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+arXiv:2301.04620v1  [eess.SY]  11 Jan 2023
+AdaptSLAM: Edge-Assisted Adaptive SLAM with
+Resource Constraints via Uncertainty Minimization
+Ying Chen∗, Hazer Inaltekin†, Maria Gorlatova∗
+∗Duke University, Durham, NC, †Macquarie University, North Ryde, NSW, Australia
+∗{ying.chen151, maria.gorlatova}@duke.edu, †[email protected]
+Abstract—Edge computing is increasingly proposed as a solu-
+tion for reducing resource consumption of mobile devices running
+simultaneous localization and mapping (SLAM) algorithms, with
+most edge-assisted SLAM systems assuming the communication
+resources between the mobile device and the edge server to be
+unlimited, or relying on heuristics to choose the information to
+be transmitted to the edge. This paper presents AdaptSLAM, an
+edge-assisted visual (V) and visual-inertial (VI) SLAM system
+that adapts to the available communication and computation re-
+sources, based on a theoretically grounded method we developed
+to select the subset of keyframes (the representative frames) for
+constructing the best local and global maps in the mobile device
+and the edge server under resource constraints. We implemented
+AdaptSLAM to work with the state-of-the-art open-source V-
+and VI-SLAM ORB-SLAM3 framework, and demonstrated that,
+under constrained network bandwidth, AdaptSLAM reduces the
+tracking error by 62% compared to the best baseline method.
+Index Terms—Simultaneous localization and mapping, edge
+computing, uncertainty quantification and minimization
+I. INTRODUCTION
+Simultaneous localization and mapping (SLAM), the pro-
+cess of simultaneously constructing a map of the environ-
+ment and tracking the mobile device’s pose within it, is an
+essential capability for a wide range of applications, such
+as autonomous driving and robotic navigation [1]. In partic-
+ular, visual (V) and visual-inertial (VI) SLAM, which use
+cameras either alone or in combination with inertial sensors,
+have demonstrated remarkable progress over the last three
+decades [2], and have become an indispensable component
+of emerging mobile applications such as drone-based surveil-
+lance [3], [4] and markerless augmented reality [5]–[8].
+Due to the high computational demands placed by the V-
+and VI-SLAM on mobile devices [9]–[12], offloading parts of
+the workload to edge servers has recently emerged as a promis-
+ing solution for lessening the loads on the mobile devices and
+improving the overall performance [9]–[18]. However, such
+approach experiences performance degradation under resource
+limitations and fluctuations. The existing edge-assisted SLAM
+solutions either assume wireless network resources to be
+sufficient for unrestricted offloading, or rely on heuristics in
+making offloading decisions. By contrast, in this paper we
+develop an edge computing-assisted SLAM framework, which
+we call AdaptSLAM, that intelligently adapts to both commu-
+nication and computation resources to maintain high SLAM
+performance. Similar to prior work [11]–[17], AdaptSLAM
+runs a real-time tracking module and maintains a local map
+on the mobile device, while offloading non-time-critical and
+computationally expensive processes (global map optimization
+and loop closing) to the edge server. However, unlike prior
+designs, AdaptSLAM uses a theoretically grounded method
+to build the local and global maps of limited size, and
+minimize the uncertainty of the maps, laying the foundation
+for the optimal adaptive offloading of SLAM tasks under the
+communication and computation constraints.
+First, we develop an uncertainty quantification model for
+the local and global maps in edge-assisted V-SLAM and VI-
+SLAM. Specifically, since these maps are built from the infor-
+mation contained in the keyframes (i.e., the most representative
+frames) [19]–[21], the developed model characterizes how the
+keyframes and the connections between them contribute to
+the uncertainty. To the best of our knowledge, this is the first
+uncertainty quantification model for V-SLAM and VI-SLAM in
+edge-assisted architectures.
+Next, we apply the developed uncertainty quantification
+model to efficiently select subsets of keyframes to build local
+and global maps under the constraints of limited computation
+and communication resources. The local and global map
+construction is formulated as NP-hard cardinality-constrained
+combinatorial optimization problems [22]. We demonstrate
+that the map construction problems are ‘close to’ submodular
+problems under some conditions, propose a low-complexity
+greedy-based algorithm to obtain near-optimal solutions, and
+present a computation reuse method to speed up map construc-
+tion. We implement AdaptSLAM in conjunction with the state-
+of-the-art open-source V- and VI-SLAM ORB-SLAM3 [20]
+framework, and evaluate the implementation with both sim-
+ulated and real-world communication and computation con-
+ditions. Under constrained bandwidth, AdaptSLAM reduces
+the tracking error by 62% compared with the best baseline
+method.
+To summarize, the main contributions of this paper are: (i)
+the first uncertainty quantification model of maps in V- and
+VI-SLAM under the edge-assisted architecture, (ii) an analyt-
+ically grounded algorithm for efficiently selecting subsets of
+keyframes to build local and global maps under computation
+and communication resource budgets, and (iii) a compre-
+hensive evaluation of AdaptSLAM on two configurations of
+mobile devices. We open-source AdaptSLAM via GitHub.1
+The rest of this paper is organized as follows. §II reviews
+the related work, §III provides the preliminaries, §IV and
+1https://github.com/i3tyc/AdaptSLAM
+
+§V introduce the AdaptSLAM system architecture and model,
+§VI presents the problem formulation, and §VII presents the
+problem solutions. We present the evaluation in §VIII and
+conclude the paper in §IX.
+II. RELATED WORK
+V- and VI-SLAM. Due to the affordability of cameras and
+the richness of information provided by them, V-SLAM has
+been widely studied in the past three decades [2]. It can be
+classified into direct approaches (LSD-SLAM [23], DSO [24]),
+which operate directly on pixel intensity values, and feature-
+based approaches (PTAM [25], ORB-SLAM2 [19], Pair-
+Navi [26]), which extract salient regions in each camera
+frame. We focus on feature-based approaches since direct
+approaches require high computing power for real-time per-
+formance [2]. To provide robustness (to textureless areas,
+motion blur, illumination changes), there is a growing trend
+of employing VI-SLAM, that assists the cameras with an
+inertial measurement unit (IMU) [20], [21], [27]; VI-SLAM
+has become the de-facto standard SLAM method for modern
+augmented reality platforms [5], [6]. In VI-SLAM, visual
+information and IMU data can be loosely [27] or tightly [20],
+[21] coupled. We implement AdaptSLAM based on ORB-
+SLAM3 [20], a state-of-the-art open-source V- and VI-SLAM
+system which tightly integrates visual and IMU information.
+Edge-assisted SLAM. Recent studies [4], [9], [11], [13]–
+[18], [28]–[30] have focused on offloading parts of SLAM
+workloads from mobile devices to edge (or cloud) servers
+to reduce mobile device resource consumption. A standard
+approach is to offload computationally expensive tasks (global
+map optimization, loop closing), while exploiting onboard
+computation for running the tasks critical to the mobile
+device’s autonomy (tracking, local map optimization) [11],
+[13]–[18]. Most edge-assisted SLAM frameworks assume
+wireless network resources to be sufficient for unconstrained
+offloading [4], [13]–[16], [29]; some use heuristics to choose
+the information to be offloaded under communication con-
+straints [9], [11], [17], [18], [28], [30]. Some frameworks only
+keep the newest keyframes in the local map to combat the
+constrained computation resources on mobile devices [14],
+[16]. Complementing this work, we propose a theoretical
+framework to characterize how keyframes contribute to the
+SLAM performance, laying the foundation for the adaptive
+offloading of SLAM tasks under the communication and
+computation constraints.
+Uncertainty quantification and minimization. Recent
+work [31]–[33] has focused on quantifying and minimizing
+the pose estimate uncertainty in V-SLAM. Since the pose
+estimate accuracy is difficult to obtain due to the lack of
+ground-truth poses of mobile devices, the uncertainty can
+guide the decision-making in SLAM systems. In [31], [32],
+it is used for measurement selection (selecting measurements
+between keyframes [31] and selecting extracted features of
+keyframes [32]); in [33], it is used for anchor selection (se-
+lecting keyframes to make their poses have ‘zero uncertainty’).
+Complementing this work, we quantify the pose estimate
+uncertainty of both V- and VI-SLAM under the edge-assisted
+architecture. After the uncertainty quantification, we study
+the problem of selecting a subset of keyframes to minimize
+the uncertainty. This problem is largely overlooked in the
+literature, but is of great importance for tackling computation
+and communication constraints in edge-assisted SLAM.
+III. PRELIMINARIES
+A. Graph Preliminaries
+A directed multigraph is defined by the tuple of sets G =
+(V, E, C), where V = {v1, · · · , v|V|} is the set of nodes, E is
+the set of edges, and C is the set of edge categories. Let e =
+((vi, vj), c) ∈ E denote the edge, where the nodes vi, vj ∈ V
+are the head and tail of e, and c ∈ C is the category of e.
+We let we be the weight of edge e. We allow multiple edges
+from vi to vj to exist, and denote the set of edges from vi to
+vj by Ei,j. Note that the edges in Ei,j are differentiated from
+each other by their category labels. The total edge weight from
+nodes vi to vj is given by wi,j =
+�
+e∈Ei,j
+we, which is the sum
+of all edge weights from vi to vj.
+The weighted Laplacian matrix L of graph G is a |V| × |V|
+matrix where the i, j-th element Li,j is given by:
+Li,j =
+� −wi,j,
+i ̸= j
+�
+e∈Ei
+we,
+i = j
+,
+where Ei ⊆ E is the set of all edges whose head is node vi.
+The reduced Laplacian matrix ˜L is obtained by removing an
+arbitrary node (i.e., removing the row and column associated
+to the node) from L.
+B. Set Function
+We define a set function f for a finite set V as a mapping
+f : 2V → R that assigns a value f (S) to each subset S ⊆ V .
+Submodularity. A set function f is submodular if f (L) +
+f (S) ⩾ f (L ∪ S) + f (L ∩ S) for all L, S ⊆ V .
+Submodularity ratio. The submodularity ratio of a set
+function f with respect to a parameter s is
+γ =
+min
+L⊆V,S⊆V,|S|⩽s,x∈V \(S∪L)
+f (L ∪ {x}) − f (L)
+f (L ∪ S ∪ {x}) − f (L ∪ S).
+(1)
+where we define 0/0 := 1.
+The cardinality-fixed maximization problem is
+max
+S⊆V,|S|=s f (S) .
+(2)
+The keyframe selection optimization is closely related to
+the cardinality-fixed maximization problem introduced above,
+which is an NP-hard problem [34]. However, for submodular
+set functions, there is an efficient greedy approach that will
+come close to the optimum value for (2), with a provable
+optimality gap. This result is formally stated in Theorem 1.
+Theorem 1.
+[34], [35] Given a non-negative and monoton-
+ically increasing set function f with a submodularity ratio γ,
+let S# be the solution produced by the greedy algorithm
+
+(Algorithm 1) and S⋆ be the solution of (2). Then, f
+�
+S#�
+⩾
+(1 − exp(−γ)) f (S⋆) .
+C. SLAM Preliminaries
+The components of SLAM systems include [2], [20], [21]:
+Tracking. The tracking module detects 2D feature points
+(e.g., by extracting SIFT, SURF, or ORB descriptors) in the
+current frame. Each feature point corresponds to a 3D map
+point (e.g., a distinguishable landmark) in the environment.
+The tracking module uses these feature points to find corre-
+spondences with a previous reference frame. It also processes
+the IMU measurements. Based on the correspondences in
+feature points and the IMU measurements, it calculates the
+relative pose change between the selected reference frame and
+the current frame. The module also determines if this frame
+should be a keyframe based on a set of criteria such as the
+similarity to the previous keyframes [20].
+Local and global mapping. It finds correspondences (of
+feature points) between the new keyframe and the other
+keyframes in the map. It then performs map optimizations,
+i.e., estimates the keyframe poses given the common feature
+points between the keyframes and the IMU measurements.
+Map optimizations are computationally expensive. In edge-
+assisted SLAM, global mapping runs on the server [11], [13]–
+[17].
+Loop closing. By comparing the new keyframe to all
+previous keyframes, the module checks if the new keyframe is
+revisiting a place. If so (i.e., if a loop is detected), it establishes
+connections between the keyframe and all related previous
+ones, and then performs global map optimizations. Loop
+closing is computationally expensive and can be offloaded to
+the edge server in the edge-assisted SLAM [11], [13]–[17].
+IV. ADAPTSLAM SYSTEM ARCHITECTURE
+The design of AdaptSLAM is shown in Fig. 1. The mobile
+device, equipped with a camera and an IMU, can communicate
+with the edge server bidirectionally. The mobile device and the
+edge server cooperatively run SLAM algorithms to estimate
+the mobile device’s pose and a map of the environment.
+AdaptSLAM optimizes the SLAM performance under compu-
+tation resource limits of the mobile device and communication
+resource limits between the mobile device and the edge server.
+We split the modules between the mobile device and the
+edge server similar to [11], [13]–[18]. The mobile device
+offloads loop closing and global map optimization modules
+to the edge server, while running real-time tracking and local
+mapping onboard. Unlike existing edge-assisted SLAM sys-
+tems [11], [13]–[18], AdaptSLAM aims to optimally construct
+the local and global maps under the computation and commu-
+nication resource constraints. The design of AdaptSLAM is
+Algorithm 1 Greedy algorithm to solve (2)
+1: S# ← ∅;
+2: while (
+��S#�� < s) do
+3:
+x⋆ ← arg max
+x
+f(S#∪{x})−f(S#). S# ← S#∪{x⋆}.
+Mobile Device
+Edge Server
+Tracking
+Local Map 
+Optimization
+Global Map
+Construction
+Global Map 
+Optimization
+Loop Closing
+IMU
+Image
+Local Map
+Construction
+Candidate 
+Keyframes
+Selected 
+Keyframes
+Detected 
+Loop
+Local 
+Map
+Global 
+Map
+Fig. 1: Overview of the AdaptSLAM system architecture.
+mainly focused on two added modules, local map construction
+and global map construction highlighted in purple in Fig. 1.
+In local map construction, due to the computation resource
+limits, the mobile device selects a subset of keyframes from
+candidate keyframes to build a local map. In global map
+construction, to adapt to the constrained wireless connection
+for uplink transmission, the mobile device also selects a subset
+of keyframes to be transmitted to the edge server to build a
+global map. The AdaptSLAM optimally selects the keyframes
+to build local and global maps, minimizing the pose estimate
+uncertainty under the resource constraints.
+Similar to [11], the selected keyframes are transmitted from
+the mobile device to the server, and the map after the global
+map optimization is transmitted from the server to the mobile
+device. For the uplink transmission, instead of the whole
+keyframe, the 2D feature points extracted from the keyframes
+are sent. For the downlink communication, the poses of the
+keyframes obtained by the global map optimization, and the
+feature points of the keyframes are transmitted.
+V. ADAPTSLAM SYSTEM MODEL
+A. The Pose Graph and the Map
+We divide time into slots of equal size of ∆t. We introduce
+the pose graph and the map at time slot t that lasts for ∆t
+seconds. For clarity of notation, we will omit the time index
+below.
+Definition 1 (Pose graph). For a given index set K =
+{1, . . ., |K|}
+(indexing
+camera
+poses
+and
+representing
+keyframes), the pose graph is defined as the undirected multi-
+graph G = (K, E, C), where K is the node set, E is the edge
+set, and C = {IMU, vis} is the category set. Here, IMU stands
+for the IMU edges, and vis stands for the covisibility edges.
+Given a pose graph G = (K, E, C), there is a camera pose
+Pn = (x, y, z, wx, wy, wz) for all n ∈ K, where the first
+three entries are the 3-D positions and the last three ones are
+the Euler angles (yaw, pitch and roll) representing the camera
+orientation. Edges in E are represented as e = ((n, m), c) for
+n, m ∈ K and c ∈ C. Two keyframes in K are connected
+by a covisibility edge if there are 3D map points observed
+in both keyframes. Two consecutive keyframes are connected
+by an IMU edge if there are accelerometer and gyroscope
+readings from one keyframe to another. There may exist both
+a covisibility edge and an IMU edge between two keyframes.
+For each e = ((n, m), c) ∈ E, we observe relative noisy
+pose measurements between n and m, which is written as
+∆e = Pm − Pn + xe, where xe is the measurement noise
+
+on edge e. The map optimization problem is to find the
+maximum likelihood estimates {˜Pn}n∈K for the actual camera
+poses {Pn}n∈K. For Gaussian distributed edge noise, the map
+optimization problem is
+min
+{˜Pn}n∈K
+�
+e∈E
+(˜xe)⊤Ie˜xe,
+(3)
+where ˜xe = ∆e − ˜Pm + ˜Pn and Ie is the information matrix
+(i.e., inverse covariance matrix) of the measurement error on
+e [36]. (˜xe)⊤ Ie˜xe is the Mahalanobis norm [20], [21] of the
+estimated measurement noise for e with respect to Ie.
+Below, we assume that the measurement noise xe is Gaus-
+sian distributed with isotropic covariance (as in [31], [37],
+[38]). We assume that the information matrix Ie can be
+characterized by a weight assigned to e [37], [39]. Specifically,
+Ie = weI, where we ⩾ 1 is the weight for e and I is the
+matrix that is constant for all measurements. We note that the
+relative measurements between keyframes n and m introduce
+the same information for them. We assume all weights we to
+be independent from each other for edges between different
+pairs of keyframes as in [20], [21], [39], [40].
+The map optimization problem in (3) is solved by standard
+methods such as Levenberg-Marquardt algorithm implemented
+in g2o [41] and Ceres solvers [42] as in [20], [21].
+Definition 2 (Anchor). We say that a node is the anchor of
+the pose graph if the pose of the node is known.
+The map (local or global) consists of the pose graph (in
+Definition 1) and map points in the environment. In this paper,
+we will use the terms map and pose graph interchangeably.
+Without loss of generality, we will also assume that the global
+(or local) map is anchored on the first node, as in [37], [39].
+This assumption is made because SLAM can only estimate
+the relative pose change based on the covisibility and inertial
+measurements, while the absolute pose estimate in the global
+coordinate system cannot be provided.
+B. The Local Map
+Local map construction. The candidate keyframes are se-
+lected from camera frames according to the selection strategy
+in ORB-SLAM3 [20], and these candidate keyframes form
+the set K. Due to the constrained computation resources,
+the mobile device selects a fixed keyframe set Kfixed and a
+local keyframe set Kloc from the candidate keyframes, where
+|Kfixed| ⩽ lf and |Kloc| ⩽ lloc. The fixed keyframe set
+Kfixed ⊆ Kg,user is selected from the global map Kg,user
+that was last transmitted from the edge server. The poses
+of keyframes in Kfixed act as fixed priors in the local map
+optimization. This is because poses of keyframes in Kg,user
+are already optimized in the global map optimization and
+hence have low uncertainty. The poses of keyframes in the
+local keyframe set Kloc ⊆ K \ Kg,user will be optimized
+according to the map optimization problem introduced above.
+The edges between keyframes in Kloc form the set Eloc, and
+the edges whose one node belongs to Kloc and another node
+belongs to Kfixed form the set El,f.
+Local map optimization. After selecting Kloc in the local
+map construction, the local map optimization is to optimize
+the estimated poses
+�
+˜Pn
+�
+n∈Kloc
+to minimize the sum of
+Mahalanobis norms
+�
+e∈Eloc∪El,f
+(˜xe)⊤Ie˜xe. Note that in the
+local pose graph optimization, the keyframes in Kfixed are
+included in the optimization with their poses fixed. The local
+map optimization to solve (3) is
+min
+{˜Pn}n∈Kloc
+�
+e∈Eloc∪El,f
+(˜xe)⊤Ie˜xe.
+(4)
+C. The Global Map
+Global map construction. Due to the limited bandwidth
+between the mobile device and the edge server, only a subset
+of candidate keyframes are offloaded to the edge server to
+build a global map. The selection of keyframes to be offloaded
+will be optimized to minimize the pose estimation uncertainty
+of the global map when considering the underlying wireless
+network constraints.
+The edge server maintains the global map, denoted as
+Kg,edge, holding all keyframes uploaded by the mobile device.
+The edges between keyframes in the global map Kg,edge
+constitute the set Eglob. Note that Kg,edge may be different
+from Kg,user, because the global map is large and it takes
+time to transmit the most up-to-date global map from the edge
+server to the mobile device.
+Global map optimization. After selecting Kg,edge in the
+global map construction, the edge server performs the global
+map optimization to estimate poses
+˜Pn in Kg,edge and
+minimize the sum of Mahalanobis norms
+�
+e∈Eglob
+(˜xe)⊤Ie˜xe.
+Specifically, the edge solves (3) when E = Eglob and K =
+Kg,edge, i.e., the global map optimization is to solve
+min
+{˜Pn}n∈Kg,edge
+�
+e∈Eglob
+(˜xe)⊤Ie˜xe.
+(5)
+VI. PROBLEM FORMULATION
+AdaptSLAM aims to efficiently select keyframes to con-
+struct optimal local and global maps, i.e., we select keyframes
+in Kloc and Kfixed for the local map and Kg,edge for the
+global map. From §IV, after constructing the optimal local and
+global maps, the map optimization can be performed using the
+standard algorithms [41], [42]. We construct optimal local and
+global maps by minimizing the uncertainty of the keyframes’
+estimated poses. Hence, we represent and quantify the uncer-
+tainty in §VI-A, and formulate the uncertainty minimization
+problems in §VI-B.
+A. Uncertainty Quantification
+Let pn
+=
+˜Pn − Pn denote the pose estimate error
+of keyframe n. The estimated measurement noise can be
+rewritten as �xe = pn − pm + xe = pn,m + xe, where
+pn,m
+=
+pn − pm. We stack all pn, n
+∈
+K and get
+a pose estimate error vector w
+=
+�
+p⊤
+1 , p⊤
+2 , · · · , p⊤
+|K|
+�
+.
+We rewrite the objective function of map optimization
+
+in (3) as
+�
+e∈E
+(˜xe)⊤Ie˜xe
+=
+�
+e=((n,m),c)∈E
+p⊤
+n,mIepn,m +
+2
+�
+e=((n,m),c)∈E
+p⊤
+n,mIexe + �
+e∈E
+x⊤
+e Iexe. If we can rewrite
+the quadratic term
+�
+e=((n,m),c)∈E
+p⊤
+n,mIepn,m in the format of
+wIww⊤, where Iw is called the information matrix of the
+pose graph, the uncertainty of the pose graph is quantified by
+− log det (Iw) according to the D-optimality [31]–[33].2
+We denote the pose estimate error vectors for the global
+and local maps as wg
+=
+�
+p⊤
+u1, · · · , p⊤
+u|Kg,edge|
+�
+and
+wl =
+�
+p⊤
+r1, · · · , p⊤
+r|Kloc|
+�
+, where u1, · · · , u|Kg,edge| are the
+keyframes in Kg,edge, and r1, · · · , r|Kloc| are the keyframes
+in Kloc. The first pose in the global and local pose graph is
+known (pu1 = 0, pr1 = 0). We rewrite the quadratic terms of
+the objective functions of global and local map optimizations
+in
+(5)
+and
+(4)
+as
+�
+e=((n,m),c)∈Eglob
+p⊤
+n,mIepn,m
+=
+wgIglob (Kg,edge) w⊤
+g (or
+�
+e=((n,m),c)∈Eloc∪El,f
+p⊤
+n,mIepn,m =
+wlIloc (Kloc, Kfixed)w⊤
+l ),
+where
+Iglob (Kg,edge)
+and
+Iloc (Kloc, Kfixed) are called the information matrices of
+the global and local maps and will be derived later (in
+Definition 3 and Lemmas 1 and 2).
+Definition 3 (Uncertainty). The uncertainty of the global (or
+local) pose graph is defined as − log det
+�
+˜Iglob (Kg,edge)
+�
+(or − log det
+�
+˜Iloc (Kloc, Kfixed)
+�
+, where ˜Iglob (Kg,edge) and
+˜Iloc (Kloc, Kfixed) are obtained by removing the first row and
+first column in the information matrices Iglob (Kg,edge) and
+Iloc (Kloc, Kfixed).
+From Definition 3, the uncertainty quantification is based
+on the global and local map optimizations introduced in §V-C
+and §V-B. After quantifying the uncertainty, we will later (in
+§VI-B) optimize the local and global map construction which
+in turn minimizes the uncertainty of poses obtained from local
+and global map optimizations.
+Lemma 1 (Uncertainty of global pose graph). For the
+global map optimization, the uncertainty is calculated as
+− log det
+�
+˜Iglob (Kg,edge)
+�
+, where ˜Iglob (Kg,edge) = ˜Lglob⊗I
+with ˜Lglob being the matrix obtained by deleting the first row
+and column in the Laplacian matrix Lglob, and ⊗ being the
+Kronecker product. The i, j-th element of Lglob is given by
+[Lglob]i,j =
+
+
+
+
+
+−
+�
+e=((ui,uj),c)∈Eg,edge
+we,
+i ̸= j
+�
+e=((ui,q),c)∈Eg,edge,ui̸=q
+we,
+i = j
+.
+(6)
+Proof. See Appendix A. Proof sketch: The proof follows from
+the global map optimization formulation in §V-C and the
+definition of ˜Iglob (Kg,edge).
+2Common approaches to quantifying uncertainty in SLAM are to use real
+scalar functions of the maximum likelihood estimator covariance matrix [43].
+Among them, D-optimality (determinant of the covariance matrix) [37], [39]
+captures the uncertainty due to all the elements of a covariance matrix and
+has well-known geometrical and information-theoretic interpretations [44].
+From Lemma 1, the uncertainty of the global pose graph
+can be calculated based on the reduced Laplacian matrix
+(˜Lglob). According to the relationship between the reduced
+Laplacian matrix and the tree structure [45], the uncertainty is
+inversely proportional to the logarithm of weighted number of
+spanning trees in the global pose graph. Similar conclusions
+are drawn for 2D pose graphs [31] and 3D pose graphs with
+only covisibility edges [37], [39], where the device can move
+in 2D plane and 3D space respectively. We extend the results
+to VI-SLAM where the global pose graph is a multigraph with
+the possibility of having both a covisibility edge and an IMU
+edge between two keyframes.
+Lemma 2 (Uncertainty of local pose graph). The uncertainty
+is − log det
+�
+˜Iloc (Kloc, Kfixed)
+�
+for the local map, where
+˜Iloc (Kloc, Kfixed) = ˜Lloc ⊗ I with ˜Lloc being the matrix
+obtained by deleting the first row and the first column in Lloc.
+The i, j-th element of Lloc (of size |Kloc| × |Kloc|) is given by
+[Lloc]i,j =
+
+
+
+
+
+−
+�
+e=((ri,rj),c)∈Eloc
+we,
+i ̸= j
+�
+e=((ri,q),c)∈El,f∪Eloc,q̸=ri
+we, i = j
+.
+(7)
+Proof. See Appendix B. Proof sketch: Setting pn
+= 0,
+n ∈ Kfixed (the fixed keyframes have poses with ‘zero
+uncertainty’), the proof follows from the local pose graph
+optimization formulation in §V-B and the definition of
+˜Iloc (Kloc, Kfixed).
+From Lemma 2, the uncertainty of the local map is propor-
+tional to the uncertainty of the pose graph G anchoring on the
+first node in Kloc and all nodes in Kfixed, where G’s node set is
+Kfixed∪Kloc and edge set includes all measurements between
+any two nodes in Kfixed ∪ Kloc. Note that keyframe poses
+in Kfixed are optimized on the edge server and transmitted
+to the mobile device, and they are considered as constants
+in the local pose graph optimization. From the uncertainty’s
+perspective, adding fixed keyframes in Kfixed is equivalent
+to anchoring these keyframe poses (i.e., deleting rows and
+columns corresponding to the anchored nodes in the Laplacian
+matrix of graph G). In addition, from Lemma 2, although poses
+are fixed, the anchored nodes still reduce the uncertainty of
+the pose graph. Hence, apart from Kloc, we will select the
+anchored keyframe set Kfixed to minimize the uncertainty.
+B. Uncertainty Minimization Problems
+We now formulate optimization problems whose objectives
+are to minimize the uncertainty of the local and global
+maps. For the local map optimization, under the computation
+resource constraints, we solve Problem 1 for each keyframe k.
+For the global map optimization, under the communication
+resource constraints, we solve Problem 2 to adaptively offload
+keyframes to the edge server.
+
+Problem 1 (Local map construction).
+max
+Kloc,Kfixed log det
+�
+�Iloc (Kloc ∪ {k} , Kfixed)
+�
+(8)
+s.t.
+|Kloc| ⩽ lloc, Kloc ⊆ K \ Kg,user
+(9)
+|Kfixed| ⩽ lf, Kfixed ⊆ Kg,user.
+(10)
+The objective of Problem 1 is equivalent to minimizing the
+uncertainty of the local map. Constraint (9) means that the size
+of Kloc is constrained to reduce the computational complexity
+in the local map optimization, and that the keyframes to
+be optimized in the local map are selected from keyframes
+that are not in Kg,user. Constraint (10) means that the size
+of Kfixed is constrained, and that the fixed keyframes are
+selected from Kg,user that were previously optimized on and
+transmitted from the edge server.
+Problem 2 (Global map construction).
+max
+K′⊆K\Kg,edge log det
+�
+�Iglob (Kg,edge ∪ K′)
+�
+(11)
+s.t. d |K′| ⩽ D.
+(12)
+The objective of Problem 2 is equivalent to minimizing the
+uncertainty of the global map. K \ Kg,edge is set of the
+keyframes that have not been offloaded to the server, and
+we select a subset of keyframes, K′, from K \ Kg,edge.
+The constraint (12) guarantees that the keyframes cannot be
+offloaded from the device to the server at a higher bitrate than
+the available channel capacity, where D is the channel capacity
+constraint representing the maximum number of bits that can
+be transmitted in a given transmission window. We assume that
+the data size d of each keyframe is the same, which is based on
+the observation that the data size is relatively consistent across
+keyframes in popular public SLAM datasets [46], [47].
+VII. LOCAL AND GLOBAL MAP CONSTRUCTION
+We analyze the properties of approximate submodularity
+in map construction problems, and propose low-complexity
+algorithms to efficiently construct local and global maps.
+A. Local Map Construction
+The keyframes in the local map include those in two disjoint
+sets Kloc and Kfixed. To efficiently solve Problem 1, we
+decompose it into two problems aiming at minimizing the
+uncertainty: Problem 3 that selects keyframes in Kloc and
+Problem 4 that selects keyframes in Kfixed. We obtain the
+optimal local keyframe set K⋆
+loc in Problem 3. Based on K⋆
+loc,
+we then obtain the optimal fixed keyframe set K⋆
+fixed in
+Problem 4. We will compare the solutions to Problems 3 and 4
+with the optimal solution to Problem 1 in §VIII to show that
+the performance loss induced by the decomposition is small.
+Problem 3.
+K⋆
+loc = arg max
+Kloc log det
+�
+˜Iloc(Kloc∪ {k}, ∅)
+�
+s.t. (9).
+Problem 4.
+K⋆
+fixed = arg max
+Kfixed log det
+�
+˜Iloc(K⋆
+loc∪ {k}, Kfixed)
+�
+s.t. (10).
+1) The Selection of Local Keyframe Set Kloc: We first
+solve Problem 3. It is a nonsubmodular optimization problem
+with constraints, which are NP-hard and generally difficult
+to be solved with an approximation ratio [22]. Hence, we
+decompose Problem 3 into subproblems (Problems 5 and 6)
+that are equivalent to the original Problem 3 and can be
+approximately solved with a low-complexity algorithm.
+In problem 5, assume that we already select a keyframe
+subset Kbase from K \ Kg,user (with the size lb ≜ |Kbase| ⩽
+lloc), and we aim to further select a keyframe set Kadd
+to be added to Kbase to minimize the local map uncer-
+tainty. Rewriting the objective as Unc (Kadd ∪ Kbase ∪ {k}) ≜
+− log det
+�
+�Iloc (Kadd ∪ Kbase ∪ {k}, ∅)
+�
+, the problem is to
+obtain the optimal Kadd (denoted as OPTadd(Kbase)) given
+Kbase:
+Problem 5.
+OPTadd (Kbase) = arg max
+Kadd −Unc (Kadd ∪ Kbase ∪ {k})
+s.t.
+|Kadd| ⩽ lloc − lb.
+After getting the solutions (i.e., OPTadd (Kbase)) to Prob-
+lem 5 for all possible Kbase of size lb, we obtain the optimal
+Kbase (denoted as K⋆
+base) in Problem 6.
+Problem 6.
+K⋆
+base = arg max
+Kbase −Unc (OPTadd(Kbase) ∪ Kbase ∪ {k})
+s.t.
+|Kbase| = lb.
+Lemma 3. Given lb, the solution to Problems 5 and 6, i.e.,
+K⋆
+base and OPTadd (K⋆
+base), will give us the solution K⋆
+loc to
+Problem 3. Specifically, K⋆
+loc = K⋆
+base ∪ OPTadd (K⋆
+base).
+Proof. The proof is straightforward and hence omitted.
+We can obtain K⋆
+loc in Problem 3 by solving Problems 5
+and 6. We will show that the objective function of Problem 5 is
+‘close to’ a submodular function when the size of the keyframe
+set Kbase is large. In this case, Problem 5 can be efficiently
+solved using a greedy algorithm with an approximation ratio.
+When |Kbase| is small, we need to compare the objective
+function for different combinations of Kbase and Kadd.
+Lemma 4. When
+wmax
+|Kbase|wmin < 1, the submodularity ratio γ
+of the objective function in Problem 5 is lower bounded by
+γ ⩾ 1 + 1
+ϑ log
+�
+1 −
+4|Kadd|2w2
+max
+|Kbase| wmin − wmax
+�
+,
+(13)
+where
+ϑ
+=
+min
+m∈Kadd
+�
+n∈Kbase
+log wn,m,
+wmax
+=
+max
+n,m∈Kbase∪Kadd wn,m, and wmin =
+min
+n,m∈Kbase∪Kadd wn,m. γ
+is close to 1 when |Kbase| is significantly larger than |Kadd|.
+
+Algorithm 2 Selecting local keyframe set Kloc in the local
+map (top-h greedy-based algorithm)
+1: Θ ← ∅;
+2: while ( |Λ| ⩽ lloc) do
+3:
+if |Λ| ⩽ lthr then h ← H else h ← 1;
+4:
+Select
+the
+top-h
+highest-scoring
+combinations
+of
+Λ, Λ ∈ Θ and n, n ∈ K \ Kg,user that minimize
+Unc (Λ ∪ {n, k}). Unc (Λ ∪ {n, k}) is calculated using
+the computation reuse algorithm in Algorithm 2;
+5:
+Update Θ as the set of h highest-scoring combinations
+of Λ and n. Each element of Θ is a set (i.e., Λ ∪ {n})
+corresponding to one combination;
+6: K⋆
+loc ← arg min
+Λ∈Θ Unc(Λ ∪ {k}).
+Proof. See Appendix C. Proof sketch: Following from the
+definition of γ in (1), we first prove that the denomina-
+tor in (1), denoted as log det(Mden), is lower bounded
+by ϑ. Denoting the numerator in (1) as log det(Mnum),
+we
+show
+that
+log det(Mnum)
+⩾
+log det(Mden) +
+log
+�
+1 −
+4|Kadd|2w2
+max
+|Kbase|wmin−wmax
+�
+, by proving that the absolute val-
+ues of all elements in Mnum are bounded.
+From Lemma 4, the objective function in Problem 5 is
+‘close to’ a submodular function when the size of the exist-
+ing keyframe set (i.e., |Kbase|) is much larger than |Kadd|.
+Hence, we can use the greedy algorithm to approximately
+solve Problem 5. According to Theorem 1, the solution
+obtained by the greedy algorithm for Problem 5, denoted
+by OPT#
+add (Kbase), has an approximation guarantee that
+OPT#
+add (Kbase) ⩾ (1 − exp(−γ)) OPTadd (Kbase).
+According to the analysis of the properties of Problems 5
+and 6, we now solve Problem 3 to select the local keyframe set
+Kloc using Algorithm 2 (top-h greedy-based algorithm). Θ is
+the set of possible keyframe sets that minimize the local map
+uncertainty, and we only maintain h keyframe sets to save
+the computation resources. Λ, Λ �� Θ, denotes the element
+in Θ and represents one possible keyframe set. When the
+size of Λ is smaller than a threshold lthr (|Λ| ⩽ lthr), we
+select the top-H (H > 1) highest-scoring combinations of
+Λ and n, n ∈ K \ Kg,user, that minimize Unc (Λ ∪ {k, n}).
+When |Λ| gets larger, we only select the highest-scoring
+combination. The reasons are as follows. Λ can be seen as the
+existing keyframe set Kbase. According to Lemma 4, when
+the size of the existing keyframe set (which is |Λ| here) is
+small, there is no guarantee that Unc (Kadd ∪ Kbase ∪ {k}) is
+close to a submodular function (i.e., the submodularity ratio
+is much smaller than 1). Hence, we need to try different
+combinations of Λ and n to search for the combination that
+minimizes the uncertainty after each iteration. As |Λ| grows,
+the submodularity ratio is close to 1, and a greedy algorithm
+can achieve η approximation (η = 1 − exp(−γ), γ → 1).
+In this case, we apply the greedy algorithm and only keep
+the combination that achieves the minimal uncertainty at each
+step.
+Algorithm 3 Computation reuse algorithm
+1: Input: det(A), A−1;
+2: B ← A−1. Calculate BiB⊤
+i , i = 1, · · · , |Λ|;
+3: Calculate (A′)−1 using (15). Calculate det (A′) using
+(16). Calculate det(˜I (Λ ∪ {n, k})) using (14).
+2) Computation Reuse Algorithm: We use the computation
+reuse algorithm (Algorithm 3) to speed up Algorithm 2. We
+observe that for different n, n ∈ K \ Kg,user, only a limited
+number (3|Λ|+1) of elements in the matrix ˜I (Λ ∪ {n, k}) are
+different. Calculating the log-determinant function of a (|Λ|+
+1)×(|Λ|+1) matrix ˜I (Λ ∪ {n, k}) has a high computational
+complexity (of O(|Λ|+1)3) [48]. Hence, instead of computing
+the objective function for each n from scratch, we reuse parts
+of computation results for different n.
+Letting A ≜ ˜I (Λ ∪ {k}) denote the information matrix of
+the local map in the |Λ|-th iteration (of Algorithm 2), the infor-
+mation matrix in the (|Λ|+1)-th iteration is ˜I (Λ ∪ {n, k}) =
+�
+A + diag (a)
+a⊤
+a
+d
+�
+, where a = (a1, a2, · · · , a|Λ|) with
+ai = wλi,n, λi is the i-th element of Λ, and d = wk,n+
+|Λ|
+�
+i=1
+ai.
+We aim to calculate det(˜I (Λ ∪ {n, k})) using the calcula-
+tion of det(A) and A−1 from the previous iteration. Letting
+A′ ≜ A + diag(a), det(˜I (Λ ∪ {n, k})) is calculated by
+det(˜I (Λ ∪ {n, k})) = (d − a(A′)−1a⊤) det(A′).
+(14)
+Next we efficiently calculate (A′)−1 and det(A′) to get
+det(˜I (Λ ∪ {n, k})). We can rewrite A′ as A′
+= A +
+|Λ|
+�
+i=1
+β⊤
+i βi where βi =
+
+0, · · · , √ai
+����
+i−th
+, · · · , 0
+
+. According to
+Sherman–Morrison formula [49], (A′)−1 is given by
+(A′)−1 ≈
+B
+����
+Reuse
+−
+|Λ|
+�
+i=1
+ai
+1 + aiBi,i
+BiB⊤
+i
+� �� �
+Reuse
+,
+(15)
+where B = A−1, Bi,i is the i, i-th element of B, and Bi is
+the i-th column vector of B. Using (15), B and BiB⊤
+i can be
+computed only once to be used for different n, n ∈ K\Kg,user,
+which greatly reduces the computational cost. According to the
+rank-1 update of determinant [49], det(A′) can be written as
+det (A′) = det (A) (1 + a1B1,1) {1(|Λ| = 1) + 1(|Λ| > 1)
+×
+|Λ|
+�
+i=2
+
+1 + ai
+
+B −
+i−1
+�
+j=1
+ajBjBT
+j
+1 + ajBj,j
+
+
+i,i
+
+
+
+
+ .
+(16)
+�
+B −
+i−1
+�
+j=1
+ajBjBT
+j
+1+ajBj,j
+�
+is already calculated in (15), which re-
+duces the computational complexity. Substituting (15) and (16)
+into (14), we get the final results of det(˜I (Λ ∪ {n, k})).
+The computation complexity of different algorithms.
+If we select keyframes in Kloc using a brute-force algo-
+rithm based on exhaustive enumeration of combinations of
+
+keyframes in Kloc, the complexity is O
+�� ρ
+lloc
+�
+l3
+loc
+�
+, where
+ρ = |K \ Kg,user| is the number of keyframes that have not
+been offloaded to the edge server. Without computation reuse,
+the computation complexity of the proposed top-h greedy-
+based algorithm is O(Hρl4
+loc). With computation reuse, it is
+reduced to O(Hl4
+loc) + O(Hρl3
+loc). Since we only keep lloc
+keyframes in Kloc of the local map and a small H in Algo-
+rithm 2 to save computation resources, i.e., ρ ≫ lloc > H,
+the proposed greedy-based algorithm with computation reuse
+significantly reduces the computational complexity.
+3) The Selection of Fixed Keyframe Set Kfixed: After
+selecting the local keyframe set Kloc by solving Problem 3,
+we solve Problem 4 to select the fixed keyframe set.
+Lemma 5. Problem 4 is non-negative, monotone and submod-
+ular with a cardinality-fixed constraint.
+Proof sketch. It is straightforward to prove the non-negativity
+and monotonicity. For the submodularity, we can prove that
+det(�Iloc(K⋆
+loc∪{k},L)) det(�Iloc(K⋆
+loc∪{k},S))
+det(�Iloc(K⋆
+loc∪{k},L∪S)) det(�Iloc(K⋆
+loc∪{k},∅)) ⩾ 1, using the
+property that det(M) ⩾ det(N) holds for positive semidefi-
+nite matrices M, N when M−N is positive semidefinite.
+Lemma 5 indicates that the problem can be approximately
+solved with greedy methods in Algorithm 1 [34]. For each
+iteration, the algorithm selects one keyframe from Kg,user to
+be added to the fixed keyframe set Kfixed. The approximation
+ratio η = 1−exp(−1) guarantees that worst-case performance
+of a greedy algorithm cannot be far from optimal.
+B. Global Map Construction
+We use a low-complexity algorithm to solve Problem 2 to
+construct the global map. The objective function of Problem 2
+can be rewritten as −Unc (Kg,edge ∪ K′), which has the same
+structure as that of Problem 3. Problems 2 and 3 both add
+keyframes to the existing keyframe sets to construct a pose
+graph and optimize the keyframe poses in the pose graph.
+Hence, Algorithms 2 and 3 can be used to solve Problem 2.
+In Algorithm 2, lloc is replaced by
+D
+d , and K \ Kg,user is
+replaced by K \ Kg,edge. Calculating the uncertainty of a
+large global map is computationally intensive, and hence the
+proposed low-complexity algorithm is essential to reducing the
+computational load on the mobile device.
+VIII. EVALUATION
+We implement AdaptSLAM on the open-source ORB-
+SLAM3 [20] framework which typically outperforms older
+SLAM methods [25], [26], with both V- and VI- configura-
+tions. The edge server modules are run on a Dell XPS 8930
+desktop with Intel (R) Core (TM) i7-9700K [email protected]
+and NVIDIA GTX 1080 GPU under Ubuntu 18.04LTS. In
+§VIII-A, the mobile device modules are run on the same
+desktop under simulated computation and network constraints.
+In §VIII-B, the mobile device modules are implemented on a
+laptop (with an AMD Ryzen 7 4800H CPU and an NVIDIA
+GTX 1660 Ti GPU), using a virtual machine with 4-core CPUs
+and 8GB of RAM. The weight we, e = ((n, m), c) is set as the
+number of common map features visible in keyframes n and
+m for covisibility edges, similar to [20], [50], and the IMU
+edge weight is set as a large value (i.e., 500) as the existence
+of IMU measurements greatly reduces the tracking error. We
+empirically set H = 5 and lthr = 30 in Algorithm 2 to ensure
+low complexity and good performance at the same time.
+Metric. We use root mean square (RMS) absolute trajectory
+error (ATE) as the SLAM performance metric which is com-
+monly used in the literature [20], [51]. ATE is the absolute
+distance between the estimated and ground truth trajectories.
+Baseline methods. We compare AdaptSLAM with 5 base-
+lines. Random selects the keyframe randomly. DropOldest
+drops the oldest keyframes when the number of keyframes
+is constrained. ORBBuf, proposed in [28], chooses the
+keyframes that maximize the minimal edge weight between
+the adjacent selected keyframes. BruteForce examines all the
+combinations of keyframes to search for the optimal one that
+minimizes the uncertainty (in Problems 1 and 2). BruteForce
+can achieve better SLAM performance than AdaptSLAM
+but is shown to have exponential computation complexity
+in §VII-A. In the original ORB-SLAM3, the local map
+includes all covisibility keyframes, and the global map in-
+cludes all keyframes. The original ORB-SLAM3 also achieves
+better SLAM performance and consumes more computation
+resources than AdaptSLAM as the numbers of keyframes in
+both local and global maps are large.
+Datasets. We evaluate AdaptSLAM on public SLAM
+datasets containing V and VI sequences, including TUM [47]
+and EuRoC [46]. The difficulty of a SLAM sequence depends
+on the extent of device mobility and scene illumination. We
+use EuRoC sequences V101 (easy), V102 (medium), and V103
+(difficult), and difficult TUM VI room1 and room6 sequences.
+We report the results over 10 trials for each sequence.
+A. Simulated Computation and Network Constraints
+First, we limit the number of keyframes in the local map
+under computation constraints, and all keyframes are used
+to build the global map without communication constraints.
+Second, we maintain local maps as in the default settings
+of ORB-SLAM3, and limit the number of keyframes in the
+global map under constrained communications, where D in
+Problem 2 is set according to the available bandwidth.
+Local map construction. We demonstrate the RMS ATE of
+different keyframe selection methods, for different V-SLAM
+(Fig. 2a) and VI-SLAM (Fig. 2b) sequences. The size of
+the local map is limited to 10 keyframes and 9 anchors
+in V-SLAM sequences, and 25 keyframes and 10 anchors
+in VI-SLAM sequences (to ensure successful tracking while
+keeping a small local map). AdaptSLAM reduces the RMS
+ATE compared with Random, DropOldest, and ORBBuf by
+more than 70%, 62%, and 42%, averaged over all sequences.
+The performance of AdaptSLAM is close to BruteForce, which
+demonstrates that our greedy-based algorithms yield near-
+optimal solutions, with substantially reduced computational
+complexity. Moreover, the performance of AdaptSLAM is close
+to the original ORB-SLAM3 (less than 0.05 m RMS ATE
+
+V101
+V102
+V103
+room1
+room6
+Sequence
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+RMS ATE (m)
+Random
+DropOldest
+ORBBuf
+BruteForce
+ORB-SLAM3
+AdaptSLAM
+(a) V-SLAM
+V101
+V102
+V103
+room1
+room6
+Sequence
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+RMS ATE (m)
+Random
+DropOldest
+ORBBuf
+BruteForce
+ORB-SLAM3
+AdaptSLAM
+(b) VI-SLAM
+Fig. 2: RMS ATE for 6 keyframe selection methods in the local map construction for 5
+sequences in EuRoC and TUM.
+Random DropOldest ORBBuf AdaptSLAM
+Method
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+RMS ATE (m)
+lloc = 10
+lloc = 20
+lloc = 30
+Fig. 3: RMS ATE for different sizes
+of local keyframe set (for EuRoC
+V102).
+V101
+V102
+V103
+room1
+room6
+Sequence
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+RMS ATE (m)
+Random
+DropOldest
+ORBBuf
+BruteForce
+ORB-SLAM3
+AdaptSLAM
+(a) V-SLAM
+V101
+V102
+V103
+room1
+room6
+Sequence
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+RMS ATE (m)
+Random
+DropOldest
+ORBBuf
+BruteForce
+ORB-SLAM3
+AdaptSLAM
+(b) VI-SLAM
+Fig. 4: RMS ATE for 6 keyframe selection methods in the global map construction for
+5 sequences in EuRoC and TUM.
+Random DropOldest ORBBuf AdaptSLAM
+Method
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+RMS ATE (m)
+40Mbps
+80Mbps
+w/o bandwidth 
+ limitation
+Fig.
+5:
+RMS
+ATE
+for different
+available bandwidth for offloading
+keyframes (for EuRoC V102).
+difference for all sequences) even though the size of the local
+map is reduced by more than 75%.
+The influence of the number lloc of keyframes in the local
+map on the RMS ATE for different methods is shown in
+Fig. 3. We present the results for EuRoC V102 (of medium
+difficulty), which are representative. When lloc is reduced
+from 30 to 10, AdaptSLAM increases the RMS ATE by only
+6.7%, to 0.09 m, as compared to 0.37, 0.16, and 0.12 m
+for, correspondingly, Random, DropOldest, and ORBBuf. This
+indicates that AdaptSLAM achieves low tracking error under
+stringent computation resource constraints.
+Global map construction. First, we examine the case
+where only half of all keyframes are offloaded to build a
+global map, for V-SLAM (Fig. 4a) and VI-SLAM (Fig. 4b)
+sequences. AdaptSLAM reduces the RMS ATE compared with
+the closest baseline ORBBuf by 27% and 46% on average for
+V- and VI-SLAM, and has small performance loss compared
+with the original ORB-SLAM3, despite reducing the number
+of keyframes by half.
+Next, in Fig. 5, we examine four methods whose perfor-
+mance is impacted by the available bandwidth, under different
+levels of communication constraints. Without bandwidth lim-
+itations, all methods have the same performance as the global
+map holds all keyframes. When the bandwidth is limited,
+Random and DropOldest have the worst performance as they
+ignore the relations of keyframes in the pose graph. The
+ORBBuf performs better, but the tracking error is increased
+by 4.0× and 9.8× when the bandwidth is limited to 80
+and 40 Mbps. AdaptSLAM achieves the best performance,
+reducing the RMS ATE compared to ORBBuf by 62% and 78%
+when network bandwidth is 80 and 40 Mbps, correspondingly.
+This highlights the superiority of AdaptSLAM in achieving
+high tracking accuracy under communication constraints.
+foot1
+foot3
+foot5
+Network trace
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+RMS ATE (m)
+Random
+DropOldest
+ORBBuf
+AdaptSLAM
+Fig. 6: RMS ATE for dif-
+ference network traces.
+Method
+Latency (ms)
+Random
+133.0±86.3
+DropOldest
+139.9±53.7
+ORBBuf
+149.3±75.6
+BruteForce
+863.4±123.5
+ORB-SLAM3
+556.4±113.7
+AdaptSLAM
+162.8±68.9
+TABLE I: The latency for local
+map construction and optimization.
+B. Real-World Computation and Network Constraints
+Following the approach of splitting modules between the
+edge server and the mobile device [11], we split the modules
+as shown in Fig. 1. The server and the device are connected
+via a network cable to minimize other factors. To ensure
+reproducibility, we replay the network traces collected from a
+4G network [52]. Focusing on mobile devices carried by users,
+we choose network traces (foot1, foot3, and foot5) collected
+by pedestrians. We set D in Problem 2 according to the traces.
+We examine the RMS ATE under the network traces in
+Fig. 6 for the EuRoC V102 sequence. The results for only
+four methods are presented because the overall time taken for
+running the SLAM modules onboard is high for BruteForce
+and the original SLAM. AdaptSLAM reduces the RMS ATE by
+65%, 61%, and 35% (averaged over all traces) compared with
+Random, DropOldest, and ORBBuf. AdaptSLAM achieves
+high tracking accuracy under real-world network traces.
+Table I shows the computation latency of mobile devices
+for all six methods. We compare the latency for running
+local map construction and optimization, which is the main
+source of latency for modules running onboard [11]. Compared
+
+with AdaptSLAM, the original ORB-SLAM3 takes 3.7× as
+much time for optimizing the local map as all covisibility
+keyframes are included in the local map without keyframe
+selection. Without the edge-assisted architecture, the original
+ORB-SLAM3 also runs global mapping and loop closing
+onboard which have even higher latency [11]. BruteForce takes
+5.3× as much time for examining all the combinations of
+keyframes to minimize the local map uncertainty. The latency
+for constructing and optimizing local maps using AdaptSLAM
+is close to that using Random and DropOldest (<12.3%
+difference). Low latency for local mapping shows that edge-
+assisted SLAM is appealing, as local mapping is the biggest
+source of delay for modules executing onboard after offloading
+the intensive tasks (loop closing and global mapping).
+IX. CONCLUSION
+We present AdaptSLAM, an edge-assisted SLAM that effi-
+ciently select subsets of keyframes to build local and global
+maps, under constrained communication and computation re-
+sources. AdaptSLAM quantifies the pose estimate uncertainty
+of V- and VI-SLAM under the edge-assisted architecture, and
+minimizes the uncertainty by low-complexity algorithms based
+on the approximate submodularity properties and computation
+reuse. AdaptSLAM is demonstrated to reduce the size of the
+local keyframe set by 75% compared with the original ORB-
+SLAM3 with a small performance loss.
+ACKNOWLEDGMENTS
+This work was supported in part by NSF grants CSR-
+1903136, CNS-1908051, and CNS-2112562, NSF CAREER
+Award IIS-2046072, by an IBM Faculty Award, and by the
+Australian Research Council under Grant DP200101627.
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+APPENDIX
+A. Proof of Lemma 1
+The quadratic term of the objective function in (5) is
+�
+e=((n,m),c)∈Eglob
+p⊤
+n,mIepn,m, where p⊤
+n,mIepn,m can be
+rewritten as
+p⊤
+n,mIepn,m
+=we
+�
+p⊤
+n , p⊤
+m
+� �
+I6I
+−I6I
+−I6I
+I6I
+�
+(pn, pm)
+=wgΞew⊤
+g ,
+(17)
+where the i, j-th block of Ξe, [Ξe]i,j, is derived as
+[Ξe]i,j =
+
+
+
+−weI,
+ui = n, uj = m
+weI,
+ui = uj = n
+0,
+otherwise
+.
+From the definition of Iglob (Kg,edge) and the global pose
+graph optimization formulation in §V-C, we can obtain that
+Iglob (Kg,edge) =
+�
+e∈Eg,edge
+Ξe. Hence, Lglob is given by (6),
+which concludes the proof.
+B. Proof of Lemma 2
+As introduced in §V-B, the local pose graph optimization is
+to solve
+min
+{ ˜Pn}n∈Kloc
+�
+e∈Eloc∪El,f
+(xe)⊤Iexe. In optimizing poses
+of keyframes in Kloc, the poses of keyframes in Kfixed are
+fixed. Hence, the quadratic term of the objective function can
+be rewritten as
+�
+e=((n,m),c)∈Eloc∪El,f
+p⊤
+n,mIepn,m
+=
+�
+e=((n,m),c)∈Eloc
+p⊤
+n,mIepn,m
++
+�
+e=((n,m),c)∈El,f,n∈Kloc,m∈Kfixed
+p⊤
+n Iepn
++
+�
+e=((n,m),c)∈El,f,n∈Kfixed,m∈Kloc
+�
+−p⊤
+m
+�
+Ie (−pm) .
+According to the above analysis, we can reformulate (4) as
+min
+{ ˜Pn}n∈Kloc
+wlΛloc (Kloc, Kfixed) w⊤
+l ,
+where Λloc (Kloc, Kfixed) is the |Kloc| × |Kloc| block matrix
+whose i, j-th block is
+[Λloc (Kloc, Kfixed)]i,j =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+−
+�
+e=((ri,rj),c)∈Eloc
+weI,
+i ̸= j
+�
+�
+e=((ri,q),c)∈El,f,q∈Kfixed
+we +
+�
+e=((ri,q),c)∈Eloc,q∈Kloc,q̸=ri
+we
+�
+I
+,
+i = j
+.
+According to the uncertainty definition in Definition 3,
+the uncertainty of the local pose graph is calculated as
+− log det
+�
+˜Iloc (Kloc, Kfixed)
+�
+, where ˜Iloc (Kloc, Kfixed) is
+given by (7).
+C. Proof of Lemma 4
+According to the definition of the submodularity ratio given
+in (1), the submodularity ratio γ of the objective function in
+Problem 5 can be calculated as (18), where (a) follows from
+the definition of the submodularity ratio. The denominator of
+(18), denoted as ς, is lower bounded by
+ς = log
+det
+�
+˜Iloc (Kbase ∪ L ∪ S ∪ {x, k} , ∅)
+�
+det
+�
+˜Iloc (Kbase ∪ L ∪ S ∪ {k}) , ∅
+�
+⩾
+�
+n∈Kbase∪L∪S
+log wx,n
+⩾
+min
+m∈Kadd
+�
+n∈Kbase
+log wn,m ≜ ϑ,
+(20)
+where the first inequality is due to the fact that the determinant
+of the reduced weighted Laplacian matrix is equal to the tree-
+connectivity of its corresponding graph [31].
+
+γ
+(a)
+=
+min
+L⊆Kadd,S⊆Kadd,|S|⩽lloc−lb,x∈Kadd,x̸∈S∪L
+−Unc (Kbase ∪ L ∪ {x}) + Unc (Kbase ∪ L)
+−Unc (Kbase ∪ L ∪ S ∪ {x}) + Unc (Kbase ∪ L ∪ S)
+=
+min
+L⊆Kadd,S⊆Kadd,|S|⩽lloc−lb,x∈Kadd,x̸∈S∪L
+log
+det(˜Iloc(Kbase∪L∪{x,k},∅))
+det(˜Iloc(Kbase∪L∪{k}),∅)
+log
+det(˜Iloc(Kbase∪L∪S∪{x,k},∅))
+det(˜Iloc(Kbase∪L∪S∪{k}),∅)
+.
+(18)
+γ = 1 +
+log
+�
+min
+L⊆Kadd,S⊆Kadd,|S|⩽lloc−lb,x∈Kadd,x̸∈S∪L
+det(˜Iloc(Kbase∪L∪{x,k},∅)) det(˜Iloc(Kbase∪L∪S∪{k}),∅)
+det(˜Iloc(Kbase∪L∪S∪{x,k},∅)) det(˜Iloc(Kbase∪L∪{k}),∅)
+�
+ς
+= 1 +
+log
+
+
+
+
+
+
+
+
+min
+L⊆Kadd,S⊆Kadd,|S|⩽lloc−lb,x∈Kadd,x̸∈S∪L det
+
+
+
+
+
+
+
+
+
+
+
+Q1+Q2+
+
+
+0z
+I|S|
+0
+
+
+
+
+
+
+
+Q1+Q3+
+
+ 0z+|S|
+1
+
+
+
+
+(Q1+Q2+Q3+Q4)
+
+Q1+
+
+ 0
+0
+0
+I|S|+1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+ς
+= 1 +
+log
+
+
+
+
+min
+L⊆Kadd,S⊆Kadd,|S|⩽lloc−lb,x∈Kadd,x̸∈S∪L det
+(Q1)2+Q1Q2+Q1Q3+(Q2+Q3)
+
+ 0
+0
+0
+I|S|+1
+
++Q2Q3
+(Q1)2+Q1Q2+Q1Q3+(Q2+Q3)
+
+ 0
+0
+0
+I|S|+1
+
++Q4
+
+
+
+
+ς
+(a)
+⩾ 1 + 1
+ϑ log
+�
+1 − g1Q−1g⊤
+1
+�
+.
+(19)
+Substituting (20) into (18), γ can be further calculated
+by (19), where Ii and 0i are the i × i identity matrix and
+zero matrix, and Q = Q1 + Q2 + Q3 + Q4. Q1, Q2, Q3 and
+Q4 are defined as follows. We express Kbase ∪ L ∪ S ∪ {x, k}
+as {si}i={1,··· ,z+|S|+2} where z = |Kbase ∪L|, si ∈ Kbase ∪L
+when i ⩽ z, si ∈ S when z < i ⩽ z + |S|, sz+|S|+1 = x,
+and sz+|S|+2 = k. For each edge e, we define a vector qe,
+and each element [qe]i = −[qe]j = we if vertexes si and
+sj are the head or tail of e and zero otherwise. We then get
+˜qe after removing the last element of qe. Q1, Q2, Q3 and
+Q4 are defined as Q1 =
+1
+2
+�
+e=((si,sj),c),si,sj∈Kbase∪L
+�qe�q⊤
+e
+( 1
+2 is used because the edges from si to sj and from sj
+to si are both included), Q2 =
+�
+e=((si,x),c),si∈Kbase∪L
+�qe�q⊤
+e ,
+Q3
+=
+�
+e=((si,sj),c),si∈Kbase∪L,sj∈S
+�qe�q⊤
+e ,
+and
+Q4
+=
+�
+e=((x,sj),c),sj∈S
+�qe�q⊤
+e . g1 is given by
+g1 =
+
+0, · · · , 0
+� �� �
+z ‘0′s
+, −wx,s1, · · · , −wx,s|S|
+|S|
+�
+i=1
+wx,si
+
+ ,
+where wmax =
+max
+n,m∈Kbase∪Kadd wn,m, and wx,si ⩽ wmax
+for si ∈ S. (a) in (19) is because for invertible positive
+semidefinite matrices M, N, det(M) ⩾ det(N) holds when
+M − N is positive semidefinite [48].
+We will prove that
+��Q−1�� ⩽
+1
+|Kbase|wmin−wmax when
+|Kbase| is significantly larger than |Kadd|, where ∥M∥ is the
+l∞ norm of M (defined as the largest magnitude among each
+element in M), and wmin =
+min
+n,m∈Kbase∪Kadd wn,m. Rewrite
+Q as Q = DQ − EQ, where DQ is a diagonal matrix
+with elements on the diagonal the same as those of Q, and
+EQ = DQ − Q. Q−1 is calculated as
+Q−1 =
+�
+Iz+|S|+1 − D−1
+Q EQ
+�−1
+D−1
+Q
+=
+� ∞
+�
+i=0
+�
+D−1
+Q EQ
+�i
+�
+D−1
+Q .
+D−1
+Q EQ has the properties that all elements in D−1
+Q EQ are
+positive and smaller than
+wmax
+|Kbase|wmin , and all row vectors has
+an l∞ norm smaller than 1. Hence, we have
+����
+�
+D−1
+Q EQ
+�i���� ⩽
+wmax
+|Kbase|wmin .
+
+Q−1 is given by
+��Q−1�� ⩽
+1
+|Kbase| wmin
+�����
+∞
+�
+i=1
+�
+D−1
+Q EQ
+�i
+�����
+⩽
+1
+|Kbase| wmin
+1
+1 −
+���D−1
+Q EQ
+���
+⩽
+1
+|Kbase| wmin
+1
+1 −
+wmax
+|Kbase|wmin
+=
+1
+|Kbase| wmin − wmax
+.
+(21)
+Substituting (21) into (19), we can derive that γ ⩾ 1 +
+1
+ϑ log
+�
+1 −
+4|Kadd|2w2
+max
+|Kbase|wmin−wmax
+�
+.
+

