery with a mass measurement at ∼ 125 GeV consistent with the supergravity grand unification prediction that the Higgs boson mass lie below 130 GeV. Thus the discovery of the Higgs boson and the measurement of its mass provide a further impetus for the search for sparticles to continue at the current and future colliders. * | 10.1088/1402-4896/aa9512 | [
"https://arxiv.org/pdf/1709.09718v1.pdf"
]
| 4,683,498 | 1709.09718 | ae276a009470bf3a12e53a0e472838a70c56ecf7 |
A Stronger Case for Superunification Post Higgs Boson Discovery
27 Sep 2017
Pran Nath [email protected]
Department of Physics
Northeastern University
02115-5000BostonMAUSA
Raza M Syed
Department of Physics
Northeastern University
02115-5000BostonMAUSA
Department of Physics
American University of Sharjah
P.O. Box 26666SharjahUAE
A Stronger Case for Superunification Post Higgs Boson Discovery
27 Sep 20173 Permanent address
Supersymmetry and more specifically supergravity grand unification allow one to extrapolate physics from the electroweak scale up to the grand unification scale consistent with electroweak data. Here we give a brief overview of their current status and show that the case for supersymmetry is stronger as a result of the Higgs boson discovery with a mass measurement at ∼ 125 GeV consistent with the supergravity grand unification prediction that the Higgs boson mass lie below 130 GeV. Thus the discovery of the Higgs boson and the measurement of its mass provide a further impetus for the search for sparticles to continue at the current and future colliders. *
Introduction
The standard model of particle physics [1] based on the gauge group SU (3) C ⊗ SU (2) L ⊗ U (1) Y and three generations of quarks and leptons is a highly successful model at low energies up to the electroweak scale. One of the basic elements of the model is that it is anomaly free. Specifically, the quarks and the leptons have the SU (3) C , SU (2) L , U (1) Y quantum numbers so that q(3, 2, 1 6 ), u c (3, 1, − 2 3 ), d c (3, 1, 1 3 ), L(1, 2, − 1 2 ), e c (1, 1, 1), where the first two entries refer to the SU (3) C , SU (2) L representations, and the last entry refers to the hypercharge defined so that Q = T 3 + Y . The anomaly free condition in this case implies that one has i f i Y i = 0, where f i is a product of multiplicity and color factor. Here one generation of quarks and leptons exactly satisfies the anomaly cancellation condition. The interesting phenomenon is that while the leptons have integral charge, Q = −1 for charged lepton, Q = 0 for the neutrino, the quarks have fractional charge, 2/3 for the up quarks and −1/3 for the down quark. The charge assignment appears intriguing and leads one to ask if there exist a larger framework within which one may understand such charge assignments.
Such a framework must be more unified and exist at a larger scale. There are other aspects which point to the possibility that a more unified framework may exist such as the product nature of standard model group. Here it requires three gauge coupling constants g 3 , g 2 , g Y to describe the interactions. One may speculate if they are remnants of a single coupling. Such issues were the subject of investigations in the early seventies. Thus in 1973- 74 Pati and Salam [2,3] proposed that the standard model was remnant of the group G(4, 2, 2) ≡ SU (4) ⊗ SU (2) L ⊗ SU (2) R . Here leptons and quarks are unified with the leptons arising as a fourth color. Soon after the work of [2] Georgi and Glashow [4] proposed the group SU (5) which in addition to unifying the quarks and the leptons, also unifies the gauge coupling constants. A group which gives an even greater unification was proposed subsequently. This is the group SO(10) [5]. It has the benefit of unifying a full generation of quarks and leptons in one irreducible representation of SO (10). The general criteria for grand unification is that one needs those unification groups which have chiral representations.
Here the relevant groups are SU (N ); SO(4N + 2), N ≥ 1; E 6 . As noted one of the constraints on model building is that of anomaly cancellation. Here the groups E 6 and SO(4N + 2), N ≥ 1 are automatically anomaly free while for SU (N ) one needs combinations of representations which are anomaly free.
However, as is well known non-supersymmetric models have a serious fine tuning problem [6]. Quantum loop corrections to the Higgs boson mass-squared give contributions which are quadratically divergent in the cutoff. In grand unified theories that cutoff would be the GUT scale ∼ 10 16 GeV, which is much larger than the electroweak scale. A cancellation of the loop term would require a fine tuning of one part in 10 28 . The cancellation of the quadratic divergence occurs naturally in supersymmetry [7]. It is desirable then to formulate unified models using supersymmetry. One persistent problem here concerns breaking of supersymmetry which is essential for building a viable phenomenology. This is problematic in global supersymmetry. In order to break supersymmetry in a phenomenologically viable way one needs local supersymmetry/supergravity [8,9,10] (For a more extensive discussion see [11]). Indeed grand unified models based on local supersymmetry provide the appropriate framework for unifying the strong and the electroweak interactions [12]. Gauge coupling unification is an important touchstone of unified model [13].
An important success of supersymmetry models is the unification of gauge coupling constants consistent with LEP data [14,15,16,17]. We note in passing that Planck scale physics could affect the predictions at the grand unification scale [18] (see [19,20]). Further, a significant feature of supergravity grand unification is that it is also the appropriate vehicle for the analysis of string based models since supergravity is the low energy limit of strings, i.e., at scales E < M Pl (see, e.g., [21]).
The outline of the rest of the paper is as follows: In section 2 we discuss the first works on unification beyond the standard model. These include the quark-lepton unification and the unification of gauge coupling constants. As noted the group SO(10) is now the preferred unification group for the unification of the electroweak and the strong interactions. This is discussed in section 3. In section 4 we discuss an alternative possibility for a unifying group, i.e., E 6 . The flavor puzzle which relates to the hierarchy of quark and lepton masses and mixings is discussed in section 5. In section 6 we discuss supergravity grand unification which provides the modern framework for realistic analyses of grand unified models and allows one to extrapolate physics from the electroweak scale to the grand unification scale. Unification in strings is discussed in section 7. One of the hallmarks of grand unified models is the prediction for the existence of monopoles and we discuss it in section 8. We note that no signature of monopole has thus far been detected and confirmation of its existence remains an outstanding experimental question. Since grand unification implies quark-lepton unification, another important prediction of grand unification is the decay of the proton. We discuss the current status of proton stability in GUTs and strings in section 9. In section 10 we argue that the discovery of the Higgs boson mass at ∼ 125 GeV lends further support for SUSY/SUGRA/Strings, and consequently for the discovery of sparticles at colliders. The role of future colliders for testing superunification is discussed in section 11. Conclusions are given in section 12.
2 G(4, 2, 2) and SU(5) unification As mentioned in section 1 a significant step toward unification beyond the Standard Model was taken by Pati and Salam [2] in 1974 when they proposed an extension of the Standard Model gauge group to the group G(4, 2, 2) ≡ SU (4) ⊗ SU (2) L ⊗ SU (2) R . Here (4, 2, 1) and (4, 1, 2) representations of G(4, 2, 2) contain a full generation of quarks and leptons. Since quarks and leptons reside in the same multiplet, G(4, 2, 2) represents unification of quarks and leptons. This phenomenon has a direct consequence in that it allows conversion of quarks into leptons and thus one might expect the proton to become unstable and decay. The feature above is shared by essentially all unified models and thus proton lifetime limits act as a strong constraint on unified models of particle unifications. We also note that in (4, 2, 1) + (4, 1, 2) one has one more particle, i.e., ν c , which does not appear in the standard model. ν c enters in the so called seesaw mechanism [22] that gives mass to the neutrinos. In G(4, 2, 2) the charge operator takes the form Q em = T 3L + T 3R + B−L 2 , where T 3L and T 3R are the generators of SU (2) L and SU (2) R and B and L are baryon and lepton numbers.
To break the G(4, 2, 2) symmetry one introduces heavy Higgs representations (
= T 3L + Y where Y = T 3R + B−L 2 .
The group G(4, 2, 2) can be broken further down to SU (3) C ⊗ U (1) em by use of (1, 2, 2) Higgs representation.
A comprehensive review of G(4, 2, 2) was recently given in [23]. 3 ). The GUT symmetry is broken by a 24-plet of heavy field Σ i j (i, j = 1, 2, · · · , 5) which breaks SU (5) down to the Standard Model gauge group. To break the symmetry further to the residual gauge group SU (3) C ⊗ U (1) em one introduces a 5 plet of Higgs H in the non-supersymmetric case. Here one immediate issue concerns the so-called doublet-triplet problems, i.e., how to keep the doublet of the 5-plet of Higgs light while making the triplet of the 5-plet superheavy. A concrete way to see this problem is to consider the scalar
potential V = M 2 TrΣ 2 + λ1 2 Tr(Σ 4 ) + λ2 2 (TrΣ 2 ) 2 + µTr(Σ 3 ) + 1 2 λ 3 H † HTr(Σ 2 ) + 1 2 λ 4 H † Σ 2 H + λ 4 (H † H) 2 .
In order to break SU (5) down to the Standard Model gauge group we need to have the VEV formation of Σ so that < Σ >= diag(2, 2, 2, −3, −3)v. Here the spontaneous breaking of the symmetry gives the constraint
M 2 + (7λ 1 + 30λ 2 )v 2 − 3 2 µv = 0.
The breaking generates a mass for the Higgs doublet which is superheavy whereas electroweak symmetry breaking requires that the Higgs doublet be light. In order to achieve a light Higgs doublet we need the constraint 10λ 3 + 3λ 4 = 0. This constraint must be satisfied to one part in 10 28 , which is a high degree of fine tuning.
In SU (5) GUT the hypercharge coupling g Y is related to the SU (5) invariant coupling g 5 so that g Y = 3/5g 5 . Thus SU (5) predicts the weak angle at the GUT scale so that sin 2 θ W = g 2 Y /(g 2 2 + g 2 Y ) = 3/8, where we set g 2 = g 5 . One of the problems of the minimal SU (5) model is that it generates undesirable relationships among the quark and the lepton masses. Thus consider the SU (5) Yukawa couplings
L Y = h dψ c i ψ ij H † j + h u ǫ ijklmψ cij ψ kl H m + h.c.,
where ψ c i and ψ ij are the5-plet and 10 -plet of fermions. In the above we have suppressed the generation index. More explicitly each generation will have their own Yukawa couplings h d and h u . After spontaneous breaking when the Higgs doublet gets a VEV one finds the following mass relation m e = m d , m µ = m s , m b = m τ . For the first two generations one has at the GUT scale the equality m e /m µ = m d /m s . These ratios are independent of the scale to one loop order and they hold at the electroweak scale to this order. However, the relation is badly in conflict with data. For the third generation m b = m τ also does not quite work for the non-supersymmetric case although it does for the supersymmetric case. This means that the minimal SU (5) We turn now to the supersymmetric version of SU (5) [24]. Here the superpotential for the minimal supersymmetric SU (5) is given by
W = λ 1 [ 1 3 TrΣ 3 + 1 2 M TrΣ 2 ] + λ 2 H 1i [Σ i j + 2M ′ δ i j ]H j 2 + f 1 H 1 ·5 · 10 + f 2 H 2 · 10 · 10.
The GUT symmetry breaks as in the non supersymmetric case with Σ developing a VEV.
Here as in the non-supersymmetric case a fine tuning is needed to get a light doublet. Specifically under the constraint M ′ = M the Higgs doublets are light while the Higgs triplets are superheavy. While a fine tuning is needed in the minimal SU (5) model to get light Higgs doublets, it is possible to achieve the doublet-triplet splitting without fine tuning in non-minimal or extended SU (5) models. One example is the so called missing partner mechanism [25,26] and the other flipped SU (5) ⊗ U (1) model [27]. In the missing partner mechanism one uses a 75-plet representation to break the GUT symmetry and additionally 50 + 50 representations to give masses to the color triplets. The superpotential involving the 50, 50 and 75 plet has the form Another method of producing light Higgs doublets while keeping the Higgs triplets heavy is to use the flipped SU (5)⊗U (1). As in the case of SU (5) one uses5⊕10 plet representations of SU (5) for each generation of quarks and leptons. However, the u and the d quarks as well as e and ν leptons are interchanged, the right handed neutrino ν c replaces e c in the 10-plet representation and e c appears in the singlet representation.
W G = λ 1 Φ lm ijk Σ ij lm H k 2 +λ 2 Φ ijk lm Σ lm ij H 1k + M Φ lm ijk Φ ijk lm + W Σ (Σ), where 50 = Φ ijk lm , 50 = Φ
To break the GUT symmetry one uses in the Higgs sector 10 ⊕ 10 rather than the 24-plet. For the breaking of the electroweak symmetry one introduces a5 ⊕ 5, i.e., H 1 and H 2 as for the standard SU (5) case. The superpotential for the Higgs sector is then W flipped = W 0 (10) + λ 1 ǫ ijklm H ij H kl H m 2 + λ 2 ǫ ijklmH ijHklH1m where W 0 (10) is the superpotential that depends only on the 10 and 10. The SU (3) ⊗ SU (2) ⊗ U (1) branchings of the 10-plet of SU (5) are given by 10 = (3, 1)(−4) ⊕ (3, 2)(1) ⊕ (1, 1)(6). When the singlet field (1, 1)(6) develops a VEV, the color triplet in the 5-plet combines with the color anti-triplet in 10-plet to become supermassive, while the SU (2) doublet and color singlet (1,2) in the 5-plet has no partner in the 10-plet. Thus one gets a natural missing partner mechanism in this case. The X, Y gauge fields of SU (5) have the SU (3) C ⊗ SU (2) L ⊗ U (1) Y quantum numbers (X, Y ) = (3, 2, 5/6), and the charges for them are Q X = 4/3, Q Y = 1/3. In contrast the (X ′ , Y ′ ) gauge bosons of the flipped SU (5) ⊗ U (1) have the quantum numbers (X ′ , Y ′ ) = (3, 2, −1/6) so that the charges for X ′ and Y ′ are given by
Q X ′ = 1/3, Q Y ′ = −2/3.
The unusual charge assignment in this case requires that the hypercharge be a linear combination of U (1) and of the generators in SU (5). A drawback of the model is that it is not fully unified since the underlying structure is a product group.
SO(10) unification
The group SO(10) as the framework for grand unification appears preferred over SU (5). The group SO (10) contains both G(4, 2, 2) and SU For SO(10) one will have Tr ({Σ µν , Σ αβ }Σ λρ ). However, there is no invariant tensor to which the above quantity can be proportional which then automatically guarantees vanishing of the anomaly for SO (10).
This analysis extends to other SO(N ) groups. One exception is SO (6) where there does exist a six index invariant tensor ǫ µναβλρ and so in this case vanishing of the anomaly is not automatic.
The group SO(10) is rank 5 where as the standard model gauge group is rank 4. The rank of the group can be reduced by either using 16 ⊕ 16 of Higgs fields or 126 ⊕ 126 of Higgs. Since under SU (5) ⊗ U (1) one has 16 ⊃ 1(−5) we see that a VEV formation for the singlet will reduce the rank of the group. Similarly 126 ⊃ 1(−10) under the above decomposition. Thus when the singlets in 16 ⊕ 16 of Higgs or 126 ⊕ 126 get VEVs, the SO(10) gauge symmetry will break reducing its rank. However, we still need to reduce the remaining group symmetry to the Standard Model gauge group. For this we need to have additional Higgs fields such as 45, 54, 210. Further to get the residual gauge group SU (3) C ⊗ U (1) em we need to have 10 -plet of Higgs fields. Thus the breaking of SO(10) down to SU (3) C ⊗ U (1) em requires at least three sets of Higgs representations: one to reduce the rank, the second to break the rest of the gauge group to the Standard Model gauge group and then at least one 10-plet to break the electroweak symmetry. As discussed above For some of the works utilizing large representations see [28].