0dFST4oBgHgl3EQfVTiq/content/tmp_files/2301.13777v1.pdf.txt

@@ -0,0 +1,1263 @@
+Computer Algebra in R Bridges a Gap Between Mathematics and Data
+in the Teaching of Statistics and Data Science
+Mikkel Meyer Andersen and Søren Højsgaard
+February 1, 2023
+Mikkel Meyer Andersen
+Department of Mathematical Sciences, Aalborg University, Denmark
+Skjernvej 4A
+9220 Aalborg Ø, Denmark
+ORCiD: 0000-0002-0234-0266
+mikl@ math. aau. dk
+Søren Højsgaard
+Department of Mathematical Sciences, Aalborg University, Denmark
+Skjernvej 4A
+9220 Aalborg Ø, Denmark
+ORCiD: 0000-0002-3269-9552
+sorenh@ math. aau. dk
+Abstract
+The capability of R to do symbolic mathematics is enhanced by the caracas package. This package
+uses the Python computer algebra library SymPy as a back-end but caracas is tightly integrated in
+the R environment, thereby enabling the R user with symbolic mathematics within R. We demonstrate
+how mathematics and statistics can benefit from bridging computer algebra and data via R. This is done
+thought a number of examples and we propose some topics for small student projects. The caracas
+package integrates well with e.g. Rmarkdown, and as such creation of scientific reports and teaching is
+supported.
+Introduction
+The caracas package [Andersen and Højsgaard, 2021] and the Ryacas package [Andersen and Højsgaard,
+2019] enhance the capability of R [R Core Team, 2023] to handle symbolic mathematics. In this paper
+we will illustrate the use of the caracas package in connection with teaching mathematics and statistics.
+Focus is on 1) treating statistical models symbolically, 2) on bridging the gap between symbolic mathe-
+matics and numerical computations and 3) on preparing teaching material in a reproducible framework
+(provided by, e.g. rmarkdown [Allaire et al., 2021, Xie et al., 2018, 2020]. The caracas package is avail-
+able from CRAN [R Core Team, 2023]. The open-source development version of caracas is available
+at https://github.com/r-cas/caracas and readers are recommended to study the online documenta-
+tion at https://r-cas.github.io/caracas/. The caracas package provides an interface from R to the
+Python package sympy [Meurer et al., 2017]. This means that SymPy is “running under the hood” of R
+via the reticulate package [Ushey et al., 2020]. The sympy package is mature and robust with many
+users and developers.
+Neither caracas nor Ryacas are as powerful as some of the larger commercial computer algebra
+systems (CAS). The virtue of caracas and Ryacas lie elsewhere: (1) Mathematical tools like equation
+solving, summation, limits, symbolic linear algebra, outputting in tex format etc. are directly available
+from within R. (2) The packages enable working with the same language and in the same environment
+as the user does for statistical analyses. (3) Symbolic mathematics can easily be combined with data
+which is helpful in e.g. numerical optimization. (4) The packages are open-source and therefore support
+1
+arXiv:2301.13777v1  [stat.AP]  31 Jan 2023
+
+e.g. education - also for people with limited economical means and thus contributing to United Nations
+sustainable development goals [United Nations General Assembly, 2015].
+The paper is organized in the following sections: The section Mathematics and documents containing
+mathematics briefly introduces the caracas package and its syntax, including how caracas can be used in
+connection with preparing texts, e.g. teaching material. More details are provided in the Section Important
+technical aspects. Several vignettes illustrating caracas are provided and they are also available online,
+see https://r-cas.github.io/caracas/. The section Statistics examples is the main section of the
+paper and here we present a sample of statistical models where we believe that a symbolic treatment is
+a valuable supplement to a numerical in connection with teaching. The section Possible topics to study
+contains suggestions about hand-on activities for students. Lastly, the section Discussion and future work
+contains a discussion of the paper.
+Mathematics and documents containing mathematics
+We start by introducing the caracas syntax on familiar topics within calculus and linear algebra.
+Calculus
+First we define a caracas symbol x (more details will follow in Section Important technical aspects) and
+subsequently a caracas polynomial p in x (p becomes a symbol because x is):
+R> library(caracas)
+R> def_sym(x) ## Declares ’x’ as a symbol
+R> p <- 1 - x^2 + x^3 + x^4/4 - 3 * x^5 / 5 + x^6 / 6
+R> p
+#> [c]:
+6
+5
+4
+#>
+x
+3*x
+x
+3
+2
+#>
+-- - ---- + -- + x
+- x
++ 1
+#>
+6
+5
+4
+The gradient of p is:
+R> grad <- der(p, x) ## ’der’ is shorthand for derivative
+R> grad
+#> [c]:
+5
+4
+3
+2
+#>
+x
+- 3*x
++ x
++ 3*x
+- 2*x
+Stationary points of p can be found by finding roots of the gradient. In this simple case we can factor
+the gradient:
+R> factor_(grad)
+#> [c]:
+2
+#>
+x*(x - 2)*(x - 1) *(x + 1)
+The factorizations shows that stationary points are −1, 0, 1 and 2. To investigate if extreme points
+are local minima, local maxima or saddle points, we compute the Hessian and evaluate the Hessian in the
+stationary points:
+R> hess <- der2(p, x)
+R> hess
+#> [c]:
+4
+3
+2
+#>
+5*x
+- 12*x
++ 3*x
++ 6*x - 2
+R> hess_ <- as_func(hess)
+R> hess_
+2
+
+#> function (x)
+#> {
+#>
+5 * x^4 - 12 * x^3 + 3 * x^2 + 6 * x - 2
+#> }
+#> <environment: 0x55e4f8ea8890>
+R> stationary_points <- c(-1, 0, 1, 2)
+R> hess_(stationary_points)
+#> [1] 12 -2
+0
+6
+Alternatively, we can create an R expression and evaluate:
+R> eval(as_expr(hess), list(x = stationary_points))
+#> [1] 12 -2
+0
+6
+The sign of the Hessian in these points gives that x = −1 and x = 12 are local minima, x = 0 is a local
+maximum and x = 1 is a saddle point. In general we can find the stationary symbolically and evaluate
+the Hessian as follows (output omitted):
+R> sol <- solve_sys(lhs = grad, vars = x) ## finds roots by default
+R> subs(hess, sol[[1]]) ## the first solution
+R> lapply(sol, function(s) subs(hess, s)) ## iterate over all solutions
+Linear algebra
+Next, we create a symbolic matrix and find its inverse:
+R> M <- as_sym(toeplitz(c("a", "b", 0))) ## as_sym() converts an R object to caracas symbol
+R> Minv <- inv(M) %>% simplify()
+Default printing of M is (Minv is shown below in next section):
+R> M
+#> [c]: [a
+b
+0]
+#>
+[
+]
+#>
+[b
+a
+b]
+#>
+[
+]
+#>
+[0
+b
+a]
+A vector is a one-column matrix, but it is printed as its transpose to save space:
+R> v <- vector_sym(3, "v")
+R> v
+#> [c]: [v1
+v2
+v3]^T
+Matrix products are computed using the %*% operator:
+R> M %*% v
+#> [c]: [a*v1 + b*v2
+a*v2 + b*v1 + b*v3
+a*v3 + b*v2]^T
+3
+
+Preparing mathematical documents
+The packages Sweave [Leisch, 2002] and Rmarkdown [Allaire et al., 2021] provide integration of LaTeX
+and other text formatting systems into R helping to produce text document with R content. In a similar
+vein, caracas provides an integration of computer algebra into R and in addition, caracas also facilitates
+creation of documents with mathematical content without e.g. typing tedious LaTeX instructions.
+A LaTeX rendering of the caracas symbol p is obtained by typing $$p(x) = `r tex(p)`$$ which
+results in the following when the document is compiled:
+p(x) = x6
+6 − 3x5
+5
++ x4
+4 + x3 − x2 + 1
+Typing $$Mˆ{-1} = `r tex(Minv)`$$ produces the result:
+M −1 =
+�
+��
+a2−b2
+a(a2−2b2)
+−
+b
+a2−2b2
+b2
+a(a2−2b2)
+−
+b
+a2−2b2
+a
+a2−2b2
+−
+b
+a2−2b2
+b2
+a(a2−2b2)
+−
+b
+a2−2b2
+a2−b2
+a(a2−2b2)
+�
+�� .
+The determinant of M is det(M) = a3 −2∗a∗b2 and this can be factored out of the matrix by dividing
+each entry with the determinant and multiplying the new matrix by the determinant which simplifies the
+appearance of the matrix:
+R> Minv_fact <- as_factor_list(1 / factor_(det(M)), simplify(Minv * det(M)))
+Typing $$Mˆ{-1} = `r tex(Minv_fact)`$$ produces this:
+M −1 =
+1
+a (a2 − 2b2)
+�
+�
+a2 − b2
+−ab
+b2
+−ab
+a2
+−ab
+b2
+−ab
+a2 − b2
+�
+� .
+Finally we illustrate creation of additional mathematical expressions:
+R> def_sym(x, n)
+R> y <- (1 + x/n)^n
+R> lim(y, n, Inf)
+#> [c]: exp(x)
+Typing $$y = `r tex(y)`$$ etc. gives
+y =
+�
+1 + x
+n
+�n
+, lim
+n−>∞ y = exp(x).
+We can also prepare unevaluated expressions using the doit argument. That helps making repro-
+ducible documents where the changes in code appears automatically in the generated formulas. This is
+done as follows:
+R> l <- lim(y, n, Inf, doit = FALSE)
+R> l
+#> [c]:
+n
+#>
+/
+x\
+#>
+lim |1 + -|
+#>
+n->oo\
+n/
+R> doit(l)
+#> [c]: exp(x)
+Typing $$`r tex(l)` = `r tex(doit(l))`$$ gives
+lim
+n→∞
+�
+1 + x
+n
+�n
+= ex.
+Several functions have the doit argument, e.g. lim(), int() and sum_().
+4
+
+Important technical aspects
+A caracas symbol is a list with a pyobj slot and the class caracas_symbol. The pyobj is an an object
+in Python (often a sympy object). As such, a symbol (in R) provides a handle to a Python object. In
+the design of caracas we have tried to make this distinction something the user should not be concerned
+with, but it is worthwhile being aware of the distinction.
+Sections Calculus and Linear algebra illustrate that caracas symbols can be created with def_sym()
+and as_sym(). Both declares the symbol in R and in Python. A symbol can also be defined in terms of
+other symbols: Define symbols s1 and s2 and define symbol s3 in terms of s1 and s2:
+R> def_sym(s1, s2) ## Note: ’s1’ and ’s2’ exist in both R and Python
+R> s1$pyobj
+#> s1
+R> s3_ <- s1 * s2
+## Note: ’s3’ is a symbol in R; no corresponding object in Python
+R> s3_$pyobj
+#> s1*s2
+The underscore in s3_ indicates that this expression is defined in terms of other symbols.
+This
+convention is used through out the paper. Next express s1 and s2 in terms of symbols u and v (which
+are created on the fly):
+R> s4_ <- subs(s3_, c("s1", "s2"), c("u+v", "u-v"))
+R> s4_
+#> [c]: (u - v)*(u + v)
+Statistics examples
+In this section we examine larger statistical examples and demonstrate how caracas can help improve
+understanding of the models.
+Linear models
+A matrix algebra approach to e.g. linear models is very clear and concise. On the other hand, it can also
+be argued that matrix algebra obscures what is being computed. Numerical examples are useful for some
+aspects of the computations but not for other. In this respect symbolic computations can be enlightening.
+Consider a two-way analysis of variance (ANOVA) with one observation per group, see Table 1.
+Table 1: Two-by-two layout of data.
+y11
+y21
+y12
+y22
+R> nr <- 2
+R> nc <- 2
+R> y <- matrix_sym(nr, nc, "y")
+R> dim(y) <- c(nr*nc, 1)
+R> y
+#> [c]: [y11
+y21
+y12
+y22]^T
+R> dat <- expand.grid(r=factor(1:nr), s=factor(1:nc))
+R> X <- model.matrix(~r+s, data=dat) |> as_sym()
+R> b <- vector_sym(ncol(X), "b")
+R> mu <- X %*% b
+5
+
+For the specific model we have random variables y = (yij). All yijs are assumed independent and
+yij ∼ N(µij, v). The corresponding mean vector µ has the form given below:
+y =
+�
+���
+y11
+y21
+y12
+y22
+�
+��� ,
+X =
+�
+���
+1
+.
+.
+1
+1
+.
+1
+.
+1
+1
+1
+1
+�
+��� ,
+b =
+�
+�
+b1
+b2
+b3
+�
+� ,
+µ = Xb =
+�
+���
+b1
+b1 + b2
+b1 + b3
+b1 + b2 + b3
+�
+��� .
+Above and elsewhere, dots represent zero. The least squares estimate of b is the vector ˆb that minimizes
+||y − Xb||2 which leads to the normal equations (X⊤X)b = X⊤y to be solved. If X has full rank, the
+unique solution to the normal equations is ˆb = (X⊤X)−1X⊤y.
+Hence the estimated mean vector is
+ˆµ = Xˆb = X(X⊤X)−1X⊤y. Symbolic computations are not needed for quantities involving only the
+model matrix X, but when it comes to computations involving y, a symbolic treatment of y is useful:
+R> XtX <- t(X) %*% X
+R> XtXinv <- inv(XtX)
+R> Xty <- t(X) %*% y
+R> b_hat <- XtXinv %*% Xty
+X⊤y =
+�
+�
+y11 + y12 + y21 + y22
+y21 + y22
+y12 + y22
+�
+� ,
+ˆb = 1
+4
+�
+�
+3y11 + y12 + y21 − y22
+−2y11 − 2y12 + 2y21 + 2y22
+−2y11 + 2y12 − 2y21 + 2y22
+�
+�
+(1)
+Hence X⊤y (a sufficient reduction of data if the variance is known) consists of the sum of all observa-
+tions, the sum of observations in the second row and the sum of observations in the second column. For
+ˆb, the second component is, apart from a scaling, the sum of the second row minus the sum of the first
+row. Likewise, the third component is the sum of the second column minus the sum of the first column.
+It is hard to give an interpretation of the first component of ˆb.
+Logistic regression
+In the following we go through details of a logistic regression model, see e.g. McCullagh and Nelder [1989]
+for a classical description of logistic regression: Observables are binomially distributed, yi ∼ bin(pi, ni).
+The probability pi is connected to a q-vector of covariates xi = (xi1, . . . , xiq) and a q-vector of regression
+coefficients b = (b1, . . . , bq) as follows: The term si = xi ·b is denoted the linear predictor. The probability
+pi can be linked to si in different ways, but the most commonly employed is via the logit link function
+which is logit(pi) = log(pi/(1 − pi)) so here logit(pi) = si.
+As an example, consider the budworm data from the doBy package [Højsgaard and Halekoh, 2023].
+The data shows the number of killed moth tobacco budworm Heliothis virescens. Batches of 20 moths of
+each sex were exposed for three days to the pyrethroid and the number in each batch that were dead or
+knocked down was recorded:
+R> data(budworm, package = "doBy")
+R> bud <- subset(budworm, sex == "male")
+R> bud
+#>
+sex dose ndead ntotal
+#> 1 male
+1
+1
+20
+#> 2 male
+2
+4
+20
+#> 3 male
+4
+9
+20
+#> 4 male
+8
+13
+20
+#> 5 male
+16
+18
+20
+#> 6 male
+32
+20
+20
+Below we focus only on male budworms and the mortality is illustrated in Figure 1 (produced with
+ggplot2 [Wickham, 2016]). On the y-axis we have the empirical logits, i.e. log((ndead + 0.5)/(ntotal −
+ndead + 0.5)). The figure suggests that logit grows linearly with log dose.
+6
+
+−2
+0
+2
+4
+0
+10
+20
+30
+dose
+Empirical logits
+−2
+0
+2
+4
+0
+1
+2
+3
+4
+5
+log2(dose)
+Empirical logits
+Figure 1: Insecticide mortality of the moth tobacco budworm.
+Each component of the likelihood
+The log-likelihood is log L = �
+i yi log(pi)+(ni−yi) log(1−pi) = �
+i log Li, say. With log(pi/(1−pi)) = si
+we have pi = 1/(1 + exp(−si)) and
+d
+dsi pi =
+exp(−si)
+(1+exp(−si))2 . With si = xi · b, we have
+d
+dbsi = xi.
+Consider the contribution to the total log-likelihood from the ith observation which is li = yi log(pi)+
+(ni − yi) log(1 − pi). Since we are focusing on one observation only, we shall ignore the subscript i in this
+section. First notice that with s = log(p/(1 − p)) we can find p as:
+R> def_sym(s, p)
+R> sol_ <- solve_sys(lhs = log(p / (1 - p)), rhs = s, vars = p)
+R> sol_[[1]]$p
+#> [c]:
+exp(s)
+#>
+----------
+#>
+exp(s) + 1
+Next, find the likelihood as a function of p, as a function of s and as a function of b. The underscore
+in logLb_ and elsewhere indicates that this expression is defined in terms of other symbols (this is in
+contrast to the free variables, e.g. y, p, and n.):
+R> def_sym(y, n, p, x, s, b)
+R> logLp_ <- y * log(p) + (n - y) * log(1 - p)
+R> p_ <- exp(s) / (exp(s) + 1)
+R> logLs_ <- subs(logLp_, p, p_)
+R> s_ <- sum(x * b)
+R> logLb_ <- subs(logLs_, s, s_)
+R> logLb_
+#> [c]:
+/
+exp(b*x)
+\
+/
+exp(b*x)
+\
+#>
+y*log|------------| + (n - y)*log|1 - ------------|
+#>
+\exp(b*x) + 1/
+\
+exp(b*x) + 1/
+The log-likelihood can be maximized using e.g. Newton-Rapson (see e.g. Nocedal and Wright [2006])
+and in this connection we need the score function, S, and the Hessian, H:
+R> Sb_ <- score(logLb_, b) |> simplify()
+R> Hb_ <- hessian(logLb_, b) |> simplify()
+R> Sb_
+#> [c]: [x*(y - (n - y)*exp(b*x))]
+#>
+[------------------------]
+#>
+[
+exp(b*x) + 1
+]
+R> Hb_
+7
+
+#> [c]: [
+2
+]
+#>
+[
+-n*x *exp(b*x)
+]
+#>
+[---------------------------]
+#>
+[exp(2*b*x) + 2*exp(b*x) + 1]
+Since x and b are vectors, the term b*x above should be read as the inner product x · b (or as x⊤b in
+matrix notation). Also, since x is a vector, the term xˆ2 above should be read as the outer product x ⊗ x
+(or as xx⊤ in matrix notation). More insight in the structure is obtained by letting b and x be 2-vectors
+(to save space, the Hessian matrix is omitted in the following):
+R> b <- vector_sym(2, "b")
+R> x <- vector_sym(2, "x")
+R> s_ <- sum(x * b)
+R> logLb_ <- subs(logLs_, s, s_)
+R> Sb_ <- score(logLb_, b) |> simplify()
+logLb_ = y log
+�
+eb1x1+b2x2
+eb1x1+b2x2 + 1
+�
++ (n − y) log
+�
+1 −
+eb1x1+b2x2
+eb1x1+b2x2 + 1
+�
+,
+(2)
+Sb_ =
+�
+�
+x1(−neb1x1+b2x2+yeb1x1+b2x2+y)
+eb1x1+b2x2+1
+x2(−neb1x1+b2x2+yeb1x1+b2x2+y)
+eb1x1+b2x2+1
+�
+� .
+(3)
+Next, insert data, e.g. x1 = 1, x2 = 2, y = 9, n = 20 to obtain a function of the regression parameters
+only. Note how the expression depending on other symbols, S_, is named S. to indicate that data has
+been inserted:
+R> nms <- c("x1", "x2", "y", "n")
+R> vls <- c(1, 2, 9, 20)
+R> logLb. <- subs(logLb_, nms, vls)
+R> Sb. <- subs(Sb_, nms, vls)
+The total score for the entire dataset can be obtained as follows:
+R> Sb_list <- lapply(seq_len(nrow(bud)), function(r){
++
+vls <- c(1, log2(bud$dose[r]), bud$ndead[r], bud$ntotal[r])
++
+subs(Sb_, nms, vls)
++ })
+R> Sb_total <- Reduce(‘+‘, Sb_list)
+This score can be used as part of an iterative algorithm for solving the score equations. If one wants to
+use Newton-Rapson, the total Hessian matrix must also be created following lines similar to those above.
+It is straight forward implement a Newton-Rapson algorithm based on these quantities, one must only
+note the distinction between the two expressions below (and it is the latter one would use in an iterative
+algorithm):
+R> subs(Sb_total, b, c(1, 2))
+R> subs(Sb_total, b, c(1, 2)) |> as_expr()
+An alternative is to construct the total log-likelihood for the entire dataset as a caracas object, convert
+this object to an R function and maximize this function using one of R’s optimization methods:
+R> logLb_list <- lapply(seq_len(nrow(bud)), function(r){
++
+vls <- c(1, log2(bud$dose[r]), bud$ndead[r], bud$ntotal[r])
++
+subs(logLb_, nms, vls)
++ })
+R> logLb_total <- Reduce(‘+‘, logLb_list)
+R> logLb_total_func <- as_func(logLb_total, vec_arg = TRUE)
+8
+
+The total likelihood symbolically
+We conclude this section by illustrating that the log-likelihood for the entire dataset can be constructed
+in a few steps (output is omitted to save space):
+R> X. <- as_sym(cbind(1, log2(bud$dose)))
+R> n. <- as_sym(bud$ntotal)
+R> y. <- as_sym(bud$ndead)
+R> N <- nrow(X.)
+R> q <- ncol(X.)
+R> X <- matrix_sym(N, q, "x")
+R> n <- vector_sym(N, "n")
+R> y <- vector_sym(N, "y")
+R> p <- vector_sym(N, "p")
+R> s <- vector_sym(N, "s")
+R> b <- vector_sym(q, "b")
+X =
+�
+�������
+x11
+x12
+x21
+x22
+x31
+x32
+x41
+x42
+x51
+x52
+x61
+x62
+�
+�������
+,
+X. =
+�
+�������
+1
+0
+1
+1
+1
+2
+1
+3
+1
+4
+1
+5
+�
+�������
+,
+n. =
+�
+�������
+20
+20
+20
+20
+20
+20
+�
+�������
+,
+n =
+�
+�������
+n1
+n2
+n3
+n4
+n5
+n6
+�
+�������
+,
+y. =
+�
+�������
+1
+4
+9
+13
+18
+20
+�
+�������
+.
+The symbolic computations are as follows:
+R> ## log-likelihood as function of p
+R> logLp
+<- sum(y * log(p) + (n-y) * log(1-p))
+R> ## log-likelihood as function of s
+R> p_ <- exp(s) / (exp(s) + 1)
+R> logLs <- subs(logLp, p, p_)
+R> ## linear predictor as function of regression coefficients:
+R> s_
+<- X %*% b
+R> ## log-Likelihood as function of regression coefficients:
+R> logLb <- subs(logLs, s, s_)
+Next, numerical values can be inserted:
+R> logLb <- subs(logLb, cbind(n, y, X), cbind(n., y., X.))
+An alternative would have been to define logLp above in terms of n. and y. and similarly define
+s_ in terms of X. If doing so, the last step where numerical values are inserted could have been avoided.
+From here, one may proceed by computing the score function and the Hessian matrix and solve the
+score equation, using e.g. Newton-Rapson. Alternatively, one might create an R function based on the
+log-likelihood, and maximize this function using one of R’s optimization methods (see the example in the
+previous section):
+R> logLb_func <- as_func(logLb, vec_arg = TRUE)
+R> optim(c(0, 0), logLb_func, control = list(fnscale = -1), hessian = TRUE)
+Maximum likelihood under constraints
+In this section we illustrate constrained optimization using Lagrange multipliers. This is demonstrated
+for the independence model for a two-way contingency table. Consider a 2 × 2 contingency table with
+cell counts yij and cell probabilities pij for i = 1, 2 and j = 1, 2, where i refers to row and j to column as
+illustrated in Table 1.
+Under multinomial sampling, the log likelihood is
+l = log L =
+�
+ij
+yij log(pij).
+9
+
+Under the assumption of independence between rows and columns, the cell probabilities have the form,
+(see e.g. Højsgaard et al. [2012], p. 32)
+pij = u · ri · sj.
+To make the parameters (u, ri, sj) identifiable, constraints must be imposed. One possibility is to
+require that r1 = s1 = 1. The task is then to estimate u, r2, s2 by maximizing the log likelihood under
+the constraint that �
+ij pij = 1. This can be achieved using a Lagrange multiplier where we instead solve
+the unconstrained optimization problem maxp Lag(p) where
+Lag(p) = −l(p) + λg(p)
+under the constraint that
+(4)
+g(p) =
+�
+ij
+pij − 1 = 0,
+(5)
+where λ is a Lagrange multiplier. In SymPy, lambda is a reserved symbol. Hence the underscore as postfix
+below:
+R> y_ <- c("y_11", "y_21", "y_12", "y_22")
+R> y
+<- as_sym(y_)
+R> def_sym(u, r2, s2, lambda_)
+R> p <- as_sym(c("u", "u*r2", "u*s2", "u*r2*s2"))
+R> logL
+<- sum(y * log(p))
+R> Lag
+<- -logL + lambda_ * (sum(p) - 1)
+R> vars <- list(u, r2, s2, lambda_)
+R> gLag <- der(Lag, vars)
+R> sol <- solve_sys(gLag, vars)
+R> print(sol, method = "ascii")
+#> Solution 1:
+#>
+lambda_ =
+y_11 + y_12 + y_21 + y_22
+#>
+r2
+=
+(y_21 + y_22)/(y_11 + y_12)
+#>
+s2
+=
+(y_12 + y_22)/(y_11 + y_21)
+#>
+u
+=
+(y_11 + y_12)*(y_11 + y_21)/(y_11 + y_12 + y_21 + y_22)^2
+R> sol <- sol[[1]]
+There is only one critical point. Fitted cell probabilities ˆpij are:
+R> p11 <- sol$u
+R> p21 <- sol$u * sol$r2
+R> p12 <- sol$u * sol$s2
+R> p22 <- sol$u * sol$r2 * sol$s2
+R> p.hat <- matrix_(c(p11, p21, p12, p22), nrow = 2)
+ˆp =
+1
+(y11 + y12 + y21 + y22)2
+�
+(y11 + y12) (y11 + y21)
+(y11 + y12) (y12 + y22)
+(y11 + y21) (y21 + y22)
+(y12 + y22) (y21 + y22)
+�
+To verify that the maximum likelihood estimate has been found, we compute the Hessian matrix which
+is negative definite (the Hessian matrix is diagonal so the eigenvalues are the diagonal entries and these
+are all negative), output omitted:
+R> H <- hessian(logL, list(u, r2, s2)) |> simplify()
+An AR(1) model
+Symbolic computations
+In this section we study the auto regressive model of order 1 (an AR(1) model), see e.g. Shumway and
+Stoffer [2016], p. 75 ff. for details: Consider random variables x1, x2, . . . , xn following a stationary zero
+mean AR(1) process:
+xi = axi−1 + ei;
+i = 2, . . . , n,
+(6)
+10
+
+where ei ∼ N(0, v) and all eis are independent. Note that v denotes the variance. The marginal
+distribution of x1 is also assumed normal, and for the process to be stationary we must have that the
+variance Var(x1) = v/(1 − a2). Hence we can write x1 =
+1
+√
+1−a2 e1.
+For simplicity of exposition, we set n = 4. All terms e1, . . . , e4 are independent and N(0, v) distributed.
+Let e = (e1, . . . , e4) and x = (x1, . . . x4). Hence e ∼ N(0, vI). Isolating error terms in (6) gives
+e =
+�
+���
+e1
+e2
+e3
+e4
+�
+��� =
+�
+���
+√
+1 − a2
+.
+.
+.
+−a
+1
+.
+.
+.
+−a
+1
+.
+.
+.
+−a
+1
+�
+���
+�
+���
+x1
+x2
+x3
+x4
+�
+��� = Lx.
+Since Var(e) = vI we have Var(e) = vI = LVar(x)L′ so the covariance matrix of x is V = Var(x) =
+vL−(L−)⊤ while the concentration matrix (the inverse covariance matrix) is K = v−1L⊤L:
+R> n <- 4
+R> L <- diff_mat(n, "-a")
+R> def_sym(a)
+R> L[1, 1] <- sqrt(1-a^2)
+R> def_sym(v)
+R> Linv <- inv(L)
+R> K <- crossprod_(L) / v
+R> V <- tcrossprod_(Linv) * v
+L−1 =
+�
+����
+1
+√
+1−a2
+.
+.
+.
+a
+√
+1−a2
+1
+.
+.
+a2
+√
+1−a2
+a
+1
+.
+a3
+√
+1−a2
+a2
+a
+1
+�
+���� ,
+(7)
+K = 1
+v
+�
+���
+1
+−a
+.
+.
+−a
+a2 + 1
+−a
+.
+.
+−a
+a2 + 1
+−a
+.
+.
+−a
+1
+�
+��� ,
+(8)
+V = v
+�
+����
+1
+1−a2
+a
+1−a2
+a2
+1−a2
+a3
+1−a2
+a
+1−a2
+a2
+1−a2 + 1
+a3
+1−a2 + a
+a4
+1−a2 + a2
+a2
+1−a2
+a3
+1−a2 + a
+a4
+1−a2 + a2 + 1
+a5
+1−a2 + a3 + a
+a3
+1−a2
+a4
+1−a2 + a2
+a5
+1−a2 + a3 + a
+a6
+1−a2 + a4 + a2 + 1
+�
+���� .
+(9)
+The zeros in the concentration matrix K implies a conditional independence restriction: If the ijth
+element of a concentration matrix is zero then xi and xj are conditionally independent given all other
+variables, see e.g. Højsgaard et al. [2012], p. 84 for details.
+Next, we take the step from symbolic computations to numerical evaluations. The joint distribution
+of x is multivariate normal distribution, x ∼ N(0, K−1). Let W = xx⊤ denote the matrix of (cross)
+products. The log-likelihood is therefore (ignoring additive constants)
+log L = n
+2 (log det(K) − x⊤Kx) = n
+2 (log det(K) − tr(KW)),
+where we note that tr(KW) is the sum of the elementwise products of K and W since both matrices
+are symmetric. Ignoring the constant n
+2 , this can be written symbolically to obtain the expression in this
+particular case:
+R> x <- vector_sym(n, "x")
+R> logL <- log(det(K)) - sum(K * (x %*% t(x))) %>% simplify()
+log L = log
+�
+−a2
+v4 + 1
+v4
+�
+− −2ax1x2 − 2ax2x3 − 2ax3x4 + x2
+1 + x2
+2
+�
+a2 + 1
+�
++ x2
+3
+�
+a2 + 1
+�
++ x2
+4
+v
+.
+11
+
+Numerical evaluation
+Next we illustrate how bridge the gap from symbolic computations to numerical computations based on
+a dataset: For a specific data vector we get:
+R> xt <- c(0.1, -0.9, 0.4, .0)
+R> logL. <- subs(logL, x, xt)
+log L = log
+�
+−a2
+v4 + 1
+v4
+�
+− 0.97a2 + 0.9a + 0.98
+v
+.
+We can use R for numerical maximization of the likelihood and constraints on the parameter values
+can be imposed e.g. in the optim() function:
+R> logL_wrap <- as_func(logL., vec_arg = TRUE)
+R> eps <- 0.01
+R> par <- optim(c(a=0, v=1), logL_wrap,
++
+lower=c(-(1-eps), eps), upper=c((1-eps), 10),
++
+method="L-BFGS-B", control=list(fnscale=-1))$par
+R> par
+#>
+a
+v
+#> -0.376
+0.195
+The same model can be fitted e.g. using R’s arima() function as follows (output omitted):
+R> arima(xt, order = c(1, 0, 0), include.mean = FALSE, method = "ML")
+It is less trivial to do the optimization in caracas by solving the score equations. There are some
+possibilities for putting assumptions on variables in caracas (see the “Reference” vignette), but it is not
+possible to restrict the parameter a to only take values in (−1, 1).
+Variance of the average of correlated data
+Consider random variables x1, . . . , xn where Var(xi) = v and Cov(xi, xj) = vr for i ̸= j, where 0 ≤ |r| ≤
+1. For n = 3, the covariance matrix of (x1, . . . , xn) is therefore
+V = vR = v
+�
+�
+1
+r
+r
+r
+1
+r
+r
+r
+1
+�
+� .
+(10)
+Let ¯x = �
+i xi/n denote the average. Suppose interest is in the variance of the average, Var(¯x), when
+n goes to infinity. One approach is as follow: Let 1 denote an n-vector of 1’s and let V be an n × n
+matrix with v on the diagonal and vr outside the diagonal. Then Var(¯x) =
+1
+n2 1⊤V 1. The answer lies in
+studying the limiting behaviour of this expression when n → ∞. First, we must calculate variance of a
+sum x. = �
+i xi which is Var(x.) = �
+i Var(xi) + 2 �
+ij:i<j Cov(xi, xj) (i.e., the sum of the elements of
+the covariance matrix). We can do this in caracas as follows:
+R> def_sym(v, r, n, j, i)
+R> var_sum <- v*(n + 2*sum_(sum_(r, j, i+1, n), i, 1, n-1)) |> simplify()
+R> var_avg <- var_sum / n^2
+Var(x.) = nv (r (n − 1) + 1) ,
+Var(¯x) = v (r (n − 1) + 1)
+n
+.
+From hereof, we can study the limiting behavior of the variance Var(¯x) in different situations:
+R> l_1 <- lim(var_avg, n, Inf)
+## when sample size n goes to infinity
+R> l_2 <- lim(var_avg, r, 0, dir=’+’)
+## when correlation r goes to zero
+R> l_3 <- lim(var_avg, r, 1, dir=’-’)
+## when correlation r goes to one
+12
+
+For a given correlation r it is instructive to investigate how many independent variables k the n
+correlated variables correspond to (in the sense of the same variance of the average), because the k can
+be seen as a measure of the amount of information in data. Moreover, one might study how k behaves
+as function of n when n → ∞. That is we must (1) solve v(1 + (n − 1)r)/n = v/k for k and (2) find
+limn→∞ k:
+R> def_sym(k)
+R> k <- solve_sys(var_avg - v / k, k)[[1]]$k
+R> l_k <- lim(k, n, Inf)
+The findings above are:
+l1 = rv,
+l2 = v
+n,
+l3 = v,
+k =
+n
+nr − r + 1,
+lk = 1
+r .
+With respect to k, it is illustrative to supplement the symbolic computations above with numerical
+evaluations, which shows that even a moderate correlation reduces the effective sample size substantially:
+R> dat <- expand.grid(r=c(.1, .2, .5), n=c(10, 50))
+R> k_fun <- as_func(k)
+R> dat$k <- k_fun(r=dat$r, n=dat$n)
+R> dat$ri <- 1/dat$r
+R> dat
+#>
+r
+n
+k ri
+#> 1 0.1 10 5.26 10
+#> 2 0.2 10 3.57
+5
+#> 3 0.5 10 1.82
+2
+#> 4 0.1 50 8.47 10
+#> 5 0.2 50 4.63
+5
+#> 6 0.5 50 1.96
+2
+Possible topics to study
+1. Related to Section Linear models:
+a) The orthogonal projection matrix onto the span of the model matrix X is P = X(X⊤X)−1X⊤.
+The residuals are r = (I − P)y. From this one may verify that these are not all independent.
+b) If one of the factors is ignored, then the model becomes a one-way analysis of variance model,
+at it is illustrative to redo the computations in Section Linear models in this setting.
+c) Likewise if an interaction between the two factors is included in the model.
+What are the
+residuals in this case?
+2. Related to Section Logistic regression:
+a) In Each component of the likelihood, Newton-Rapson can be implemented to solve the likelihood
+equations and compared to the output from glm().
+Note how sensitive Newton-Rapson is
+to starting point.
+This can be solved by another optimisation scheme, e.g.
+Nelder-Mead
+(optimising the log likelihood) or BFGS (finding extreme for the score function).
+b) The example is done as logistic regression with the logit link function. Try other link functions
+such as cloglog (complementary log-log).
+3. Related to Section Maximum likelihood under constraints:
+a) Identifiability of the parameters was handled by not including r1 and s1 in the specification
+of pij. An alternative is to impose the restrictions r1 = 1 and s1 = 1, and this can also be
+handled via Lagrange multipliers. Another alternative is to regard the model as a log-linear
+model where log pij = log u + log ri + log sj = ˜u + ˜ri + ˜sj. This model is similar in its structure
+to the two-way ANOVA for Section Linear models. This model can be fitted as a generalized
+linear model with a Poisson likelihood and log as link function. Hence, one may modify the
+results in Section Logistic regression to provide an alternative way of fitting the model.
+b) A simpler task is to consider a multinomial distribution with four categories, counts yi and cell
+probabilities pi, i = 1, 2, 3, 4 where �
+i pi = 1. For this model, find the maximum likelihood
+estimate for pi (use the Hessian to verify that the critical point is a maximum).
+13
+
+4. Related to Section An AR(1) model:
+a) Compare the estimated parameter values with those obtained from the arima() function.
+b) Modify the model in Equation (6) by setting x1 = axn + e1 (“wrapping around”) and see what
+happens to the pattern of zeros in the concentration matrix.
+c) Extend the AR(1) model to an AR(2) model (“wrapping around”) and investigate this model
+along the same lines. Specifically, where are the conditional independencies (try at least n = 6)?
+5. Related to Section Variance of the average of correlated data: It is illustrative to study such be-
+haviours for other covariance functions. Replicate the calculations for the covariance matrix of the
+form
+V = vR = v
+�
+���
+1
+r
+0
+0
+r
+1
+r
+0
+0
+r
+1
+r
+0
+0
+r
+1
+�
+��� ,
+(11)
+i.e., a special case of a Toeplitz matrix. How many independent variables, k, do the n correlated
+variables correspond to?
+Discussion and future work
+We have presented the caracas package and argued that the package extends the functionality of R sig-
+nificantly with respect to symbolic mathematics. One practical virtue of caracas is that the package
+integrates nicely with Rmarkdown, Allaire et al. [2021], (e.g. with the tex() functionality) and thus sup-
+ports creating of scientific documents and teaching material. As for the usability in practice we await
+feedback from users.
+Another related package we mentioned in the introduction is Ryacas. This package has existed for
+many years and is still of relevance. Ryacas probably has fewer features than caracas. On the other hand,
+Ryacas does not require Python (it is compiled), is faster for some computations (like matrix inversion).
+Finally, the Yacas language [Pinkus and Winitzki, 2002, Pinkus et al., 2016] is extendable (see e.g. the
+vignette “User-defined yacas rules” in the Ryacas package).
+One possible future development could be an R package which is designed without a view towards the
+underlying engine (SymPy or Yacas) and which then draws more freely from SymPy and Yacas. In this
+connection we mention that there are additional resources on CRAN such as calculus [Guidotti, 2022].
+Lastly, with respect to freely available resources in a CAS context, we would like to draw attention
+to WolframAlpha, see https://www.wolframalpha.com/, which provides an online service for answering
+(mathematical) queries.
+Acknowledgements
+We would like to thank the R Consortium for financial support for creating the caracas package, users
+for pin pointing points that can be improved in caracas and Ege Rubak (Aalborg University, Denmark),
+Malte Bødkergaard Nielsen (Aalborg University, Denmark), and Poul Svante Eriksen (Aalborg University,
+Denmark) for comments on this manuscript.
+References
+J. Allaire, Y. Xie, J. McPherson, J. Luraschi, K. Ushey, A. Atkins, H. Wickham, J. Cheng, W. Chang,
+and R. Iannone. rmarkdown: Dynamic Documents for R, 2021. URL https://github.com/rstudio/
+rmarkdown. R package version 2.7. 1, 4, 14
+M. M. Andersen and S. Højsgaard. Ryacas: A computer algebra system in R. Journal of Open Source
+Software, 4(42), 2019. URL https://doi.org/10.21105/joss.01763. 1
+M. M. Andersen and S. Højsgaard. caracas: Computer algebra in R. Journal of Open Source Software, 6
+(63):3438, 2021. doi: 10.21105/joss.03438. URL https://doi.org/10.21105/joss.03438. 1
+E. Guidotti. calculus: High-Dimensional Numerical and Symbolic Calculus in R. Journal of Statistical
+Software, 104(1):1–37, 2022. doi: 10.18637/jss.v104.i05. URL https://www.jstatsoft.org/index.
+php/jss/article/view/v104i05. 14
+14
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+S. Højsgaard, D. Edwards, and S. Lauritzen. Graphical Models with R. Springer, New York, 2012. doi:
+10.1007/978-1-4614-2299-0. ISBN 978-1-4614-2298-3. 10, 11
+S. Højsgaard and U. Halekoh. doBy: Groupwise Statistics, LSmeans, Linear Estimates, Utilities, 2023.
+URL https://github.com/hojsgaard/doby. R package version 4.6.16. 6
+F. Leisch. Sweave: Dynamic generation of statistical reports using literate data analysis. In W. Härdle
+and B. Rönz, editors, Compstat, pages 575–580, Heidelberg, 2002. Physica-Verlag HD. 4
+P. McCullagh and J. A. Nelder. Generalized Linear Models. Chapman & Hall/CRC Monographs on
+Statistics and Applied Probability. Chapman & Hall/CRC, Philadelphia, PA, 2 edition, Aug. 1989.
+ISBN 9780412317606. 6
+A. Meurer, C. P. Smith, M. Paprocki, O. Čertík, S. B. Kirpichev, M. Rocklin, A. Kumar, S. Ivanov, J. K.
+Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller, F. Bonazzi, H. Gupta, S. Vats,
+F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel, v. Roučka, A. Saboo, I. Fernando, S. Kulal,
+R. Cimrman, and A. Scopatz. Sympy: symbolic computing in python. PeerJ Computer Science, 3:
+e103, Jan. 2017. ISSN 2376-5992. doi: 10.7717/peerj-cs.103. URL https://doi.org/10.7717/peerj-
+cs.103. 1
+J. Nocedal and S. J. Wright. [Numerical Optimization. Springer New York, 2006. doi: 10.1007/978-0-
+387-40065-5. URL https://doi.org/10.1007/978-0-387-40065-5. 7
+A. Pinkus, S. Winnitzky, and G. Mazur. Yacas - yet another computer algebra system. Technical report,
+2016. URL https://yacas.readthedocs.io/en/latest/. 14
+A. Z. Pinkus and S. Winitzki. YACAS: A Do-It-Yourself Symbolic Algebra Environment. In Proceedings
+of the Joint International Conferences on Artificial Intelligence, Automated Reasoning, and Symbolic
+Computation, AISC ’02/Calculemus ’02, pages 332–336, London, UK, UK, 2002. Springer-Verlag. ISBN
+3-540-43865-3. doi: 10.1007/3-540-45470-5_29. URL http://doi.org/10.1007/3-540-45470-5_29.
+14
+R Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical
+Computing, Vienna, Austria, 2023. URL http://www.R-project.org/. ISBN 3-900051-07-0. 1
+R. H. Shumway and D. S. Stoffer. Time Series Analysis and Its Applications. Springer, fourth edition
+edition, 2016. 10
+United Nations General Assembly. Sustainable development goals, 2015. https://sdgs.un.org/. 2
+K. Ushey, J. Allaire, and Y. Tang.
+reticulate: Interface to ’Python’, 2020.
+URL https://CRAN.R-
+project.org/package=reticulate. R package version 1.18. 1
+H. Wickham.
+ggplot2: Elegant Graphics for Data Analysis.
+Springer-Verlag New York, 2016.
+ISBN
+978-3-319-24277-4. URL https://ggplot2.tidyverse.org. 6
+Y. Xie, J. Allaire, and G. Grolemund. R Markdown: The Definitive Guide. Chapman and Hall/CRC,
+Boca Raton, Florida, 2018. URL https://bookdown.org/yihui/rmarkdown. ISBN 9781138359338. 1
+Y. Xie, C. Dervieux, and E. Riederer. R Markdown Cookbook. Chapman and Hall/CRC, Boca Raton,
+Florida, 2020. URL https://bookdown.org/yihui/rmarkdown-cookbook. ISBN 9780367563837. 1
+15
+

19E0T4oBgHgl3EQf_wLw/content/tmp_files/2301.02832v1.pdf.txt

@@ -0,0 +1,503 @@
+Prepared for submission to JINST
+Proc. of the 23rd International Workshop on Radiation Imaging Detectors
+J-PET detection modules based on plastic scintillators for
+performing studies with positron and positronium beams
+S. Sharma,1,2,3,∗ J. Baran,1,2,3 R.S. Brusa,4,5 R. Caravita,5 N. Chug,1,2,3 A. Coussat,1,2,3 C.
+Curceanu,6 E. Czerwiński,1,2,3 M. Dadgar,1,2,3 K. Dulski,1,2,3 K. Eliyan,1,2,3 A. Gajos,1,2,3 B.C.
+Hiesmayr,7 K. Kacprzak,1,2,3 Ł. Kapłon,1,2,3 K. Klimaszewski,8 P. Konieczka,9 G. Korcyl,1,2 T.
+Kozik,1 W. Krzemień,9 D. Kumar,1,2,3 S. Mariazzi,4,5 S. Niedźwiecki,1,2,3 L. Panasa,4,5 S.
+Parzych,1,2,3 L. Povolo,4,5 E. Perez del Rio,1,2,3 L. Raczyński,9 Shivani,1,2,3 R.Y. Shopa,9 M.
+Skurzok,1,2,3 E.Ł. Stępień,1,2,3 F. Tayefi,1,2,3 K. Tayefi,1,2,3 W. Wiślicki,9 P. Moskal,1,2,3
+1Faculty of Physics, Astronomy and Applied Computer Science, Jagiellonian University, Krakow, Poland
+2Total-Body Jagiellonian-PET Laboratory, Jagiellonian University, Kraków, Poland
+3Center for Theranostics, Jagiellonian University, Cracow, Poland
+4Department of Physics, University of Trento, via Sommarive 14, 38123 Povo,Trento, Italy
+5TIFPA/INFN, via Sommarive 14, 38123 Povo,Trento, Trento, Italy
+6INFN, Laboratori Nazionali di Frascati, Frascati, Italy
+7Faculty of Physics, University of Vienna, Vienna, Austria
+8Department of Complex Systems, National Centre for Nuclear Research, Otwock-Świerk, Poland
+9High Energy Physics Division, National Centre for Nuclear Research, Otwock-Świerk, Poland
+E-mail: [email protected]
+Abstract: The J-PET detector, which consists of inexpensive plastic scintillators, has demon-
+strated its potential in the study of fundamental physics. In recent years, a prototype with 192
+plastic scintillators arranged in 3 layers has been optimized for the study of positronium decays.
+This allows performing precision tests of discrete symmetries (C, P, T) in the decays of positronium
+atoms. Moreover, thanks to the possibility of measuring the polarization direction of the photon
+based on Compton scattering, the predicted entanglement between the linear polarization of anni-
+hilation photons in positronium decays can also be studied. Recently, a new J-PET prototype was
+commissioned, based on a modular design of detection units. Each module consists of 13 plastic
+scintillators and can be used as a stand-alone, compact and portable detection unit. In this paper,
+the main features of the J-PET detector, the modular prototype and their applications for possible
+studies with positron and positronium beams are discussed. Preliminary results of the first test
+experiment performed on two detection units in the continuous positron beam recently developed
+at the Antimatter Laboratory (AML) of Trento are also reported.
+Keywords: J-PET, modular J-PET, positron and positronium beam, entanglement, inertial sensing
+1Corresponding author.
+arXiv:2301.02832v1  [physics.ins-det]  7 Jan 2023
+
+Contents
+1
+Introduction
+1
+2
+J-PET tomograph as multi-photon detector
+2
+2.1
+Fundamental studies in decays of Ps using J-PET
+2
+3
+Modular J-PET and its possible applications with positron and positronium beams
+4
+4
+Performance study of two J-PET detection units with positron beam
+5
+5
+Summary and perspectives
+6
+1
+Introduction
+Jagiellonian Positron Emission Tomograph (J-PET) is the first tomograph based on the idea of
+using plastic scintillators instead of crystals as currently used in commercial tomographs [1, 2].
+Plastic scintillators have excellent time resolution and are therefore good candidates for building
+TOF-PET tomographs [3]. The novelty that distinguishes J-PET from other tomographs is its
+potential to conduct studies of fundamental physics problems [4] and positronium imaging [5–7].
+Therefore, it can be used in a dual role, both as a PET scanner and as a multimodule detector. The
+J-PET detector is optimised for studying the decay of positronium atoms (Ps, the bound state of
+electron (𝑒−) and positron (𝑒+)) [8, 9]. In recent years, data have been collected with the 3-layer
+prototype of J-PET [10], which consists of 192 detection modules, demonstrating its applicability
+not only in medical physics [6, 11, 12] but also in the study of fundamental physics [13]. Following
+the J-PET technology, a new prototype was recently built based on a modular design consisting
+of 24 individual detection units [14]. Thanks to the modular design, the detection units can be
+conveniently transported to other research facilities to perform experiments. At the University
+of Trento, a continuous positron beam has been put into operation in the Antimatter Laboratory
+(AML). With the know-how to fabricate efficient positron/positronium converters [15, 16] and to
+manipulate positronium atoms into a metastable state with increased lifetime [17], the generation
+of Ps beams is envisaged [18]. Two detection modules with a supported data acquisition chain
+have recently been moved to the AML to perform studies with positrons and positronium beams to
+investigate fundamental physics not yet explored. The outline of the draft is divided as follows. In
+the next section, an introduction to the 3-layer protoype of J-PET is given, followed by the studies
+that are currently being performed. Then, the modular detection units are briefly described and their
+possible applications with 𝑒+ and Ps beams are discussed. Finally, the preliminary results of the
+first experiment using a continuous positron beam to reconstruct the beam spot with two detection
+modules are reported.
+– 1 –
+
+2
+J-PET tomograph as multi-photon detector
+The 3-layer prototype is developed to test the concept of constructing a cost-effective total-body
+PET from plastic scintillators [10, 14, 19]. To obtain a longer axial field of view (AFOV), strips of
+500×19×7 mm3 plastic scintillators are used. In the design of J-PET, 192 plastic scintillators (EJ-
+230) are arranged in 3 concentric cylinders with radii 42.5 cm, 46.75 cm, and 57.5 cm, respectively.
+Signals from each scintillator are read out with an R9800 Hamamatsu photomultiplier on each
+side. Data are measured and stored in triggerless mode, which can handle data streams of up to 8
+Gbps [20, 21]. To take advantage of the excellent time resolution, Time Over Threshold (TOT) is
+measured instead of the charge collection. The energy deposition in a given interaction of photons
+within the scintillator is estimated based on the established relationship between TOT and the energy
+deposition [22]. A special framework based on advanced C++ routines and ROOT (Data Analysis
+Framework from CERN) was also developed to analyse the measured data [23]. Hit position and
+hit time of a photon interaction (hit) within the scintillator are calculated based on the measured
+light signals read from both ends of a scintillator [24]. The hit time is calculated as the average of
+the times of the light signals arriving at both ends, while the hit position is calculated as the product
+of the difference in the arrival times of the light signal at both ends multiplied by half of their
+effective speed [25, 26]. Fig. 1 shows the pictures of the J-PET detector (left), the installation of the
+Figure 1. (A) Shows an image of J-PET in the laboratory. 3-Layers (blue, yellow, red) represent the angular
+arrangement of the scintillator strips and their cross-section in the plane. (B) shows the installation of hollow
+cylindrical chamber in the centre of J-PET, while (C) represents the small aluminium chamber.
+hollow cylindrical chamber (centre), and the small aluminium chamber (right). Before turning to
+the modular prototype of J-PET, the next section briefly discusses the physics aspects of the J-PET
+detector in studying the decays of positronium atoms (Ps) [4].
+2.1
+Fundamental studies in decays of Ps using J-PET
+Ps, being a pure leptonic system of electron and positron, is an excellent probe to test the bound state
+of quantum electrodynamics [27]. Ps can be formed in one of two ground states: Singlet state (1S0:
+para-positronium (p-Ps)) with lifetime 125 ps or in the triplet state (3S1: ortho-positronium(o-Ps))
+with lifetime 142 ns. Under the charge conjugation condition, p-Ps and o-Ps decay into an even
+– 2 –
+
+A
+Bnumber (2n𝛾; n=1,2,.), while o-Ps decays into an odd number ((2n+1)𝛾; n=1,2,.) of photons,
+predominantly 2𝛾 and 3𝛾, respectively. In studying the decays of Ps atoms, several fundamental
+problems can be investigated, e.g., quantum entanglement, violation of discrete symmetries, etc.
+We briefly discuss here the studies that are currently being carried out with the J-PET detector.
+Photon polarization and quantum entanglement: The measurement of entanglement in the
+annihilation photons emitted in Ps is one of the important aspects that can be studied with the J-PET
+detector. It is predicted that the linear polarizations of back-to-back photons emitted in the singlet
+state (p-Ps) of Ps are orthogonally correlated [28, 29]. Measurement of the correlation between
+the polarizations of annihilation photons can be used to observe the entangled state [30–33], a
+subject of fundamental importance that has direct application in medical imaging [34, 35]. There
+are no mechanical polarizers to measure the polarization of 511 keV photons. However, Compton
+scattering can be used as a polarizer for such measurements [36]. J-PET is capable of registering
+both annihilation photons before and after their scattering with angular resolution≈1𝑜 [37]. Based
+on the fact that in Compton scattering, a photon is scattered most likely at right angles to the
+direction of linear polarization of the incident photon, the polarization of the photon can be defined
+as �𝜖 = �𝑘 × �𝑘
+′ [4, 38].
+By measuring the polarization of each photon, one can measure the
+azimuthal correlation between their polarizations and test the theoretically predicted claims for
+entanglement [30–33]. Entanglement studies can be extended to the o-Ps →3𝛾 case [33].
+Symmetry violation - test of discrete symmetries in decays of Ps atoms: Ps exhibits interesting
+properties that make it an exotic atom for performing the tests on discrete symmetries. For example,
+it is an eigenstate of the parity operator (P) since it is bound by a central potential. Moreover, Ps is a
+system of a particle and its antiparticle that remains symmetric in their exchange and thus it is also an
+eigenstate of the charge conjugation operator C. Therefore, positronium is also an eigenstate of the
+CP operator. In conjunction with the CPT theorem, one can test the T violation effects separately or
+in combination for CPT test. Therefore, Ps can serve as an excellent laboratory to perform tests for
+C, P, CP and CPT violations. In 1988, studying the o-Ps→3𝛾 decays, Bernreuther and coworkers
+suggested that to test the discrete symmetries, odd-symmetry operators can be constructed using
+the spin of the o-Ps atom and momenta of annihilation photons [39]. Non vanishing expectation
+value of these operators will be the confirmation of the symmetry violation. Table 1 represents the
+list of operators for the the discrete symmetries test [4]. Minus sign represents the odd-symmetric
+operators which are sensitive to observe the violation effects for discrete symmetries. In addition,
+with the ability of J-PET detector to measure the polarization direction of photons [38], additional
+operators are also constructed utilizing the photon’s polarization direction which are unique and
+currently possible only with the J-PET [4].
+Table 1. Odd-symmetry operators constructed of the photons momenta (�𝑘𝑖) and polarization (�𝜖𝑖), and spin
+of o-Ps (�𝑆) [4]
+Odd symmetric
+C
+P
+T
+CP
+CPT
+Odd symmetric
+C
+P
+T
+CP
+CPT
+�𝑆 . �𝑘 1
++
+-
++
+-
+-
+�𝑘 1 . �𝜖 2
++
+-
+-
+-
++
+�𝑆 . (�𝑘 1 × �𝑘 2)
++
++
+-
++
+-
+�𝑆 . �𝜖 1
++
++
+-
++
+-
+( �𝑆 . �𝑘 1)( �𝑆 . (�𝑘 1 × �𝑘 2)
++
+-
+-
+-
++
+�𝑆 . (�𝑘 2 × �𝜖 1)
++
+-
++
+-
+-
+In table 1, one can see that none of the operators is sensitive to the C symmetry test. However,
+– 3 –
+
+tests for violation of C symmetry can be performed by examining C-prohibited Ps-decays (e.g. p-
+Ps→3𝛾, o-Ps→2𝛾, 4𝛾,.). In symmetry studies with J-PET, the estimate of the o-Ps spin is based on
+the intrinsic polarization of positrons emitted in 𝛽+ decays [13]. These are longitudinally polarized
+due to parity violation, which is proportional to the velocity of the positrons. Different types of
+chambers can be used depending on the specifics of the operators. For the operators involving the
+spin of o-Ps, large hollow chambers (see Fig. 1 (B)) are used, the inner wall of which are coated with
+porous materials to increase the probability of Ps formation. The annihilation points of the o-Ps in
+the chamber wall can be reconstructed using the trilateration method [40], which gives the direction
+of the positrons with respect to the known position of the source and thus allows the o-Ps spin to be
+estimated [4, 13]. Small chambers (Fig. 1 (C)) are used for the other operators, in particular with
+photon polarization.
+3
+Modular J-PET and its possible applications with positron and positronium beams
+The modular J-PET is a new prototype, which consists of 24 independent detection modules.
+Each module consists of 13 plastic scintillators of size 500×24×6 mm3, read out at each end by a
+SiPM matrix, together with their front-end electronics housed in a single module. Thanks to their
+modular design and FPGA-based compact data acquisition, they can be easily transported to be
+used as potential detectors in different laboratories. In this context, we have explored the possibility
+of using the modular detection units for studies with positron and positronium beams at the AML in
+Trento [41]. A continuous positron beam was recently commissioned at the AML. In the future, the
+continuous beam will be injected into a Surko trap [42] where positrons will be trapped, stored for
+fraction of seconds and then bunched to form pulses containing up to 104 positrons. Implantation
+of positron pulses in efficient positron/positronium converters [15, 43, 44] allows producing dense
+Ps clouds [45, 46]. In particular, in recent years the possibility to populate the long-lived 23S
+state of positronium via spontaneous [17] and stimulated [47] decay from the 33P level (previously
+reached via 13S→33P laser excitation) has been demonstrated [48]. A monochromatic pulsed 23S
+positronium beam with low angular divergence can then be produced by placing an iris diaphragm
+in front of the target [18]. By employing properly polarized laser pulses, the production of Ps
+in 23S with fully controlled quantum numbers looks feasible. In studying the annihilation of Ps
+in 3-photons, an interesting fundamental problem, the experimental measurement of the quantum
+entanglement of the polarization of the annihilation photons could be addressed for the first time.
+Theoretical studies predict that the entanglement type of the 3-photons depends on the quantum
+numbers of the annihilating positronium [33]. Ps in the 23S state is also of interest for direct
+measurements of gravitational interaction on antimatter [49, 50]. Indeed, Ps excited in a long-lived
+state [51] together with antihydrogen [52–54] and muonium [55], have been proposed as a probe for
+test of weak equivalence principle on antimatter. A possible experimental scheme consists in the
+employment of a Ps beam in the metastable 23S state crossing a deflectometer or an interferometer
+to form a fringe pattern [49, 50]. In presence of an external force, the fringe pattern shows a
+displacement that is proportional to the acceleration experienced by the Ps [50]. In order to detect
+such a fringe pattern shift, Ps atom distribution on a plane could be scanned by using a slit or a
+material grating [50]. Ps annihilating on the obstacles and the ones crossing it can then be counted
+as a function of the position of the slit/grating and the Ps spatial distribution on the plane can be
+– 4 –
+
+reconstructed. A detector able to resolve the annihilation points of Ps along the beam direction (to
+distinguish the annihilations on the obstacles from the ones occurring forward) is needed. To verify
+the applicability of the J-PET detection units for this purpose, two such units with complete readout
+electronics were transported to AML. A test run was performed to measure the spatial resolution
+that can be achieved with only 2 detection units with e+ beam. The details and first results of the
+test can be found in the next section.
+4
+Performance study of two J-PET detection units with positron beam
+To investigate the performance of the J-PET modules, 511 keV photons emitted by the annihilations
+of 𝑒+ implanted with the AML beam into a stainless-steel flange have been recorded. Two modules
+were placed 20 cm apart from the e+ beam spot (red dot) as shown in the left panel of Fig. 2. Binary
+data registered by the FPGA cards were processed using framework software developed by the
+J-PET collaboration [23]. Hit time and hit position are reconstructed as described above. Signals
+from each SiPM are sampled at two thresholds in the voltage domain (30 mV and 70 mV). TOT as
+a measure of the energy deposition by photons interacting in a scintillator (hit) is calculated as the
+sum of the TOTs at both thresholds of the connected SiPMs. In the right panel of Fig. 2, the upper
+left inset shows the measured TOT spectra. Since TOT is the measure of energy deposition, higher
+Figure 2. The photo of the experimental setup (left). Two modules are 20 cm apart and centered around
+the e+ annihilation points. The X,Y,Z directional frame ( width(24 mm), thickness(6 mm), length(500 mm)
+) of the modules is such that the Y-axis is along the direction of the beam, while the X- and Z-directions are
+perpendicular and parallel to the plane of the J- PET module, respectively. On the right is the preliminary
+measured TOT distribution (upper left inset) and the 3D (X,Y,Z) projections of the reconstructed vertices.
+TOT values are expected with increasing energy deposition [22]. The structure with two peaks
+corresponds to the energy depositions by 511 keV photons and their scattered photons. The first
+peak results from the interactions of the scattered photons, while the second enhancement indicated
+in orange color represents the contribution by the 511 keV photons. In the analysis of events with
+– 5 –
+
+X103
+[a.u]
+3500
+4000
+Entries46841
+Counts
+3000
+Mean0.1685
+Std Dev 1.084
+2500
+3000
+2059 832280
+2000
+2000
+1500
+20 cm
+1000
+1000
+reference
+500
+detector
+×103
+-15 -10-50
+510
+15
+2000
+4000
+6000
+Xposition[cm]
+Time over Threshold [ps]
+u
+Entries 46841
+Entries46841
+Mean
+0.01471
+Mean-0.00574
+3000
+StdDev1.325
+Std Dev 0.2632
+15000
+2000
+DAQ
+10000
+&
+Controller Board
+1000
+STORAGE
+5000
+(ability to handle 6 modules)
+-15 -10-50
+51015
+510 15
+Y position [cml
+Z position [cm]2 hits by 511 keV photons, the annihilation points of 𝑒+ are reconstructed. For the selection of 511
+keV interactions, the first criterion is based on the measured TOT values for both hits. Only those
+events for which the TOT values are lying in the shaded region (orange) were selected. The second
+criterion is based on their angular correlation, i.e., the photons that caused 2 hits are considered only
+if emitted in back-to-back directions. After the selection of the 511 keV photons, the annihilation
+vertices are reconstructed. The projections of the reconstructed vertices on each axis are shown
+in the upper right and lower insets of Fig. 2. Preliminary results of the analysis performed over a
+set of data measured with a stainless-steel flange are presented. The obtained spatial resolutions in
+X, Y, and Z coordinates are 1.01 cm, 0.26 cm, and 1.33 cm, respectively (right panel in Fig. 2).
+These results are very promising. The ability to resolve the annihilation points along the beam
+(𝜎(Y)=0.26) justifies the use of J-PET modules for inertial sensing measurements on 23S Ps as
+described in [50]. Analysis of the complete measured data with both flanges is in progress. A
+detailed analysis, including the procedure for calibration of the detection modules, description of
+the analysis algorithm, and final results in terms of achievable spatial resolutions and reconstruction
+performance of the detectors will be reported in a separate article.
+5
+Summary and perspectives
+In this article, we have discussed several research problems that can be studied with the modular
+detection units of J-PET in the positron beam facility at AML in Trento. The new experimental
+system at AML can deliver a velocity-moderated, continuous e+ beam. In the next phase, the
+generation of a monochromatic 23S Ps beam will be developed. In addition, it is expected that
+the Ps beam can be produced in a defined quantum state. With the availability of the long-lived
+23S Ps beam, it is planned to use of atomic interferometry to study inertial sensing to measure the
+gravitational acceleration on Ps. Moreover, the ability to produce Ps in a defined quantum state
+will enrich studies of entanglement in Ps decays [33]. These studies will require the registration of
+multiphotons emitted in Ps decays with good angular and spatial resolution. Modular units based
+on J-PET technology can be used as potential detectors to perform such studies. A first test with
+two such modules has already been performed. Preliminary results show that the resolution in
+spatial coordinates is promising for performing the planned studies. The modular detector units
+used were developed primarily for tomographic purposes. In the future, a new design with a shorter
+scintillator length could be considered to meet the specific beam conditions for performing studies
+with positronium beam at AML.
+Acknowledgments
+The authors gratefully acknowledge support from the Foundation for Polish Science through
+the program TEAM/POIR.04.04.00-00-4204/17; the National Science Centre of Poland through
+grant no.
+2019/35/B/ST2/03562; the Ministry of Education and Science through grant no.
+SPUB/SP/490528/2021; the SciMat and qLIFE Priority Research Areas budget under the pro-
+gram Excellence Initiative - Research University at the Jagiellonian University, and Jagiellonian
+University project no. CRP/0641.221.2020. The authors also gratefully acknowledge the support
+– 6 –
+
+of Q@TN, the joint laboratory of the University of Trento, FBK-Fondazione Bruno Kessler, INFN-
+National Institute of Nuclear Physics, and CNR-National Research Council, as well as support
+from the European Union’s Horizon 2020 research and innovation programme under the Marie
+Sklodowska-Curie Grant Agreement No.754496 -FELLINI and Canaletto project for the Executive
+Programme for Scientific and Technological Cooperation between Italian Republic and the Republic
+of Poland 2019-2021.
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+1
+Nonlinear THz Control of the Lead Halide Perovskite Lattice
+Maximilian Frenzel1,*, Marie Cherasse1,2,*, Joanna M. Urban1, Feifan Wang3,‡ , Bo Xiang3,
+Leona Nest1, Lucas Huber3,§, Luca Perfetti2, Martin Wolf1, Tobias Kampfrath1,4, X.-Y. Zhu3,
+Sebastian F. Maehrlein1,†
+1 Fritz Haber Institute of the Max Planck Society, Department of Physical Chemistry, Berlin, Germany
+2 LSI, CEA/DRF/IRAMIS, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, Palaiseau,
+France
+3 Columbia University, Department of Chemistry, New York City, USA
+4 Freie Universität Berlin, Berlin, Germany
+* Authors contributed equally
+‡ Present address: Department of Materials, ETH Zurich, 8093 Zürich, Switzerland
+§ Present address: Sensirion AG, Staefa, Switzerland
+† Corresponding author. Email: [email protected]
+Abstract
+Lead halide perovskites (LHPs) have emerged as an excellent class of semiconductors for
+next-generation solar cells and optoelectronic devices. Tailoring physical properties by
+fine-tuning the lattice structures has been explored in these materials by chemical
+composition or morphology. Nevertheless, its dynamic counterpart, phonon-driven
+ultrafast material control, as contemporarily harnessed for oxide perovskites, has not
+been established yet. Here we employ intense THz electric fields to obtain direct lattice
+control via nonlinear excitation of coherent octahedral twist modes in hybrid
+CH3NH3PbBr3 and all-inorganic CsPbBr3 perovskites. These Raman-active phonons at
+0.9 – 1.3 THz are found to govern the ultrafast THz-induced Kerr effect in the low-
+temperature orthorhombic phase and thus dominate the phonon-modulated
+polarizability with potential implications for dynamic charge carrier screening beyond
+the Fröhlich polaron. Our work opens the door to selective control of LHP’s vibrational
+degrees of freedom governing phase transitions and dynamic disorder.
+Introduction
+During the last decade, lead halide perovskites (LHPs) emerged as promising semiconductors
+for efficient solar cells, light-emitting diodes, and other optoelectronic devices (1-3). Key
+prerequisites for the high LHP device efficiencies are the long charge carrier diffusion lengths
+and lifetimes (4, 5), often explained by the unusual defect physics (6, 7) and/or dynamic charge
+carrier screening (8, 9). The latter relies on delicate electron-phonon coupling, established by
+the dominant role of the static structure and dynamics of the lead-halide framework (10, 11).
+However, the exact mechanisms of the carrier-lattice interaction in the highly polarizable and
+anharmonic LHP lattices remain debated (12, 13). The sensitivity of the physical properties to
+structural distortions is a common feature in the extensive family of perovskites. In particular,
+for oxide perovskites, the control of specific lattice modes leads to ultrafast material control
+and nonlinear phononics (14, 15). Successful examples include, among others, light-induced
+superconductivity (16), magnetization switching (17), access to hidden quasi-equilibrium spin
+states (18), ferroelectricity (19, 20) and insulator-metal transitions (21) in perovskite or similar
+garnet structures.
+The crystal structure of LHPs features a large A-site cation surrounded by PbX6 octahedra
+consisting of lead (Pb) and halide (X) ions in the ABX3 crystal structure (see Fig. 1A). The
+
+2
+electronic band structure is mainly determined by the identities of metal and halide but is also
+highly sensitive to the Pb-X-Pb bond angle, which can be controlled through the steric
+hindrance of the A-cation (22). Changing the Pb-X-Pb bond angle is equivalent to tilting of the
+PbX6octahedra, which serves as an order parameter for the cubic  tetragonal  orthorhombic
+phase transitions (23, 24). Octahedral tilting is also an important factor governing structural
+stability (25), dynamic disorder (26, 27), and potential ferroelectricity (26, 27) in LHPs. A
+recent study using resonant excitation of the ~1 THz octahedral twist mode (Pb-I-Pb bending)
+revealed modulation of the bandgap of CH3NH3PbI3 at room temperature (28). A similar
+observation of dynamic bandgap modulation due twist modes was made at 80K for off-resonant
+impulsive Raman excitation (29). These twist modes are also believed to contribute to the
+formation of a polaronic state (30). All of these findings indicate an intriguing role of carrier
+coupling to Raman-active non-polar phonons in addition to the polar LO phonons in the
+conventional Fröhlich polaron picture (11, 31). In addition, the application of the Fröhlich
+polaron picture to LHPs has been questioned (9, 26), because of the limited applicability of the
+harmonic approximation in these soft lattices (13).
+Accordingly, the dynamic screening picture in LHPs is incomplete and its microscopic
+mechanism continues to be debated (32, 33). Furthermore, identifying and characterizing
+polaronic behavior is experimentally difficult (31, 33-37). Optical Kerr effect (OKE) in LHPs
+(38, 39) did not succeed in unveiling a lattice response and can be explained by an instantaneous
+electronic polarization (due to hyperpolarizability) instead (40). Moreover, previous strong
+field THz excitation could not directly detect the driven vibrational modes (28, 31) and coherent
+control of the phonons remained elusive. Here, we turn to the THz-induced Kerr effect (TKE)
+(41, 42) to investigate lattice-modulated polarization dynamics in the electronic ground state.
+We employ intense THz electric fields (Fig. 1B) that broadly cover most of the inorganic cage
+modes (Fig. 1C) and may nonlinearly probe the THz polarizability. The rapidly changing
+single-cycle THz field macroscopically mimics the sub-picosecond variation of local electric
+fields following electron-hole separation (43, 44) and elucidates the isolated lattice response.
+Experiment
+Generally, the polarizability describes the tendency of matter to form an electric dipole moment
+when subject to an electric field, such as the local field from a mobile charge carrier in a
+semiconductor. In the presence of an electric field 𝑬, the microscopic dipole moment is given
+by 𝒑(𝑬) = 𝝁0 + 𝛼𝑬, where 𝜇0 is the static dipole moment and 𝛼 is the polarizability tensor. In
+LHPs 𝛼 originates from three contributions: instantaneous electronic response (𝛼e), lattice
+distortion (𝛼lat), and molecular A cation reorientation (𝛼mol). For small perturbations of the
+respective collective coordinate 𝑄 (charge distribution, molecular orientation, or lattice mode)
+a Taylor expansion yields
+𝒑(𝑬, 𝑄) = 𝝁0  + 𝜕𝝁𝟎
+𝜕𝑄 𝑄   +
+𝜕𝛼
+𝜕𝑄 𝑄𝑬 ,
+(1)
+where the two partial derivatives correspond to the mode effective charge 𝑍∗ and the Raman
+𝑅𝑖𝑗 tensor, respectively. Macroscopically, the two terms lead to lattice polarization 𝑍∗𝑄IR and
+phonon-modulated susceptibility 𝜒eq
+(1) + (𝜕𝜒eq
+(1)/𝜕𝑄R)𝑄R for polar, 𝑄IR, and non-polar, 𝑄R,
+modes, respectively. The latter relates 𝜕𝛼/𝜕𝑄 to a transient dielectric function and change in
+refractive index of the material. This relation thus enables studying microscopic polarizability
+through the observation of macroscopic transient birefringence induced by a pump pulse and
+experienced by a weak probe pulse (41, 45). Collective polarization dynamics are induced by
+the driving force 𝐹 = −𝜕𝑊int/𝜕𝑄, where 𝑊int = −𝑷(𝑬, 𝑄) ∙ 𝑬 is the potential energy of the
+macroscopic polarization 𝑷 = ∑ 𝒑𝑖
+𝑖
+ interacting with an electric field 𝑬 (from a local charge
+
+3
+carrier or through light-matter coupling in the electric dipole approximation). Thus, two 𝐸THz
+interactions lead to THz polarizability-induced transient birefringence in TKE (42), which is
+linearly probed by a weak probe pulse 𝐸pr in an effective 3rd order nonlinear process
+proportional to 𝜒(3)𝐸THz𝐸THz𝐸pr (see Methods) (41, 46).
+To induce polarization dynamics, we use intense THz single-cycle pulses with a 1.0 THz center
+frequency (> 1.5 THz spectral width, see Fig. 1C), delivering sub-picosecond peak electric
+fields exceeding 1.5 MV/cm generated by optical rectification in LiNbO3 (47). We probe the
+resulting transient birefringence, i.e. anisotropic four-wave mixing signals, by stroboscopic
+sampling with a synchronized 20 fs pulse (800 nm center wavelength) in a balanced detection
+scheme, see Fig. 1A. We therefore effectively measure a 3rd order nonlinear signal field
+heterodyned with the transmitted probe field. The probe pulses are linearly polarized at 45°
+with respect to the vertically polarized THz pulses. As representative LHPs, we investigate
+hybrid organic-inorganic CH3NH3PbBr3 (MAPbBr3) and fully inorganic CsPbBr3. The
+freestanding single crystal samples (200 – 500 µm thickness) were solution grown by an
+antisolvent diffusion method (48, 49) (see Methods). Complementary polycrystalline thin films
+(~ 400 nm thickness) were spin-coated on 500 µm BK7 substrates, being particularly
+technologically relevant as most state-of-the-art LHP solar cells are fabricated in a similar way
+(50).
+Results
+Fig. 2A shows the THz induced transient birefringence in MAPbBr3 single crystals at room
+temperature. The signal (blue line) initially follows 𝐸THz
+2
+ (grey area, measured via electrooptic
+sampling), but then transitions into a nearly mono-exponential decay for time delays 𝑡 >
+500 fs. The transient birefringence peak at 𝑡 = 0 clearly scales quadratically with the THz-field
+amplitude as found by the pump fluence dependence in Fig. 2B. With the exponential decay
+dynamics remaining also constant for different fluences (Fig. S2), we can infer the Kerr-type
+origin of the full signal and thus conclude a strong THz polarizability. Furthermore, the peak
+amplitude’s (Fig. 2C) and the exponential tail’s (Fig. S3) dependence on the azimuthal angle
+between probe polarization direction and crystal axes perfectly obeys the expected 4-fold
+rotational symmetry of the 𝜒(3) tensor and TKE dependence of 𝜒𝑖𝑗���𝑙
+(3) 𝐸𝑗
+THz𝐸𝑘
+THz𝐸𝑙
+pr. We quantify
+the THz polarizability of MAPbBr3 by a nonlinear THz refractive index 𝑛2 of about 2 × 10−14
+cm2/W (see details in SI), being on the same order as in the optical region (51) and roughly 80
+times larger than 𝑛2 of Diamond (52), which is known as a suitable material for THz nonlinear
+optics (53).
+The small oscillatory deviations from the exponential tail in MAPbBr3 (Fig. 2A), become more
+pronounced and qualitatively different in CsPbBr3 in the form of a bumpy, non-trivial shape
+(Fig. 2D). This stark difference between MAPbBr3 and CsPbBr3 is reminiscent of 2D-OKE
+results (40), where the oscillatory signal of CsPbBr3 was found to be mainly due to anisotropic
+light propagation, since CsPbBr3 is orthorhombic and thus birefringent at room temperature.
+The fluence (Fig. 2E) and azimuthal dependences (Fig. 2F) are consistent with the pure 3rd
+order nonlinearity of the signal. However, fits to the azimuthal angle dependences in Figs. 2C,F
+(black lines) yield different ratios of the off-diagonal 𝜒𝑖𝑗𝑘𝑙
+(3)  to diagonal 𝜒𝑖𝑖𝑖𝑖
+(3) tensor elements for
+the two materials: 1.6 for MAPbBr3 and 1.0 for CsPbBr3.  A similar polarization dependence of
+static Raman spectra was recently attributed to additional isotropic disorder from the rotational
+freedom of the polar MA+ cation in MAPbBr3 (54).
+
+4
+Figs. 3A,B show a comparison of the temperature dependent TKE in MAPbBr3 single crystals
+and polycrystalline thin films. At room temperature (both top traces), it stands out that the thin
+film TKE signal lacks the exponential decay seen in the single crystals, providing a first
+evidence that the tail stems from dispersion effects and is not due to intrinsic molecular
+relaxation dynamics as previously interpreted (55). A strong and sophisticated THz dispersion,
+as seen in Fig 1C, is a general, but often overlooked, phenomenon in broadband high-field THz
+pump-probe spectroscopy. In analogy to the OKE (40), the features of the room temperature
+TKE in both MAPbBr3 and CsPbBr3 might therefore be dominated by dispersive and
+anisotropic light propagation. Hence, we assign the main contribution of the TKE response at
+room temperature to the instantaneous electronic polarizability (hyperpolarizability), which
+may overwhelm possible lattice contributions. This interpretation will be further supported by
+the modeling below.
+From here on, we mainly focus on the TKE of MAPbBr3, especially at low temperatures at
+which increased phonon lifetimes should facilitate the observation of a coherent lattice response
+(54, 56). For the single crystal (Fig. 3A), the TKE dynamics at 180 K are different than at room
+temperature, which might reflect the change of structural phase from cubic to tetragonal. At
+180K, an oscillatory signal at short times (< 2 ps) appears, suggesting the presence of a coherent
+phonon which was overdamped in the cubic phase at room temperature (54). The coherent
+oscillations become much stronger for the single crystal at 80 K, where MAPbBr3 is in the
+orthorhombic phase. Less pronounced, but clear oscillations are also visible in the thin film
+sample at 80 K (Fig. 3A, lowest trace). We extract the oscillatory parts, Fig. 3C, of both single
+crystal and thin film samples at 80 K by subtracting incoherent backgrounds, using a
+convolution of the squared THz field with a bi-exponential function. The respective Fourier
+transforms in Fig. 3D reveal the same oscillations frequency of 1.15 ± 0.05 THz for both
+samples. This clearly rules out anisotropic propagation effects as the origin of these oscillations
+(40), as the 400 nm film is too thin for significant walk-off between pump and probe (shown in
+simulations later) and the different thicknesses of the two samples rule out a Fabry-Pérot
+resonance effect. Thus, we can clearly assign the signal to a lattice-modulation of the THz
+polarizability dominated by a single 1.15 THz phonon in MAPbBr3. We now turn to THz-THz-
+VIS four-wave-mixing simulations to understand the origins of TKE from MAPbBr3.
+Modelling
+For dispersive and birefringent materials, the Kerr signal cannot be decomposed into an
+effective birefringence change observed by an independent probe beam (46). Instead, the Kerr-
+effect induced nonlinear polarization 𝑃(3) needs to be captured in a full four-wave-mixing
+(FWM) picture. To separate the three polarizability contributions (instantaneous electronic,
+molecular and lattice) and to take anisotropic light propagation across dispersive phonon
+resonances into account, we simulate the 3rd order nonlinear polarization by
+𝑃𝑖
+(3)(𝑡, 𝑧)
+= 𝜖0 �
+𝑑𝑡′ �
+𝑑𝑡′′
+𝑡′
+−∞
+�
+𝑑𝑡′′′
+𝑡′′
+−∞
+𝑡
+−∞
+𝑅�𝑖𝑗𝑘𝑙𝑅(𝑡, 𝑡′, 𝑡′′, 𝑡′′′)𝐸𝑗
+THz(𝑡′, 𝑧)𝐸𝑘
+THz(𝑡′′, 𝑧)𝐸𝑙
+pr(𝑡′′′, 𝑧),      (2)
+where 𝑅 is the time-domain 𝜒(3) response function (46) and 𝐸THz and 𝐸pr are the pump and
+probe electric fields, respectively. The time-independent 𝑅�𝑖𝑗𝑘𝑙 tensor constitutes the respective
+𝜒(3) symmetry for the different crystalline phases, in agreement with the ratios of the tensor
+elements obtained from the azimuthal fits in Fig. 2C. For the instantaneous electronic
+polarizability 
+(hyperpolarizability), 
+we 
+assume 
+temporal 
+Dirac 
+delta 
+functions
+𝑅e(𝑡, 𝑡′, 𝑡′′, 𝑡′′′) = 𝑅e,0𝛿(𝑡 − 𝑡′)𝛿(𝑡′ − 𝑡′′)𝛿(𝑡′′ − 𝑡′′′). For a lattice response, we model the
+driven phonon response by a Lorentz oscillator
+
+5
+𝑅ph(𝑡, 𝑡′, 𝑡′′, 𝑡′′′) = 𝑅ph,0𝛿(𝑡′ − 𝑡′′)𝛿(𝑡′′ − 𝑡′′′)𝑒−Γ�𝑡−𝑡′� sin ���𝜔ph
+2 − Γ2�(𝑡 − 𝑡′)�,   (3)
+where 𝜔ph/2𝜋 is the frequency and 1/2Γ the lifetime of the phonon (46). The driving force for
+Raman-active phonons is hereby 𝐸𝑗
+THz𝐸𝑘
+THz, which contains difference- and sum-frequency
+terms (57, 58). The latter is a unique distinction to the OKE. For 𝐸THz we can directly use the
+experimental THz electric field, as measured in amplitude and phase resolved electro-optic
+sampling. After we determine the complex refractive indices (Fig. 1C) and extrapolate the static
+birefringence (see Methods and SI), we calculate and propagate all involved fields from Eq. (2)
+incl. signal fields 𝐸𝑖
+s(𝑡, 𝑧) emitted from 𝑃𝑖
+(3)(𝑡, 𝑧), followed by our full detection scheme,
+including balanced detection, to obtain the pump-probe signal (see details in Methods).
+Fig. 4A shows the simulated TKE signal (grey) compared to the experimental data (blue) at
+room temperature for a 500 µm thick MAPbBr3 single crystal. It unveils the formation of a long
+exponential tail produced by walk-off, dispersion, and absorption effects, even for only an
+instantaneous electronic response 𝑅e. This confirms that the electronic polarizability dominates
+the TKE signal at room temperature. It contrasts a previous interpretation of a TKE
+measurement in thick single crystal MAPbBr3, which neglected propagation effects entirely
+(55). At 80 K, MAPbBr3 is orthorhombic. We therefore need to include additional static
+birefringence. Instantaneous hyperpolarizability alongside static birefringence and dispersion
+can cause the appearance of oscillatory features (40). Nevertheless, our modelling finds these
+features to be too short-lived (see Fig. S14) to explain our experimental observation at 80 K.
+Thus, we need to account for both hyperpolarizability 𝑅e and lattice-modulated polarizability
+𝑅ph responses (fit parameters: 𝜔ph/2𝜋 = 1.14 THz, Γ = (2 ∙ 1.7ps)−1, 𝑅e,0/𝑅ph,0  = 2.4) to
+describe the low-temperature TKE signals in the time- and frequency domain (Figs. 4B,C). In
+contrast to OKE at 80K (40), the oscillations in TKE are therefore due to coherent phonon
+modes and we hence finally observe an ultrafast lattice response to a sub-picosecond electric
+field transient.
+The simulation assuming only instantaneous hyperpolarizability for a 400 nm thin film agrees
+well with the experimental TKE at room temperature (see Fig. 4D). As expected, the simulation
+lacks the clear tail seen in the thick single crystals, thereby additionally confirming that the tail
+is due to light propagation effects. Also here, at 80 K, we need to include both instantaneous
+electronic and phonon contributions (𝜔ph/2𝜋 = 1.14 THz, Γ = (2 ∙ 1.7ps)−1, 𝑅e,0/𝑅ph,0  =
+24) to describe the experimental signals for the thin films in Figs. 4E,F. Here, a purely
+instantaneous electronic contribution alongside static birefringence does not lead to oscillatory
+features (see Figs. S14A,C). This provides direct proof that the observed oscillations in Figs.
+3C,D originate from a coherent phonon. Therefore, through comparison of single crystals with
+thin films and by rigorous FWM simulation, we prove to witness a coherent lattice-driven
+dynamic polarization response.
+Interpretation
+Besides potential rotational disorder, our rigorous modeling shows that we do not observe a
+TKE contribution that we can unambiguously relate to an ultrafast cation reorientation in the
+form of a liquid-like exponential decay (41, 42). We rather find MAPbBr3‘s TKE tail at room
+temperature to be most likely overwhelmed by the instantaneous hyperpolarizability 𝑅e in
+conjunction with dispersive light propagation. This might be also explained by the THz pump
+spectrum being far off the cation rotational resonances around the 100 GHz frequency range
+(59). The cation species nevertheless influences the static and dynamic properties of the
+inorganic lattice, highlighting the importance of the interplay between the organic and inorganic
+
+6
+sub-lattices for the LHPs equilibrium structure (56). This fact shows up e.g. as a single
+dominating PbBr6 cage mode in MAPbBr3 but two dominating modes in CsPbBr3 (see Fig. S1);
+in agreement with static Raman spectra (54). The various templating mechanisms by which the
+cation influences these properties (60) are through its steric size (22), lone-pair effects (27, 61),
+or hydrogen bonding (62).
+For MAPbBr3, we find a single phonon mode dominating the Raman-active lattice dynamics in
+response to a sub-ps electric field spike. The observed phonon at 1.15 THz is consistent with
+static Raman spectra in the visible range, where this mode also exhibits the highest scattering
+amplitude (54, 63). Thus, we can assign it to a dynamic change in the Pb-Br-Pb bond angle
+corresponding to a twisting of the PbBr6 octahedra (twist mode) (64). Based on theory work for
+MAPbI3 (65), we assign this to Ag symmetry, which matches the experimental observations that
+the mode is still present when we rotate the single crystal by 45° (see Fig. S7) and that we also
+observe the same mode in polycrystalline thin films (Figs. 3C,D). We suggest that at room
+temperature this mode also strongly modulates the THz dielectric response, even though its
+oscillations are potentially overdamped as inferred from the broad Raman spectra (54, 56). To
+distinguish whether this twist mode only dominates the ultrafast lattice response in MAPbBr3,
+or is of wider relevance for other LHPs, we analyze the TKE response of CsPbBr3, where we
+observe two modes at 0.9 and 1.3 THz at 80K (see Fig. S1), corresponding to two octahedra
+twist modes as observed in static Raman spectra (54). We thus conclude that the transient THz
+polarizability (𝜕𝜒eq
+(1)/𝜕𝑄) 𝑄 is generally dominated by the octahedra twist modes in LHPs.
+We now consider the excitation mechanism of the coherent phonon. Fig. 5A shows that the
+1.15 THz oscillations at 80K scale with the square of the THz electric field amplitude,
+suggesting nonlinear excitation with a Raman-type driving force. This is consistent with the
+Kerr effect being also a Raman-type probing mechanism. Generally, there are four types of
+Raman-active THz excitation mechanisms: Difference- or sum-frequency excitation via Ionic
+Raman Scattering (IRS) or Stimulated Raman Scattering, corresponding to nonlinear ionic
+(=phononic) or nonlinear electronic (= photonic) pathways, respectively (58). Indeed, the Ag
+symmetry of the observed modes permits IRS, where a resonantly driven IR-active phonon
+couples anharmonically to a Raman-active mode (14, 58). However, this phononic pathway
+requires phonon anharmonicity, whereas the photonic pathway requires electronic THz
+polarizability. The sum-frequency (SF) and difference-frequency (DF) photonic force spectra
+in Fig. 5B indicate a comparable probability for both photonic mechanisms to drive the 1.15
+THz mode (dashed line). For the phononic pathways in Fig. 5C, the DF excitation requires a
+primarily driven IR-active phonon with a bandwidth of ≳ 1 THz, which exists in our excitation
+range even at 80 K (66). On the other hand, there are also IR-active modes, which provide
+roughly half the frequency of the Raman-active mode ΩIR = ΩR/2 enabling phononic SF-IRS
+(58). Accordingly, none of the four nonlinear excitation pathways can be neglected, but the
+observed strong electronic THz polarizability in conjunction with a longer penetration depth
+for lower THz frequencies favors a SF nonlinear photonic mechanism. We leave the
+determination of the exact excitation pathway to further studies, e.g. by two-dimensional TKE
+(67) or more narrowband THz excitation (68).
+Discussion
+Independent of the precise excitation pathway and in contrast to optical Raman or transient
+absorption studies, we unambiguously observe strong electron-phonon coupling of the
+octahedral twist modes via a pure THz polarizability (electronic or ionic). This explains the
+mode’s dominating influence on the electronic bandgap in MAPbI3 previously observed by Kim
+et al. (28). The twist mode’s half-cycle period of ~0.5 ps is short enough to contribute to
+
+7
+electron-phonon coupling within the estimated polaron formation time (69), even in the
+overdamped case at room temperature. We can understand carrier screening by non-polar
+modes as follows. As shown in Eq. (1), the THz polarizability contains two lattice
+contributions: From polar lattice modes 𝑃IR(𝜔) ∝ 𝑍∗𝑄IR(𝜔) ∝ 𝑍∗𝐸THz(𝜔) and from the non-
+resonant electron cloud moving at THz speeds (sub-ps time scales):
+𝑃𝑒(𝜔) = 𝜖0[ 𝜒𝑒
+(1)(𝜔) + 𝜕𝜒𝑒
+(1)
+𝜕𝑄R
+(𝜔, 𝛺) 𝑄R(𝛺) ]𝐸THz(𝜔),
+(4)
+where the latter is modulated in the presence of a Raman-active phonon 𝑄R. Thus, excited
+Raman-active modes lead to a transient dielectric response 𝜖(ω) = 𝜖𝑒𝑞(ω) + Δ𝜖(𝜔, Ω) at THz
+frequencies 𝜔 with Δ𝜖 =
+𝜕𝜒𝑒
+(1)
+𝜕𝑄R 𝑄R, which constitutes an additional contribution of higher order
+screening due to a fluctuating lattice. In the macroscopic incoherent case, Δ𝜖 averages out. On
+time and length scales relevant to electron-hole separation and localization (< 1 nm and < 1ps)
+(43, 44), collective octahedral tilting produces (70) an additional THz polarizability, which
+might add to the conventional Fröhlich picture of carrier screening. We speculate that a local
+non-zero twist angle 𝑄R could be either already present due to dynamic disorder (see discussion
+below), or might be nonlinearly excited by the transient local charge field 𝐸loc
+2 , easily exceeding
+1 MV/cm (9) (analog to the excitation pathways above). The latter scenario agrees with
+MAPbBr3’s unusually large optical 𝜒(3), previously attributed to local confinement effects (51).
+The observed 1.15 THz mode is therefore a good candidate for contributing to strong electron-
+phonon coupling beyond the polar Fröhlich picture.
+The driven twist mode is similar to soft modes in oxide perovskites, where the tilting angle of
+adjacent oxygen octahedra is an order parameter for phase transitions (71). Recently, TKE was
+similarly employed to drive and detect ultrafast field-induced ferroelectricity in the quantum
+paramagnet SrTiO3 (19). In Eu and Sr doped La2CuO4, driving the tilt of oxygen octahedra was
+found to induce signatures of superconductivity persisting over a few ps above the critical
+temperature (16). Consistent with these observations in oxide perovskites, the tilting angle of
+the PbX6 octahedra (twist mode) was found to act as an order parameter for phase transitions in
+LHPs (23, 24) and in the double-perovskite Cs2AgBiBr6 (72). Especially for MAPbBr3, the
+Raman scattering intensity of the 1.1 THz peak was recently shown as measure of the
+orthorhombic-tetragonal phase transition (63) and its spectral evolution in Raman (56) and
+neutron scattering (73) is indicative of a soft mode phase transition. Yet, the LHP lattice
+properties were previously mainly tuned in a static and chemical manner, e.g. by acting on the
+octahedral tilting angle through the steric size of the A-site cation (22). The coherent lattice
+control demonstrated here allows dynamic tuning of the structure and thus ultrafast phonon-
+driven steering of LHP’s optoelectronic properties, e.g. for integrated photonic devices
+operating at GHz to THz clock-rates (74).
+In addition, imposing a coherence on the octahedral tilting should directly influence the
+dynamic disorder (75), which is considered one of the key components determining the
+optoelectronic properties of LHPs (12, 54, 76). Dynamic disorder means that the effective
+crystallographic structure (e.g. cubic at 300 K) only exist in spatial and temporal average.
+Specifically, in LHPs with a Goldschmidt tolerance factor below 1, such as MAPbBr3 and
+CsPbBr3, the disorder mainly arises from the lattice instability associated with octahedral tilting
+(61, 75, 77), evidenced by X-ray total scattering in CsPbBr3 (78), inelastic X-ray scattering in
+MAPbI3 (70) and Raman spectroscopy in MAPbBr3, CsPbBr3 and MAPbI3 (54, 77). The
+resulting fluctuating lattice potential and polar nanodomains have been suggested as underlying
+mechanisms for dynamic charge carrier screening in the form of preferred current pathways
+
+8
+(79, 80) and ferroelectric polarons (26, 81), respectively. All these phenomena might be
+potentially controlled or temporally lifted by the THz control of octahedral motion.
+Overall, we find that the octahedral tilting motion, which serves as an order parameter for phase
+transitions (23, 24) and contributes significantly to dynamic disorder (54, 77), shows a strong
+nonlinear coupling to a rapidly varying electric field on sub ps-timescales that are relevant to
+local electron-hole separation polaron formation. Our results thus indicate that the TO
+octahedral twist mode contributes to strong electron-phonon coupling and dynamic carrier
+screening in LHPs, which may be inherently linked to a local and transient phase instability as
+suggested by the ferroelectric polaron picture (26, 81).
+Conclusion
+By investigating 3rd order nonlinear polarization dynamics in hybrid and all-inorganic LHPs,
+we reveal that the room temperature TKE response stems predominantly from a strong THz
+hyperpolarizability, leading to a nonlinear THz refractive index on the order of 10-14 cm2/W. In
+analogy to previous OKE studies (40), we explain and model the appearance of retarded TKE
+dynamics by dispersion, absorption, walk-off, and anisotropy effects (46). These effects are of
+crucial relevance to contemporary THz pump-probe experiments, such as TKE or THz-MOKE
+studies (82, 83). For sufficiently long phonon lifetimes at lower temperatures, we can
+nonlinearly drive and observe a coherent lattice response of the ~1 THz octahedral twist
+mode(s). These phonons couple most strongly to the THz polarizability, which means they must
+be highly susceptible to transient local fields on the 100s fs time scale, relevant to electron-
+phonon coupling and carrier localization. We find this ultrafast non-polar lattice response to be
+mediated by anharmonic phonon-phonon coupling and/or by the strong nonlinear electronic
+THz polarizability. The same octahedral twist mode serving as a sensitive order parameter for
+structural phase transitions (63, 73) is likely the origin of significant intrinsic dynamic disorder
+in LHPs (54, 75). Thus, our findings suggest that the microscopic mechanism of the unique
+defect tolerance (39, 84) and long carrier diffusion lengths (4, 5) of LHPs might also rely on
+small phase instabilities accompanying the polaronic effects.
+Our work demonstrates the possibility of coherent control over the twist modes via nonlinear
+THz excitation. Since the octahedral twist modes are the dynamic counterparts to steric
+engineering of the metal-halide-metal bond angle, our work paves the way to study charge
+carriers in defined modulated lattice potentials, to control dynamic lattice disorder, or to
+macroscopically switch polar nanodomains leading to the emergence of transient
+ferroelectricity.
+
+9
+Materials and Methods
+Sample Growth
+The single crystal samples were synthesized based on our previous published method (40). For
+MAPbBr3, the precursor solution (0.45 M) was prepared by dissolving equal molar ratio of
+MABr (Dyesol, 98%) and PbBr2 (Aldrich, ≥98%) in dimethylformamide (DMF, Aldrich,
+anhydrous 99.8%). After filtration, the crystal was allowed to grow using a mixture of
+dichloromethane (Aldrich, ≥99.5%) and nitromethane (Aldrich, ≥96%) as the antisolvent (48).
+Similar method was used for CsPbBr3 crystal growth (49). The precursor solution (0.38 M) was
+formed by dissolving equal molar ratio of CsBr (Aldrich, 99.999%) and PbBr2 in dimethyl
+sulfoxide (EMD Millipore Co., anhydrous ≥99.8%). The solution was titrated by methanol till
+yellow precipitates show up and did not redissolve after stirring at 50 °C for a few hours. The
+yellow supernatant was filtered and used for the antisolvent growth. Methanol was used for the
+slow vapor diffusion. All solid reactants were dehydrated in a vacuum oven at 150 °C overnight
+and all solvents were used without further purification.
+Thin films. Before spin-coating, the substrate was rinsed by acetone, methanol and isopropanol
+and treated under oxygen plasma for 10 min. The freshly prepared substrate was transferred to
+the spin coater within a short time. For MAPbBr3, the precursor DMSO (Aldrich, ≥99.9%)
+solution (2M) containing the equimolar ratio of MABr and PbBr2 was used for the one-step
+coating method. The film was formed by spin-coating at 2000 rpm for 45 s and annealed at 110
+°C for 10 min. For CsPbBr3, a two-step method was implemented. First, the PbBr2 layer was
+obtained by spin-coating the 1 M PbBr2/DMF precursor solution at 2000 rpm for 45 s and dried
+at 80 °C for 30 min. Subsequently, the PbBr2 film was immersed in a 70 mM CsBr/methanol
+solution for 20 min. Following the rinsing by isopropanol, the film was annealed at 250 °C for
+5 min to form the uniform perovskite phase.
+THz-induced Kerr effect
+THz pulses with 1.0 THz center frequency and field strength exceeding 1.5 MV/cm (Fig. 1B,C),
+were generated by optical rectification in LiNbO3 with the tilted pulse front technique (47). To
+that end, LiNbO3 was driven by laser pulses from an amplified Ti:sapphire laser system (central
+wavelength 800 nm, pulse duration 35 fs  FWHM, pulse energy 5 mJ, repetition rate 1 kHz).
+The probe pulses came from a synchronized Ti:sapphire oscillator (center wavelength 800nm,
+repetition rate 80 MHz) and were collinearly aligned and temporarily delayed with respect to
+the THz pulse. The probe polarization was set at 45 degrees with respect to the vertically-
+polarized THz pulses. The THz pulses induced a change in birefringence (TKE) in the sample
+(41). This birefringence causes the probe field to acquire a phase difference between
+polarization components parallel and perpendicular to THz pulse polarization. The phase
+difference is detected via a half- and quarter-waveplate (HWP and QWP) followed by a
+Wollaston prism to spatially separate perpendicularly polarized probe beam components. The
+intensity of the two beams is detected by two photodiodes in a balanced detection configuration.
+Four-wave-mixing simulation
+The 3rd order nonlinear polarization 𝑃(3)(𝑡, 𝑧) is simulated using the general four-wave mixing
+equation (Eq. (2)) and according to Ref. (46). To compute 𝑃(3)(𝑡, 𝑧), all three contributing light
+fields, 𝐸𝑗
+THz, 𝐸𝑘
+THz and 𝐸𝑙
+pr, are propagated through the crystal on a time-space grid. The three
+fields inside the sample are calculated at any location 𝑧 using
+𝐸𝑖(𝑡, 𝑧) = �
+𝑡𝑖(𝜔)𝐴𝑖(𝜔)𝑒−𝑖(𝜔𝑡−𝑘𝑖(𝜔)𝑧)(1 − 𝑅𝑖(𝜔, 𝑧))𝑑𝜔
+∞
+−∞
+,
+(5)
+with
+
+10
+𝑅𝑖(𝜔, 𝑧) = 𝑟𝑖�1 + 𝑒2𝑖𝑧𝑘𝑖(𝜔)�
+𝑒2𝑖(𝑑−𝑧)𝑘𝑖(𝜔)
+1 − 𝑟𝑖
+2(𝜔)𝑒2𝑖𝑑𝑘𝑖(𝜔) ,
+(6)
+where 𝐴𝑖(𝜔) is the spectral amplitude of the field and 𝑡𝑖 and 𝑟𝑖 denote the Fresnel transmission
+and reflection coefficients respectively. As the input pump field 𝐸THz we use the full
+experimental THz electric field generated via optical rectification in LiNbO3 as measured using
+electro-optic sampling in Quartz (85). For the probe field 𝐸pr we assume a Fourier limited
+Gaussian spectrum with center wavelength 800 nm and pulse duration 20 fs, experimentally
+measured by a spectrometer and a commercial SPIDER. For both non-birefringent and
+birefringent simulations, we use the THz refractive index for MAPbBr3 as calculated from its
+dielectric function based on the experimental work by Sendner et al. (10) (Fig. S11). In the
+optical region, the precise anisotropic refractive index of CsPbBr3 is used as measured using
+the 2D-OKE (46). For the birefringent lead halide perovskite simulation, the static birefringence
+of CsPbBr3 is used and interpolated to THz region (Fig. S12). For the isotropic cubic perovskite,
+the static birefringence is set to zero. In the shown simulation results, the time-grid had a finite
+element size of Δ𝑡′ = 16.6 fs and the spatial grid had a finite element size of Δ𝑧 = 10 μm for
+the single crystal and Δ𝑧 = 0.1 μm for the thin film simulations respectively. These values were
+chosen for the sake of computational efficiency and did not qualitatively affect the simulation
+results. The pump-probe delay finite element size was chosen to be Δ𝑡 = 16.6 fs.
+We assume that the nonlinear polarization 𝑃(3) emits an electric field 𝐸(4) at every slice 𝑧
+according to the inhomogeneous wave equation
+[∇2 + 𝑘𝑖
+2(𝜔)]𝐸𝑖
+(4)(𝜔, 𝑡, 𝑧) = − 𝜔2
+𝜖0𝑐2 𝑃𝑖
+(3)(𝜔, 𝑡, 𝑧),
+(7)
+which then co-propagates with the probe field 𝐸pr. The transmitted probe field 𝐸pr and emitted
+field 𝐸(4) are projected on two orthogonal polarization components by propagating through a
+half-wave plate, quarter-wave plate and Wollaston prism. The combined effect of these optical
+devices are captured by the Jones matrices 𝐽1 and 𝐽2 for the two separated polarization
+component channels. A balanced detection scheme allows observation of 𝐸(4) by interfering
+with 𝐸pr. Under balanced conditions, the detected non-equilibrium signal is
+𝑆 ∝ ∫ 𝑅𝑒�(𝐽1𝐸𝑝𝑟) ∙ �𝐽1𝐸(4)∗� − (𝐽2𝐸𝑝𝑟) ∙ �𝐽2𝐸(4)∗��𝑑𝜔.  (8)
+Our simulation therefore mimics the balancing conditions of the experiment. A detailed
+description of this calculation is given in (46).
+To model the response of the system, we assume the response function 𝑅e(𝑡, 𝑡′, 𝑡′′, 𝑡′′′) for an
+instantaneous electronic response and 𝑅ph(𝑡, 𝑡′, 𝑡′′, 𝑡′′′) for a phonon response. The expressions
+for 𝑅e and 𝑅lat are given in the main paper. In the frequency domain, 𝑅(𝜔) = 𝜒(3)(𝜔). For
+normal incidence on the (101) crystal surface, the orthorhombic space group Pnma allows Kerr
+signals from 𝜒𝑥𝑥𝑥𝑥
+(3) , 𝜒𝑦𝑦𝑦𝑦
+(3)
+, 𝜒𝑥𝑥𝑦𝑦
+(3)
+= 𝜒𝑥𝑦𝑦𝑥
+3
+= 𝜒𝑥𝑦𝑥𝑦
+(3)   and 𝜒𝑦𝑦𝑥𝑥
+(3)
+= 𝜒𝑦𝑥𝑥𝑦
+(3)
+= 𝜒𝑦𝑥𝑦𝑥
+(3)  (40). While
+the cubic space group Pm3m and allows for 𝜒𝑥𝑥𝑥𝑥
+(3)
+= 𝜒𝑦𝑦𝑦𝑦
+(3)  and 𝜒𝑥𝑥𝑦𝑦
+(3)
+= 𝜒𝑥𝑦𝑦𝑥
+(3)
+= 𝜒𝑥𝑦𝑥𝑦
+(3)
+=
+𝜒𝑦𝑦𝑥𝑥
+(3)
+= 𝜒𝑦𝑥𝑥𝑦
+(3)
+= 𝜒𝑦𝑥𝑦𝑥
+(3) . The Pnma space group applies to CsPbBr3 in its orthorhombic phase,
+which is the case for all temperatures considered in this work. The Pm3m space applies to
+MAPbBr3 for its room temperature cubic phase.  All allowed tensor elements were assumed to
+have the same magnitude.
+Simulations for an electronic response only and without optical anisotropy are shown for
+various thicknesses in Fig. S13. This applies to MAPbBr3 single crystals at room temperature
+when this material is in the cubic phase. Simulations for an electronic response only and with
+optical anisotropy are shown for various thicknesses in Fig. S14. This applies to MAPbBr3 in
+its low-temperature orthorhombic phase and CsPbBr3, which is orthorhombic for all
+temperatures considered in this work. The effect of optical anisotropy on the TKE is very
+
+11
+dependent on the azimuthal angle of the crystal. Results are shown for two different azimuthal
+angles: 0° and 45° angle between crystal axis and probe polarization in Fig. S14.
+Fig. S14 shows that the oscillatory features due to propagation effects and static birefringence
+cannot explain the oscillations observed in low temperature MAPbBr3 single crystals and thin
+films. To simulate this oscillatory signal, we have to consider both electronic and phonon
+contributions to 𝑅 = 𝑅e + 𝑅ph alongside static birefringence. The chosen simulation
+parameters are given in the main text and were chosen in accordance with the Lorentzian fits to
+our experimental data in Fig. S8. The instantaneous contribution to 𝑅 is 𝑅e,0 𝑅ph,0
+⁄
+times larger
+than the phononic contribution, when the respective spectral amplitude of the responses are
+integrated in the 0 - 10 THz range.
+References
+1.
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+
+15
+Figures
+Fig. 1 | THz fields for nonlinear lattice control in lead halide perovskites. A. Sketch of the
+experimental pump-probe configuration. An intense THz electric field causes a transient change
+of birefringence, leading to an altered probe pulse polarization. This change in polarization is
+read out using a balanced detection scheme, consisting of balancing optics (BO), Wollaston
+prism (WP) and two photodiodes (PD1, PD2). B. Employed pump THz electric field measured
+using electro-optic sampling. C. Complex refractive index of MAPbBr3 (blue curves) obtained
+from (10) and Fourier transform of THz field (red area) in B.
+
+A
+MAPbBr
+PD1
+dwnd ZH
+BO
+WP
+PD2
+VIS probezelectricfield(Mv/cm
+8
+B
+THz amplitude (norm.)
+Re(n)
+Refractiveindexn
+6
+Im(n)
+0.5
+0
+0
+2
+0
+2
+0
+L
+2
+3
+4
+5
+Time (ps)
+Freguency(THz)16
+Fig. 2 | THz-induced birefringence in MAPbBr3 and CsPbBr3 at room temperature. A.
+Room temperature TKE in MAPbBr3 and D. CsPbBr3 single crystals. B. and E., THz fluence
+dependence of transient birefringence peak amplitude with quadratic fit (black line),
+demonstrating the Kerr effect nature of the signals. C. and F., azimuth angle (between probe
+beam polarization and crystal facets) dependence of main TKE peak with fit (black line) to
+expected 𝜒(3) tensor geometries in cubic and orthorhombic phase, respectively.
+
+A
+B
+c
+Kerr signal
+MAPbBr,
+1.0
+MAPbBr,
+90
+peak (norm.)
+norr
+I peak (norm.)
+Quadratic fit
+135
+45
+0.8
+THz
+gence
+0.8
+0.6
+0.6
+ signal
+180
+0
+0.4
+0.4
+ransient
+0.2
+Ker
+0.2
+225
+315
+0.0
+0.0
+-2
+0
+2
+4
+6
+0
+0.5
+1
+270
+Time (ps)
+Max E-field amp. (norm.)
+Azimuthangle(o)D
+E
+F
+Kerr signal
+1.0
+1.0
+CsPbBr3
+90
+CsPbBr,
+peak (norm.)
+nori
+ peak (norm.)
+Quadratic fit
+135
+45
+0.8
+Jence
+TH7
+0.8
+0.6
+0.6
+9
+ signal
+180
+0
+0.4
+0.4
+ransient
+0.2
+0.2
+Ker
+225
+315
+0.0
+0.0
+-2
+0
+2
+4
+6
+0
+0.5
+1
+270
+Time (ps)
+Max E-field amp. (norm.)
+Azimuth angle (°)17
+Fig. 3 | TKE temperature dependence of single crystal vs. thin film MAPbBr3. A.
+Temperature-dependent TKE for single crystal and B. thin film samples. C. Oscillatory signal
+components at 80K extracted by subtracting an exponential tail (dashed lines) and starting after
+the main peak (bottom black arrow) in A, B. D. Respective Fourier transforms (blue and red)
+of C and incident THz pump spectrum (gray area).
+
+A
+B
+c
+Single crystal
+Thin film (polycryst.)
+80K
+(Cn
+3.0
+d= 500 μm
+3.0
+d = 0.4 μm
+(arb.
+10
+latory signal
+300K
+2.5
+2.5
+300K
+cub.
+ingence (norm.)
+A
+0.0
+Oscilla
+2.0
+2.0
+0
+5
+10
+180K
+1.5
+180K
+Time (ps)
+fetbirefr
+Transient birefr
+D
+Transient
+80K
+1.0
+10
+1.0
+norm.
+80K
+THz spectrum
+0.8
+0.5
+0.5
+80K
+orth.
+Spectrum
+0.6
+0.4
+0.0
+S
+0.0
+0.2
+0.0
+0
+5
+10
+15
+0
+5
+10
+0
+1
+2
+3
+4
+5
+Time (ps)
+Time(ps)
+Freguency(THz)18
+Fig. 4 | Four-wave mixing simulations vs. experimental results in MAPbBr3. Isotropic cubic
+phase (300 K): Simulated TKE signals for A. single crystal (500 µm thickness) and D. thin film
+(400 nm thickness) assuming only instantaneous electronic response 𝑅e(𝑡) (gray lines).
+Anisotropic orthorhombic phase (80 K): B. Single crystal and E. thin film TKE vs simulation
+for model system with static birefringence, instantaneous electronic 𝑅e(𝑡) and Lorentz
+oscillator 𝑅ph(𝑡) phonon response (purple lines). C., F. Fourier transforms of experimental data
+(blue and red) and simulation results (purple) from B., E., respectively, normalized to the
+phonon amplitude at 1.15 THz.
+
+A
+Cubic (300K)
+B
+c
+Orthorhombic (80K)
+1.0
+1.0
+1.0
+Exp.Singlecrystal
+Sim. R。 + Rpr
+Exp. Single crystal
+Sim. R.
+Sim. R。 + Rph
+(norm.)
+fringence (norm.)
+0.5
+0.5
+T= 2.7ps
+0.5
+lence
+0.0
+(norm.
+T=0.8ps
+0.0
+-0.5
+0.0
+-bire
+Transient bire
+1.0
+1.0
+0.05
+1.0
+pe
+Transient
+Exp. Thin film
+Exp. Thin film
+S
+Sim. R.
+0.5
+0.5
+0.5
+4
+8
+Sim. R。+ Rph
+0.0
+0.0
+0.0
+-2
+0
+2
+4
+6
+8
+0
+5
+10
+0
+1
+2
+3
+4
+5
+Time (ps)
+Time (ps)
+Freauencv(THz)19
+Fig. 5 | Nonlinear excitation pathways for the 1.15 THz Raman-active twist mode. A.
+Time domain coherent phonon oscillations (normalized to 𝑡 = 0 TKE main peak) at 80 K for
+different THz field strengths (left panel) and respective coherent phonon amplitude (right
+panel) obtained from Fourier transform; both unveiling a 𝐸THz
+2
+ scaling law and thus
+demonstrating a nonlinear excitation. B. Possible nonlinear photonic excitation pathways for
+the 𝜔ph = 1.15 THz mode (dashed line) mediated via a THz electronic polarizability. The
+nonlinearly coupled 𝐸THz spectrum (gray area) leads to difference-frequency 𝐸THz𝐸THz
+∗
+ (DF,
+red area) and sum-frequency 𝐸THz𝐸THz (SF, blue area) driving forces. The octahedral twist
+mode is schematically sketched on the right hand side. C. Possible phononic pathways via a
+directly driven IR-active phonon 𝑄IR, which nonlinearly couples to the Raman-active mode
+𝑄𝑅 via anharmonic 𝑄R𝑄IR
+2  coupling.
+
+Phonon Amp.
+1.0
+An (norm.
+0.0
+0.5
+0.0
+0
+5
+10
+15
+20
+0
+0.5
+1
+Time (ps)
+B
+PhotonicPathways
+1.0
+EE
+Driving force
+0.8
+arb.
+VE
+0.6
+wn
+0.4DF
+SF
+0.2
+S
+-EE
+Wph
+0.0
+0
+1
+2
+3
+4
+5
+6
+Frequency (THz)
+c
+Phononic Pathways
+Difference-frequency
+Sum-frequency
+hQIR
+QIR
+h2R
+hIR
+0
+0
+IR-active
+Raman-active
+IR-active
+Raman-active20
+Acknowledgments
+We thank A. Paarmann, M. S. Spencer, M. Chergui, A. Mattoni, and H. Seiler for fruitful
+discussion.
+Funding: S.F.M. acknowledges funding for his Emmy Noether group from the Deutsche
+Forschungsgemeinschaft (DFG, German Research Foundation, Nr. 469405347). S.F.M and
+L.P acknowledge support of the 2D-HYPE project from the Deutsche
+Forschungsgemeinschaft (DFG, German Research Foundation, Nr. 490867834) and Agence
+Nationale de la Recherche (ANR, Nr. ANR-21-CE30-0059), respectively. XYZ acknowledges
+support by the Vannevar Bush Faculty Fellowship through Office of Naval Research Grant #
+N00014-18-1-2080. M.C. was supported by the DAAD Scholarship 57507869.
+Author contributions: S.F.M. conceived the experimental idea; M.F., M.C., and S.F.M.
+designed the research; M.F., M.C., L.N. performed experiments; F.W., B.X. prepared
+samples; M.F., M.C. analyzed data; J.U., L.H., S.F.M. contributed theory/analytic tools. L.H.
+developed FWM model and M.F. carried out FWM simulations. M.F., X.-Y.Z., and S.F.M.
+wrote the manuscript. All authors read, discussed and commented the manuscript. M.F. and
+M.C. contributed equally to this work.
+Competing interests: The authors declare that they have no competing interests.
+Data and materials availability: All data and simulation codes will be uploaded to a public
+repository after publication of the manuscript.
+
+21
+Supplementary materials
+Supplementary information
+1.1 CsPbBr3 TKE temperature dependence
+Fig. S1 | TKE temperature evolution in CsPbBr3. a. TKE in CsPbBr3 at RT and 80K. In
+contrast to MAPbBr3, where the structural phase changes for lower temperatures (from cubic
+to tetragonal to orthorhombic), CsPbBr3 remains in the orthorhombic phase as the temperature
+is lowered. This is also reflected in the overall TKE shape. However, additional oscillations are
+visible on the longer timescales at 80K. b. Fourier transforming the oscillations after the time
+indicated by the arrow reveals two main frequency components of 0.9 and 1.3 THz. These
+frequencies agree well with the two dominating phonon modes in the static Raman spectra of
+CsPbBr3 (54). c, e. THz fluence dependence reveals that both oscillation amplitudes (for 0.9
+and 1.3 THz) scale quadratically with the THz electric field. d. Comparison between simulation
+for an anisotropic material (100 µm thick and 22.5° azimuthal angle between crystal axis and
+probe polarization) considering an electronic response only and experimental room temperature
+CsPbBr3 TKE. This shows that the complex CsPbBr3 TKE signal may be understood in terms
+of an instantaneous electronic polarization response alongside anisotropic light propagation.
+
+a
+0.4
+(norm.)
+1.3
+(arb. u.)
+2
+0.9
+2
+300K
+0.3
+Orth.
+peak (
+efringence (norm.)
+Spectrum
+0.2
+0.5
+ ZHI
+1.5
+0.1
+3.
+0
+0
+1
+2
+3
+Frequency (THz)Transient bire
+ou
+Orth
+(norm.)
+(wou)
+Sim.
+_2 
+0.5
+birefr.
+ak
+0.5
+pea
+0.5
+ZHI
+Trans.
+0
+0.9
+0
+0
+0
+5
+10
+15
+20
+-2
+0
+2
+4
+6
+8
+0
+0.5
+1
+Time (ps)
+Time (ps)22
+1.2 Estimating the THz nonlinear refractive index of MAPbBr3
+Fig. S5 shows a comparison between the TKE in MAPbBr3 and Diamond. The measured TKE
+signal strength 𝑆(𝑑) = Δ𝐼/𝐼0, where 𝐼0 is the total probe intensity measured by the photodiodes
+and Δ𝐼 is the intensity difference, is proportional to Δ𝑛𝜔pr𝑑/𝑐0 in Diamond, where 𝑑 is the
+sample thickness and 𝜔pr is the probing frequency.
+This simple relation holds because there is no significant THz dispersion in Diamond. However,
+due to significant THz absorption and dispersion, this relation does not hold in MAPbBr3 as
+seen in Fig. S4b. For MAPbBr3, 𝑆(𝑑) may rather be approximated by
+Δ𝑛𝜔pr
+𝑐0
+𝑓(𝑑), where
+𝑓(𝑑) = ∫ 𝑑𝑧
+𝑑
+0
+∫
+𝑑𝜔𝐸THz
+2
+(𝜔)exp(−𝛼(𝜔)𝑧)
+∞
+0
+/ ∫
+𝑑𝜔𝐸THz
+2
+(𝜔)
+∞
+0
+.
+S1
+Here, 𝐸THz(ω) is the THz pump spectrum and α is the absorption of MAPbBr3 as extracted
+from the complex refractive index data in Fig. S11.
+Since Δ𝑛 = 𝑛2𝑐0𝜖0𝐸THz
+2
+, we may estimate 𝑛2 of MAPbBr3 using:
+𝑛2
+MA =
+𝑆MA(𝑑MA)𝑑D
+𝑆D(𝑑D)𝑓(𝑑MA) 𝑛2
+D.
+S2
+𝑛2
+D of Diamond has been measured to be 3 × 10−16 cm2/W for 1 THz pump and 800 nm optical
+probing (52).  Based on
+𝑆MA
+𝑆D = 9.4, 𝑓(𝑑𝑀𝐴 = 500 µm ) = 47µm , we therefore estimate 𝑛2
+MA
+to be 2 × 10−14 cm2/W, roughly 80 times higher than 𝑛2
+D for 1 THz pump and 800 nm optical
+probing.
+For comparison, 𝑛2
+MA has been previously measured in the near-infrared spectral region using
+the Z-scan technique (51). They found a similar order of magnitude of 𝑛2
+MA = 9.5 × 10−14
+cm2/W at 1000 nm wavelength.
+
+23
+Supplementary figures
+Experimental figures
+Fig. S2 | MAPbBr3 TKE temporal dependence on THz fluence. Normalised experimental
+TKE of MAPbBr3 at RT for various THz fluences showing that the temporal evolution is not
+affected by the THz-field strength. Fig. 2 in the main text already showed that the 𝑡 = 0 ps peak
+scales quadratically with the THz field amplitude.
+
+1.2
+Max E-Field Amplitude (norm.)
+0.13
+Trans. birefringence (norm.)
+0.17
+0.21
+0.8
+0.26
+0.3
+0.38
+0.6
+0.47
+0.56
+0.4
+0.85
+0.95
+1
+0.2
+0
+-2
+0
+2
+4
+6
+Time (ps)24
+Fig. S3 | MAPbBr3 TKE azimuthal angle dependence at RT. a. TKE signal showing the 4-
+fold rotational symmetry of the measured signal. b, c. TKE signal is normalized to show that
+the time constant of the tail is independent of azimuthal angle. This agrees with the simulations
+for an isotropic material in Fig. S13, where the origin of the exponential tail is high absorption,
+dispersion and pump-probe walkoff, which do not depend on the crystal azimuthal angle. Note
+that the azimuthal angle is not calibrated with respect to the crystal axes in this figure.
+
+b
+a
+c
+Trans.
+Transient
+birefringence (arb. u.)
+birefringence (norm.)
+1.2
+1.2
+MAPbBrs (RT)
+MAPbBr3 (RT)
+0.9
+Azimuthangle(°)
+50
+50
+10
+130
+250
+0.8
+20
+140
+260
+C
+100
+C
+100
+30
+150
+270
+0.7
+ngle
+ngle
+40
+160
+280
+150
+0.8
+nce
+0.6
+50
+170
+290
+150Azimuth
+0.6
+Azimuth 
+0.5
+70
+190
+310
+200
+200
+80
+200
+320
+0.4
+0.4
+90
+210
+330
+250
+0.4
+250
+0.3
+ns.
+100
+220
+340
+ 0.2
+110
+230
+350
+300
+0.2
+120
+240
+360
+0.2
+300
+0.1
+0
+350
+350
+-1
+2
+3
+0
+3
+-2
+0
+2
+4
+-1
+6
+1
+2
+Time (ps)
+Time (ps)
+Time (ps)25
+Fig. S4 | MAPbBr3 TKE dependence on sample thickness. a. Normalised experimental TKE
+thickness dependence of MAPbBr3 at RT. The results agree well with the simulations in Fig.
+S13. b. shows the measured TKE peak signal as a function of sample thickness. The black line
+shows the expected signal dependence when accounting for strong THz absorption and
+dispersion 
+using 
+the 
+formula
+𝑆(𝑑) = ∫ 𝑑𝑧
+𝑑
+0
+∫
+𝑑𝜔𝐸THz
+2
+(𝜔)exp(−𝛼(𝜔)𝑧)
+∞
+0
+/
+∫
+����𝑧
+1000
+0
+∫
+𝑑𝜔𝐸THz
+2
+(𝜔)exp(−𝛼(𝜔)𝑧)
+∞
+0
+, where 𝐸𝑇𝐻𝑧(𝜔) is the THz pump spectrum and 𝛼 is
+the absorption of MAPbBr3 as extracted from the complex refractive index data in Fig. S11.
+
+a
+b
+1.2
+THz field squared
+d=972μm
+d=547μm
+d=132μm
+0.8
+(norm.)
+0.6birefrin
+pea
+0.4
+0.4
+TKE 
+0.2
+0
+0
+-2
+0
+2
+4
+6
+0
+200
+400
+600
+800
+1000
+Time (ps)
+Thickness d (um)26
+Fig. S5 | Comparison between TKE in MAPbBr3 and Diamond for estimating the THz
+nonlinear refractive index n2. Diamond has already been shown to have a strong THz-induced
+Kerr nonlinearity and be a good nonlinear material in the THz range (51). 𝑛2 of Diamond has
+been measured to be 3 × 10−16 cm2/W for 1 THz pump and 800 nm optical probing (52). For
+a 500 µm thick MAPbBr3 single crystal, the TKE peak signal is about 10 times bigger than for
+a 400 µm thick Diamond.
+
+MAPbBr,(d=500um)
+0.8
+Transient birefringence (
+Diamond (d=400um)
+0.6
+0.4
+0.2
+0
+-2
+0
+2
+4
+Time (ps)27
+Fig. S6 | CsPbBr3 TKE azimuthal angle dependence at RT. Although the main peak exhibits
+a 4-fold symmetry, the temporal evolution as a function of azimuthal angle is more complex
+than for MAPbBr3. As CsPbBr3 is in the orthorhombic phase at room temperature, this extra
+complexity might be explained by additional static birefringence and resulting anistropic light
+propagation as can be seen in Fig. S14. Note that the azimuthal angle is not calibrated with
+respect to the crystal axes in this figure.
+
+Transient
+birefringence (arb. u.)
+0
+0.4
+50
+CsPbBr3 (RT)
+0.3
+Azimuth angle (°)
+100
+0.2
+150
+0.1
+200
+0
+250
+-0.1
+-0.2
+300
+-0.3
+350
+-1
+0
+1
+2
+3
+Time (ps)28
+Fig. S7 | MAPbBr3 TKE at 45° azimuthal angle at 80K. a. MAPbBr3 TKE at 80K for 0° and
+about 45° azimuthal angle. MAPbBr3 is orthorhombic at 80K, which might explain the different
+overall signal shape for both orientations. However, in both TKEs we can see a strong
+oscillatory signal. b. By subtracting off fits to the tails (dotted line in a) for the TKEs at 0° and
+45°, we extract the oscillatory signals. c. Fourier transforming the oscillatory signals in b.
+reveals that the same 1.1 THz mode dominates the oscillatory response at 0° and 45° azimuthal
+angle.
+
+a
+b
+C
+ingence (norm.)
+1.5
+Azimuth angle (°)
+('n
+1.1THz
+0°
+(arb.
+1.1THz
+3
+45°
+0.8
+0.5
+(arb.
+ignal
+0.6
+0.5
+mTransient birefr
+Oscillatory
+0.4
+-0.5
+Exponentialfit
+0.2
+-0.5
+0
+5
+10
+15
+20
+0
+5
+10
+15
+20
+0
+1
+2
+3
+4
+5
+Time (ps)
+Time (ps)
+Frequency (THz)29
+Fig. S8 | Lorentzian fits to spectral peaks in MAPbBr3 at 180K and 80K. a,c., Oscillatory
+signals extracted from the MAPbBr3 TKE at 180K and 80K in Fig. 3A respectively. The signals
+are extracted by subtracting off exponential fits to the tails from the main TKE signals. b. The
+modulus squared of the Fourier transform of the oscillatory signal at 180K shows a broad peak
+at 1.5 THz, which we fit with a Lorentzian. The FWHM of the Lorentzian amplitude is
+𝛥𝜈FWHM =0.58 THz. This corresponds to a phonon lifetime of 𝜏 = 1/(2𝜋𝛥𝜈FWHM ) = 0.27
+ps. d.  The modulus squared of the Fourier transform of the oscillatory signal at 80K shows two
+peaks at 1.14 THz and 1.39 THz.  By fitting Lorentzians, we obtain phonon lifetimes of 1.7(4)
+ps and 1.5(1) ps for the two peaks respectively.
+
+a
+b
+X10-3
+0.2
+. units)
+(arb. units)
+MAPbBr3 Single Crystal (180K)
+Lorentzianfit
+Wp = 1.5THz,
+0.8
+Oscillatory signal (arb.
+1 OscillatorModel
+T = 0.27(5)ps
+spectrum
+0.6
+0.4
+Power
+0.2
+0
+2
+6
+0
+2
+30.3
+Oscillatory signal (arb. units)
+units
+MAPbBr3SingleCrystal(80K)
+Lorentzian fit
+0.06
+wp = 1.14THz,
+2 Oscillator Model
+t = 1.7(4)ps
+(arb.
+spectrum
+0.04
+0
+Lorentzian fit
+Wp = 1.39THz,
+0.02
+t = 1.5(1)ps
+0.1
+Power
+0
+0
+5
+10
+15
+20
+25
+0
+2
+3
+Time (ps)
+Frequency (THz)30
+Fig. S9 | BK7 TKE at room temperature. a. TKE of a BK7 substrate with 0.5 mm thickness.
+b. shows the BK7 TKE relative to the TKE of a MAPbBr3 thin film on top of a BK7 substrate
+with 0.5 mm thickness at room temperature.
+
+a
+b
+X10-3
+BK7 (0.5mm)
+6
+BK7 (0.5mm)
+MAPbBr3 0n BK7
+0.8
+5
+(norm.)
+0.6Difference
+2
+0.2
+.1
+-2
+0
+2
+4
+6
+-2
+0
+2
+4
+6
+Time (ps)
+Time (ps)31
+Fig. S10 | BK7 TKE for various temperatures. a. TKE temperature dependence of BK7
+substrate with 0.5 mm thickness. b. shows the relative strength and shape compared to
+MAPbBr3 thin film on top of a BK7 substrate for room temperature and 80K.
+
+BK7 (0.5mm)
+a
+b
+BK7 (0.5mm)
+2
+MAPbBr3 0on BK7
+300K
+3.5
+1.5(noi
+250K
+1/IV
+2.
+Difference signal 
+145K
+ransient 
+0.580K
+80K
+0.5
+0
+-2
+0
+2
+4
+6
+0
+5
+10
+Time (ps)
+Time (ps)32
+Simulation figures
+Fig. S11 | Dispersion of MAPbBr3 in the THz region. a. Refractive index 𝑛 and extinction
+coefficient 𝜅 of MAPbBr3 calculated using the dielectric function from Sendner et al. (10). b.
+Absorption coefficient is calculated using relation 𝛼 = 4𝜋𝜅/𝜆. The penetration depth is equal
+to 1/𝛼.
+
+a
+b
+10
+1000
+6000
+5
+fficient α (1/cm)
+oefficient k
+5000
+Depth (μm)
+8
+800
+e Index n
+4
+4000
+6
+600
+3Refractiv
+Extinction
+Coe
+4
+400
+2
+2000
+Absorption
+2
+200
+1000
+0
+0
+0
+0
+0
+1
+2
+3
+4
+5
+0
+1
+2
+3
+4
+5
+Freguency (THz)
+Freguency(THz)33
+Fig. S12 | Extrapolated static birefringence of MAPbBr3 for the simulations of the low
+temperature orthorhombic phase. In the optical region, the refractive index of CsPbBr3 is
+used as measured using the 2D-OKE (46). The static birefringence of CsPbBr3 is then
+extrapolated to the THz region. 𝑛f and 𝑛s correspond to the refractive index along the fast and
+slow crystal axes and static birefringence is defined as the difference between 𝑛f and 𝑛s.
+
+8
+0.03
+Re(n.)
+Re(n.)
+0.025
+Refractive Index n
+6
+Birefringence
+0.02
+0.015
+2
+0.01
+0
+0.005
+0
+1
+2
+3
+4
+5
+Freguency (THz)34
+Fig. S13 | Four-wave-mixing simulation for cubic MAPbBr3 for various thicknesses
+assuming an instantaneous electronic hyperpolarizability response only. a. shows the
+normalised TKE signal for various thickness. For thicknesses larger than 100 µm, we can see
+an exponential tail with a decay time constant largely independent of thickness. b. The
+normalised TKE signals for various thicknesses are plotted on top of each other. On top of the
+exponential tail, there are small modulations, whose onset depends on the thickness. The onset
+time can be roughly estimated by the 𝑡1 time (𝑡1 = (𝑛𝑔,𝑓(𝜔𝑇𝐻𝑧) − 𝑛𝑔,𝑓(𝜔𝑝𝑟))𝑑/𝑐0), where 𝑑
+is the sample thickness and 𝑛𝑔 is the group velocity refractive index (40).
+
+a
+6
+wn006
+ringence (norm.)
+ringence (norm.)
+500μm
+0.8
+300μm
+0.6Transient biret
+200μm
+Transient biref
+0.4
+100μm
+0.2
+0.4μm
+0
+0
+2
+4
+6
+8
+10
+-2
+0
+2
+4
+6
+8
+10
+Time (ps)
+Time (ps)35
+Fig. S14 | Four-wave-mixing simulation for orthorhombic CsPbBr3 assuming an
+instantaneous electronic hyperpolarizability response only. In contrast to the isotropic
+simulation in Fig. S13, the azimuthal angle of the model crystal matters for the temporal TKE
+shape. a-d. Results for 0° azimuthal angle of crystal with respect to probe pulse polarization are
+shown for various thicknesses. For this angle, the birefringence experienced by the probe is
+maximized. For 0° azimuthal angle and for all thicknesses larger than 200 µm, we can see the
+appearance of a short-lived oscillatory signal of around 1.4 THz in (a-d). These oscillations
+arise due to static birefringence, but are too short-lived to explain our experimental observation
+at 80K as shown in (b, d). For 0.4 µm thickness, these oscillations due to static birefringence
+disappear. e-f. Results for 45° azimuthal angle for various thicknesses. For this angle, the
+birefringence experienced by the pump is maximized. The peak t = 0 ps is diminishingly small
+in comparison to the oscillatory features that happen at later times, which is due to the input
+tensor symmetry of 𝑅. The small oscillatory features correspond to internal reflections - similar
+to the small modulations on top of the tail in Fig. S13. The onset time for these oscillatory
+features can be roughly estimated by the 𝑡1 time.
+
+o° azimuth angle
+o° azimuth angle
+a
+c
+rans. birefringence (norm.)
+wno06
+Spectrum (norm.)
+0.8
+M
+500μm
+0.6
+200μm
+0.4
+0.2
+0.4μm0
+N.W
+0
+-2
+0
+2
+4
+6
+8
+10
+0
+1
+2
+3
+4
+5
+Time (ps)
+Frequency (THz)
+b
+d
+ringence (norm.)
+Exp. MAPbBr3 Single
+Crystal (80K)
+0.8
+500μm Simulation
+(norm.
+0.5
+THzFieldSguared
+0.6biref
+0.4
+Trans.
+0.2
+0.5
+0
+0
+5
+10
+15
+0
+1
+2
+3
+4
+5
+Time (ps)
+Frequency (THz)
+45° azimuth angle
+45° azimuth angle
+e
+MTrans. birefringence (no
+0.8
+Spectrum (norm.)
+500um
+0.6
+0.4
+200μm
+0.2
+0.4um
+0
+-2
+0
+2
+4
+6
+8
+10
+0
+1
+2
+3
+4
+5
+Time (ps)
+Frequency (THz)