As discussed above the conventional models have the drawback that one needs several sets of Higgs fields to accomplish complete breaking. One recent suggestion to overcome this drawback is to use 144 ⊕ 144 of Higgs. This combination can allow one to break SO(10) symmetry all the way to SU
that V = −M 2 Tr(ΣΣ † ) + κ1 2 Tr(Σ 2 Σ †2 ) + κ2 2 (Tr(ΣΣ † )) 2 + κ3 2 Tr(Σ 2 )Tr(Σ †2 ) + κ4 2 Tr(ΣΣ † ΣΣ † )
. Symmetry breaking allows for local minima which lead to the Standard Model gauge group for the vacuum structure
so that < Σ >=< Σ † >= v diag(2, 2, 2, −3, −3), where v 2 = M 2 /(7(κ 1 + κ 4 ) + 30(κ 2 + κ 3 )
). This local minimum will be a global minimum for some some range of the parameters of the potential which implies that spontaneous symmetry breaking does indeed break SO (10)
down to SU (3) C ⊗SU (2) L ⊗U (1) Y . Identical
conclusions can be arrived at for the case of supersymmetric SO(10) model. Here the potential will become the superpotential (with Σ † replaced by a chiral superfield Σ), the couplings κ i will have inverse dimensions of mass, and the mass term M 2 will be replaced by M . The analysis of [29,30] shows that in SUSY SO(10) a 144 ⊕ 144 pair of chiral superfields do indeed break SO(10) in one step down to the supersymmetric standard model gauge group. Techniques for the computation of SO(10) couplings using oscillator algebra [31] and SU (5) ⊗ U (1) decomposition are discussed in [32,33,34,35]. Specifically this technique is very useful for analyses involving vector-spinors, i.e., 144 ⊕ 144. For alternative techniques see [36].
As mentioned in sec 2 grand unified models typically have the doublet-triplet problem. In SU (5) aside from fine tuning two other avenues to overcome this problem were mentioned, one being the missing partner mechanism and the other flipped SU (5)⊗ U (1). In SO(10) one early suggestion to resolve the doublet-triplet problem is the missing VEV method where using a 45-plet of Higgs fields, one breaks the SO(10) gauge symmetry in the (B − L)-preserving direction, which results in generation of Higgs triplet masses while the Higgs doublets from a 10-plet remain massless. Another possibility is a missing partner mechanism in SO(10) which in spirit is similar to the one for SU (5). In [37] a missing partner mechanism for SO(10) was constructed and the heavy fields used were 126 ⊕ 126 ⊕ 210 along with a set of light fields. The reason for this choice is to simulate the missing partner mechanism of SU (5) in the following way: 126 ⊕ 126 are chosen because they contain 50 ⊕ 50 of SU (5) and 210 is chosen because it contains the 75 of SU (5). This parallels the analysis in SU (5) and leads to a pair of light Higgs doublets and heavy Higgs triplets.
In [38] a more comprehensive analysis of the missing partner mechanism for SO(10) was given. Here in addition to the missing partner model consisting of (i) Heavy
{126 ⊕ 126 ⊕ 210} + Light {2 × 10 ⊕ 120} one has others: (ii) Heavy {126 ⊕ 126 ⊕ 45} + Light {10 ⊕ 120}, (iii) Heavy {126 ⊕ 126} + Light {10 ⊕ 120}, (iv) Heavy {560 ⊕ 560} + Light {2 × 10 ⊕ 320}
. Models (i), (ii) and (iii) are anchored in the heavy fields 126+126 since they contain the 50⊕50 of SU (5). However, model (iv) is anchored by a pair of 560⊕560 Higgs fields which also contain 50⊕50 of SU (5). Interestingly these are the next higher dimensional representations in SO (10) . We also note that 45
of SU (5) under SU (3) C ⊗ SU (2) L × U (1) has the decomposition so that 45 = (1, 2)(3) ⊕ (3, 1)(−2) ⊕ (3, 3)(−2) ⊕ (3, 1)(8) + (3, 2)(−7) ⊕ (6, 1)(−2) ⊕ (8, 2)(3),
which exhibits the doublet/triplet content of the 45-plet. One now finds that the exotic light modes become superheavy by mixing with the heavy exotics in 560 ⊕ 560 and only a pair of light Higgs doublets remain.
Also remarkably the gauge group SO(10) breaks directly to the Standard Model gauge group right at the GUT scale so that we have one step breaking of the gauge symmetry.
Higher dimensional operators appear in effective theories and allow one to explore the nature of physics beyond the standard model. They have been explored in significant depth in the literature. These operators also include the ones that violate B − L. Such operators appear in n −n oscillations and more recently they have gained further attention as they may be helpful in generating baryogenesis [39]. It is interesting to investigate the type of B − L violating operators that arise in grand unified theories. The minimal SU (5) grand unification under the assumption of R parity conservation and renormalizable interactions does not possess any B − L violating operators. However, SO(10) models do generate B − L violation. Recently an analysis of B − L = −2 operators has been given in [40] for a class of SO(10) models where there is doublettriplet splitting using the missing partner mechanism [38] (for related work see [41]). The ∆(B − L) = ±2 operators lead to some unconventional proton decay modes such as p → νπ + , n → e − K + and p → e − π + .
The ∆(B − L) = ±2 operators also allow for GUT scale baryogenesis. The baryogengesis produced by such operators is not wiped out by sphaleron processes and survives at low temperatures [39].
E 6 unification
Among the exceptional groups only E 6 , E 7 , E 8 are possible candidates for unification. However, symmetry breaking schemes E 6 can produce a low energy theory consistent with data (see, e.g., [42,43]).
Investigation of E 6 as the unification group within string theory has a long history. In models of this type E 6 is broken down to the standard model gauge group by a combination of flux breaking and breaking by VEVs of Higgs fields. In one breaking sequence one has
E 6 → SO(10) ⊗ U (1) ψ , SO(10) → SU (5) ⊗ U (1) ξ , SO(10) → SU (3) C ⊗ SU (2) L ⊗ U (1) Y .
Phenomenology of such breaking and of other scenarios leads to some distinctive signatures. More recently E 6 unification has also been investigated within F-theory (see e.g., [44] and the references therein).
The Flavor Puzzle
The observed structure of quarks and leptons exhibits some very interesting properties. There are at least two broad features which may be summarized as follows: (i) The quarks and leptons show a hierarchy in masses for different flavors; (ii) The mixing among quarks in going from flavor to mass diagonal basis is small, while the mixing among neutrinos is large. The constraints on the neutrino mixing come from the solar and the atmospheric neutrino oscillation data which is sensitive only to the differences of neutrino mass squares, i.e., ∆m 2 ij = |m 2 i − m 2 j |. A fit to the neutrino data gives [45]
∆m 2 sol = ||m 2 | 2 − |m 1 | 2 |, ∆m 2 atm = ||m 3 | 2 − |m 2 | 2 |.
Here the mixing angles θ 12 and θ 23 are large while θ 13 is small. The constraints on the absolute value of the masses themselves arise from neutrino less double beta decay and from cosmology. Thus from cosmology one has i |m νi | < (0.7 − 1)eV .
One avenue to decode the flavor structure is to assume that while it looks complex at low scales, there could be simplicity at high scales. An example of this is the ratio of the mass of the b quark vs the mass of the τ lepton which is approximately three at low energy but one can explain this ratio starting from the equality of the b and τ Yukawa couplings at the GUT scale. This result holds in supergravity GUTs but not in non-supersymmetric unification. In fact, one can also explain the ratio of the top mass to the b-quark mass starting with equality of the bottom and the top Yukawa coupling at the GUT scale if one assumes large tan β [49]. One of the early works in correlating low energy data on quark and lepton flavors with textures at the GUT scale is by Georgi and Jarlskog (GJ) [46]. Aside from the GJ textures there are several other textures that can generate the desired flavor structure at low energy (for early works see [47,48]). The question then is what are the underlying models which can produce the desired textures. One possibility is that they arise from cubic and higher dimensional operators. In the analysis of [50] it was shown that in SU (5), higher dimensions operators in an expansion in Σ/M , where Σ is the 24-dimensional field that breaks the GUT symmetry, can produce the GJ textures. This analysis also revealed that similar textures exist in dimension five operators which enter in proton decay which are different from the ones that appear in the Yukawa couplings.
Within renormalizable interactions one approach is to expand the superpotential in all allowed couplings and try to fit the data with the assumed set of couplings. For example, for SO(10) one could use the Higgs fields 10 s , 120 a , 126 s and assume general form of Yukawas consistent with these couplings, use renormalization group evolution from the GUT scale down to low energy and fit the quark and lepton masses and mixings [51].
Other approaches involve flavor symmetries. One of the common flavor symmetry used is S 4 of which there are a large number of variants., see. e.g., [52] and [53] and the references therein.
The flavor structure can be understood in another way in a class of SO(10) models within the unified Higgs sector [29,35]. As discussed in sec 3 in unified Higgs models, one uses a single pair of vector-spinor representation 144 + 144 which breaks the SO(10) gauge symmetry to the standard model gauge group. Here the matter fields can couple to the Higgs sector only at the quartic level, i.e., the interaction involves terms such as 16 · 16 · 144 · 144. This must be suppressed by a heavy mass. For this reason the Yukawa couplings arising from this interaction are naturally small and can provide an appropriate description of the masses and mixings of the first two generations. However, for the third generation, one needs cubic couplings and one can obtain much larger couplings in a natural way by including additional matter in 10, 45 and 120.
The additional matter representations allow one to have couplings of the type 16 · 144 · 10, 16 · 144 · 45, and 16 · 144 · 120. Specifically using 16 · 144 · 45, and 16 · 144 · 120 one finds that cubic couplings of size appropriate for the third generation arise [35]. One can obtain b − τ as well as b − t unification even at low values of tan β. The formalism also correctly generates the charged lepton and neutrino masses which arise from a type I see-saw mechanism [35]. There are also a variety of other approaches. For some recent ones see [54].
Supergravity grand unification
As mentioned in the introduction, supersymmetry which is a global symmetry, cannot be broken in a phenomenologically viable fashion. Supergravity grand unification overcomes this problem and allows one to build realistic models with spontaneous breaking of supersymmetry which lead to sparticles with calculable masses which can be searched for at colliders. The formulation of supergravity grand unification utilizes the framework of applied suerrgravity where one couples an arbitrary number of N = 1 chiral superfields, which we denote by Z, to N = 1 vector superfield belonging to the adjoint representation of the gauge group and then couple them to N = 1 supergravity [12,57,58,59].
The applied N = 1 supergravity lagrangian depends on three arbitrary functions consisting of the superpotential W (Z), the Kahler potential potential K(Z, Z † ) and the gauge kinetic energy function f αβ which transforms like the symmetric product of adjoint representations. However, before discussing the breaking of supersymmetry we begin by considering the breaking of a grand unified group with no breaking of supersymmetry in the framework of supergravity grand unification. As an illustration let us consider the simplest case where one has a 24-plet of SU (5) field Σ b a (a, b = 1 − 5) and a superpotential that is given by W σ = [ 1 2 M T rΣ 2 + 1 3 T rΣ 3 ]. After a spontaneous breaking of SU (5) occurs and Σ b a develops a VEV, one has three possibilities:
(i) < Σ a b >= 0, (ii) < Σ a b >= 1 3 M δ a b − 5δ a 5 δ 5 b , and (iii) < Σ a b >= M 2δ a b − 5(δ a 5 δ 5 b + δ a 4 δ 4 b ) .
Here (i) gives no breaking of gauge symmetry, (ii) gives the breaking of SU (5) to SU (4) ⊗ U (1), and (iii) gives the breaking to SU (3) ⊗ SU (2) ⊗ U (1).
In global supersymmetry these are flat directions and the potential vanishes for all the three cases. Thus all three vacua are degenerate. However, for the case of supergravity the potential does not vanish and one finds that the potential at the minimum is given by [12]
V 0 (Σ 0 , Σ * 0 ) = − 3 4 κ 2 e ( 1 2 κ 2 Σ0Σ * 0 ) |W (Σ 0 )| 2 .
We note now that unlike the case of global supersymmetry the potential is no longer degenerate for the three vacua [55,12]. Suppose we add a term to the superpotential and make the vacuum energy for one case vanish. In that case it is easy to check that the vacuum energy for other cases will be negative, i.e., that the vacua will be anti-deSitter. This would imply that the Minkowskian vacuum would be unstable in each case.
However, it turns out that vacuum stability is helped by gravity [56]. In fact it has been shown that the Minkowski vacuum will be stable against any finite size perturbations [55]. It should be noted that vacuum degeneracy is not lifted in all cases when gauge symmetry breaks even in the presence of gravity.
We turn now to the breaking of supersymmetry. In supergravity grand unification the breaking of supersymmetry can be generated by a superhiggs potential similar to the breaking of a gauge symmetry by a Higgs potential. A general form of the superpotential for the superhiggs is given by
W SH (Z) = m 2 κ −1 f s (κZ)
where the function f s (κZ) depends on the dimensionless product κZ. Here the breaking of supersymmetry To shield the visible sector from such large mass growths the breaking of supersymmetry is communicated to the visible sector by gravity mediation. A simple way to see this is to write the superpotential including superhiggs and matter fields so that [12,60]
W tot = W (Z a ) + W SH (Z), where W (Z a )
depends on the matter fields Z a and W SH (Z) depends only on the superhiggs field Z which breaks supersymmetry. The two are, however, connected via the supergravity scalar potential [12,58] , i.e., V =
e κ 2 K [(K −1 ) i j ( ∂W ∂zi + κ 2 K ,i W )( ∂W ∂zi + κ 2 K ,i W ) † − 3κ 2 |W | 2 ] + V D ,
where V D is the D-term potential. Integrating out the superhiggs field, one finds that the low energy theory in the visible sector does contains soft breaking generated by gravity mediation which are size m s ∼ κm 2 and thus the soft breaking is free of the Planck mass. However, supergravity models with grand unification contain another heavy mass, i.e., the GUT mass M G in addition to the Planck mass. Including M G in the analysis in a grand unified supergravity model brings in another type of hierarchy problem, i.e., one would have mass terms of the form m s M G , αm s M G · · · , α n m s M G , where α ≡ (κM G ). A remarkable aspect of supergravity unified models is that all items of the type above cancel or vanish [12]. In spontaneously broken supergravity the sum rule for masses is given by [57,59] 3/2 J=0 (−1) 2J (2J + 1)m 2 J = 2(N − 1)m 2 3/2 , where N is the number of chiral superfields.
In the universal supergravity models the effective low energy theory after integration over the heavy fields is thus given by [12] (2) and W 3) are parts of the superpotential which are bilinear and trilinear in the scalar fields. Here the soft parameters consist of the universal scalar mass m 0 , and the dimensioned parameters A 0 and B 0 . In the presence of nonuniversalities, one may have flavor dependence on the scalar masses as well as on the trilinear couplings. In the analysis above the vacuum energy has been adjusted to vanish. We note in passing that there exist models in which this is not the case and the minimum has a large negative energy [61]. The class of supergravity unification is rather large, as it encompasses a vast variety of particle physics models depending on the choice of the Kahler potential, the superpotential and the gauge kinetic energy function. Further, strings in low energy limit are describable by supergravity models (for a review see [11]). Similarly M-theory [62] in low energy limit can be described by 11D supergravity [63] which can be reduced further to 4D. Some examples of low energy string based supergravity models are KKLT [64] and the Large Volume Scenario [65].
V ef f = | ∂W ∂Zα | 2 + m 2 0 Z † α Z α + (B 0 W (2) + A 0 W (3) + h.c) + D-term, where W
Soft terms also arise for gaugino masses. One such source is via loops through their gauge interactions with the fields in the heavy sector [66,67]. Thus the gaugino massesm i , i = 1, 2, 3 for the U (1), SU (2) Here f αβ transforms like the symmetric product of adjoint representations. For the case when f αβ transforms like a singlet of the gauge group L λ m = − 1 2 m 1/2λ λ. In general, however, one would have non-universality of gaugino masses when f αβ does not transform as a pure singlet [68,69,70,71]. SUSY breaking can also arise from gaugino condensation such that < λγ 0 λ > = 0 [72]. This kind of breaking is often used in string model building. The Higgs boson mass measurement at 125 GeV gives strong support to supergravity models with gravity breaking (see, e.g., [73]).