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+arXiv:2301.02617v1  [math.FA]  6 Jan 2023
+LINEAR TOPOLOGICAL INVARIANTS FOR KERNELS OF
+DIFFERENTIAL OPERATORS BY SHIFTED FUNDAMENTAL
+SOLUTIONS
+A. DEBROUWERE1 AND T. KALMES2
+Abstract. We characterize the condition (Ω) for smooth kernels of partial differen-
+tial operators in terms of the existence of shifted fundamental solutions satisfying
+certain properties.
+The conditions (PΩ) and (PΩ) for distributional kernels are
+characterized in a similar way. By lifting theorems for Fr´echet spaces and (PLS)-
+spaces, this provides characterizations of the problem of parameter dependence for
+smooth and distributional solutions of differential equations by shifted fundamen-
+tal solutions.
+As an application, we give a new proof of the fact that the space
+{f ∈ E (X) | P(D)f = 0} satisfies (Ω) for any differential operator P(D) and any
+open convex set X ⊆ Rd.
+Keywords: Partial differential operators; Fundamental solutions; Linear topological
+invariants.
+MSC 2020: 46A63, 35E20, 46M18
+1. Introduction
+In their seminal work [16] Meise, Taylor and Vogt characterized the constant co-
+efficient linear partial differential operators P(D) = P(−i ∂
+∂x1, . . . , −i ∂
+∂xd) that have a
+continuous linear right inverse on E (X) and/or D′(X) (X ⊆ Rd open) in terms of
+the existence of certain shifted fundamental solutions of P(D). Later on, Frerick and
+Wengenroth [8,27] gave a similar characterization of the surjectivity of P(D) on E (X),
+D′(X), and D′(X)/E (X) as well as of the existence of right inverses of P(D) on the
+latter space. Roughly speaking, these results assert that P(D) satisfies some condition
+(e.g. being surjective on E (X)) if and only if for each compact subset K of X and
+ξ ∈ X far enough away from K there is a shifted fundamental solution E for δξ such
+that E satisfies a certain property on K. Of course, this property depends on the
+condition one wants to characterize. Results of the same type have also been shown for
+spaces of non-quasianalytic ultradifferentiable functions and ultradistributions [13,14]
+and for spaces of real analytic functions [15]. The aim of this paper is to complement
+1Department of Mathematics and Data Science, Vrije Universiteit Brussel, Plein-
+laan 2, 1050 Brussels, Belgium
+2Faculty of Mathematics, Chemnitz University of Technology, 09107 Chemnitz,
+Germany
+E-mail addresses: [email protected], [email protected].
+1
+
+2
+A. DEBROUWERE AND T. KALMES
+the above results by characterizing several linear topological invariants for smooth and
+distributional kernels of P(D) by means of shifted fundamental solutions.
+The study of linear topological invariants for kernels of P(D) goes back to the work
+of Petzsche [19] and Vogt [23] and was reinitiated by Bonet and Doma´nski [1, 2, 5].
+It is motivated by the question of surjectivity of P(D) on vector-valued function and
+distribution spaces, as we now proceed to explain.
+We assume that the reader is
+familiar with the condition (Ω) for Fr´echet spaces [18] and the conditions (PΩ) and
+(PΩ) for (PLS)-spaces [1, 5] (see also the preliminary Section 2). Set EP(X) = {f ∈
+E (X) | P(D)f = 0} and D′
+P(X) = {f ∈ D′(X) | P(D)f = 0}. Suppose that P(D) is
+surjective on E (X), respectively, D′(X). Given a locally convex space E, it is natural to
+ask whether P(D) : E (X; E) → E (X; E), respectively, P(D) : D′(X; E) → D′(X; E)
+is still surjective.
+If E is a space of functions or distributions, this question is a
+reformulation of the well-studied problem of parameter dependence for solutions of
+partial differential equations; see [1, 2, 5] and the references therein.
+The splitting
+theory for Fr´echet spaces [24] implies that the mapping P(D) : E (X; E) → E (X; E)
+for E = D′(Y ) (Y ⊆ Rn open) or S ′(Rn) is surjective if and only if EP(X) satisfies
+(Ω). Similarly, as an application of their lifting results for (PLS)-spaces, Bonet and
+Doma´nski showed that the mapping P(D) : D′(X; E) → D′(X; E) for E = D′(Y ) or
+S ′(Rn) is surjective if and only if DP(X) satisfies (PΩ) [1], while it is surjective for
+E = A (Y ) if and only if D′
+P(X) satisfies (PΩ) [5].
+Petzsche [19] showed that EP(X) satisfies (Ω) for any convex open set X1, while
+Vogt proved that this is the case for an arbitrary open set X if P(D) is elliptic [23].
+Similarly, D′
+P(X) satisfies (PΩ) for any convex open set X [1] and for an arbitrary
+open set X if P(D) is elliptic [1, 9, 23]. On the negative side, the second author [12]
+constructed a differential operator P(D) and an open set X ⊆ Rd such that P(D) is
+surjective on D′(X) (and thus also on E (X)) but EP(X) and D′
+P(X) do not satisfy (Ω),
+respectively, (PΩ). Furthermore, D′
+P(X) does not satisfy (PΩ) for any convex open
+set X if P(D) is hypoelliptic and for an arbitrary open set X if P(D) is elliptic [5,25].
+We refer to [3,5] and the references therein for further results concerning (Ω) for EP(X)
+and (PΩ) and (PΩ) for DP(X).
+Apart from this classical application to the problem of surjectivity of P(D) on spaces
+of vector-valued smooth functions and distributions, in our recent article [4], the linear
+topological invariant (Ω) for EP(X) played an important role to establish quantitative
+approximation results of Runge type for several classes of partial differential operators.
+See [7,20,21] for other works on this topic.
+In the present note, we characterize the condition (Ω) for EP(X) and the conditions
+(PΩ) and (PΩ) for D′
+P(X) in terms of the existence of certain shifted fundamental
+solutions for P(D). By the above mentioned results from [1, 5], the latter provides
+characterizations of the problem of distributional and real analytic parameter depen-
+dence for distributional solutions of the equation P(D)f = g by shifted fundamental
+solutions. This answers a question of Doma´nski [6, Problem 7.5] for distributions.
+1Petzsche actually showed this result under the additional hypothesis that P(D) is hypoelliptic.
+However, as observed in [3], a careful inspection of his proof shows that this hypothesis can be omitted.
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+3
+We now state our main result. Set N = {0, 1, 2, . . .}. Let Y ⊆ Rd be relatively
+compact and open. For N ∈ N we define
+∥f∥Y ,N =
+max
+x∈Y ,|α|≤N |f (α)(x)|,
+f ∈ CN(Y ),
+and
+∥f∥∗
+Y ,N = sup{|⟨f, ϕ⟩| | ϕ ∈ DY , ∥ϕ∥Y ,N ≤ 1},
+f ∈ (DY )′,
+where DY denotes the Fr´echet space of smooth functions with support in Y .
+Theorem 1.1. Let P ∈ C[ξ1, . . . , ξd], let X ⊆ Rd be open, and let (Xn)n∈N be an
+exhaustion by relatively compact open subsets of X.
+(a) P(D) : E (X) → E (X) is surjective and EP(X) satisfies (Ω) if and only if
+∀ n ∈ N ∃ m ≥ n, N ∈ N ∀ k ≥ m, ξ ∈ X\Xm ∃ K ∈ N, s, C > 0 ∀ ε ∈ (0, 1)
+∃ Eξ,ε ∈ D′(Rd) with P(D)Eξ,ε = δξ in Xk such that
+∥Eξ,ε∥∗
+Xn,N ≤ ε
+and
+∥Eξ,ε∥∗
+Xk,K ≤ C
+εs.
+(1.1)
+(b) P(D) : D′(X) → D′(X) is surjective and D′
+P(X) satisfies (PΩ) if and only if
+∀ n ∈ N ∃ m ≥ n ∀ k ≥ m, N ∈ N, ξ ∈ X\Xm ∃ K ∈ N, s, C > 0 ∀ ε ∈ (0, 1)
+∃ Eξ,ε ∈ D′(Rd) ∩ CN(Xn) with P(D)Eξ,ε = δξ in Xk such that
+∥Eξ,ε∥Xn,N ≤ ε
+and
+∥Eξ,ε∥∗
+Xk,K ≤ C
+εs.
+(1.2)
+(c) P(D) : D′(X) → D′(X) is surjective and D′
+P(X) satisfies (PΩ) if and only if
+(1.2) with “∃ K ∈ N, s, C > 0′′ replaced by “∀s > 0 ∃ K ∈ N, C > 0′′ holds.
+The proof of Theorem 1.1 will be given in Section 3. Interestingly, Theorem 1.1 is
+somewhat of a different nature than the above mentioned results from [8,16,27] in the
+sense that the characterizing properties on the shifted fundamental solutions Eξ,ε are
+not only about the behavior of Eξ,ε on the Xn but also on the larger set Xk. In this
+regard, we mention that P(D) is surjective on E (X), respectively, D′(X) if and only
+if (1.1), respectively, (1.2) without the assumption ∥Eξ,ε∥∗
+Xk,K ≤ C
+εs holds [27].
+It would be interesting to evaluate the conditions in Theorem 1.1 in specific cases
+in order to obtain concrete necessary and sufficient conditions on X and P for EP(X)
+to satisfy (Ω) and for D′
+P(X) to satisfy (PΩ) and (PΩ) (cf. [14, 16]).
+We plan to
+study this in the future. As a first result in this direction, we show in Section 4 that
+EP(X) satisfies (Ω) for any differential operator P(D) and any open convex set X by
+combining Theorem 1.1(a) with a powerful method to construct fundamental solutions
+due to H¨ormander [10, Proof of Theorem 7.3.2]. As mentioned above, this result is
+originally due to Petzsche [19], who proved it with the aid of the fundamental principle
+of Ehrenpreis. A completely different proof was recently given by the authors in [3].
+Finally, we would like to point out that Theorem 1.1 implies that surjectivity of
+P(D) on E (X) and (Ω) for EP(X) as well as surjectivity of P(D) on D′(X) and (PΩ),
+respectively, (PΩ), for D′
+P(X) are preserved under taking finite intersections of open
+
+4
+A. DEBROUWERE AND T. KALMES
+sets. For (PΩ) this also follows from [1, Proposition 8.3] and the fact that surjectivity
+of P(D) is preserved under taking finite intersections. However, for (Ω) and (PΩ) we
+do not see how this may be shown without Theorem 1.1.
+2. Linear topological invariants
+In this preliminary section we introduce the linear topological invariants (Ω) for
+Fr´echet spaces and (PΩ) and (PΩ) for (PLS)-spaces. We refer to [1, 5, 18] for more
+information about these conditions and examples of spaces satisfying them.
+Throughout, we use standard notation from functional analysis [18] and distribution
+theory [10,22]. In particular, given a locally convex space E, we denote by U0(E) the
+filter basis of absolutely convex neighborhoods of 0 in E and by B(E) the family of
+all absolutely convex bounded sets in E.
+2.1. Projective spectra. A projective spectrum (of locally convex spaces)
+E = (En, ̺n
+n+1)n∈N
+consists of locally convex spaces En and continuous linear maps ̺n
+n+1 : En+1 → En,
+called the spectral maps. We define ̺n
+n = idEn and ̺n
+m = ̺n
+n+1 ◦ · · · ◦ ̺m−1
+m
+: Em → En
+for n, m ∈ N with m > n. The projective limit of E is defined as
+Proj E =
+�
+(xn)n∈N ∈
+�
+n∈N
+En | xn = ̺n
+n+1(xn+1), ∀n ∈ N
+�
+.
+For n ∈ N we write ̺n : Proj E → En, (xj)j∈N �→ xn. We always endow Proj E with
+its natural projective limit topology. For a projective spectrum E = (En, ̺n
+n+1)n∈N of
+Fr´echet spaces, the projective limit Proj E is again a Fr´echet space. We will implicitly
+make use of the derived projective limit Proj1 E. We refer to [26, Sections 2 and 3] for
+more information. In particular, see [26, Theorem 3.1.4] for an explicit definition of
+Proj1 E.
+2.2. The condition (Ω) for Fr´echet spaces. A Fr´echet space E is said to satisfy
+the condition (Ω) [18] if
+∀U ∈ U0(E) ∃V ∈ U0(E) ∀W ∈ U0(E) ∃s, C > 0 ∀ε ∈ (0, 1) : V ⊆ εU + C
+εsW.
+The following result will play a key role in the proof of Theorem 1.1(a).
+Lemma 2.1. [3, Lemma 2.4] Let E = (En, ̺n
+n+1)n∈N be a projective spectrum of Fr´echet
+spaces. Then, Proj1 E = 0 and Proj E satisfies (Ω) if and only if
+∀n ∈ N, U ∈ U0(En) ∃m ≥ n, V ∈ U0(Em) ∀k ≥ m, W ∈ U0(Ek) ∃s, C > 0 ∀ε ∈ (0, 1) :
+̺n
+m(V ) ⊆ εU + C
+εs̺n
+k(W).
+(2.1)
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+5
+2.3. The conditions (PΩ) and (PΩ) for (PLS)-spaces. A locally convex space E
+is called a (PLS)-space if it can be written as the projective limit of a spectrum of
+(DFS)-spaces.
+Let E = (En, ̺n
+n+1)n∈N be a spectrum of (DFS)-spaces. We call E strongly reduced if
+∀n ∈ N ∃m ≥ n : ̺n
+m(Em) ⊆ ̺n(Proj E).
+The spectrum E is said to satisfy (PΩ) if
+∀n ∈ N ∃m ≥ n ∀k ≥ m ∃B ∈ B(En) ∀M ∈ B(Em) ∃K ∈ B(Ek), s, C > 0 ∀ε ∈ (0, 1) :
+̺n
+m(M) ⊆ εB + C
+εs̺n
+k(K).
+(2.2)
+The spectrum E is said to satisfy (PΩ) if (2.2) with “∃K ∈ B(Ek), s, C > 0” replaced
+by “∀s > 0 ∃K ∈ B(Ek), C > 0” holds.
+A (PLS)-space E is said to satisfy (PΩ), respectively, (PΩ) if E = Proj E for some
+strongly reduced spectrum E of (DFS)-spaces that satisfies (PΩ), respectively, (PΩ).
+This notion is well-defined as [26, Proposition 3.3.8] yields that all strongly reduced
+projective spectra E of (DFS)-spaces with E = Proj E are equivalent (in the sense
+of [26, Definition 3.1.6]).
+The bipolar theorem and [1, Lemma 4.5] imply that the
+above definitions of (PΩ) and (PΩ) are equivalent to the original ones from [1].
+3. Proof of Theorem 1.1
+This section is devoted to the proof of Theorem 1.1. We fix P ∈ C[ξ1, . . . , ξd]\{0},
+an open set X ⊆ Rd, and an exhaustion by relatively compact open subsets (Xn)n∈N of
+X. For n, N ∈ N we write ∥ · ∥n,N = ∥ · ∥Xn,N and ∥ · ∥∗
+n,N = ∥ · ∥∗
+Xn,N. For ξ ∈ Rd and
+r > 0 we denote by B(ξ, r) the open ball in Rd with center ξ and radius r. Moreover,
+for p ∈ {1, ∞} and N ∈ N we set
+∥ϕ∥Lp,N = max
+|α|≤N ∥ϕ(α)∥Lp,
+ϕ ∈ D(Rd).
+We fix χ ∈ D(B(0, 1)) with χ ≥ 0 and
+�
+Rd χ(x)dx = 1, and set χε(x) = ε−dχ(x/ε) for
+ε > 0.
+3.1. Proof of Theorem 1.1(a). We write E (X) for the space of smooth functions in
+X endowed with its natural Fr´echet space topology. We set
+EP(X) = {f ∈ E (X) | P(D)f = 0}
+and endow it with the relative topology induced by E (X).
+Let n ∈ N.
+We write E (Xn) for the space of smooth functions in Xn endowed
+with its natural Fr´echet space topology, i.e, the one induced by the sequence of norms
+(∥ · ∥n,N)N∈N. We define
+EP(Xn) = {f ∈ E (Xn) | P(D)f = 0}
+
+6
+A. DEBROUWERE AND T. KALMES
+and endow it with the relative topology induced by E (Xn). Since EP(Xn) is closed in
+E (Xn), it is a Fr´echet space. For N ∈ N we set
+Un,N = {f ∈ EP(Xn) | ∥f∥n,N ≤ 1}.
+Note that
+�
+1
+N+1Un,N
+�
+N∈N is a decreasing fundamental sequence of absolutely convex
+neighborhoods of 0 in E (Xn).
+Consider the projective spectrum (EP(Xn), ̺n
+n+1)n∈N with ̺n
+n+1 the restriction map
+from EP(Xn+1) to EP(Xn). Then,
+EP(X) = Proj(EP(Xn), ̺n
+n+1)n∈N.
+By [3, Lemma 3.1(i)] (see also [26, Section 3.4.4]), P(D) : E (X) → E (X) is surjective
+if and only if
+Proj1(EP(Xn), ̺n
+n+1)n∈N = 0.
+Hence, Lemma 2.1 and a simple rescaling argument yield that P(D) : E (X) → E (X)
+is surjective and EP(X) satisfies (Ω) if and only if
+∀n, N ∈ N ∃m ≥ n, M ≥ N ∃k ≥ m, K ≥ M ∃s, C > 0 ∀ε ∈ (0, 1) :
+Um,M ⊆ εUn,N + C
+εsUk,K,
+(3.1)
+where we did not write the restriction maps explicitly, as we shall not do in the sequel
+either. We are ready to show Theorem 1.1(a).
+Sufficiency of (1.1). It suffices to show (3.1). Let n, N ∈ N be arbitrary. Choose �m, �N
+according to (1.1) for n + 1. Set m = �m + 1 and M = N + �N + deg P + 1. Let
+k ≥ m, K ≥ M be arbitrary. Choose ψ ∈ D(Xm) such that ψ = 1 in a neighborhood
+of X �m. Pick ε0 ∈ (0, 1] such that ψ = 1 on X �m + B(0, ε0), supp ψ + B(0, ε0) ⊆ Xm,
+Xn + B(0, ε0) ⊆ Xn+1, Xk + B(0, ε0) ⊆ Xk+1.
+Cover the compact set Xk\X �m by
+finitely many balls B(ξj, ε0), j ∈ J, with ξj ∈ X\X �m, and choose ϕj ∈ D(B(ξj, ε0)),
+j ∈ J, such that �
+j∈J ϕj = 1 in a neighborhood of Xk\X �m. As J is finite, (1.1) for
+k + 1 implies that there are �K ∈ N, �s, �C > 0 such that for all ε ∈ (0, ε0) there exist
+Eξj,ε ∈ D′(Rd), j ∈ J, with P(D)Eξj,ε = δξj in Xk+1 such that
+(3.2)
+∥Eξj,ε∥∗
+n+1, ˜
+N ≤ ε
+and
+∥Eξj,ε∥∗
+k+1, �
+K ≤
+�C
+ε�s.
+Let f ∈ EP(Xm) be arbitrary. For ε ∈ (0, ε0) we define fε = (ψf) ∗ χε ∈ EP(X �m) and
+hε =
+�
+j∈J
+Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)fε).
+Since δ−ξj ∗ (ϕjP(D)fε) = (ϕjP(D)fε)(· + ξj) ∈ D(B(0, ε0)), j ∈ J, it holds that
+P(D)hε = �
+j∈J ϕjP(D)fε in a neighborhood of Xk. As �
+j∈J ϕj = 1 in a neighborhood
+of Xk\X �m and fε ∈ EP(X �m), we obtain that P(D)hε = P(D)fε in a neighborhood of
+Xk and thus hε ∈ EP(X �m) and fε − hε ∈ EP(Xk). We decompose f as follows
+f = (f − fε + hε) + (fε − hε) ∈ EP(Xn) + EP(Xk).
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+7
+We claim that there are s, Ci > 0, i = 1, 2, 3, 4, such that for all f ∈ EP(Xm) and
+ε ∈ (0, ε0)
+∥f − fε∥n,N ≤ C1ε∥f∥m,M,
+∥hε∥n,N ≤ C2ε∥f∥m,M,
+∥fε∥k,K ≤ C3
+εs ∥f∥m,M,
+∥hε∥k,K ≤ C4
+εs ∥f∥m,M,
+which implies (3.1). Let f ∈ EP(Xm) and ε ∈ (0, ε0) be arbitrary. By the mean value
+theorem, we find that
+∥f − fε∥n,N ≤ ε
+√
+d∥f∥n+1,N+1 ≤ ε
+√
+d∥f∥m,M.
+Furthermore, it holds that
+∥fε∥k,K ≤ ∥χ∥L1,K
+εK
+∥ψf∥L∞ ≤ ∥χ∥L1,K∥ψ∥L∞
+εK
+∥f∥m,M.
+By the first inequality in (3.2), we obtain that
+∥hε∥n,N
+≤
+�
+j∈J
+∥Eξj,ε∥∗
+n+1, �
+N∥(ϕjP(D)fε)(· + ξj)∥L∞,N+ �
+N
+≤
+ε
+�
+j∈J
+∥ϕj((P(D)(ψf)) ∗ χε)∥L∞,N+ �
+N
+≤
+C′
+2ε∥P(D)(ψf)∥L∞,N+ �
+N
+≤
+C2ε∥f∥m,N+ �
+N+deg P ≤ C2ε∥f∥m,M,
+for some C′
+2, C2 > 0. Similarly, by the second inequality in (3.2), we find that
+∥hε∥k,K
+≤
+�
+j∈J
+∥Eξj,ε∥∗
+k+1, �
+K∥(ϕjP(D)fε)(· + ξj)∥L∞,K+ �
+K
+≤
+�C
+ε�s
+�
+j∈J
+∥ϕj((P(D)(ψf)) ∗ χε)∥L∞,K+ �
+K
+≤
+C′
+4
+ε�s ∥P(D)(ψf)∥L∞∥χε∥L1,K+ �
+K
+≤
+C4
+ε�s+K+ �
+K ∥f∥m,deg P ≤
+C4
+ε�s+K+ �
+K ∥f∥m,M,
+for some C′
+4, C4 > 0.This proves the claim with s = �s + K + �K.
+□
+Necessity of (1.1). As explained above, condition (3.1) holds.
+Let F ∈ D′(Rd) be
+a fundamental solution for P(D) of finite order q. Let n ∈ N be arbitrary. Choose
+m, �
+M ∈ N according to (3.1) for n and 0. Set N = q + 1. Let k ≥ m and ξ ∈ X\Xm
+be arbitrary. Set K = q + 1. (3.1) for k + 1 and 0 implies that there are �C, �s > 0 such
+that for all δ ∈ (0, 1)
+(3.3)
+Um,�
+M ⊆ δUn,0 +
+�C
+δ�sUk+1,0.
+
+8
+A. DEBROUWERE AND T. KALMES
+Let ε0 ∈ (0, 1] be such that B(ξ, ε0) ⊆ X\Xm. Set Fξ = F ∗ δξ ∈ D′(Rd). For all
+ε ∈ (0, ε0) it holds that Fξ ∗ χε ∈ EP(Xm) and
+∥Fξ ∗ χε∥m,�
+M ≤
+C′
+εd+�
+M+q
+with C′ = ∥Fξ∥∗
+Xm+B(0,ε0),q. Hence, (3.3) with δ = εd+�
+M+q+1 implies that
+Fξ ∗ χε ∈
+C′
+εd+�
+M+q Um,�
+M ⊆ C′εUn,0 + C′ �C
+εs Uk+1,0,
+with s = d + �
+M + q + �s(d + �
+M + q + 1). Let fξ,ε ∈ C′εUn,0 and hξ,ε ∈ C′ �Cε−sUk+1,0 be
+such that
+(3.4)
+Fξ ∗ χε = fξ,ε + hξ,ε.
+Choose ψ ∈ D(Xk+1) such that ψ = 1 in a neighborhood of Xk and define Eξ,ε =
+Fξ − ψhξ,ε ∈ D′(Rd). Then, P(D)Eξ,ε = δξ in Xk. Moreover, for all ε ∈ (0, ε0) it holds
+that
+∥Eξ,ε∥∗
+n,q+1
+≤
+∥Fξ − Fξ ∗ χε∥∗
+n,q+1 + ∥Fξ ∗ χε − ψhξ,ε∥∗
+n,q+1
+≤
+∥Fξ∥∗
+Xn+B(0,ε0),q
+√
+dε + ∥fξ,ε∥n,0
+≤
+(∥Fξ∥∗
+Xn+B(0,ε0),q
+√
+d + C′)ε,
+where we used the mean value theorem, and
+∥Eξ,ε∥∗
+k,q+1
+≤
+∥Fξ∥∗
+k,q+1 + ∥ψhξ,ε∥∗
+k,q+1
+≤
+∥Fξ∥∗
+k,q+1 + |Xk|∥hξ,ε∥k,0
+≤
+∥Fξ∥∗
+k,q+1 + C′ �C|Xk|
+εs
+,
+where |Xk| denotes the Lebesgue measure of Xk. This completes the proof.
+□
+3.2. Proof of Theorem 1.1(b) and (c). We write D′(X) for the space of distribu-
+tions in X endowed with its strong dual topology. We set
+D′
+P(X) = {f ∈ D′(X) | P(D)f = 0}
+and endow it with the relative topology induced by D′(X).
+In [27, Theorem (5)] it is shown that the mapping P(D) : D′(X) → D′(X) is
+surjective if and only if
+∀ n ∈ N ∃ m ≥ n ∀ k ≥ m, N ∈ N, ξ ∈ X\Xm, ε ∈ (0, 1)
+∃ Eξ,ε ∈ D′(Rd) ∩ CN(Xn) with P(D)Eξ,ε = δξ in Xk such that
+∥Eξ,ε∥Xn,N ≤ ε.
+(3.5)
+Let n ∈ N. We endow the space DXn of smooth functions with support in Xn with
+the relative topology induced by E (Xn). We write D′(Xn) for the strong dual of DXn.
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+9
+Then, D′(Xn) is a (DFS)-space. We define
+D′
+P(Xn) = {f ∈ D′(Xn) | P(D)f = 0}
+and endow it with the relative topology induced by D′(Xn). Since D′
+P(Xn) is closed
+in D′(Xn), it is a (DFS)-space. For N ∈ N we set
+Bn,N = {f ∈ D′
+P(Xn) | ∥f∥∗
+n,N ≤ 1}.
+Note that (NBn,N)N∈N is an increasing fundamental sequence of absolutely convex
+bounded sets in D′
+P(Xn).
+Consider the projective spectrum (D′
+P(Xn), ̺n
+n+1)n∈N with ̺n
+n+1 the restriction map
+from D′
+P(Xn+1) to D′
+P(Xn). Then,
+D′
+P(X) = Proj(D′
+P(Xn), ̺n
+n+1)n∈N.
+By [3, Lemma 3.1(ii)] (see also [26, Section 3.4.5]), P(D) : D′(X) → D′(X) is surjective
+if and only if
+Proj1(D′
+P(Xn), ̺n
+n+1)n∈N = 0.
+The latter condition implies that (D′
+P(Xn), ̺n
+n+1)n∈N is strongly reduced [26, Theorem
+3.2.9]. Hence, if P(D) : D′(X) → D′(X) is surjective, D′
+P(X) satisfies (PΩ), respec-
+tively, (PΩ) if and only if (D′
+P(Xn), ̺n
+n+1)n∈N does so. A simple rescaling argument
+yields that (D′
+P(Xn), ̺n
+n+1)n∈N satisfies (PΩ) if and only if
+∀n ∈ N ∃m ≥ n ∀k ≥ m ∃N ∈ N ∀M ∈ N ∃K ∈ N, s, C > 0 ∀ε ∈ (0, 1) :
+Bm,M ⊆ εBn,N + C
+εsBk,K,
+(3.6)
+where, as before, we did not write the restriction maps explicitly. Similarly, the spec-
+trum (D′
+P(Xn), ̺n
+n+1)n∈N satisfies (PΩ) if and only if (3.6) with “∃K ∈ N, s, C > 0”
+replaced by “∀s > 0 ∃K ∈ N, C > 0” holds. We now show Theorem 1.1(b).
+Sufficiency of (1.2). Clearly, (1.2) implies (3.5) and thus that P(D) : D′(X) → D′(X)
+is surjective. Hence, by the above discussion, it suffices to show (3.6). Let n ∈ N
+be arbitrary. Choose m according to (1.2) for n + 1. Let k ≥ m be arbitrary. Set
+N = 0. Let M ∈ N be arbitrary. Pick ε0 ∈ (0, 1] such that Xn + B(0, ε0) ⊆ Xn+1
+and Xk + B(0, ε0) ⊆ Xk+1.
+Cover the compact set Xk\Xm by finitely many balls
+B(ξj, ε0), j ∈ J, with ξj ∈ X\Xm, and choose ϕj ∈ D(B(ξj, ε0)), j ∈ J, such that
+�
+j∈J ϕj = 1 in a neighborhood of Xk\Xm. As J is finite, (1.2) for k+1 and M +deg P
+implies that there are �K ∈ N, �s, �C > 0 such that for all ε ∈ (0, ε0) there exist Eξj,ε ∈
+D′(Rd) ∩ CM+deg P(Xn+1), j ∈ J, with P(D)Eξj,ε = δξj in Xk+1 such that
+(3.7)
+∥Eξj,ε∥Xn+1,M+deg P ≤ ε
+and
+∥Eξj,ε∥∗
+k+1, �
+K ≤
+�C
+ε�s.
+
+10
+A. DEBROUWERE AND T. KALMES
+Pick ψ ∈ D(Xm) with ψ = 1 in a neighborhood of Xn.
+Let f ∈ D′
+P(Xm) with
+∥f∥∗
+m,M < ∞ be arbitrary. For ε ∈ (0, ε0) we define
+hε =
+�
+j∈J
+Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)(ψf)).
+By the same reasoning as in the proof of part (a) it follows that hε ∈ D′
+P(Xn) and ψf −
+hε ∈ D′
+P(Xk). Furthermore, as Eξj,ε ∈ D′(Rd) ∩ CM+deg P(Xn+1) and the distributions
+δ−ξj ∗ (ϕjP(D)(ψf)) = ϕjP(D)(ψf)(· + ξj), j ∈ J, have order at most M + deg P and
+are supported in B(0, ε0), it holds that hε ∈ C(Xn). We decompose f as follows in Xn
+f = ψf = hε + (ψf − hε) ∈ (D′
+P(Xn) ∩ C(Xn)) + D′
+P(Xk).
+We claim that there are K ∈ N, s, C1, C2 > 0 such that for all f ∈ D′
+P(Xm) with
+∥f∥∗
+m,M < ∞ and ε ∈ (0, ε0)
+∥hε∥∗
+n,0 ≤ C1ε∥f∥∗
+m,M,
+∥ψf − hε∥∗
+k,K ≤ C2
+εs ∥f∥∗
+m,M,
+which implies (3.6). Let f ∈ D′
+P(Xm) with ∥f∥∗
+m,M < ∞ and ε ∈ (0, ε0) be arbitrary.
+Choose ρ ∈ D(Xm) with ρ = 1 in a neighborhood of supp ψ. The first inequality in
+(3.7) implies that
+∥hε∥∗
+n,0
+≤
+|Xn|∥hε∥n,0
+≤
+|Xn|
+�
+j∈J
+∥P(D)(ψf)∥∗
+m,M+deg P sup
+x∈Xn
+∥(ϕjρ)Eξj,ε(x + ξj − ·)∥L∞,M+deg P
+≤
+C1∥f∥∗
+m,M∥Eξj,ε∥n+1,M+deg P
+≤
+C1ε∥f∥∗
+m,M
+for some C1 > 0. Next, by the second inequality in (3.7), we obtain that for all ϕ ∈ DXk
+|⟨hε, ϕ⟩|
+≤
+�
+j∈J
+|⟨Eξj,ε ∗ δ−ξj ∗ (ϕjP(D)(ψf)), ϕ⟩|
+=
+�
+j∈J
+|⟨Eξj,ε, (δ−ξj ∗ (ϕjP(D)(ψf)))∨ ∗ ϕ⟩|
+≤
+�
+j∈J
+∥Eξj,ε∥∗
+k+1, �
+K∥(δ−ξj ∗ (ϕjP(D)(ψf)))∨ ∗ ϕ∥L∞, �
+K
+≤
+�C
+ε�s
+�
+j∈J
+∥P(D)(ψf)∥∗
+m,M+deg P sup
+x∈Rd ∥(ϕjρ)ϕ(· − ξj − x)∥L∞, �
+K+M+deg P
+≤
+C′
+2
+ε�s ∥f∥∗
+m,M∥ϕ∥L∞, �
+K+M+deg P,
+for some C′
+2 > 0, whence
+∥hε∥∗
+k, �
+K+M+deg P ≤ C′
+2
+ε�s ∥f∥∗
+m,M.
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+11
+Furthermore,
+∥ψf∥∗
+k, �
+K+M+deg P ≤ C′′
+2∥f∥∗
+m,M,
+for some C′′
+2 > 0. Therefore,
+∥ψf − hε∥∗
+k, �
+K+M+deg P ≤ C′′
+2 + C′
+2
+ε�s
+∥f∥∗
+m,M.
+This shows the claim with K = �K + M + deg P and s = �s.
+□
+Necessity of (1.2). As explained above, conditions (3.5) and (3.6) hold. Let n ∈ N
+be arbitrary. Choose �m according to (3.6) for n + 1 and m according to (3.5) for �m.
+Let k ≥ m, N ∈ N, and ξ ∈ X\Xm be arbitrary. (3.5) for k and N + 1 implies that
+there exist Fξ ∈ D′(Rd) ∩ CN+1(X �m) with P(D)Fξ = δξ in Xk such that ∥Fξ∥ �m,N+1 ≤
+min{1, 1/|X �m|}. By (3.6) for k + 1, we obtain that there are �N, �K ∈ N, �s, �C > 0 such
+that for all δ ∈ (0, 1)
+(3.8)
+B �m,0 ⊆ δBn+1, �
+N +
+�C
+δ�sBk+1, �
+K.
+Note that Fξ ∈ D′
+P(X �m) and ∥Fξ∥∗
+�m,0 ≤ |X �m|∥Fξ∥ �m,0 ≤ 1, whence Fξ ∈ B �m,0. There-
+fore, (3.8) yields that for all δ ∈ (0, 1) there are Gξ,δ ∈ δBn+1, �
+N and Hξ,δ ∈ �Cδ−�sBk+1, �
+K
+such that Fξ = Gξ,δ + Hξ,δ in Xn+1.
+Let ψ ∈ D(Xk+1) be such that ψ = 1 on
+a neighborhood of Xk.
+Choose ε0 ∈ (0, 1] such that ψ = 1 on Xk + B(0, ε0) and
+Xn + B(0, ε0) ⊆ Xn+1. For δ ∈ (0, 1) and ε ∈ (0, ε0) we define
+Eξ,ε,δ = Fξ − (ψHξ,δ) ∗ χε ∈ D′(Rd).
+Since Hξ,δ ∈ D′
+P(Xk+1), we have that (ψHξ,δ) ∗ χε ∈ EP(Xk). This implies that Eξ,ε,δ ∈
+CN(Xn) and P(D)Eξ,ε,δ = δξ in Xk. As ψ = 1 on Xn+1, it holds that (ψHξ,δ) ∗ χε =
+Hξ,δ ∗ χε on Xn. Hence, we obtain that
+∥Eξ,ε,δ∥n,N
+≤
+∥Fξ − Fξ ∗ χε∥n,N + ∥Fξ ∗ χε − Hξ,δ ∗ χε∥n,N
+≤
+√
+dε + ∥Gξ,δ ∗ χε∥n,N
+≤
+√
+dε + ∥χ∥L∞,N+ �
+N
+δ
+εN+ �
+N+d,
+where we used the mean value theorem. Let r ∈ N be the order of Fξ in Xk and set
+K = max{r, �K}. Then,
+∥Eξ,ε,δ∥∗
+k,K ≤ ∥Fξ∥∗
+k,r + ∥(ψHξ,δ) ∗ χε∥∗
+k, �
+K ≤ ∥Fξ∥∗
+k,r + C′
+δ�s
+,
+for some C′ > 0. For ε ∈ (0, ε0) we set δε = εN+ �
+N+d+1 and Eξ,ε = Eξ,ε,δε. We obtain
+that Eξ,ε ∈ CN(Xn) with P(D)Eξ,ε = δξ in Xk. Furthermore, there are C1, C2 > 0
+such that for all ε ∈ (0, ε0)
+∥Eξ,ε,δ∥n,N ≤ C1ε
+and
+∥Eξ,ε,δ∥∗
+k,K ≤ C2
+εs
+with s = �s(N + �
+N + d + 1). This completes the proof.
+□
+
+12
+A. DEBROUWERE AND T. KALMES
+Theorem 1.1(c) can be shown in the same way as Theorem 1.1(b) (see in particular
+the values of s in terms of �s in the above proof). We leave the details to the reader.
+4. The condition (Ω) for EP(X) if X is convex
+In this final section we use Theorem 1.1(a) to prove that P(D) : E (X) → E (X) is
+surjective and EP(X) satisfies (Ω) for any non-zero differential operator P(D) and any
+open convex set X ⊆ Rd. To this end, we show that (1.1) holds for any exhaustion by
+relatively compact open convex subsets (Xn)n∈N of X. The latter is a consequence of
+the following lemma.
+Lemma 4.1. Let P ∈ C[ξ1, . . . , ξd]\{0}. Let K ⊆ Rd be compact and convex, and
+let ξ ∈ Rd be such that ξ /∈ K. For all ε ∈ (0, 1) there exists Eξ,ε ∈ D′(Rd) with
+P(D)Eξ,ε = δξ in Rd such that
+∥Eξ,ε∥∗
+K,d+1 ≤ ε
+and for every L ⊂ Rd compact and convex there are s, C > 0 such that
+∥Eξ,ε∥∗
+L,d+1 ≤ C
+εs.
+The rest of this section is devoted to the proof of Lemma 4.1, which is based on a
+construction of fundamental solutions due to H¨ormander [10, proof of Theorem 7.3.10].
+We need some preparation. For Q ∈ C[ξ1, . . . , ξd] we define
+�Q(ζ) =
+� �
+α∈Nd
+|Q(α)(ζ)|2
+�1/2
+,
+ζ ∈ Cd.
+Let m ∈ N. We denote by Pol◦(m) the finite-dimensional vector space of non-zero
+polynomials in d variables of degree at most m with the origin removed. By [10, Lemma
+7.3.11 and Lemma 7.3.12] there exists a non-negative Φ ∈ C∞(Pol◦(m)×Cd) such that
+(i) For all Q ∈ Pol◦(m) it holds that Φ(Q, ζ) = 0 if |ζ| > 1 and
+�
+Cd Φ(Q, ζ)dζ = 1.
+(ii) For all entire functions F on Cd and Q ∈ Pol◦(m) it holds that
+�
+Cd F(ζ)Φ(Q, ζ)dζ = F(0).
+(iii) There is A > 0 such that for all Q ∈ Pol◦(m) and ζ ∈ Cd with Φ(Q, ζ) ̸= 0 it
+holds that
+�Q(0) ≤ A|Q(ζ)|.
+Let K ⊆ Rd be compact and convex.
+As customary, we define the supporting
+function of K as
+HK(η) = sup
+x∈K
+η · x,
+η ∈ Rd.
+
+LINEAR TOPOLOGICAL INVARIANTS BY SHIFTED FUNDAMENTAL SOLUTIONS
+13
+Note that HK is subadditive and positive homogeneous of degree 1. Furthermore, it
+holds that [10, Theorem 4.3.2]
+(4.1)
+K = {x ∈ Rd | η · x ≤ HK(η), ∀η ∈ Rd}.
+We define the Fourier transform of ϕ ∈ D(Rd) as
+�ϕ(ζ) =
+�
+Rd ϕ(x)e−iζ·xdx,
+ζ ∈ Cd.
+Then, �ϕ is an entire function on Cd. For all N ∈ N there is C > 0 such that for all
+ϕ ∈ D(Rd)
+(4.2)
+|�ϕ(ζ)| ≤ C∥ϕ∥L1,N
+eHch supp ϕ(Imζ)
+(2 + |ζ|)N ,
+ζ ∈ Cd,
+where ch supp ϕ denotes the convex hull of supp ϕ. We are ready to show Lemma 4.1.
+Proof. We may assume without loss of generality that ξ = 0.
+Since 0 /∈ K, (4.1)
+implies that there is η ∈ Rd such that HK(−η) < 0. For t > 0 and σ ∈ Rd we define
+Pt,σ = P(σ +itη + · ) ∈ Pol◦(m). Note that there is c > 0 such for all t > 0 and σ ∈ Rd
+(4.3)
+�
+Pt,σ(0) = �P(σ + itη) ≥ c.
+Let Φ be as above. We define Ft ∈ D′(Rd) via (cf. [10, proof of Theorem 7.3.10])
+⟨Ft, ϕ⟩ =
+1
+(2π)d
+�
+Rd
+�
+Cd
+�ϕ(−σ − itη − ζ)
+P(σ + itη + ζ) Φ(Pt,σ, ζ)dζdσ,
+ϕ ∈ D(Rd).
+Let L be an arbitrary compact convex subset of Rd. By properties (i) and (iii) of Φ,
+(4.2) (with N = d + 1) and (4.3) we have that for all ϕ ∈ DL
+|⟨Ft, ϕ⟩|
+≤
+1
+(2π)d
+�
+Rd
+�
+|ζ|≤1
+|�ϕ(−σ − itη − ζ)|
+|Pt,σ(ζ)|
+Φ(Pt,σ, ζ)dζdσ
+≤
+AC∥ϕ∥L1,d+1
+(2π)d
+�
+Rd
+�
+|ζ|≤1
+eHL(−tη−Im ζ)
+(2 + |σ + itη + ζ|)d+1�
+Pt,σ(0)
+Φ(Pt,σ, ζ)dζdσ
+≤
+AC∥ϕ∥L1,d+1
+(2π)dc
+�
+Rd
+�
+|ζ|≤1
+etHL(−η)eHL(− Im ζ)
+(1 + |σ|)d+1
+Φ(Pt,σ, ζ)dζdσ
+≤
+C′
+L∥ϕ∥L∞,d+1etHL(−η),
+(4.4)
+where
+C′
+L = AC|L|
+(2π)dc max
+|ζ|≤1 eHL(− Im ζ)
+�
+Rd
+1
+(1 + |σ|)d+1dσ.
+In particular, Ft is a well-defined distribution. Property (ii) of Φ and Cauchy’s integral
+formula yield that for all ϕ ∈ D(Rd)
+⟨P(D)Ft, ϕ⟩
+=
+⟨Ft, P(−D)ϕ⟩
+=
+1
+(2π)d
+�
+Rd
+�
+Cd �ϕ(−σ − itη − ζ)Φ(Pt,σ, ζ)dζdσ
+=
+1
+(2π)d
+�
+Rd �ϕ(−σ − itη)dσ
+
+14
+A. DEBROUWERE AND T. KALMES
+=
+1
+(2π)d
+�
+Rd �ϕ(σ)dσ = ϕ(0)
+and thus P(D)Ft = δ. For ε ∈ (0, 1) we set tε = log ε/HK(−η) > 0 and E0,ε = Ftε.
+Then, P(D)E0,ε = δ. By (4.4), we obtain that for all ε ∈ (0, 1)
+∥E0,ε∥∗
+K,d+1 ≤ C′
+Kε
+and, for any L ⊆ Rd compact and convex,
+∥E0,ε∥∗
+L,d+1 ≤ C′
+L
+εs ,
+with s = |HL(−η)/HK(−η)|. This gives the desired result.
+□
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+