Astrophysical evidence suggests the existence of dark matter in great abundance in our universe. Thus up to 95% of the universe may be constituted of dark matter or dark energy. Here we will focus on dark matter. Some leading candidates for dark matter include the weakly interacting massive particles (WIMPs), the extra weakly interacting particles [74], and axions among many others. In supergravity models several neutral particles exist which could be possible candidates for dark matter, such as the neutralino, the sneutrino, and the gravitino [75,76]. We will focus here on neutralino type WIMP which is an odd R parity particle. It has been shown that such a particle turns out to be the lightest supersymmetric particle in a large part of the parameter space of supergravity models [77] and if R -parity is conserved, it becomes a candidate for dark matter. Indeed the neutralino was proposed as a possible candidate for dark matter soon after the formulation of supergravity grand unified models [78].
The relic density of dark matter turns out to be an important constraint on model building. In supergravity grand unified models under the constraint that the weak SUSY scale is high perhaps lying in the several TeV region, the neutralino turns out to be most often a bino. The annihilation cross section for bino-like neutralino is rather small which implies that the neutralinos cannot efficiently annihilate to produce the desired relic density consistent with the data from WMAP [79] and from PLANCK [80] experiments. In this case we need coannihilation [81,82,83,84]. Coannihilation is a process in which one or more sparticles other than the neutralino enter in the annihilation process modifying the Boltzman equation that controls the relic density. Analysis shows that if the next to the lightest supersymmetric particle (NLSP) lies close to the LSP in mass then there can be a significant enhancement in the annihilation of the LSP allowing one to satisfy the relic density constraints consistent with the WMAP and the PLANCK data. Coannihilation, however, makes the detection of supersymmetric signals more difficult because the decay of the NLSP leads to soft final states which often do not pass the kinematical cuts for the conventional signal regions.
Cosmological models with cold dark matter such as ΛCDM have been pretty successful in explaining the large-scale structure of the universe. At small scales, however, some features arise which require attention [85]. The first of these is the cusp-core problem. This relates to the fact that the observed galaxy rotation curves are better fit by constant dark matter density cores which is the Berkert profile, ρ(r) = ρ 0 r 3 0 /[(r + r 0 )(r 2 + r 2 0 )] rather than the NFW profile ρ(r) = ρ 0 δ/[ r rs (1 + r rs ) 2 ] which is produced in N-body simulations using CDM. The second is the so called "missing satellite" problem where CDM predicts too many dwarf galaxies. More detailed analyses indicate that both problems could be solved by inclusion of complex dynamics and baryonic physics [86]. Aside from complex dynamics and baryonic matter, another possibility involves particle physics models such as (repulsive) self-interactions. or a dark particle of de
Unification in strings
As discussed in section 6, N=1 supergravity grand unified models with a hidden sector lead to the breaking of supersymmetry in a phenomenologically viable manner and with three generations of quarks and leptons allow an extrapolation of physics from the electroweak scale to the grand unification scale. Above this scale one expects quantum gravity effects to become important. The next step then is to look for a theory of quantum gravity whose low energy limit is supergravity grand unification. Possible candidates are string theories which come in several varieties: Type I, Type IIA, Type IIB, heterotic SO(32) and heterotic E 8 ⊗E 8 [90] . The Type I and Type II strings contain D branes. The D branes can support gauge groups and chiral matter can exist at the intersection of D branes. The various string Types can arise from the so called M-theory whose low energy limit is 11D supergravity. Most of the model building in string theory has occurred in heterotic E 8 ⊗ E 8 /Horava -Witten theory [91], where the Horava-Witten theory arising from the low energy limit of M theory on R 10 × S 1 /Z 2 may be viewed as the strong coupling limit of E 8 ⊗ E 8 , and Type IIB/F theory, where F theory [92] may be viewed as the strong coupling limit of Type IIB strings.
One can also compactify M-theory on other manifolds such as on a manifold X of G 2 holonomy [93,94,95].
One problem in working with string models is that they possess a huge number of vacuum states [96], as many as 10 500 , and the search for the right vacuum state that describes our universe is a daunting task.
One possible way to proceed then is to thin out the landscape of vacuum states by imposing phenomenological constraints. The most obvious ones are the emergence of groups which support chiral matter which limits the groups to SU (N ), SO(4N + 2), N ≥ 1 and E 6 after reduction to four dimensions, an N = 1 supersymmetry in 4D, a hidden sector that allows for breaking of supersymmetry, and three generation of chiral fermions which correspond to the three observed generations of quarks and leptons. For the emergence of supergravity grand unified model it is also of relevance that the grand unification scale M G emerge in some natural way from the Planck scale where M G is expected to be close to the scale where the 10-dimensional theory reduces to four dimensions.
The E 8 ⊗ E 8 heterotic string was one of the first to be investigated at significant length [97]. After compactification to four dimensions it has a rank which can be up to 22. This allows for many possibilities for model building which have been pursued in the context of using free field constructions, orbifolds, and Calabi-Yau compactification (for a sample of the early phenomenological analyses see [98]). Most of the analyses are within Kac-Moody level 1. Here one can achieve unified groups such as [99] SU (5), SO(10) or E(6). However, one does not have scalars in the adjoint representation to break the gauge symmetry. At level 2, adjoint scalars are achievable but three massless generations are not easy to get. At level 3, it is possible to have scalars in the adjoint representation and also 3 massless generations. However, here one finds that the quark-lepton textures are rank 3 and thus difficult to get realistic quark-lepton masses [100].
In addition to the heterotic string constructions, a large number of string model constructions have since appeared, and a significant number within Type II (for reviews see [101,102,103,104]). Of course in such models which have an effective theory with N = 1 supersymmetry, one would need to break supersymmetry to make contact with observable physics (see, e.g., [105]).
We note in passing that while it is desirable that grand unification arise from strings (for a recent review see [106]) it is not essential that it do so since string theory is already a unified theory and it is not necessary for us to insist on grand unification in the effective low energy theory. Rather we may have the standard model gauge group emerging directly from string theory without going through grand unification. In this case we will have g 2 i k i = g 2 string = 8πG N /α ′ , where G N is the Newtonian constant, α ′ is the Regge slope, and k i are the Kac-Moody levels of the subgroups. There are positive integers for non-abelian gauge groups [107] while for U (1) the normalization of k is arbitrary.
Of course string theory is supposed to unify gravity along with other forces, and one may look at the evolution of the dimensionless parameter α GR = G N E 2 along with the other three couplings (for a review see [108]). In the normal evolution of the gauge and gravity couplings using the spectrum of MSSM, one finds that while the gauge couplings do unify at the scale M G ∼ 10 16 GeV, the gravity coupling α GR does not. There are several possible ways to address this lack of unification. One possibility is that as we evolve the couplings above the scale M G , α G and α GR will unify. Another possible way is that at some scale below M G , a new dimension of space opens up. Here if matter resides on the 4 dimensional wall while gravity resides in the bulk, the evolution of α GR will be much rapid and there may be unification of gauge and gravity couplings at a scale much below the Planck scale. A further modification of this idea is strings where the string scale is lowered to lie at the TeV scale. In this case gravity can get strong at a much lower scale and unification of gauge and gravity coupling can occur in the few TeV region [109,110,111]. More recent progress in model building has come from F-theory which as noted earlier can be viewed as a strong coupling limit of Type IIB string. For some recent work in F-theory model building see, e.g. [112,113,114,115,44].
Monopoles
It was shown by Dirac [116] that in Maxwell electrodynamics the existence of a monopole of magnetic charge g would imply a quantization of the electric charge so that e.g. = 1 2 n . In unified theories while SU (2)×U (1) does not possess a magnetic monopole, SO(3) does and it arises as a consequence of solution to the field equations. The quantization condition in this case is e.g. = n which is the Schwinger quantization [117].