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+Beyond spectral gap (extended):
+The role of the topology in decentralized learning
+Thijs Vogels*
+[email protected]
+Hadrien Hendrikx*
+[email protected]
+Martin Jaggi
+[email protected]
+Machine Learning and Optimization Laboratory
+EPFL
+Lausanne, Switzerland
+Abstract
+In data-parallel optimization of machine learning models, workers collaborate to improve
+their estimates of the model: more accurate gradients allow them to use larger learning
+rates and optimize faster. In the decentralized setting, in which workers communicate over a
+sparse graph, current theory fails to capture important aspects of real-world behavior. First,
+the ‘spectral gap’ of the communication graph is not predictive of its empirical performance
+in (deep) learning. Second, current theory does not explain that collaboration enables larger
+learning rates than training alone. In fact, it prescribes smaller learning rates, which further
+decrease as graphs become larger, failing to explain convergence dynamics in infinite graphs.
+This paper aims to paint an accurate picture of sparsely-connected distributed optimization.
+We quantify how the graph topology influences convergence in a quadratic toy problem and
+provide theoretical results for general smooth and (strongly) convex objectives. Our theory
+matches empirical observations in deep learning, and accurately describes the relative merits
+of different graph topologies. This paper is an extension of the conference paper by Vogels
+et al. (2022). Code: github.com/epfml/topology-in-decentralized-learning.
+Keywords:
+Decentralized Learning, Convex Optimization, Stochastic Gradient Descent,
+Gossip Algorithms, Spectral Gap
+1. Introduction
+Distributed data-parallel optimization algorithms help us tackle the increasing complexity of
+machine learning models and of the data on which they are trained. We can classify those
+training algorithms as either centralized or decentralized, and we often consider those settings
+to have different benefits over training ‘alone’. In the centralized setting, workers compute
+gradients on independent mini-batches of data, and they average those gradients between all
+workers. The resulting lower variance in the updates enables larger learning rates and faster
+training. In the decentralized setting, workers average their models with only a sparse set of
+‘neighbors’ in a graph instead of all-to-all, and they may have private datasets sampled from
+different distributions. As the benefit of decentralized learning, we usually focus only on the
+(indirect) access to other worker’s datasets, and not of faster training.
+Homogeneous (i.i.d.) setting. While decentralized learning is typically studied with
+heterogeneous datasets across workers, sparse (decentralized) averaging between them is also
+useful when worker’s data is identically distributed (i.i.d.) (Lu and Sa, 2021). As an example,
+©2022 Thijs Vogels and Hadrien Hendrikx and Martin Jaggi. *: Equal contribution.
+Preprint. Under Review. License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/.
+arXiv:2301.02151v1  [cs.LG]  5 Jan 2023
+
+Vogels, Hendrikx, Jaggi
+10
+100
+1000
+↑ Steps until loss < 0.01
+0.001
+0.01
+0.1
+1
+Learning rate →
+Fully connected
+Ring
+Alone (disconnected)
+Current theory uses
+lower learning rates
+but decentralized averaging
+enables higher learning rates
+0.001
+0.01
+0.1
+1
+Learning rate →
+1-ring (spectral gap 1)
+2-ring (spectral gap 1)
+4-ring (spectral gap 0.67)
+8-ring (s.g. 0.20)
+∞-ring (s.g. 0)
+↮ Instead of a speedup,
+current theory predicts a slowdown with ring size
+Figure 1: ‘Time to target’ for D-SGD (Lian et al., 2017) with constant learning rates on
+an i.i.d. isotropic quadratic dataset (Section 3.1). The noise disappears at the
+optimum. Compared to optimizing alone, 32 workers in a ring (left) are faster for
+any learning rate, but the largest improvement comes from being able to use a
+large learning rate. This benefit is not captured by current theory, which prescribes
+a smaller learning rate than training alone. On the right, we see that rings of
+increasing size enable larger learning rates and faster optimization. Because a
+ring’s spectral gap goes to zero with the size of the ring, this cannot be explained
+by current theory.
+sparse averaging is used in data centers to mitigate communication bottlenecks (Assran et al.,
+2019). When the data is i.i.d. (or heterogeneity is mild), the goal of sparse averaging is to
+optimize faster, just like in centralized (all-to-all) graphs. Yet, current decentralized learning
+theory poorly explains this speed-up. Analyses typically show that, for small enough learning
+rates, training with sparse averaging behaves the same as with all-to-all averaging (Lian
+et al., 2017; Koloskova et al., 2020) and so it reduces the gradient variance by the number of
+workers compared to training alone with the same small learning rate. In practice, however,
+such small learning rates would never be used. In fact, a reduction in variance should allow
+us to use a larger learning rate than training alone, rather than imposing a smaller one.
+Contrary to current theory, we show that (sparse) averaging lowers variance throughout all
+phases of training (both initially and asymptotically), allowing to take higher learning rates,
+which directly speeds up convergence. We characterize how much averaging with various
+communication graphs reduces the variance, and show that centralized performance (variance
+divided by the number of workers) is not always achieved when using optimal large learning
+rates. The behavior we explain is illustrated in Figure 1.
+Heterogeneous (non-i.i.d.) setting. In standard analyses, heterogeneity affects convergence
+in a very worst-case manner. Standard guarantees intuitively correspond to the pessimistic
+case in which the most distant workers have the most different functions. These guarantees are
+typically loose in the settings where workers have different finite datasets sampled i.i.d. from
+the same distribution, or if each worker has a lot of diversity in its close neighbors. In this
+work, we characterize the impact of heterogeneity together with the communication graph,
+2
+
+Beyond spectral gap
+enabling non-trivial guarantees even for infinite graphs under non-adversarial heterogeneity
+patterns.
+Spectral gap. In both the homogeneous and heterogeneous settings, the graph topology
+appears in current convergence rates through the spectral gap of its averaging (gossip) matrix.
+The spectral gap poses a conservative lower bound on how much one averaging step brings
+all worker’s models closer together. The larger, the better. If the spectral gap is small, a
+significantly smaller learning rate is required to make the algorithm behave close to SGD
+with all-to-all averaging with the same learning rate. Unfortunately, we experimentally
+observe that, both in deep learning and in convex optimization, the spectral gap of the
+communication graph is not predictive of its performance under tuned learning rates.
+The problem with the spectral gap quantity is clearly illustrated in a simple example. Let
+the communication graph be a ring of varying size. As the size of the ring increases to infinity,
+its spectral gap goes to zero since it becomes harder and harder to achieve consensus between
+all the workers. This leads to the optimization progress predicted by current theory to go to
+zero as well. In some cases, when the worker’s objectives are adversarially heterogeneous
+in a way that requires workers to obtain information from all others, this is indeed what
+happens. In typical cases, however, this view is overly pessimistic. In particular, this view
+does not match the empirical behavior with i.i.d. data. With i.i.d. data, as the size of the
+ring increases, the convergence rate actually improves (Figure 1), until it saturates at a point
+that depends on the problem.
+In this work, we aim to accurately describe the behavior of distributed learning algorithms
+with sparse averaging, both in theory and in practice. We aim to do so both in the high learning
+rate regime, which was previously studied in the conference version of this paper Vogels et al.
+(2022), as well as in the small learning rate regime, in which we characterize the interplay
+between topology and data heterogeneity, as well as stochastic noise.
+• We quantify the role of the graph in a quadratic toy problem designed to mimic the
+initial phase of deep learning (Section 3.1), showing that averaging enables a larger
+learning rate.
+• From these insights, we derive a problem-independent notion of ‘effective number of
+neighbors’ in a graph that is consistent with time-varying topologies and infinite graphs,
+and is predictive of a graph’s empirical performance in both convex and deep learning.
+• We provide convergence proofs for (strongly) convex objectives that do not depend on
+the spectral gap of the graph (Section 4), and consider finer spectral quantities instead.
+Our rates disentangle the homogeneous and heterogeneous settings, and highlight that
+all problems behave as if they were homogeneous when the iterates are far from the
+optimum.
+At its core, our analysis does not enforce global consensus, but only between workers that are
+close to each other in the graph. Our theory shows that sparse averaging provably enables
+larger learning rates and thus speeds up optimization. These insights prove to be relevant in
+deep learning, where we accurately describe the performance of a variety of topologies, while
+their spectral gap does not (Section 5).
+3
+
+Vogels, Hendrikx, Jaggi
+2. Related work
+Decentralized SGD.
+This paper studies decentralized SGD. Koloskova et al. (2020)
+obtain the tightest bounds for this algorithm in the general setting where workers optimize
+heterogeneous objectives. They show that gossip averaging reduces the asymptotic variance
+suffered by the algorithm at the cost of a degradation (depending on the spectral gap of
+the gossip matrix) of the initial linear convergence term. This key term does not improve
+through collaboration and gives rise to a smaller learning rate than training alone. Besides,
+as discussed above, this implies that optimization is not possible in the limit of large graphs,
+even in the absence of heterogeneity: for instance, the spectral gap of an infinite ring is zero,
+which would lead to a learning rate of zero as well.
+These rates suggest that decentralized averaging speeds up the last part of training
+(dominated by variance), at the cost of slowing down the initial (linear convergence) phase.
+Beyond the work of Koloskova et al. (2020), many papers focus on linear speedup (in the
+variance phase) over optimizing alone, and prove similar results in a variety of settings (Lian
+et al., 2017; Tang et al., 2018; Lian et al., 2018). All these results rely on the following insight:
+while linear speedup is only achieved for small learning rates, SGD eventually requires such
+small learning rates anyway (because of, e.g., stochastic noise, or non-smoothness). This
+observation leads these works to argue that “topology does not matter”. This is the case
+indeed, but only for very small learning rates, as shown in Figure 1. Besides, while linear
+speedup might be achievable indeed for very small learning rates, some level of variance
+reduction should be obtained by averaging for any learning rate. In practice, averaging
+speeds up both the initial and last part of training and in a possibly non-linear way. This is
+what we show in this work, both in theory and in practice.
+Another line of work studies decentralized SGD under statistical assumptions on the local
+data. In particular, Richards and Rebeschini (2020) show favorable properties for D-SGD
+with graph-dependent implicit regularization and attain optimal statistical rates. Their
+suggested learning rate does depend on the spectral gap of the communication network, and
+it goes to zero when the spectral gap shrinks. Richards and Rebeschini (2019) also show that
+larger (constant) learning rates can be used in decentralized GD, but their analysis focuses
+on decentralized kernel regression. Their analysis relies on statistical concentration of local
+objectives rather, while the analysis in this paper relies on the notion of local neighborhoods.
+Gossiping in infinite graphs.
+An important feature of our results is that they do not
+depend on the spectral gap, and so they apply independently of the size of the graph. Instead,
+our results rely on new quantities that involve a combination of the graph topology and
+the heterogeneity pattern. These may depend on the spectral gap in extreme cases, but
+are much better in general. Berthier et al. (2020) study acceleration of gossip averaging in
+infinite graphs, and obtain the same conclusions as we do: although spectral gap is useful
+for asymptotics (how long does information take to spread in the whole graph), it fails to
+accurately describe the transient regime of gossip averaging, i.e., how quickly information
+spreads over local neighborhoods in the first few gossip rounds. This is especially limiting
+for optimization (compared to just averaging), as new local updates need to be averaged at
+every step. The averaging for latest gradient updates always starts in the transient regime,
+implying that the transient regime of gossip averaging deeply affects the asymptotic regime
+of decentralized SGD. In this work, we build on tools from Berthier et al. (2020) to show
+4
+
+Beyond spectral gap
+how the effective number of neighbors, a key quantity we introduce, is related to the graph’s
+spectral dimension.
+The impact of the graph topology.
+Lian et al. (2017) argue that the topology of
+the graph does not matter. This is only true for asymptotic rates in specific settings, as
+illustrated in Figure 1. Neglia et al. (2020) investigate the impact of the graph on decentralized
+optimization, and contradict this claim. Similarly to us, they show that the graph has an
+impact in the early phases of training. Their analysis of the heterogeneous setting, their
+analysis depends on how gradient heterogeneity spans the eigenspace of the Laplacian. Their
+assumptions, however, differ from ours, and they retain an unavoidable dependence on the
+spectral gap of the graph. Our results are different in nature, and show the benefits of
+averaging and the impact of the graph through the choice of large learning rates, and a better
+dependence on the noise and the heterogeneity for a given learning rate. Even et al. (2021)
+also consider the impact of the graph on decentralized learning. They focus on non-worst-case
+dependence on heterogeneous delays, and still obtain spectral-gap-like quantities but on a
+reweighted gossip matrix.
+Another line of work studies the interaction of topology with particular patterns of data
+heterogeneity (Le Bars et al., 2022; Dandi et al., 2022), and how to optimize graphs with this
+heterogeneity in mind. Our analysis highlights the role of heterogeneity through a different
+quantity than these works, that we believe is tight. Besides, both works either try to reduce
+this heterogeneity all along the trajectory, or optimize for both the spectral gap of the graph
+and the heterogeneity term. Instead, we show that heterogeneity changes the fixed-point of
+the algorithm but not the global dynamics.
+Time-varying topologies.
+Time-varying topologies are popular for decentralized deep
+learning in data centers due to their strong mixing (Assran et al., 2019; Wang et al., 2019).
+The benefit of varying the communication topology over time is not easily explained through
+standard theory, but requires dedicated analysis (Ying et al., 2021). While our proofs
+only cover static topologies, the quantities that appear in our analysis can be computed
+for time-varying schemes, too. With these quantities, we can empirically study static and
+time-varying schemes in the same framework.
+Conference version.
+This paper is an extension of Vogels et al. (2022), which focused on
+the homogeneous setting where all workers share the same global optimum. In this extension,
+we introduce a simpler analysis that strictly improves and generalizes the previous one,
+extending the results to the important heterogeneous setting. In the conference version, it
+remained unclear if larger learning rates could only be achieved thanks to homogeneity. We
+also connect the quantities we introduce to the spectral dimension of a graph, and use this
+connection to derive explicit formulas for the optimal learning rates based on the spectral
+dimension. This allows us to accurately compare with previous bounds (for instance Koloskova
+et al. (2020)) and show that we improve on them in all settings.
+3. Measuring collaboration in decentralized learning
+Both this paper’s analysis of decentralized SGD for general convex objectives and its deep
+learning experiments revolve around a notion of ‘effective number of neighbors’ that we
+would introduce in Section 3.2. The aim of this section is to motivate the quantity based on
+5
+
+Vogels, Hendrikx, Jaggi
+a simple toy model for which we can exactly characterize the convergence (Section 3.1). We
+then connect this quantity to the typical graph metrics such as spectral gap and spectral
+dimensions in Section 3.3.
+3.1 A toy problem: D-SGD on isotropic random quadratics
+The aim of this section is to provide intuition while avoiding the complexities of general
+analysis. To keep this section light, we omit any derivations. The appendix of (Vogels et al.,
+2022) contains a longer version of this section that includes derivations and proofs.
+We consider n workers that jointly optimize an isotropic quadratic Ed∼N d(0,1)
+1
+2(d⊤x)2 =
+1
+2∥x∥2 with a unique global minimum x⋆ = 0. The workers access the quadratic through
+stochastic gradients of the form g(x) = dd⊤x, with d ∼ N d(0, 1). This corresponds to
+a linear model with infinite data, and where the model can fit the data perfectly, so that
+stochastic noise goes to zero close to the optimum. We empirically find that this simple model
+is a meaningful proxy for the initial phase of (over-parameterized) deep learning (Section 5).
+A benefit of this model is that we can compute exact rates for it. These rates illustrate the
+behavior that we capture more generally in the theory of Section 4.
+The stochasticity in this toy problem can be quantified by the noise level
+ζ = sup
+x∈Rd
+Ed∥g(x)∥2
+∥x∥2
+= sup
+x∈Rd
+Ed∥dd⊤x∥2
+∥x∥2
+,
+(1)
+which is equal to ζ = d + 2, due to the random normal distribution of d.
+The workers run the D-SGD algorithm (Lian et al., 2017). Each worker i has its own
+copy xi ∈ Rd of the model, and they alternate between local model updates xi ← xi − ηg(xi)
+and averaging their models with others: xi ← �n
+j=1 wijxj. The averaging weights wij are
+summarized in the gossip matrix W ∈ Rn×n. A non-zero weight wij indicates that i and
+j are directly connected. In the following, we assume that W is symmetric and doubly
+stochastic: �n
+j=1 wij = 1 ∀i.
+On our objective, D-SGD either converges or diverges linearly. Whenever it converges,
+i.e., when the learning rate is small enough, there is a convergence rate r such that
+E∥x(t)
+i ∥2 ≤ (1 − r)∥x(t−1)
+i
+∥2,
+with equality as t → ∞. When the workers train alone (W = I), the convergence rate for a
+given learning rate η reads:
+ralone = 1 − (1 − η)2 − (ζ − 1)η2.
+(2)
+The optimal learning rate η⋆ = 1
+ζ balances the optimization term (1 − η)2 and the stochastic
+term (ζ − 1)η2. In the centralized (fully connected) setting (wij = 1
+n ∀i, j), the rate is simple
+as well:
+rcentralized = 1 − (1 − η)2 − (ζ − 1)η2
+n
+.
+(3)
+Averaging between n workers reduces the impact of the gradient noise, and the optimal
+learning rate grows to η⋆ =
+n
+n+ζ−1. We find that D-SGD with a general gossip matrix W
+interpolates those results.
+6
+
+Beyond spectral gap
+3.2 The effective number of neighbors
+To quantify the reduction of the (ζ − 1)η2 term in general, we introduce the problem-
+independent notion of effective number of neighbors nW(γ) of the gossip matrix W and
+decay parameter γ.
+Definition 1 (Effective number of neighbors) The effective number of neighbors nW(γ) =
+limt→∞
+�n
+i=1 Var[y(t)
+i
+]
+�n
+i=1 Var[z(t)
+i
+] measures the ratio of the asymptotic variance of the processes
+y(t+1) = √γ · y(t) + ξ(t),
+where y(t) ∈ Rn and ξ(t) ∼ N n(0, 1)
+(4)
+and
+z(t+1) = W(√γ · z(t) + ξ(t)),
+where z(t) ∈ Rn and ξ(t) ∼ N n(0, 1).
+(5)
+We call y and z random walks because workers repeatedly add noise to their state, somewhat
+like SGD’s parameter updates. This should not be confused with a ‘random walk’ over nodes
+in the graph.
+Since averaging with W decreases the variance of the random walk by at most n, the
+effective number of neighbors is a number between 1 and n. The decay γ modulates the
+sensitivity to communication delays. If γ = 0, workers only benefit from averaging with their
+direct neighbors. As γ increases, multi-hop connections play an increasingly important role.
+As γ approaches 1, delayed and undelayed noise contributions become equally weighted, and
+the reduction tends to n for any connected topology.
+Proposition 2 For regular doubly-stochastic symmetric gossip matrices W with eigenvalues
+λ1, . . . , λn, nW(γ) has a closed-form expression
+nW(γ) =
+1
+1−γ
+1
+n
+�n
+i=1
+λi2
+1−λ2
+i γ
+.
+(6)
+This follows from unrolling the recursions for y and z, using the eigendecomposition of W,
+and the limit lim t → ∞ �t
+k=1 xk =
+x
+1−x.
+While this closed-form expression only covers a restricted set of gossip matrices, the notion
+of variance reduction in random walks, however, naturally extends to infinite topologies or
+time-varying averaging schemes. Figure 2 illustrates nW for various topologies.
+In our exact characterization of the convergence of D-SGD on the isotropic quadratic toy
+problem, we find that the effective number of neighbors appears in place of the number of
+workers n in the fully-connected rate of Equation 3. The rate r is the unique solution to
+r = 1 − (1 − η)2 −
+(ζ − 1)η2
+nW
+� (1−η)2
+1−r
+�.
+(7)
+For fully-connected and disconnected W, nW(γ) = n or 1 respectively, irrespective of γ, and
+Equation 7 recovers Equations 2 and 3. For other graphs, the effective number of workers
+depends on the learning rate. Current theory only considers the case where nW ≈ n, but
+7
+
+Vogels, Hendrikx, Jaggi
+↑ Effective number of neighbors (variance reduction in a ‘random walk’)
+1
+4
+8
+16
+24
+32
+0.9999
+0.999
+0.99
+0.9
+0
+Decay γ of the ‘random walk’ →
+(Think “lower learning rate” or “iterates moving slower”) →
+Fully connected
+Two cliques
+Time-varying
+exponential
+Ring
+Alone (disconnected)
+· · ·
+Figure 2: The effective number of neighbors for several topologies measured by their variance
+reduction in (5). The point γ on the x-axis that matters depends on the learning
+rate and the task. Which topology is ‘best’ varies from problem to problem. For
+large decay rates γ (corresponding small learning rates), all connected topologies
+achieve variance reduction close to a fully connected graph. For small decay rates
+(large learning rates), workers only benefit from their direct neighbors (e.g. 3 in
+a ring). These curves can be computed explicitly for constant topologies, and
+simulated efficiently for the time-varying exponential scheme (Assran et al., 2019).
+the small learning rates this requires can make the term (1 − η)2 too large, defeating the
+purpose of collaboration.
+Beyond this toy problem, we find that the proposed notion of effective number of neighbors
+is also meaningful in the analysis of general objectives (Section 4) and in deep learning
+(Section 5).
+3.3 Links between the effective number of neighbors and other graph quantities
+In general, the effective number of neighbors function nW(γ) cannot be summarized by a
+single scalar. Figure 2 demonstrates that the behavior of this function varies from graph to
+graph. We can, however, bound the effective number of neighbors by known graph quantities
+such as its spectral gap or spectral dimension.
+We aim to create bounds for both finite and infinite graphs. To allow for this, we introduce
+a generalization of Proposition 2 as an integral over the spectral measure dσ of the gossip
+matrix, instead of a sum over its eigenvalues:
+nW(γ)−1 = (1 − γ)
+� 1
+0
+λ2
+1 − γλ2 dσ(λ).
+(8)
+For finite graphs, dσ is a sum of Dirac deltas of mass 1
+n at each eigenvalue of matrix W,
+recovering Equation (6).
+8
+
+Beyond spectral gap
+3.3.1 Upper and lower bounds
+We can use the fact that there all eigenvalues λ are ≤ 1, leading to:
+nW(γ)−1 ≤ (1 − γ)
+� 1
+0
+1
+1 − γ dσ(λ) = 1,
+(9)
+This lower bound to the ‘effective number of neighbors’ corresponds to a disconnected graph.
+On the other hand, for finite graphs, we can use the fact that σ(λ) contains a series of n
+Diracs. The peak at λ = 1, corresponding to the fully-averaged state, has value 1
+n, while the
+other peaks have values ≥ 0. Using this bound, we obtain
+nW(γ)−1 ≥ 1 − γ
+1 − γ
+1
+n = 1
+n.
+(10)
+This upper bound to the ‘effective number of neighbors’ is tight for a fully-connected graph.
+3.3.2 Bounding by spectral gap
+If the graph has a spectral gap α, this means that σ(λ) contains a Dirac delta with mass 1
+n
+at λ = 1, corresponding to the fully-averaged state. The rest of σ(λ) has mass n−1
+n
+and is
+contained in the subdomain λ ∈ [0, 1 − α]. In this setting, we obtain
+nW(γ)−1 ≤ 1
+n + n − 1
+n
+(1 − γ)(1 − α)2
+1 − γ(1 − α)2 .
+(11)
+This lower bound to the ‘effective number of neighbors’ is typically pessimistic, but it is tight
+for the finite gossip matrix W = (1 − α)I + α
+n11⊤.
+3.3.3 Bounding by spectral dimension
+Next, we will link the notion of ‘effective number of neighbors’ to the spectral dimension ds
+of the graph (Berthier, 2021, e.g. Definition 1.9), which controls the decay of eigenvalues
+near 1. This notion is usually linked with the spectral measure of the Laplacian of the
+graph. However, to avoid introducing too many graph-related quantities, we define spectral
+dimension with respect to the gossip matrix W. Standard definitions using the Laplacian
+LW = I − W are equivalent. In the remainder of this paper, the ‘graph’ will always refer to
+the communication graph implicitly induced by W of Laplacian LW.
+Definition 3 (Spectral Dimension) A gossip matrix has a spectral dimension at least ds
+if there exists cs > 0 such that for all λ ∈ [0, 1], the density of its eigenvalues is bounded by
+σ((λ, 1)) ≤ c−1
+s (1 − λ)
+ds
+2 .
+(12)
+The notation σ((λ, 1)) here refers to the integral
+� 1
+λ σ(l) dl. The spectral dimension of a
+graph has a natural geometric interpretation. For instance, the line (or ring) are of spectral
+dimension ds = 1, whereas 2-dimensional grids are of spectral dimension 2. More generally,
+a d-dimensional torus is of spectral dimension d. Besides, the spectral dimension describes
+macroscopic topological features and are robust to microscopic changes. For instance, random
+geometric graphs are of spectral dimension 2.
+9
+
+Vogels, Hendrikx, Jaggi
+Note that since finite graphs have a spectral gap, σ((λ2(W), 1)) = 0 and so finite graphs
+verify (12) for any spectral dimension ds. However, the notion of spectral dimension is still
+relevant for finite graphs, since the constant cs blows up when ds is bigger than the actual
+spectral dimension of an infinite graph with similar topology. Alternatively, it is sometimes
+helpful to explicitly take the spectral gap into account in (12), as in Berthier et al. (2020,
+Section 6).
+We now proceed to bounding nW(γ) using the spectral dimension. Since λ �→ λ2(1 −
+γλ2)−1 is a non-negative non-decreasing function on [0, 1], we can use Berthier et al. (2020,
+Lemma C.1) to obtain that:
+nW(γ)−1 ≤ 1
+n + c−1
+s (1 − γ)
+� 1
+0
+λ2
+1 − γλ2 (1 − λ)
+ds
+2 −1dλ.
+(13)
+The term 1
+n comes from the fact that for finite graphs, the density dσ includes a Dirac delta
+with mass 1
+n at eigenvalue 1. This Dirac is not affected by spectral dimension, and is required
+for consistency, as it ensures that nW(γ) ≤ n for any finite graph. To evaluate the integral,
+we then distinguish three cases.
+Case ds > 2.
+Since γλ < 1, then 1 − λ ≤ 1 − γλ2. In particular we use integration by parts
+to get:
+nW(γ)−1 − n−1 ≤ c−1
+s (1 − γ)
+� 1
+0
+λ2(1 − γλ2)
+ds
+2 −2dλ
+≤ − (1 − γ)c−1
+s
+2γ(ds/2 − 1)
+� 1
+0
+−2γλ(ds/2 − 1)(1 − γλ2)
+ds
+2 −2dλ
+= (1 − γ)c−1
+s
+γ(ds − 2)
+�
+1 − (1 − γ)
+ds
+2 −1�
+.
+This leads to a scaling of:
+nW(γ) ≥
+� 1
+n +
+(1 − γ)
+γ(ds − 2)cs
+�−1
+.
+(14)
+For large enough n, we obtain the same scaling of (1 − γ)−1 as in the previous section, thus
+indicating that for networks that are well-enough connected (ds > 2), the spectral dimension
+only affects the constants, and not the scaling in γ.
+Case ds = 2.
+When ds = 2, only the primitive of the integrand changes, leading to:
+nW(γ) ≥
+� 1
+n − (1 − γ) ln(1 − γ)
+2γcs
+�−1
+(15)
+Case ds < 2.
+In this case, we start by splitting the integral as:
+(1 − γ)
+� 1
+0
+λ2(1 − λ)
+ds
+2 −1
+(1 − γλ2)
+dλ = (1 − γ)
+� γ
+0
+λ2(1 − λ)
+ds
+2 −1
+(1 − γλ2)
+dλ + (1 − γ)
+� 1
+γ
+λ2(1 − λ)
+ds
+2 −1
+(1 − γλ2)
+dλ
+10
+
+Beyond spectral gap
+For the first term, note that γλ ≤ 1, so (1 − γλ2)−1 ≤ (1 − λ)−1, leading to:
+(1 − γ)
+� γ
+0
+λ2(1 − λ)
+ds
+2 −1
+(1 − γλ2)
+dλ ≤ (1 − γ)
+� γ
+0
+(1 − λ)
+ds
+2 −2dλ
+= 2(1 − γ)
+2 − ds
+�
+(1 − γ)
+ds
+2 −1 − 1
+�
+≤
+2
+2 − ds
+(1 − γ)
+ds
+2 .
+For the second term, note that λ2 ≤ 1, so (1 − γλ2)−1 ≤ (1 − γ)−1, leading to:
+(1 − γ)
+� 1
+γ
+λ2(1 − λ)
+ds
+2 −1
+(1 − γλ2)
+dλ ≤
+� 1
+γ
+(1 − λ)
+ds
+2 −1dλ = 2
+ds
+(1 − γ)
+ds
+2 .
+(16)
+In the end, we obtain that nW(γ)−1 − 1
+n ≤ 2
+cs
+�
+1
+2−ds + 1
+ds
+�
+(1 − γ)
+ds
+2 , and so:
+nW(γ) ≥
+�
+1
+n + 4(1 − γ)
+ds
+2
+ds(2 − ds)cs
+�−1
+.
+(17)
+In this case, scaling in γ is impacted by the spectral dimension. Better-connected graphs
+benefit more from higher γ.
+4. Convergence analysis
+4.1 Notations and Definitions
+In the previous section, we have derived exact rates for a specific function. Now we present
+convergence rates for general (strongly) convex functions that are consistent with our
+observations in the previous section. We obtain rates that depend on the level of noise, the
+hardness of the objective, and the topology of the graph. More formally, we assume that we
+would like to solve the following problem:
+min
+θ∈Rd
+n
+�
+i=1
+fi(θ) =
+min
+x∈Rnd,xi=xj
+n
+�
+i=1
+fi(xi).
+(18)
+In this case, xi ∈ Rd represents the local variable of node i, and x ∈ Rnd the stacked variables
+of all nodes. We will assume the following iterations for D-SGD:
+(D-SGD):
+x(t+1)
+i
+=
+n
+�
+j=1
+wijx(t)
+j
+− η∇fξ(t)
+i (x(t)
+i )
+(19)
+where fξ(t)
+i
+represent sampled data points and the gossip weights wij are elements of W.
+Denoting LW = I − W, we rewrite this expression in matrix form as:
+x(t+1) = x(t) −
+�
+η∇Fξ(t)(x(t)) + LWx(t)�
+,
+(20)
+where (∇Fξ(t)(x(t)))i = ∇fξ(t)
+i (x(t)
+i ). We abuse notations in the sense that W ∈ Rnd×nd is
+now the Kronecker product of the standard n×n gossip matrix and the d×d identity matrix.
+11
+
+Vogels, Hendrikx, Jaggi
+This definition is a slight departure from the conference version of this work (Vogels
+et al., 2022), which alternated randomly between gossip steps and gradient updates instead
+of in turns. The analysis of the randomized setting is still possible, but with heterogeneous
+objectives xi ̸= �n
+j=1 wijxj, even for the fixed points of D-SGD (19), and randomizing the
+updates adds undesirable variance. Similarly, it is also possible to analyze the popular variant
+x(t+1) = W[x(t) − η∇Fξ(t)(x(t))], which locally averages the stochastic gradients before they
+are applied. Yet, the D-SGD algorithm in (19) allows communications and computations to
+be performed in parallel, and leads to a simpler analysis. We analyze this model under the
+following assumptions, where Df(x, y) = f(x) − f(y) − ∇f(y)⊤(x − y) denotes the Bregman
+divergence of f between points x and y.
+Assumption 4 The stochastic gradients are such that: ( i) the sampled data points ξ(t)
+i
+and
+ξ(ℓ)
+j
+are independent across times t, ℓ and nodes i ̸= j. ( ii) stochastic gradients are locally
+unbiased: E [fξ(t)
+i ] = fi for all t, i ( iii) the objectives fξ(t)
+i
+are convex and ζξ-smooth for all t, i,
+with E
+�
+ζξDfξ(x, y)
+�
+≤ ζDf(x, y) for all x, y. ( iv) all local objectives fi are µ-strongly-convex
+for µ ≥ 0 and L-smooth.
+Large learning rates.
+The smoothness constant ζ of the stochastic functions fξ defines
+the level of noise in the problem (the lower, the better) in the transient regime. The ratio
+ζ/L compares the difficulty of optimizing with stochastic gradients to the difficulty with the
+true global gradient before reaching the ‘variance region’ in which the iterates of D-SGD with
+a constant learning rate lie almost surely as t → ∞. This ratio is thus especially important
+in interpolating settings when all fξ(t)
+i
+have the same minimum, so that the ‘variance region’
+is reduced to the optimum x⋆. Assuming better smoothness for the global average objective
+than for the local functions is key to showing that averaging between workers allows for larger
+learning rates. Without communication, convergence to the ‘variance region’ is ensured for
+learning rates η ≤ 1/ζ. If ζ ≈ L, there is little noise and cooperation only helps to reduce
+the final variance, and to get closer to the global minimum (instead of just your own). Yet,
+in noisy regimes (ζ ≫ L), such as in Section 3.1 in which ζ = d + 2 ≫ 1 = L, averaging
+enables larger learning rates up to min(1/L, n/ζ), greatly speeding up the initial training
+phase. This is precisely what we will prove in Theorem 6.
+If the workers always remain close (xi ≈ 1
+n(x1 +. . .+xn) ∀i, or equivalently 1
+n11⊤x ≈ x),
+D-SGD behaves the same as SGD on the average parameter 1
+n
+�n
+i=1 xi, and the learning rate
+depends on max(ζ/n, L), showing a reduction of variance by n. Maintaining “ 1
+n11⊤x ≈ x”,
+however, requires a small learning rate. This is a common starting point for the analysis of
+D-SGD, in particular for the proofs in Koloskova et al. (2020). On the other extreme, if we
+do not assume closeness between workers, “Ix ≈ x” always holds. In this case, there is no
+variance reduction, but no requirement for a small learning rate either. In Section 3.1, we
+found that, at the optimal learning rate, workers are not close to all other workers, but they
+are close to others that are not too far away in the graph.
+We capture the concept of ‘local closeness’ by defining a neighborhood matrix M ∈ Rn×n.
+It allows us to consider semi-local averaging beyond direct neighbors, but without fully
+averaging with the whole graph. We ensure that “Mx ≈ x”, leading to an improvement in the
+smoothness somewhere between ζ (achieved alone) and ζ/n (achieved when global consensus
+12
+
+Beyond spectral gap
+is maintained). Each neighborhood matrix M implies a requirement on the learning rate, as
+well as an improvement in smoothness.
+While we can conduct our analysis with any M, those matrices that strike a good balance
+between the learning rate requirement and improved smoothness are most interesting. Based
+on Section 3.1, we therefore focus on a specific construction of matrices: We choose M as
+the covariance of a decay-γ ‘random walk process’ with the graph, as in (5), meaning that
+M = (1 − γ)
+∞
+�
+k=1
+γk−1W2k = (1 − γ)W2(I − γW2)−1.
+(21)
+Varying γ induces a spectrum of averaging neighborhoods from M = W2 (γ = 0) to
+M = 1
+n11⊤ (γ = 1). γ also implies an effective number of neighbors nW(γ): the larger γ,
+the larger nW(γ). We make the following assumption on the neighborhood matrix M:
+Assumption 5 The neighborhood matrix M is of the form of (21), and all the diagonal
+elements have the same value, i.e., Mii = Mjj for all i, j.
+Assumption 5 implies that Mii−1 = nW(γ): the effective number of neighbors defined in (6)
+is equal to the inverse of the self-weights of M. This comes from the fact that the trace of
+M is equal to the sum of its eigenvalues. Otherwise, all results that require Assumption 5
+hold by replacing nW(γ) with mini Mii−1. Besides this interesting relationship with the
+effective number of neighbors nW(γ), we will be interested in another spectral property
+of M, namely the constant β(γ) (which only depends on γ through M, but we make this
+dependence explicit), which is such that:
+LM ≼ β(γ)−1LWW
+(22)
+This constant can be interpreted as the strong convexity of the semi-norm defined by LWW
+relatively to the one defined by LM. Due to the form of M, we have 1 − λ2(W) ≤ β(γ) ≤ 1,
+and the lower bound is tight for γ → 1.
+However, the specific form of M (involving
+neighborhoods as defined by W) and the use of γ < 1 ensure a much larger constant β(γ) in
+general.
+Fixed points of D-(S)GD.
+In Vogels et al. (2022), we consider a homogeneous setting,
+in which E fξ(t)
+i
+= f for all i. We now go beyond this analysis, and consider a setting in
+which local functions fi might be different. In this case, constant-learning-rate Decentralized
+Gradient Descent (the deterministic version of D-SGD) does not converge to the minimizer
+of the average function but to a different one. Let us now consider this fixed point x⋆
+η, which
+verifies:
+η∇F(x⋆
+η) + LWx⋆
+η = 0.
+(23)
+Note that x⋆
+η crucially depends on the learning rate η (which we emphasize in the notation)
+and that it is generally not at consensus (LWx⋆
+η ̸= 0). In the presence of stochastic noise,
+D-SGD will oscillate in a neighborhood (proportional to the gradients’ variance) of this fixed
+point x⋆
+η, and so from now on we will refer to x⋆
+η as the fixed point of D-SGD.
+In the remainder of this section, we show that the results from Vogels et al. (2022) still
+hold as long as we replace the global minimizer x⋆ (solution of Problem (18)) by this fixed
+13
+
+Vogels, Hendrikx, Jaggi
+point x⋆
+η. More specifically, we measure convergence by ensuring the decrease of the following
+Lyapunov function:
+Lt = ∥x(t) − x⋆
+η∥2
+M + ω∥x(t) − x⋆
+η∥2
+LM = (1 − ω)∥x(t) − x⋆
+η∥2
+M + ω∥x(t) − x⋆
+η∥2,
+(24)
+for some parameter ω ∈ [0, 1], and where LM = I − M. Then, we will show how these results
+imply convergence to a neighborhood of x⋆
+η, and that this neighborhood shrinks with smaller
+learning rates η. More specifically, the section unrolls as follows:
+1. Theorem 6 first proves a general convergence result to x⋆
+η, the fixed point of D-(S)GD.
+2. Theorem 9 then bounds the distance to the true optimum for general learning rates.
+3. Corollary 10 finally gives a full convergence result with optimized learning rates.
+Readers interested in quickly comparing our results with state-of-the art ones can skip
+to this result.
+4.2 General convergence result
+Theorem 6 provides convergence rates for any choice of the parameter γ that determines the
+neighborhood matrix M, and for any Lyapunov parameter ω. The best rates are obtained
+for specific γ and ω that balance the benefit of averaging with the constraint it imposes on
+closeness between neighbors. We will discuss these choices more in depth in the next section.
+Theorem 6 If Assumptions 4 and 5 hold and if η is such that
+η ≤ min
+�
+�β(γ)ω
+L
+,
+1
+4
+��
+nW(γ)−1 + ω
+�
+ζ + L
+�
+�
+� ,
+(25)
+then the Lyapunov function defined in (24) verifies the following:
+L(t+1) ≤ (1 − ηµ)L(t) + η2σ2
+M,
+where σ2
+M = 2[(1 − ω)nW(γ)−1 + ω] E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
+.
+This theorem shows convergence (up to a variance region) to the fixed point x⋆
+η of D-SGD,
+regardless of the ‘true’ minimizer x⋆. Although converging to x⋆
+η might not be ideal depending
+on the use case (but do keep in mind that x⋆
+η → x⋆ as η shrinks), this is what D-SGD does,
+and so we believe it is important to start by stating this clearly. The homogeneous case did
+not have this problem since x⋆
+η = x⋆ for all η for η that implied convergence.
+Parameter ω ∈ [0, 1] is free, and it is often convenient to choose it as ω = ηL/β(γ) to
+get rid of the first condition on η. However, we present the result with a free parameter ω
+since, as we will see in the remainder of this section, setting ω = nW(γ)−1 allows for simple
+corollaries.
+Proof
+We now detail the proof, which is both a simplification and generalization of
+Theorem IV from Vogels et al. (2022).
+14
+
+Beyond spectral gap
+1 - General decomposition
+We first analyze the first term in the Lyapunov (24), and
+use the fixed-point conditions of (23) to write:
+E
+�
+∥x(t+1) − x⋆
+η∥2
+M
+�
+= ∥x(t) − x⋆
+η∥2
+M + ∥η∇Fξt(x(t)) + LWx(t)∥2
+M
+− 2η
+�
+∇F(x(t)) − ∇F(x⋆
+η)
+�⊤
+M(x(t) − x⋆
+η) − 2∥x(t) − x⋆
+η∥2
+LWM.
+(26)
+The second term is the same with M in place of I.
+2 - Error terms
+We start by bounding the error terms, and use the optimality conditions
+to obtain:
+E
+�
+∥η∇Fξt(x(t)) + LWx(t)∥2
+M
+�
+= E
+�
+∥η
+�
+∇Fξt(x(t)) − ∇F(x⋆
+η)
+�
++ LW(x(t) − x⋆
+η)∥2
+M
+�
+= E
+�
+∥η
+�
+∇Fξt(x(t)) − ∇Fξt(x⋆
+η)
+�
++
+�
+η
+�
+∇Fξt(x⋆
+η) − ∇F(x⋆
+η)
+�
++ LW(x(t) − x⋆
+η)
+�
+∥2
+M
+�
+≤ 2η2 E
+�
+∥∇Fξt(x(t)) − ∇Fξt(x⋆
+η)∥2
+M
+�
++ 2η2 E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2
+M
+�
++ 2∥x(t) − x⋆
+η∥2
+LWMLW,
+where the last inequality comes from the bias-variance decomposition. The second term
+corresponds to variance, whereas the first and last one will be canceled by descent terms.
+Stochastic gradient noise.
+To bound the first term, we crucially use that stochastic
+noises are independent for two different nodes, so in particular:
+E
+�
+∥∇Fξt(x(t)) − ∇Fξt(x⋆
+η)∥2
+M
+�
+= nW(γ)−1 E
+�
+∥∇Fξt(x(t)) − ∇Fξt(x⋆
+η)∥2�
++ ∥∇F(x(t)) − ∇F(x⋆
+η)∥2
+M−nW(γ)−1I
+≤ 2nW(γ)−1 E
+�
+ζξtDFξt(x⋆
+η, x(t))
+�
++ ∥∇F(x(t)) − ∇F(x⋆
+η)∥2
+≤ 2
+�
+nW(γ)−1ζ + L
+�
+DF (x(t), x⋆
+η),
+where we used that M ≼ I, the L-cocoercivity of F, and the noise assumption, i.e.,
+E
+�
+ζξtDFξt
+�
+≤ ζDF .
+The effective number of neighbors nW(γ) kicks in since Assump-
+tion 5 implies that the diagonal of M is equal to nW(γ)−1I. Using independence again, we
+obtain:
+E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2
+M
+�
+= nW(γ)−1 E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
+(27)
+Performing the same computations for the E
+�
+∥∇Fξt(x(t)) − ∇F(x⋆
+η)∥2�
+term and adding
+consensus error leads to:
+E
+�
+∥η∇Fξt(x(t)) + LWx(t)∥2
+(1−ω)M+ωI
+�
+≤ 4
+��
+(1 − ω)nW(γ)−1 + ω
+�
+ζ + (1 − ω)L
+�
+DF (x(t), x⋆
+η)
++ 2η2((1 − ω)nW(γ)−1 + ω) E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
++ 2∥x(t) − x⋆
+η∥2
+LW[M+ωLM]LW
+(28)
+Here, the first term will be controlled by the descent obtained through the gradient terms,
+and the second one through communication terms.
+15
+
+Vogels, Hendrikx, Jaggi
+3 - Descent terms
+Gradient terms
+We first analyze the effect of all gradient terms. In particular, we use
+that (1 − ω)M + ωI = I − (1 − ω)LM. Then, we use that
+�
+∇F(x(t)) − ∇F(x⋆
+η)
+�⊤
+(x(t) − x⋆
+η) = DF (x(t), x⋆
+η) + DF (x⋆
+η, x(t)),
+and:
+2
+�
+∇F(x(t)) − ∇F(x⋆
+η)
+�⊤
+LM(x(t) − x⋆
+η) ≤ 2∥∇F(x(t)) − ∇F(x⋆
+η)∥∥LM(x(t) − x⋆
+η)∥
+≤ 1
+2L∥∇F(x(t)) − ∇F(x⋆
+η)∥2 + 2L∥x(t) − x⋆
+η∥LM2
+≤ DF (x(t), x⋆
+η) + 2L∥x(t) − x⋆
+η∥LM2.
+Overall, the gradient terms sum to:
+− 2
+�
+∇F(x(t)) − ∇F(x⋆
+η)
+�⊤
+(x(t) − x⋆
+η) + 2(1 − ω)
+�
+∇F(x(t)) − ∇F(x⋆
+η)
+�⊤
+LM(x(t) − x⋆
+η)
+≤ −2DF (x⋆
+η, x(t)) − (1 + ω)DF (x(t), x⋆
+η) + 2(1 − ω)L∥x(t) − x⋆
+η∥LM2
+≤ −µ∥x(t) − x⋆
+η∥2 − DF (x(t), x⋆
+η) + 2L∥x(t) − x⋆
+η∥LM2
+≤ −(1 − ω)µ∥x(t) − x⋆
+η∥2
+M − ω∥x(t) − x⋆
+η∥2 − DF (x(t), x⋆
+η) + 2β(γ)−1L∥x(t) − x⋆
+η∥LMLWW,
+(29)
+where we used that LM ≼ β(γ)−1LWW.
+Gossip terms.
+We simply recall the gossip terms we use for descent here, which write:
+−2∥x(t) − x⋆
+η∥2
+LWM − 2ω∥x(t) − x⋆
+η∥2
+LWLM.
+(30)
+4 - Putting everything together.
+We now add all the descent and error terms together.
+More specifically, using Equations (28), (29) and (30) we obtain:
+L(t+1) ≤ (1 − ηµ)L(t)
+− 2∥x(t) − x⋆
+η∥2
+LWM(I−LW)
+− 2ω [1 − ηL/(ωβ(γ))] ∥x(t) − x⋆
+η∥2
+LWLMW
+− η
+�
+1 − 4η
+��
+(1 − ω)nW(γ)−1 + ω
+�
+ζ + (1 − ω)L
+��
+DF (x(t), x⋆
+η)
++ 2η2 �
+(1 − ω)nW(γ)−1 + ω
+�
+E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
+.
+The conditions in the theorem are chosen so that the terms from lines 3 and 4 are positive
+(which is automatically true for line 2), and using that 1 − ω ≤ 1 (since ω is small anyway).
+16
+
+Beyond spectral gap
+5
+10
+15
+20
+25
+30
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+↑ Learning rate given by Theorem 1 (L = 1.0, ζ = 2000)
+Effective number of neighbors nW(γ) →
+5
+10
+15
+20
+25
+30
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+Effective number of neighbors nW(γ) →
+Ring
+Torus (4x8)
+Hypercube
+Restricted by noise
+Restricted by consensus
+M
+Figure 3: Maximum learning rates prescribed by Theorem 6, varying the parameter γ that
+implies an effective neighborhood size (x-axis) and an averaging matrix M (drawn
+as heatmaps). On the left, we show the details for a 32-worker ring topology, and
+on the right, we compare it to more connected topologies. Increasing γ (and with
+it nW(γ)) initially leads to larger learning rates thanks to noise reduction. At the
+optimum, the cost of consensus exceeds the benefit of further reduced noise.
+4.3 Main corollaries
+4.3.1 Large learning rate: speeding up convergence for large errors
+We now investigate Theorem 6 in the case in which both the noise σ2 and the heterogeneity
+∥∇F(x⋆)∥2
+LW† are small (compared to L(0)), and so we would like to have the highest
+possible learning rate in order to ensure fast decrease of the objective (which is consistent
+with Figure 1). Using (25), we obtain a rate for each parameter γ that controls the local
+neighborhood size (remember that β(γ) depends on γ). The task that remains is to find
+the γ parameter that gives the best convergence guarantees (the largest learning rate). As
+explained before, one should never reduce the learning rate in order to be close to others,
+because the goal of collaboration (in this regime in which we are not affected by variance
+and heterogeneity) is to increase the learning rate.
+We illustrate this in Figure Figure 3, that we obtain by choosing ω = nW(γ)−1, and
+evaluating the two terms of (25) for different values of γ. The expression for the linear part
+of the curve (before consensus dominates) is given in Corollary 7.
+Corollary 7 Consider that Assumptions 4 and 5 hold, then the largest (up to constants)
+learning rate is obtained as:
+η = (8ζ/nW(γ) + 4L)−1 , for γ such that 4nW(γ)−1β(γ)(2nW(γ)−1ζ + L) ≤ L
+(31)
+We see that the learning rate scales linearly with the number of effective neighbors in this case
+(which is equivalent to taking a mini-batch of size linear in nW(γ)) until a certain number
+17
+
+Vogels, Hendrikx, Jaggi
+of neighbors is reached (condition on the right), or centralized performance is achieved
+(ζ = nW(γ)L). The condition on γ always has a solution since when γ ≈ 0, both β(γ)
+and nW(γ)−1 are close to 1, and they both decrease when γ grows. This corollary directly
+follows from taking ω = nW(γ)−1 in Theorem 6. Note that a slightly tighter choice could be
+obtained by setting ω = ηβ(γ)/L.
+Investigating β(γ).
+We now evaluate β(γ) in order to obtain more precise bounds. In
+particular, choosing M as in (21), the eigenvalues of LM are equal to:
+λLM
+i
+= 1 − λ2
+i
+1 − γλ2
+i
+,
+(32)
+where λi are the eigenvalues of W. In particular, β(γ)LM ≼ WLW translates into the fact
+that for all i such that λi ̸= 1 (automatically verified in this case), we want for all i:
+β(γ) ≤ 1 − γλ2
+i
+1 − λ2
+i
+(1 − λi)λi = λi(1 − γλ2
+i )
+1 + λi
+.
+(33)
+We now make the simplifying assumption that λmin(W) ≥ 1
+2 (which we can always enforce
+by taking W′ = (I + W)/2), but note that the theory holds regardless. We motivate this
+simplifying assumption by the fact that the for arbitrarily small spectral gaps, the right
+side of (33) will always be minimized for λ2(W) assuming γ is large enough, so the actual
+value of λmin(W) < 1 does not matter. In particular, in this case, neglecting the effect of the
+spectral gap, we can just take:
+β(γ) = 1 − γλ2(W)
+4
+≥ 1 − γ
+4
+,
+(34)
+Note that β(γ) allows for large γ when the spectral gap 1 − λ2(W) is large, but we allow
+non-trivial learning rates η > 0 even when λ2(W) = 1 (infinite graphs) as long as γ < 1.
+Optimal choice of nW(γ).
+Leveraging the spectral dimension results from Section 3.1,
+we obtain the following corollary:
+Corollary 8 Under Assumption 4 and 5, and assuming that λmin(W) ≥ 1
+2, that the commu-
+nication graph has spectral dimension ds > 2, and that ζ ≫ L, the highest possible learning
+rate is
+η = 1
+8
+�cs(ds − 2)
+ζ2L
+� 1
+3
+, obtained for nW(γ) =
+�
+cs(ds − 2) ζ
+L
+� 1
+3
+(35)
+This result follows from Corollary 7, which, if ζ ≫ L, writes:
+L
+ζ ≥ 8nW(γ)−2β(γ) = nW(γ)−3cs(ds − 2),
+(36)
+where the right part is obtained by plugging in the expressions for β(γ) from (34) into
+nW(γ)−1 ≤
+2(1−γ)
+cs(ds−2) from (14) (assuming γ ≥ 1/2). Then, one can solve for 1 − γ. Assump-
+tions besides Assumption 4 allow to give a simple result in this specific case, but similar
+expressions can easily be obtained for ds ≤ 2 and ζ < LnW(γ).
+18
+
+Beyond spectral gap
+4.3.2 Small learning rate: approaching the optimum arbitrarily closely
+Theorem 6 gives a convergence result to x⋆
+η, the fixed point of D-SGD, and we have investigated
+in the previous section the behavior of D-SGD for large learning rates. In Theorem 9, we
+focus on small error levels, for which the variance and heterogeneity terms dominate, and we
+would like to take small learning rates η. In this setting, we bound the distance between the
+current iterate and the true minimizer x⋆ instead of x⋆
+η. We also provide a result that gets
+rid of all dependence on x⋆
+η, and only explicitly depends on the learning rate η.
+Theorem 9 Under the same assumptions and conditions on the learning rate as Theo-
+rem 6 and Corollary 8, we have that:
+∥x(t) − x⋆∥M ≤ 2(1 − ηµ)tL(0) + 2ησ2
+M
+µ
++ 2η2(1 + κ)∥LW†∇F(x⋆
+η)∥2
+(37)
+We can further remove x⋆
+η from the bound, and obtain:
+∥x(t) − x⋆∥M ≤ 2(1 − ηµ)tL(0) +
+6ησ2
+M,⋆
+µ
++ 6η2κp−1∆2
+W,
+where σ2
+M,⋆ = (nW(γ)−1 + ω) E
+�
+∥∇Fξ(x⋆) − ∇F(x⋆)∥2�
+and p−1 = maxη
+∥LW†∇F(x⋆
+η)∥2
+∥∇F(x⋆η)∥2
+LW† ,
+so that p ≥ 1 − λ2(W), and ∆2
+W = ∥∇F(x⋆)∥2
+LW†
+The norm ∥x(t) − x⋆∥2
+M considers convergence of locally averaged neighborhoods, but ∥x(t) −
+x⋆∥2
+M ≥ ∥x(t) −x⋆∥2 since 1 is an eigenvector of M with eigenvalue 1. We now briefly discuss
+the various terms in this corollary, and then prove it.
+Heterogeneity term.
+The term due to heterogeneity only depends on the distance between
+the true optimum x⋆ and the fixed point x⋆
+η, which we then transform into a condition on
+∥∇F(x⋆)∥2
+LW†. In particular, it is not influenced by the choice of M (and thus of γ).
+Constant p.
+We introduce constant p to get rid of the explicit dependence on x⋆
+η. Indeed,
+p−1 intuitively denotes how large LW† is in the direction of ∇F(x⋆
+η). For instance, if ∇F(x⋆
+η)
+is an eigenvector of W associated with eigenvalue λ, then we have p = 1−λ. In the worst case,
+we have that p = 1 − λ2(W), but p can be much better in general, when the heterogeneity is
+spread evenly, instead of having very different functions on distant nodes.
+Variance term.
+In this case, the largest variance reduction (of order n) is obtained by
+taking ω and nW(γ)−1 as small as possible. For learning rates that are too large to imply
+nW(γ)−1 ≈ n−1, decreasing it decreases the variance term in two ways: (i) directly, through
+the η term, (ii) indirectly, by allowing to take smaller values of nW(γ)−1.
+For very large (infinite) graphs, we can take ω = nW(γ)−1, and in this case Theorem 6
+gives that the smallest nW(γ)−1 is given by nW(γ)−1β(γ) = ηL. Using spectral dimension
+results (for instance with ds > 2), we obtain (similarly to Corollary 8) that we can take
+19
+
+Vogels, Hendrikx, Jaggi
+β(γ) = nW(γ)−1cs(ds − 2)/8, and so:
+nW(γ)−1 =
+�
+8ηL
+cs(ds − 2),
+(38)
+so the residual variance term for this choice of nW(γ)−1 is of order:
+O
+�
+η
+3
+2
+µ
+�
+L
+cs(ds − 2) E
+�
+∥∇Fξ(x⋆) − ∇F(x⋆)∥2�
+�
+(39)
+In particular, we obtain super-linear scaling when reducing the learning rate η thanks to the
+added benefit of gaining more effective neighbors. Note that again, the cases ds ≤ 2 can be
+treated in the same way.
+Proof [Theorem 9] We start by writing:
+∥x(t) − x⋆∥2
+M ≤ 2∥x(t) − x⋆
+η∥2
+M + 2∥x⋆
+η − x⋆∥2
+M ≤ 2L(t) + 2∥x⋆
+η − x⋆∥2.
+(40)
+Theorem 6 ensures that L(t) becomes small, and so we are left with bounding the distance
+between x⋆
+η and x⋆.
+1 - Distance to the global minimizer.
+We define x⋆η = 11⊤x⋆
+η/n. Using the fact that
+both x⋆η and x⋆ are at consensus, and 1⊤∇F(x⋆
+η) = 0 (immediate from (23)), we write:
+DF (x⋆, x⋆
+η) = F(x⋆) − F(x⋆
+η) − ∇F(x⋆
+η)⊤(x⋆ − x⋆
+η)
+= F(x⋆η) − F(x⋆
+η) − ∇F(x⋆
+η)⊤(x⋆η − x⋆
+η) + F(x⋆) − F(x⋆η)
+≤ DF (x⋆η, x⋆
+η),
+(41)
+where the last line comes from the fact that x⋆ is the minimizer of F on the consensus space.
+Therefore:
+∥x⋆
+η − x⋆∥2 = ∥x⋆η − x⋆∥2 + ∥x⋆
+η − x⋆η∥2
+≤ 1
+µDF (x⋆, x⋆
+η) + ∥x⋆
+η − x⋆η∥2
+≤ 1
+µDF (x⋆η, x⋆
+η) + ∥x⋆
+η − x⋆η∥2
+≤
+�
+1 + L
+µ
+���
+∥x⋆η − x⋆
+η∥2 = η2
+�
+1 + L
+µ
+�
+∥LW†∇F(x⋆
+η)∥2.
+Note that the result depends on the heterogeneity pattern of the gradients at the fixed point,
+and might be bounded (and even small) even when W has no spectral gap. However, this
+quantity is proportional to the squared inverse spectral gap in the worst case.
+2 - Monotonicity in η.
+We now prove that ∥∇F(x⋆
+η)∥2
+LW† decreases when η increases,
+and so is maximal for η = 0, corresponding to x⋆
+η = x⋆. More specifically:
+d∥∇F(x⋆
+η)∥2
+LW†
+dη
+=
+d
+�
+η−2∥x⋆
+η∥2
+LW
+�
+dη
+= −
+2∥x⋆
+η∥2
+LW
+η3
++ 2η−2(x⋆
+η)⊤LW
+dx⋆
+η
+dη
+20
+
+Beyond spectral gap
+Differentiating the fixed-point conditions, we obtain that
+η∇2F(x⋆
+η)dx⋆
+η
+dη + ∇F(x⋆
+η) + LW
+dx⋆
+η
+dη = 0,
+(42)
+so that:
+dx⋆
+η
+dη = −
+�
+η∇2F(x⋆
+η) + LW
+�−1 ∇F(x⋆
+η) = η−1 �
+η∇2F(x⋆
+η) + LW
+�−1 LWx⋆
+η.
+(43)
+Plugging this into the previous expression and using that ∇2F(x⋆
+η) is positive semi-definite,
+we obtain:
+d∥∇F(x⋆
+η)∥2
+LW†
+dη
+= − 2
+η3 (x⋆
+η)⊤ �
+LW − LW
+�
+LW + η∇2F(x⋆
+η)
+�−1 LW
+�
+x⋆
+η
+≤ − 2
+η3 (x⋆
+η)⊤ �
+LW − LWLW†LW
+�
+x⋆
+η = 0.
+3 - Getting rid of x⋆
+η.
+By definition of p, we can write:
+∥LW†∇F(x⋆
+η)∥2 ≤ p−1∥∇F(x⋆
+η)∥2
+LW† ≤ p−1∥∇F(x⋆)∥2
+LW†.
+(44)
+Note that we have to bound this constant p in order to use the monotonicity in η of
+∥∇F(x⋆
+η)∥2
+LW† since this result does not hold for ∥LW†∇F(x⋆
+η)∥2. For the variance, we write
+that:
+E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
+≤ 3 E
+�
+∥∇Fξt(x⋆
+η) − ∇Fξt(x⋆)∥2�
++ 3 E
+�
+∥∇Fξt(x⋆) − ∇F(x⋆)∥2�
++ 3∥∇F(x⋆
+η) − ∇F(x⋆)∥2
+≤ 3σ2
+M,⋆ + 3 (ζ + L) DF (x⋆, x⋆
+η).
+From here, we use Equation (41) and obtain that:
+E
+�
+∥∇Fξt(x⋆
+η) − ∇F(x⋆
+η)∥2�
+≤ 3σ2
+M,⋆ + 3L (ζ + L) η2∥LW†∇F(x⋆
+η)∥2.
+(45)
+To obtain the final result, we use that η(nW(γ)−1 +ω)(ζ +L) ≤ 1/4 thanks to the conditions
+on the learning rate.
+4.3.3 Comparison with existing work.
+Expressed in the form of Koloskova et al. (2020), we can summarize the previous corollaries
+into the following result by taking either η as the largest possible constant (as indicated in
+Corollary 8) or η = ˜O(1/(µT)). Here, ˜O denotes inequality up to logarithmic factors, and
+recall that ∥x(t) − x⋆∥2
+M ≥ ∥x(t) − x⋆∥2. We recall that L is the smoothness of the global
+objective f, ζ is the smoothness of the stochastic functions fξ, µ is the strong convexity
+parameter, ds is the spectral dimension of the gossip matrix W (and we assume ds > 2) and
+cs is the associated constant.
+21
+
+Vogels, Hendrikx, Jaggi
+Corollary 10 (Final result.) Under the same assumptions as Corollary 8, there exists
+a choice of learning rate (and, equivalently, of decay parameters γ∗
+large and γ∗
+small) such that
+the expected squared distance to the global optimum after T steps of D-SGD ∥x(t) − x⋆∥2
+is of order:
+˜O
+�
+σ2
+µ2TnW(γ∗
+small) + L∆2
+W
+µ3pT 2 + exp
+�
+−nW(γ∗
+large)µ
+ζ T
+��
+,
+(46)
+where ∆2
+W and p are defined in Theorem 9, and x(t) is the average parameter. The
+optimal effective number of neighbors in respectively the small and large learning rate
+settings are:
+nW(γ∗
+small) = min
+��
+dsT
+Lcs
+, n
+�
+and nW(γ∗
+large) = min
+��csdsζ
+L
+� 1
+3
+, n
+�
+.
+(47)
+This result can be contrasted with the result from Koloskova et al. (2020), which writes:
+˜O
+� σ2
+µ2T
+� 1
+n +
+L
+µ(1 − λ2(W))T
+�
++
+L∆2
+µ3(1 − λ2(W))2T 2 + exp
+�
+−
+µ
+(1 − λ2(W))ζ T
+��
+, (48)
+We can now make the following observations.
+Scheduling the learning rate.
+Here, the learning rate is either chosen as ηlarge =
+nW(γ∗
+large)/ζ, or as ηsmall = ˜O((µT)−1). In practice, one would start with the large learning
+rate, and switching to η when training does not improve anymore (heterogeneity/variance
+terms dominate).
+Exponential decrease term.
+We first show a significant improvement in the exponential
+decrease term. Indeed, nW(γ∗
+large)/(1 − λ2(W)), the ratio between the largest learning rate
+permitted in our analysis versus existing ones, is always large since nW(γ∗
+large) ≥ 1 and
+1 − λ2(W) ≤ 1. Besides, the exponential decrease term is no longer affected by the spectral
+gap in our analysis, which only affects how big nW(γ) can be. This improvement holds even
+when ζ = L (in this case nW(γ) = 1 is enough), and is due to the fact that heterogeneity
+only affects lower-order terms, so that when cooperation brings nothing it doesn’t hurt
+convergence either.
+Impact of heterogeneity.
+The improvement in the heterogeneous case does not depend
+on some γ, and relies on bounding heterogeneity in a non-worst case fashion. Indeed, ζW and
+p capture the interplay between how heterogeneity is distributed among nodes, and the actual
+topology of the graph. Note that this does not contradict the lower bound from Koloskova
+et al. (2020), since ∆2
+W/p = ∆2/(1 − λ2(W))2 in the worst case. In the worst case, the
+heterogeneity pattern of ∇F(x⋆) is aligned with the smallest eigenvalue of LW, i.e., very
+distant nodes have very different objectives. The quantity p, however, gives more fine-grained
+bounds that depend on the actual heterogeneity pattern in general.
+Variance term.
+One key difference between the analyses is on the variance term that
+involves σ2. Both analyses depend on the variance of a single node, σ2/(µT), which is
+22
+
+Beyond spectral gap
+then multiplied by a ‘variance reduction’ term. In both cases, this term is of the form
+nW(γ)−1+ηLβ(γ)−1. However, the standard analysis implicitly use γ = 1, and so nW(γ) = n,
+and β(γ) = 1 − λ2(W). Then, the form from (48) follows from taking η = ˜O(1/(µT)). Our
+analysis on the other hands relies on tuning γ such that nW(γ)−1 + ηLβ(γ)−1 is the smallest
+possible, and is therefore strictly better than just considering γ = 1. Assuming a given
+spectral dimension ds > 2 for the graph leads to (46), but any assumption that precisely
+relates nW(γ) and γ would allow getting similar results.
+While the ˜O(T −2) in the variance term of Koloskova et al. (2020) seems better than our
+˜O(T −3/2) term, this is misleading because constants are very important in this case. Our
+rate is optimized by over γ, which accounts for the fact that if the ˜O(T −2) term dominates,
+then it is better to just consider a smaller neighborhood. In that case, we would not benefit
+from n−1 variance reduction anyway. Our result optimally balances the two variance terms
+from (48) instead. Thanks to this balancing, we obtain that in graphs of spectral dimension
+ds > 2, the variance decreases as ˜O(T − 3
+2 ) with a learning rate of ˜O(T −1) due to the combined
+effect of a smaller learning rate and adding more effective neighbors. In finite graphs, this
+effect caps at nW(γ) = n.
+Finally, note that our analysis and the analysis of Koloskova et al. (2020) allow for
+different generalizations of the standard framework: our analysis applies to arbitrarily large
+(infinite) graphs, while Koloskova et al. (2020) can handle time-varying graphs with weak
+(multi-round) connectivity assumptions.
+5. Empirical relevance in deep learning
+While the theoretical results in this paper are for convex functions, the initial motivation for
+this work comes from observations in deep learning. First, it is crucial in deep learning to
+use a large learning rate in the initial phase of training (Li et al., 2019). Contrary to what
+current theory prescribes, we do not use smaller learning rates in decentralized optimization
+than when training alone (even when data is heterogeneous.) And second, we find that the
+spectral gap of a topology is not predictive of the performance of that topology in deep
+learning experiments.
+In this section, we experiment with a variety of 32-worker topologies on Cifar-10 (Krizhevsky
+et al.) with a VGG-11 model (Simonyan and Zisserman, 2015). Like other recent works (Lin
+et al., 2021; Vogels et al., 2021), we opt for this older model, because it does not include
+BatchNorm (Ioffe and Szegedy, 2015) which forms an orthogonal challenge for decentralized
+SGD. Please refer to Appendix E of (Vogels et al., 2022) for full details on the experimental
+setup. Our set of topologies includes regular graphs like rings and toruses, but also irregular
+graphs such as a binary tree (Vogels et al., 2021) and social network Davis et al. (1930),
+and a time-varying exponential scheme (Assran et al., 2019). We focus on the initial phase
+of training, 25k steps in our case, where both train and test loss converge close to linearly.
+Using a large learning rate in this phase is found to be important for good generalization (Li
+et al., 2019).
+Figure 4 shows the loss reached after the first 2.5k SGD steps for all topologies and for a
+dense grid of learning rates. The curves have the same global structure as those for isotropic
+quadratics Figure 1: (sparse) averaging yields a small increase in speed for small learning
+rates, but a large gain over training alone comes from being able to increase the learning
+23
+
+Vogels, Hendrikx, Jaggi
+2.3
+0.2
+1.55
+1.15
+0.5
+0.001
+0.01
+0.1
+↑ Cifar-10 training loss after 2.5k steps (∼25 epochs)
+Learning rate →
+Binary tree
+Fully connected
+Hypercube
+Ring
+Social network
+Solo
+Star
+Time-varying exponential
+Torus (4x8)
+Two cliques
+Figure 4: Training loss reached after 2.5k SGD steps with a variety of graph topologies. In
+all cases, averaging yields a small increase in speed for small learning rates, but a
+large gain over training alone comes from being able to increase the learning rate.
+While the star has a better spectral gap (0.031) than the ring (0.013), it performs
+worse, and does not allow large learning rates. For reference, similar curves for
+fully-connected graphs of varying sizes are in the appendix of Vogels et al. (2022).
+rate. The best schemes support almost the same learning rate as 32 fully-connected workers,
+and get close in performance.
+We also find that the random walks introduced in Section 3.1 are a good model for
+variance between workers in deep learning. Figure 5 shows the empirical covariance between
+the workers after 100 SGD steps. Just like for isotropic quadratics, the covariance is accurately
+modeled by the covariance in the random walk process for a certain decay rate γ.
+Finally, we observe that the effective number of neighbors computed by the variance
+reduction in a random walk (Section 3.1) accurately describes the relative performance under
+tuned learning rates of graph topologies on our task, including for irregular and time-varying
+topologies. This is in contrast to the topology’s spectral gaps, which we find to be not
+predictive. We fit a decay rate γ = 0.951 that seems to capture the specifics of our problem,
+and show the correlation in Figure 6.
+Appendix F of (Vogels et al., 2022) replicates the same experiments in a different setting.
+There, we use larger graphs (of 64 workers), a different model and data set (an MLP on
+Fashion MNIST Xiao et al. (2017)), and no momentum or weight decay. The results in this
+setting are qualitatively comparable to the ones presented above.
+6. Conclusion
+We have shown that the sparse averaging in decentralized learning allows larger learning rates
+to be used, and that it speeds up training. With the optimal large learning rate, the workers’
+models are not guaranteed to remain close to their global average. Enforcing global consensus
+is often unnecessary and the small learning rates it requires can be counter-productive. Indeed,
+24
+
+Beyond spectral gap
+Gossip matrix
+Measured cov.
+on Cifar-10
+Covariance in
+random walk
+Two cliques
+nW(γ := 0.948)
+= 17.8
+Torus (4x8)
+nW(γ := 0.993)
+= 29.4
+Star
+nW(γ := 0.986)
+= 5.1
+Social network
+nW(γ := 0.992)
+= 27.3
+Ring
+nW(γ := 0.983)
+= 13.9
+Hypercube
+nW(γ := 0.997)
+= 31.3
+Binary tree
+nW(γ := 0.984)
+= 12.3
+Figure 5: Measured covariance in Cifar-10 (second row) between workers using various graphs
+(top row). After 10 epochs, we store a checkpoint of the model and train repeatedly
+for 100 SGD steps, yielding 100 models for 32 workers. We show normalized
+covariance matrices between the workers. These are very well approximated by
+the covariance in the random walk process of Section 3.1 (third row). We print
+the fitted decay parameters and corresponding ‘effective number of neighbors’.
+↑ Cifar-10 training loss after 2.5k steps (∼25 epochs)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+Spectral gap →
+×
+×
+×
+×
+×
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1 2
+4
+8
+16
+32
+Effective num. neighbors (γ = 0.951, tuned) →
+×
+×
+×
+×
+×
+Figure 6: Cifar-10 training loss after 2.5k steps for all studied topologies with their optimal
+learning rates. Colors match Figure 4, and × indicates fully-connected graphs
+with varying number of workers. After fitting a decay parameter γ = 0.951 that
+captures problem specifics, the effective number of neighbors (left) as measured
+by variance reduction in a random walk (like in Section 3.1) explains the relative
+performance of these graphs much better than the spectral gap of these topologies
+(right).
+25
+
+Vogels, Hendrikx, Jaggi
+models do remain close to some local average in a weighted neighborhood around them
+even with high learning rates. The workers benefit from a number of ‘effective neighbors’,
+potentially smaller than the whole graph, that allow them to use larger learning rates while
+retaining sufficient consensus within the ‘local neighborhood’.
+Similar insights apply when nodes have heterogeneous local functions: there is no need
+to enforce global averaging over the whole network when heterogeneity is small across local
+neighborhoods. Besides, there is no need to compensate for heterogeneity in the early phases
+of training, when models are all far from the global optimum.
+Based on our insights, we encourage practitioners of sparse distributed learning algorithms
+to look beyond the spectral gap of graph topologies, and to investigate the actual ‘effective
+number of neighbors’ that is used. We also hope that our insights motivate theoreticians to
+be mindful of assumptions that artificially limit the learning rate, even though they are tight
+in worst cases. Indeed, the spectral gap is omnipresent in the decentralized litterature, which
+sometimes hides some subtle phenomena such as the superlinear decrease of the variance in
+the learning rate, that we highlight.
+We show experimentally that our conclusions hold in deep learning, but extending our
+theory to the non-convex setting is an important open direction that could reveal interesting
+new phenomena. Another interesting direction would be to better understand (beyond the
+worst-case) the effective number of neighbors for irregular graphs.
+Acknowledgments and Disclosure of Funding
+This project was supported by SNSF grant 200020_200342.
+We thank Lie He for valuable conversations and for identifying the discrepancy between
+a topology’s spectral gap and its empirical performance.
+We also thank Raphaël Berthier for helpful discussions that allowed us to clarify the links
+between effective number of neighbors and spectral dimension.
+We also thank Aditya Vardhan Varre, Yatin Dandi and Mathieu Even for their feedback
+on the manuscript.
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+28
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+Absence of off-diagonal long-range order in hcp 4He dislocation cores
+Maurice de Koning
+Instituto de F´ısica Gleb Wataghin, Universidade Estadual de Campinas,
+UNICAMP, 13083-859, Campinas, S˜ao Paulo, Brazil and
+Center for Computing in Engineering & Sciences, Universidade Estadual de Campinas,
+UNICAMP, 13083-861, Campinas, S˜ao Paulo, Brazil∗
+Wei Cai
+Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040†
+Claudio Cazorla and Jordi Boronat
+Departament de F´ısica, Universitat Polit`ecnica de Catalunya, Campus Nord B4-B5, 08034 Barcelona, Spain‡
+The mass transport properties along dislocation cores in hcp 4He are revisited by considering two types
+of edge dislocations as well as a screw dislocation, using a fully correlated quantum simulation approach.
+Specifically, we employ the zero-temperature path-integral ground state (PIGS) method together with er-
+godic sampling of the permutation space to investigate the fundamental dislocation core structures and
+their off-diagonal long-range order properties. It is found that the Bose-Einstein condensate fraction of
+such defective 4He systems is practically null (≤ 10−6), just as in the bulk defect-free crystal. These re-
+sults provide compelling evidence for the absence of intrinsic superfluidity in dislocation cores in hcp 4He
+and challenge the superfluid dislocation-network interpretation of the mass-flux-experiment observations,
+calling for further experimental investigation.
+Although torsional oscillator experiments on hcp 4He
+by Kim and Chan in 2004 [1, 2] initially pointed at the
+existence of superfluidity in a solid-phase system, also
+known as supersolidity [3, 4], posterior examination un-
+ambiguously established that, instead, the observed phe-
+nomenology was a consequence of its anomalous mechan-
+ical behavior. Specifically, it was found to be caused by
+the obstructing influence of 3He impurities on the low-
+temperature mobility of lattice dislocations [5–10], the
+one-dimensional defects whose motion induces plastic de-
+formation in crystalline solids [11, 12].
+Still, the possibility of intrinsic supersolidity in hcp
+4He has not been discarded, in particular due to a vari-
+ety of mass flux experiments that report the flow of mat-
+ter across solid 4He samples [13–22]. However, the in-
+terpretation of these observations remains controversial.
+On the one hand, it has been proposed that the matter
+flow is transmitted through a superfluid network of in-
+terconnected, one-dimensional dislocation cores [20–22].
+This view relies fundamentally on the results of com-
+putational grand-canonical finite-temperature path-integral
+Monte Carlo (PIMC) studies of one group [23–25], which
+conclude that the cores of dislocations with Burgers vec-
+tors along the c-axis, b = [0001], are superfluid at ultralow
+temperatures of ∼ 0.1 K. In contrast, other authors argue
+that the mass flow is not dislocation-based but, rather, in-
+volves interfacial disorder effects within the samples, in-
+cluding at cell walls and grain boundaries [18, 19]. This
+account is supported by the fact that large amounts of 3He
+impurities, much larger than required to saturate typical
+dislocation networks and their intersections, are required
+to block the flow at low temperatures [19]. In either case,
+dislocations play a central role in this controversy and, in
+view of the scarce computational evidence, further theoret-
+ical scrutiny of their properties is pressingly needed.
+In this Letter we do so, revisiting the basic properties of
+dislocations in hcp 4He using first-principles quantum sim-
+ulations. However, the employed computational approach
+differs significantly from that applied in Refs. [23], [24]
+and [25]. First, instead of finite-temperature PIMC cal-
+culations, we resort to the zero-temperature path-integral
+ground state (PIGS) approach, a generalization of the
+PIMC method to zero temperature [26–28], that has shown
+to converge to exact ground-state results regardless of the
+initially chosen wave function for condensed phases of
+4He [27, 29].
+Like in Refs. [23–25], permutation sam-
+pling is carried out using the worm algorithm [30, 31] to
+guarantee ergodicity in permutation space [28]. Second,
+we adopt different boundary conditions for the computa-
+tional cells [32]. The results of the previous PIMC cal-
+culations [23, 24] are based on tube-like setups, in which
+only atoms within a cylindrical (or pencil-shaped in the
+case of Ref. [23]) region are treated explicitly while fix-
+ing a set of atoms outside of it to their classical positions,
+applying periodic boundary conditions (PBC) only along
+the dislocation line. Such an arrangement can give rise
+to lateral incompatibility stresses [33–37] that may result
+in incorrect dislocation core structures if these are not ad-
+equately relieved, e.g., by using Green’s function bound-
+ary conditions [36, 37]. Here, we employ different config-
+urations, including a dislocation-dipole arrangement em-
+ploying fully three-dimensional PBC [32, 38], as well as
+slab configurations containing a single dislocation subject
+to two-dimensional PBC [39]. Finally, we focus on the fun-
+damental atomic lattice structure of the dislocation cores,
+without considering processes that require the addition or
+arXiv:2301.02854v1  [cond-mat.other]  7 Jan 2023
+
+2
+removal of material through a grand-canonical (GC) ap-
+proach as used in Refs. [23], [24] and [25]. Indeed, if one
+does not adequately thermalize, changing particle num-
+bers may introduce artificial disorder, possibly leading to
+spurious the appearance of long-winding permutation cy-
+cles [23]. By applying this computational scheme to edge
+dislocations with their Burgers vectors both perpendicular
+and parallel to the c axis and to the screw dislocation with
+its Burgers vector along the c-axis, we find that, at zero
+temperature, the off-diagonal long-range order (ODLRO)
+is practically null (≤ 10−6), just as in the defect-free hcp
+crystal. This result contrasts previous claims [23–25] and
+signals the absence of quantum mass transport through dis-
+location cores in hcp 4He, instead lending support to the
+interpretation that the mass-flow observations are due to
+interfacial disorder effects rather than dislocation-mediated
+superfluidity.
+The integral Schr¨odinger equation for a system of N in-
+teracting particles can be expressed in imaginary time as
+Ψ(R, τ) =
+�
+dR′ G(R, R′; τ)Ψ(R′, 0) ,
+(1)
+where G(R, R′; τ) ≡ ⟨R|e−Hτ|R′⟩ is the correspond-
+ing Green’s function, with H the system Hamiltonian,
+Ψ(R, τ) the system wave function at imaginary time τ and
+|R⟩ = |r1, r2, . . . , rN⟩, with ri the particle positions.
+In the path-integral ground state (PIGS) approach [26–
+28], one exploits the formal identity between G(R, R′; τ)
+and the thermal density matrix of the system at an in-
+verse temperature of ϵ ≡ 1/T (we measure energy in
+units of Kelvins according to 1 K = 8.617×10−5 eV, such
+that ℏ2/2m = 6.059615 K ˚A2), namely, ρ(R, R′; ϵ).
+In
+this manner, the ground-state wave function of the system,
+Ψ0(R), can be asymptotically projected out of a trial wave
+function, ΨT(R), according to
+Ψ0(RM) =
+�
+M−1
+�
+i=0
+dRi ρ(Ri, Ri+1; ϵ) ΨT(R0) . (2)
+Likewise, the ground-state average value of any physical
+observable can be written in terms of a multidimensional
+integral that can be calculated exactly, within statistical
+uncertainties, independently of whether the corresponding
+operator commutes or not with the Hamiltonian of the sys-
+tem. The only requirement for the trial wave function ΨT
+is to satisfy the symmetry conditions imposed by the statis-
+tics of the simulated quantum many-body system. In this
+work, since we are dealing with boson particles, we con-
+sider a symmetrized trial wave function of the Jastrow type
+that typically is employed in quantum Monte Carlo (QMC)
+simulation of quantum liquids [28].
+The central physical quantity in our PIGS study is the
+one-body density matrix (OBDM), which is defined as
+ρ1(r1, r′
+1) = 1
+Z
+�
+dr2 . . . drN ρ(R, R′) ,
+(3)
+a)
+[1210]
+[0001]
+[1010]
+[1210]
+[0001]
+[1010]
+b)
+[1210]
+[0001]
+[1010]
+c)
+FIG. 1. Computational cells employed in the zero-temperature
+PIGS simulations of edge and screw dislocations in hcp 4He as
+visualized using the OVITO package [41]. Atoms shown in red
+and green are located in hcp and fcc surroundings, respectively.
+In all panels, the total Burgers vector b is indicated by the black
+arrow a) Dipole arrangement for the edge dislocations with Burg-
+ers vector in the basal plane, with PBC applied in all directions,
+following Ref. [38]. Each dislocation is dissociated into Shock-
+ley partial dislocations separated by a ribbon of stacking fault.
+b) Setup for the single edge dislocation with Burgers vector ori-
+ented along the c-axis dissociated into two Frank partials, with
+PBC applied along the dislocation line as well as the c-axis. The
+blue spheres in the upper and lower regions depict frozen atoms.
+c) Setup for single screw dislocation with Burgers vector oriented
+along the c-axis, with PBC applied along the dislocation line as
+well as the [1010] directions. The blue spheres in the upper and
+lower regions depict frozen atoms. Blue atoms in the central re-
+gion are close to the dislocation core.
+where the two configurations |R⟩ = |r1, r2, . . . , rN⟩ and
+|R′⟩ = |r′
+1, r2, . . . , rN⟩ differ only in one particle co-
+ordinate, and Z represents the quantum partition function
+of the system. In PIGS, ρ1(r1, r′
+1) is computed by track-
+ing the distances between the two extremities of one open
+chain (worm) during the QMC sampling [40]. Importantly,
+the condensate fraction of a N-boson system, n0, can be
+deduced from the long-range asymptotic behavior of the
+OBDM,
+n0 =
+lim
+|r1−r′
+1|→∞ ρ1(r1, r′
+1) .
+(4)
+We carried out PIGS simulations of hcp 4He crystals
+containing edge dislocations with their lines in the basal
+plane and with Burgers vectors oriented in the basal plane
+and along the c-axis, respectively, as well as for the screw
+dislocation with Burgers vector parallel to the c axis. The
+interactions between He atoms were modeled using the
+pairwise Aziz potential [42]. The computational cells em-
+ployed in the calculations are shown in Fig. 1. Depending
+on the type of edge dislocation, two different setups were
+employed. Fig. 1 a) displays the arrangement utilized for
+the basal edge (BE) dislocation. It is analogous to that used
+in Ref. [38], containing a pair of edge dislocations with
+
+3
+opposite Burgers vectors of the type b =
+1
+3[1210] disso-
+ciated into Shockley partials [11] with Burgers vectors of
+the kind b =
+1
+3[1100] separated by a stacking-fault rib-
+bon. PBC were applied in all three directions and the cell
+contained 1872 atoms. As shown in Fig. 1 b), a different
+approach was adopted for the c-axis edge (CE) dislocation
+with Burgers vector b = [0001]. While a dipole setup
+would also be possible, it would require simulating num-
+bers of atoms that are prohibitively large for the excessively
+demanding PIGS calculations. Therefore, we employed a
+cell containing only a single CE dislocation, applying PBC
+along the dislocation-line direction and the c-axis while fix-
+ing the top and bottom two layers in the [1010] directions.
+This is a standard approach that has been routinely used
+in atomistic simulations of dislocations [39, 43, 44] and
+preserves translational symmetry along the glide direction.
+The cell contains a total of 2280 atoms, of which 2052 were
+treated explicitly, whereas the remaining 228 atoms were
+fixed in the top and bottom layers. The CE dislocation
+dissociates into two Frank partial dislocations with Burg-
+ers vectors of the type b = 1
+6[2023] (Ref. [11], pg. 361)
+separated by a ribbon of stacking fault. A similar single-
+dislocation setup was also employed for the c-axis screw
+(CS) dislocation, as shown in Fig. 1 c), with a cell con-
+taining 1920 of which 228 atoms in the surface layers were
+held fixed. For all dislocation cells the atomic number den-
+sity was held fixed at ρ = 0.0287 ˚A−3, which corresponds
+to a lattice parameter of a = 3.67 ˚A. The number of time-
+slices used in Eq. (2) was M = 25 and an imaginary-time
+step of τ = 0.0125 K−1. We have verified that larger
+values of M and smaller values of τdo not modify our
+results within the statistical uncertainties (see Supporting
+Information [45]). Finally, for comparison with the defect-
+cell results, we also carried out subsidiary calculations for
+defect-free hcp 4He at the same density, employing a fully
+periodic cell containing 180 atoms.
+The red circles in Figs. 2 a) and the red and grey circles
+in Fig. 2 b) show the PIGS results for the zero-temperature
+OBDM of hcp 4He crystals containing, respectively, the
+BE, CS and CE dislocations. In all cases, ρ1 clearly ex-
+hibits a generally decreasing tendency under increasing ra-
+dial distance r ≡ |r1 − r′
+1| (note the logarithmic y-scale
+in the graphs). For the BE dislocation, the steady OBDM
+reduction is slightly smaller than for the CS and CE dis-
+locations; for example, at a radial distance of ∼ 7 ˚A the
+one-body density matrix has reduced to ∼ 10−5 in the for-
+mer case compared to ∼ 10−6 for the latter. Nevertheless,
+the slope of all ρ1 asymptotes are manifestly negative. This
+is clear evidence that the Bose-Einstein condensate fraction
+(Eq.4) of bulk hcp 4He containing these types of disloca-
+tions is negligible in practice (≤ 10−6) as ρ1 tends to zero
+in the limit of long radial distances. For further compari-
+son, the blue circles in Figs. 2 a) and b) display the PIGS
+OBDM calculations carried out for the defect-free hcp 4He
+cell at the same density.
+The results for these dislocation systems display the
+FIG. 2. PIGS one-body density matrix results obtained at zero
+temperature for hcp 4He for the cells containing (a) a BE dislo-
+cation (red circles) and (b) CS (grey circles) and CE dislocation
+(red circles). The y-axis is in logarithmic scale. For compari-
+son, PIGS results obtained for defect-free bulk hcp 4He at the
+same density (blue circles) as well as the liquid at a density of
+0.0227 ˚A−3 (black circles) are also shown.
+same general trend as seen for the defect-free crystal, pro-
+viding further support for our conclusion of negligible n0
+in the presence of these types of dislocations. As a final
+consistency check, we carried out an additional simulation
+starting from the CE dislocation cell, but reducing its den-
+sity to 0.0227 ˚A−3 to induce a transition into the liquid
+phase. The corresponding ODLRO, obtained after reach-
+ing the equilibrated liquid, is shown as the black circles in
+Fig. 2 b). The Bose-Einstein condensate fraction obtained
+in this case, employing the same PIGS approach applied to
+the solid-phase systems, is found to be n0 ∼ 0.02. This is
+in agreement with the known value corresponding to bulk
+liquid 4He at that density at ultralow temperatures [46], at-
+testing to the numerical reliability of our zero-temperature
+computational approach.
+The fact that the zero-temperature OBDM results in
+Fig. 2 display a practically null Bose-Einstein condensate
+
+a)
+0.1
+0.01
+0.001
+0.0001
+1*10-5
+BE dislocation
+Solid (perfect)
+Liquid
+1*10
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+b)
+0.1
+0.01
+0.001
+(a)id
+0.0001
+CS dislocation
+CE dislocation
+1*10
+.6
+Solid (perfect)
+Liquid
+1*10-7
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+r (A)4
+FIG. 3. Visualization of the 4He system containing the dissociated CE dislocation at the beginning and end of the PIGS simulations;
+quantum polymers “centroids” are represented in both cases. The initial configuration was obtained after equilibrating the system at
+T = 1 K with the PIMC method. A few quantum polymers located at a similar distance within the dislocation core are represented in
+the inset of b); long chains of atomic exchanges involving several quantum polymers are absent.
+fraction (i.e., ≲ 10−6) in both the defect-free as well as de-
+fected 4He crystal is compelling evidence that the cores of
+the considered types of dislocations are in fact insulating
+in nature. The lack of quantum mass flux along the dis-
+location cores can be further verified by visual inspection
+of the quantum polymers during the simulation. A repre-
+sentative example is depicted in Fig. 3 for the case of the
+dissociated CE dislocation. Fig. 3 a) and the main panel of
+Fig. 3 b) display the centroids (i.e., the “centers-of-mass”
+of the quantum polymers) for the initial and final configu-
+rations of the PIGS simulation, respectively. Both pictures
+qualitatively demonstrate the prevalence of atomic order,
+including the regions of the partial dislocation cores. Fur-
+thermore, when visualizing entire quantum polymers in the
+core region as depicted in the expanded view, there are no
+evident traces of long-winding quantum exchanges [40],
+thus corroborating the absence of superfluidity in these dis-
+location cores.
+While the absence of superfluidity for the BE disloca-
+tions is consistent with the PIMC calculations reported in
+Ref. [38] and the unpublished data referred to in Ref. [47],
+the present PIGS results for the CS and CE dislocations
+are at odds with the findings in Refs. [23] and
+[24] as
+well as the proposed mechanism of “superclimb” of dis-
+locations [24, 25]. Accordingly, our results are incompat-
+ible with the superfluid dislocation network interpretation
+of the mass flux experiments, and lend support to the alter-
+nate view that effects related to disordered regions at inter-
+nal interfaces, including vessel walls and grain boundaries,
+are responsible for the observations [18, 19].
+A further issue with the superfluid-network interpreta-
+tion is that, given the consensus that dislocations with
+Burgers vectors in the basal plane are insulating [38, 47],
+it relies fundamentally on the presence of a spanning net-
+work consisting entirely of dislocations with c-axis Burg-
+ers vectors. Such an arrangement of dislocations, however,
+is geometrically impossible due to the requirement of con-
+servation of Burgers vector at network nodes [11]. In con-
+trast, there is abundant experimental evidence [6, 8, 48–
+51] for the existence of networks of nonsuperfluid basal-
+plane Burgers-vector dislocations, which drive the domi-
+nant mode of basal slip in hcp 4He [50, 51] and play a cen-
+tral role in the phenomenon of giant plasticity [8], as well
+as in the nonsupersolid explanation of the original torsion-
+oscillator observations by Kim and Chan [6]. This premise
+is also consistent with findings in other hcp-structured ma-
+terials such as Zn [52] and Mg [53] in which observed
+dislocation networks display the characteristic hexagonal
+structure of basal-plane Burgers vector dislocations.
+In
+this light, the present results further challenge the super-
+fluid dislocation-network interpretation of the mass-flux-
+experiment observations and call for further experimental
+investigation.
+M.K. acknowledges support from CNPq, Fapesp grant
+no. 2016/23891-6 and the Center for Computing in En-
+gineering & Sciences - Fapesp/Cepid no.
+2013/08293-
+7.
+W.C. acknowledges support from the U.S. Depart-
+ment of Energy, Office of Basic Energy Sciences, Divi-
+sion of Materials Sciences and Engineering under Award
+No.
+DE-SC0010412.
+J.B. acknowledges financial sup-
+
+a
+b
+.
+8
+a
+8
+.
+.
+Quantum
+polymers
+Initial configuration
+Final configuration5
+port from the Secretaria d’Universitats i Recerca del De-
+partament d’Empresa i Coneixement de la Generalitat de
+Catalunya, co-funded by the European Union Regional De-
+velopment Fund within the ERDF Operational Program
+of Catalunya (project QuantumCat, Ref. 001-P-001644),
+and the MINECO (Spain) Grant PID2020-113565GB-C21.
+C.C. acknowledges financial support from the MINECO
+(Spain) under the “Ram´on y Cajal” fellowship (RYC2018-
+024947-I).
+∗ [email protected]
+† [email protected]
+‡ [email protected]
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+

9NAyT4oBgHgl3EQfdPda/content/tmp_files/2301.00298v1.pdf.txt

@@ -0,0 +1,1810 @@
+arXiv:2301.00298v1  [math.NT]  31 Dec 2022
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA
+SERIES
+T. WAKHARE1 AND C. VIGNAT2
+Abstract. An unpublished identity of Gosper restates a hypergeometric identity for odd
+zeta values in terms of an infinite product of matrices. We show that this correspondence
+runs much deeper, and show that many examples of WZ-accelerated series for zeta values lift
+to infinite matrix products. We also introduce a new matrix subgroup, the Gosper group,
+which all of our matrix products fall into.
+1. Introduction
+In his famous book “Mathematical Constants” [3], Finch cites an unpublished result by
+Gosper [2]:
+(1.1)
+∞
+�
+k=1
+� −
+k
+2(2k+1)
+5
+4k2
+0
+1
+�
+=
+�
+0
+ζ (3)
+0
+1
+�
+,
+and its (N + 1) × (N + 1) extension, for N ⩾ 2,
+(1.2)
+∞
+�
+k=1
+
+
+−
+k
+2(2k+1)
+1
+2k(2k+1)
+0
+. . .
+0
+1
+k2N
+0
+−
+k
+2(2k+1)
+1
+2k(2k+1)
+. . .
+1
+k2N−2
+...
+...
+...
+...
+...
+0
+0
+0
+. . .
+1
+2k(2k+1)
+1
+k4
+0
+0
+0
+. . .
+−
+k
+2(2k+1)
+5
+4k2
+0
+0
+0
+. . .
+0
+1
+
+
+=
+
+
+0
+. . .
+0
+ζ (2N + 1)
+0
+. . .
+0
+ζ (2N − 1)
+...
+...
+...
+0
+. . .
+0
+ζ (5)
+0
+. . .
+0
+ζ (3)
+0
+. . .
+0
+1
+
+
+.
+We will show that this formula is in fact equivalent to Koecher’s identity [4, Eq. (3)]
+(1.3)
+∞
+�
+n=0
+1
+n(n2 − x2) = 1
+2
+∞
+�
+k=1
+(−1)k−1
+�2k
+k
+�
+k3
+5k2 − x2
+k2 − x2
+k−1
+�
+m=1
+�
+.1 − x2
+m2
+�
+.
+By extracting coefficients of 1 and x2 in Koecher’s identity, we recover Markov’s series ac-
+celeration identity [5]
+ζ (3) = 5
+2
+�
+n⩾1
+(−1)n−1
+n3�2n
+n
+�
+and its higher order counterpart
+ζ (5) = 2
+∞
+�
+n=1
+(−1)n−1
+n5�2n
+n
+� − 5
+2
+∞
+�
+n=1
+(−1)n−1 H(2)
+n−1
+n3�2n
+n
+�
+.
+1
+
+2
+T. WAKHARE1 AND C. VIGNAT2
+These are efficiently encoded by the matrix product. By extracting other coefficients of
+xn in Koecher’s identity, we recover counterparts for ζ(2n + 1) which are again encoded by
+the matrix product.
+This correspondence runs much deeper, and we will show that several hypergeometric-type
+series for the zeta function at small integers are equivalent to infinite products for N × N
+matrices. The fact that these identities support an expression in terms of matrix products is
+already interesting. The pattern of entries of some small matrices suggest the general form
+of the relevant n × n generalizations, which would then be equivalent to new accelerated
+series for zeta values.
+2. Background
+2.1. Special Functions. The Riemann zeta function, absolutely convergent for s ∈ C, ℜs >
+1 is given by
+(2.1)
+ζ(s) :=
+∞
+�
+n=1
+1
+ns.
+This straightforwardly extends to the Hurwitz zeta function with the addition of a parameter
+z ∈ C, z ̸= 0, −1, −2, . . .:
+(2.2)
+ζ(s|z) :=
+∞
+�
+n=1
+1
+(n + z)s ,
+so that ζ(s) = ζ(s|1).
+The harmonic numbers are given by H0 := 0 and
+(2.3)
+Hn :=
+n
+�
+k=1
+1
+k,
+n ⩾ 1.
+The hyper-harmonic numbers are defined similar.
+We will also consider the elementary
+symmetric functions
+(2.4)
+e(s)
+ℓ (k) := [tℓ]
+k−1
+�
+j=1
+�
+1 + t
+js
+�
+=
+�
+1⩽j1<j2<···<jℓ⩽k−1
+1
+(j1 · · · jℓ)s,
+which reduce to the harmonic numbers at e1
+1(n) = Hn−1.
+3. The Gosper Group
+Each Gosper matrix in the product (1.2) has the form
+Mk =
+�
+Ak
+uk
+0
+1
+�
+where Ak is square (N × N), uk is a (N × 1) vector and 0 is the (1 × N) vector of zeros.
+Matrices of this kind form a group, which we shall name the Gosper group. With IN the
+(N × N) identity matrix, the unit element of the group is
+�
+IN
+0
+0
+1
+�
+, and the inverse of
+
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES
+3
+an element M =
+�
+A
+u
+0
+1
+�
+is M−1 =
+�
+A−1
+−A−1u
+0
+1
+�
+. Closure follows from M1M2 =
+�
+A1A2
+A1u2 + u1
+0
+1
+�
+. We can inductively verify that
+M1M2 . . . Mn =
+�
+A1A2 . . . An
+�n
+k=1 A1 . . . Ak−1uk
+0
+1
+�
+.
+3.1. Toeplitz Matrices. Moreover, each Ak in Gosper’s identity has the simple form
+Ak = αkI + βkJ
+where J is the (N × N) matrix with a first superdiagonal of ones.
+Hence JN = 0 and, for p ⩾ N, we have
+A1A2 . . . Ap = (α1I + β1J) (α2I + β2J) . . . (αpI + βpJ)
+=
+� p
+�
+i=1
+αi
+� 
+I +
+p
+�
+j=1
+βj
+αj
+J + · · · +
+�
+1⩽j1<···<jN−1⩽p
+βj1 . . . βjN−1
+αj1 . . . αjN−1
+JN−1
+
+ .
+For p < N the summation is instead truncated at Jp.
+The general form of the components of the limiting infinite product case can be deduced
+by induction.
+Lemma 3.1. The components of
+(3.1)
+∞
+�
+k=1
+�
+Ak
+uk
+0
+1
+�
+=
+� �∞
+k=1 Ak
+v∞
+0
+1
+�
+,
+with
+�
+v(N)
+∞ , . . . , v(1)
+∞
+�T := v∞ =
+∞
+�
+p=1
+A1 . . . Ap−1up,
+are
+v(1)
+∞ =
+∞
+�
+p=1
+(α1 · · · αp−1) u(1)
+p ,
+v(2)
+∞ =
+∞
+�
+p=1
+(α1 · · · αp−1)
+�
+u(2)
+p
++
+�p−1
+�
+j=1
+βj
+αj
+�
+u(1)
+p
+�
+,
+...
+v(ℓ)
+∞ =
+∞
+�
+p=1
+(α1 · · · αp−1)
+
+u(ℓ)
+p +
+�p−1
+�
+j=1
+βj
+αj
+�
+u(ℓ−1)
+p
++ · · · +
+
+
+�
+1⩽j1<···<jℓ−1⩽p−1
+βj1 . . . βjℓ−1
+αj1 . . . αjℓ−1
+
+ u(1)
+p
+
+ ,
+with 1 ⩽ ℓ ⩽ N.
+Already the connection to zeta series and hyperharmonic numbers is clear: with the correct
+choice of α and β, the multiple sums will reduce to multiple zeta type functions.
+These matrix products also exhibit a stability phenomenon, where increasing the dimen-
+sion of the matrix does not impact any entries in v∞ except the top right one, since mapping
+N → N + 1 only changes the formula for v(N+1)
+∞
+.
+
+4
+T. WAKHARE1 AND C. VIGNAT2
+We will consistently refer to the N = 1 and N = 2 cases. Explicitly, when N = 1 so that
+both Ak (denoted αk to avoid confusion) and uk are scalars, we have
+Lemma 3.2. For N = 1,
+(3.2)
+n
+�
+k=1
+�
+αk
+βk
+0
+1
+�
+=
+� �n
+k=1 αk
+�n
+k=1 α1 . . . αk−1βk
+0
+1
+�
+.
+Although we will only need the n → ∞ limit, let us note that this identity holds for finite
+n.
+4. Koecher’ Identity
+Theorem 4.1. Identity (1.1) and Koecher’s identity are equivalent.
+Proof. Begin with Koecher’s identity (1.3). By extracting coefficients of x2n, in general we
+obtain
+(4.1)
+ζ(2n + 3) = 5
+2
+∞
+�
+k=1
+(−1)k−1
+k3�2k
+k
+� (−1)ne(2)
+n (k) + 2
+n
+�
+j=1
+∞
+�
+k=1
+(−1)k−1
+k2j+3�2k
+k
+�(−1)n−je(2)
+n−j(k).
+Take αk = −
+k
+2(2k+1), βk =
+1
+2k(2k+1), u(1)
+k
+=
+5
+4k2, and u(ℓ)
+k
+=
+1
+k2ℓ+2 for 2 ⩽ ℓ ⩽ N. This
+corresponds to the Gosper matrix
+�
+Ak
+uk
+0
+1
+�
+=
+
+
+−
+k
+2(2k+1)
+1
+2k(2k+1)
+0
+. . .
+0
+1
+k2N
+0
+−
+k
+2(2k+1)
+1
+2k(2k+1)
+. . .
+1
+k2N−2
+...
+...
+...
+...
+...
+0
+0
+0
+. . .
+1
+2k(2k+1)
+1
+k4
+0
+0
+0
+. . .
+−
+k
+2(2k+1)
+5
+4k2
+0
+0
+0
+. . .
+0
+1
+
+
+.
+Then
+p
+�
+i=1
+αi = (−1)p
+p
+�
+i=1
+i2
+(2i)(2i + 1) = (−1)p
+p!2
+(2p + 1)!,
+and (for 2 ⩽ ℓ ⩽ N)
+�
+j1<···<jℓ−1⩽p−1
+βj1 . . . βjℓ−1
+αj1 . . . αjℓ−1
+= (−1)ℓ
+�
+j1<···<jℓ−1⩽p−1
+1
+(j1 · · · jℓ−1)2 = (−1)ℓe(2)
+ℓ−1(p).
+We deduce
+lim
+p→∞ α1 · · · αp = 0,
+while
+lim
+p→∞
+�
+j1<···<jk⩽p−1
+1
+(j1 · · · jk)2 ⩽ lim
+p→∞
+p
+�
+j1=1
+1
+j2
+1
+= ζ(2).
+Hence, applying Lemma 3.1, we deduce
+∞
+�
+i=1
+Ai = 0.
+
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES
+5
+The components in the right column are then explicitly given as
+v(ℓ)
+∞ =
+∞
+�
+p=1
+(α1 · · · αp−1)
+
+u(ℓ)
+p +
+�p−1
+�
+j=1
+βj
+αj
+�
+u(ℓ−1)
+p
++ · · · +
+
+
+�
+1⩽j1<···<jℓ−1⩽p−1
+βj1 . . . βjℓ−1
+αj1 . . . αjℓ−1
+
+ u(1)
+p
+
+
+=
+∞
+�
+p=1
+(−1)p−1(p − 1)!2
+(2p − 1)!
+�
+1
+p2ℓ+2 − e(2)
+1 (p)
+p2ℓ
++ · · · + (−1)ℓ−15
+4
+e(2)
+ℓ−1(p)
+p2
+�
+= 5
+2
+∞
+�
+p=1
+(−1)p−1
+p3�2p
+p
+� e(2)
+ℓ−1(p) + 2
+ℓ−1
+�
+j=1
+∞
+�
+p=1
+(−1)p−1
+p3+2j�2p
+p
+�e(2)
+ℓ−1−j(p)(−1)ℓ−1−j.
+We see that this is exactly the formula from Koecher’s identity, hence equals ζ(2ℓ + 1) for
+1 ⩽ ℓ ⩽ N.
+■
+5. Leschiner’s identity
+Begin with the Leschiner identity
+�
+n⩾1
+(−1)n−1
+n2 − z2 = 1
+2
+�
+k⩾1
+1
+�2k
+k
+�
+k2
+3k2 + z2
+k2 − z2
+k−1
+�
+j=1
+�
+1 − z2
+j2
+�
+,
+so that
+˜ζ (2) = 3
+2
+�
+k⩾1
+1
+�2k
+k
+�
+k2,
+and
+¯ζ (4) = 3
+2
+�
+k⩾1
+1
+�2k
+k
+�
+k2
+� 4
+k2 − H(2)
+k−1
+�
+,
+and in general (I think I made a mistake here)
+˜ζ(2n + 2) = 3
+2
+∞
+�
+k=1
+1
+k2�2k
+k
+�(−1)ne(2)
+n (k) + 6
+n
+�
+j=1
+∞
+�
+k=1
+1
+k2j+2�2k
+k
+�(−1)n−je(2)
+n−j(k).
+A Gosper representation for ¯ζ (2) and ¯ζ (4) is
+�
+n⩾1
+
+
+n
+2(2n+1)
+−1
+2n(2n+1)
+1
+n3
+0
+n
+2(2n+1)
+3
+4n
+0
+0
+1
+
+ =
+
+
+0
+0
+¯ζ (4)
+0
+0
+¯ζ (2)
+0
+0
+1
+
+ .
+This will generalize using the same method as Koecher.
+6. Borwein’s Identity
+6.1. the infinite product case. Extracting coefficient of z2n from Borwein’s identity [1]
+(6.1)
+�
+n⩾1
+1
+n2 − z2 = 3
+�
+k⩾1
+1
+�2k
+k
+�
+1
+k2 − z2
+k−1
+�
+j=1
+j2 − 4z2
+j2 − z2 .
+
+6
+T. WAKHARE1 AND C. VIGNAT2
+gives
+�
+k⩾1
+1
+�2k
+k
+�
+1
+k2 − z2
+k−1
+�
+j=1
+j2 − 4z2
+j2 − z2 =
+�
+k⩾1
+1
+k2�2k
+k
+�
+k−1
+�
+j=1
+�
+1 − 4z2
+j2
+�
+k
+�
+j=1
+1
+1 − z2
+j2
+=
+�
+k⩾1
+1
+k2�2k
+k
+�
+�
+ℓ⩾0
+z2ℓ4ℓe(2)
+ℓ (k)
+�
+m⩾0
+z2mh(2)
+m (k + 1),
+where hm is the complete symmetric function. This gives us a formula for the coefficient
+of z2n as a convolution over hm and em. How do we encode this in the matrix, in terms of
+αk, βk, uk?
+Theorem 6.1. A Gosper representation for ζ (2) is obtained as
+�
+n⩾1
+�
+n
+2(2n+1)
+3
+2n
+0
+1
+�
+=
+�
+0
+ζ (2)
+0
+1
+�
+.
+Proof. Identifying the constant term produces
+ζ (2) = 3
+�
+k⩾1
+1
+�2k
+k
+�
+k2.
+With αk =
+k
+2(2k+1) and βk =
+3
+2k, we have
+�
+n⩾1
+�n−1
+�
+k=1
+αk
+�
+βn = 3
+2
+�
+n⩾1
+2
+n2�2n
+n
+� = ζ (2) .
+■
+Identifying the linear term in (6.1) produces
+ζ (4) = 3
+�
+k⩾1
+1
+�2k
+k
+�
+k2
+� 1
+k2 − 3H(2)
+k−1
+�
+.
+This suggests the following result.
+Theorem 6.2. A Gosper representation for ζ (2) and ζ (4) is obtained as
+�
+n⩾1
+
+
+n
+2(2n+1)
+−3
+2n(2n+1)
+3
+2n3
+0
+n
+2(2n+1)
+3
+2n
+0
+0
+1
+
+ =
+
+
+0
+0
+ζ (4)
+0
+0
+ζ (2)
+0
+0
+1
+
+ .
+Proof. Denote
+Mn =
+
+
+δn
+γn
+u(1)
+n
+0
+δn
+u(2)
+n
+0
+0
+1
+
+ =
+�
+An
+un
+0
+1
+�
+
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES
+7
+with An =
+�
+δn
+γn
+0
+δn
+�
+= δnI + γnJ and δn =
+2
+n(2n+1) so that, with I =
+�
+1
+0
+0
+1
+�
+, J =
+�
+0
+1
+0
+0
+�
+, un =
+�
+u(1)
+n
+u(2)
+n =
+3
+2n
+�
+,
+A1 . . . Ai−1 =
+2
+i
+�2i
+i
+�
+�
+I + J
+i−1
+�
+j=1
+γj
+δj
+�
+.
+We know that
+M1 . . . Mn =
+�
+A1 . . . An
+vn
+0
+1
+�
+with
+vn =
+n
+�
+i=1
+A1 . . . Ai−1ui
+so that
+vn =
+n
+�
+i=1
+2
+i
+�2i
+i
+�
+�
+ui +
+i−1
+�
+j=1
+γj
+δj
+�
+3
+2i
+0
+��
+=
+n
+�
+i=1
+2
+i
+�2i
+i
+�
+��
+u(1)
+i3
+2i
+�
++
+i−1
+�
+j=1
+γj
+δj
+�
+3
+2i
+0
+��
+.
+=
+
+
+�n
+i=1
+2
+i(2i
+i )u(1)
+i
++ �i−1
+j=1
+γj
+δj
+3
+2i
+�n
+i=1
+2
+i(2i
+i)
+3
+2i
+
+
+This produces
+v(2)
+∞ = ζ (2) =
+∞
+�
+i=1
+3
+i2�2i
+i
+�
+and
+v(1)
+∞ = ζ (4) =
+∞
+�
+i=1
+2
+i
+�2i
+i
+�u(1)
+i
++
+∞
+�
+i=1
+2
+i
+�2i
+i
+� 3
+2i
+i−1
+�
+j=1
+γj
+δj
+.
+Identifying with
+ζ (4) = 3
+�
+k⩾1
+1
+�2k
+k
+�
+k2
+� 1
+k2 − 3H(2)
+k−1
+�
+produces
+u(1)
+i
+= 3
+2i3, γj =
+−3
+2j (2j + 1).
+■
+Unfortunately, the case that includes ζ (6) is not as straightforward.
+Theorem 6.3. A Gosper representation for ζ (2) , ζ (4) and ζ (6) is obtained as
+�
+n⩾1
+
+
+n
+2(2n+1)
+−
+3
+2n(2n+1)
+0
+3
+2n5 −
+9H(4)
+n−1
+2n
+0
+n
+2(2n+1)
+−
+3
+2n(2n+1)
+3
+2n3
+0
+0
+n
+2(2n+1)
+3
+2n
+0
+0
+0
+1
+
+ =
+
+
+0
+0
+0
+ζ (6)
+0
+0
+0
+ζ (4)
+0
+0
+0
+ζ (2)
+0
+0
+0
+1
+
+ .
+
+8
+T. WAKHARE1 AND C. VIGNAT2
+For example, the truncated product from n = 1 up to n = 200 is
+
+
+2.4222.10−122
+−1.1917.10−121
+1.7517.10−121
+1.01734
+0.
+2.4222.10−122
+-1.1917.10−121
+1.08232
+0.
+0.
+2.4222.10−122
+1.64493
+0.
+0.
+0.
+1.
+
+ .
+Proof. Identifying the coefficient of z2 in Borwein’s identity (6.1) produces
+ζ (6) = 3
+�
+k⩾1
+1
+�2k
+k
+�
+k2
+�
+17H(2,2)
+k−1 + H(4)
+k−1 − 4
+�
+H(2)
+k−1
+�2
+− 3H(2)
+k−1
+k2
++ 1
+k4
+�
+.
+Moreover, the vector vn is computed as
+vn =
+n
+�
+i=1
+A1 . . . Ai−1ui =
+n
+�
+i=1
+2
+�2i
+i
+�
+i
+
+ui − 3H(2)
+i−1
+
+
+ui (2)
+ui (3)
+0
+
+ + 9H(2,2)
+i−1
+
+
+ui (3)
+0
+0
+
+
+
+ .
+Hence
+v(1)
+∞ =
+∞
+�
+i=1
+2
+�2i
+i
+�
+i
+�
+ui (1) − 3H(2)
+i−1ui (2) + 9H(2,2)
+i−1 ui (3)
+�
+=
+∞
+�
+i=1
+2
+�2i
+i
+�
+i
+�
+ui (1) − 3H(2)
+i−1
+3
+2i3 + 9H(2,2)
+i−1
+3
+2i
+�
+.
+Using
+3
+n
+�
+4H(2,2)
+n−1 + 1
+2H(4)
+n−1 − 2
+�
+H(2)
+n−1
+�2�
+= − 9
+2nH(4)
+n−1
+and identifying v(1)
+∞ = ζ (6) produces the result.
+■
+6.2. A finite matrix product for H(3)
+N . From the identity
+H(3)
+N =
+N
+�
+n=1
+(−1)n−1
+n3�2n
+n
+�
+�
+5
+2 −
+1
+2
+�N+n
+2n
+�
+�
+,
+we deduce the following finite product representation
+N
+�
+n=1
+
+ −
+n
+2 (2n + 1)
+5
+4n2
+�
+1 −
+1
+5
+�N+n
+2n
+�
+�
+0
+1
+
+ =
+
+
+2 (−1)N
+(N + 1)
+�2N+2
+N+1
+�
+H(3)
+N
+0
+1
+
+ .
+7. Gosper Representation of Markov’s identity for ζ(2) and ζ(z + 1, 3)
+7.1. Markov’s identity for ζ(z + 1, 3). Markov’s identity reads
+(7.1)
+ζ (z + 1, 3) =
+∞
+�
+n=1
+1
+(n + z)3 = 1
+4
+∞
+�
+k=1
+(−1)k−1 (k − 1)!6
+(2k − 1)!
+5k2 + 6kz + 2z2
+((z + 1) (z + 2) . . . (z + k))4.
+
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES
+9
+Theorem 7.1. A Gosper’s representation for Markov’s identity is
+∞
+�
+n=1
+
+ −
+n6
+2n (2n + 1) (z + n + 1)4
+5k2 + 6kz + 2z2
+0
+1
+
+ =
+�
+0
+4 (z + 1)4 ζ (z + 1, 3)
+0
+1
+�
+or equivalently
+∞
+�
+n=1
+
+ −
+n6
+2n (2n + 1) (z + n + 1)4
+5k2 + 6kz + 2z2
+4 (z + 1)4
+0
+1
+
+ =
+�
+0
+ζ (z + 1, 3)
+0
+1
+�
+.
+Proof. Rewrite Markov’s identity as
+4ζ (z + 1, 3) =
+�
+k⩾1
+(−1)k−1 (k − 1)!6
+(2k − 1)!
+5k2 + 6kz + 2z2
+((z + 1) . . . (z + k))4,
+define
+uk = 5k2 + 6kz + 2z2
+and notice that writing
+4ζ (z + 1, 3) = u1 + α1u2 + α1α2u3 + . . .
+requires that the coefficient of u1 should be equal to 1; as it is equal to
+1
+(z+1)4, consider the
+variation
+4 (z + 1)4 ζ (z + 1, 3) =
+�
+k⩾1
+(−1)k−1 (k − 1)!6
+(2k − 1)!
+(z + 1)4
+((z + 1) . . . (z + k))4uk
+which now satisfies this constraint. Then identifying
+α1 . . . αk−1 = (−1)k−1 (k − 1)!6
+(2k − 1)!
+(z + 1)4
+((z + 1) . . . (z + k))4
+provides
+αk =
+−k6
+2k (2k + 1) (z + k + 1)4.
+Notice that the constant term (z + 1)4 disappears from αk.
+■
+Another identity [6] due to Tauraso is
+(7.2)
+�
+n⩾1
+1
+n2 − an − b2 =
+�
+k⩾1
+3k − a
+�2k
+k
+�
+k
+1
+k2 − ak − b2
+k−1
+�
+j=1
+j2 − a2 − 4b2
+j2 − aj − b2 .
+Theorem 7.2. A Gosper’s matrix representation for identity (7.2) is
+∞
+�
+n=1
+�
+k
+2(2k+1)
+k2−a2−4b2
+k2−ak−b2
+3k−a
+k2−ak−b2
+0
+1
+�
+=
+� 0
+�
+n⩾1
+2
+n2−an−b2
+0
+1
+�
+.
+Notice that
+�
+n⩾1
+2
+n2 − an − b2 =
+2
+√
+a2 + 4b2
+�
+ψ
+�
+1 − a
+2 +
+√
+a2 + 4b2
+2
+�
+− ψ
+�
+1 − a
+2 −
+√
+a2 + 4b2
+2
+��
+.
+
+10
+T. WAKHARE1 AND C. VIGNAT2
+Proof. Choose
+uk =
+3k − a
+k2 − ak − b2.
+The first term in (7.2) is
+3 − a
+2
+1
+1 − a − b2 = 1
+2u1
+so that we consider twice identity (7.2), and choose
+α1 . . . αk−1 =
+1
+�2k
+k
+�
+k
+k−1
+�
+j=1
+j2 − a2 − 4b2
+j2 − aj − b2
+so that
+αk = k2 − a2 − 4b2
+k2 − ak − b2
+k
+2 (2k + 1).
+■
+A quartic version reads
+�
+n⩾1
+n
+n4 − a2n2 − b4 = 1
+2
+�
+k⩾1
+(−1)k−1
+�2k
+k
+�
+k
+5k2 − a2
+k4 − a2k2 − b4
+k−1
+�
+j=1
+(j2 − a2)2 + 4b4
+j4 − a2j2 − b4 .
+The same approach as above produces
+∞
+�
+n=1
+�
+−
+k
+2(2k+1)
+(k2−a2)
+2+4b4
+k4−a2k2−b4
+5k2−a2
+k4−a2k2−b4
+0
+1
+�
+=
+� 0
+�
+n=1
+4n
+n4−a2n2−b4
+0
+1
+�
+.
+Amdeberhan-Zeilberger’s ultra-fast series representation [7]
+ζ (3) =
+�
+n⩾1
+(−1)n−1
+(n − 1)!10
+64 (2n − 1)!5
+�
+205n2 − 160n + 32
+�
+can be realized as
+∞
+�
+n=1
+�
+−
+�
+k
+2(2k+1)
+�5
+205k2 − 160k + 32
+0
+1
+�
+=
+�
+0
+64ζ (3)
+0
+1
+�
+or equivalently
+∞
+�
+n=1
+�
+−
+�
+k
+2(2k+1)
+�5
+205k2−160k+32
+64
+0
+1
+�
+=
+�
+0
+ζ (3)
+0
+1
+�
+.
+The resemblance with (3.1) is interesting and suggests the generalization
+∞
+�
+n=1
+
+
+−
+�
+k
+2(2k+1)
+�5
+�
+1
+2k(2k+1)
+�5
+P (k)
+0
+−
+�
+k
+2(2k+1)
+�5
+205k2−160k+32
+64
+0
+0
+1
+
+ =
+
+
+0
+0
+ζ (5)
+0
+0
+ζ (3)
+0
+0
+1
+
+
+where P (k) is to be determined. Another fast representation due to Amdeberhan [8] is
+ζ (3) = 1
+4
+∞
+�
+n=1
+(−1)n−1 (56n2 − 32n + 5)
+n3 (2n − 1)2 �3n
+n
+��2n
+n
+�
+
+INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES
+11
+and produces
+∞
+�
+n=1
+
+ −
+k3
+(3k + 3) (3k + 2) (3k + 1)
+�2k − 1
+2k + 1
+�2
+56k2 − 32k + 5
+24
+0
+1
+
+ =
+�
+0
+ζ (3)
+0
+1
+�
+.
+References
+[1] J.M. Borwein, D.M. Bradley and D.J. Broadhurst, Evaluations of k-fold Euler/Zagier sums: a com-
+pendium of results for arbitrary k, Electron. J. Combin., 4-2, 1-21, 1997
+[2] R. W. Gosper, Analytic identities from path invariant matrix multiplication, unpublished manuscript,
+1976
+[3] S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications 94, Cambridge
+University Press, 2003
+[4] M. Koecher, Letters, Math. Intelligencer 2 (1980), no. 2, 62-64.
+[5] A. A. Markoff, M´emoire sur la transformation des s´eries peu convergentes en s´eries tr`es convergentes,
+M´em. de l’Acad. Imp. Sci. de St. P´etersbourg, t. XXXVII, No. 9 (1890)
+[6] R.
+Tauraso,
+A
+bivariate
+generating
+function
+for
+zeta
+values
+and
+related
+supercongruences,
+arXiv:1806.00846
+[7] T. Amdeberhan and D. Zeilberger, Hypergeometric Series Acceleration via the WZ Method, THe Elec-
+tronic Journal of Combinatorics, 4-2, The Wilf Festchrift Volume, 1997
+[8] T. Amdeberhan, Faster and faster convergent series for z(3), Electronic Journal of Combinatorics, Volume
+3 Issue 1, 1996
+1 Department of Electrical Engineering and Computer Science, Massachusetts Institute
+of Technology, Cambridge, Massachusetts, USA
+Email address: [email protected]
+2 Department of Mathematics, Tulane University, New Orleans, Louisiana, USA
+Email address: [email protected]
+

9NAyT4oBgHgl3EQfdPda/content/tmp_files/load_file.txt

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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='00298v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='NT]  31 Dec 2022  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA  SERIES  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' An unpublished identity of Gosper restates a hypergeometric identity for odd  zeta values in terms of an infinite product of matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' We show that this correspondence  runs much deeper, and show that many examples of WZ-accelerated series for zeta values lift  to infinite matrix products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' We also introduce a new matrix subgroup, the Gosper group,  which all of our matrix products fall into.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Introduction  In his famous book “Mathematical Constants” [3], Finch cites an unpublished result by  Gosper [2]:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  ∞  �  k=1  � −  k  2(2k+1)  5  4k2  0  1  �  =  �  0  ζ (3)  0  1  �  ,  and its (N + 1) × (N + 1) extension, for N ⩾ 2,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2)  ∞  �  k=1  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  −  k  2(2k+1)  1  2k(2k+1)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
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+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1  2k(2k+1)  1  k4  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  ζ (2N + 1)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  ζ (2N − 1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  ζ (5)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  ζ (3)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  We will show that this formula is in fact equivalent to Koecher’s identity [4, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (3)]  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='3)  ∞  �  n=0  1  n(n2 − x2) = 1  2  ∞  �  k=1  (−1)k−1  �2k  k  �  k3  5k2 − x2  k2 − x2  k−1  �  m=1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1 − x2  m2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  By extracting coefficients of 1 and x2 in Koecher’s identity, we recover Markov’s series ac-  celeration identity [5]  ζ (3) = 5  2  �  n⩾1  (−1)n−1  n3�2n  n  �  and its higher order counterpart  ζ (5) = 2  ∞  �  n=1  (−1)n−1  n5�2n  n  � − 5  2  ∞  �  n=1  (−1)n−1 H(2)  n−1  n3�2n  n  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1  2  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  These are efficiently encoded by the matrix product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' By extracting other coefficients of  xn in Koecher’s identity, we recover counterparts for ζ(2n + 1) which are again encoded by  the matrix product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  This correspondence runs much deeper, and we will show that several hypergeometric-type  series for the zeta function at small integers are equivalent to infinite products for N × N  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' The fact that these identities support an expression in terms of matrix products is  already interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' The pattern of entries of some small matrices suggest the general form  of the relevant n × n generalizations, which would then be equivalent to new accelerated  series for zeta values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Background  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Special Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' The Riemann zeta function, absolutely convergent for s ∈ C, ℜs >  1 is given by  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  ζ(s) :=  ∞  �  n=1  1  ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  This straightforwardly extends to the Hurwitz zeta function with the addition of a parameter  z ∈ C, z ̸= 0, −1, −2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' :  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2)  ζ(s|z) :=  ∞  �  n=1  1  (n + z)s ,  so that ζ(s) = ζ(s|1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The harmonic numbers are given by H0 := 0 and  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='3)  Hn :=  n  �  k=1  1  k,  n ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The hyper-harmonic numbers are defined similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  We will also consider the elementary  symmetric functions  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='4)  e(s)  ℓ (k) := [tℓ]  k−1  �  j=1  �  1 + t  js  �  =  �  1⩽j1<j2<···<jℓ⩽k−1  1  (j1 · · · jℓ)s,  which reduce to the harmonic numbers at e1  1(n) = Hn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' The Gosper Group  Each Gosper matrix in the product (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2) has the form  Mk =  �  Ak  uk  0  1  �  where Ak is square (N × N), uk is a (N × 1) vector and 0 is the (1 × N) vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Matrices of this kind form a group, which we shall name the Gosper group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' With IN the  (N × N) identity matrix, the unit element of the group is  �  IN  0  0  1  �  , and the inverse of  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES  3  an element M =  �  A  u  0  1  �  is M−1 =  �  A−1  −A−1u  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Closure follows from M1M2 =  �  A1A2  A1u2 + u1  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' We can inductively verify that  M1M2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Mn =  �  A1A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' An  �n  k=1 A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ak−1uk  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Toeplitz Matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Moreover, each Ak in Gosper’s identity has the simple form  Ak = αkI + βkJ  where J is the (N × N) matrix with a first superdiagonal of ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Hence JN = 0 and, for p ⩾ N, we have  A1A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ap = (α1I + β1J) (α2I + β2J) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (αpI + βpJ)  =  � p  �  i=1  αi  � \uf8eb  \uf8edI +  p  �  j=1  βj  αj  J + · · · +  �  1⩽j1<···<jN−1⩽p  βj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' βjN−1  αj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αjN−1  JN−1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  For p < N the summation is instead truncated at Jp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The general form of the components of the limiting infinite product case can be deduced  by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' The components of  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  ∞  �  k=1  �  Ak  uk  0  1  �  =  � �∞  k=1 Ak  v∞  0  1  �  ,  with  �  v(N)  ∞ , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' , v(1)  ∞  �T := v∞ =  ∞  �  p=1  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ap−1up,  are  v(1)  ∞ =  ∞  �  p=1  (α1 · · · αp−1) u(1)  p ,  v(2)  ∞ =  ∞  �  p=1  (α1 · · · αp−1)  �  u(2)  p  +  �p−1  �  j=1  βj  αj  �  u(1)  p  �  ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  v(ℓ)  ∞ =  ∞  �  p=1  (α1 · · · αp−1)  \uf8eb  \uf8edu(ℓ)  p +  �p−1  �  j=1  βj  αj  �  u(ℓ−1)  p  + · · · +  \uf8eb  \uf8ed  �  1⩽j1<···<jℓ−1⩽p−1  βj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' βjℓ−1  αj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αjℓ−1  \uf8f6  \uf8f8 u(1)  p  \uf8f6  \uf8f8 ,  with 1 ⩽ ℓ ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Already the connection to zeta series and hyperharmonic numbers is clear: with the correct  choice of α and β, the multiple sums will reduce to multiple zeta type functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  These matrix products also exhibit a stability phenomenon, where increasing the dimen-  sion of the matrix does not impact any entries in v∞ except the top right one, since mapping  N → N + 1 only changes the formula for v(N+1)  ∞  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  4  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  We will consistently refer to the N = 1 and N = 2 cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Explicitly, when N = 1 so that  both Ak (denoted αk to avoid confusion) and uk are scalars, we have  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' For N = 1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2)  n  �  k=1  �  αk  βk  0  1  �  =  � �n  k=1 αk  �n  k=1 α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αk−1βk  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Although we will only need the n → ∞ limit, let us note that this identity holds for finite  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Koecher’ Identity  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1) and Koecher’s identity are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Begin with Koecher’s identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' By extracting coefficients of x2n, in general we  obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  ζ(2n + 3) = 5  2  ∞  �  k=1  (−1)k−1  k3�2k  k  � (−1)ne(2)  n (k) + 2  n  �  j=1  ∞  �  k=1  (−1)k−1  k2j+3�2k  k  �(−1)n−je(2)  n−j(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Take αk = −  k  2(2k+1), βk =  1  2k(2k+1), u(1)  k  =  5  4k2, and u(ℓ)  k  =  1  k2ℓ+2 for 2 ⩽ ℓ ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' This  corresponds to the Gosper matrix  �  Ak  uk  0  1  �  =  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0  −  k  2(2k+1)  1  2k(2k+1)  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  1  k2N  0  −  k  2(2k+1)  1  2k(2k+1)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1  k2N−2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1  2k(2k+1)  1  k4  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  −  k  2(2k+1)  5  4k2  0  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0  1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Then  p  �  i=1  αi = (−1)p  p  �  i=1  i2  (2i)(2i + 1) = (−1)p  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2  (2p + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=',  and (for 2 ⩽ ℓ ⩽ N)  �  j1<···<jℓ−1⩽p−1  βj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' βjℓ−1  αj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αjℓ−1  = (−1)ℓ  �  j1<···<jℓ−1⩽p−1  1  (j1 · · · jℓ−1)2 = (−1)ℓe(2)  ℓ−1(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  We deduce  lim  p→∞ α1 · · · αp = 0,  while  lim  p→∞  �  j1<···<jk⩽p−1  1  (j1 · · · jk)2 ⩽ lim  p→∞  p  �  j1=1  1  j2  1  = ζ(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Hence, applying Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1, we deduce  ∞  �  i=1  Ai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES  5  The components in the right column are then explicitly given as  v(ℓ)  ∞ =  ∞  �  p=1  (α1 · · · αp−1)  \uf8eb  \uf8edu(ℓ)  p +  �p−1  �  j=1  βj  αj  �  u(ℓ−1)  p  + · · · +  \uf8eb  \uf8ed  �  1⩽j1<···<jℓ−1⩽p−1  βj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' βjℓ−1  αj1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αjℓ−1  \uf8f6  \uf8f8 u(1)  p  \uf8f6  \uf8f8  =  ∞  �  p=1  (−1)p−1(p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2  (2p − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  �  1  p2ℓ+2 − e(2)  1 (p)  p2ℓ  + · · · + (−1)ℓ−15  4  e(2)  ℓ−1(p)  p2  �  = 5  2  ∞  �  p=1  (−1)p−1  p3�2p  p  � e(2)  ℓ−1(p) + 2  ℓ−1  �  j=1  ∞  �  p=1  (−1)p−1  p3+2j�2p  p  �e(2)  ℓ−1−j(p)(−1)ℓ−1−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  We see that this is exactly the formula from Koecher’s identity, hence equals ζ(2ℓ + 1) for  1 ⩽ ℓ ⩽ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Leschiner’s identity  Begin with the Leschiner identity  �  n⩾1  (−1)n−1  n2 − z2 = 1  2  �  k⩾1  1  �2k  k  �  k2  3k2 + z2  k2 − z2  k−1  �  j=1  �  1 − z2  j2  �  ,  so that  ˜ζ (2) = 3  2  �  k⩾1  1  �2k  k  �  k2,  and  ¯ζ (4) = 3  2  �  k⩾1  1  �2k  k  �  k2  � 4  k2 − H(2)  k−1  �  ,  and in general (I think I made a mistake here)  ˜ζ(2n + 2) = 3  2  ∞  �  k=1  1  k2�2k  k  �(−1)ne(2)  n (k) + 6  n  �  j=1  ∞  �  k=1  1  k2j+2�2k  k  �(−1)n−je(2)  n−j(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  A Gosper representation for ¯ζ (2) and ¯ζ (4) is  �  n⩾1  \uf8eb  \uf8ed  n  2(2n+1)  −1  2n(2n+1)  1  n3  0  n  2(2n+1)  3  4n  0  0  1  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  0  0  ¯ζ (4)  0  0  ¯ζ (2)  0  0  1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  This will generalize using the same method as Koecher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Borwein’s Identity  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' the infinite product case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Extracting coefficient of z2n from Borwein’s identity [1]  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  �  n⩾1  1  n2 − z2 = 3  �  k⩾1  1  �2k  k  �  1  k2 − z2  k−1  �  j=1  j2 − 4z2  j2 − z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  6  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  gives  �  k⩾1  1  �2k  k  �  1  k2 − z2  k−1  �  j=1  j2 − 4z2  j2 − z2 =  �  k⩾1  1  k2�2k  k  �  k−1  �  j=1  �  1 − 4z2  j2  �  k  �  j=1  1  1 − z2  j2  =  �  k⩾1  1  k2�2k  k  �  �  ℓ⩾0  z2ℓ4ℓe(2)  ℓ (k)  �  m⩾0  z2mh(2)  m (k + 1),  where hm is the complete symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' This gives us a formula for the coefficient  of z2n as a convolution over hm and em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' How do we encode this in the matrix, in terms of  αk, βk, uk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A Gosper representation for ζ (2) is obtained as  �  n⩾1  �  n  2(2n+1)  3  2n  0  1  �  =  �  0  ζ (2)  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Identifying the constant term produces  ζ (2) = 3  �  k⩾1  1  �2k  k  �  k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  With αk =  k  2(2k+1) and βk =  3  2k, we have  �  n⩾1  �n−1  �  k=1  αk  �  βn = 3  2  �  n⩾1  2  n2�2n  n  � = ζ (2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  Identifying the linear term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1) produces  ζ (4) = 3  �  k⩾1  1  �2k  k  �  k2  � 1  k2 − 3H(2)  k−1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  This suggests the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A Gosper representation for ζ (2) and ζ (4) is obtained as  �  n⩾1  \uf8eb  \uf8ed  n  2(2n+1)  −3  2n(2n+1)  3  2n3  0  n  2(2n+1)  3  2n  0  0  1  \uf8f6  \uf8f8 =  \uf8eb  \uf8ed  0  0  ζ (4)  0  0  ζ (2)  0  0  1  \uf8f6  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Denote  Mn =  \uf8eb  \uf8ed  δn  γn  u(1)  n  0  δn  u(2)  n  0  0  1  \uf8f6  \uf8f8 =  �  An  un  0  1  �  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES  7  with An =  �  δn  γn  0  δn  �  = δnI + γnJ and δn =  2  n(2n+1) so that, with I =  �  1  0  0  1  �  , J =  �  0  1  0  0  �  , un =  �  u(1)  n  u(2)  n =  3  2n  �  ,  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ai−1 =  2  i  �2i  i  �  �  I + J  i−1  �  j=1  γj  δj  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  We know that  M1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Mn =  �  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' An  vn  0  1  �  with  vn =  n  �  i=1  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ai−1ui  so that  vn =  n  �  i=1  2  i  �2i  i  �  �  ui +  i−1  �  j=1  γj  δj  �  3  2i  0  ��  =  n  �  i=1  2  i  �2i  i  �  ��  u(1)  i3  2i  �  +  i−1  �  j=1  γj  δj  �  3  2i  0  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  =  \uf8ee  \uf8f0  �n  i=1  2  i(2i  i )u(1)  i  + �i−1  j=1  γj  δj  3  2i  �n  i=1  2  i(2i  i)  3  2i  \uf8f9  \uf8fb  This produces  v(2)  ∞ = ζ (2) =  ∞  �  i=1  3  i2�2i  i  �  and  v(1)  ∞ = ζ (4) =  ∞  �  i=1  2  i  �2i  i  �u(1)  i  +  ∞  �  i=1  2  i  �2i  i  � 3  2i  i−1  �  j=1  γj  δj  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Identifying with  ζ (4) = 3  �  k⩾1  1  �2k  k  �  k2  � 1  k2 − 3H(2)  k−1  �  produces  u(1)  i  = 3  2i3, γj =  −3  2j (2j + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  Unfortunately, the case that includes ζ (6) is not as straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A Gosper representation for ζ (2) , ζ (4) and ζ (6) is obtained as  �  n⩾1  \uf8ee  \uf8ef\uf8ef\uf8ef\uf8f0  n  2(2n+1)  −  3  2n(2n+1)  0  3  2n5 −  9H(4)  n−1  2n  0  n  2(2n+1)  −  3  2n(2n+1)  3  2n3  0  0  n  2(2n+1)  3  2n  0  0  0  1  \uf8f9  \uf8fa\uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8ef\uf8ef\uf8f0  0  0  0  ζ (6)  0  0  0  ζ (4)  0  0  0  ζ (2)  0  0  0  1  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  8  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  For example, the truncated product from n = 1 up to n = 200 is  \uf8ee  \uf8ef\uf8ef\uf8f0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='4222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−122  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−121  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='7517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−121  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='01734  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='4222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−122  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1917.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−121  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='08232  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='4222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10−122  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='64493  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  \uf8f9  \uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Identifying the coefficient of z2 in Borwein’s identity (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1) produces  ζ (6) = 3  �  k⩾1  1  �2k  k  �  k2  �  17H(2,2)  k−1 + H(4)  k−1 − 4  �  H(2)  k−1  �2  − 3H(2)  k−1  k2  + 1  k4  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Moreover, the vector vn is computed as  vn =  n  �  i=1  A1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Ai−1ui =  n  �  i=1  2  �2i  i  �  i  \uf8ee  \uf8f0ui − 3H(2)  i−1  \uf8ee  \uf8f0  ui (2)  ui (3)  0  \uf8f9  \uf8fb + 9H(2,2)  i−1  \uf8ee  \uf8f0  ui (3)  0  0  \uf8f9  \uf8fb  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Hence  v(1)  ∞ =  ∞  �  i=1  2  �2i  i  �  i  �  ui (1) − 3H(2)  i−1ui (2) + 9H(2,2)  i−1 ui (3)  �  =  ∞  �  i=1  2  �2i  i  �  i  �  ui (1) − 3H(2)  i−1  3  2i3 + 9H(2,2)  i−1  3  2i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Using  3  n  �  4H(2,2)  n−1 + 1  2H(4)  n−1 − 2  �  H(2)  n−1  �2�  = − 9  2nH(4)  n−1  and identifying v(1)  ∞ = ζ (6) produces the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A finite matrix product for H(3)  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' From the identity  H(3)  N =  N  �  n=1  (−1)n−1  n3�2n  n  �  �  5  2 −  1  2  �N+n  2n  �  �  ,  we deduce the following finite product representation  N  �  n=1  \uf8ee  \uf8ef\uf8f0 −  n  2 (2n + 1)  5  4n2  �  1 −  1  5  �N+n  2n  �  �  0  1  \uf8f9  \uf8fa\uf8fb =  \uf8ee  \uf8f0  2 (−1)N  (N + 1)  �2N+2  N+1  �  H(3)  N  0  1  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Gosper Representation of Markov’s identity for ζ(2) and ζ(z + 1, 3)  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Markov’s identity for ζ(z + 1, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Markov’s identity reads  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1)  ζ (z + 1, 3) =  ∞  �  n=1  1  (n + z)3 = 1  4  ∞  �  k=1  (−1)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='6  (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  5k2 + 6kz + 2z2  ((z + 1) (z + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (z + k))4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES  9  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A Gosper’s representation for Markov’s identity is  ∞  �  n=1  \uf8ee  \uf8f0 −  n6  2n (2n + 1) (z + n + 1)4  5k2 + 6kz + 2z2  0  1  \uf8f9  \uf8fb =  �  0  4 (z + 1)4 ζ (z + 1, 3)  0  1  �  or equivalently  ∞  �  n=1  \uf8ee  \uf8f0 −  n6  2n (2n + 1) (z + n + 1)4  5k2 + 6kz + 2z2  4 (z + 1)4  0  1  \uf8f9  \uf8fb =  �  0  ζ (z + 1, 3)  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Rewrite Markov’s identity as  4ζ (z + 1, 3) =  �  k⩾1  (−1)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='6  (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  5k2 + 6kz + 2z2  ((z + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (z + k))4,  define  uk = 5k2 + 6kz + 2z2  and notice that writing  4ζ (z + 1, 3) = u1 + α1u2 + α1α2u3 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  requires that the coefficient of u1 should be equal to 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' as it is equal to  1  (z+1)4, consider the  variation  4 (z + 1)4 ζ (z + 1, 3) =  �  k⩾1  (−1)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='6  (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  (z + 1)4  ((z + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (z + k))4uk  which now satisfies this constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Then identifying  α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αk−1 = (−1)k−1 (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='6  (2k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  (z + 1)4  ((z + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' (z + k))4  provides  αk =  −k6  2k (2k + 1) (z + k + 1)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Notice that the constant term (z + 1)4 disappears from αk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  Another identity [6] due to Tauraso is  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2)  �  n⩾1  1  n2 − an − b2 =  �  k⩾1  3k − a  �2k  k  �  k  1  k2 − ak − b2  k−1  �  j=1  j2 − a2 − 4b2  j2 − aj − b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A Gosper’s matrix representation for identity (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2) is  ∞  �  n=1  �  k  2(2k+1)  k2−a2−4b2  k2−ak−b2  3k−a  k2−ak−b2  0  1  �  =  � 0  �  n⩾1  2  n2−an−b2  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Notice that  �  n⩾1  2  n2 − an − b2 =  2  √  a2 + 4b2  �  ψ  �  1 − a  2 +  √  a2 + 4b2  2  �  − ψ  �  1 − a  2 −  √  a2 + 4b2  2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  10  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' WAKHARE1 AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' VIGNAT2  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Choose  uk =  3k − a  k2 − ak − b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The first term in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2) is  3 − a  2  1  1 − a − b2 = 1  2u1  so that we consider twice identity (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='2), and choose  α1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' αk−1 =  1  �2k  k  �  k  k−1  �  j=1  j2 − a2 − 4b2  j2 − aj − b2  so that  αk = k2 − a2 − 4b2  k2 − ak − b2  k  2 (2k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  ■  A quartic version reads  �  n⩾1  n  n4 − a2n2 − b4 = 1  2  �  k⩾1  (−1)k−1  �2k  k  �  k  5k2 − a2  k4 − a2k2 − b4  k−1  �  j=1  (j2 − a2)2 + 4b4  j4 − a2j2 − b4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The same approach as above produces  ∞  �  n=1  �  −  k  2(2k+1)  (k2−a2)  2+4b4  k4−a2k2−b4  5k2−a2  k4−a2k2−b4  0  1  �  =  � 0  �  n=1  4n  n4−a2n2−b4  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Amdeberhan-Zeilberger’s ultra-fast series representation [7]  ζ (3) =  �  n⩾1  (−1)n−1  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='10  64 (2n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='5  �  205n2 − 160n + 32  �  can be realized as  ∞  �  n=1  �  −  �  k  2(2k+1)  �5  205k2 − 160k + 32  0  1  �  =  �  0  64ζ (3)  0  1  �  or equivalently  ∞  �  n=1  �  −  �  k  2(2k+1)  �5  205k2−160k+32  64  0  1  �  =  �  0  ζ (3)  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  The resemblance with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='1) is interesting and suggests the generalization  ∞  �  n=1  \uf8ee  \uf8ef\uf8ef\uf8f0  −  �  k  2(2k+1)  �5  �  1  2k(2k+1)  �5  P (k)  0  −  �  k  2(2k+1)  �5  205k2−160k+32  64  0  0  1  \uf8f9  \uf8fa\uf8fa\uf8fb =  \uf8ee  \uf8f0  0  0  ζ (5)  0  0  ζ (3)  0  0  1  \uf8f9  \uf8fb  where P (k) is to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Another fast representation due to Amdeberhan [8] is  ζ (3) = 1  4  ∞  �  n=1  (−1)n−1 (56n2 − 32n + 5)  n3 (2n − 1)2 �3n  n  ��2n  n  �  INFINITE MATRIX PRODUCTS AND HYPERGEOMETRIC ZETA SERIES  11  and produces  ∞  �  n=1  \uf8ee  \uf8f0 −  k3  (3k + 3) (3k + 2) (3k + 1)  �2k − 1  2k + 1  �2  56k2 − 32k + 5  24  0  1  \uf8f9  \uf8fb =  �  0  ζ (3)  0  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Borwein, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Bradley and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Broadhurst, Evaluations of k-fold Euler/Zagier sums: a com-  pendium of results for arbitrary k, Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=', 4-2, 1-21, 1997  [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Gosper, Analytic identities from path invariant matrix multiplication, unpublished manuscript,  1976  [3] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications 94, Cambridge  University Press, 2003  [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Koecher, Letters, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Intelligencer 2 (1980), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' 2, 62-64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Markoff, M´emoire sur la transformation des s´eries peu convergentes en s´eries tr`es convergentes,  M´em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' de l’Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Imp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' de St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' P´etersbourg, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' XXXVII, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' 9 (1890)  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='  Tauraso,  A  bivariate  generating  function  for  zeta  values  and  related  supercongruences,  arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='00846  [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Amdeberhan and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Zeilberger, Hypergeometric Series Acceleration via the WZ Method, THe Elec-  tronic Journal of Combinatorics, 4-2, The Wilf Festchrift Volume, 1997  [8] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content=' Amdeberhan, Faster and faster convergent series for z(3), Electronic Journal of Combinatorics, Volume  3 Issue 1, 1996  1 Department of Electrical Engineering and Computer Science, Massachusetts Institute  of Technology, Cambridge, Massachusetts, USA  Email address: twakhare@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='edu  2 Department of Mathematics, Tulane University, New Orleans, Louisiana, USA  Email address: cvignat@tulane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NAyT4oBgHgl3EQfdPda/content/2301.00298v1.pdf'}