Grand unified theories also possess monopoles and they appear again as solutions to field equations. Unlike the Maxwell electrodynamics where the monopole may or may not exist, in grand unified theories where a U (1) arises as a reduced symmetry, the monopole is a necessity and a prediction. However, the monopoles in grand unified theories will be superheavy with a mass of size the GUT scale. One problem associated with these monopoles is that monopoles produced in the early universe would contribute a matter density in excess of the critical relic density which would over close the universe. Inflationary cosmology solves this problem. In some models the magnetic monopoles can be much lighter. Monopoles also appear in intersecting D-brane models where they appear along with color singlet fractionally charged states [118]. For the current experimental status of magnetic monopoles see [119].
Proton decay in GUTs and Strings
One of the hallmarks of most unified models is the prediction that proton will decay. It is also a possible discriminant of models based on GUTs vs strings. One of the predictions in both supersymmetric and nonsupersymmetric grand unification is the proton decay mode p → e + π 0 . This mode arises from dimension six operators and involves the exchange of leptoquarks (for reviews see Refs. [120,121,122]
(p → e + π 0 ) ≃ α 2 G m 5 p M 4 V
where α G = g 2 G /4π with g G being the unified coupling constant, and M V the lepto-quark mass. It leads to a partial lifetime of τ (p → e + π 0 ) ≃ 10 36±1 yrs. This mode allows the possibility of discrimination among models based on GUTs vs those based on strings. Thus a generic analysis of D brane models allows only 10 2 · 10 2 SU(5) type dimension six operators which leads to the decay p → π 0 e + L [123]. In SU (5) grand unification, one has in addition the operator 10 · 10 · 5 · 5 which allows p → π 0 e + R . Further, generic D brane models do not allow the decay p → π + ν while SU (5) grand unification does. It has been pointed out, however, that special regions of intersecting D brane models exist which allow the operator 10 · 10 · 5 · 5 and the purely stringy proton decay rate can be of the same order as the one from SU(5) GUTs including the mode p → π + ν [124]. The current experimental status of proton decay for this partial decay mode is the following: Superkamiokande gives the limit τ (p → e + π 0 ) > 2 × 10 34 yrs [125] while in the future Hyper-K is expected to achieve a sensitivity of τ (p → e + π 0 ) > 1 × 10 35 yrs [126].
In supersymmetric unified models there are additional operators that can generate proton decay. Thus if R parity is not conserved, proton decay can proceed with baryon and lepton number violating dimension four operators. Such a decay is too rapid and must be eliminated by imposition of R parity conservation. In this case we still have baryon and lepton number violating dimension five operators arising from the exchange of Higgs triplets, which give rise to proton decay [127,120]. Here the dominant decay mode is p → K +ν and could also be dangerous in terms of proton stability [128,129,130,131,132,133,134,135]. The SUSY decay modes depend sensitively on the sparticle spectrum as well as on CP phases [136,137]. The current experimental limit from Super-Kamiokande is τ (p →νK + ) > 4 × 10 33 yrs, while in the future it is expected that Hyper-K may reach a sensitivity of τ (p →νK + ) > 2 × 10 34 yrs.
It is known that proton decay lifetime from baryon and lepton number violating dimension five operators in SUSY GUTs with a low sparticle spectrum would lie below the current experimental limits. One possibility for stabilizing the proton is via a cancellation mechanism [120,138]. The other possibility is via using a heavy sparticle spectrum which enters in the loop diagrams. Very roughly the proton decay from dimension five operators has the sparticle mass dependence of m 2
χ ± 1 /m 4 q where χ ± 1
is the chargino and theq is the squark. This means that a suppression of proton decay can be achieved with a large sfermion mass. As will be discussed in section 10, the discovery of the Higgs boson with a mass of ∼ 125 GeV points to a high SUSY scale and a high SUSY scale implies a larger proton decay lifetime for the SUSY modes. In fact one finds a direct correlation between the proton lifetime and the Higgs boson mass [139] which shows that the experimental lower limit on the proton lifetime for the SUSY mode can be easily met under the Higgs boson mass constraint. As discussed in sec 6 the Higgs boson mass measurement at 125 GeV [142,143] indicates that the loop correction to the Higgs boson mass in supersymmetry is large. This in turn indicates that the weak SUSY scale, and specifically the scalar masses, must be large lying in the several TeV region [144,73,145,146].
It turns out that large scalar masses arise naturally in the radiative breaking of supergravity. Thus the radiative breaking in supergravity unified models obeys the relation (for a review see [147]) M 2 Z + 2µ 2 ≃ (1 − 3D)m 2 0 + C(m 0 , A 0 , tan β) C > 0, where C is a polynomial in m 0 and A 0 and D depends on loop corrections, Yukawas and the weak SUSY scale. For the case when D < 1/3 there is an upper bound on sparticles masses for fixed µ. This the case of ellipsoidal geometry (EB) where for a given µ, one has an upper limit on how large the soft parameters can get. When D ≥ 1/3 the geometry becomes hyperbolic (HB) and for a fixed µ the scalar masses get large [149].
The HB contains focal curves and focal surfaces (see Akula etal. in [149]). The transition point between the two branches, i.e., between EB and HB, is D = 1/3 referred to as the focal point. Here µ essentially becomes independent of m 0 . Thus in general on the focal point, focal curves and focal surfaces, µ can remain small while scalar masses get large. On HB the gauginos can be light and discoverable at the LHC even if the SUSY scale M S is in the TeV region. Further, the weak SUSY scale is approximated by M S = √ mt 1 mt 2 . wheret 1 ,t 2 are the stop masses. Thus mt 1 may be a few hundred GeV while mt 2 is order several TeV which leads to M S in the TeV region [150]. This means that even for the case when the weak SUSY scale is large, one of the stops could have a mass that lies in the few hundred GeV region and may be discoverable.
Of course, for non-universal supergravity models the sleptons can be much lighter than the squarks. Here we note that in searches for supersymmetry optimization of signal analysis beyond what is employed in simplified models is often necessary as important regions of the parameter space can otherwise be missed (see, e.g., [151,152,153,154]).
It is possible, however, that the weak scale of SUSY is much larger than previously thought and could be upwards of 10 TeV. A weak SUSY scale of this size is consistent with radiative breaking of the electroweak symmetry, the Higgs boson mass constraint and the relic density constraint [157]. It can also resolve the so called gravitino decay problem in supergravity and string theories. Thus an intrinsic element of supergravity unified models is the existence of a gravitino which is a supersymmetric partner of the graviton and becomes massive after spontaneous breaking of supersymmetry. The gravitino could be either stable or unstable. If it is stable it would be the lowest mass odd R parity object and thus contribute to dark matter. In this case one finds that the mass of the gravitino must be less than 1KeV in order that it not over close the universe. If the gravitino is not the LSP, it would be unstable and decay and there are strong constraints on the gravitino in this case. Here one needs to make certain that the gravitino which has only gravitational interactions does not decay too late, i.e., does not occur after the BBN time, i.e., (1 − 10 2 )s. which would upset the successes of the BBN. As already noted one of the implications of the Higgs boson measurement, is that the weak SUSY scale which is typically set by the gravitino mass is high lying in the several TeV region. However, the scalar masses though scaled by the gravitino mass need not be equal to the gravitino mass.
We note that supergravity models have a large landscape of soft parameters [155] which include nonuniversalities in the gaugino sector and as well as non-universalities in the matter and Higgs sectors [156].
First there could be non-universalities which are model dependent and split the scalar masses and second that renormalization group evolution has significant effect on the scalar masses when evolved down to the electroweak scale. Thus in general the scalar masses could be much lower than the gravitino mass. Further, the gaugino masses could be significantly smaller than the scalar masses. This would allow the gravitino to have decay modesgg,χ ± 1 W ∓ ,χ 0 1 γ,χ 0 1 Z. We note in passing that supergravity theories arising from string compactification with stabilized moduli often lead to a gravitino mass which may lie in the range 10 TeV and above (see, e.g., [89,158,159]).
For the gravitino mass lying in the 50-100 TeV region, the decay of the gravitino occurs significantly before the BBN time [157]. There is, however, one more constraint that one needs to attend to, which is that under the assumption of R parity conservation, each of the gravitino decay will result in an LSP neutralino which will contribute to the relic density of the universe. Thus here the total relic density of the neutralinos will be given by Ωχ0
1 = Ω th χ 0 1 + ΩG χ 0 1
where Ω th
χ 0 1
is the regular relic density arising from the thermal production of neutralinos after the freeze out and ΩG χ 0 1 is the relic density arising from the decay of the gravitinos. We may write ΩG χ 0 1 so that ΩG χ 0 1 = (mχ0 1 /mG)ΩG, which implies that we need to compute the quantity ΩG. We assume that the gravitinos produced in the early universe before the start of inflation have been inflated away and the relevant relic density of gravitinos is the one that is produced after inflation during the reheating period.