9dAyT4oBgHgl3EQfQ_am/content/tmp_files/2301.00058v1.pdf.txt

@@ -0,0 +1,1127 @@
+Detecting TCP Packet Reordering in the Data Plane
+YUFEI ZHENG, Princeton University, USA
+HUACHENG YU, Princeton University, USA
+JENNIFER REXFORD, Princeton University, USA
+Network administrators are often interested in detecting TCP-level packet reordering to diagnose performance
+problems and neutralize attacks. However, packet reordering is expensive to measure, because each packet
+must be processed relative to the TCP sequence number of its predecessor in the same flow. Due to the volume
+of traffic, the detection of packet reordering should take place in the data plane of the network devices as
+the packets fly by. However, restrictions on the memory size and the number of memory accesses per packet
+make it impossible to design an efficient algorithm for pinpointing the flows with heavy packet reordering.
+In practice, packet reordering is typically a property of a network path, due to a congested or flaky link.
+Flows traversing the same path are correlated in their out-of-orderness, and aggregating out-of-order statistics
+at the IP prefix level would provide useful diagnostic information. In this paper, we present efficient algorithms
+for identifying IP prefixes with heavy packet reordering under memory restrictions. First, we analyze as much
+of the traffic as possible by going after the largest flows. Next, we sample as many flows as possible, regardless
+of their sizes. To achieve the best of both worlds, we also combine these two methods. In all algorithms, we
+resolve the challenging interplay between measuring at the flow level and aggregating at the prefix level by
+allocating memory using prefix information. Our simulation experiments using packet traces from a campus
+network show that our algorithms are effective at identifying IP prefixes with heavy packet reordering using
+moderate memory resources.
+1
+INTRODUCTION
+Transmission Control Protocol (TCP) performance problems are often associated with packet
+reordering. Packet loss, commonly caused by congested links, triggers TCP senders to retransmit
+packets, leading the retransmitted packets to appear out of order. Also, the network itself can cause
+packet reordering, due to malfunctioning equipment or traffic splitting over multiple links [17].
+TCP overreacts to inadvertent reordering by retransmitting packets that were not actually lost and
+erroneously reducing the sending rate [5, 17]. In addition, reordering of acknowledgment packets
+muddles TCP’s self-clocking property, and induces bursts of traffic [4]. Perhaps more strikingly,
+reordering can be a form of denial-of-service (DoS) attack. In this scenario, an adversary persistently
+reorders existing packets, or injects malicious reordering into the network, to make the goodput
+low or even close to zero, despite delivering all of the packets [1, 7].
+To diagnose performance problems and neutralize attacks, it is therefore crucial to detect packet
+reordering quickly and efficiently, e.g., on the order of minutes. Due to the sheer volume of traffic,
+the detection of packet reordering should take place in the data plane of network devices as the
+packets fly by. This is because each packet must be processed in conjunction with its predecessor
+in the same flow, which renders simple packet sampling insufficient. In many cases, it suffices to
+report reordering at a coarser level, such as to identify the IP prefixes associated with performance
+problems. In this paper, we focus on an edge network. Since routing is determined at the IP prefix
+level, by identifying heavily reordered source prefixes in the incoming traffic, we can locate the
+part of network experiencing trouble. However, this does not obviate the need to maintain state for
+at least some flows, as packet reordering is still a flow-level phenomenon.
+The emergence of programmable data planes makes it possible to keep simple reordering statistics
+directly in the packet-processing pipeline. With flexible parsing, we can extract the header fields
+we need to analyze the packets in a flow, including the TCP flow identifier (source and destination
+IP addresses and port numbers) and the TCP sequence number for each packet. Using register
+1
+arXiv:2301.00058v1  [cs.NI]  30 Dec 2022
+
+Zheng, Yu and Rexford
+arrays, we can keep state across successive packets of the same flow. In addition, simple arithmetic
+operations allow us to detect reordering and count the number of out-of-order packets in a flow.
+However, the limited memory in the programmable data plane usually needs to be shared among
+several monitoring tasks, and keeping per-flow state is taxing on the memory resources. To see that,
+simply storing the flow signatures for a 5-minute traffic trace from a campus network could take
+more than 228 bits of memory, which is already a significant fraction of the total register memory
+in bits available on a Tofino switch, without even accounting for the memory necessary to keep
+the statistics for each flow. Moreover, to keep up with line rate, we can only access memory a
+small constant number of times for each packet, which limits our choice of data structures. As in
+many previous works [3, 19], we turn to the hash-indexed array for our algorithms. Due to the
+number of processing stages present in the hardware, it is also impossible to use more than a small
+constant number of arrays. Furthermore, since the data-plane hardware has limited bandwidth for
+communicating with the control-plane software, we cannot offload monitoring tasks to the control
+plane. As such, we need to design compact data structures that work within these constraints.
+In this paper, we present data structures that detect and report packet-reordering statistics to the
+control plane. As packet reordering is typically a property of a network path, packets traversing the
+same path at the same time are often correlated in their out-of-orderness. This hints that we can
+identify prefixes with heavy packet reordering without needing to sieve through all of the flows
+in that prefix. To work with the limited memory, we approach this problem from two different
+directions:
+• Study as much of the traffic as possible by going after the heavy flows, since it is memory
+efficient to identify heavy hitters, even in the data plane [3, 19, 21]. However, with heavy
+hitters not being the only flows of interest, this method is not robust if the traffic distribution
+contains heavy-reordering prefixes with only non-heavy flows.
+• Sample as many flows as possible, regardless of their sizes. The minor drawback is that the
+amount of communication necessary from the data plane to the control plane, though not to
+the extent of overwhelming hardware resources, could be significantly larger than that of
+the first approach.
+To achieve the best of both worlds, we also propose a combination of these two approaches.
+The interplay between measuring at the flow level and acting at the prefix level lies in the heart
+of this problem. To decide which set of flows to monitor, we need to incorporate prefix identity in
+managing the data structures, which gives rise to the idea of allocating memory on the prefix level.
+In what follows, § 2 formulates the reordering problem and shows hardness of identifying out-
+of-order heavy flows. We elaborate on the two approaches for finding heavily reordered prefixes in
+§ 3, and briefly discuss a combination of the two. In § 4, we verify the correlation among flows from
+the same prefix through measurement results, and demonstrate that our algorithms are extremely
+memory-efficient. We discuss related work in §5 and then conclude our paper in § 6.
+2
+PROBLEM FORMULATION: IDENTIFY HEAVY OUT-OF-ORDER IP PREFIXES
+Consider a switch close to the receiving hosts, where we observe a stream of incoming packets
+(Figure 1). Our goal is to identify the senders whose paths to the receivers are experiencing
+performance problems, through counting out-of-order packets. In § 2.1, we first introduce notations
+and definitions at the flow level, and show that identifying flows with heavy reordering is hard,
+even with randomness and approximation. Later, in § 2.2, we extend the definitions to the prefix
+level, then discuss possible directions to identify heavy out-of-order prefixes.
+2
+
+Detecting TCP Packet Reordering in the Data Plane
+Fig. 1. Different source prefixes send packets over different paths. Packets on a path are colored differently
+to show that traffic from a single prefix has a mix of packets from different flows. While flows from a single
+prefix may split over parallel subpaths, they do share many portions of their network resources.
+2.1
+Flow-level reordering statistics
+2.1.1
+Definitions at the flow level. Consider a stream 𝑆 of TCP packets from different remote senders
+to the local receivers. In practice, TCP packets may contain payloads, and sequence numbers advance
+by the length of payload in bytes. But, to keep the discussions simple, we assume sequence numbers
+advance by 1 at a time, and we ignore sequence number rollovers. We note that these assumptions
+can be easily adjusted to reflect the more realistic scenarios. Then, a packet can be abstracted as a
+3-tuple (𝑓 ,𝑠,𝑡), with 𝑓 ∈ F being its flow ID, 𝑠 ∈ [𝐼] the sequence number and 𝑡 the timestamp. In
+this case, a flow ID is a 4-tuple of source and destination IP addresses, and the source and destination
+TCP port numbers.
+Let 𝑆𝑓 = {(𝑓 ,𝑠𝑖,𝑡𝑖)}
+𝑁𝑓
+𝑖=1 ⊆ 𝑆 be the set of packets corresponding to some flow 𝑓 , sorted by time 𝑡𝑖
+in ascending order. We say the packets of flow 𝑓 are perfectly in-order if 𝑠𝑖+1 = 𝑠𝑖 + 1 for all 𝑖 in
+[𝑁𝑓 − 1]. By commonly used definitions, the 𝑖th packet in flow 𝑓 is out-of-order if it has:
+Definition 1 a lower sequence number than its predecessor in 𝑓 , 𝑠𝑖 < 𝑠𝑖−1.
+Definition 2 a sequence number larger than that expected from its predecessor in 𝑓 , 𝑠𝑖 > 𝑠𝑖−1 + 1.
+Definition 3 a smaller sequence number than the maximum sequence number seen in 𝑓 so far,
+𝑠𝑖 < max𝑗 ∈[���−1] 𝑠𝑗.
+When 𝑠𝑖 < 𝑠𝑖−1 in flow 𝑓 , we sometimes say an out-of-order event occurs at packet 𝑖 with respect
+to Definition 1. Out-of-order events with respect to other definitions are similarly defined. Under
+each definition, denote the number of out-of-order packets in flow 𝑓 as 𝑂𝑓 , a flow 𝑓 is said to be
+out-of-order heavy if 𝑂𝑓 > 𝜀𝑁𝑓 for some small 𝜀 > 0.
+In practice, none of these three definitions is a clear winner. Rather, different applications may
+call for different metrics. From an algorithmic point of view, Definition 1 and Definition 2 are
+essentially identical, in that detecting the out-of-order events only requires comparing adjacent
+pairs of packets. An out-of-order event with respect to Definition 3, however, is far more difficult to
+uncover, as looking at pairs of packets is no longer enough—the algorithm always has to record the
+maximum sequence number (over a potentially large number of packets) in order to report such
+events. In this paper, we focus on Definition 1 and show that easy modifications to the algorithms
+can be effective for Definition 2.
+2.1.2
+A strawman solution for identifying out-of-order heavy flows. A naive algorithm that identifies
+out-or-order heavy flows would memorize, for every flow, the flow ID 𝑓 , the sequence number 𝑠 of
+the latest arriving packet from 𝑓 when using Definition 1, and the number of out-of-order packets
+𝑜. When a new packet of 𝑓 arrives, we go to its flow record, and compare its sequence number 𝑠′
+with 𝑠. If 𝑠′ < 𝑠, the new packet is out-of-order and we increment 𝑜 by 1.
+3
+
+vantage
+point
+区区
+区
+XZheng, Yu and Rexford
+For Definition 2, we simply save the expected sequence number 𝑠 + 1 of the next packet when
+maintaining the flow record, and compare it to that of the new packet, according to Definition 2.
+We see that different definitions only slightly altered the sequence numbers saved in memory, and
+we always decide whether an out-of-order event has happened based on the comparison.
+2.1.3
+Memory lower bound for identifying out-of-order heavy flows. To show that identifying
+out-of-order heavy flows is fundamentally expensive, we want to construct a worst-case packet
+stream, for which detecting heavy reordering requires a lot of memory. For simplicity, we consider
+the case where heavy reordering occurs in only one of the |F | flows, and let this flow be 𝑓 . If 𝑓
+is also heavy in size, it suffices to use a heavy-hitter data structure to identify 𝑓 . Problems arise
+when 𝑓 is not that heavy on any timescale, and yet is not small enough to be completely irrelevant.
+A low-rate, long-lived flow fits such a profile. Unless given a lot of memory, a heavy-hitter data
+structure is incapable of identifying 𝑓 . Moreover, since the packet inter-arrival times for a low-rate
+flow are large, to see more than one packet from 𝑓 , the record of 𝑓 would need to remain in memory
+for a longer duration, relative to other short-lived or high-rate flows.
+Next we formalize this intuition, and show that given some flow 𝑓 , it is infeasible for a streaming
+algorithm to always distinguish whether 𝑂𝑓 is large or not, with memory sublinear in the total
+number of flows |F |, even with randomness and approximation.
+Claim 1. Divide a stream with at most |F | flows into 𝑘 time-blocks 𝐵1, 𝐵2, . . . , 𝐵𝑘. It is guaranteed
+that one of the following two cases holds:
+(1) For any pair of blocks 𝐵𝑖 and 𝐵𝑗 with 𝑖 ≠ 𝑗, there does not exist a flow that appears in both 𝐵𝑖
+and 𝐵𝑗.
+(2) There exists a unique flow 𝑓 that appears in Θ(𝑘) blocks.
+Then distinguishing between the the two cases is hard for low-memory algorithms. Specifically, a
+streaming algorithm needs Ω(min (|F | , |F|
+𝑘 log 1
+𝛿 )) bits of space to identify 𝑓 with probability at least
+1 − 𝛿, if 𝑓 exists.
+Claim 1 follows from reducing the communication problem MostlyDisjoint stated in [10], by
+treating elements of the sets as flow IDs in a packet stream.
+Claim 1 implies the hardness of identifying out-of-order heavy flows, as the unique flow 𝑓 may
+have many packets, but not be heavy enough for a heavy-hitter algorithm to detect it efficiently.
+Deciding whether such a flow exists is already difficult, identifying it among other flows is at least
+as difficult. Consequently, checking whether it has many out-of-order packets is difficult as well.
+The same reduction also implies that detecting duplicated packets requires Ω(|F |) space. In
+fact, Claim 1 corroborates the common perception that measuring performance metrics such as
+round-trip delays, reordering, and retransmission in the data plane is generally challenging, as it is
+hard to match tuples of packets that span a long period of time, with limited memory.
+2.2
+Prefix-level reordering statistics
+2.2.1
+Problem statement. Identifying out-of-order heavy flows is hard; fortunately, we do not
+always need to report individual flows. Since reordering is typically a property of a network path,
+and routing decisions are made at the prefix level, it is natural to focus on heavily reordered prefixes.
+Throughout this paper, we consider 24-bit source IP prefixes, as they achieve a reasonable level of
+granularity. The same methods apply if prefixes of a different length are more suitable in other
+applications.
+By common definitions of the flow ID, the prefix 𝑔 of a packet (𝑓 ,𝑠,𝑡) is encoded in 𝑓 . To simplify
+notations, we think of a prefix 𝑔 as the set of flows with that prefix, and when context is clear, 𝑆
+also refers to the set of all prefixes in the stream. Let 𝑂𝑔 = �
+𝑓 ∈𝑔 𝑂𝑓 be the number of out-of-order
+4
+
+Detecting TCP Packet Reordering in the Data Plane
+packets in prefix 𝑔. A prefix 𝑔 is out-of-order heavy if 𝑂𝑔 > 𝜀𝑁𝑔 for some small 𝜀 > 0, where 𝑁𝑔 is
+the number of packets in prefix 𝑔.
+For localizing attacks and performance problems, it is not always sensible to catch prefixes with
+the highest fraction of out-of-order packets. When a prefix is small, even a single out-of-order
+packet would lead to a large fraction, but it might just be caused by a transient loss. In addition,
+with the control plane being more computationally powerful yet less efficient in packet processing,
+there is an apparent trade-off between processing speed and the amount of communication from
+the data plane to the control plane. As a result, we also want to limit the communication overhead
+incurred.
+Therefore, for some 𝜀, 𝛼, 𝛽, our goals can be described as:
+(1) Report prefixes 𝑔 with 𝑁𝑔 ≥ 𝛽 and 𝑂𝑔 > 𝜖𝑁𝑔.
+(2) Avoid reports of prefixes with at most 𝛼 packets.
+(3) Keep the communication overhead from the data plane to the control plane small.
+2.2.2
+Bypassing memory lower bound. As a consequence of Claim 1, it is evidently infeasible
+to study all flows from a prefix and aggregate all of that information to determine whether to
+report the prefix. So why would reporting at the prefix level circumvent the lower bound? In
+practice, packets are often reordered due to a congested or flaky link that causes lost, reordered,
+or retransmitted packets at the TCP level. Therefore, flows traversing the same path at the same
+time are positively correlated in their out-of-orderness. This effectively means that we only need
+to study a few flows from a prefix to estimate the extent of reordering this prefix suffers. We state
+the correlation assumption that all of our algorithms are based on as follows, and postpone its
+verification to §4.1:
+Assumption 1. Let 𝑓 be a flow chosen uniformly at random from all flow in prefix 𝑔. If 𝑁𝑔 > 𝛼,
+and 𝑔 has at least two flows,
+𝑂𝑔−𝑂𝑓
+𝑁𝑔−𝑁𝑓 and
+𝑂𝑓
+𝑁𝑓 are positively correlated.
+3
+DATA-PLANE DATA STRUCTURES FOR OUT-OF-ORDER MONITORING
+At a high level, a data-plane algorithm generates reports of flows with potentially heavy packet
+reordering on the fly, and a simple program that sits on the control plane parses through the reports
+to return their prefixes. Each report includes the prefix, the number of packets monitored, and
+the number of out-of-order packets of a suspicious flow. At the end of the time interval, we can
+also scan the data-plane data structure to generate reports for highly-reordered flows remaining in
+memory. On seeing reports, a control-plane program simply aggregates counts from reports of the
+same prefix, and outputs a prefix when its count exceeds a threshold.
+One of the challenges of designing the data-plane data structures is working with complex
+real-world traffic. A huge number of small flows or prefixes only represent a small fraction of the
+traffic, while a small number of very large flows or prefixes makes up a large fraction. Moreover,
+the distribution of flows within each prefix is quite varied.
+In the data plane, we keep state at the flow level, and consider prefix information in allocating
+memory. Assuming a positive correlation between the out-of-orderness of a prefix and that of the
+flows from that prefix, we do not have to monitor all flows to gain enough information about a
+prefix, which leads us to two different threads of thought. In § 3.1, we mainly consider heavy flows,
+and monitor them for long periods each time a flow enters the memory, while in § 3.2, we sample
+as many flows as possible, regardless of their sizes, and for a constant number of packets at a time.
+§ 3.3 introduces hybrid scheme that combines these approaches.
+5
+
+Zheng, Yu and Rexford
+Fig. 2. A modification of PRECISION for tracking out-of-order packets.
+3.1
+Track heavy flows over long periods
+3.1.1
+Track reordering using a heavy-hitter data structure. To capture out-of-orderness in heavy
+flows, we want a data structure that is capable of simultaneously tracking heaviness and reordering.
+The SpaceSaving [14] data structure fits naturally for the task, as we can maintain extra state
+for each flow record, while the data structure gradually identifies the flows with heavy volume
+by keeping estimates of their traffic counts. However, when overwriting a flow record to admit
+a new flow, SpaceSaving needs to go over all entries to locate the flow with the smallest traffic
+count, which makes it infeasible for the data plane due to the constraint on the number of memory
+accesses per packet.
+Thus, we opt for PRECISION [3], the data-plane adaptation of SpaceSaving, which checks only a
+small number of 𝑑 entries when overwriting a flow record. We emphasize that the specifics about
+how PRECISION works are not, in fact, important in this context. It is enough to bear in mind that
+with a suitable data-plane friendly heavy-hitter algorithm, tracking reordering is exactly the same
+as in the strawman solution (§ 2.1.2), but applied only to heavy flows.
+Figure 2 shows the modified PRECISION for tracking out-of-order packets using 𝑑 stages. To set
+the stage for later discussions, throughout this paper, we refer to the unit of memory allocated to
+keep one flow record as a bucket. Depending on the algorithm, a bucket might include different
+information about a flow. Here a bucket stores the flow ID 𝑓 , the estimated size of 𝑓 , the sequence
+number of the last arriving packet from 𝑓 , and the number of out-of-order packets of 𝑓 .
+3.1.2
+Allocate memory by prefix. On the one hand, we want to track the heavy flows, on the other
+hand, we do not want some very large prefix to have its many heavy flows dominate the data
+structure. To this end, we assign flows from the same prefix to the same set of buckets, by hashing
+prefixes instead of flow IDs, a technique we use in all our algorithms. In a PRECISION data structure
+with 𝑑 stages, at the end of the stream, at most 𝑑 heaviest flows from each prefix 𝑔 would remain in
+memory. Doing so effectively frees up buckets that used to be taken by a few prefixes with many
+heavy flows, and allows more prefixes to have their heaviest flows measured.
+3.2
+Sample flows over short periods
+Previously, we worked with the limited memory essentially by having each bucket track one heavy
+flow. However, as we shall see in § 4.1.1, some prefixes do not have any large flow, and some of
+these prefixes experience heavy reordering. Unless the reordering is concentrated on large flows,
+the method of tracking reordering only for heavy hitters inevitably suffers from poor performance.
+Now that large flows are not representative enough in terms of out-of-orderness, we have to sieve
+through more flows regardless of their sizes, and with limited memory. Rather than one bucket per
+flow, the main idea is to use one bucket to check multiple flows in turn.
+6
+
+Packet
+910234510
+srclP pref = A
+fID = 10
+t= 100
+SEQ# = 5Detecting TCP Packet Reordering in the Data Plane
+3.2.1
+Flow sampling with buckets. Under the strict memory access constraints, we again opt for a
+hash-indexed array as a natural choice of data structure, where each row in the array corresponds
+to a bucket, and all buckets behave independently. Similar to § 3.1, we use the IP prefix as the
+hash key, so that all flows from the same prefix are assigned to the same bucket, which effectively
+prevents a prefix with a huge number of flows from consuming many buckets. Therefore, we fix a
+bucket 𝔟, and consider the substream of packets hashed to 𝔟. When a packet (𝑓 ,𝑠,𝑡) arrives at 𝔟,
+there are three cases:
+(1) If 𝔟 is empty, we always admit the packet, that is, we save its flow record 𝑓 , sequence number
+𝑠, timestamp 𝑡 in 𝔟, together with the number of packets 𝑛 and the number of out-of-oder
+packets 𝑜, both initilized to 0.
+(2) If flow 𝑓 ’s record is already in 𝔟, we update the record as in the strawman solution (§ 2.1.2),
+and update the timestamp in memory to 𝑡.
+(3) If 𝔟 is occupied by another flow’s record (𝑓 ′,𝑠′,𝑡 ′,𝑛′,𝑜′), we only admit 𝑓 if 𝑓 ′ has been
+monitored in memory for a sufficient period specified by parameters 𝑇 and 𝐶, or the prefix
+of 𝑓 ′ could be potentially heavily reordered with respect to another parameter 𝑅. That is, 𝑓
+overwrites 𝑓 ′ with record (𝑓 ,𝑠,𝑡,𝑛 = 0,𝑠 = 0) only if one of the following holds:
+(a) 𝑓 ′ is stale: 𝑡 − 𝑡 ′ > 𝑇.
+(b) 𝑓 ′ has been hogging 𝔟 for too long: 𝑛′ > 𝐶.
+(c) 𝑓 ′ might belong to a prefix with heavy reordering: 𝑜′ > 𝑅.
+In Case 3c, the algorithm sends a 3-tuple report (𝑔′,𝑛′,𝑜′) to the control plane, where 𝑔′ is the
+prefix of flow 𝑓 ′. On seeing reports from the data plane, a simple control-plane program keeps a
+tally for each reported prefix 𝑔. Let {(𝑔,𝑛𝑖,𝑜𝑖)}𝑟
+𝑖=1 be the set of all reports corresponding to a prefix
+𝑔. The control-plane program outputs 𝑔 if �𝑟
+𝑖=1 𝑛𝑖 ≥ 𝛼, for the same 𝛼 in § 2.2.1. In the following
+sections, we refer to the data-plane component together with the simple control-plane program as
+the flow-sampling algorithm.
+Lazy expiration of flow records in memory. Due to memory access constraints, many data-plane
+algorithms lazily expire records in memory on collisions with other flows, as opposed to actively
+searching for stale records in the data structure. We again adopt the same technique in the algorithm
+above, though here it is more nuanced. We could imagine a variant of the algorithm where a flow
+is monitored for up to 𝐶 + 1 packets at a time. That is, when the (𝐶 + 1)st packet arrives, we check
+whether to report this flow, and evict its record. Compared to this variant, lazy expiration helps in
+preventing a heavy flow being admitted into the data structure consecutively, so that the heavy
+flow can be evicted before a integer multiple of (𝐶 + 1) packets, should another flow appears in the
+meantime.
+Robustness of flow sampling. For the flow-sampling method to be effective, the data structure
+needs to sample as many flows as possible. Therefore, it is not desirable to keep a large flow in
+memory when we have already seen many of its packets, and learned enough information about its
+packet reordering. This means that the packet count threshold 𝐶 should not be too large. Neither
+do we want to keep a flow, regardless of its size, that has long been finished. We can eliminate such
+cases by setting a small inter-arrival timeout 𝑇.
+Now the question is, how small can these parameters be. Real-world traffic can be bursty, meaning
+that sometimes there are packets from the same flow arriving back-to-back. In this case, even if
+we overwrite the existing flow record on every hash collision (𝑇 = 0 and 𝐶 = 1), the algorithm
+still generates meaningful samples. When the memory is not too small compared to the number of
+prefixes, and hash collisions are rare, the algorithm might even have good performance. However,
+setting small 𝑇 > 0 and 𝐶 > 1 makes the algorithm more robust against worst-case streams.
+7
+
+Zheng, Yu and Rexford
+Consider a stream of packets where no adjacent pairs of packets come from the same flow. On
+seeing such a stream, a flow-sampling algorithm that overwrites existing records on every hash
+collision with another flow will no doubt collect negligible samples. In contrast, small 𝑇 > 0 and
+𝐶 > 1 allow a small period of time for a flow in memory to be monitored, and hence gives a better
+chance of capturing packet reordering.
+3.2.2
+Performance guarantee. In this section, we analyze the number of times a flow with a certain
+size is sampled. Consider a prefix 𝑔 when the hash function is fixed. Let 𝔟 be the bucket prefix 𝑔
+is hashed to, and we know all the flows as well as the prefixes that are hashed to 𝔟. With a slight
+abuse of notation, we write 𝑔 ∈ 𝔟 when the bucket with index ℎ(𝑔) is 𝔟. We also write 𝑓 ∈ 𝔟 when
+𝑓 ’s prefix is hashed to 𝔟. To capture the essence of the flow-sampling algorithm without excessive
+details, we make the following assumptions:
+(1) Each packet in 𝑆 is sampled i.i.d. from distribution (𝑝𝑓 )𝑓 ∈F, that is, each packet belongs to
+some flow 𝑓 ∈ F independently with probability 𝑝𝑓 . Consequently, each packet belongs to
+some prefix 𝑔 independently with probability 𝑝𝑔 = �
+𝑓 ∈𝑔 𝑝𝑓 .
+(2) Let 𝑝𝑓 |𝔟 =
+𝑝𝑓
+�
+𝑓 ′∈𝔟 𝑝𝑓 ′ , 𝑝𝑔|𝔟 can be similarly defined. Only a flow 𝑓 with 𝑝𝑓 |𝔟 greater than
+some 𝑝min will get checked, where we think of 𝑝min as a fixed threshold depending on the
+inter-arrival time threshold 𝑇 and distribution (𝑝𝑓 )𝑓 ∈F.
+(3) A flow is checked exactly 𝐶 + 1 packets at a time.
+Note that Assumption (2) is a way to approximate the effect of𝑇, where we assume a low-frequency
+flow would soon be overwritten by some other flow on hash collision. In contrast to Assumption
+(3), the flow sampling algorithm does not immediately evict a flow record with 𝐶 + 1 packets, if
+there is no hash collision. In this way, though 𝑓 is monitored beyond its original 𝐶 + 1 packets,
+once a hash collision occurs, the collided flow would seize 𝑓 ’s bucket. By imposing Assumption (3),
+the heavier flows would likely benefit by getting more checks, while the smaller flows would likely
+suffer. Empirically, the eviction scheme of the flow-sampling algorithm (§ 3.2.1) achieves better
+performance in comparison to Assumption (3).
+Lemma 3.1. Given the total length of stream |𝑆|, distributions (𝑝𝑓 )𝑓 ∈F, with the assumptions above,
+for a fixed hash function ℎ and any 𝜀,𝛿 ∈ (0, 1), a prefix 𝑔 in bucket 𝔟 is checked at least (1 − 𝛿)𝑡1𝑝𝑔|𝔟
+times with probability at least 1 − 𝑒−𝑝min𝑡1𝐶𝐹𝔟 · 𝜀2
+24 − 𝑒−
+𝜀2|𝑆| �
+𝑔∈𝔟 𝑝𝑔
+3
+− 𝑒−
+𝛿2𝑡1𝑝𝑔|𝔟
+2
+, where 𝑡1 =
+� |𝑆 | �
+𝑔∈𝔟 𝑝𝑔
+(1+ 𝜀
+2 )𝐶𝐹𝔟
+�
+and 𝑝𝑔|𝔟 =
+�
+𝑓 ∈𝑔:𝑝𝑓 |𝔟≥𝑝min 𝑝𝑓
+�
+𝑓 ′∈𝔟 𝑝𝑓 ′
+.
+Proof. Let 𝑆𝔟 the substream of 𝑆 that is hashed to 𝔟. Given |𝑆|, the length |𝑆𝔟| of substream 𝑆𝔟 is
+a random variable, E |𝑆𝔟| = |𝑆| �
+𝑔∈𝔟 𝑝𝑔, then by Chernoff bound,
+P[|𝑆𝔟| < (1 − 𝜀) E |𝑆𝔟|] < 𝑒− 𝜀2 E|𝑆𝔟 |
+3
+= 𝑒−
+𝜀2|𝑆| �
+𝑔∈𝔟 𝑝𝑔
+3
+.
+(1)
+Let 𝑡 be a random variable denoting the number of checks in 𝔟. Let random variable 𝑋𝑖,𝑗 be
+the number of packets hashed to 𝔟 after seeing the 𝑗th packet till receiving the (𝑗 + 1)st packet
+from the currently monitored flow, where 𝑖 ∈ [𝑡] and 𝑗 ∈ [𝐶]. 𝑋𝑖,𝑗s are independent geometric
+random variables, and 𝑋𝑖,𝑗 ∼ 𝐺𝑒𝑜(𝑝𝑓𝑖 |𝑏), where 𝑓𝑖 is the flow under scrutiny during the 𝑖th check,
+by Assumption 2, 𝑝𝑓𝑖 |𝑏 ≥ 𝑝min. Next we look at 𝑋 = �𝑡
+𝑖=1
+�𝐶
+𝑗=1 𝑋𝑖,𝑗, the length of the substream in 𝔟
+after 𝑡 checks,
+E𝑋 =
+𝑡∑︁
+𝑖=1
+𝐶
+∑︁
+𝑗=1
+E𝑋𝑖,𝑗 =
+𝑡∑︁
+𝑖=1
+𝐶
+∑︁
+𝑗=1
+∑︁
+𝑓 ∈𝔟:
+𝑝𝑓 |𝑏 ≥𝑝min
+𝑝𝑓 |𝑏 ·
+𝐶
+𝑝𝑓 |𝑏
+= 𝑡𝐶𝐹𝔟,
+(2)
+8
+
+Detecting TCP Packet Reordering in the Data Plane
+where 𝐹𝔟 =
+��{𝑓 ∈ 𝑏 | 𝑝𝑓 |𝑏 ≥ 𝑝min}
+��. By the Chernoff-type tail bound for independent geometric
+random variables (Theorem 2.1 in [8]), for any 𝜀 ∈ (0, 1),
+P[𝑋 > (1 + 𝜀
+2) E𝑋] < 𝑒−𝑝min E𝑋 ( 𝜀
+2 −ln (1+ 𝜀
+2 )) ≤ 𝑒−𝑝min𝑡𝐶𝐹𝔟 · 𝜀2
+24 .
+(3)
+Let 𝑡1 be the largest 𝑡 such that (1 + 𝜀
+2) E𝑋 < E |𝑆𝔟|, we have 𝑡1 =
+� |𝑆 | �
+𝑔∈𝔟 𝑝𝑔
+(1+ 𝜀
+2 )𝐶𝐹𝔟
+�
+. Consider two
+events:
+(i) The number of checks 𝑡 on seeing 𝑆𝔟 is less than 𝑡1.
+Applying 3 on 𝑡1, we have that with probability at most 𝑒−𝑝min𝑡1𝐶𝐹𝔟 · 𝜀2
+24 , after seeing (1−𝜀) E |𝑆𝑏|
+packets, the number of checks is at most 𝑡1. Together with 1, by union bound,
+P[𝑡 < 𝑡1] < 𝑒−𝑝min𝑡1𝐶𝐹𝔟 · 𝜀2
+24 + 𝑒−
+𝜀2|𝑆| �
+𝑔∈𝔟 𝑝𝑔
+3
+.
+(4)
+(ii) Prefix 𝑔 is checked less than (1 − 𝛿)𝑡1𝑝𝑔|𝔟 times. By Chernoff bound, this event holds with
+probability at most 𝑒−
+𝛿2𝑡1𝑝𝑔|𝔟
+2
+.
+The Lemma follows from applying the union bound over these two events.
+□
+Counterintuitively, the proof of Lemma 3.1 suggests hash collisions are in fact harmless in the
+flow-sampling algorithm. To see that, suppose we add another heavy flow to bucket 𝔟, E |𝑆𝔟| would
+increase by some factor 𝑥, which means E𝑋 would increase by the same factor. Since 𝐹𝔟 would
+only increase by 1, if 𝐹𝔟 is large enough, by (2), 𝑡 would also increase by roughly a factor of 𝑥, while
+𝑝𝑓 |𝔟 decreases by roughly a factor of 𝑥. Then 𝑡 · 𝑝𝑓 |𝔟 is about the same with or without the added
+heavy flow. Therefore colliding with heavy flows does not decrease the number of checks of a
+smaller flow, as long as the total number of flows in a bucket is large enough, which is usually the
+case in practice.
+3.2.3
+Decrease the number of false positives. Since the parameters of the flow-sampling algorithm
+are chosen so that many flows are sampled, and some might get sampled multiple times, it is possible
+for the algorithm to capture many out-of-order events, but not every one of them indicates that the
+prefix is out-of-order heavy. After all, there is only a weak correlation between the out-of-orderness
+of flows and that of their prefixes, not to mention that even if the correlation is stronger, we are
+inferring the extent of reordering on a scale much larger than the snippets of flows that we observe.
+In such cases, the algorithm could output many false positives.
+To reduce the number of false positives, we could imagine feeding the control plane more
+information, so that the algorithm can make a more informed decision about whether the fraction
+of out-of-order packets exceeds 𝜀, for each reported prefix. To this end, we modify the flow-sampling
+algorithm to always report before eviction, even if the number of out-of-order packets is below
+threshold 𝑅. Again denote {(𝑔,𝑛𝑖,𝑜𝑖)}𝑟
+𝑖=1 as the set of all reports corresponding to a prefix 𝑔, the
+control plane outputs 𝑔 if �𝑟
+𝑖=1 𝑛𝑖 ≥ 𝛼, and
+�𝑟
+𝑖=1 𝑜𝑖
+�𝑟
+𝑖=1 𝑛𝑖 > 𝑐 · 𝜀, for some tunable parameter 0 < 𝑐 ≤ 1.
+The parameter 𝑐 compensates for the fact that we only monitor a subset of the traffic, so the exact
+fraction of out-of-order packets we observe might not directly align with 𝜀.
+3.3
+Separate large flows
+When heavy reordering is concentrated in heavy flows, PRECISION (§ 3.1) performs well. The flow-
+sampling algorithm (§ 3.2.1) generally has good performance, regardless of where the reordering
+occurs. However, compared to PRECISION, the flow-sampling algorithm generates more false
+positives, and sends more reports to the control plane. We can reduce the number of false positives
+(§ 3.2.3), but doing so leads to sending even more reports.
+9
+
+Zheng, Yu and Rexford
+To combine the best of both approaches, we introduce a hybrid scheme, where the packets first go
+through a heavy-hitter data structure, and the array only admits flows whose prefixes are not being
+monitored in the heavy-hitter data structure. On the one hand, the array component makes the
+hybrid scheme robust when reordering is no longer concentrated in heavy flows. On the other, by
+keeping some heavy flows in the heavy-hitter data structure, the hybrid scheme avoids repeatedly
+admitting large flows into the array hence potentially reducing the number of reports sent to the
+control plane. Moreover, through monitoring heavy flows in a more continuous manner, the hybrid
+scheme extracts more accurate reordering statistics for heavy flows, which reduces the number of
+false positives at the prefix level.
+In any practical setting, the correct memory allocation between the heavy-hitter data structure
+and the array in the hybrid scheme depends on the workload properties: the relationship of flows
+to prefixes, the heaviness of flows and prefixes, and where the reordering actually occurs. Next we
+understand how these algorithms behave under real-world workloads.
+4
+EVALUATION
+This section presents both measurement results and performance evaluations. First, we show
+several traffic traits that drive our algorithm design (§ 4.1). Next, we evaluate the flow-sampling
+algorithm in § 4.2, and the hybrid scheme in § 4.3, through running a Python simulator on a
+5-minute real-world traffic trace. We recognize that the optimal parameters for our algorithms are
+often workload dependent. Therefore, we do not attempt to always find the optimum, but instead,
+we show in § 4.2 and § 4.3 that arbitrarily chosen parameters already give good performance. Finally
+in § 4.4, we see that these parameters we used previously for evaluations are indeed representative,
+and the algorithms are robust against small perturbations.
+4.1
+Traffic workload characterization
+For all of our measurement and evaluation, we use a 5-minute anonymized packet trace, collected
+ethically from a border router on a university campus network. This study has been conducted
+with necessary approvals from our university, including its Institutional Review Board (IRB). Note
+that only packets with payloads are relevant for our application, as TCP sequence numbers must
+advance for our algorithms to detect reordering events. We therefore preprocess the trace to only
+contain flows from external servers to campus hosts, with the rationale that these senders are
+more likely to generate continuous streams of traffic. After preprocessing, the trace consists of
+82, 359, 630 packets, which come from 546, 126 flows and 17, 097 24-bit source IP prefixes.
+4.1.1
+Heavy-tailed traffic volume and out-of-orderness. It has long been observed that in real-world
+traffic, a small fraction of flows and prefixes account for a large fraction of the traffic (Figure 3a).
+Out-of-orderness in prefixes is similarly heavy-tailed; only a small fraction of prefixes are out-
+of-order heavy (Figure 3b). If heavy reordering happens to occur in heavy flows and prefixes,
+detecting heavy reordering would be easy, by solely focusing on large flows and prefixes using
+heavy-hitter data structures. However, what happens in reality is quite the opposite. Figure 4 shows
+the wide variation of flow sizes in prefixes with heavy reordering, and the sizes of such prefixes
+can be orders-of-magnitude different. Thus, by zooming in on large flows and prefixes, we would
+inevitably miss out on many prefixes of interest without any large flow.
+Fortunately, to report a prefix with a significant amount of reordering, we need not measure
+every flow in that prefix, as flows in the same prefix have some correlation in their out-of-orderness.
+As it turns out, the fraction of out-of-order packets in a prefix is positively correlated with that of a
+flow within the prefix, which we verify next.
+10
+
+Detecting TCP Packet Reordering in the Data Plane
+24
+210
+216
+222
+Number of packets
+0.00
+0.25
+0.50
+0.75
+1.00
+Cumulative probability
+Flow sizes
+Prefix sizes
+(a) Flow and prefix size distribu-
+tions are heavy-tailed. A small frac-
+tion of flows and prefixes account
+for a large fraction of the traffic.
+2
+1
+2
+5
+2
+9
+2
+13
+2
+17
+Fraction of OoO packets in a prefix
+0.00
+0.25
+0.50
+0.75
+1.00
+Cumulative probability
+Def 1
+Def 2
+(b) Out-of-order heavy prefixes are
+rare. Here we consider prefixes
+with at least 𝛽 = 27 packets, and
+𝜀1 = 0.01, 𝜀2 = 0.02 (§ 2.2.1) for
+Definitions 1 and 2 respectively.
+2
+15
+2
+8
+2
+1
+26
+Inter-arrival times (s)
+0.00
+0.25
+0.50
+0.75
+1.00
+Cumulative probability
+in order
+Def 2
+Def 1
+(c) In-order packets tend to have
+smaller inter-arrival times, while
+out-of-order events defined by Def-
+inition 1 exhibit the highest inter-
+arrival times.
+Fig. 3. Heavy-tailed distributions in real-world workload.
+5
+12
+16
+19
+38
+55
+113
+131
+206
+249
+376
+447
+763 1047 1999 2221 2420 2689 4022 5110
+20
+24
+28
+212
+216
+220
+Number of packets in a flow
+222.5
+219
+216
+213
+210
+27
+Number of packets in a prefix
+Rank in the number of packets in a 24-bit IP prefix
+Fig. 4. A violin of rank 𝑟 shows the flow-size distribution of the 𝑟-th largest prefix in the trace, and each
+violin corresponds to a heavily reordered prefix with at least 𝛽 = 27 packets, using Definition 2 with 𝜀 = 0.02.
+All violins are scaled to the same width, and where colors indicate the prefix size. When a prefix consists of
+flow(s) with only one size, its violin degenerates into a horizontal segment. We see that many prefixes do not
+have any large flow. And the many prefixes beyond rank 22644 (not shown in plot) consist only of small flows.
+4.1.2
+Correlation among flows with the same prefix. Let 𝑓 be a flow drawn uniformly at random
+from a set of flows. Let 𝑋 be the random variable representing the fraction of out-of-order packets in
+flow 𝑓 , 𝑋 =
+𝑂𝑓
+𝑁𝑓 . Denote 𝑔 as the prefix of flow 𝑓 , let 𝑌 be the random variable denoting the fraction
+of out-of-order packets among all flows in prefix 𝑔 excluding 𝑓 , that is, 𝑌 =
+𝑂𝑔−𝑂𝑓
+𝑁𝑔−𝑁𝑓 , where 𝑁𝑔 is the
+number of packets in prefix 𝑔. To ensure that 𝑁𝑔 > 𝑁𝑓 , the prefixes we sample from must have
+at least two flows. Since we are less interested in small prefixes, we focus only on prefixes of size
+greater than 𝛼. We use the Pearson correlation coefficient (PCC) to show that 𝑋 and 𝑌 are positively
+correlated, which implies that the out-of-orderness of a flow 𝑓 is statistically representative of
+other flows in the prefix of 𝑓 . Essentially a normalized version of Cov(𝑋,𝑌), PCC always lies in the
+interval [−1, 1], and a positive PCC indicates a positive linear correlation. Lacking a better reason
+to believe the correlation between 𝑋 and 𝑌 is of higher order, we shall see that PCC suffices for our
+analysis.
+Let 𝑆 be a set of flows whose prefixes are of size greater than 𝛼, and have at least two flows. We
+compute the PCC as follows:
+11
+
+Zheng, Yu and Rexford
+(1) Draw 𝑛 flows from 𝑆, independently and uniformly at random.
+(2) For each of the 𝑛 flows 𝑓𝑖, let 𝑥𝑖 =
+𝑂𝑓𝑖
+𝑁𝑓𝑖 , 𝑦𝑖 =
+�
+𝑓 ′∈𝑔,𝑓 ′≠𝑓𝑖 𝑂𝑓 ′
+�
+𝑓 ′∈𝑔,𝑓 ′≠𝑓𝑖 𝑁𝑓 ′ ,
+(3) The PCC 𝑟 =
+�𝑛
+𝑖=1(𝑥𝑖−¯𝑥) (𝑦𝑖− ¯𝑦)
+√�𝑛
+𝑖=1(𝑥𝑖−¯𝑥)2√�𝑛
+𝑖=1(𝑦𝑖− ¯𝑦)2 , where ¯𝑥 = 1
+𝑛
+�𝑛
+𝑖=1 𝑥𝑖, ¯𝑦 = 1
+𝑛
+�𝑛
+𝑖=1 𝑦𝑖.
+We perform 𝑚 = 100 tests on 𝑆, generating 𝑚 PCCs {𝑟 𝑗}𝑚
+𝑗=1 with 𝑛 = 5000 and 𝛼 = 24. Using
+Definition 1, we obtain the average PCC 𝜇1({𝑟 𝑗}𝑚
+𝑗=1) = 0.2348, and the variance 𝜎1({𝑟 𝑗}𝑚
+𝑗=1) = 0.0524.
+Definition 2 of reordering yields even larger PCCs, with 𝜇2({𝑟 𝑗}𝑚
+𝑗=1) = 0.4028 and 𝜎2({𝑟 𝑗}𝑚
+𝑗=1) =
+0.0189.
+We conclude that there is a positive correlation between 𝑋 and 𝑌. Moreover, for reasons not yet
+clear, the correlation is stronger when using Definition 2. Since we heavily rely on the correlation
+assumption in all of our algorithms, this suggests that the performance of the algorithms might be
+worse under Definition 1.
+4.1.3
+Inter-arrival time of packets within a flow. We also study the inter-arrival time of packets
+within a flow to understand how efficient the flow sampling algorithm can be. Due to TCP window-
+ing dynamics, where the sender transmits a window of data and then waits for acknowledgments,
+in-order packets tend to have small inter-arrival times. Depending on the definition, reordering
+can be a result of gaps in transmission of non-consecutive packets (Definition 2), or worse yet the
+retransmissions of lost packets (Definition 1), which often lead to larger inter-arrival times.
+Indeed, Figure 3c shows that the inter-arrival times of out-of-order packets using Definition 2
+tend to be smaller than that of the out-of-order packets using Definition 1, with the inter-arrival
+times of in-order packets being the smallest. This implies, that to detect the reordering events in
+Definition 2, the flow-sampling algorithm (§ 3.2) can afford to use a shorter waiting period 𝑇.
+4.2
+Evaluate flow sampling
+Following § 4.1.2 and § 4.1.3, capturing out-of-order heavy prefixes under Definition 1 appears to be
+more difficult. Next, we show that even when using this more difficult definition of reordering, the
+flow-sampling algorithm is extremely memory efficient. We start by introducing the three metrics
+we use throughout this section to evaluate our algorithms.
+Let ˆ𝐺 denote the set of prefixes output by an algorithm A.
+• Let 𝐺 ≥𝛽 = {𝑔∗ ∈ 𝑆 | 𝑁𝑔∗ ≥ 𝛽,𝑂𝑔∗ > 𝜀 �
+𝑔∈𝑆 𝑂𝑔} be the ground truth set of heavily reordered
+prefixes with at least 𝛽 packets. Define the accuracy 𝐴 of algorithm A to be the fraction of
+ground-truth prefixes output by A, that is,
+𝐴(A) =
+�� ˆ𝐺 ∩ 𝐺 ≥𝛽
+��
+��𝐺 ≥𝛽
+��
+.
+• Let 𝐺>𝛼 = {𝑔∗ ∈ 𝑆 | 𝑁𝑔∗ > 𝛼,𝑂𝑔∗ > 𝜀 �
+𝑔∈𝑆 𝑂𝑔}, then the false-positive rate of A is defined
+as
+𝐹𝑃(A) =
+�� ˆ𝐺 \ 𝐺 ≥𝛼
+��
+|𝐺 ≥𝛼 |
+.
+• The communication overhead from the data plane to the control plane is defined as the
+number of reports sent by A, divided by the length of stream 𝑆, where the number of reports
+also accounts for the flow records in the data structure that exceed the reporting thresholds.
+Unless otherwise specified, each experiment is repeated five times with different seeds to the
+hash functions. Whenever using Definition 1, we are interested in identifying prefixes with at least
+𝛽 = 27 packets, with more than 𝜀 = 0.01 fraction of their packets reordered. And we do not wish to
+output prefixes with at most 𝛼 = 24 packets, irrespective of their out-of-orderness.
+12
+
+Detecting TCP Packet Reordering in the Data Plane
+25
+28
+211
+214
+217 219
+(1)
+0.4
+0.6
+0.8
+1.0
+Accuracy
+25
+28
+211
+214
+217 219
+(2)
+0.00
+0.25
+0.50
+0.75
+False positive
+25
+28
+211
+214
+217 219
+(3)
+0.00
+0.05
+0.10
+0.15
+Communication
+Number of buckets B
+Report all c=0.5
+Report reorder
+Report all c=1
+Fig. 5. Performance of the array sampling algorithm and its variant, with 𝑇 = 2−15,𝐶 = 24, and 𝑅 = 1.
+4.2.1
+Performance evaluation. Figure 5 evaluates the performance of the “Report reorder” version
+of the flow-sampling algorithm (§ 3.2.1), and the “Report all” version (§ 3.2.3) using two different
+values of parameter 𝑐. Recall that in the “Report all” version, we output a prefix if more than 𝑐𝜀
+fraction of its packets observed is out-of-order.
+Note that our trace (§ 4.1) contains more than 219 flows and more than 214 prefixes, and using
+only 25 buckets, the original version of the flow-sampling algorithm is already capable of reporting
+half of the out-of-order prefixes. To put it into perspective, reordering happens at the flow level,
+and assigning even one bucket per prefix to detect reordering already requires a nontrivial solution,
+while the flow-sampling algorithm achieves good accuracy using memory orders-of-magnitude
+smaller.
+If we are willing to generate reports for more than 10% of the traffic, with a increased communi-
+cation overhead comes a reduced false-positive rate. Moreover, with a better chosen parameter 𝑐,
+the extra information sent to the control plane helps in further improving the accuracy.
+4.3
+Evaluate the hybrid scheme
+To fairly compare the hybrid scheme with the flow-sampling algorithm, we need to determine the
+optimal memory allocation between Precision and the array. Lacking a better way to optimize the
+memory allocation, we turn to experiments with our packet trace. Given a total of 𝐵 buckets, we
+assign ⌊𝑥 · 𝐵⌋ buckets to PRECISION, 𝐵 − ⌊𝑥𝐵⌋ buckets to the array, and conduct a grid search on
+𝑥 ∈ 𝐼 = {0.1, . . . , 0.9} to find the value of 𝑥 that maximizes the performance of the hybrid scheme.
+We evaluate the hybrid scheme using the optimal 𝑥 we found for each 𝐵, for both Definition
+1 (Figure 6a) and Definition 2 (Figure 6b). Admittedly, the grid 𝐼 might not be fine enough to
+reveal the true optimal allocation, it nonetheless conveys the main idea. When the memory is
+small compared to the number of prefixes (214), the performance of the flow-sampling algorithm
+significantly dominates that of the heavy-hitter data structure. The optimal hybrid scheme then
+only allocates a small fraction of the memory to the heavy-hitter data structure. However, compared
+to the flow-sampling algorithm, filtering even a small number of large flows helps in significantly
+reducing the communication overhead, while not deteriorating the accuracy. As we approach the
+memory range where there is roughly one bucket per prefix, the heavy-hitter data structure starts
+to perform well, and more memory is devoted to it in the optimal hybrid scheme. In this case, the
+hybrid scheme also reduces the false-positive rate, in comparison to the flow-sampling algorithm.
+13
+
+Zheng, Yu and Rexford
+210
+212
+214
+216
+218
+(a.1)
+0.0
+0.2
+0.4
+0.6
+0.8
+Accuracy
+210
+212
+214
+216
+218
+(a.2)
+0.0
+0.2
+0.4
+0.6
+0.8
+False positive
+212
+215
+218
+(a.3)
+0.00%
+0.02%
+0.04%
+0.06%
+0.08%
+Communication
+Hybrid
+Array
+PRECISION
+(a) Performance using Definition 1, with 𝑅 = 0.01 for PRECISION.
+210
+212
+214
+216
+218
+(b.1)
+0.00
+0.25
+0.50
+0.75
+1.00
+Accuracy
+210
+212
+214
+216
+218
+(b.2)
+0.0
+0.1
+0.2
+0.3
+False positive
+210
+212
+214
+216
+218
+(b.3)
+0.0%
+0.1%
+0.2%
+0.3%
+0.4%
+Communication
+Number of buckets B
+(b) Performance using Definition 2, with 𝑅 = 0.02 for PRECISION.
+Fig. 6. Performance of all proposed algorithms using two definitions of reordering, with 𝑑 = 2 for PRECISION.
+4.4
+Parameter robustness
+We started the evaluation using arbitrarily picked parameters. Now we verify that all parameters
+in our algorithms are either easily set, or robust to changes.
+4.4.1
+Thresholds 𝑇,𝐶, 𝑅 for flow sampling. To reveal how thresholds 𝑇 and 𝐶 individually affect
+the accuracy of the flow-sampling algorithm, ideally we want to fix one of them to infinity, and
+vary the other. In this way, only one of them governs the frequency of evictions. Applying this
+logic, when studying the effect of 𝑇 (Figure 7a), we fix 𝐶 to a number larger than the length of the
+entire trace. We see that as long as 𝑇 is small, the algorithm samples enough flows, and has high
+accuracy.
+Evaluating the effects on a varying 𝐶 turns out to be less straight-forward. If we make 𝑇 too
+large, the algorithm generally suffers from extremely poor performance, which makes it impossible
+to observe any difference that changing 𝐶 might bring. If 𝑇 is too small, the frequency of eviction
+would be primarily driven by𝑇, and 𝐶 would not have any impact. And it is not as simple as setting
+𝑇 larger than all inter-arrival times, since eviction only occurs on hash collisions, inter-arrival
+time alone only paints part of the picture. All evidence above points to the fact that 𝑇 is the more
+important parameter. Once we have a good choice of 𝑇, the boost from optimizing 𝐶 is secondary.
+Armed with this knowledge, we fix a 𝑇 = 25, an ad hoc choice that is by no means perfect. Yet it is
+enough to observe (Figure 7b) that having a small 𝐶 is slightly more beneficial.
+However, 𝐶 cannot be too small, as inserting a new flow record into the array requires recircula-
+tion in the hardware implementation. Programmable switches generally support recirculating up to
+3% − 10% of packets without penalty. Here we set 𝐶 to be 16, which allows us to achieve line rate.
+Given that each non-small flow is continuously monitored for roughly 𝐶 = 16 packets at a time,
+we report its prefix to the control plane when we encounter any out-of-order packet, that is, 𝑅 = 1.
+14
+
+Detecting TCP Packet Reordering in the Data Plane
+20
+2
+3
+2
+6
+2
+9
+2
+12
+2
+15
+2
+18
+Inter-arraival timeout T (s)
+0.0
+0.2
+0.4
+0.6
+Accuracy
+(a) The accuracy of the flow-
+sampling algorithm with varying
+𝑇, and fixed 𝐵 = 28, 𝑅 = 1 and
+𝐶 = 108.
+20
+22
+24
+26
+28
+210 212
+Packet count threshold C
+0.00
+0.01
+0.02
+0.03
+Accuracy
+(b) The accuracy of the flow-
+sampling algorithm with varying
+𝐶, with fixed 𝐵 = 28, 𝑅 = 1 and
+𝑇 = 25.
+212
+215
+218
+Number of buckets B
+0.00
+0.25
+0.50
+0.75
+1.00
+Accuracy
+d = 2
+d = 3
+d = 4
+d = 5
+(c) The accuracy of the flow-
+sampling algorithm with varying
+𝑑, with fixed 𝑅 = 0.01.
+Fig. 7. The effect of changing parameters on the accuracy of the flow-sampling algorithm and PRECISION.
+4.4.2
+The number of stages 𝑑 in PRECISION. It is observed in [3] that a small constant 𝑑 > 1
+only incurs minimal accuracy loss in finding heavy flows. Increasing 𝑑 leads to diminishing gains
+in performance, and adds the number of pipeline stages when implemented on the hardware.
+Therefore, 𝑑 = 2 is preferable for striking a balance between accuracy and hardware resources.
+Building on [3], we evaluate PRECISION for𝑑 = 2, 3, 4, 5, for reporting out-of-order heavy prefixes.
+The results in Figure 7c show that when the total memory is small, using fewer stages provides a
+slight benefit. The opposite holds when there is ample memory. However, as the performance gap
+using different 𝑑 is insignificant, we also suggest using 𝑑 = 2 for hardware implementations.
+5
+RELATED WORK
+Characterization of out-of-orderness on the Internet. Packet reordering is first studied in the
+seminal work by Paxson [17]. It has since been well understood that packet reordering can be
+caused by parallel links, routing changes, and the presence of adversaries [4]. In typical network
+conditions, only a small fraction of packets are out-of-order [17, 20]. However, when the network
+reorders packets, TCP endpoints may wrongly infer that the network is congested, harming end-
+to-end performance by retransmitting packets and reducing the sending rate [4, 11, 12]. Metrics
+for characterizing reordering are intensively studied in [15] and [9], though many of the proposed
+metrics are more suitable for offline analysis. In addition to the network causing packet reordering,
+the stream of packets in the same TCP connection can appear out of order because congestion along
+the path leads to packet losses and subsequent retransmissions. Our techniques for identifying IP
+prefixes with heavy reordering of TCP packets are useful for pinpointing network paths suffering
+from both kinds of reordering—whether caused by the network devices themselves or induced by
+the TCP senders in response to network congestion.
+Data-plane efficient data structures for volume-based metrics. For heavy-hitter queries,
+HashPipe [19] adapts SpaceSaving [14] to work with the data-plane constraints, using a multi-
+stage hash-indexes array. PRECISION [3] further incorporates the idea of Randomized Admission
+Policy [2] to better deal with the massive number of small flows generally found in network traffic.
+We extend PRECISION to keep reordering statistics for large flows. However, such an extension
+cannot be used to detect flows with a large number of out-of-order packets with a reasonable
+amount of memory.
+Data-plane efficient data structures for performance metrics. Liu et al. [13] proposes
+memory-efficient algorithms for identifying flows with high latency, or lost, reordered, and retrans-
+mitted packets. Several solutions for measuring round-trip delay in the data-plane [6, 18, 22] have
+15
+
+Zheng, Yu and Rexford
+a similar flavor to identifying out-of-order heavy prefixes, as in both cases keeping at least some
+state is necessary, with the difference that for reordering we generally need to match more than a
+pair of packets.
+Detect heavy reordering in the data plane. Several existing systems can detect TCP packet
+reordering in the data plane. Marple is a general-purpose network telemetry platform with a
+database-like query language [16]. While Marple can analyze out-of-order packets, the compiler
+generates a data-plane implementation that requires per-flow state. Unfortunately, such methods
+consume more memory than the programmable switch can offer in practice. The algorithm proposed
+by Liu et al. [13] for detecting flows with a large number of out-of-order packets remains the work
+most related to ours. We note that our lower bound on memory consumption in § 2.1.3 is stronger
+than a similar lower bound (Lemma 10) in [13], as we also allow randomness and approximation. Liu
+et al. [13] considers out-of-order events specified by Definition 3, and works around the lower bound
+by assuming out-of-order packets always arrive within some fixed period of time. In contrast, we
+circumvent the lower bound using the more natural observation that out-of-orderness is correlated
+among flows within a prefix, and identify heavily reordered prefixes instead of flows.
+6
+CONCLUSION
+In this paper, we introduce three algorithms for identifying out-of-order prefixes in the data plane.
+In particular, the flow-sampling algorithm achieves good accuracy empirically, even with a memory
+orders-of-magnitude smaller than the number of prefixes, let alone the number of flows. When given
+memory comparable to the number of prefixes, the hybrid scheme using both a heavy-hitter data
+structure and flow sampling gives similar accuracy, while significantly reducing the false-positive
+rate and the communication overhead.
+Next, we plan to build prototypes of the flow-sampling array and the hybrid scheme for the
+Intel Tofino high-speed programmable switch. Moreover, notice that measuring reordering is
+fundamentally memory-expensive, yet we leverage the correlation of out-of-orderness among flows
+in the same prefix so that compact data structures can be effective. In fact, there is nothing special
+about out-of-orderness. Other properties of a network path could very well lead to an analogous
+correlation. For many performance metrics that suffer from similar memory lower bounds, it would
+be intriguing to look into whether such correlation helps in squeezing good performance out of
+limited memory. We leave that for future work.
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+17
+

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+arXiv:2301.03099v1  [cs.AI]  8 Jan 2023
+Fully Dynamic Online Selection through Online Contention Resolution Schemes
+Vashist Avadhanula*, Andrea Celli1, Riccardo Colini-Baldeschi2,
+Stefano Leonardi3, Matteo Russo3
+1Department of Computing Sciences, Bocconi University, Milan, Italy
+2 Core Data Science, Meta, London, UK
+3Department of Computer, Control and Management Engineering, Sapienza University, Rome, Italy
+[email protected], [email protected], [email protected], [email protected], [email protected]
+Abstract
+We study fully dynamic online selection problems in an ad-
+versarial/stochastic setting that includes Bayesian online se-
+lection, prophet inequalities, posted price mechanisms, and
+stochastic probing problems subject to combinatorial con-
+straints. In the classical “incremental” version of the problem,
+selected elements remain active until the end of the input se-
+quence. On the other hand, in the fully dynamic version of the
+problem, elements stay active for a limited time interval, and
+then leave. This models, for example, the online matching of
+tasks to workers with task/worker-dependent working times,
+and sequential posted pricing of perishable goods. A success-
+ful approach to online selection problems in the adversarial
+setting is given by the notion of Online Contention Resolution
+Scheme (OCRS), that uses a priori information to formulate
+a linear relaxation of the underlying optimization problem,
+whose optimal fractional solution is rounded online for any
+adversarial order of the input sequence. Our main contribu-
+tion is providing a general method for constructing an OCRS
+for fully dynamic online selection problems. Then, we show
+how to employ such OCRS to construct no-regret algorithms
+in a partial information model with semi-bandit feedback and
+adversarial inputs.
+1
+Introduction
+Consider the case where a financial service provider receives
+multiple operations every hour/day. These operations might
+be malicious. The provider needs to assign them to human
+reviewers for inspection. The time required by each reviewer
+to file a reviewing task and the reward (weight) that is ob-
+tained with the review follow some distributions. The distri-
+butions can be estimated from historical data, as they depend
+on the type of transaction that needs to be examined and on
+the expertise of the employed reviewers. To efficiently solve
+the problem, the platform needs to compute a matching be-
+tween tasks and reviewers based on the a priori information
+that is available. However, the time needed for a specific re-
+view, and the realized reward (weight), is often known only
+after the task/reviewer matching is decided.
+A multitude of variations to this setting are possible. For
+instance, if a cost is associated with each reviewing task, the
+total cost for the reviewing process might be bounded by a
+budget. Moreover, there might be various kinds of restric-
+tions on the subset of reviewers that are assigned at each
+*Research performed while the author was working at Meta.
+time step. Finally, the objective function might not only be
+the sum of the rewards (weights) we observe, if, for example,
+the decision maker has a utility function with “diminishing
+return” property.
+To model the general class of sequential decision prob-
+lems described above, we introduce fully dynamic online se-
+lection problems. This model generalizes online selection
+problems (Chekuri, Vondr´ak, and Zenklusen 2011), where
+elements arrive online in an adversarial order and algorithms
+can use a priori information to maximize the weight of the
+selected subset of elements, subject to combinatorial con-
+straints (such as matroid, matching, or knapsack).
+In the classical version of the problem (Chekuri, Vondr´ak,
+and Zenklusen 2011), once an element is selected, it will af-
+fect the combinatorial constraints throughout the entire input
+sequence. This is in sharp contrast with the fully dynamic
+version, where an element will affect the combinatorial con-
+straint only for a limited time interval, which we name ac-
+tivity time of the element. For example, a new task can be
+matched to a reviewer as soon as she is done with previ-
+ously assigned tasks, or an agent can buy a new good as
+soon as the previously bought goods are perished. A large
+class of Bayesian online selection (Kleinberg and Weinberg
+2012), prophet inequality (Hajiaghayi, Kleinberg, and Sand-
+holm 2007), posted price mechanism (Chawla et al. 2010),
+and stochastic probing (Gupta and Nagarajan 2013) prob-
+lems that have been studied in the classical version of on-
+line selection can therefore be extended to the fully dynamic
+setting. Note that in the dynamic algorithms literature, fully
+dynamic algorithms are algorithms that deal with both adver-
+sarial insertions and deletions (Demetrescu et al. 2010). We
+could also interpret our model in a similar sense since ele-
+ments arrive online (are inserted) according to an adversarial
+order, and cease to exist (are deleted) according to adversar-
+ially established activity times.
+A successful approach to online selection problems is
+based on Online Contention Resolution Schemes (OCRSs)
+(Feldman, Svensson, and Zenklusen 2016). OCRSs use a
+priori information on the values of the elements to formu-
+late a linear relaxation whose optimal fractional solution up-
+per bounds the performance of the integral offline optimum.
+Then, an online rounding procedure is used to produce a so-
+lution whose value is as close as possible to the fractional re-
+laxation solution’s value, for any adversarial order of the in-
+
+put sequence. The OCRS approach allows to obtain good ap-
+proximations of the expected optimal solution for linear and
+submodular objective functions. The existence of OCRSs for
+fully dynamic online selection problems is therefore a natu-
+ral research question that we address in this work.
+The OCRS approach is based on the availability of a pri-
+ori information on weights and activity times. However, in
+real world scenarios, these might be missing or might be
+expensive to collect. Therefore, in the second part of our
+work, we study the fully dynamic online selection problem
+with partial information, where the main research question
+is whether the OCRS approach is still viable if a priori in-
+formation on the weights is missing. In order to answer this
+question, we study a repeated version of the fully dynamic
+online selection problem, in which at each stage weights are
+unknown to the decision maker (i.e., no a priori informa-
+tion on weights is available) and chosen adversarially. The
+goal in this setting is the design of an online algorithm with
+performances (i.e., cumulative sum of weights of selected
+elements) close to that of the best static selection strategy in
+hindsight.
+Our Contributions
+First, we introduce the fully dynamic online selection prob-
+lem, in which elements arrive following an adversarial or-
+dering, and revealed one-by-one their weights and activ-
+ity times at the time of arrival (i.e., prophet model), or
+after the element has been selected (i.e., probing model).
+Our model describes temporal packing constraints (i.e.,
+downward-closed), where elements are active only within
+their activity time interval. The objective is to maximize the
+weight of the selected set of elements subject to temporal
+packing constraints. We provide two black-box reductions
+for adapting classical OCRS for online (non-dynamic) se-
+lection problems to the fully dynamic setting under full and
+partial information.
+Blackbox reduction 1: from OCRS to temporal OCRS.
+Starting from a (b, c)-selectable greedy OCRS in the clas-
+sical setting, we use it as a subroutine to build a (b, c)-
+selectable greedy OCRS in the more general temporal setting
+(see Algorithm 1 and Theorem 1). This means that competi-
+tive ratio guarantees in one setting determine the same guar-
+antees in the other. Such a reduction implies the existence
+of algorithms with constant competitive ratio for online opti-
+mization problems with linear or submodular objective func-
+tions subject to matroid, matching, and knapsack constraints,
+for which we give explicit constructions. We also extend the
+framework to elements arriving in batches, which can have
+correlated weights or activity times within the batch, as de-
+scribed in the appendix of the paper.
+Blackbox reduction 2: from temporal OCRS to no-α-
+regret algorithm. Following the recent work by Gergatsouli
+and Tzamos (2022) in the context of Pandora’s box prob-
+lems, we define the following extension of the problem to the
+partial-information setting. For each of the T stages, the al-
+gorithm is given in input a new instance of the fully dynamic
+online selection problem. Activity times are fixed before-
+hand and known to the algorithm, while weights are chosen
+by an adversary, and revealed only after the selection at the
+current stage has been completed. In such setting, we show
+that an α-competitive temporal OCRS can be exploited in
+the adversarial partial-information version of the problem, in
+order to build no-α-regret algorithms with polynomial per-
+iteration running time. Regret is measured with respect to the
+cumulative weights collected by the best fixed selection pol-
+icy in hindsight. We study three different settings: in the first
+setting, we study the full-feedback model (i.e., the algorithm
+observes the entire utility function at the end of each stage).
+Then, we focus on the semi-bandit-feedback model, in which
+the algorithm only receives information on the weights of the
+elements it selects. In such setting, we provide a no-α-regret
+framework with ˜O(T 1/2) upper bound on cumulative regret
+in the case in which we have a “white-box” OCRS (i.e., we
+know the exact procedure run within the OCRS, and we are
+able to simulate it ex-post). Moreover, we also provide a no-
+α-regret algorithm with ˜O(T 2/3) regret upper bound for the
+case in which we only have oracle access to the OCRS (i.e.,
+the OCRS is treated as a black-box, and the algorithm does
+not require knowledge about its internal procedures).
+Related Work
+In the first part of the paper, we deal with a setting where
+the algorithm has complete information over the input but is
+unaware of the order in which elements arrive. In this con-
+text, Contention resolution schemes (CRS) were introduced
+by Chekuri, Vondr´ak, and Zenklusen (2011) as a powerful
+rounding technique in the context of submodular maximiza-
+tion. The CRS framework was extended to online contention
+resolution schemes (OCRS) for online selection problems
+by Feldman, Svensson, and Zenklusen (2016), who provided
+constant competitive OCRSs for different problems, e.g. in-
+tersections of matroids, matchings, and prophet inequalities.
+We generalize the OCRS framework to a setting where ele-
+ments are timed and cease to exist right after.
+In the second part, we lift the complete knowledge as-
+sumption and work in an adversarial bandit setting, where at
+each stage the entire set of elements arrives, and we seek
+to select the “best” feasible subset. This is similar to the
+problem of combinatorial bandits (Cesa-Bianchi and Lugosi
+2012), but unlike it, we aim to deal with combinatorial se-
+lection of timed elements. In this respect, blocking bandits
+(Basu et al. 2019) model situations where played arms are
+blocked for a specific number of stages. Despite their con-
+textual (Basu et al. 2021), combinatorial (Atsidakou et al.
+2021), and adversarial (Bishop et al. 2020) extensions, re-
+cent work on blocking bandits only addresses specific cases
+of the fully dynamic online selection problem (Dickerson
+et al. 2018), which we solve in entire generality, i.e. adver-
+sarially and for all packing constraints.
+Our problem is also related to sleeping bandits (Klein-
+berg, Niculescu-Mizil, and Sharma 2010), in that the adver-
+sary decides which actions the algorithm can perform at each
+stage t. Nonetheless, a sleeping bandit adversary has to com-
+municate all available actions to the algorithm before a stage
+starts, whereas our adversary sets arbitrary activity times for
+each element, choosing in what order elements arrive.
+
+2
+Preliminaries
+Given a finite set X ⊆ Rn and Y ⊆ 2X , let 1Y ∈ {0, 1}|X|
+be the characteristic vector of set X, and co X be the convex
+hull of X. We denote vectors by bold fonts. Given vector x,
+we denote by xi its i-th component. The set {1, 2, . . . , n},
+with n ∈ N>0, is compactly denoted as [n]. Given a set X
+and a scalar α ∈ R, let αX := {αx : x ∈ X}. Finally, given
+a discrete set X, we denote by ∆X the |X|-simplex.
+We start by introducing a general selection problem in the
+standard (i.e., non-dynamic) case as studied by Kleinberg
+and Weinberg (2012) in the context of prophet inequalities.
+Let E be the ground set and let m := |E|. Each element e ∈ E
+is characterized by a collection of parameters ze. In gen-
+eral, ze is a random variable drawn according to an element-
+specific distribution ζe, supported over the joint set of pos-
+sible parameters. In the standard (i.e., non-dynamic) setting,
+ze just encodes the weight associated to element e, that is
+ze = (we), for some we ∈ [0, 1].1 In such case distributions
+ζe are supported over [0, 1]. Random variables {ze : e ∈ E}
+are independent, and ze is distributed according to ζe. An in-
+put sequence is an ordered sequence of elements and weights
+such that every element in E occurs exactly once in the se-
+quence. The order is specified by an arrival time se for each
+element e. Arrival times are such that se ∈ [m] for all e ∈ E,
+and for two distinct e, e′ we have se ̸= se′. The order of
+arrival of the elements is a priori unknown to the algorithm,
+and can be selected by an adversary. In the standard full-
+information setting the distributions ζe can be chosen by an
+adversary, but they are known to the algorithm a priori. We
+consider problems characterized by a family of packing con-
+straints.
+Definition 1 (Packing Constraint). A family of constraints
+F = (E, I), for ground set E and independence family I ⊆
+2E, is said to be packing (i.e., downward-closed) if, taken
+A ∈ I, and B ⊆ A, then B ∈ I.
+Elements of I are called independent sets. Such family of
+constraints is closed under intersection, and encompasses
+matroid, knapsack, and matching constraints.
+Fractional LP formulation
+Even in the offline setting, in
+which the ordering of the input sequence (se)e∈E is known
+beforehand, determining an independent set of maximum
+cumulative weight may be NP-hard in the worst-case (Feige
+1998). Then, we consider the relaxation of the problem
+in which we look for an optimal fractional solution. The
+value of such solution is an upper bound to the value of
+the true offline optimum. Therefore, any algorithm guar-
+anteeing a constant approximation to the offline fractional
+optimum immediately yields the same guarantees with re-
+spect to the offline optimum. Given a family of packing con-
+straints F = (E, I), in order to formulate the problem of
+computing the best fractional solution as a linear program-
+ming problem (LP) we introduce the notion of packing con-
+straint polytope PF ⊆ [0, 1]m which is such that PF :=
+co ({1S : S ∈ I}) . Given a non-negative submodular func-
+tion f : [0, 1]m → R≥0, and a family of packing constraints
+1This is for notational convenience. In the dynamic case ze will
+contain other parameters in addition to weights.
+F, an optimal fractional solution can be computed via the
+LP maxx∈PF f(x). If the goal is maximizing the cumula-
+tive sum of weights, the objective of the optimization prob-
+lem is ⟨x, w⟩, where w := (w1, . . . , wm) ∈ [0, 1]m is a
+vector specifying the weight of each element. If we assume
+access to a polynomial-time separation oracle for PF such
+LP yields an optimal fractional solution in polynomial time.
+Online selection problem. In the online version of the prob-
+lem, given a family of packing constraints F, the goal is se-
+lecting an independent set whose cumulative weight is as
+large as possible. In such setting, the elements reveal one
+by one their realized ze, following a fixed prespecified order
+unknown to the algorithm. Each time an element reveals ze,
+the algorithm has to choose whether to select it or discard it,
+before the next element is revealed. Such decision is irrevo-
+cable. Computing the exact optimal solution to such online
+selection problems is intractable in general (Feige 1998), and
+the goal is usually to design approximation algorithms with
+good competitive ratio.2 In the remainder of the section we
+describe one well-known framework for such objective.
+Online contention resolution schemes. Contention resolu-
+tion schemes were originally proposed by Chekuri, Vondr´ak,
+and Zenklusen (2011) in the context of submodular function
+maximization, and later extended to online selection prob-
+lems by Feldman, Svensson, and Zenklusen (2016) under
+the name of online contention resolution schemes (OCRS).
+Given a fractional solution x ∈ PF, an OCRS is an online
+rounding procedure yielding an independent set in I guar-
+anteeing a value close to that of x. Let R(x) be a random
+set containing each element e independently and with prob-
+ability xe. The set R(x) may not be feasible according to
+constraints F. An OCRS essentially provides a procedure to
+construct a good feasible approximation by starting from the
+random set R(x). Formally,
+Definition 2 (OCRS). Given a point x ∈ PF and the set of
+elements R(x), elements e ∈ E reveal one by one whether
+they belong to R(x) or not. An OCRS chooses irrevocably
+whether to select an element in R(x) before the next element
+is revealed. An OCRS for PF is an online algorithm that
+selects S ⊆ R(x) such that 1S ∈ PF.
+We will focus on greedy OCRS, which were defined by Feld-
+man, Svensson, and Zenklusen (2016) as follows.
+Definition 3 (Greedy OCRS). Let PF ⊆ [0, 1]m be the fea-
+sibility polytope for constraint family F. An OCRS π for PF
+is called a greedy OCRS if, for every ex-ante feasible solu-
+tion x ∈ PF, it defines a packing subfamily of feasible sets
+Fπ,x ⊆ F, and an element e is selected upon arrival if, to-
+gether with the set of already selected elements, the resulting
+set is in Fπ,x.
+A greedy OCRS is randomized if, given x, the choice of
+Fπ,x is randomized, and deterministic otherwise. For b, c ∈
+[0, 1], we say that a greedy OCRS π is (b, c)-selectable if,
+for each e ∈ E, and given x ∈ bPF (i.e., belonging to a
+2The competitive ratio is computed as the worst-case ratio be-
+tween the value of the solution found by the algorithm and the value
+of an optimal solution.
+
+down-scaled version of PF),
+Prπ,R(x) [S ∪ {e} ∈ Fπ,x
+∀S ⊆ R(x), S ∈ Fπ,x] ≥ c.
+Intuitively, this means that, with probability at least c, the
+random set R(x) is such that an element e is selected no
+matter what other elements I of R(x) have been selected
+so far, as long as I ∈ Fπ,x. This guarantees that an ele-
+ment is selected with probability at least c against any ad-
+versary, which implies a bc competitive ratio with respect
+to the offline optimum (see Appendix A for further details).
+Now, we provide an example due to Feldman, Svensson, and
+Zenklusen (2016) of a feasibility constraint family where
+OCRSs guarantee a constant competitive ratio against the
+offline optimum. We will build on this example throughout
+the paper in order to provide intuition for the main concepts.
+Example 1 (Theorem 2.7 in (Feldman, Svensson, and Zen-
+klusen 2016)). Given a graph G = (V, E), with |E| = m
+edges, we consider a matching feasibility polytope PF =
+�
+x ∈ [0, 1]m : �
+e∈δ(u) xe ≤ 1, ∀u ∈ V
+�
+, where δ(u) de-
+notes the set of all adjacent edges to u ∈ V. Given b ∈ [0, 1],
+the OCRS takes as input x ∈ bPF, and samples each edge e
+with probability xe to build R(x). Then, it selects each edge
+e ∈ R(x), upon its arrival, with probability (1 − e−xe)/xe
+only if it is feasible. Then, the probability to select any edge
+e = (u, v) (conditioned on being sampled) is
+1 − e−xe
+xe
+·
+�
+e′∈δ(u)∪δ(v)\{e}
+e−xe′
+= 1 − e−xe
+xe
+· e− �
+e′∈δ(u)∪δ(v)\{e} xe′ ≥ 1 − e−xe
+xe
+· e−2b
+≥ e−2b,
+where the inequality follows from xe
+∈
+bPF, i.e.,
+�
+e′∈δ(u)\{e} xe′ ≤ b−xe, and similarly for δ(v). Note that
+in order to obtain an unconditional probability, we need to
+multiply the above by a factor xe.
+We remark that this example resembles closely our in-
+troductory motivating application, where financial transac-
+tions need to be assigned to reviewers upon their arrival.
+Moreover, Feldman, Svensson, and Zenklusen (2016) give
+explicit constructions of (b, c)-selectable greedy OCRSs for
+knapsack, matching, matroidal constraints, and their inter-
+section. We include a discussion of their feasibility poly-
+topes in Appendix B. Ezra et al. (2020) generalize the above
+online selection procedure to a setting where elements arrive
+in batches rather than one at a time; we provide a discussion
+of such setting in Appendix C.
+3
+Fully Dynamic Online Selection
+The fully dynamic online selection problem is characterized
+by the definition of temporal packing constraints. We gen-
+eralize the online selection model (Section 2) by introduc-
+ing an activity time de ∈ [m] for each element. Element e
+arrives at time se and, if it is selected by the algorithm, it re-
+mains active up to time se + de and “blocks” other elements
+from being selected. Elements arriving after that time can
+be selected by the algorithm. In this setting, each element
+e ∈ E is characterized by a tuple of attributes ze := (we, de).
+Let Fd := (E, Id) be the family of temporal packing fea-
+sibility constraints where elements block other elements in
+the same independent set according to activity time vector
+d = (de)e∈E. The goal of fully dynamic online selection is
+selecting an independent set in Id whose cumulative weight
+is as large as possible (i.e., as close as possible to the offline
+optimum). We can naturally extend the expression for pack-
+ing polytopes in the standard setting to the temporal one for
+every feasibility constraint family, by exploiting the follow-
+ing notion of active elements.
+Definition 4 (Active Elements). For element e ∈ E and
+given {ze}e∈E, we denote the set of active elements as
+Ee := {e′ ∈ E : se′ ≤ se ≤ se′ + de′}.3
+In this setting, we don’t need to select an independent set
+S ∈ F, but, in a less restrictive way, we only require that for
+each incoming element we select a feasible subset of the set
+of active elements.
+Definition 5 (Temporal packing constraint polytope). Given
+F = (E, I), a temporal packing constraint polytope Pd
+F ⊆
+[0, 1]m is such that Pd
+F := co ({1S : S ∩ Ee ∈ I, ∀e ∈ E}) .
+Observation 1. For a fixed element e, the temporal polytope
+is the convex hull of the collection containing all the sets
+such that S ∩ Ee is feasible. This needs to be true for all
+e ∈ E, meaning that we can rewrite the polytope and the
+feasibility set as Pd
+F = co
+��
+e∈E {1S : S ∩ Ee ∈ F}
+�
+, and
+Id = �
+e∈E {S : S ∩ Ee ∈ I}. Moreover, when d and d′
+differ for at least one element e, that is de < d′
+e, then Ee ⊆
+E′
+e. Then, Pd
+F ⊇ Pd′
+F , Id ⊇ Id′.
+We now extend Example 1 to account for activity times.
+In Appendix B we also work out the reduction from stan-
+dard to temporal packing constraints for a number of exam-
+ples, including rank-1 matroids (single-choice), knapsack,
+and general matroid constraints.
+Example 2. We consider the temporal extension of the
+matching polytope presented in Example 1, that is
+Pd
+F =
+
+
+y ∈ [0, 1]m :
+�
+e∈δ(u)∩Ee
+xe ≤ 1, ∀u ∈ V, ∀e ∈ E
+
+
+ .
+Let us use the same OCRS as in the previous example, but
+where “feasibility” only concerns the subset of active edges
+in δ(u) ∪ δ(v). The probability to select an edge e = (u, v)
+is
+1 − e−xe
+xe
+·
+�
+e′∈δ(u)∪δ(v)∩Ee\{e}
+e−xe′ ≥ 1 − e−xe
+xe
+· e−2b ≥ e−2b,
+which is obtained in a similar way to Example 1.
+The above example suggests to look for a general reduc-
+tion that maps an OCRS for the standard setting, to an OCRS
+for the temporal setting, while achieving at least the same
+competitive ratio.
+3Note that, since for distinct elements e, e′, we have se′ ̸= se,
+we can equivalently define the set of active elements as Ee :=
+{e′ ∈ E : se′ < se ≤ se′ + de′} ∪ {e}.
+
+Algorithm 1: Greedy OCRS Black-box Reduction
+Input: Feasibility families F and Fd, polytopes PF
+and Pd
+F, OCRS π for F, a point x ∈ bPd
+F;
+Initialize Sd ← ∅;
+Sample R(x) such that Pr [e ∈ R(x)] = xe;
+for e ∈ E do
+Upon arrival of element e, compute the set of
+currently active elements Ee;
+if (Sd ∩ Ee) ∪ {e} ∈ Fπ,y then
+Execute the original greedy OCRS π(x);
+Update Sd accordingly;
+else
+Discard element e;
+return set Sd;
+4
+OCRS for Fully Dynamic Online Selection
+The first black-box reduction which we provide consists
+in showing that a (b, c)-selectable greedy OCRS for stan-
+dard packing constraints implies the existence of a (b, c)-
+selectable greedy OCRS for temporal constraints. In partic-
+ular, we show that the original greedy OCRS working for
+x ∈ bPF can be used to construct another greedy OCRS
+for y ∈ bPd
+F. To this end, Algorithm 1 provides a way of
+exploiting the original OCRS π in order to manage temporal
+constraints. For each element e, and given the induced sub-
+family of packing feasible sets Fπ,y, the algorithm checks
+whether the set of previously selected elements Sd which
+are still active in time, together with the new element e, is
+feasible with respect to Fπ,y. If that is the case, the algo-
+rithm calls the OCRS π. Then, if the OCRS π for input y
+decided to select the current element e, the algorithm adds it
+to Sd, otherwise the set remains unaltered. We remark that
+such a procedure is agnostic to whether the original greedy
+OCRS is deterministic or randomized. We observe that, due
+to a larger feasibility constraint family, the number of in-
+dependent sets have increased with respect to the standard
+setting. However, we show that this does not constitute a
+problem, and an equivalence between the two settings can
+be established through the use of Algorithm 1. The follow-
+ing result shows that Algorithm 1 yields a (b, c)-selectable
+greedy OCRS for temporal packing constraints.
+Theorem 1. Let F, Fd be the standard and temporal pack-
+ing constraint families, respectively, and let their corre-
+sponding polytopes be PF and Pd
+F. Let x ∈ bPF and
+y ∈ bPd
+F, and consider a (b, c)-selectable greedy OCRS π
+for Fπ,x. Then, Algorithm 1 equippend with π is a (b, c)-
+selectable greedy OCRS for Fd
+π,y.
+Proof. Let us denote by ˆπ the procedure described in Algo-
+rithm 1. First, we show that ˆπ is a greedy OCRS for Fd.
+Greedyness. It is clear from the setting and the construc-
+tion that elements arrive one at a time, and that ˆπ irrevoca-
+bly selects an incoming element only if it is feasible, and
+before seeing the next element. Indeed, in the if statement
+of Algorithm 1, we check that the active subset of the el-
+ements selected so far, together with the new arriving ele-
+ment e, is feasible against the subfamily Fπ,x ⊆ F. Con-
+straint subfamily Fπ,x is induced by the original OCRS
+π, and point x belongs to the polytope bPd
+F. Note that
+we do not necessarily add element e to the running set
+Sd, even though feasible, but act as the original greedy
+OCRS would have acted. All that is left to be shown is
+that such a procedure defines a subfamily of feasibility
+constraints Fd
+π,x ⊆ Fd. By construction, on the arrival
+of each element e, we guarantee that Sd is a set such that
+its subset of active elements is feasible. This means that
+Sd ∩ Ee ∈ Fπ,x ⊆ F. Then,
+Sd ∈ Fd
+π,x :=
+�
+e∈E
+{S : S ∩ Ee ∈ Fπ,x}.
+Finally, Fπ,x ⊆ F implies that Fd
+π,x ⊆ Fd, which shows
+that ˆπ is greedy. With the above, we can now turn to
+demonstrate (b, c)-selectability.
+Selectability. Upon arrival of element e ∈ E, let us con-
+sider S and Sd to be the sets of elements already selected
+by π and ˆπ, respectively. By the way in which the con-
+straint families are defined, and by construction of ˆπ, we
+can observe that, given x ∈ bPd
+F and y ∈ bPF, for all
+S ⊆ R(y) such that S ∪{e} ∈ Fπ,y, there always exists a
+set Sd ⊆ R(x) such that (Sd ∩Ee)∪{e} ∈ Fπ,x. This es-
+tablishes an injection between the selected set under stan-
+dard constraints, and its counterpart under temporal con-
+straints. We observe that, for all e ∈ E and x ∈ bPd
+F,
+Pr
+�
+Sd ∪ {e} ∈ Fd
+π,x
+∀Sd ⊆ R(x), Sd ∈ Fd
+π,x
+�
+=
+Pr
+�
+(Sd ∩ Ee) ∪ {e} ∈ Fπ,x ∀Sd ⊆ R(x), Sd ∩ Ee ∈ Fd
+π,x
+�
+.
+Hence, since for greedy OCRS π and y ∈ bPF, we have
+that Pr [S ∪ {e} ∈ Fπ,y ∀S ⊆ R(y), S ∈ Fπ,y] ≥ c, we
+can conclude by the injection above that
+Pr
+�
+(Sd ∩ Ee) ∪ {e} ∈ Fπ,x
+∀Sd ⊆ R(x), Sd ∩ Ee ∈ Fπ,x
+�
+≥ c.
+The theorem follows.
+We remark that the above reduction is agnostic to the
+weight scale, i.e., we need not assume that we ∈ [0, 1] for
+all e ∈ E. In order to further motivate the significance of
+Algorithm 1 and Theorem 1, in the Appendix we explic-
+itly reduce the standard setting to the fully dynamic one for
+single-choice, and provide a general recipe for all packing
+constraints.
+5
+Fully Dynamic Online Selection under
+Partial Information
+In this section, we study the case in which the decision-
+maker has to act under partial information. In particular,
+we focus on the following online sequential extension of
+the full-information problem: at each stage t ∈ [T ], a de-
+cision maker faces a new instance of the fully dynamic
+
+online selection problem. An unknown vector of weights
+wt ∈ [0, 1]|E| is chosen by an adversary at each stage t,
+while feasibility set Fd is known and fixed across all T
+stages. This setting is analogous to the one recently stud-
+ied by Gergatsouli and Tzamos (2022) in the context of
+Pandora’s box problems. A crucial difference with the on-
+line selection problem with full-information studied in Sec-
+tion 4 is that, at each step t, the decision maker has to de-
+cide whether to select or discard an element before observ-
+ing its weight. In particular, at each t, the decision maker
+takes an action at := 1Sd
+t , where Sd
+t
+∈ Fd is the fea-
+sible set selected at stage t. The choice of at is made be-
+fore observing wt. The objective of maximizing the cumu-
+lative sum of weights is encoded in the reward function
+f : [0, 1]2m ∋ (a, w) �→ ⟨a, w⟩ ∈ [0, 1], which is the
+reward obtained by playing a with weights w = (we)e∈E. 4
+In this setting, we can think of Fd as the set of super-arms
+in a combinatorial online optimization problem. Our goal is
+designing online algorithms which have a performance close
+to that of the best fixed super-arm in hindsight.5 In the analy-
+sis, as it is customary when the online optimization problem
+has an NP-hard offline counterpart, we resort to the notion
+of α-regret. In particular, given a set of feasible actions X,
+we define an algorithm’s α-regret up to time T as
+Regretα(T ) := α max
+x∈X
+� T
+�
+t=1
+f(x, wt)
+�
+−E
+� T
+�
+t=1
+f(xt, wt)
+�
+,
+where α ∈ (0, 1] and xt is the strategy output by the online
+algorithm at time t. We say that an algorithm has the no-α-
+regret property if Regretα(T )/T → 0 for T → ∞.
+The main result of the section is providing a black-box re-
+duction that yields a no-α-regret algorithm for any fully dy-
+namic online selection problem admitting a temporal OCRS.
+We provide no-α-regret frameworks for three scenarios:
+• full-feedback model: after selecting at the decision-maker
+observes the exact reward function f(·, wt).
+• semi-bandit feedback with white-box OCRS: after taking a
+decision at time t, the algorithm observes wt,e for each el-
+ement e ∈ Sd
+t (i.e., each element selected at t). Moreover,
+the decision-maker has exact knowledge of the procedure
+employed by the OCRS, which can be easily simulated.
+• semi-bandit feedback with oracle access to the OCRS: the
+decision maker has semi-bandit feedback and the OCRS is
+given as a black-box which can be queried once per step t.
+Full-feedback Setting
+In this setting, after selecting at, the decision-maker gets
+to observe the reward function f(·, wt). In order to achieve
+performance close to that of the best fixed super-harm in
+hindsight the idea is to employ the α-competitive OCRS de-
+signed in Section 4 by feeding it with a fractional solution
+4The analysis can be easily extended to arbitrary functions lin-
+ear in both terms.
+5As we argue in Appendix D it is not possible to be competitive
+with respect to more powerful benchmarks.
+Algorithm 2: FULL-FEEDBACK ALGORITHM
+Input: T , Fd, temporal OCRS ˆπ, subroutine RM
+Initialize RM for strategy space Pd
+F
+for t ∈ [T ] do
+xt ← RM.RECOMMEND()
+at ← execute OCRS ˆπ with input xt
+Play at, and subsequently observe f(·, wt)
+RM.UPDATE(f(·, wt))
+xt computed by considering the weights selected by the ad-
+versary up to time t − 1.6
+Let us assume to have at our disposal a no-α-regret algo-
+rithm for decision space Pd
+F. We denote such regret min-
+imizer as RM, and we assume it offers two basic opera-
+tions: i) RM.RECOMMEND() returns a vector in Pd
+F; ii)
+RM.UPDATE(f(·, w)) updates the internal state of the re-
+gret minimizer using feedback received by the environment
+in the form of a reward function f(·, w). Notice that the
+availability of such component is not enough to solve our
+problem since at each t we can only play a super-arm at ∈
+{0, 1}m feasible for Fd, and not the strategy xt ∈ Pd
+F ⊆
+[0, 1]m returned by RM. The decision-maker can exploit the
+subroutine RM together with a temporal greedy OCRS ˆπ by
+following Algorithm 2. We can show that, if the algorithm
+employs a regret minimizer for Pd
+F with a sublinear cumu-
+lative regret upper bound of RT , the following result holds.
+Theorem 2. Given a regret minimizer RM for decision
+space Pd
+F with cumulative regret upper bound RT , and an
+α-competitive temporal greedy OCRS, Algorithm 2 provides
+α max
+S∈Id
+T
+�
+t=1
+f(1S, wt) − E
+� T
+�
+t=1
+f(at, wt)
+�
+≤ RT .
+Since we are assuming the existence of a polynomial-
+time separation oracle for the set Pd
+F, then the LP
+arg maxx∈Pd
+F f(x, w) can be solved in polynomial time
+for any w. Therefore, we can instantiate a regret minimizer
+for Pd
+F by using, for example, follow-the-regularised-leader
+which yields RT ≤ ˜O(m
+√
+T) (Orabona 2019).
+Semi-Bandit Feedback with White-Box OCRS
+In this setting, given a temporal OCRS ˆπ, it is enough to
+show that we can compute the probability that a certain
+super-arm a is selected by ˆπ given a certain order of ar-
+rivals at stage t and a vector of weights w. If that is the
+case, we can build a no-α-regret algorithm with regret upper
+bound of ˜O(m
+√
+T) by employing Algorithm 2 and by in-
+stantiating the regret minimizer RM as the online stochastic
+mirror descent (OSMD) framework by Audibert, Bubeck,
+and Lugosi (2014). We observe that the regret bound ob-
+tained is this way is tight in the semi-bandit setting (Audib-
+ert, Bubeck, and Lugosi 2014). Let qt(e) be the probability
+6We remark that a (b, c)-selectable OCRS yields a bc competi-
+tive ratio. In the following, we let α := bc.
+
+Algorithm 3: SEMI-BANDIT-FEEDBACK ALGO-
+RITHM WITH ORACLE ACCESS TO OCRS
+Input: T , Fd, temporal OCRS ˆπ, full-feedback
+algorithm RM for decision space Pd
+F
+Let Z be initialized as in Theorem 3, and initialize
+RM appropriately
+for τ = 1, . . . , Z do
+Iτ ←
+�
+(τ − 1) T
+Z + 1, . . . , τ T
+Z
+�
+Choose a random permutation p : [m] → E, and
+t1, . . . , tm stages at random from Iτ
+xτ ← RM.RECOMMEND()
+for t = (τ − 1) T
+Z + 1, . . . , τ T
+Z do
+if t = tj for some j ∈ [m] then
+xt ← 1Sd for a feasible set Sd
+containing p(j)
+elsext ← xτ
+Play at obtained from the OCRS ˆπ executed
+with fractional solution xt
+Compute estimators ˜fτ(e) of
+fτ(e) :=
+1
+|Iτ |
+�
+t∈Iτ f(1e, wt) for each e ∈ E
+RM.UPDATE
+�
+˜fτ(·)
+�
+with which our algorithm selects element e at time t. Then,
+we can equip OSMD with the following unbiased estimator
+of the vector of weights: ˆwt,e := wt,eat,e/qt(e). 7 In order to
+compute qt(·) we need to have observed the order of arrival
+at stage t, the weights corresponding to super-arm at, and
+we need to be able to compute the probability with which
+the OCRS selected e at t. This the reason for which we talk
+about “white-box” OCRS, as we need to simulate ex post
+the procedure followed by the OCRS in order to compute
+qt(·). When we know the procedure followed by the OCRS,
+we can always compute qt(e) for any element e selected at
+stage t, since at the end of stage t we know the order of
+arrival, weights for selected elements, and the initial frac-
+tional solution xt. We provide further intuition as for how to
+compute such probabilities through the running example of
+matching constraints.
+Example 3. Consider Algorithm 2 initialized with the OCRS
+of Example 1. Given stage t, we can safely limit our atten-
+tion to selected edges (i.e., elements e such that at,e = 1).
+Indeed, all other edges will either be unfeasible (which im-
+plies that the probability of selecting them is 0), or they were
+not selected despite being feasible. Consider an arbitrary
+element e among those selected. Conditioned on the past
+choices up to element e, we know that e ∈ at will be fea-
+sible with certainty, and thus the (unconditional) probability
+it is selected is simply qt(e) = 1 − e−yt,e.
+Semi-Bandit Feedback and Oracle Access to OCRS
+As in the previous case, at each stage t the decision maker
+can only observe the weights associated to each edge se-
+7We observe that ˆwt,e is equal to 0 when e has not been selected
+at stage t because, in that case, at,e = 0.
+lected by at. Therefore, they have no counterfactual infor-
+mation on their reward had they selected a different feasi-
+ble set. On top of that, we assume that the OCRS is given
+as a black-box, and therefore we cannot compute ex post
+the probabilities qt(e) for selected elements. However, we
+show that it is possible to tackle this setting by exploiting a
+reduction from the semi-bandit feedback setting to the full-
+information feedback one. In doing so, we follow the ap-
+proach first proposed by Awerbuch and Kleinberg (2008).
+The idea is to split the time horizon T into a given num-
+ber of equally-sized blocks. Each block allows the decision
+maker to simulate a single stage of the full information set-
+ting. We denote the number of blocks by Z, and each block
+τ ∈ [Z] is composed by a sequence of consecutive stages
+Iτ. Algorithm 3 describes the main steps of our procedure.
+In particular, the algorithm employs a procedure RM, an al-
+gorithm for the full feedback setting as the one described
+in the previous section, that exposes an interface with the
+two operation of a traditional regret minimizer. During each
+block τ, the full-information subroutine is used to compute
+a vector xτ. Then, in most stages of the window Iτ, the de-
+cision at is computed by feeding xτ to the OCRS. A few
+stages are chosen uniformly at random to estimate utilities
+provided by other feasible sets (i.e., exploration phase). Af-
+ter the execution of all the stages in the window Iτ, the al-
+gorithm computes estimated reward functions and uses them
+to update the full-information regret minimizer.
+Let p : [m] → E be a random permutation of elements in
+E. Then, for each e ∈ E, by letting j be the index such that
+p(j) = e in the current block τ, an unbiased estimator ˜fτ(e)
+of fτ(e) :=
+1
+|Iτ |
+�
+t∈Iτ f(1e, wt) can be easily obtained by
+setting ˜fτ(e) := f(1e, wtj). Then, it is possible to show that
+our algorithm provides the following guarantees.
+Theorem 3. Given a temporal packing feasibility set Fd,
+and an α-competitive OCRS ˆπ, let Z = T 2/3, and the full
+feedback subroutine RM be defined as per Theorem 2. Then
+Algorithm 3 guarantees that
+α max
+S∈Id
+T
+�
+t=1
+f(1S, wt) − E
+� T
+�
+t=1
+f(at, wt)
+�
+≤ ˜O(T 2/3).
+6
+Conclusion and Future Work
+In this paper we introduce fully dynamic online selection
+problems in which selected items affect the combinatorial
+constraints during their activity times. We presented a gen-
+eralization of the OCRS approach that provides near opti-
+mal competitive ratios in the full-information model, and no-
+α-regret algorithms with polynomial per-iteration running
+time with both full- and semi-bandit feedback. Our frame-
+work opens various future research directions. For example,
+it would be particularly interesting to understand whether a
+variation of Algorithms 2 and 3 can be extended to the case
+in which the adversary changes the constraint family at each
+stage. Moreover, the study of the bandit-feedback model re-
+mains open, and no regret bound is known for that setting.
+
+Acknowledgements
+The authors of Sapienza are supported by the Meta Re-
+search grant on “Fairness and Mechanism Design”, the ERC
+Advanced Grant 788893 AMDROMA “Algorithmic and
+Mechanism Design Research in Online Markets”, the MIUR
+PRIN project ALGADIMAR “Algorithms, Games, and Dig-
+ital Markets”.
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+
+A
+Contention Resolution Schemes and Online Contention Resolution Schemes
+As explained at length in Section 2, our goal in general is that of finding the independent set of maximum weight for a
+given feasibility constraint family. However, doing this directly might be intractable in general and we need to aim for a
+good approximation of the optimum. In particular, given a non-negative submodular function f : [0, 1]m → R≥0, and a family
+of packing constraints F, we start from an ex ante feasible solution to the linear program maxx∈PF f(x), which upper bounds
+the optimal value achievable. An ex ante feasible solution is simply a distribution over the independent sets of F, given by
+a vector x in the packing constraint polytope of F. A key observation is that we can interpret the ex ante optimal solution
+to the above linear program as a vector x∗ of fractional values, which induces distribution over elements such that x∗
+e is the
+marginal probability that element e ∈ E is included in the optimum. Then, we use this solution to obtain a feasible solution that
+suitably approximates the optimum. The random set R(x∗) constructed by ex ante selecting each element independently with
+probability x∗
+e can be infeasible. Contention Resolution Schemes (Chekuri, Vondr´ak, and Zenklusen 2011) are procedures that,
+starting from the random set of sampled elements R(x∗), construct a feasible solution with good approximation guarantees
+with respect to the optimal solution of the original integer linear program.
+Definition 6 (Contention Resolution Schemes (CRSs) (Chekuri, Vondr´ak, and Zenklusen 2011)). For b, c ∈ [0, 1], a (b, c)-
+balanced Contention Resolution Scheme (CRS) π for F = (E, I) is a procedure such that, for every ex-ante feasible solution
+x ∈ bPF (i.e., the down-scaled version of polytope PF), and every subset S ⊆ E, returns a random set π(x, S) ⊆ S satisfying
+the following properties:
+1. Feasibility: π(x, S) ∈ I.
+2. c-balancedness: Prπ,R(x) [e ∈ π(x, R(x)) | e ∈ R(x)] ≥ c, ∀e ∈ E.
+When elements arrive in an online fashion, Feldman, Svensson, and Zenklusen (2016) extend CRS to the notion of OCRS,
+where R(x) is obtained in the same manner, but elements are revealed one by one in adversarial order. The procedure has to
+decide irrevocably whether or not to add the current element to the final solution set, which needs to be feasible and a competitive
+against the offline optimum. The idea is that adding a sampled element e ∈ E to the set of already selected elements S ⊆ R(x)
+maintains feasibility with at least constant probability, regardless of the element and the set. This originates Definition 3 and
+the subsequent discussion.
+B
+Examples
+In this section, we provide some clarifying examples for the concepts introduced in Section 2 and 3.
+Polytopes
+Example 4 provides the definition of the constraint polytopes of some standard problems, while Example 5 describes their
+temporal version. For a set S ⊆ E and x ∈ Rm, we define, with a slight abuse of notation, x(S) := �
+e∈S xe.
+Example 4 (Standard Polytopes). Given a ground set E,
+• Let K = (E, I) be a knapsack constraint. Then, given budget B > 0 and a vector of elements’ sizes c ∈ Rm
+≥0, its feasibility
+polytope is defined as
+PK = {x ∈ [0, 1]m : ⟨c, x⟩ ≤ B} .
+• Let G = (E, I) be a matching constraint. Then, its feasibility polytope is defined as
+PG = {x ∈ [0, 1]m : x(δ(u)) ≤ 1, ∀u ∈ V } ,
+where δ(u) denotes the set of all adjacent edges to u ∈ V. Note that the ground set in this case is the set of all edges of
+graph G = (V, E).
+• Let M = (E, I) be a matroid constraint. Then, its feasibility polytope is defined as
+PM = {x ∈ [0, 1]m : x(S) ≤ rank(S), ∀S ⊆ E} .
+Here, rank(S) := max {|I| : I ⊆ S, I ∈ I}, i.e., the cardinality of the maximum independent set contained in S.
+We can now rewrite the above polytopes under temporal packing constraints.
+Example 5 (Temporal Polytopes). For ground set E,
+• Let K = (E, I) be a knapsack constraint. Then, for B > 0 and cost vector c ∈ Rm
+≥0, its feasibility polytope is defined as
+Pd
+K = {x ∈ [0, 1]m : ⟨c, x⟩ ≤ B, ∀e ∈ E} .
+• Let G = (E, I) be a matching constraint. Then, its feasibility polytope is defined as
+Pd
+G = {x ∈ [0, 1]m : x(δ(u) ∩ Ee) ≤ 1, ∀u ∈ V, ∀e ∈ E} .
+• Let M = (E, I) be a matroid constraint. Then, its feasibility polytope is defined as
+Pd
+M = {x ∈ [0, 1]m : x(S ∩ Ee) ≤ rank(S), ∀S ⊆ E, ∀e ∈ E} .
+We also note that, for general packing constraints, if de = ∞ for all e ∈ E, then Ee = E, P∞
+F = PF, and similarly for the
+constraint family F∞ = F.
+
+From Standard OCRS to Temporal OCRS for Rank-1 Matroids, Matchings, Knapsacks, and General
+Matroids
+In this section, we explicitly derive a (1, 1/e)-selectable (randomized) temporal greedy OCRS for the rank-1 matroid feasibility
+constraint, from a (1, 1/e)-selectable (randomized) greedy OCRS in the standard setting (Livanos 2021), which is also tight.
+Let us denote this standard OCRS as πM, where M is a rank-1 matroid.
+Corollary 1. For the rank-1 matroid feasibility constraint family under temporal constraints, Algorithm 1 produces a (1, 1/e)-
+selectable (randomized) temporal greedy OCRS ˆπM from πM.
+Proof. Since it is clear from context, we drop the dependence on M and write π, ˆπ. We will proceed by comparing side-by-side
+what happens in π and in ˆπ. Let us recall from Examples 4, 5 that the polytopes can respectively be written as
+PM = {x ∈ [0, 1]m : x(S) ≤ 1, ∀S ⊆ E} ,
+Pd
+M = {y ∈ [0, 1]m : y(S ∩ Ee) ≤ 1, ∀S ⊆ E, ∀e ∈ E} .
+The two OCRSs perform the following steps, on the basis of Algorithm 1. On one hand, π defines a subfamily of constraints
+Fπ,x := {{e} : e ∈ H(x)}, where e ∈ E is included in random subset H(x) ⊆ E with probability 1−e−xe
+xe
+. Then, it selects
+the first sampled element e ∈ R(x) such that {e} ∈ Fπ,x. On the other hand, πy defines a subfamily of constraints Fd
+π,y :=
+{{e} : e ∈ H(y)}, where e ∈ E is included in random subset H(y) ⊆ E with probability qe(y) = 1−e−ye
+ye
+. The feasibility
+family Fd
+π,y induces, as per Observation 1, a sequence of feasibility families Fπ,y(e) := {{e} : e ∈ H(y) ∩ Ee}, for each
+e ∈ E. For all e′ ∈ E, the OCRS selects the first sampled element e ∈ R(y) such that {e} ∈ Fπ,y(e). In other words, the
+temporal OCRS selects a sampled element that is active only if no other element in its active elements set has been selected
+earlier. It is clear that both are randomized greedy OCRSs.
+We will now proceed by showing that each element e is selected with probability at least 1/e in both π, ˆπ. In π element e is
+selected if sampled, and no earlier element has been selected before (i.e. its singleton set belongs to the subfamily Fπ,x). An
+element e′ is not selected with probability 1 − xe′ · 1−e−xe′
+xe′
+= e−xe′. This means that the probability of e being selected is
+1 − e−xe
+xe
+·
+�
+se′ <se
+e−xe′ = 1 − e−xe
+xe
+· e
+− �
+se′ <se xe′ ≥ (1 − e−xe) exe−1
+xe
+≥ 1
+e,
+where the first inequality is justified by �
+se′ <se xe′ ≤ 1, and the second follows because the expression is minimized for
+xe = 0. Similarly, in ˆπ element e is selected if sampled, and no earlier element that is still active has been selected before (i.e.
+its singleton set belongs to the subfamily Fπ,y(e)). We have that the probability of e being selected is
+1 − e−ye
+ye
+·
+�
+se′ <se:e′∈Ee
+e−ye′ = 1 − e−ye
+ye
+· e
+− �
+se′ <se:e′∈Ee ye′ ≥ (1 − e−ye) eye−1
+ye
+≥ 1
+e.
+Again, the first inequality is justified by �
+se′ <se:e′∈Ee ye′ ≤ 1 by the temporal feasibility constraints, and the second follows
+because the expression is minimized for ye = 0. Selectability is thus shown.
+Remark 1. Adapting the OCRSs in Theorem 1.8 of (Feldman, Svensson, and Zenklusen 2016) for general matroids, matchings
+and knapsacks, by following Algorithm 1 step-by-step, we get the same selectability guarantees in the temporal settings as in
+the standard ones: respectively, (b, 1−b), (b, e−2b), (b, (1−2b)/(2−2b)). There are two crucial steps to map a standard OCRS
+into a temporal one, as exemplified by Corollary 1:
+1. We first need to define the temporal constraints based on the standard ones. This is done simply by enforcing the constraint
+in standard setting only for the current set of active elements, i.e. transforming Fπ,x into Fπ,y(e) for all elements e ∈ E.
+Such a transformation is analogous to the one used to go from Example 4 to Example 5.
+2. When proving selectability, the probability of feasibility is only calculated on elements e′ belonging the same independent
+set as e (which arrives later), that are still active. This means that the probability computation is confined to only e′ ∈ Ee
+such that se′ < se, rather than all e′ ∈ E such that se′ < se.
+C
+Batched Arrival: Matching Constraints
+As mentioned in Section 1, Ezra et al. (2020) generalize the one-by-one online selection problem to a setting where elements
+arrive in batches. The existence of batched greedy OCRSs implies a number of results, as for instance Prophet Inequalities
+under matching constraints where, rather than edges, vertices with all the edges adjacent to them arrive one at a time. This can
+be viewed as an incoming batch of edges, for which Ezra et al. (2020) explicitly construct a (1, 1/2)-selectable batched greedy
+OCRS.
+
+Indeed, we let the ground set E be partitioned in k disjoint subsets (batches) arriving in the order B1, . . . , Bk, and where
+elements in each batch appear at the same time. Such batches need to belong to a feasible family of batches B: for example,
+all batches could be required to be singletons, or they could be required to be all edges incident to a given vertex in a graph,
+and so on. Similarly to the traditional OCRS, we sample a random subset Rj(x) ⊆ Bj, for all j ∈ [k], so as to form R(x) :=
+�
+j∈[k] Rj(x) ⊆ E, where Rj’s are mutually independent. The fundamental difference with greedy OCRSs is that, within a
+given batch, weights are allowed to be correlated.
+Definition 7 (Batched Greedy OCRSs (Ezra et al. 2020)). For b, c ∈ [0, 1], let PF ⊆ [0, 1]m be F’s feasibility polytope.
+An OCRS π for bPF is called a batched greedy OCRS with respect to R if, for every ex-ante feasible solution x ∈ bPF, π
+defines a packing subfamily of feasible sets Fπ,x ⊆ F, and it selects a sampled element e ∈ Bj when, together with the
+set of already selected elements, the resulting set is in Fπ,x. We say that a batched greedy OCRS π is (b, c)-selectable if
+Prπ,R(x) [Sj ∪ {e} ∈ Fπ,x
+∀Sj ⊆ Rj(x), Sj ∈ Fπ,x] ≥ c, for each j ∈ [k], e ∈ Sj. The output feasible set will be S :=
+�
+j∈[m] Sj ∈ Fπ,x.
+Naturally, Theorem 1 extends to batched OCRSs.
+Corollary 2. Let F, Fd be respectively the standard and temporal packing constraint families, with their corresponding
+polytopes PF, Pd
+F. Let x ∈ bPF and y ∈ bPd
+F, and consider a (b, c)-selectable batched greedy OCRS π for Fπ,x, with
+batches B1, . . . , Bk ∈ B. We can construct a batched greedy OCRS ˆπ that is also (b, c)-selectable for Fd
+π,y, with batches
+B1, . . . , Bk ∈ B.
+The proof of this corollary is identical to that of Theorem 1: we can indeed define a set of active elements Ej for each batch
+Bj, and ˆπ is essentially in Algorithm 1 but with incoming batches rather than elements, and the necessary modifications in the
+sets. We will demonstrate the use of batched greedy OCRSs in the graph matching setting, where vertices come one at a time
+together with their contiguous edges. This allows us to solve the problem of dynamically assigning tasks to reviewers for the
+reviewing time, and to eventually match new tasks to the same reviewers, so as to maximize the throughput of this procedure.
+Details are presented in Appendix C.
+By Corollary 2 together with Theorem 4.1 in Ezra et al. (2020), which gives an explicit construction of a (1, 1/2)-selectable
+batched greedy OCRS under matching constraints, we immediately have that (1, 1/2)-selectable batched greedy OCRS exists
+even under temporal constraints. For clarity, and in the spirit of Appendix B, we work out how to derive from scratch an
+online algorithm that is 1/2-competitive with respect to the offline optimal matching when the graph is bipartite and temporal
+constraints are imposed. We do not use of Corollary 2, but we follow the proof of this general statement for the specific setting
+of bipartite graph matching. Batched OCRSs in the non-temporal case are not specific to the bipartite matching case but extend
+in principle to arbitrary packing constraints. Nevertheless, the only known constant competitive batched OCRS is the one for
+general graph matching by Ezra et al. (2020). Finally, we note that our results closely resemble the ones of Dickerson et al.
+(2018), with the difference that their arrival order is assumed to be stochastic, whereas ours is adversarial.
+This is motivated for instance by the following real-world scenario: there are |U| = m “offline” beds (machines) in an
+hospital, and |V | = n “online” patients (jobs) that arrive. Once a patient v ∈ V comes, the hospital has to irrevocably assign it
+to one of the beds, say u ∈ U, and occupy it for a stochastic time equal to duv := dv[u], for dv ∼ Dv, i.e., the u-th component
+of random vector dv. The sequence of arrivals is adversarial, but with known ex-ante distributions (Wv, Dv). Moreover, the
+patient’s healing can be thought of as a positive reward/weight equal to wuv := wv[u], for wv ∼ Wv, i.e., the u-th component
+of random vector wv, whose distributions are known to the algorithm. The hospital’s goal is that of maximizing the sum of the
+healing weights over time, i.e., over a discrete time period of length |V | = n. Across v’s, both wv’s and dv’s are independent.
+However, within the vector itself, components duv and du′v could be correlated, and the same holds for wv’s.
+Linear Programming Formulation
+First, we construct a suitable linear-programming formulation whose fractional solution yields an upper bound on the
+expected optimum offline algorithm. Then, we devise an online algorithm that achieves an α-competitive ratio with re-
+spect to the linear programming fractional solution. We follow the temporal LP Definition , and let f(x) := ⟨w, x⟩,
+for x
+∈
+Pd
+G being a feasible fractional solution in the matching polytope. Since the matching polytope is Pd
+G
+=
+{x ∈ [0, 1]m : x(δ(u) ∩ Ee) ≤ 1, ∀u ∈ V, ∀e ∈ E}, we can equivalently write the temporal linear program as
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+max
+x∈[0,1]m
+�
+u∈U
+�
+v∈V
+wuv · xuv
+⊳ Objective
+s.t.
+�
+u∈U
+xuv ≤ 1, ∀v ∈ V
+⊳ Constr. 1
+�
+v′:sv′ <sv
+xuv′ · Pr [duv′ ≥ sv − sv′] + xuv ≤ 1, ∀u ∈ U, v ∈ V
+⊳ Constr. 2
+xuv ≥ 0, ∀u ∈ U, v ∈ V
+⊳ Constr. 3
+(1)
+
+where wuv := Ewv∼Wv [wuv], when wuv is a random variable; when instead, it is deterministic, we simply have wuv = wuv.
+Furthermore, as we argued in Section 2, we can think of xuv to be the probability that edge uv is inserted in the offline (frac-
+tional) optimal matching. We now show why the above linear program yields an upper bound to the offline optimal matching.
+Lemma 1. Cosider solution x∗ to linear program (1). Then, x∗ is such that ⟨w, x∗⟩ ≥ Ew,d [⟨w, 1OPT⟩], where 1OPT ∈
+{0, 1}m is the vector denoting which of the elements have been selected by the integral offline optimum.
+Proof. The proof follows from analyzing the constraints. The meaning of Constraint 1 is that upon the arrival of vertex v, v
+must be matched at most once in expectation. In fact, for each job v ∈ V , at most one machine u ∈ U can be selected by the
+optimum, which yields
+�
+u∈U
+xuv ≤ 1.
+This justifies Constraint 1. Constraint 2, on the other hand, has the following simple interpretation: machine u is unavailable
+when job v arrives if it has been matched earlier to a job v′ such that the activity time is longer than the difference of v, v′ arrival
+times. Otherwise, u can in fact be matched to v, and this probability is of course lower than the probability of being available.
+This implies that for each machine u ∈ U and each job v ∈ V ,
+�
+v′:sv′ <sv
+xuv′ · Pr [duv′ ≥ sv − sv′] + xuv ≤ 1.
+We have shown that all constraints are less restrictive for the linear program as they would be for the offline optimum. Since
+the objective function is the same for both, a solution for the integral optimum is also a solution for the linear program, while
+the converse does not necessarily hold. The statement follows.
+A simple algorithm
+Inspired by the algorithm by Dickerson et al. (2018) (which deals with stochastic rather than adversarial arrivals), we propose
+Algorithm 4. In the remainder, let Navail(v) denote the set of available vertices u ∈ U when v ∈ V arrives.
+Algorithm 4: Bipartite Matching Temporal OCRS
+Data: Machine set U, job set V , and distributions Wv, Dv
+Result: Matching M ⊆ U × V
+Solve LP (1) and obtain fractional solution solution x∗;
+M ← ∅;
+for v ∈ V do
+if Navail(v) = ∅ then
+Reject v;
+else
+Select u ∈ Navail(v) with probability α ·
+x∗
+uv
+Pr[u∈Navail(v)];
+M ← M ∪ {uv};
+Lemma 2. Algorithm 4 makes every vertex u ∈ U available with probability at least α. Moreover, such probability is maximized
+for α = 1/2.
+Proof. We will prove the claim by induction. For the first incoming job v = 1, Pr [u ∈ Navail(v)] = 1 ≥ α for all machines
+u ∈ U, no matter what the values of wuv, duv are. To complete the base case, we only need to check that the probability of
+selecting one machine is in fact no larger than one: for this purpose, let us name the event u is selected by Algorithm 4 when v
+comes as u ∈ ALG(v).
+Pr [∃u ∈ Navail(v) : u ∈ ALG(v)] =
+�
+u∈U
+α ·
+x∗
+uv
+Pr [u ∈ Navail(v)] ≤ α,
+where the first equality follows from the fact the events within the existence quantifier are disjoint, and recalling that Navail(v) =
+U for the first job. Consider all vertices v′ arriving before vertex v (sv′ < sv), and assume that Pr [u ∈ Navail(v′)] ≥ α always.
+This means that the algorithm is makes each u available with probability at least α for all vertex arrivals before v. This, in turn,
+implies that each u is selected with probability α · x∗
+uv′. Let us observe that a machine u ∈ U will not be available for the
+
+incoming job v ∈ V only if the algorithm has matched it to an earlier job v′ with activity time larger than sv − sv′. Formally,
+the probability that u is available for v is
+Pr [u ∈ Navail(v)] = 1 − Pr [u /∈ Navail(v)]
+= 1 − Pr [∃v′ ∈ V : sv′ < sv, u ∈ ALG(v′), duv′ > sv − sv′]
+≥ 1 − α ·
+�
+v′:sv′ <sv
+x∗
+uv′Pr [duv′ ≥ sv − sv′]
+≥ α + α · x∗
+uv
+≥ α
+The second to last inequality follows from Constraint 2, and by observing the following simple implication for all r, z ∈ R: if
+r + z ≤ 1, then 1 − αr ≥ α + αz, so long as α ≤ 1
+2. Since we would like to choose α as large as possible, we choose α = 1
+2.
+What is left to be shown is that the probability of selecting one machine is at most one:
+Pr [∃u ∈ Navail(v) : u ∈ ALG(v)] =
+�
+u∈U
+α ·
+x∗
+uv
+Pr [u ∈ Navail(v)] ≤ 1.
+The statement, thus, follows.
+A direct consequence of the above two lemmata is the following theorem. Indeed, if every u is available with at least
+probability 1/2, then the algorithm will select it, regardless of what the previous algorithm actions. In turn, the optimum will
+be approximated with the same factor.
+Theorem 4. Algorithm 4 is 1
+2-competitive with respect to the expected optimum Ew,d [⟨w, 1OPT⟩].
+Various applications such as prophet and probing inequalities for the batched temporal setting can be derived from the
+above theorem. Solving them with a constant competitive ratio yields a solution for the review problem illustrated in the
+introduction, where multiple financial transactions arriving over time could be assigned to one of many potential reviewers, and
+these reviewers can be “reused” once they have completed their review time.
+D
+Benchmarks
+The need for stages
+We argue that, for the results in Section 5, stages are necessary in order for us to be able to compare our algorithm against any
+meaningful benchmark. Suppose, in contrast, that we chose to compare against the optimum (or an approximation of it) within
+a single stage where n jobs arrive to a single arm. A non-adaptive adversary could simply run the following procedure, with
+each job having weight 1: with probability 1/2, jobs with odd arrival order have activity time 1, and jobs with even arrival order
+have activity time ∞, with probability 1/2 the opposite holds. To be precise, let us recall that ∞ is just a shorthand notation to
+mean that all future jobs would be blocked: indeed, the activity time of a job arriving at time se is not unbounded but can be at
+most n − se. As activity times are revealed after the algorithm has made a decision for the current job, the algorithm does not
+know whether taking the current job will prevent it from being blocked for the entire future. The best thing the algorithm can
+do is to pick the first job with probability 1/2. Indeed, if the algorithm is lucky and the activity time is 1 then it knows to be
+in the first scenario and gets n. Otherwise, it only gets 1. Hence, the regret would be Rn = n − n+1
+2
+∈ Ω(n), which is linear.
+Note that n and T here represent two different concepts: the first is the number of elements sent within a stage; the second is the
+number of stages. In the case outlined above, T = 1, since it is a single stage scenario. Thus, there is no hope that in a single
+stage we could do anything meaningful, and we turn to the framework where an entire instance of the problem is sent at each
+stage t ∈ [T ].
+Choosing the right benchmark
+Now, we motivate why the Best-in-Hindsight policy introduced at the beginning of Section 5 is a strong and realistic benchmark,
+for an algorithm that knows the feasibility polytopes a priori. In fact, when we want to measure regret, we need to find a
+benchmark to compare against, which is neither too trivial nor unrealistically powerful compared to the information we have at
+hand. Below, we provide explicit lower bounds which show that the dynamic optimum is a too powerful benchmark even when
+the polytope is known. In particular, the next examples prove that it is impossible to achieve sublinear (α-)Regret against the
+dynamic optimum. In the remainder, we always assume full feedback and that the adversary is non-adaptive.
+In the remainder, we denote by aOPT
+t
+and aALG
+t
+the action chosen at time t by the optimum and the algorithm respectively.
+Lemma 3. Every algorithm has RT = �
+t∈[T ] E[ft(aOPT
+t
+)] − �
+t∈[T ] E[ft(aALG
+t
+)] ∈ Ω(T ) against the dynamic optimum.
+Proof. Consider the case of a single arm and the arrival of 3 jobs at each stage (on at a time within the stage, revealed from
+top to bottom), with the constraint that at most 1 active job can be selected. The (non-adaptive) adversary simply tosses T fair
+
+coins independently at each stage: if the tth coin lands heads, then all 3 jobs at the tth stage have activity times 1 and weights
+1, otherwise all jobs have activity time ∞, the first job has weight ǫ and the last two have weight 1 (recall that ∞ is just a
+shorthand notation to mean that all future jobs would be blocked). Figure 1 shows a possible realization of the T stages: at each
+stage the expected reward of the optimal policy is 3
+2, since the optimal value is 1 or 2 with equal probability. By linearity of
+expectation, �
+t∈[T ] E[ft(aOPT
+t
+)] = T · E[f(aOPT)] ≥ 3
+2T .
+1, 1
+1, 1
+1, 1
+ǫ, ∞
+1, ∞
+1, ∞
+ǫ, ∞
+1, ∞
+1, ∞
+. . . . . . . . .
+Figure 1: Three jobs per stage: w.p. 1/2, either {(1, 1), (1, 1), (1, 1)} or {(ǫ, ∞), (1, ∞), (1, ∞)}.
+On the other hand, the algorithm will discover which scenario it has landed into only after the value of the first job has been
+revealed. If it does not pick it and it results in a weight of 1, then the algorithm can get at most 1 from the remaining jobs. If
+instead it decides to pick it but it realizes in an ǫ value, it will only get ǫ. Even if the algorithm is aware of such a stochastic
+input beforehand, it knows that stages are independent and, hence, cannot be adaptive before a given stage begins. Then, it
+observes the first job weight without taking it, but it may already be too late. Any algorithm in this setting can be described by
+deciding to accept the first job with probability p (and reject it with 1 − p), and then act adaptively. Then, again by linearity of
+expectation,
+�
+t∈[T ]
+E[ft(aALG
+t
+)] = T · E[f(aALG)] = T ·
+�1
+2 (2p + (1 − p)) + 1
+2 (ǫp + (1 − p))
+�
+= 2 + ǫp
+2
+· T ≤ (1 + ǫ) · T.
+Thus, RT ≥ (1 − ǫ) · T ∈ Ω(T ).
+Now, we ask whether there exists a similar lower bound on approximate regret. Similarly to the previous lemma, we denote
+by aOCRS
+t
+the action chosen at time t by the OCRS.
+Lemma 4. Every algorithm has RT = α · �
+t∈[T ] E[ft(aOPT
+t
+)] − �
+t∈[T ] E[ft(aALG
+t
+)] ∈ Ω(T ) against an α-approximation of
+the dynamic optimum, for α ∈ (0, 1].
+Proof. Let all the activity times be infinite, and define (for a given stage) the constraint to be picking a single job irrevocably.
+We know that, for the single-choice problem, a tight OCRS achieves α = 1/2 competitive ratio. However, such OCRS is not
+greedy. Livanos (2021) constructs a tight greedy OCRS for single-choice, which is α = 1/e competitive. For our purposes,
+nonetheless, we only require the trivial inequality α ≤ 1. The non-adaptiveadversary could run the following a priori procedure,
+for each of the T stages: let δ = α−1/n
+2
+be a constant, sample k ∼ [n] uniformly at random, and send jobs in order of weights
+δk, δk−1, . . . , δ, 0, . . . , 0 (ascending until δ and then all 0s).8 We know that, by Theorem 1,
+�
+t∈[T ]
+E[ft(aOCRS
+t
+)] ≥ α ·
+�
+t∈[T ]
+E[ft(aOPT
+t
+)].
+This is possible because the greedy OCRS has access full-information about the current stage a priori (it knows δ and the
+sampled k at each stage), unlike the algorithm, which is unaware of how the T stages are going to be presented. It is easy to
+see that what the best the algorithm can do within a given stage is to randomly guess what the drawn k has been, i.e., where
+δ will land. We now divide the time horizon in T/n intervals, each composed of n stages. In each interval, since no stage is
+predictive of the next, we know that the algorithm cannot be adaptive across stages, nor can it be within a stage, since all possible
+sequences have the same prefix. By construction, we expect the algorithm to catch δ once per time interval, and otherwise get
+8This construction is inspired by the notes of Kesselheim and Mehlhorn (2016).
+
+at most δ2, optimistically for all remaining n − 1 stages. In other words, let us index each interval by I ∈ [T/n] and rewrite the
+algorithm and the OCRS expected rewards as
+�
+t∈[T ]
+E[ft(aALG
+t
+)] ≤
+�
+I∈[T/n]
+�
+δ + (n − 1)δ2�
+≤
+� δ
+n + δ2
+�
+· T,
+�
+t∈[T ]
+E[ft(aOCRS
+t
+)] ≥
+�
+I∈[T/n]
+(αδ · n) = αδ · T.
+Hence,
+RT =
+�
+t∈[T ]
+E[ft(aOCRS
+t
+)] −
+�
+t∈[T ]
+E[ft(aALG
+t
+)]
+≥
+�
+αδ − δ
+n − δ2
+�
+· T
+= (α − 1/n)2
+4
+· T ∈ Ω(T ).
+The last step follows from the fact that 1/n ∈ o(α), and α ∈ o(T ).
+E
+Omitted proofs from Section 5
+Theorem 2. Given a regret minimizer RM for decision space Pd
+F with cumulative regret upper bound RT , and an α-competitive
+temporal greedy OCRS, Algorithm 2 provides
+α max
+S∈Id
+T
+�
+t=1
+f(1S, wt) − E
+� T
+�
+t=1
+f(at, wt)
+�
+≤ RT .
+Proof. We assume to have access to a regret minimizer for the set Pd
+F guaranteeing an upper bound on the cumulative regret
+up to time T of RT . Then,
+E
+� T
+�
+t=1
+f(at, wt)
+�
+≥ α
+T
+�
+t=1
+f(xt, wt)
+≥ α
+�
+max
+x∈Pd
+F
+T
+�
+t=1
+f(x, wt) − RT
+�
+= α
+�
+max
+a
+T
+�
+t=1
+f(a, wt) − RT
+�
+,
+where the first inequality follows from the fact that Algorithm 2 employs a suitable temporal OCRS ˆπ to select at: for each
+e ∈ E, the probability with which the OCRS selects e is at least α · xt,e, and since f is a linear mapping (in particular, it is
+defined as the scalar product between a vector of weights and the choice at t) the above inequality holds. The second inequality
+is by no-regret property of the regret minimizer for decision space Pd
+F. This concludes the proof.
+Theorem 3. Given a temporal packing feasibility set Fd, and an α-competitive OCRS ˆπ, let Z = T 2/3, and the full feedback
+subroutine RM be defined as per Theorem 2. Then Algorithm 3 guarantees that
+α max
+S∈Id
+T
+�
+t=1
+f(1S, wt) − E
+� T
+�
+t=1
+f(at, wt)
+�
+≤ ˜O(T 2/3).
+Proof. We start by computing a lower bound on the average reward the algorithm gets. Algorithm 3 splits its decisions into Z
+blocks, and, at each τ ∈ [Z], chooses the action xτ suggested by the RM, unless the stage is one of the randomly sampled
+
+exploration steps. Then, we can write
+1
+T · E
+� T
+�
+t=1
+f(at, wt)
+�
+≥ α
+T ·
+�
+τ∈[Z]
+�
+t∈Iτ
+f(xt, wt)
+≥ α
+T
+�
+τ∈[Z]
+�
+t∈Iτ
+f(xτ, wt) − αm2Z
+T
+= α
+T ·
+�
+τ∈[Z]
+�
+e∈E
+xτ,e
+�
+t∈Iτ
+f(1e, wt) − αm2Z
+T
+= α
+Z ·
+�
+τ∈[Z]
+�
+e∈E
+xτ,e · E
+�
+˜fτ(e)
+�
+− αm2Z
+T
+,
+where the first inequality is by the use of a temporal OCRS to select at, and the second inequality is obtained by subtracting
+the worst-case costs incurred during exploration; note that the m2 factor in the second inequality is due to the fact that at each
+of the m exploration stages, we can lose at most m. The last equality is by definition of the unbiased estimator, since the value
+of f is observed T/Z times (once for every block) in expectation.
+We can now bound from below the rightmost expression we just obtained by using the guarantees of the regret-minimizer.
+α
+Z ·
+�
+τ∈[Z]
+�
+e∈E
+xτ,e · E
+�
+˜fτ(e)
+�
+− αm2Z
+T
+≥ α
+Z · E
+
+ max
+x∈Pd
+F
+�
+τ∈[Z]
+�
+e∈E
+xe ˜fτ(e) − RZ
+
+ − αm2Z
+T
+= α
+Z max
+x∈Pd
+F
+�
+τ∈[Z]
+�
+e∈E
+xe · E
+�
+˜fτ(e)
+�
+− α
+Z RZ − αm2Z
+T
+= α
+T max
+a∈F d
+�
+τ∈[Z]
+�
+e∈E
+xe
+�
+t∈Iτ
+f(1e, wt) − α
+Z RZ − αm2Z
+T
+= α
+T max
+a∈F d
+T
+�
+t=1
+f(at, wt) − α
+Z RZ − αm2Z
+T
+,
+where we used unbiasedness of ˜fτ(e), and the fact that the value of optimal fractional vector in the polytope is the same value
+provided by the best superarm (i.e., best vertex of the polytope) by convexity. The third equality follows from expanding the
+expectation of the unbiased estimator (i.e. E
+�
+˜fτ(e)
+�
+:= Z
+T · �
+t∈Iτ f(1e, wt)). Let us now rearrange the last expression and
+compute the cumulative regret:
+α max
+a∈F d
+T
+�
+t=1
+f(at, wt) − E
+� T
+�
+t=1
+f(at, wt)
+�
+≤ α
+Z
+RZ
+����
+≤ ˜
+O(
+√
+Z)
+T + αm2Z
+≤ ˜O(T 2/3),
+where in the last step we set Z = T 2/3 and obtain the desired upper bound on regret (the term αm2 is incorporated in the ˜O
+notation). The theorem follows.
+F
+Further Related Works
+CRS and OCRS.
+Contention resolution schemes (CRS) were introduced by Chekuri, Vondr´ak, and Zenklusen (2011) as a
+powerful rounding technique in the context of submodular maximization. The CRS framework was extended to online con-
+tention resolution schemes (OCRS) for online selection problems by Feldman, Svensson, and Zenklusen (2016), who provided
+OCRSs for different problems, including intersections of matroids, matchings, and prophet inequalities. Ezra et al. (2020) re-
+cently extended OCRS to batched arrivals, providing a constant competitive ratio for stochastic max-weight matching in vertex
+and edge arrival models.
+Combinatorial Bandits.
+The problem of combinatorial bandits was first studied in the context of online shortest paths (Awer-
+buch and Kleinberg 2008; Gy¨orgy et al. 2007), and the general version of the problem is due to Cesa-Bianchi and Lugosi (2012).
+Improved regret bounds can be achieved in the case of combinatorial bandits with semi-bandit feedback (see, e.g., (Chen, Wang,
+and Yuan 2013; Kveton et al. 2015; Audibert, Bubeck, and Lugosi 2014)). A related problem is that of linear bandits (Awer-
+buch and Kleinberg 2008; McMahan and Blum 2004), which admit computationally efficient algorithms in the case in which
+the action set is convex (Abernethy, Hazan, and Rakhlin 2009).
+
+Blocking bandits.
+In blocking bandits (Basu et al. 2019) the arm that is played is blocked for a specific number of stages.
+Blocking bandits have recently been studied in contextual (Basu et al. 2021), combinatorial (Atsidakou et al. 2021), and ad-
+versarial (Bishop et al. 2020) settings. Our bandit model differs from blocking bandits since we consider each instance of the
+problem confined within each stage. In addition, the online full information problems that are solved in most blocking bandits
+papers (Atsidakou et al. 2021; Basu et al. 2021; Dickerson et al. 2018) only addresses specific cases of the fully dynamic online
+selection problem, which we solve in entire generality.
+Sleeping bandits.
+As mentioned, our problem is similar to that of sleeping bandits (see (Kleinberg, Niculescu-Mizil, and
+Sharma 2010) and follow-up papers), but at the same time the two models differ in a number of ways. Just like the sleeping
+bandits case, the adversary in our setting decides which actions we can perform by setting arbitrary activity times at each t.
+The crucial difference between the two settings is that, in sleeping bandits, once an adversary has chosen the available actions
+for a given stage, they have to communicate them all at once to the algorithm. In our case, instead, the adversary can choose
+the available actions within a given stage as the elements arrive, so it is, in some sense, “more dynamic”. In particular, in the
+temporal setting there are two levels of adaptivity for the adversary: on one hand, the adversary may or may not be adaptive
+across stages (this is the classic bandit notion of adaptivity). On the other hand, the adversary may or may not be adaptive within
+the same stage (which is the notion of online algorithms adaptivity).
+