Detailed analyses of this production exists in the literature and one finds that the relic density of neutralinos generated by the decay of the gravitinos produced by reheating is given by ΩG χ 0 11 Testing supergravity unification at future colliders As mentioned in section 10, the case for SUSY/SUGRA is much stronger as a consequence of the Higgs boson discovery and the measurement of its mass at 125 GeV. Currently there is no other paradigm that can replace supergravity grand unification as we extrapolate physics from the electroweak scale to the GUT scale.
1 h 2 = 3 i=1 ω i g 2 i 1 + m 2 i /3m 2 G ln (k i /g i ) mχ0 1 /100GeV T R /
For these reasons the search for sparticles must continue. There is a good chance that we will find sparticles at the LHC by the time all its runs are over. The discovery of even one sparticle will open up a new era for sparticle spectroscopy including the discovery of the remaining sparticles, and precision measurement of their masses and couplings. For these higher energy colliders are essential. For the future several proposals are under consideration both for high energy e + e − colliders as well as for high energy proton-proton colliders.
Conclusion
There are a variety of reasons why supersymmetry is desirable when we think of high scale physics. One reason is the well known hierarchy problem [6] arising from loop corrections to the scalar masses. Thus while a loop correction to a fermion mass is proportional to the fermion mass, i.e., δm f ∝ m f , for the scalars the corrections to the scalar mass m is of the form δm 2 ∝ Λ 2 , where Λ is a cutoff scale. In a quantum field theory the cutoff scale could be order the Planck mass and thus the correction is very large. One of the ways to overcome this problem is supersymmetry where the loop correction from the squark exchange cancels the loop correction from the quark exchange. This leads to a natural cancellation of 1 part in 10 28 . The situation is similar to the ∆S = 1 neutral current case in the Standard Model where the charm quark loop cancels the u quark loop consistent with experiment Br(K 0 → µ + µ − )/Br(K + → µ + ν µ ) = (6.84 ± 0.11) × 10 −9 [161]. In this case the cancellation required is order one part in 10 9 and it leads to the discovery of the charm quark.
In comparison for the case of the Higgs boson mass, the cancellation is 1 part in 10 28 between the squark loop and the quark loop and thus there is an overwhelming reason for supersymmetry to be discovered.
In addition to the above let us recount some of the other successes of the supersymmetry/supergravity models. One of the early ones includes the fact that supersymmetry/supergravity models provide the right number of extra particles needed for the unification of couplings using the LEP data as one moves from the electroweak scale to the grand unification scale. Supergravity models provide an explanation for the tachyonic Higgs mass term that is central to accomplish spontaneous breaking of the electroweak symmetry in the Standard Model. Supergravity grand unification predicted the Higgs boson mass to lie below 130
GeV which is consistent with the current measurement of the Higgs boson. Currently there is no good alternative to supergravity grand unification if we wish to extrapolate physics from the electroweak scale to the grand unification scale. So the search for sparticles must continue. The most likely sparticle candidates for discovery are the neutralinoχ 0 , the charginoχ ± , the gluinog, the stopt 1 , and the stauτ 1 . There is also the possibility of discovering the additional Higgs bosons that appear in SUSY extensions of the standard model. There is a good chance that with the full capability of the LHC (L = 3000 fb −1 , √ s = 14 TeV) one discovers sparticles and/or additional Higgs boson. At the same time a good idea to look ahead and plan for a 100 TeV pp super collider.
Acknowledgments: This research was supported in part by NSF Grant PHY-1620526.
SU ( 5 )
5[4] is the lowest rank unified group which can accommodate the standard model gauge group. Here a full generation of quarks and leptons can be accommodated in its representations5⊕10. Under the Standard Model gauge group they decompose so that 5 identifies (ν L , e L ) with (1, 2, − 1 2 ), e c L with (1, 1, 1), (u L , d L ) with (
model is not sufficient to explain the flavor structure of the Standard Model. In SU (5) there are 24 gauge bosons of which 12 are the gauge bosons of the Standard Model consisting of the gluon, W ± , Z, γ. The remaining gauge bosons are superheavy leptons-quarks (X 4/3 , Y 1/3 ), (X −4/3 , Y −1/3 ) where the super-scripts give the charge assignments. However, non-supersymmetric SU (5) does not produce unification of gauge coupling constants consistent with LEP data.
lm ijk , and 75 = Σ ij lm . The superpotential W Σ (Σ) is chosen to produce a GUT symmetry breaking when Σ ij kl develops a non-vanishing VEV of O(M ). The SU (3) ⊗ SU (2) ⊗ U (1) decomposition of the 75 plet contains the singlet field (1, 1)(0) and its VEV formation breaks the SU (5) GUT symmetry. The Higgs fields 5 and 50-plet have the SU (3) ⊗ SU (2) ⊗ U (1) decomposition so that 5 = (1, 2)(3) ⊕ (3, 1)(−2) and 50 = (1, 1)(−12) ⊕ (3, 1)(−2) ⊕ (3, 2)(−7) ⊕(6, 3)(−2) ⊕ (6, 1)(8) ⊕ (8, 2)(3). Here it is interesting to note that the 5-plet contains a Higgs triplet and similarly5 contains an anti-triplet, while 50-plet contains a Higgs triplet and similarly 50 contains a Higgs anti-triplet. Thus the interaction 5 · 50 · 75 gives superheavy masses to the triplets and anti-triplet by matching the triplet in 5 to the anti-triplet in 50 and similar tying the anti-triple in5 with the triplet in 50. On the other hand, 50 and 50 do not have Higgs doublets and thus the Higgs doublets of 5 and5 remain light.
( 5 )
5⊗U (1) as subgroups, i.e., SO(10) has the branchings SO(10) → SU (4) C ⊗ SU (2) L ⊗ SU (2) R and SO(10) → SU (5) ⊗ U (1). It possesses a spinor representation which is 2 5 = 32 dimensional and which splits into 16 ⊕ 16. A full generation of quarks and leptons can be accommodated in a single 16 plet representation. Thus the 16-plet has the decomposition in SU (5) ⊗ U (1) so that 16 = 10(−1) ⊕ 5(3) ⊕ 1(−5). As noted the combination 5 ⊕ 10 in SU (5) is anomaly free and further 1(−5) in the 16-plet decomposition is a right handed neutrino which is a singlet of the standard model gauge group and thus the 16-plet of matter in SO(10) is anomaly free. The absence of anomaly in this case is the consequence of a more general result for SO(N ) gauge theories. Thus in general anomalies arise due to the non-vanishing of the trace over the product of three group generators in some given group representation Tr ({T a , T b }T c ).
one can do this by a combination of fields from the set: 10, 16 ⊕ 16, 45, 54, 120, 126 ⊕ 126, 210. To generate quark and lepton masses we need to couple two 16-plets of matter with Higgs fields. To see which Higgs fields couple we expand the product 16 ⊗ 16 as a sum over the irreducible representations of SO(10). Here we have 16 ⊗ 16 = 10 s ⊕ 120 a ⊕ 126 s , where the s(a) refer to symmetric (anti-symmetric) under the interchange of the two 16-plets. The array of Higgs bosons available lead to a large number of possible SO(10) models.
( 3 )
3C ⊗ U (1). This can be seen as follows: In SU (5) ⊗ U (1) decomposition one finds that 144 ⊃ 24(−5) which implies that spontaneous symmetry breaking which gives VEV to the Standard Model singlet of 24 would also reduce the rank of the group. Thus in one step one can break SO(10) gauge symmetry down to the Standard Model gauge group. Further, there exist several Higgs doublets in 144 which have the quantum numbers of the Standard Model Higgs and one may use fine tuning to make one of the Higgs doublets light which is needed to give masses to the quarks and the leptons. A VEV formation that breaks the SO(10) symmetry can be achieved by taking a combination of dimension two and dimension four terms in the potential so
after 126 ⊕ 126 which contain an excess of color triplets over SU (2) L doublets. Further, it turns out that 560 also contains in the SU (5) decomposition the SU (5) representations 1(−5) ⊕ 24(−5) ⊕ 75(−5). After spontaneous breaking these fields acquire VEVs. They reduce the rank of the group from 5 to 4 and further break the gauge symmetry down to the symmetry of the Standard Model gauge group. That means that the SO(10) gauge symmetry breaks to the Standard Model gauge group in one step. This is reminiscent of the unified Higgs case discussed earlier in this section where 144 ⊕ 144 break the SO(10) gauge symmetry in one step. Next we explain how the missing partner mechanism works in this case. To this end we exhibit the complete SU (5)⊗U (1) decomposition of 560. Here one has 560 = 1(−5)⊕5(3)⊕10(−9)⊕10(−1) 1 ⊕10(−1) 2 ⊕ 24(−5) + 40(−1) +45(7) ⊕ 45(3) ⊕ 50(3) ⊕ 70(3) ⊕ 75(−5) ⊕ 175(−1). Regarding the light sector it turns out that we have a unique choice in this case, i.e., the light fields must be in (2 × 10 ⊕ 320) representations. A very stringent condition for the missing partner mechanism to work is that all the exotic fields must become heavy and thus in the entire array of Higgs fields only a pair of Higgs doublet fields must survive and remain light. To exhibit that this indeed is the case let us consider the SU (5) ⊗ U (1) decomposition of 320 and of 10's. For the 320 we have 320 = 5(2) + 5(−2) ⊕ 40(−6) ⊕ 40(6) ⊕ 45(2) ⊕ 45(−2) ⊕ 70(2) ⊕ 70(−2), and for the 10-plet we have 10 = 5(2) ⊕ 5(−2)
E 7 , E 8 are eliminated as they do not have chiral representations which leaves E 6 as the only possible candidate for unification among the exceptional groups. The lowest representation of E 6 is 27-plet and 27⊗27 = 1⊕78⊕650 where 78 -plet is the adjoint representation. One notes that 27 ⊗ 27 = 27 s ⊕ 351 a ⊕ 351 ′ s . This result leads to the remarkable feature of E 6 models that the only cubic interaction one can have for 27 is W 27 = λ 27⊗27⊗27.
E 6
6has many possible breaking schemes such as E 6 → SO(10)⊗U (1) ψ , SU (3) C ⊗SU (3) L ⊗SU (3) R , SU (6)⊗ SU (2). One of the most investigated symmetry breaking schemes is SU (3) 3 . One can exhibit the spectrum of E 6 in terms of representations of SU (3) 3 so that 27 = (1, 3,3) ⊕ (3,3, 1) ⊕ (3, 1, 3) and thus the particle content of the model consists of nonets of leptons, quarks and conjugate quarks where L = (1, 3, 3) , Q = (3, 3, 1) and Q c = (3, 1, 3). The symmetry of the group can be broken by the combination of Higgs fields 27 ⊕ 27 ⊕ 351 ′ ⊕ 351 ′ . Extensive analyses exist in the literature and it is shown that with appropriate
∆m 2 sol = (5.4 − 9.5) × 10 −5 eV 2 , ∆m 2 atm = (1.4 − 3.7) × 10 −3 eV 2 , sin 2 θ 12 = (0.23 − 0.39), sin 2 θ 23 = (0.31 − 0.72), sin 2 θ 13 < 0.054, where
gives < Z >∼ O(κ −1 ) and f s (κZ) ∼ O(1). The gravitino mass in this case is m 3/2 ∼ O(κm 2 ). If we take m to be intermediate scale of size 10 10−12 GeV, then m 3/2 lies in the range 1 − 100 TeV. In early works a simple choice for W SH was made, i.e., W SH = m 2 (z + B). The VEV of the field Z which breaks supersymmetry has no direct interaction with the visible sector since < Z >∼ κ −1 and thus any direct interaction between the superhiggs field Z and the visible sector fields would generate a mass for the visible sector fields which would be O(κ −1 ).
, SU (3) gauge groups are given bym i = αi 4π m 3/2 CD(R)/D(A), where D(R) is the dimensionality of the exchanged representation, D(A) is the dimensionality of the adjoint representation and C is the quadratic Casimir of the fields that contribute. One can also generate gaugino masses at the tree level via a field dependent gauge kinetic energy function f αβ after spontaneous breaking of supersymmetry and one has L λ m = −(1/2)m αβλ α λ β .
Broglie wavelength of ∼ 1 kpc and a mass which lies in the range of 10 −21 − 10 −22 eV[87].
10 A stronger case for SUSY/SUGRA after Higgs boson discoveryThe measurement of the Higgs boson at 125 GeV gives further support for supersymmetry. One reason for that is vacuum stability. For large field configurations where h >> v the Higgs potential is governed by the quartic term V h ∼λ ef f 4 h 4 . For vacuum stability λ ef f must be positive. In the Standard Model analyses done using NLO and NNLO corrections show that with a 125 GeV Higgs boson, the vacuum can be stable only up to about 10 10 −10 11 GeV [140]. For vacuum stability up to the Planck scale one needs m h > 129.4±1.8 GeV. Vacuum stability depends critically on the top mass. A larger top mass makes the vacuum more unstable. An advanced precision analysis [141] including two-loop matching, three-loop renormalization group evolution, and pure QCD corrections through four loops gives an upper bound on the top pole mass for the stability of the standard model vacuum up to the Planck mass scale of m cri t = (171.54 ± 0.30 +0.26 −0.41 ) GeV, while the experimental value of the top is m exp t = (173.21 ± 0.51 ± 0.71) GeV. Though an improvement on previous analyses the stability of the Standard Model vacuum is still not fully guaranteed. In models based on supersymmetry with a Higgs mass of 125 GeV, the vacuum is stable up to the Planck scale. Of course one has to choose the parameter space of supergravity models appropriately.
10 10 GeV . Here g i , m i (i = 1, 2, 3) are the gauge couplings and the gaugino masses for the gauge groups U (1) Y , SU (2) L and SU (3) C which are evaluated at the reheat value T R using renormalization group to evolve their values from the GUT scale. Further, ω i (i = 1, 2, 3) = (0.018, 0.044, 0.177)[160]. Analysis shows that negligible amount of the relic density arises from the decay of the gravitino up to reheat temperatures of 10 10 GeV[157].
For
the e + e − colliders the possibilities are: (a) ILC: International Linear Collider, (b) CEPC: Circular Collider, (c) FCC-ee (TLEP): Future Circular Collider. The ILC is under consideration in Japan, CEPC in China and FCC-ee at CERN. These colliders are essentially Higgs factories which are likely to run at an energy around 240 GeV which gives the optimal cross section for Higgsstrahlung, i.e., e + e − → Zh. The Zh final state is the preferred mode rather than hh since Z can be efficiently detected via Z → ℓ + ℓ − . The Higgs factories can do precision physics related to the Higgs boson specifically the couplings of the Higgs bosons to fermions and other electroweak parameters with great accuracy. Some supersymmetric effects could show up in these high precision experiments. The sparticles most likely to be probed would be electroweak gauginos and light sleptons if they are low mass. For the study for most other sparticles one needs proton-proton colliders. Here the possibilities are (i) SppC: 70-100 TeV pp collider, (ii) VLHC: 100 TeV pp collider. The SppC if built would be in China and VLHC is a possibility for the future at CERN.
4, 2, 1) H ⊕ (4, 1, 2) H . In this case when ν H and ν c H develop VEVs, i..e, < ν H >=< ν c H >= M νH , G(4, 2, 2) breaks to the Standard Model gauge group and the charge operator takes the familiar form Q em
). A rough estimate of the decay width is Γ
G(4, 2, 2) and SU (5) unification
G(4, 2, 2) and SU (5) unification
SO(10) unification
SO(10) unification
SO(10) unification
A stronger case for SUSY/SUGRA after Higgs boson discovery
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