B9E0T4oBgHgl3EQfQABU/content/tmp_files/2301.02186v1.pdf.txt

@@ -0,0 +1,1989 @@
+MNRAS 000, 1–16 (0000)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Inferring the impact of feedback on the matter distribution
+using the Sunyaev Zel’dovich effect: Insights from
+CAMELS simulations and ACT+DES data
+Shivam Pandey,1,2 Kai Lehman,3,4 Eric J. Baxter,3 Yueying Ni,5
+Daniel Angl´es-Alc´azar,6,7 Shy Genel,7,1 Francisco Villaescusa-Navarro,7
+Ana Maria Delgado,5 Tiziana di Matteo9
+1Department of Physics, Columbia University, New York, NY, USA 10027
+2Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104, USA
+3Institute for Astronomy, University of Hawai‘i, 2680 Woodlawn Drive, Honolulu, HI 96822, USA
+4Universit¨ats-Sternwarte M¨unchen, Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at, Scheinerstr. 1, 81679 M¨unchen, Germany
+5Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA 02138, US
+6Department of Physics, University of Connecticut, 196 Auditorium Road, U-3046, Storrs, CT, 06269, USA
+7Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, New York, NY, 10010, USA
+9McWilliams Center for Cosmology, Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213
+6 January 2023
+ABSTRACT
+Feedback from active galactic nuclei and stellar processes changes the matter distri-
+bution on small scales, leading to significant systematic uncertainty in weak lensing
+constraints on cosmology. We investigate how the observable properties of group-scale
+halos can constrain feedback’s impact on the matter distribution using Cosmology and
+Astrophysics with MachinE Learning Simulations (CAMELS). Extending the results
+of previous work to smaller halo masses and higher wavenumber, k, we find that the
+baryon fraction in halos contains significant information about the impact of feed-
+back on the matter power spectrum. We explore how the thermal Sunyaev Zel’dovich
+(tSZ) signal from group-scale halos contains similar information. Using recent Dark
+Energy Survey (DES) weak lensing and Atacama Cosmology Telescope (ACT) tSZ
+cross-correlation measurements and models trained on CAMELS, we obtain 10% con-
+straints on feedback effects on the power spectrum at k ∼ 5 h/Mpc. We show that
+with future surveys, it will be possible to constrain baryonic effects on the power spec-
+trum to O(< 1%) at k = 1 h/Mpc and O(3%) at k = 5 h/Mpc using the methods
+that we introduce here. Finally, we investigate the impact of feedback on the matter
+bispectrum, finding that tSZ observables are highly informative in this case.
+Key words: large-scale structure of Universe – methods: statistical
+1
+INTRODUCTION
+The statistics of the matter distribution on scales k ≳
+0.1 hMpc−1 are tightly constrained by current weak lensing
+surveys (e.g. Asgari et al. 2021; Abbott et al. 2022). How-
+ever, modeling the matter distribution on these scales to ex-
+tract cosmological information is complicated by the effects
+of baryonic feedback (Rudd et al. 2008). Energetic output
+from active galactic nuclei (AGN) and stellar processes (e.g.
+winds and supernovae) directly impacts the distribution of
+gas on small scales, thereby changing the total matter dis-
+tribution (e.g. Chisari et al. 2019).1 The coupling between
+these processes and the large-scale gas distribution is chal-
+lenging to model theoretically and in simulations because of
+the large dynamic range involved, from the scales of individ-
+ual stars to the scales of galaxy clusters. While it is generally
+agreed that feedback leads to a suppression of the matter
+power spectrum on scales 0.1 hMpc−1 ≲ k ≲ 20 hMpc−1,
+the amplitude of this suppression remains uncertain by tens
+of percent (van Daalen et al. 2020; Villaescusa-Navarro et al.
+1 Changes to the gas distribution can also gravitationally influ-
+ence the dark matter distribution, further modifying the total
+matter distribution.
+© 0000 The Authors
+arXiv:2301.02186v1  [astro-ph.CO]  5 Jan 2023
+
+2
+Pandey et al.
+2021) (see also Fig. 1). This systematic uncertainty limits
+constraints on cosmological parameters from current weak
+lensing surveys (e.g. Abbott et al. 2022; Asgari et al. 2021).
+For future surveys, such as the Vera Rubin Observatory
+LSST (The LSST Dark Energy Science Collaboration et al.
+2018) and Euclid (Euclid Collaboration et al. 2020), the
+problem will become even more severe given expected in-
+creases in statistical precision. In order to reduce the sys-
+tematic uncertainties associated with feedback, we would
+like to identify observable quantities that carry information
+about the impact of feedback on the matter distribution and
+develop approaches to extract this information (e.g. Nicola
+et al. 2022).
+Recently, van Daalen et al. (2020) showed that the halo
+baryon fraction, fb, in halos with M ∼ 1014 M⊙ carries sig-
+nificant information about suppression of the matter power
+spectrum caused by baryonic feedback. They found that the
+relation between fb and matter power suppression was ro-
+bust to at least some changes in the subgrid prescriptions
+for feedback physics. Note that fb as defined by van Daalen
+et al. (2020) counts baryons in both the intracluster medium
+as well as those in stars. The connection between fb and feed-
+back is expected, since one of the main drivers of feedback’s
+impact on the matter distribution is the ejection of gas from
+halos by AGN. Therefore, when feedback is strong, halos will
+be depleted of baryons and fb will be lower. The conversion
+of baryons into stars — which will not significantly impact
+the matter power spectrum on large scales — does not im-
+pact fb, since fb includes baryons in stars as well as the ICM.
+van Daalen et al. (2020) specifically consider the measure-
+ment of fb in halos with 6 × 1013M⊙ ≲ M500c ≲ 1014 M⊙.
+In much more massive halos, the energy output of AGN is
+small compared to the binding energy of the halo, preventing
+gas from being expelled. In smaller halos, van Daalen et al.
+(2020) found that the correlation between power spectrum
+suppression and fb is less clear.
+Although fb carries information about feedback, it is
+somewhat unclear how one would measure fb in practice.
+Observables such as the kinematic Sunyaev Zel’dovich (kSZ)
+effect can be used to constrain the gas density; combined
+with some estimate of stellar mass, fb could then be in-
+ferred. However, measuring the kSZ is challenging, and cur-
+rent measurements have low signal-to-noise (Hand et al.
+2012; Hill et al. 2016; Soergel et al. 2016). Moreover, van
+Daalen et al. (2020) consider a relatively limited range of
+feedback prescriptions. It is unclear whether a broader range
+of feedback models could lead to a greater spread in the
+relationship between fb and baryonic effects on the power
+spectrum. In any case, it is worthwhile to consider other
+potential observational probes of feedback.
+Another potentially powerful probe of baryonic feed-
+back is the thermal SZ (tSZ) effect. The tSZ effect is caused
+by inverse Compton scattering of CMB photons with a pop-
+ulation of electrons at high temperature. This scattering pro-
+cess leads to a spectral distortion in the CMB that can be
+reconstructed from multi-frequency CMB observations. The
+amplitude of this distortion is sensitive to the line-of-sight
+integral of the electron pressure. Since feedback changes the
+distribution and thermodynamics of the gas, it stands to rea-
+son that it could impact the tSZ signal. Indeed, several works
+using both data (e.g Pandey et al. 2019, 2022; Gatti et al.
+2022a) and simulations (e.g. Scannapieco et al. 2008; Bhat-
+tacharya et al. 2008; Moser et al. 2022; Wadekar et al. 2022)
+have shown that the tSZ signal from low-mass (group scale)
+halos is sensitive to feedback. Excitingly, the sensitivity of
+tSZ measurements is expected to increase dramatically in
+the near future due to high-sensitivity CMB measurements
+from e.g. SPT-3G (Benson et al. 2014), Advanced ACTPol
+(Henderson et al. 2016), Simons Observatory (Ade et al.
+2019), and CMB Stage 4 (Abazajian et al. 2016).
+The goal of this work is to investigate what informa-
+tion the tSZ signals from low-mass halos contain about the
+impact of feedback on the small-scale matter distribution.
+The tSZ signal, which we denote with the Compton y pa-
+rameter, carries different information from fb. For one, y is
+sensitive only to the gas and not to stellar mass. Moreover,
+y carries sensitivity to both the gas density and tempera-
+ture, unlike fb which depends only on the gas density. The
+y signal is also easier to measure than fb, since it can be
+estimated simply by cross-correlating halos with a tSZ map.
+The signal-to-noise of such cross-correlation measurements
+is already high with current data, on the order of 10s of σ
+(Vikram et al. 2017; Pandey et al. 2019, 2022; S´anchez et al.
+2022).
+In this paper, we investigate the information content
+of the tSZ signal from group-scale halos using the Cosmol-
+ogy and Astrophysics with MachinE Learning Simulations
+(CAMELS) simulations. As we describe in more detail in
+§2, CAMELS is a suite of many hydrodynamical simula-
+tions run across a range of different feedback prescriptions
+and different cosmological parameters. The relatively small
+volume of the CAMELS simulations ((25/h)3 Mpc3) means
+that we are somewhat limited in the halo masses and scales
+that we can probe. We therefore view our analysis as an ex-
+ploratory work that investigates the information content of
+low-mass halos for constraining feedback and how to extract
+this information; more accurate results over a wider range
+of halo mass and k may be obtained in the future using the
+same methods applied to larger volume simulations.
+By training statistical models on the CAMELS sim-
+ulations, we explore what information about feedback ex-
+ists in tSZ observables, and how robust this information is
+to changes in subgrid feedback prescriptions. We consider
+three very different prescriptions for feedback based on the
+SIMBA (Dav´e et al. 2019), Illustris-TNG (Pillepich et al.
+2018, henceforth TNG) and Astrid (Bird et al. 2022; Ni et al.
+2022) models across a wide range of possible parameter val-
+ues, including variations in cosmology. The flexibility of the
+statistical models we employ means that it is possible to
+uncover more complex relationships between e.g. fb, y, and
+the baryonic suppression of the power spectrum than consid-
+ered in van Daalen et al. (2020). The work presented here
+is complementary to Delgado et al. (2023) which explores
+the information content in the baryon fraction of halos en-
+compassing broader mass range (M > 1010M⊙/h), finding
+a broad correlation with the matter power suppression.
+Finally, we apply our trained statistical models to recent
+measurements of the y signal from low-mass halos by Gatti
+et al. (2022a) and Pandey et al. (2022). These analyses in-
+ferred the halo-integrated y signal from the cross-correlation
+of galaxy lensing and the tSZ effect using lensing data from
+the Dark Energy Survey (DES) (Amon et al. 2022; Secco
+et al. 2022) and tSZ measurements from the Atacama Cos-
+mology Telescope (ACT) (Madhavacheril et al. 2020). In
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+3
+addition to providing interesting constraints on the impact
+of feedback, these results highlight the potential of future
+similar analyses with e.g. Dark Energy Spectroscopic Ex-
+periment (DESI; DESI Collaboration et al. 2016), Simons
+Observatory (Ade et al. 2019), and CMB Stage 4 (Abaza-
+jian et al. 2016).
+Two recent works — Moser et al. (2022) and Wadekar
+et al. (2022) — have used the CAMELS simulations to ex-
+plore the information content of the tSZ signal for constrain-
+ing feedback. These works focus on the ability of tSZ ob-
+servations to constrain the parameters of subgrid feedback
+models in hydrodynamical simulations. Here, in contrast, we
+attempt to connect the observable quantities directly to the
+impact of feedback on the matter power spectrum and bis-
+pectrum. Additionally, unlike some of the results presented
+in Moser et al. (2022) and Wadekar et al. (2022), we consider
+the full parameter space explored by the CAMELS simula-
+tions rather than the small variations around a fiducial point
+that are relevant to calculation of the Fisher matrix. Finally,
+we only focus on the intra-halo gas profile of the halos in
+the mass range captured by the CAMELS simulations (c.f.
+Moser et al. 2022). We do not expect the inter-halo gas pres-
+sure to be captured by the small boxes used here as it may
+be sensitive to higher halo masses (Pandey et al. 2020).
+Nonlinear evolution of the matter distribution induces
+non-Gaussianity, and hence there is additional information
+to be recovered beyond the power spectrum. Recent mea-
+surements detect higher-order matter correlations at cos-
+mological scales at O(10σ)(Secco et al. 2022; Gatti et al.
+2022b), and the significance of these measurements is ex-
+pected to rapidly increase with up-coming surveys (Pyne
+& Joachimi 2021). Jointly analyzing two-point and three-
+point correlations of the matter field can help with self-
+calibration of systematic parameters and improve cosmolog-
+ical constraints. As described in Foreman et al. (2020), the
+matter bispectrum is expected to be impacted by baryonic
+physics at O(10%) over the scales of interest. With these
+considerations in mind, we also investigate whether the SZ
+observations carry information about the impact of baryonic
+feedback on the matter bispectrum.
+The plan of the paper is as follows. In §2 we discuss the
+CAMELS simulation and the data products that we use in
+this work. In §3, we present the results of our explorations
+with the CAMELS simulations, focusing on the information
+content of the tSZ signal for inferring the impact of feedback
+on the matter distribution. In §4, we apply our analysis to
+the DES and ACT measurements. We summarize our results
+and conclude in §5.
+2
+CAMELS SIMULATIONS AND
+OBSERVABLES
+2.1
+Overview of CAMELS simulations
+We investigate the use of SZ signals for constraining the
+impact of feedback on the matter distribution using approx-
+imately 3000 cosmological simulations run by the CAMELS
+collaboration (Villaescusa-Navarro et al. 2021). One half of
+these are gravity-only N-body simulations and the other half
+are hydrodynamical simulations with matching initial con-
+ditions. The simulations are run using three different hy-
+drodynamical sub-grid codes, TNG (Pillepich et al. 2018),
+SIMBA (Dav´e et al. 2019) and Astrid (Bird et al. 2022; Ni
+et al. 2022). As detailed in Villaescusa-Navarro et al. (2021),
+for each sub-grid implementation six parameters are varied:
+two cosmological parameters (Ωm and σ8) and four param-
+eters dealing with baryonic astrophysics. Of these, two deal
+with supernovae feedback (ASN1 and ASN2) and two deal
+with AGN feedback (AAGN1 and AAGN2). The meanings of
+the feedback parameters for each subgrid model are summa-
+rized in Table 1.
+Note that the astrophysical parameters have somewhat
+different physical meanings for the different subgrid pre-
+scriptions, and there is usually a complex interplay between
+the parameters and their impact on the properties of galax-
+ies and gas. For example, the parameter ASN1 approximately
+corresponds to the pre-factor for the overall energy output in
+galactic wind feedback per-unit star-formation in both the
+TNG (Pillepich et al. 2018) and Astrid (Bird et al. 2022) sim-
+ulations. However, in the SIMBA simulations it corresponds
+to the to the wind-driven mass outflow rate per unit star-
+formation calibrated from the Feedback In Realistic Envi-
+ronments (FIRE) zoom-in simulations (Angl´es-Alc´azar et al.
+2017b). Similarly, the AAGN2 parameter controls the bursti-
+ness and the temperature of the heated gas during the AGN
+bursts in the TNG simulations (Weinberger et al. 2017). In
+the SIMBA suite, it corresponds to the speed of the kinetic
+AGN jets with constant momentum flux (Angl´es-Alc´azar
+et al. 2017a; Dav´e et al. 2019). However, in the Astrid suite,
+it corresponds to the efficiency of thermal mode of AGN
+feedback. As we describe in § 3.2, this can result in counter-
+intuitive impact on the matter power spectrum in the Astrid
+simulation, relative to TNG and SIMBA.
+For each of the sub-grid physics prescriptions, three va-
+rieties of simulations are provided. These include 27 sims
+for which the parameters are fixed and initial conditions are
+varied (cosmic variance, or CV, set), 66 simulations varying
+only one parameter at a time (1P set) and 1000 sims varying
+parameters in a six dimensional latin hyper-cube (LH set).
+We use the CV simulations to estimate the variance expected
+in the matter power suppression due to stochasticity (see
+Fig. 1). We use the 1P sims to understand how the matter
+suppression responds to variation in each parameter individ-
+ually. Finally we use the full LH set to effectively marginalize
+over the full parameter space varying all six parameters. We
+use publicly available power spectrum and bispectrum mea-
+surements for these simulation boxes (Villaescusa-Navarro
+et al. 2021).2 Where unavailable, we calculate the power
+spectrum and bispectrum, using the publicly available code
+Pylians.3
+2.2
+Baryonic effects on the power spectrum and
+bispectrum
+The left panel of Fig. 1 shows the measurement of the
+power spectrum suppression caused by baryonic effects in
+the TNG, SIMBA, and Astrid simulations for their fiducial
+feedback settings. The right two panels of the figure show the
+impact of baryonic effects on the bispectrum for two different
+2 See also https://www.camel-simulations.org/data.
+3 https://github.com/franciscovillaescusa/Pylians3
+MNRAS 000, 1–16 (0000)
+
+4
+Pandey et al.
+Simulation
+Type/Code
+Astrophysical parameters varied
+& its meaning
+TNG
+Magneto-hydrodynamic/
+AREPO
+ASN1: (Energy of Galactic winds)/SFR
+ASN2: Speed of galactic winds
+AAGN1: Energy/(BH accretion rate)
+AAGN2: Jet ejection speed or burstiness
+SIMBA
+Hydrodynamic/GIZMO
+ASN1 : Mass loading of galactic winds
+ASN2 : Speed of galactic winds
+AAGN1 : Momentum flux in QSO and jet mode of feedback
+AAGN2 : Jet speed in kinetic mode of feedback
+Astrid
+Hydrodynamic/pSPH
+ASN1: (Energy of Galactic winds)/SFR
+ASN2: Speed of galactic winds
+AAGN1: Energy/(BH accretion rate)
+AAGN2: Thermal feedback efficiency
+Table 1. Summary of the three feedback models used in this analysis. For each model, four feedback parameters are varied: AAGN1,
+AAGN2, ASN1, and ASN2. The meanings of these parameters are different for each model, and are summarized in the rightmost column.
+In addition to these four astrophysical parameters, the cosmological parameters Ωm and σ8 were also varied.
+100
+101
+k (h/Mpc)
+−0.6
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+∆P/PDMO
+Illustris-TNG
+Illustris-TNG (LH suite)
+SIMBA
+Astrid
+100
+101
+keq (h/Mpc)
+−0.6
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+∆Beq/Beq;DMO
+100
+101
+ksq (h/Mpc)
+−0.6
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+∆Bsq/Bsq;DMO
+Figure 1. Far left: Baryonic suppression of the matter power spectrum, ∆P/PDMO, in the CAMELS simulations. The dark-blue, red
+and orange shaded regions correspond to the 1σ range of the cosmic variance (CV) suite of TNG, SIMBA and Astrid simulations,
+respectively. The light-blue region corresponds to the 1σ range associated with the latin hypercube (LH) suite of TNG, illustrating the
+range of feedback models explored across all parameter values. Middle and right panels: the impact of baryonic feedback on the matter
+bispectrum for equilateral and squeezed triangle configurations, respectively.
+tringle configurations (equilateral and squeezed). To com-
+pute these quantitites, we use the matter power spectra and
+bispectra of the hydrodynamical simulations (hydro) and the
+dark-matter only (DMO) simulations generated at varying
+initial conditions (ICs). For each of the 27 unique IC runs,
+we calculate the ratios ∆P/PDMO = (Phydro−PDMO)/PDMO
+and ∆B/BDMO = (Bhydro − BDMO)/BDMO. As the hydro-
+dynamical and the N-body simulations are run with same
+initial conditions, the ratios ∆P/PDMO and ∆B/BDMO are
+roughly independent of sample variance.
+It is clear that the amplitude of suppression of the
+small-scale matter power spectrum can be significant: sup-
+pression on the order of tens of percent is reached for all
+three simulations. It is also clear that the impact is sig-
+nificantly different between the three simulations. Even for
+the simulations in closest agreement (TNG and Astrid), the
+measurements of ∆P/PDMO disagree by more than a fac-
+tor of two at k = 5 h/Mpc. The width of the curves in
+Fig. 1 represents the standard deviation measured across
+the cosmic variance simulations, which all have the same
+parameter values but different initial conditions. For the bis-
+pectrum, we show both the equilateral and squeezed trian-
+gle configurations with cosine of angle between long sides
+fixed to µ = 0.9. Interestingly, the spread in ∆P/PDMO
+and ∆B/BDMO increases with increasing k over the range
+0.1 h/Mpc ≲ k ≲ 10 h/Mpc. This increase is driven by
+stochasticity arising from baryonic feedback. The middle
+and right panels show the impact of feedback on the bis-
+pectrum for the equilateral and squeezed triangle configura-
+tions, respectively.
+Throughout this work, we will focus on the regime
+0.3 h/Mpc < k < 10 h/Mpc. Larger scales modes are not
+present in the (25Mpc/h)3 CAMELS simulations, and in
+any case, the impact of feedback on large scales is typically
+small. Much smaller scales, on the other hand, are difficult to
+model even in the absence of baryonic feedback (Schneider
+et al. 2016). In Appendix A we show how the matter power
+suppression changes when varying the resolution and volume
+of the simulation boxes. When comparing with the original
+TNG boxes, we find that while the box sizes do not change
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+5
+the measured power suppression significantly, the resolution
+of the boxes has a non-negligible impact. This is expected
+since the physical effect of feedback mechanisms depend on
+the resolution of the simulations. Note that the errorbars
+presented in Fig. 1 will also depend on the default choice of
+feedback values assumed.
+2.3
+Measuring gas profiles around halos
+We use 3D grids of various fields (e.g. gas density and pres-
+sure) made available by the CAMELS team to extract the
+profiles of these fields around dark matter halos. The grids
+are generated with resolution of 0.05 Mpc/h. Following van
+Daalen et al. (2020), we define fb as (Mgas + Mstars)/Mtotal,
+where Mgas, Mstars and Mtotal are the mass in gas, stars and
+all the components, respectively. The gas mass is computed
+by integrating the gas number density profile around each
+halo. We typically measure fb within the spherical overden-
+sity radius r500c.4
+The SZ effect is sensitive to the electron pressure. We
+compute the electron pressure profiles, Pe, using Pe =
+2(XH + 1)/(5XH + 3)Pth, where Pth is the total thermal
+pressure, and XH = 0.76 is the primordial hydrogen frac-
+tion. Given the electron pressure profile, we measure the
+integrated SZ signal within r500c as:
+Y500c =
+σT
+mec2
+� r500c
+0
+4πr2 Pe(r) dr,
+(1)
+where, σT is the Thomson scattering cross-section, me is the
+electron mass and c is the speed of light.
+We normalize the SZ observables by the self-similar ex-
+pectation (Battaglia et al. 2012b),5
+Y SS = 131.7h−1
+70
+�
+M500c
+1015h−1
+70 M⊙
+�5/3 Ωb
+0.043
+0.25
+Ωm kpc2,
+(2)
+where, M200c is mass inside r200c and h70 = h/0.7. This cal-
+culation, which scales as M 5/3, assumes hydrostatic equilib-
+rium and that the baryon fraction is equal to cosmic bary-
+onic fraction. Hence deviations from this self-similar scaling
+provide a probe of the effects of baryonic feedback. Our final
+SZ observable is defined as Y500c/Y SS. On the other hand,
+the amplitude of the pressure profile approximately scales
+as M 2/3. Therefore, when considering the pressure profile
+as the observable, we factor out a M 2/3 scaling.
+3
+RESULTS I: SIMULATIONS
+3.1
+Inferring feedback parameters from fb and y
+We first consider how the halo Y signal can be used to con-
+strain the parameters describing the subgrid physics mod-
+els. This question has been previously investigated using the
+CAMELS simulations by Moser et al. (2022) and Wadekar
+4 We define spherical overdensity radius (r∆c, where ∆
+=
+200, 500) and overdensity mass (M∆c) such that the mean density
+within r∆ is ∆ times the critical density ρcrit, M∆ = ∆ 4
+3 πr3
+∆ρcrit.
+5 Note that we use spherical overdensity mass corresponding to
+∆ = 500 and hence adjust the coefficients accordingly, while keep-
+ing other approximations used in their derivations as the same.
+et al. (2022). The rest of our analysis will focus on constrain-
+ing changes to the power spectrum and bispectrum, and our
+intention here is mainly to provide a basis of comparison for
+those results.
+Similar to Wadekar et al. (2022), we treat the mean
+log(Y500c/M 5/3) value of all the halos in two mass bins
+(1012 < M(M⊙/h) < 5 × 1012 and 5 × 1012 < M(M⊙/h) <
+1014) as our observable; we refer to this observable as ⃗d.
+In this section, we restrict our analysis to only the TNG
+simulations. Here and throughout our investigations with
+CAMELS we ignore the contributions of measurement un-
+certainty since our intention is mainly to assess the infor-
+mation content of the SZ signals. We therefore use the CV
+simulations to determine the covariance, C, of the ⃗d. Note
+that the level of cosmic variance will depend on the volume
+probed, and can be quite large for the CAMELS simulations.
+Given this covariance, we use the Fisher matrix formalism
+to forecast the precision with which the feedback and cos-
+mological parameters can be constrained.
+The Fisher matrix, Fij, is given by
+Fij = ∂ ⃗dT
+∂θi C−1 ∂ ⃗d
+∂θi ,
+(3)
+where θi refers to the ith parameter value. Calculation of
+the derivatives ∂ ⃗d/∂θi is complicated by the large amount
+of stochasticity between the CAMELS simulations. To per-
+form the derivative calculation, we use a radial basis function
+interpolation method based on Moser et al. (2022); Cromer
+et al. (2022). We show an example of the derivative calcu-
+lation in Appendix B. We additionally assume a Gaussian
+prior on parameter p with σ(ln p) = 1 for the feedback pa-
+rameters and σ(p) = 1 for the cosmological parameters. The
+forecast parameter covariance matrix, Cp, is then related to
+the Fisher matrix by Cp = F−1.
+The parameter constraints corresponding to our calcu-
+lated Fisher matrix are shown in Fig. 2. We show results only
+for Ωm, ASN1 and AAGN2, but additionally marginalize over
+σ8, ASN2 and AAGN1. The degeneracy directions seen in our
+results are consistent with those in Wadekar et al. (2022).
+We we find a weaker constraint on AAGN2, likely owing to
+the large sample variance contribution to our calculation.
+It is clear from Fig. 2 that the marginalized constraints
+on the feedback parameters are weak. If information about
+Ωm is not used, we effectively have no information about
+the feedback parameters. Even when Ωm is fixed, the con-
+straints on the feedback parameters are not very precise.
+This finding is consistent with Wadekar et al. (2022), for
+which measurement uncertainty was the main source of vari-
+ance rather than sample variance. Part of the reason for the
+poor constraints is the degeneracy between the AGN and SN
+parameters. As we show below, the impacts of SN and AGN
+feedback can have opposite impacts on the Y signal; more-
+over, even AAGN1 and AAGN2 can have opposite impacts on
+Y . These degeneracies, as well as degeneracies with cosmo-
+logical parameters like Ωm, make it difficult to extract tight
+constraints on the feedback parameters from measurements
+of Y . However, for the purposes of cosmology, we are ul-
+timately most interested in the impact of feedback on the
+matter distribution, and not the values of the feedback pa-
+rameters themselves. These considerations motivate us to
+instead explore direct inference of changes to the statistics
+MNRAS 000, 1–16 (0000)
+
+6
+Pandey et al.
+0
+1
+Ωm
+−2
+0
+2
+log(AAGN2)
+−2
+0
+2
+log(ASN1)
+−2
+0
+2
+log(ASN1)
+−2
+0
+2
+log(AAGN2)
+Free Ωm and σ8
+Fixed Ωm and σ8
+Figure 2. Forecast constraints on the feedback parameters when
+log Y500c/Y SS in two halo mass bins is treated as the observable.
+Even when the cosmological model is fixed (red contours), the
+AGN parameters (e.g. AAGN2) remain effectively unconstrained
+(note that we impose a Gaussian prior with σ(ln p) = 1 on all feed-
+back parameters, p). When the cosmological model is free (blue
+contours), all feedback parameters are unconstrained. We assume
+that the only contribution to the variance of the observable is
+sample variance coming from the finite volume of the CAMELS
+simulations.
+of the matter distribution from the Y observables. This will
+be the focus of the rest of the paper.
+3.2
+fb and y as probes of baryonic effects on the
+matter power spectrum
+As discussed above, van Daalen et al. (2020) observed a tight
+correlation between suppression of the matter power spec-
+trum and the baryon fraction, fb, in halos with 6×1013M⊙ ≲
+M500c ≲ 1014 M⊙. That relation was found to hold regard-
+less of the details of the feedback implementation, suggest-
+ing that by measuring fb, one could robustly infer the im-
+pact of baryonic feedback on the power spectrum. We be-
+gin by investigating the connection between matter power
+spectrum suppression and integrated tSZ parameter in low-
+mass, M ∼ 1013 M⊙, halos to test if similar correlation ex-
+ists (c.f. Delgado et al. (2023) for a similar figure between fb
+and ∆P/PDMO). We also consider a wider range of feedback
+models than van Daalen et al. (2020), including the SIMBA
+and Astrid models.
+Fig. 3 shows the impact of cosmological and feed-
+back parameters on the relationship between the power
+spectrum suppression (∆P/PDMO) and the ratio Y500c/Y SS
+for the SIMBA simulations. Each point corresponds to a
+single simulation, taking the average over all halos with
+1013 < M(M⊙/h) < 1014 when computing Y500c/Y SS. Note
+that since the halo mass function rapidly declines at high
+masses, the average will be dominated by the low mass ha-
+los. We observe that the largest suppression (i.e. more nega-
+tive ∆P/PDMO) occurs when AAGN2 is large. This is caused
+by powerful AGN jet-mode feedback ejecting gas from halos,
+−0.4
+−0.2
+0.0
+∆P/PDMO
+−0.4
+−0.2
+0.0
+∆P/PDMO
+−0.4
+−0.2
+0.0
+∆P/PDMO
+−0.4
+−0.2
+0.0
+∆P/PDMO
+−0.4
+−0.2
+0.0
+∆P/PDMO
+0.0
+0.2
+0.4
+0.6
+0.8
+Y500c/YSS
+−0.4
+−0.2
+0.0
+∆P/PDMO
+0.2
+0.3
+0.4
+Ωm
+0.7
+0.8
+0.9
+σ8
+1
+2
+3
+ASN1
+1
+2
+3
+AAGN1
+0.75
+1.00
+1.25
+1.50
+1.75
+ASN2
+0.75
+1.00
+1.25
+1.50
+1.75
+AAGN2
+Figure 3. We show the relation between matter power suppres-
+sion at k = 2h/Mpc and the integrated tSZ signal, Y500c/Y SS,
+of halos in the mass range 1013 < M (M⊙/h) < 1014 in the
+SIMBA simulation suite. In each of six panels, the points are col-
+ored corresponding to the parameter value given in the associated
+colorbar.
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+7
+leading to a significant reduction in the matter power spec-
+trum, as described by e.g. van Daalen et al. (2020); Borrow
+et al. (2020); Gebhardt et al. (2023). For SIMBA, the pa-
+rameter AAGN2 controls the velocity of the ejected gas, with
+higher velocities (i.e. higher AAGN2) leading to gas ejected
+to larger distances. On the other hand, when ASN2 is large,
+∆P/PDMO is small. This is because efficient supernovae feed-
+back prevents the formation of massive galaxies which host
+AGN and hences reduces the strength of the AGN feedback.
+The parameter AAGN1, on the other hand, controls the radia-
+tive quasar mode of feedback, which has slower gas outflows
+and thus a smaller impact on the matter distribution.
+It is also clear from Fig. 3 that increasing Ωm reduces
+|∆P/PDMO|, relatively independently of the other parame-
+ters. By increasing Ωm, the ratio Ωb/Ωm decreases, meaning
+that halos of a given mass have fewer baryons, and the im-
+pact of feedback is therefore reduced. We propose a very
+simple toy model for this effect in §3.3.
+The impact of σ8 in Fig. 3 is less clear. For halos in
+the mass range shown, we find that increasing σ8 leads to a
+roughly monotonic decrease in Y500c (and fb), presumably
+because higher σ8 means that there are more halos amongst
+which the same amount of baryons must be distributed. This
+effect would not occur for cluster-scale halos because these
+are rare and large enough to gravitationally dominate their
+local environments, giving them fb ∼ Ωb/Ωm, regardless of
+σ8. In any case, no clear trend with σ8 is seen in Fig. 3
+because σ8 does not correlate strongly with ∆P/PDMO.
+Fig. 4 shows the relationship between ∆P/PDMO at
+k = 2 h/Mpc and fb or Y500c in different halo mass bins
+and for different amounts of feedback, colored by the value
+of AAGN2. As in Fig. 3, each point represents an average
+over all halos in the indicated mass range for a particular
+CAMELS simulation (i.e. at fixed values of cosmological and
+feedback parameters). Note that the meaning of AAGN2 is
+not exactly the same across the different feedback models,
+as noted in §2. For TNG and SIMBA we expect increasing
+AAGN2 to lead to stronger AGN feedback driving more gas
+out of the halos, leading to more power suppression with-
+out strongly regulating the growth of black holes. However,
+for Astrid, increasing AAGN2 parameter would more strongly
+regulate and suppress the black hole growth in the box since
+controls the efficiency of thermal mode of AGN feedback
+(Ni et al. 2022). This drastically reduces the number of high
+mass black holes and hence effectively reducing the amount
+of feedback that can push the gas out of the halos, leading
+to less matter power suppression. We see this difference re-
+flected in Fig. 4 where for the Astrid simulations the points
+corresponding to high AAGN2, result in ∆P/PDMO ∼ 0, in
+contrast to TNG and SIMBA suite of simulations.
+For the highest mass bin (1013 < M(M⊙/h) < 1014,
+rightmost column of Fig. 4) our results are in agreement with
+van Daalen et al. (2020): we find that there is a robust corre-
+lation between between fb/(Ωb/Ωm) and the matter power
+suppression (also see Delgado et al. (2023)). This relation is
+roughly consistent across different feedback subgrid models,
+although the different models appear to populate different
+parts of this relation. Moreover, varying AAGN2 appears to
+move points along this relation, rather than broadening the
+relation. This is in contrast to Ωm, which as shown in Fig. 3,
+tends to move simulations in the direction orthogonal to the
+narrow Y500c-∆P/PDMO locus. For this reason, and given
+current constraints on Ωm, we restrict Fig. 4 to simulations
+with 0.2 < Ωm < 0.4. The dashed curves shown in Fig. 4
+correspond to the toy model discussed in §3.3.
+At low halo mass, the relation between fb/(Ωb/Ωm)
+and ∆P/PDMO appears similar to that for the high-mass
+bin, although it is somewhat flatter at high fb, and some-
+what steeper at low fb. Again the results are fairly consistent
+across the different feedback prescriptions, although points
+with high fb/(Ωb/Ωm) are largely absent for SIMBA. This
+is because the feedback mechanisms are highly efficient in
+SIMBA, driving the gas out of their parent halos.
+The relationships between Y and ∆P/PDMO appear
+quite similar to those between ∆P/PDMO and fb/(Ωb/Ωm).
+This is not too surprising because Y is sensitive to the gas
+density, which dominates fb/(Ωb/Ωm). However, Y is also
+sensitive to the gas temperature. Our results suggest that
+variations in gas temperature are not significantly impact-
+ing the Y500c-∆P/PDMO relation. The possibility of using
+the tSZ signal to infer the impact of feedback on the matter
+distribution rather than fb/(Ωb/Ωm) is therefore appealing.
+This will be the focus of the remainder of the paper.
+Fig. 5 shows the same quantities as Fig. 4, but now for
+a fixed halo mass range (1013 < M/(M⊙/h) < 1014), fixed
+subgrid prescription (TNG), and varying values of k. We
+find roughly similar results when using the different sub-
+grid physics prescriptions. At low k, we find that there is
+a regime at high fb/(Ωb/Ωm) for which ∆P/PDMO changes
+negligibly. It is only when fb/(Ωb/Ωm) becomes very low
+that ∆P/PDMO begins to change. On the other hand, at
+high k, there is a near-linear relation between fb/(Ωb/Ωm)
+and ∆P/PDMO.
+3.3
+A toy model for power suppression
+We now describe a simple model for the effects of feedback
+on the relation between fb or Y and ∆P/PDMO that ex-
+plains some of the features seen in Figs. 3, 4 and 5. We
+assume in this model that it is removal of gas from halos by
+AGN feedback that is responsible for changes to the matter
+power spectrum. SN feedback, on the other hand, can pre-
+vent gas from accreting onto the SMBH and therefore reduce
+the impact of AGN feedback (Angl´es-Alc´azar et al. 2017c;
+Habouzit et al. 2017). This scenario is consistent with the
+fact that at high SN feedback, we see that ∆P/PDMO goes
+to zero (second panel from the bottom in Fig. 3). Stellar
+feedback can also prevent gas from accreting onto low-mass
+halos (Pandya et al. 2020, 2021). In some sense, the dis-
+tinction between gas that is ejected by AGN and gas that is
+prevented from accreting onto halos by stellar feedback does
+not matter for our simple model. Rather, all that matters
+is that some amount of gas that would otherwise be in the
+halo is instead outside of the halo as a result of feedback
+effects, and it is this gas which is responsible for changes to
+the matter power spectrum.
+We identify three relevant scales: (1) the halo radius,
+Rh, (2) the distance to which gas is ejected by the AGN,
+Rej, and (3) the scale at which the power spectrum is mea-
+sured, 2π/k. If Rej ≪ 2π/k, then there will be no impact
+on ∆P at k: this corresponds to a rearrangement of the
+matter distribution on scales well below where we measure
+the power spectrum. If, on the other hand, Rej ≪ Rh, then
+there will be no impact on fb or Y , since the gas is not
+MNRAS 000, 1–16 (0000)
+
+8
+Pandey et al.
+−0.4
+−0.2
+0.0
+∆P/PDMO
+Illustris-TNG
+5 × 1012 < M(M⊙/h) < 1013
+Illustris-TNG
+5 × 1012 < M(M⊙/h) < 1013
+Illustris-TNG
+1013 < M(M⊙/h) < 1014
+Illustris-TNG
+1013 < M(M⊙/h) < 1014
+−0.4
+−0.2
+0.0
+∆P/PDMO
+SIMBA
+5 × 1012 < M(M⊙/h) < 1013
+SIMBA
+5 × 1012 < M(M⊙/h) < 1013
+SIMBA
+1013 < M(M⊙/h) < 1014
+SIMBA
+1013 < M(M⊙/h) < 1014
+0.00
+0.25
+0.50
+0.75
+Y500c/YSS
+−0.4
+−0.2
+0.0
+∆P/PDMO
+Astrid
+5 × 1012 < M(M⊙/h) < 1013
+0.0
+0.5
+1.0
+fb/(Ωb/Ωm)
+Astrid
+5 × 1012 < M(M⊙/h) < 1013
+0.0
+0.5
+1.0
+Y500c/YSS
+Astrid
+1013 < M(M⊙/h) < 1014
+0.0
+0.5
+1.0
+fb/(Ωb/Ωm)
+Astrid
+1013 < M(M⊙/h) < 1014
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+AAGN2
+Figure 4. Impact of baryonic physics on the matter power spectrum at k = 2h/Mpc for the TNG, SIMBA and Astrid simulations
+(top, middle, and bottom rows). Each point corresponds to an average across halos in the indicated mass ranges in a different CAMELS
+simulation. We restrict the figure to simulations that have 0.2 < Ωm < 0.4. The dashed curves illustrate the behavior of the model
+described in §3.3 when the gas ejection distance is large compared to the halo radius and 2π/k.
+0.5
+1.0
+fb/(Ωb/Ωm)
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+∆P/PDMO
+k = 0.6 h/Mpc
+0.5
+1.0
+fb/(Ωb/Ωm)
+k = 1.0 h/Mpc
+0.5
+1.0
+fb/(Ωb/Ωm)
+k = 5.0 h/Mpc
+0.5
+1.0
+fb/(Ωb/Ωm)
+k = 10.0 h/Mpc
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+1.8
+AAGN2
+Figure 5. Similar to Fig. 4, but for different values of k. For simplicity, we show only the TNG simulations for halos in the mass range
+1013 < M(M⊙/h) < 1014. The dashed curves illustrate the behavior of the model described in §3.3 in the regime that the radius to
+which gas is ejected by AGN is larger than the halo radius, and larger than 2π/k. As expected, this model performs best in the limit of
+high k and large halo mass.
+ejected out of the halo. We therefore consider four regimes
+defined by the relative amplitudes of Rh, Rej, and 2π/k, as
+described below. Note that there is not a one-to-one corre-
+spondence between physical scale in configuration space and
+2π/k; therefore, the inequalities below should be considered
+as approximate. The four regimes are:
+• Regime 1: Rej < Rh and Rej < 2π/k. In this regime,
+changes to the feedback parameters have no impact on fb or
+∆P.
+• Regime 2: Rej > Rh and Rej < 2π/k. In this regime,
+changes to the feedback parameters result in movement
+along the fb or Y axis without changing ∆P. Gas is be-
+ing removed from the halo, but the resultant changes to the
+matter distribution are below the scale at which we measure
+the power spectrum. Note that Regime 2 cannot occur when
+Rh > 2π/k (i.e. high-mass halos at large k).
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+9
+• Regime 3: Rej > Rh and Rej > 2π/k. In this regime, chang-
+ing the feedback amplitude directly changes the amount of
+gas ejected from halos as well as ∆P/PDMO.
+• Regime 4: Rej < Rh and Rej > 2π/k. In this regime, gas is
+not ejected out of the halo, so fb and Y should not change.
+In principle, the redistribution of gas within the halo could
+lead to changes in ∆P/PDMO. However, as we discuss below,
+this does not seem to happen in practice.
+Let us now consider the behavior of ∆P/PDMO and fb
+or Y as the feedback parameters are varied in Regime 3. A
+halo of mass M is associated with an overdensity δm in the
+absence of feedback, which is changed to δ′
+m due to ejec-
+tion of baryons as a result of feedback. In Regime 3, some
+amount of gas, Mej, is completely removed from the halo.
+This changes the size of the overdensity associated with the
+halo to
+δ′
+m
+δm
+=
+1 − Mej
+M .
+(4)
+The change to the power spectrum is then
+∆P
+PDMO
+∼
+�δ′
+m
+δm
+�2
+− 1 ≈ −2Mej
+M ,
+(5)
+where we have assumed that Mej is small compared to M.
+We have ignored the k dependence here, but in Regime 3,
+the ejection radius is larger than the scale of interest, so the
+calculated ∆P/PDMO should apply across a range of k in
+this regime.
+The ejected gas mass can be related to the gas mass
+in the absence of feedback. We write the gas mass in the
+absence of feedback as fc(Ωb/Ωm)M, where fc encapsulates
+non-feedback processes that result in the halo having less
+than the cosmic baryon fraction. We then have
+Mej
+=
+fc(Ωb/Ωm)M − fbM − M0,
+(6)
+where M0 is the mass that has been removed from the
+gaseous halo, but that does not change the power spectrum,
+e.g. the conversion of gas to stars. Substituting into Eq. 5,
+we have
+∆P
+PDMO = −2fcΩb
+Ωm
+�
+1 − fbΩm
+fcΩb − ΩmM0
+fcΩbM
+�
+.
+(7)
+In other words, for Regime 3, we find a linear relation be-
+tween ∆P/PDMO and fbΩm/Ωb. For high mass halos, we
+should have fc ≈ 1 and M0/M ≈ 0. In this limit, the rela-
+tionship between fb and ∆P/PDMO becomes
+∆P
+PDMO = −2 Ωb
+Ωm
+�
+1 − fbΩm
+Ωb
+�
+,
+(8)
+which
+is
+linear
+between
+endpoints
+at
+(∆P/PDMO, fbΩm/Ωb)
+=
+(−2Ωb/Ωm, 0)
+and
+(∆P/PDMO, fbΩm/Ωb)
+=
+(0, 1). We show this relation
+as the dashed line in the fb columns of Figs. 4 and Fig. 5.
+We can repeat the above argument for Y . Unlike the
+case with fb, processes other than the removal of gas may
+reduce Y ; these include, e.g., changes to the gas temperature
+in the absence of AGN feedback, or nonthermal pressure sup-
+port. We account for these with a term Y0, defined such that
+when Mej = M0 = 0, we have Y + Y0 = fc(Ωb/Ωm)MT/α,
+where we have assumed constant gas temperature, T, and
+α is a dimensionful constant of proportionality. We ignore
+detailed modeling of variation in the temperature of the gas
+due to feedback and departures from hydro-static equilib-
+rium (Ostriker et al. 2005). We then have
+α(Y + Y0)
+T
+= fc(Ωb/Ωm)M − Mej − M0.
+(9)
+Substituting the above equation into Eq. 5 we have
+∆P
+PDMO
+=
+−2fcΩb
+Ωm
+�
+1 − α(Y + Y0)Ωm
+fcTMΩb
+− ΩmM0
+fcΩbM
+�
+.
+(10)
+Following Eq. 2, we define the self-similar value of Y , Y SS,
+via
+αY SS/T = (Ωb/Ωm)M,
+(11)
+leading to
+∆P
+PDMO
+=
+−2fcΩb
+Ωm
+�
+1 − (Y + Y0)
+fcY SS
+− ΩmM0
+fcΩbM
+�
+. (12)
+Again taking the limit that fc ≈ 1 and M0/M ≈ 0, we have
+∆P
+PDMO
+=
+−2 Ωb
+Ωm
+�
+1 − (Y + Y0)
+Y SS
+�
+.
+(13)
+Thus, we see that in Regime 3, the relation between Y/Y SS
+and ∆P/PDMO is linear. The Y/Y SS columns of Figs. 4 show
+this relationship, assuming Y0 = 0.
+In summary, we interpret the results of Figs. 4 and 5 in
+the following way. Starting at low feedback amplitude, we
+are initially in Regime 1. In this regime, the simulations clus-
+ter around fbfcΩm/Ωb ≈ 1 (or Y ≈ Y0) and ∆P/PDMO ≈ 0
+since changing the feedback parameters in this regime does
+not impact fb or ∆P/PDMO. For high mass halos, we have
+fc ≈ 1 and Y0 ≈ 0 (although SIMBA appears to have Y0 > 0,
+even at high mass); for low mass halos, fc < 1 and Y0 > 0.
+As we increase the AGN feedback amplitude, the behavior
+is different depending on halo mass and k:
+• For low halo masses or low k, increasing the AGN feed-
+back amplitude leads the simulations into Regime 2. Increas-
+ing the feedback amplitude in this regime moves points to
+lower Y/Y SS (or fbΩm/Ωb) without significantly impacting
+∆P/PDMO. Eventually, when the feedback amplitude is suf-
+ficiently strong, these halos enter Regime 3, and we see a
+roughly linear decline in ∆P/PDMO with decreasing Y/Y SS
+(or fbΩm/Ωb), as discussed above.
+• For high mass halos and high k, we never enter Regime 2
+since it is not possible to have Rej > Rh and Rej < 2π/k
+when Rh is very large. In this case, we eventually enter
+Regime 3, leading to a linear trend of decreasing ∆P/PDMO
+with decreasing Y/Y SS or fbΩm/Ωb, as predicted by the
+above discussion. This behavior is especially clear in Fig. 5:
+at high k, the trend closely follows the predicted linear rela-
+tion. At low k, on the other hand, we see a more prominent
+Regime 2 region. The transition between these two regimes
+is expected to occur when k ∼ 2π/Rh, which is roughly
+5 h−1Mpc for the halo mass regime shown in the figure. This
+expectation is roughly confirmed in the figure.
+Interestingly, we never see Regime 4 behavior: when the halo
+mass is large and k is large, we do not see rapid changes
+in ∆P/PDMO with little change to fb and Y . This could
+be because this regime corresponds to movement of the gas
+entirely within the halo. If the gas has time to re-equilibrate,
+it makes sense that we would see little change to ∆P/PDMO
+in this regime.
+MNRAS 000, 1–16 (0000)
+
+10
+Pandey et al.
+3.4
+Predicting the power spectrum suppression
+from the halo observables
+While the toy model described above roughly captures the
+trends between Y (or fb) and ∆P/PDMO, it of course does
+not capture all of the physics associated with feedback. It
+is also clear that there is significant scatter in the relation-
+ships between observable quantities and ∆P. It is possible
+that this scatter is reduced in some higher dimensional space
+that includes more observables. To address both of these
+issues, we now train statistical models to learn the rela-
+tionships between observable quantities and ∆P/PDMO. We
+will focus on results obtained with random forest regression
+(Breiman 2001). We have also tried using neural networks to
+infer these relationships, but have not found any significant
+improvement with respect to the random forest results, pre-
+sumably because the space is low-dimensional (i.e. we con-
+sider at most about five observable quantities at a time). We
+leave a detailed comparison with other decision tree based
+approaches, such as gradient boosted trees (Friedman 2001)
+to a future study.
+We train a random forest model to go from observable
+quantities (e.g. fb/(Ωb/Ωm) and Y500c/Y SS) to a prediction
+for ∆P/PDMO at multiple k values. The random forest model
+uses 100 trees with a maxdepth = 10.6 In this section we an-
+alyze the halos in the mass bin 5 × 1012 < Mhalo(M⊙/h) <
+1014 but we also show the results for halos with lower masses
+in Appendix D. We also consider supplying the values of Ωm
+as input to the random forest, since it can be constrained
+precisely through other observations (e.g. primary CMB ob-
+servations), and as we showed in §3.2, the cosmological pa-
+rameters can impact the observables.7
+Ultimately, we are interested in making predictions for
+∆P/PDMO using observable quantities. However, the sam-
+ple variance in the CAMELS simulations limits the precision
+with which we can measure ∆P/PDMO. It is not possible
+to predict ∆P/PDMO to better than this precision. We will
+therefore normalize the uncertainties in the RF predictions
+by the cosmic variance error. In order to obtain the un-
+certainty in the predictions, we randomly split the data into
+70% training and 30% test set. After training the RF regres-
+sor using the training set and a given observable, we make
+compute the 16th and 84th percentile of the distribution of
+prediction errors evaluated on the test set. This constitutes
+our assessment of prediction uncertainty.
+Fig. 6 shows the accuracy of the RF predictions for
+∆P/PDMO when trained on stacked fb (for halos in 5 ×
+1012 < Mhalo(M⊙/h) < 1014) and Ωm, normalized to the
+sample variance error in ∆P/PDMO. As we will show later
+in this section, this combination of inputs results in pre-
+6 We use a publicly available code: https://scikit-learn.
+org/stable/modules/generated/sklearn.ensemble.
+RandomForestRegressor.html.
+We
+also
+verified
+that
+our
+conclusions are robust to changing the settings of the random
+forest.
+7 One might worry that using cosmological information to con-
+strain ∆P/PDMO defeats the whole purpose of constraining
+∆P/PDMO in order to improve cosmological constraints. How-
+ever, observations, such as those of CMB primary anisotropies,
+already provide precise constraints on the matter density with-
+out using information in the small-scale matter distribution.
+cise constraints on the matter power suppression. Specifi-
+cally to obtain the constraints, after training the RF regres-
+sor on the train simulations, we predict the ∆P/PDMO on
+test simulation boxes at four scales. Thereafter, we create
+a histogram of the difference between truth and predicted
+∆P/PDMO, normalized by the variance obtained from the
+CV set of simulations, for each respective suite of simula-
+tions (see Fig. 1). In Fig. 6, each errorbar corresponds to the
+16th and 84th percentile from this histogram and the marker
+corresponds to its peak. We show the results of training and
+testing on a single simulation suite, and also the results of
+training/testing across different simulation suites. It is clear
+that when training and testing on the same simulation suite,
+the RF learns a model that comes close to the best possi-
+ble uncertainty on ∆P/PDMO (i.e. cosmic variance). When
+training on one or two simulation suites and testing another,
+however, the predictions show bias at low k. This suggests
+that the model learned from one simulation does not gen-
+eralize very well to another in this regime. This result is
+somewhat different from the findings of van Daalen et al.
+(2020), where it was found that the relationship between fb
+and ∆P/PDMO did generalize to different simulations. This
+difference may result from the fact that we are considering
+a wider range of feedback prescriptions than in van Daalen
+et al. (2020), as well as considering significant variations in
+cosmological parameters.
+Fig. 6 also shows the results of testing and training on
+all three simulations (black points with errorbars). Encour-
+agingly, we find that in this case, the predictions are of
+comparable accuracy to those obtained from training and
+predicting on the same simulation suite. This suggests that
+there is a general relationship across all feedback models
+that can be learned to go from Ωm and fb to ∆P/PDMO.
+Henceforth, we will show results trained on all simulation
+suites and tested on all simulations suites. Of course, this
+result does not imply that our results will generalize to some
+completely different feedback prescription.
+In Fig. 7 we show the results of training the random
+forest on different combinations of fb, Y500c and Ωm. Con-
+sistent with the findings of van Daalen et al. (2020), we
+find that fb/(Ωb/Ωm) results in robust constraints on the
+matter power suppression (blue points with errors). These
+constraints come close to the cosmic variance limit across a
+wide range of k.
+We additionally find that providing fb and Ωm as sepa-
+rate inputs to the RF improves the precision of the predic-
+tions for ∆P/PDMO relative to using just the combination
+fb/(Ωb/Ωm), with the largest improvement coming at small
+scales. This is not surprising given the predictions of our
+simple model, for which it is clear that ∆P/PDMO can be
+impacted by both Ωm and fb/(Ωb/Ωb) independently. Sim-
+ilarly, it is clear from Fig. 3 that changing Ωm changes the
+relationship between ∆P/PDMO and the halo gas-derived
+quantities (like Y and fb).
+We next consider a model trained on Y500c/Y SS (orange
+points in Fig. 7). This model yields reasonable predictions
+for ∆P/PDMO, although not quite as good as the model
+trained on fb/(Ωb/Ωm). The Y/Y SS model yields somewhat
+larger errorbars, and the distribution of ∆P/PDMO predic-
+tions is highly asymmetric. When we train the RF model
+jointly on Y500c/Y SS and Ωm (green points), we find that
+the predictions improve considerably, particularly at high k.
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+11
+100
+101
+k (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆P/PDMO prediction relative to CV (σ)
+CV
+Train:TNG, Test:TNG
+Train:SIMBA, Test:SIMBA
+Train:Astrid, Test:Astrid
+Train: TNG, SIMBA
+Test: Astrid
+Train: TNG, Astrid
+Test: SIMBA
+Train: SIMBA, Astrid
+Test: TNG
+Train: TNG, SIMBA, Astrid
+Test: TNG, SIMBA, Astrid
+Figure 6. We show the results of the random forest regressor predictions for the baryonic power suppression, represented by ∆P/PDMO,
+across the LH suite of simulations at four different scales k using the subgrid physics models for TNG, SIMBA, and Astrid. The model
+was trained using the average fb of halos with masses between 5 × 1012 < M(M⊙/h) < 1014 and the cosmological parameter Ωm. The
+errorbars indicate the uncertainty in the predictions normalized by the uncertainity in the CV suite at each scale, showing the 16-84
+percentile error on the test set. The gray band represents the expected 1σ error from the CV suite. The model performs well when the
+training and test simulations are the same. When tested on an independent simulation, it remains robust at high k but becomes biased
+at low k. The results presented in the remainder of the paper are based on training the model on all three simulations. The data points
+at each scale are staggered for clarity.
+100
+101
+k (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆P/PDMO prediction relative to CV (σ)
+CV
+fb/(Ωb/Ωm)
+fb, Ωm
+Y500c/YSS
+Y500c/YSS, Ωm
+fb, Y500c/YSS, Ωm
+Figure 7. Similar to Fig. 6, but showing results when training the RF model on different observables from all three simulations (TNG,
+SIMBA and Astrid) to predict ∆P/PDMO of a random subset of the the three simulations not used in training. We find that jointly
+training on the deviation of the integrated SZ profile from the self-similar expectation, Y500c/Y SS and Ωm results in inference of power
+suppression that is comparable to cosmic variance errors, with small improvements when additionally adding the baryon fraction (fb) of
+halos in the above mass range.
+In this case, the predictions are typically symmetric around
+the true ∆P/PDMO, have smaller uncertainty compared to
+the model trained on fb/(Ωb/Ωm), and comparable uncer-
+tainty to the model trained on {fb/(Ωb/Ωm),Ωm}. We thus
+conclude that when combined with matter density informa-
+tion, Y/Y SS provides a powerful probe of baryonic effects
+on the matter power spectrum.
+Above we have considered the integrated tSZ signal
+from halos, Y500c. Measurements in data, however, can po-
+tentially probe the tSZ profiles rather than only the inte-
+grated tSZ signal (although the instrumental resolution may
+limit the extent to which this is possible). In Fig. 8 we con-
+sider RF models trained on the stacking the full electron
+density and pressure profiles in the halo mass range instead
+of just the integrated quantities. The electron pressure and
+number density profiles are measured in eight logarithmi-
+MNRAS 000, 1–16 (0000)
+
+12
+Pandey et al.
+100
+101
+k (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆P/PDMO prediction relative to CV (σ)
+CV
+Pe(r)/PSS
+e
+Pe(r)/PSS
+e , Ωm
+Pe(r)/PSS
+e , ne(r), Ωm
+Pe(r)/PSS
+e , Ωm
+Low+High mass bins
+Figure 8. Same as Fig. 7 but showing results from using the full pressure profile, Pe(r), and electron number density profiles, ne(r),
+instead of the integrated quantities. We again find that with pressure profile and Ωm information we can recover robust and precise
+constraints on the matter power suppression.
+cally spaced bins between 0.1 < r/r200c < 1. We find that
+while the ratio Pe(r)/P SS results in robust predictions for
+∆P/PDMO, simultaneously providing Ωm makes the predic-
+tions more precise. Similar to the integrated profile case, we
+find that additionally providing the electron density profile
+information only marginally improves the constraints. We
+also show the results when jointly using the measured pres-
+sure profiles for both the low and high mass halos to infer
+the matter power suppression. We find that this leads to
+only a marginal improvements in the constraints.
+Note that we have input the 3D pressure and electron
+density profiles in this case. Even though observed SZ maps
+are projected quantities, we can infer the 3D pressure profiles
+from the model used to analyze the projected correlations.
+3.5
+Predicting baryonic effects on the bispectrum
+with fb and the electron pressure
+In Fig. 9, we repeat our analysis from above to make
+predictions for baryonic effects on the matter bispectrum,
+∆B(k)/B(k). Similar to the matter power spectrum, we
+train and test our model on a combination of the three
+simulations. We train and test on equilateral triangle bis-
+pectrum configurations with different scales k. We again see
+that information about the electron pressure and Ωm results
+in precise and unbiased constraints on the impact of bary-
+onic physics on the bispectrum. The constraints improve as
+we go to the small scales. In Appendix E we show similar
+methodology applied to squeezed bispectrum configurations.
+However, there are several important caveats to these
+results. The bispectrum is sensitive to high-mass (M >
+5 × 1013M⊙/h) halos (Foreman et al. 2020) which are miss-
+ing from the CAMELS simulations. Consequently, our mea-
+surements of baryonic effects on the bispectrum can be bi-
+ased when using CAMELS. The simulation resolution can
+also impact the bispectrum significantly. A future analysis
+with larger volume sims at high resolution could use the
+methodology introduced here to obtain more robust results.
+Finally, there would is likely to be covariance between the
+power spectrum suppression and baryonic effects on the bis-
+pectrum, as they both stem from same underlying physics.
+We defer a complete exploration of these effects to future
+work.
+4
+RESULTS II: ACTXDES MEASUREMENTS
+AND FORECAST
+Our analysis above has resulted in a statistical model (i.e.
+a random forest regressor) that predicts the matter power
+suppression ∆P/PDMO given values of Y500c for low-mass
+halos. This model is robust to significant variations in the
+feedback prescription, at least across the SIMBA, TNG and
+Astrid models. We now apply this model to constraints on
+Y500c coming from the cross-correlation of galaxy lensing
+shear with tSZ maps measured using Dark Energy Survey
+(DES) and Atacama Cosmology Telescope (ACT) data.
+Gatti et al. (2022a) and Pandey et al. (2022) measured
+the cross-correlations of DES galaxy lensing with Compton y
+maps from a combination of Advanced ACT (Madhavacheril
+et al. 2020) and Planck data (Planck Collaboration et al.
+2016) over an area of 400 sq. deg. They analyze these cross-
+correlations using a halo model framework, where the pres-
+sure profile in halos was parameterized using a generalized
+Navarro-Frenk-White profile (Navarro et al. 1996; Battaglia
+et al. 2012a). This pressure profile is described using four
+free parameters, allowing for scaling with mass, redshift and
+distance from halo center. The constraints on the parame-
+terized pressure profiles can be translated directly into con-
+straints on Y500c for halos in the mass range relevant to our
+random forest models.
+We use the parameter constraints from Pandey et al.
+(2022) to generate 400 samples of the inferred 3D profiles
+of halos at z = 0 (i.e. the redshift at which the RF models
+are trained) in ten logarithmically-spaced mass bins in range
+12.7 < log10(M/h−1M⊙) < 14. We then perform the volume
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+13
+100
+101
+keq (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆Beq/Beq;DMO prediction relative to CV (σ)
+CV
+Y500c/YSS, Ωm
+Pe(r)/PSS
+e
+Pe(r)/PSS
+e , Ωm
+Pe(r)/PSS
+e , ne(r), Ωm
+Figure 9. Same as Fig. 7, but for the impact of feedback on the bispectrum in equilateral triangle configurations. We find that the
+inclusion of pressure profile information results in unbiased constraints on feedback effects on the bispectrum.
+100
+k (h/Mpc)
+−0.40
+−0.35
+−0.30
+−0.25
+−0.20
+−0.15
+−0.10
+−0.05
+0.00
+∆P/PDMO
+Chen et al 2022
+w/ cosmology prior
+Schneider et al 2022
+OWLS
+BAHAMAS
+BAHAMAS
+highAGN
+TNG-300
+DESxACT; Data
+DESIxS4; Forecast
+Figure 10. Constraints on the impact of feedback on the matter power spectrum obtained using our trained random forest model applied
+to measurements of Y500c/Y SS from the DESxACT analysis of Pandey et al. (2022) (black points with errorbars). We also show the
+expected improvements from future halo-y correlations from DESIxSO using the constraints in Pandey et al. (2020). We compare these
+to the inferred constraints obtained using cosmic shear (Chen et al. 2023) and additionally including X-ray and kSZ data (Schneider
+et al. 2022). We also compare with the results from larger simulations: OWLS (Schaye et al. 2010), BAHAMAS (McCarthy et al. 2017)
+and TNG-300 (Springel et al. 2018).
+integral of these profiles to infer the Y500c(M, z) (see Eq. 1).
+Next, we generate a halo-averaged value of Y500c/Y SS for the
+jth sample by integrating over the halo mass distribution in
+CAMELS:
+�Y500c
+Y SS
+�j
+= 1
+¯nj
+�
+dM
+� dn
+dM
+�j
+CAMELS
+Y j
+500c(M)
+Y SS
+(14)
+where ¯nj =
+�
+dM(dn/dM)j
+CAMELS and (dn/dM)j
+CAMELS are
+a randomly chosen halo mass function from the CV set of
+boxes of TNG, SIMBA or Astrid. This procedure allows us
+to incorporate the impact and uncertainties of the CAMELS
+box size on the halo mass function. Note that due to the
+small box size of CAMELS, there is a deficit of high mass
+halos and hence the functional form of the mass function
+MNRAS 000, 1–16 (0000)
+
+14
+Pandey et al.
+differs somewhat from other fitting functions in literature,
+e.g. Tinker et al. (2008).
+Fig. 10 shows the results feeding the Y500c/Y SS values
+calculated above into our trained RF model to infer the im-
+pact of baryonic feedback on the matter power spectrum
+(black points with errorbars). The RF model used is that
+trained on the TNG, SIMBA and Astrid simulations. The
+errorbars represent the 16th and 84th percentile of the recov-
+ered ∆P/PDMO distribution using the 400 samples described
+above. Note that in this inference we fix the matter density
+parameter, Ωm = 0.3, same value as used by the CAMELS
+CV simulations as we use these to estimate the halo mass
+function.
+In the same figure, we also show the constraints from
+Chen et al. (2023) and Schneider et al. (2022) obtained using
+the analysis of complementary datasets. Chen et al. (2023)
+analyze the small scale cosmic shear measurements from
+DES Year-3 data release using a baryon correction model.
+Note that in this analysis, they only use a limited range of
+cosmologies, particularly restricting to high σ8 due to the
+requirements of emulator calibration. Moreover they also
+impose cosmology constraints from the large scale analy-
+sis of the DES data. Note that unlike the procedure pre-
+sented here, their modeling and constraints are sensitive to
+the priors on σ8. Schneider et al. (2022) analyze the X-ray
+data (as presented in Giri & Schneider 2021) and kSZ data
+from ACT and SDSS (Schaan et al. 2021) and the cosmic
+shear measurement from KiDS (Asgari et al. 2021), using
+another version of baryon correction model. A joint analysis
+from these complementary dataset leads to crucial degen-
+eracy breaking in the parameters. It would be interesting
+to include the tSZ observations presented here in the same
+framework as it can potentially make the constraints more
+precise.
+Several caveats about our analysis with data are in or-
+der. First, the lensing-SZ correlation is most sensitive to
+halos in the mass range of Mhalo ≥ 1013M⊙/h. However,
+our RF model operates on halos with mass in the range
+of 5 × 1012 ≥ Mhalo ≤ 1014M⊙/h, with the limited vol-
+ume of the simulations restricting the number of halos above
+1013M⊙/h. We have attempted to account for this selection
+effect by using the halo mass function from the CV sims of
+the CAMELS simulations when calculating the stacked pro-
+file. However, using a larger volume simulation suite would
+be a better alternative (also see discussion in Appendix A).
+Moreover, the CAMELS simulation suite also fix the value
+of Ωb. There may be a non-trivial impact on the inference
+of ∆P/PDMO when varying that parameter. Note, though,
+that Ωb is tightly constrained by other cosmological obser-
+vations. Lastly, the sensitivity of the lensing-SZ correlations
+using DES galaxies is between 0.1 < z < 0.6. However, in
+this study we extrapolate those constraints to z = 0 using
+the pressure profile model of Battaglia et al. (2012a). We
+note that inference obtained at the peak sensitivity redshift
+would be a better alternative but we do not expect this to
+have a significant impact on the conclusions here.
+In order to shift the sensitivity of the data correlations
+to lower halo masses, it would be preferable to analyze the
+galaxy-SZ and halo-SZ correlations. In Pandey et al. (2020)
+we forecast the constraints on the inferred 3D pressure pro-
+file from the future halo-SZ correlations using DESI and
+CMB-S4 SZ maps for a wide range of halo masses. In Fig. 10
+we also show the expected constraints on the matter power
+suppression using the halo-SZ correlations from halos in the
+range M500c > 5 × 1012M⊙/h. We again follow the same
+methodology as described above to create a stacked normal-
+ized integrated pressure (see Eq. 14). Moreover, we also fix
+Ω = 0.3 to predict the matter power suppression. Note that
+we shift the mean value of ∆P/PDMO to the recovered value
+from BAHAMAS high-AGN simulations (McCarthy et al.
+2017). As we can see in Fig. 10, we can expect to obtain
+significantly more precise constraints from these future ob-
+servations.
+5
+CONCLUSIONS
+We have shown that the tSZ signals from low-mass halos
+contain significant information about the impacts of bary-
+onic feedback on the small-scale matter distribution. Using
+models trained on hydrodynamical simulations with a wide
+range of feedback implementations, we demonstrate that in-
+formation about baryonic effects on the power spectrum and
+bispectrum can be robustly extracted. By applying these
+same models to measurements with ACT and DES, we have
+shown that current tSZ measurements already constrain the
+impact of feedback on the matter distribution. Our results
+suggest that using simulations to learn the relationship be-
+tween halo gas observables and baryonic effects on the mat-
+ter distribution is a promising way forward for constraining
+these effects with data.
+Our main findings from our explorations with the
+CAMELS simulations are the following:
+• In agreement with van Daalen et al. (2020), we find that
+baryon fraction in halos correlates with the power spectrum
+suppression. We find that the correlation is especially robust
+at small scales.
+• We find (in agreement with Delgado et al. 2023) that there
+can be significant scatter in the relationship between baryon
+fraction and power spectrum suppression at low halo mass,
+and that the relationship varies to some degree with feed-
+back implementation. However, the bulk trends appear to
+be consistent regardless of feedback implementation.
+• We propose a simple model that qualitatively (and in some
+cases quantitatively) captures the broad features in the re-
+lationships between the impact of feedback on the power
+spectrum, ∆P/PDMO, at different values of k, and halo gas-
+related observables like fb and Y500c at different halo masses.
+• Despite significant scatter in the relations between Y500c
+and ∆P/PDMO at low halo mass, we find that simple ran-
+dom forest models yield tight and robust constraints on
+∆P/PDMO given information about Y500c in low-mass ha-
+los and Ωm.
+• Using
+the
+pressure
+profile
+instead
+of
+just
+the
+inte-
+grated Y500c signal provides additional information about
+∆P/PDMO, leading to 20-50% improvements when not us-
+ing any cosmological information. When additionally pro-
+viding the Ωm information, the improvements in constraints
+on baryonic changes to the power spectrum or bispectrum
+are modest when using the full pressure profile relative to
+integrated quantities like Y500c.
+• The pressure profiles and baryon fractions also carry infor-
+mation about baryonic effects on the bispectrum.
+MNRAS 000, 1–16 (0000)
+
+Probing feedback with the SZ
+15
+Our main results from our analysis of constraints from
+the DESxACT shear-y correlation analysis are
+• We have used the DES-ACT measurement of the shear-
+tSZ correlation from Gatti et al. (2022a) and Pandey et al.
+(2022) to infer Y500c for halos in the mass range relevant to
+our random forest models. Feeding the measured Y500c into
+these models, we have inferred the impact of baryonic effects
+on the power spectrum, as shown in Fig. 10.
+• We show that constraints on baryonic effects on the power
+spectrum will improve significantly in the future, particu-
+larly using halo catalogs from DESI and tSZ maps from
+CMB-S4.
+With data from future galaxy and CMB surveys, we
+expect constraints on the tSZ signal from halos across a
+wide mass and redshift range to improve significantly. These
+improvements will come from both the galaxy side (e.g. halos
+detected over larger areas of the sky, down to lower halo
+masses, and out to higher redshifts) and the CMB side (more
+sensitive tSZ maps over larger areas of the sky). Our forecast
+for DESI and CMB Stage 4 in Fig. 10 suggests that very
+tight constraints can be obtained on the impact of baryonic
+feedback on the matter power spectrum. We expect that
+these constraints on the impact of baryonic feedback will
+enable the extraction of more cosmological information from
+the small-scale matter distribution.
+6
+ACKNOWLEDGEMENTS
+DAA acknowledges support by NSF grants AST-2009687
+and AST-2108944, CXO grant TM2-23006X, and Simons
+Foundation award CCA-1018464.
+7
+DATA AVAILABILITY
+The TNG and SIMBA simulations used in this work are
+part of the CAMELS public data release (Villaescusa-
+Navarro et al. 2021) and are available at https://camels.
+readthedocs.io/en/latest/. The Astrid simulations used
+in this work will be made public before the end of the year
+2023. The data used to make the plots presented in this
+paper are available upon request.
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+2424
+APPENDIX A: IMPACT OF LIMITED
+VOLUME OF CAMELS SIMULATIONS
+In order to analyze the impact of varying box sizes and res-
+olution on the matter power suppression, we use the TNG
+simulations as presented in Springel et al. (2018). Partic-
+ularly we use their boxes with side lengths of 210 Mpc/h,
+75 Mpc/h and 35 Mpc/h (which they refer to as TNG-300,
+TNG-100 and TNG-50 as it corresponds to side length in the
+units of Mpc). We then make the comparison to 25 Mpc/h
+TNG boxes run from CAMELS. We use the CV set of
+simulations and use them to infer the expected variance
+due to stochasticity induced by changing initial conditions.
+Note that the hydrodynamical model is identical between
+CAMELS CV runs and the bigger TNG boxes. In Fig. A1,
+we show the power suppression for these boxes, including
+the runs at varying resolution. We find that while changing
+box sizes gives relatively robust values of power suppression,
+changing resolution can have non-negligible impact. How-
+ever, all the TNG boxes are consistent at 2-3σ level relative
+to the CAMELS boxes.
+APPENDIX B: EXAMPLE OF EMULATION
+We present an example of the constructed emulator from
+§3.1 for the AAGN1 parameter in Fig. B1. This shows how
+we estimate the derivative of the observable (Y500c/M 5/3) in
+a way that is robust to stochasticity.
+APPENDIX C: ROBUSTNESS OF RESULTS TO
+DIFFERENT TRAIN SIMULATIONS
+In Fig. C1, we test the impact of changing the simulations
+used to train the random forest regressor. We then use these
+different trained models to infer the constraints on the mat-
+ter power suppression from the same stacked ⟨Y500c/Y SS⟩ as
+described in § 4. We see that our inferred constraints remain
+consistent when changing the simulations.
+APPENDIX D: TEST WITH LOWER HALO
+MASSES
+In Fig. D1, we show the constraints on the power suppres-
+sion obtained by analyzing the observables obtained from
+halos with lower masses, 1 × 1012 < M(M⊙/h) < 5 × 1012.
+We see that remarkably, even these lower halo masses pro-
+vide unbiased constraints on the matter power suppression
+with robust inference especially at smaller scales. However,
+when compared to the results descibed in § 3.2, we obtain
+less precise constraints. This is expected as lower halos with
+−0.3
+−0.2
+−0.1
+0.0
+∆P/PDMO
+TNG-210
+NDM = 25003
+TNG-70
+NDM = 9103
+TNG-35
+NDM = 5403
+TNG-25
+NDM = 2563
+100
+101
+k (h/Mpc)
+−0.3
+−0.2
+−0.1
+0.0
+∆P/PDMO
+TNG70
+NDM = 18203
+TNG 70
+NDM = 4553
+Figure A1. Comparison of the suppression of matter power in
+the CAMELS TNG simulation and simulations using the same
+sub-grid prescription but larger box sizes (Springel et al. 2018).
+We also show 1σ and 2σ uncertainty due to cosmic variance. In
+the top panel we change the TNG box sizes, while preserving the
+resolution, where as in the bottom panel we preserve the TNG
+box size while changing the resolution.
+lower masses are more susceptible to environmental effects
+which induces a larger scatter in the relation between their
+observables (such as fb or Y500c) and their halo masses gov-
+erning feedback processes.
+APPENDIX E: TEST WITH OTHER
+BISPECTRUM CONFIGURATIONS
+In Fig. E1, we show the constraints obtained on the sup-
+pression of the squeezed bispectrum configurations. We fix
+the the angle between the long sides of the triangle to corre-
+spond to µ = 0.9. We again find robust inference of baryonic
+effects on the bispectrum when using either the integrated
+pressure profile or full radial pressure profile.
+This paper has been typeset from a TEX/LATEX file prepared by
+the author.
+MNRAS 000, 1���16 (0000)
+
+Probing feedback with the SZ
+17
+−0.6
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+log(AAGN1)
+−5.5
+−5.4
+log(Y500c/M 5/3)
+Resulting Emulator
+Resulting Derivative
+Figure B1. The constructed emulator and resulting derivative
+for the AAGN1 parameter in the mass bin 1012 < M(M⊙/h) <
+5 × 1012.
+100
+101
+k
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+∆P/PDMO
+Train
+TNG+SIMBA+Astrid
+Train
+TNG+SIMBA
+Train
+SIMBA+Astrid
+Train
+TNG+Astrid
+Figure C1. In this figure we change the simulations used to
+train the RF when inferring the power suppression from the data
+measurements.
+MNRAS 000, 1–16 (0000)
+
+18
+Pandey et al.
+100
+101
+k (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆P/PDMO prediction relative to CV (σ)
+CV
+Y500c/YSS, Ωm
+Pe(r)/PSS
+e
+Pe(r)/PSS
+e , Ωm
+Pe(r)/PSS
+e , ne(r), Ωm
+Figure D1. Same as Fig. 7 and Fig. 8, but obtained on lower halo masses, 1 × 1012 < M(M⊙/h) < 5 × 1012. We find that having
+pressure profile information results in unbiased constraints here as well, albeit with a larger errorbars
+100
+101
+ksq (h/Mpc)
+−4
+−3
+−2
+−1
+0
+1
+2
+3
+4
+Error ∆Bsq/Bsq;DMO prediction relative to CV (σ)
+CV
+Y500c/YSS, Ωm
+Pe(r)/PSS
+e
+Pe(r)/PSS
+e , Ωm
+Pe(r)/PSS
+e , ne(r), Ωm
+Figure E1. Same as Fig. 9, but for squeezed triangle configurations (µ = 0.9).
+MNRAS 000, 1–16 (0000)
+

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+arXiv:2301.01270v1  [math.GM]  9 Dec 2022
+Maurer-Cartan characterization, cohomology and
+deformations of equivariant Lie superalgebras
+RB Yadav1,∗, Subir Mukhopadhyay
+Sikkim University, Gangtok, Sikkim, 737102, INDIA
+Abstract
+In this article, we give Maurer-Cartan characterizations of equivariant Lie superalgebra
+structures. We introduce equivariant cohomology and equivariant formal deformation
+theory of Lie superalgebras. As an application of equivariant cohomology we study
+the equivariant formal deformation theory of Lie superalgebras. As another applica-
+tion we characterize equivariant central extensions of Lie superalgebras using second
+equivariant cohomology. We give some examples of Lie superalgebras with an action
+of a group and equivariant formal deformations of a classical Lie superalgebras.
+Keywords: Lie superalgebra, cohomology, extension, formal
+deformations, Maurer-Cartan equation
+2020 MSC: 17A70, 17B99, 16S80, 13D10, 13D03, 16E40
+1. Introduction
+Graded Lie algebras have been a topic of interest in physics in the context of ”su-
+persymmetries” relating particles of differing statistics. In mathematics, graded Lie
+algebras have been studied in the context of deformation theory, [1].
+Lie superalgebras were studied and a classification was given by Kac [2]. Leits
+[3] introduced a cohomology for Lie superalgebras. Lie superalgebras are also called
+Z2-graded Lie algebras by physicists.
+∗Corresponding author
+Email addresses: [email protected] (RB Yadav ), [email protected] (Subir
+Mukhopadhyay)
+Preprint submitted to ...
+January 4, 2023
+
+Algebraic deformation theory was introduced by Gerstenhaber for rings and al-
+gebras [4],[5],[6], [7], [8]. Deformation theory of Lie superalgebras was introduced
+and studied by Binegar [9]. Maurer-Cartan characterization was given for Lie algebra
+structures by Nijenhuis and Richardson in [10] and for associative algebra structures by
+Gerstenhaber in [11]. Such characterization for Lie superalgebra structures was given
+in [12]. Deformation theory of Lie superalgebras was studied in [9].
+Aim of the present paper is to give Maurer-Cartan characterization, introduce equiv-
+ariant cohomology, do some equivariant cohomology computations in lower dimen-
+sions, introduce equivariant formal deformation theory of Lie superalgebras and give
+some examples. Organization of the paper is as follows. In Section 2, we recall def-
+inition of Lie superalgebra and give some examples. In Section 4, we give Maurer-
+Cartan characterization of equivariant Lie superalgebras. In this section we construct
+a Z × Z2-graded Lie algebra from a Z2-graded G-vector space. We show that class of
+Maurer-Cartan elements of this Z×Z2-graded Lie algebra is the class of G-equivariant
+Lie superalgebra structures on V. In Section 5, we introduce equivariant chain complex
+and equivariant cohomology of Lie superalgebras. In Section 6, we compute coho-
+mology of Lie superalgebras in degree 0 and dimension 0, 1 and 2. In Section 7, we
+introduce equivariant deformation theory of Lie superalgebras. In this section we see
+that infinitesimals of equivariant deformations are equivariant cocycles . Also, in this
+section we give an example of an equivariant formal deformation of a Lie superalge-
+bras. In Section 8, we study equivalence of two equivariant formal deformations and
+prove that infinitesimals of any two equivalent equivariant deformations are cohomol-
+ogous.
+2. Lie Superalgebras
+In this section, we recall definitions of Lie superalgebras and modules over a Lie
+superalgebras. We recall some examples of Lie superalgebras. Throughout the paper
+we denote a fixed field by K. Also, we denote the ring of formal power series with
+coefficients in K by K[[t]]. In any Z2-graded vector space V we use a notation in
+which we replace degree deg(a) of an element a ∈ V by ‘a′ whenever deg(a) appears
+2
+
+in an exponent; thus, for example (−1)ab = (−1)deg(a)deg(b).
+Definition 2.1. Let V = V0 ⊕ V1 and W = W0 ⊕ W1 be Z2-graded vector spaces
+over a field K. A linear map f : V → W is said to be homogeneous of degree α if
+f(Vβ) ⊂ Wα+β, for all β ∈ Z2 = {0, 1}. We write (−1)deg(f) = (−1)f. Elements of
+Vβ are called homogeneous of degree β.
+Definition 2.2. A superalgebra is a Z2-graded vector space A = A0 ⊕ A1 together
+with a bilinear map m : A × A → A such that m(a, b) ∈ Aα+β, for all a ∈ Aα,
+b ∈ Aβ.
+Definition 2.3. A Lie superalgebra is a superalgebra L = L0 ⊕ L1 over a field K
+equipped with an operation [−, −] : L × L → L satisfying the following conditions:
+1. [a, b] = −(−1)αβ[b, a],
+2. [[a, [b, c]] = [[a, b], c] + (−1)αβ[b, [a, c]],
+(Jacobi identity)
+for all a ∈ Lαand b ∈ Lβ. Let L1 and L2 be two Lie superalgebras. A homomorphism
+f : L1 → L2 is a K-linear map such that f([a, b]) = [f(a), f(b)]. Given a Lie
+superalgebra L [L, L] is the vector subspace of L spanned by the set {[x, y] : x, y ∈
+L}. A Lie superalgebra L is called abelian if [L, L] = 0.
+Example 2.1. Let V = V¯0 ⊕ V¯1 be a Z2-graded vector space, dimV¯0 = m, dimV¯1 =
+n. Consider the associative algebra EndV of all endomorphisms of V . Define
+Endi V = {a ∈ End V | aVs ⊆ Vi+s} , i, s ∈ Z2
+(1)
+One can easily verify that End V = End¯0 V ⊕ End¯1 V . The bracket [a, b] =
+ab − (−1)¯a¯bba makes EndV into a Lie superalgebra, denoted by ℓ(V ) or ℓ(m, n). In
+some (homogeneous) basis of V , ℓ(m, n) consists of block matrices of the form
+� α β
+γ δ
+�
+,
+where α, β, γ, δ are matrices of order m × m, m × n, n × m and n × n, respectively.
+Example 2.2. Define a linear function str : ℓ(V ) → k, by str([a, b]) = 0, a, b ∈
+ℓ(V ), and str idV = m − n. str(a) is called a supertrace of a ∈ ℓ(V ). Consider the
+subspace
+sℓ(m, n) = {a ∈ ℓ(m, n) | str a = 0}.
+3
+
+Clearly, sℓ(m, n) is an ideal of ℓ(m, n) of codimension 1. Therefore sℓ(m, n) is a
+subalgebra of ℓ(m, n).
+For any
+� α β
+γ δ
+�
+in ℓ(m, n) str
+� α β
+γ δ
+�
+= tr α − tr δ. sℓ(n, n) contains the one-
+dimensional ideal {λI2n : λ ∈ K}.
+Definition 2.4. [13] Let L = L0 ⊕ L1 be a Lie superalgebra. A Z2-graded vector
+space M = M0 ⊕ M1 over the field K is called a module over L if there exists a
+bilinear map [−, −] : L × M → M such that following condition is satisfied
+[a, [b, m]] = [[a, b], m] + (−1)ab[b, [a, m]].
+for all a ∈ Lα, b ∈ Lβ, α, β ∈ {0, 1}.
+Clearly, every Lie superalgebra is a module over itself.
+3. Z2-graded Groups and their Actions on a Lie Superalgebra
+Definition 3.1. We define a Z2-graded group as a group G having a subgroup G¯0 and
+a subset G¯1 such that for all x ∈ Gi, y ∈ Gj, xy ∈ Gi+j, where i, j, i + j ∈ Z2.
+Example 3.1. Consider Z6 = {¯0, ¯1, ¯2, ¯3, ¯4, ¯5}. Take G = Z6, G¯0 = {¯0, ¯2, ¯4}, G¯1 =
+{¯1, ¯3, ¯5}. Clearly, with this choice of G¯0 and G¯1, G is a Z2-graded group.
+Example 3.2. Every group G can be seen as Z2-graded group with G¯0 = G and
+G¯1 = ∅.
+Definition 3.2. A Z2-graded group G is said to act on a Lie superalgebra L = L0⊕L1
+if there exits a map
+ψ : G × L → L, (g, x) �→ ψ(g, x) = gx
+satisfying following conditions
+1. ex = x, for all x ∈ L. Here e ∈ G is the identity element of G.
+2. ∀g ∈ Gi, i ∈ Z2 ψg : L → L given by ψg(x) = ψ(g, x) = gx is a homogeneous
+linear map of degree i.
+3. ∀g1, g2 ∈ G, ψ(g1g2, x) = ψ(g1, ψ(g2, x)), that is (g1g2)x = g1(g2x).
+4
+
+4. For x, y ∈ L, g ∈ G, [gx, gy] = g[x, y].
+We denote an action as above by (G, L).
+Proposition 3.1. Let G be a finite Z2-graded group and L be a Lie superalgebra. Then
+G acts on L if and only if there exists a group homomorphism of degree 0
+φ : G → Iso(L, L), g �→ φ(g) = ψg
+from the group G to the group of homogeneous Lie superalgebra isomorphisms from L
+to L.
+Proof. For an action (G, L), we define a map φ : G → Iso(L, L) by φ(g) = ψg. One
+can verify easily that φ is a group homomorphism. Now, let φ : G → Iso(L, L) be
+a group homomorphism. Define a map G × L → L by (g, a) �→ φ(g)(a). It can be
+easily seen that this is an action of G on L.
+Note: In this article we consider action of groups G, that is those Z2-graded groups G
+for which groups G0 = G and G1 = ∅. We call a Lie Superalgebra L = L0 ⊕ L1 with
+an action of a group G G-Lie Superalgebra.
+Example 3.3. Super-Poincare algebra: The (N = 1) Super-Poincare algebra L =
+L0 ⊕ L1 is given by1
+i[Jµν, Jρσ] = ηνρJµσ − ηµρJνσ − ησµJρν + ησνJρµ,
+i[P µ, Jρσ] = ηµρP σ − ηµσP ρ,
+[P µ, P ρ] = 0,
+[Qα, Jµν] = (σµν) β
+α Qβ,
+[ ¯Q ˙α, Jµν] = (¯σµν) ˙α
+˙β ¯Q
+˙β
+[Qα, P µ] = 0,
+[ ¯Q ˙α, P µ] = 0
+{Qα, Qβ} = 0,
+{ ¯Q ˙α, ¯Q
+˙β} = 0,
+{Qα, ¯Q
+˙β} = 2(σµ)α ˙βPµ.
+1Here we have used the following notation. µ, ν, ρ, ... = 0, 1, 2, 3.; σi, i = 1, 2, 3 represent Pauli spin
+matrices and one introduces σµ = (1, σi)
+and
+¯σµ = (1, −σi),
+(σµν) β
+α = − i
+4(σµ¯σν − σν¯σµ) β
+α , (¯σµν) β
+α = − i
+4(¯σµσν − ¯σνσµ) ˙α
+˙β. Spinor indices are denoted by
+α, β, ˙α, ˙β, they take values from the set {1, 2} and are being raised and lowered by ǫαβ (ǫ ˙α ˙β), and ǫαβ
+(ǫ ˙α ˙β). They are antisymmetric and we have chosen ǫ12 = ǫ˙1˙2 = +1
+5
+
+Here L0 is generated by the set {Jµν : µ, ν = 0, 1, 2, 3} ∪ {P µ : µ = 0, 1, 2, 3} over
+C. L1 is generated by the set {Qα : α = 1, 2} ∪ { ¯Q ˙α : ˙α = 1, 2} over C. Consider the
+group Zm = {gn : n = 0, 1, . . . , m − 1}, where g = e
+2πi
+m . There exists an action of
+Zm on the Super-Poincare algebra L = L0 ⊕ L1 given by
+(gn, Jµν) �→ Jµν,
+(gn, P µ) �→ P µ,
+(gn, Qα) �→ gnQα,
+(gn, ¯Q ˙α) �→ gm−n ¯Q ˙α,
+for every n = 0, 1, . . . , m − 1.
+Example 3.4. Let eij denote a 2 × 2 matrix with (i, j)th entry 1 and all other entries
+0. Consider L0 = span{e11, e22}, L1 = span{e12, e21}. Then L = L0 ⊕ L1 is a Lie
+superalgebra with the bracket [ , ] defined by
+[a, b] = ab − (−1)¯a¯bba.
+Define a function ψ : Z2 × L → L by ψ(0, x) = x, ∀x ∈ L, ψ(1, e11) = e22,
+ψ(1, e22) = e11, ψ(1, e12) = e21, ψ(1, e21) = e12. Obviously Conditions 1 − 3
+hold for (Z2, L) to be an action. To verify condition 4 it is enough to verify for basis
+elements of L0 and L1. We have
+1. 1[eii, eii] = 0 = [1eii, 1eii], ∀ i = 1, 2.
+2. 1[eii, ejj] = 0 = [ejj, eii] = [1eii, 1ejj], ∀ i, j = 1, 2, i ̸= j.
+3. 1[eij, eji] = 1(eii − (−1)1ejj) = ejj + eii = [1eij, 1eji], ∀ i, j = 1, 2, i ̸= j.
+4. 1[eij, eij] = 0 = [eji, 1eji] = [1eij, 1eij], ∀ i, j = 1, 2, i ̸= j.
+5. 1[eii, eij] = 1(eij) = eji = [ejj, eji] = [1eii, 1eij], ∀ i, j = 1, 2, i ̸= j.
+6. 1[ejj, eij] = 1(−eij) = −eji = [eii, eji] = [1ejj, 1eij], ∀ i, j = 1, 2, i ̸= j.
+From above it is clear that (Z2, L) is an action.
+Definition 3.3. Let L = L0 ⊕ L1 be a Lie superalgebra. Let G be a finite group which
+acts on L. A Z2-graded vector space M = M0 ⊕ M1 with an action of G is called a
+G-module over L if there exists a G-equivariant bilinear map [−, −] : L × M → M
+such that following condition is satisfied
+[a, [b, m]] = [[a, b], m] + (−1)ab[b, [a, m]],
+for all a ∈ Lα, b ∈ Lβ, α, β, ∈ {0, 1}.
+6
+
+Example 3.5. Every G-Lie superalgebra is a G-module over itself.
+Example 3.6. Let L = L0 ⊕ L1 be the (N = 1) Super-Poincare algebra, Example
+3.3. Let M0 be the span of {P µ : µ = 0, 1, 2, 3} and M1 be the span of the set
+{Qα : α = 1, 2} ∪ { ¯Q ˙α : ˙α = 1, 2}. Then clearly M = M0 ⊕ M1 is a Zm-module
+over L = L0 ⊕ L1.
+4. Maurer-Cartan Characterization of Equivariant Lie Superalgebra Structures
+Definition 4.1. A finite group G is said to act on a Z2-graded vector space V = V0⊕V1
+if there exits a map
+ψ : G × V → V, (g, x) �→ ψ(g, x) = gx
+satisfying following conditions
+1. ex = x, for all x ∈ V . Here e is the identity element of G.
+2. ∀g ∈ G, ψg : V → V given by ψg(x) = ψ(g, x) = gx is a homogeneous linear
+map of degree 0.
+3. ∀g1, g2 ∈ G, ψ(g1g2, x) = ψ(g1, ψ(g2, x)), that is (g1g2)x = g1(g2x).
+A Z2-graded vector space V = V0⊕V1 with an action of a group G is called a G-vector
+space.
+Let V = V0 ⊕ V1 and W = W0 ⊕ W1 be vector spaces over a field F. An n-linear
+map f : V × · · · ×
+� �� �
+n times
+V → W is said to be homogeneous of degree α if f(x1, · · · , xn) is
+homogeneous in W and deg(f(x1, · · · , xn)) − �n
+i=1 deg(xi) = α, for homogeneous
+xi ∈ V , 1 ≤ i ≤ n. We denote the degree of a homogeneous f by deg(f). We write
+(−1)deg(f) = (−1)f.
+Consider the permutation group Sn. For any X = (X1, . . . , Xn) with Xi ∈ Vxi
+and σ ∈ Sn, define
+K(σ, X) = card{(i, j) : i < j, Xσ(i) ∈ V1, Xσ(j) ∈ V1, σ(j) < σ(i)},
+ǫ(σ, X) = ǫ(σ)(−1)K(σ,X),
+where cardA denotes cardinality of a set A, ǫ(σ) is the signature of σ. Also, define
+σ.X = (Xσ−1(1), . . . , Xσ−1(n)). We have following Lemma [12]
+7
+
+Lemma 4.1.
+1. K(σσ′, X) = K(σ, X) + K(σ′, σ−1X)
+(mod2).
+2. ǫ(σσ′, X) = ǫ(σ, X)ǫ(σ′, σ−1X).
+For each n ∈ N, define Fn,α(V, W) as the vector space of all homogeneous n-
+linear mappings f : V × · · · ×
+� �� �
+n times
+V → W of degree α. Define Fn(V, W) = Fn,0(V, W)⊕
+Fn,1(V, W), F0(V, W) = W and F−n(V, W) = 0, ∀n ∈ N. Take F(V, W) =
+�
+n∈Z Fn(V, W).
+For F ∈ Fn(V, W), X ∈ V n, σ ∈ Sn, define
+(σ.F)(X) = ǫ(σ, X)F(σ−1X).
+By using Lemma 4.1, one concludes that this defines an action of Sn on the Z2-
+graded vector space Fn(V, W). Define En for n ∈ Z as follows:
+Set En = {F ∈ Fn+1(V, V ) : σ.F = F, ∀ σ ∈ Sn+1}, for n ≥ 0 and
+En =
+
+
+
+
+
+V
+if n = −1
+0
+if n < −1
+.
+Write E = �
+∈Z En. Define a product ◦ on E as follows: For F ∈ En,f, F ′ ∈ En′,f ′ set
+F ◦ F ′ =
+�
+σ∈S(n,n′+1)
+σ.(F ∗ F ′),
+where
+F∗F ′(X1, . . . , Xn+n′+1) = (−1)f ′(x1+···+xn)F(X1, . . . , Xn, F ′(Xn+1, . . . , Xn+n′+1)),
+for Xi ∈ Vxi, and S(n,n′+1) consists of permutations σ ∈ Sn+n′+1 such that σ(1) <
+· · · < σ(n), σ(n + 1) < · · · < σ(n + n′ + 1). Clearly, F ◦ F ′ ∈ E(n+n′,f+f ′). We
+have following Lemma [12].
+Lemma 4.2. For F ∈ En,f, F ′ ∈ En′,f ′, F ′′ ∈ En′′,f ′′
+(F ◦ F ′) ◦ F ′′ − F ◦ (F ′ ◦ F ′′) = (−1)n′n′′+f ′f ′′{(F ◦ F ′′) ◦ F ′ − F ◦ (F ′′ ◦ F ′)}.
+Using Lemma 4.2, we have following theorem [14], [12]
+8
+
+Theorem 4.1. E is a Z × Z2-graded Lie algebra with the bracket [ , ] defined by
+[F, F ′] = F ◦ F ′ − (−1)nn′+ff ′F ′ ◦ F,
+for F ∈ En,f, F ′ ∈ En′,f ′
+Let G be a finite group acting on the vector spaces V = V0⊕V1 and W = W0⊕W1.
+Denote by FG
+n (V, W) the vector space of G-equivariant elements of Fn(V, W), that
+is F(gX1, . . . , gXn) = gF(X1, . . . , Xn), for each F ∈ FG
+n (V, W), (X1, . . . , Xn) ∈
+V n. Write FG(V, W) = �
+n∈Z FG
+n (V, W). For σ ∈ Sn, g ∈ G, (X1, . . . , Xn) ∈ V n,
+we have
+σ.(gX1, . . . , gXn)
+=
+(gXσ−1(1), . . . , gXσ−1(n))
+=
+g(Xσ−1(1), . . . , Xσ−1(n))
+=
+g(σ.(X1, . . . , Xn)).
+(1)
+Let F ∈ EG
+n,f, F ′ ∈ EG
+n′,f ′. Clearly, F ∗ F ′ ∈ EG
+n+n′,f+f ′. Using Equation 1, we
+conclude that F ◦ F ′ ∈ EG
+n+n′,f+f ′. This implies that [ , ] defines a product in EG.
+Hence using Theorem 4.1, we have following theorem.
+Theorem 4.2. EG is a Z × Z2-graded Lie algebra with with the bracket [ , ] defined
+by
+[F, F ′] = F ◦ F ′ − (−1)nn′+ff ′F ′ ◦ F,
+for F ∈ En,f, F ′ ∈ En′,f ′
+Using [12], Proposition (3.1), we get following theorem.
+Theorem 4.3. Given F0 ∈ EG
+(1,0), F0 defines on a Z2-graded G-vector space V a
+G-Lie superalgebra structure if and only if [F0, F0] = 0.
+Remark 4.1. An element F0 ∈ EG
+(1,0) which satisfies the equation
+[F, F] = 0
+(2)
+is called a Maurer-Cartan element and the Equation 2 is called Maurer-Cartan equa-
+tion. Thus the class of Maurer-Cartan elements is the class of G-Lie superalgebra
+structures on a Z2-graded G-vector space V .
+9
+
+5. Equivariant Cohomology of Lie Superalgebras
+Let L = L0 ⊕ L1 be a Lie superalgebra and M = M0 ⊕ M1 be a module over L.
+For each n ≥ 0, a K-vector space Cn(L; M) is defined as follows: C0(L; M) = M
+and for n ≥ 1, Cn(L; M) consists of those n-linear maps f from Ln to M which are
+homogeneous and
+f(x1, . . . , xi, xi+1, . . . , xn) = −(−1)xixi+1f(x1, . . . , xi+1, xi . . . , xn).
+Clearly, Cn(L; M) = Cn
+0 (L; M) ⊕ Cn
+1 (L; M), where Cn
+0 (L; M) and Cn
+1 (L; M) are
+vector subspaces of Cn(L; M) containing elements of degree 0 and 1, respectively. A
+linear map δn : Cn(L; M) → Cn+1(L; M) is defined by ([9], [3])
+δnf(x1, · · · , xn+1)
+=
+�
+i<j
+(−1)i+j+(xi+xj)(x1+···+xi−1)+xj(xi+1+···+xj−1)
+f([xi, xj], x1, . . . , ˆxi, . . . , ˆxj, . . . , xn+1)
++
+n+1
+�
+i=1
+(−1)i−1+xi(f+x1+···+xi−1)[xi, f(x1, . . . , ˆxi, . . . , xn+1)],
+(3)
+(4)
+for all f ∈ Cn(L; M), n ≥ 1, and δ0f(x1) = (−1)x1f[x1, f], for all f ∈ C0(L; M) =
+M. Clearly, for each f ∈ Cn
+G(L; M), n ≥ 0, deg(δf) = deg(f). From [9], [3], we
+have following theorem:
+Theorem 5.1. δn+1◦δn = 0, that is, (C∗(L; M), δ), where C∗(L; M) = ⊕nCn(L; M),
+δ = ⊕nδn, is a cochain complex.
+Let G be a finite group which acts on L. Let M be a G-module over L. For each
+n ≥ 0, we define a K-vector space Cn
+G(L; M) as follows: C0
+G(L; M) = M and for
+n ≥ 1, Cn
+G(L; M) consists of those f ∈ Cn(L, M) which are G-equivariant, that
+is, f(ga1, . . . , gan) = gf(a1, . . . , an), for all (a1, . . . , an) ∈ Ln, g ∈ G. Clearly,
+Cn
+G(L; M) = (Cn
+G)0(L; M) ⊕ (Cn
+G)1(L; M), where (Cn
+G)0(L; M) and (Cn
+G)1(L; M)
+are vector subspaces of Cn
+G(L; M) containing elements of degree 0 and 1, respectively.
+We define a K-linear map δn
+G : Cn
+G(L; M) → Cn+1
+G
+(L; M) by
+δn
+Gf(x1, . . . , xn+1) = δnf(x1, . . . , xn+1).
+10
+
+Clearly, δn
+Gf(gx1, . . . , gxn+1) = gδnf(x1, . . . , xn+1) for each f ∈ Cn
+G(L; M),
+g ∈ G. Thus δn
+G is well defined. Write C∗
+G(L; M) = ⊕nCn
+G(L; M), δG = ⊕nδn
+G.
+Using Theorem 5.1 we have following theorem:
+Theorem 5.2. (C∗
+G(L; M), δG) is a cochain complex.
+We denote ker(δn
+G) by Zn
+G(L; M) and image of (δn−1
+G
+) by Bn
+G(L; M). We call
+the n-th cohomology Zn
+G(L; M)/Bn
+G(L; M) of the cochain complex {Cn
+G(L; M), δn
+G}
+as the n-th equivariant cohomology of L with coefficients in M and denote it by
+Hn
+G(L; M). Since L is a module over itself. So we can consider cohomology groups
+Hn
+G(L; L). We call Hn
+G(L; L) as the n-th equivariant cohomology group of L. We
+have
+Zn
+G(L; M) = (Zn
+G)0(L; M)⊕(Zn
+G)1(L; M), Bn
+G(L; M) = (Bn
+G)0(L; M)⊕(Bn
+G)1(L; M),
+where (Zn
+G)i(L; M)and (Bn
+G)i(L; M) are submodules of (Cn
+G)i(L; M), i = 0, 1.
+Since boundary map δn
+G : Cn
+G(L; M) → Cn+1
+G
+(L; M) is homogeneous of degree 0,
+we conclude that Hn
+G(L; M) is Z2-graded and
+Hn
+G(L; M) = (Hn
+G)0(L; M) ⊕ (Hn
+G)1(L; M),
+where (Hn
+G)i(L; M) = (Zn
+G)i(L; M)/(Bn
+G)i(L; M), i = 0, 1.
+6. Equivariant Cohomology of Lie Superalgebras in Low Degrees
+Let G be a finite group and L = L0 ⊕ L1 be a Lie superalgebra with an action
+of G. Let M = M0 ⊕ M1 be a G-module over L. For m ∈ M0 = (C0
+G)0(L; M),
+f ∈ (C1
+G)0(L; M) and g ∈ (C2
+G)0(L; M)
+δ0
+Gm(x) = [x, m],
+(5)
+δ1f(x1, x2) = −f([x1, x2]) + [x1, f(x2)] − (−1)x2x1[x2, f(x1)],
+(6)
+δ2g(x1, x2, x3)
+=
+−g([x1, x2], x3) + (−1)x3x2g([x1, x3], x2) − (−1)x1(x2+x3)g([x2, x3], x1)
++[x1, g(x2, x3)] − (−1)x2x1[x2, g(x1, x3)]
++(−1)x3x1+x3x2[x3, g(x1, x2)].
+(7)
+11
+
+The set {m ∈ M0|[x, m] = 0, ∀x ∈ L} is called annihilator of L in M0 and is denoted
+by annM0L. We have
+(H0
+G)0(L; M)
+=
+{m ∈ M0|[x, m] = 0, for all x ∈ L}
+=
+annM0L.
+A G-equivariant homogeneous linear map f : L → M is called derivation from L to
+M if f([x1, x2]) = (−1)fx1[x1, f(x2)] �� (−1)fx2+x2x1[x2, f(x1)], that is δ1
+Gf = 0.
+For every m ∈ M0 the map x �→ [x, m] is called an inner derivation from L to M. We
+denote the vector spaces of equivariant derivations and equivariant inner derivations
+from L to M by DerG(L; M) and DerG
+Inn(L; M) respectively. By using 5, 6 we have
+(H1
+G)0(L; M) = DerG(L; M)/DerG
+Inn(L; M).
+Let L be a Lie superalgebra with an action of a finite group G and M be a G-
+module over L. We regard M as an abelian Lie superalgebra with an action of G. An
+extension of L by M is an exact sequence
+0
+� M
+i
+� E
+π
+� L
+� 0
+(*)
+of Lie superalgebras such that
+[x, i(m)] = [π(x), m].
+The exact sequence (∗) regarded as a sequence of K-vector spaces, splits. Therefore
+without any loss of generality we may assume that E as a K-vector space coincides
+with the direct sum L ⊕ M and that i(m) = (0, m), π(x, m) = x. Thus we have
+E = E0 ⊕ E1, where E0 = L0 ⊕ M0, E1 = L1 ⊕ M1. The multiplication in E = L ⊕ M
+has then necessarily the form
+[(0, m1), (0, m2)] = 0, [(x1, 0), (0, m1)] = (0, [x1, m1]),
+[(0, m2), (x2, 0)] = −(−1)m2x2(0, [x2, m2]), [(x1, 0), (x2, 0)] = ([x1, x2], h(x1, x2)),
+for some h ∈ (C2
+G)0(L; M), for all homogeneous x1, x2 ∈ L, m1, m2 ∈ M. Thus, in
+general, we have
+[(x, m), (y, n)] = ([x, y], [x, n] − (−1)my[y, m] + h(x, y)),
+(8)
+12
+
+for all homogeneous (x, m), (y, n) in E = L ⊕ M.
+Conversely, let h : L × L → M be a bilinear G-equivariant homogeneous map of
+degree 0. For homogeneous (x, m), (y, n) in E we define multiplication in E = L⊕M
+by Equation 8. For homogeneous (x, m), (y, n) and (z, p) in E we have
+[[(x, m), (y, n)], (z, p)]
+=
+([[x, y], z], [[x, y], p] − (−1)zx+zn[z[x, n]] + (−1)ym+zy+zm[z, [y, m]] + [h(x, y), z] + h([x, y], z))
+(9)
+[(x, m), [(y, n), (z, p)]]
+=
+([x, [y, z]], [x, [y, p]] − (−1)nz[x, [z, n]] − (−1)my+mz[[y, z], m] + [x, h(y, z)] + h(x, [y, z])
+(10)
+[(y, n), [(x, m), (z, p)]]
+=
+([y, [x, z]], [y, [x, p]] − (−1)mz[y, [z, m]] − (−1)nx+nz[[x, z], n] + [y, h(x, z)] + h(y, [x, z]))
+(11)
+From Equations 9, 10, 11 we conclude that E = L ⊕ M is a Lie superalgebra with
+product given by Equation 8 if and only if δ2
+Gh = 0. We denote the Lie superalgebra
+given by Equation 8 using notation Eh. Thus for every cocycle h ∈ (C2
+G)0(L; M) there
+exists an extension
+Eh : 0
+� M
+i
+� Eh
+π
+� L
+� 0
+of L by M, where i and π are inclusion and projection maps, that is, i(m) = (0, m),
+π(x, m) = x. We say that two extensions
+0
+� M
+� Ei
+� L
+� 0 (i = 1, 2)
+of L by M are equivalent if there is a G-equivariant Lie superalgebra isomorphism
+ψ : E1 → E2 such that following diagram commutes:
+0
+� M
+IdM
+�
+� E1
+ψ
+�
+� L
+IdL
+�
+� 0
+0
+� M
+� E2
+� L
+� 0
+(**)
+13
+
+We use F(L, M)to denote the set of all equivalence classes of extensions of L by
+M. Equation 8 defines a mapping of (Z2
+G)0(L; M) onto F(L, M). If for h, h′ ∈
+(Z2
+G)0(L; M) Eh is equivalent to Eh′, then commutativity of diagram (∗∗) is equiva-
+lent to
+ψ(x, m) = (x, m + f(x)),
+for some f ∈ (C1
+G)0(L; M). We have
+ψ([(x1, m1), (x2, m2)])
+=
+ψ([x1, x2], [x1, m2] + [m1, x2] + h(x1, x2))
+=
+([x1, x2], [x1, m2] + [m1, x2] + h(x1, x2) + f([x1, x2])),
+(12)
+[ψ(x1, m1), ψ(x2, m2)]
+=
+[(x1, m1 + f(x1)), (x2, m2 + f(x2))]
+=
+([x1, x2], [x1, m2 + f(x2)] + [m1 + f(x1), x2] + h′(x1, x2)).
+(13)
+Since ψ([(x1, m1), (x2, m2)]) = [ψ(x1, m1), ψ(x2, m2)], we have
+h(x1, x2) − h′(x1, x2)
+=
+−f([x1, x2]) + [x1, f(x2)] + [f(x1), x2]
+=
+−f([x1, x2]) + [x1, f(x2)] − (−1)x1x2[x2, f(x1)]
+=
+δ1(f)(x1, x2)
+(14)
+Thus two extensions Eh and Eh′ are equivalent if and only if there exists some f ∈
+(C1
+G)0(L; M) such that δ1f = h − h′. We thus have following theorem:
+Theorem 6.1. The set F(L, M) of all equivalence classes of extensions of L by M is
+in one to one correspondence with the cohomology group (H2
+G)0(L; M). This corre-
+spondence ω : (H2
+G)0(L; M) → F(L, M) is obtained by assigning to each cocycle
+h ∈ (Z2
+G)0(L; M), the extension given by multiplication 8.
+7. Equivariant Deformation of Lie Superalgebras
+Let L = L0 ⊕ L1 be a Lie superalgebra. We denote the ring of all formal power
+series with coefficients in L by L[[t]]. Clearly, L[[t]] = L0[[t]] ⊕ L1[[t]]. So every
+at ∈ L[[t]] is of the form at = (at)0⊕(at)1, where (at)0 ∈ L0[[t]] and (at)1 ∈ L1[[t]].
+14
+
+Definition 7.1. Let L = L0 ⊕L1 be a Lie superalgebra with an action of a finite group
+G. An equivariant formal one-parameter deformation of L is a K[[t]]-bilinear map
+µt : L[[t]] × L[[t]] → L[[t]]
+satisfying the following properties:
+(a) µt(a, b) = �∞
+i=0 µi(a, b)ti, for all a, b ∈ L, where µi : L×L → L, i ≥ 0 are G-
+equivariant bilinear homogeneous mappings of degree zero and µ0(a, b) = [a, b]
+is the original product on L.
+(b) µt(a, b) = −(−1)abµt(b, a), for all homogeneous a, b ∈ L.
+(c)
+µt(a, µt(b, c)) = µt(µt(a, b), c) + (−1)abµt(b, µt(a, c)),
+(15)
+for all homogeneous a, b, c ∈ L.
+The Equation 15 is equivalent to following equation:
+�
+i+j=r
+µi(a, µj(b, c))
+=
+�
+i+j=r
+{µi(µj(a, b), c) − (−1)abµi(b, µj(a, c))},
+(16)
+for all homogeneous a, b, c ∈ L.
+Now we define a formal deformation of finite order of a Lie superalgebra L.
+Definition 7.2. Let L be a Lie superalgebra with an action of a group G. A formal
+one-parameter deformation of order n of L is a K[[t]]-bilinear map
+µt : L[[t]] × L[[t]] → L[[t]]
+satisfying the following properties:
+(a) µt(a, b) = �n
+i=0 µi(a, b)ti, ∀a, b, c ∈ L, where µi : L×L → T , 0 ≤ i ≤ n, are
+equivariant K-bilinear homogeneous maps of degree 0, and µ0(a, b) = [a, b] is
+the original product on L.
+15
+
+(b) µi(a, b) = −(−1)abµi(b, a), for all homogeneous a, b ∈ L, i ≥ 0.
+(c)
+µt(a, µt(b, c)) = µt(µt(a, b), c) + (−1)abµt(b, µt(a, c)),
+(17)
+for all homogeneous a, b, c ∈ L.
+Remark 7.1.
+• For r = 0, conditions 16 is equivalent to the fact that L is a Lie
+superalgebra.
+• For r = 1, conditions 16 is equivalent to
+0
+=
+−µ1(a, [b, c]) − [a, µ1(b, c)]
++µ1([a, b], c) + (−1)abµ1(b, [a, c]) + [µ1(a, b), c] + (−1)ab[b, µ1(a, c)]
+=
+δ2µ1(a, b, c); for all homogeneous a, b, c ∈ L.
+Thus for r = 1, 16 is equivalent to saying that µ1 ∈ C2
+0(L; L) is a cocycle. In
+general, for r ≥ 0, µr is just a 2-cochain, that is, in µr ∈ C2
+0(L; L).
+Definition 7.3. The cochain µ1 ∈ C2
+0(L; L) is called infinitesimal of the deformation
+µt. In general, if µi = 0, for 1 ≤ i ≤ n − 1, and µn is a nonzero cochain in C2
+0(L; L),
+then µn is called n-infinitesimal of the deformation µt.
+Proposition 7.1. The infinitesimal µ1 ∈ C2
+0(L; L) of the deformation µt is a cocycle.
+In general, n-infinitesimal µn is a cocycle in C2
+0(L; L).
+Proof. For n=1, proof is obvious from the Remark 7.1. For n > 1, proof is similar.
+8. Equivalence of Equivariant Formal Deformations and Cohomology
+Let µt and ˜µt be two formal deformations of a Lie superalgebra L − L0 ⊕ L1.
+A formal isomorphism from the deformation µt to ˜µt is a K[[t]]-linear automorphism
+Ψt : L[[t]] → L[[t]] of the form Ψt = �∞
+i=0 ψiti, where each ψi is a homogeneous
+K-linear map L → L of degree 0, ψ0(a) = a, for all a ∈ T and
+˜µt(Ψt(a), Ψt(b)) = Ψt ◦ µt(a, b),
+for all a, b ∈ L.
+16
+
+Definition 8.1. Two deformations µt and ˜µt of a Lie superalgebra L are said to be
+equivalent if there exists a formal isomorphism Ψt from µt to ˜µt.
+Formal isomorphism on the collection of all formal deformations of a Lie superal-
+gebra L is an equivalence relation.
+Definition 8.2. Any formal deformation of T that is equivalent to the deformation µ0
+is said to be a trivial deformation.
+Theorem 8.1. The cohomology class of the infinitesimal of a deformation µt of a Lie
+Superalgebra L is determined by the equivalence class of µt.
+Proof. Let Ψt be a formal isomorphism from µt to ˜µt. So we have, for all a, b ∈ L,
+˜µt(Ψta, Ψtb) = Ψt ◦ µt(a, b). This implies that
+(µ1 − ˜µ1)(a, b)
+=
+[ψ1a, b] + [a, ψ1b] − ψ1([a, b])
+=
+δ1ψ1(a, b).
+So we have µ1 − ˜µ1 = δ1ψ1. This completes the proof.
+9. Some Examples of Equivariant Deformations
+In this section we discuss some examples of equivariant formal deformations of Lie
+superalgebras.
+Example 9.1. Let eij denote a 2 × 2 matrix with (i, j)th entry 1 and all other entries
+0. Consider L0 = span{e11, e22}, L1 = span{e12, e21}. Then L = L0 ⊕ L1 is a Lie
+superalgebra with the bracket [, ] defined by
+[a, b] = ab − (−1)¯a¯bba.
+Define a function ψ : Z2 × L → L by ψ(0, x) = x, ∀x ∈ L, ψ(1, e11) = e22,
+ψ(1, e22) = e11, ψ(1, e12) = e21, ψ(1, e21) = e12. In Example 3.4, we have seen that
+this gives an action of Z2 on L. Define a bilinear map ∗ : L × L → L by
+eij ∗ ekl =
+
+
+
+
+
+eli, if j = k
+0, otherwise
+.
+17
+
+Now define µ1 : L × L → L by
+µ1(a, b) = a ∗ b − (−1)abb ∗ a,
+for all homogeneous a , b in L. We have
+1. 1µ1(eii, eii) = 0 = µ1(1eii, 1eii), ∀ i = 1, 2.
+2. 1µ1(eii, ejj) = 0 = µ1(ejj, eii) = µ1(1eii, 1ejj), ∀ i, j = 1, 2, i ̸= j.
+3. 1µ1(eij, eji) = 1(ejj − (−1)1eii) = eii + ejj = µ1(1eij, 1eji), ∀ i, j =
+1, 2, i ̸= j.
+4. 1µ1(eij, eij) = 0 = (eji, eji) = µ1(1eij, 1eij), ∀ i, j = 1, 2, i ̸= j.
+5. 1µ1(eii, eij) = 1(eji) = eij = µ1(ejj, eji) = µ1(1eii, 1eij), ∀ i, j = 1, 2, i ̸=
+j.
+6. 1µ1(ejj, eij) = 1(−eji) = −eij = µ1(eii, eji] = µ1(1ejj, 1eij), ∀ i, j =
+1, 2, i ̸= j.
+Hence µ1 is Z2 equivariant. Define µt = µ0 + µ1t, where µ0(a, b) = [a, b]. We shall
+show that µt is an equivariant deformation of L of order 1. To conclude this only thing
+that we need to show is that
+δ2µ1(a, b, c)
+=
+−µ1(a, [b, c]) − [a, µ1(b, c)]
++µ1([a, b], c) + (−1)abµ1(b, [a, c]) + [µ1(a, b), c] + (−1)ab[b, µ1(a, c)]
+=
+0;
+for all homogeneous a, b, c ∈ L.
+We have
+δ2µ1(b, c, a)
+=
+−µ1(b, [c, a]) − [b, µ1(c, a)]
++µ1([b, c], a) + (−1)bcµ1(c, [b, a]) + [µ1(b, c), a] + (−1)bc[c, µ1(b, a)]
+=
+(−1)acµ1(b, [a, c]) + (−1)ac[b, µ1(a, c)]
+−(−1)ab+acµ1(a, [b, c]) + (−1)ab+acµ1([a, b], c)
+−(−1)ab+ac[a, µ1(b, c)] + (−1)ab+ac[µ1(a, b), c]
+=
+(−1)ab+ac{−µ1(a, [b, c]) − [a, µ1(b, c)]
++µ1([a, b], c) + (−1)abµ1(b, [a, c]) + [µ1(a, b), c] + (−1)ab[b, µ1(a, c)]}
+=
+(−1)ab+acδ2µ1(a, b, c)
+(18)
+18
+
+δ2µ1(e11, e12, e21)
+=
+−µ1(e11, e11 + e22) − [e11, e22 + e11]
++µ1(e12, e21) + µ1(e12, −e21) + [e21, e21] + [e12, −e12]
+=
+0
+(19)
+δ2µ1(e11, e21, e12)
+=
+−µ1(e11, e11 + e22) − [e11, e22 + e11]
++µ1(−e21, e12) + µ1(e21, e12) + [−e12, e12] + [e21, e21]
+=
+0
+(20)
+δ2µ1(e11, e12, e22)
+=
+−µ1(e11, e12) − [e11, e21]
++µ1(e12, e22) + µ1(e12, 0) + [e21, e22] + [e12, 0]
+=
+0
+(21)
+δ2µ1(e11, e22, e12)
+=
+−µ1(e11, −e12) − [e11, −e21]
++µ1(0, e12) + µ1(e22, −e12) + [0, e12] + [e22, e21]
+=
+0
+(22)
+δ2µ1(e11, e22, e21)
+=
+−µ1(e11, e21) − [e11, e12]
++µ1(0, e21) + µ1(e22, −e21) + [0, e21] + [e22, −e12]
+=
+0
+(23)
+δ2µ1(e11, e21, e22)
+=
+−µ1(e11, −e21) − [e11, −e12]
++µ1(−e21, e22) + µ1(e21, 0) + [−e12, e22] + [e21, 0]
+=
+0
+(24)
+δ2µ1(e22, e12, e21)
+=
+−µ1(e22, e11 + e22) − [e22, e22 + e11]
++µ1(−e12, e21) + µ1(e12, e21) + −[e21, e21] + [e12, e12]
+=
+0
+(25)
+19
+
+δ2µ1(e22, e21, e12)
+=
+−µ1(e22, e11 + e22) − [e22, e22 + e11]
++µ1(e21, e12) + µ1(e21, −e12) + [e12, e12] + [e21, −e21]
+=
+0
+(26)
+Using Equations 18, 19, 20, 21, 22, 23, 24, 25 and 26 we conclude that Equation 18
+holds. Hence µt is an equivariant deformation of L of order 1.
+References
+[1] L. Corwin, Y. Ne’eman, S. Sternberg, Graded Lie algebras in mathematics and
+physics (Bose-Fermi symmetry), Rev. Modern Phys. 47 (1975) 573–603.
+[2] V. G. Kac, Lie superalgebras, Advances in Math. 26 (1) (1977) 8–96.
+[3] D. A. Le˘ıtes, Cohomology of Lie superalgebras, Funkcional. Anal. i Priloˇzen.
+9 (4) (1975) 75–76.
+[4] M. Gerstenhaber, On the deformation of rings and algebras, Ann. of Math. (2) 79
+(1964) 59–103.
+[5] M. Gerstenhaber, On the deformation of rings and algebras. II, Ann. of Math. 84
+(1966) 1–19.
+[6] M. Gerstenhaber, On the deformation of rings and algebras. III, Ann. of Math. (2)
+88 (1968) 1–34.
+[7] M. Gerstenhaber, On the deformation of rings and algebras. IV, Ann. of Math.
+(2) 99 (1974) 257–276.
+[8] M. Gerstenhaber, S. D. Schack, On the deformation of algebra morphisms and
+diagrams, Trans. Amer. Math. Soc. 279 (1) (1983) 1–50.
+[9] B. Binegar, Cohomology and deformations of Lie superalgebras, Lett. Math.
+Phys. 12 (4) (1986) 301–308.
+[10] A. Nijenhuis, R. W. Richardson, Jr., Deformations of Lie algebra structures, J.
+Math. Mech. 17 (1967) 89–105.
+20
+
+[11] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math.
+(2) 78 (1963) 267–288.
+[12] H. Benamor, G. Pinczon, The graded Lie algebra structure of Lie superalgebra
+deformation theory, Lett. Math. Phys. 18 (4) (1989) 307–313.
+[13] D. Liu, N. Hu, Leibniz superalgebras and central extensions, J. Algebra Appl.
+5 (6) (2006) 765–780.
+[14] A. Nijenhuis, R. W. Richardson, Jr., Cohomology and deformations in graded Lie
+algebras, Bull. Amer. Math. Soc. 72 (1966) 1–29.
+21
+

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+Exploring Machine Learning Techniques to Identify Important
+Factors Leading to Injury in Curve Related Crashes
+Mehdi Moeinaddinia, Mozhgan Pourmoradnasseria, Amnir Hadachia and Mario Coolsb,c,d
+aITS Lab, Institute of Computer Science, University of Tartu, Narva mnt 18, 51009 Tartu, Estonia
+bLEMA Research Group, Urban & Environmental Engineering Department, University of Liège, Liège, Belgium
+cDepartment of Informatics, Simulation and Modelling, KULeuven Campus Brussels, Brussels, Belgium
+dFaculty of Business Economics, Hasselt University, Diepenbeek, Belgium
+A R T I C L E I N F O
+Keywords:
+Pre-Crash Events, Machine Learning,
+Number of Vehicles with or without
+Injury, Curve Related Crashes, Most
+Effective Variables
+A B S T R A C T
+Different factors have effects on traffic crashes and crash-related injuries. These factors include
+segment characteristics, crash-level characteristics, occupant level characteristics, environment
+characteristics, and vehicle level characteristics. There are several studies regarding these fac-
+tors’ effects on crash injuries. However, limited studies have examined the effects of pre-crash
+events on injuries, especially for curve-related crashes. The majority of previous studies for
+curve-related crashes focused on the impact of geometric features or street design factors. The
+current study tries to eliminate the aforementioned shortcomings by considering important pre-
+crash events related factors as selected variables and the number of vehicles with or without
+injury as the predicted variable. This research used CRSS data from the National Highway Traf-
+fic Safety Administration (NHTSA), which includes traffic crash-related data for different states
+in the USA. The relationships are explored using different machine learning algorithms like the
+random forest, C5.0, CHAID, Bayesian Network, Neural Network, C&R Tree, Quest, etc. The
+random forest and SHAP values are used to identify the most effective variables. The C5.0 al-
+gorithm, which has the highest accuracy rate among the other algorithms, is used to develop the
+final model. Analysis results revealed that the extent of the damage, critical pre-crash event, pre-
+impact location, the trafficway description, roadway surface condition, the month of the crash,
+the first harmful event, number of motor vehicles, attempted avoidance maneuver, and roadway
+grade affect the number of vehicles with or without injury in curve-related crashes.
+1. Introduction
+Globally, more than 1.25 million people die per year as a result of road traffic crashes (WHO, 2018), and 20-50
+million people suffer minor and major injuries due to motor vehicle crashes (WHO, 2018). Crashes involve the potential
+loss of human life and damage to vehicles in addition to causing extra travel costs as a result of delays in traffic (Alireza,
+2002). Road traffic crashes impose considerable economic and social losses on vehicle manufacturers, society, and
+transportation agencies (Haghighi, Liu, Zhang and Porter, 2018). Although during the last decade policymakers and
+planners have tried to reduce these losses, still more research and studies are needed to identify factors that have effects
+on crash injuries to reduce these social and economic losses.
+The number of injuries is twice as high on curves in comparison to straight roads (Chen, 2010). Some of these
+curve-related crashes occurred due to drivers that cannot recognize the sharpness and presence of upcoming curves
+(Wang, Hallmark, Savolainen and Dong, 2017). The probability of a fatal crash at horizontal curves is significantly
+higher than in other segments (Wang et al., 2017). In 2008, around 27 percent of fatal crashes in the United States
+occurred at horizontal curves and most of these curve-related fatalities (over 80 percent) were in roadway departures
+(FHWA, 2018). So, annually more than one-quarter of all motor-vehicle fatalities in the United States are related to
+curve-related crashes (Wang et al., 2017). Because of this huge number of fatalities and injuries, the interest in curve-
+related crashes is significantly high. Thus, there is a need to examine the relationship between injuries and factors that
+have important effects on injuries in these crashes.
+There are five categories of factors that have effects on traffic crash injuries. These factors include crash level factors
+such as crash time, crash type, cause of crash and speed (Hao, Kamga and Wan, 2016; Qin, Ivan, Ravishanker, Liu
+and Tepas, 2006), vehicle level factors such as vehicle age and type (Richter, Pape, Otte and Krettek, 2005; Bedard,
+ORCID(s): 0000-0002-0679-3537 (M. Moeinaddini); 0000-0002-2092-816X (M. Pourmoradnasseri); 0000-0001-9257-3858 (A.
+Hadachi)
+Page 1 of 14
+arXiv:2301.01771v1  [cs.LG]  4 Jan 2023
+
+Guyatt, Stones and Hirdes, 2002; Langley, Mullin, Jackson and Norton, 2000), occupant level factors such as the
+number of occupants, driver attention and alcohol involvement (Movig, Mathijssen, Nagel, Van Egmond, De Gier,
+Leufkens and Egberts, 2004; Petridou and Moustaki, 2000), roadway design and environmental level factors such as
+the number of lanes, traffic control, road curvature, road grade and pavement surface (Moeinaddini, Asadi-Shekari
+and Shah, 2014; Moeinaddini, Asadi-Shekari, Sultan and Shah, 2015; RENGARASU, Hagiwara and Hirasawa, 2007;
+Aarts and Van Schagen, 2006; Karlaftis and Golias, 2002; Ahmed, Abdel-Aty and Yu, 2012; Brijs, Karlis and Wets,
+2008; Golob and Recker, 2003).
+There are various studies regarding the effects of the aforementioned five categories on crash severities. (Duddu,
+Penmetsa and Pulugurtha, 2018) examined the effects of road characteristics, environmental conditions, and driver
+characteristics on driver injury severity (for both at-fault and not-at-fault drivers), using a partial proportional odds
+model. The results of this study show that the age of the driver, physical condition, gender, vehicle type, and the
+number and type of traffic rule violations have significantly higher impacts on injury severity in traffic crashes for
+not-at-fault drivers compared to at-fault drivers. In addition, road characteristics, weather conditions, and geometric
+characteristics were observed to have similar effects on injury severity for at-fault and not-at-fault drivers. Driving
+inattention and distracted driving behavior are two important causes of traffic crashes (Bakhit, Osman, Guo and Ishak,
+2019). (Guo and Fang, 2013) predicted high-risk drivers using personality, demographic, and driving characteristic
+data. The results of their study show that the driver’s age, personality, and critical incident rate have significant effects
+on crash and near-crash risk. Three inter-related variables, including failures of driver attention, misperceptions of
+speed and curvature, and poor lane positioning, are important reasons for driver errors associated with horizontal
+curves (Charlton, 2007).
+(Charlton, 2007) used simulation to find the level of driver attention by comparing advance warning, delineation,
+and road marking. The results of this study show rumble strips can produce appreciable reductions in speed compared
+to advance warning signs. (Haghighi et al., 2018) examined the effects of different roadway geometric features (e.g.
+curve rate, lane width, narrow shoulder, shoulder width, and driveway density) on the severity outcomes in two-lane
+highways in rural areas, using data from 2007 to 2009 in Illinois. In their research, the effects of environmental
+conditions and geometric features on crash severity were analyzed using a multilevel ordered logit model. The results
+showed that the presence of a 10-ft lane and/or narrow shoulders, lower roadside hazard rate, higher driveway density,
+longer barrier length, and shorter barrier offset have lower severe crash risk.
+(Wang et al., 2017) considered the effects of variables such as driver demographic and behavioral characteristics,
+traffic environment characteristics, and roadway design characteristics on the odds of a safety-critical event (e.g., using
+a cell phone, interaction with passengers, external distraction, talking or singing, reaching or moving objects, and
+drinking or eating) in curve related crash and near-crash events using a logistic regression model. However, some
+important variables, such as variables that can explain the critical events and reactions or maneuvers of drivers during
+the crash were missing in this study. Moreover, this study also did not consider the effects of these factors on injury
+severity. Some limited studies examine driver behavior on traffic safety negotiated with curve using simulators (e.g.,
+(Jeong and Liu, 2017); (Yotsutsuji, Kita, Xing and Hirai, 2017); (Abele and Møller, 2011); (Charlton, 2007)) but these
+studies represent just a simplified real-life situation based on some certain assumptions and validation is a challenging
+process in these studies.
+Various studies focused on curve characteristics and crash risk (Elvik, 2013). The majority of these studies in-
+vestigated the effects of speed and speed limit (e.g., (Yotsutsuji et al., 2017), (Wang et al., 2017); (Dong, Nambisan,
+Richards and Ma, 2015); (Vayalamkuzhi and Amirthalingam, 2016)) and road design characteristics such as radius of
+the curve, curve rate and curve length on traffic safety (e.g., (Yotsutsuji et al., 2017); (Haghighi et al., 2018); (Wang
+et al., 2017); (Khan, Bill, Chitturi and Noyce, 2013); (Schneider IV, Savolainen and Moore, 2010)). In addition to
+speed and road design factors, some researchers investigated the effects of socio-demographic factors for drivers (age,
+gender, income, etc.) on traffic safety in curve-related crashes (e.g., (Wang et al., 2017)). However, only a few efforts
+have been undertaken to quantify the effects of pre-crash events on traffic safety and crash injuries, specifically in
+curve crashes (Wang et al., 2017). To our knowledge, little is known in the related transportation literature regarding
+the impacts of pre-crash events on traffic injuries for curve-related crashes (Bärgman, Boda and Dozza, 2017). The
+current study tries to eliminate these shortcomings by considering pre-crash events related factors as selected variables
+and the number of vehicles with or without injury as the predicted variable in curve-related crashes.
+Page 2 of 14
+
+2. Data and Methodology
+This research focuses on the relationship between pre-crash events related factors and the number of vehicles with
+or without injury in different states of the United States for curve-related crashes in 2020. In this study, the data
+are extracted from Crash Report Sampling System (CRSS) in the National Highway Traffic Safety Administration
+(NHTSA) report. This database has data for 94718 vehicles involved in crashes in different states of the United States.
+The data are extracted from all reliable police-reported motor vehicle traffic crashes. The database includes data for
+pedestrians, cyclists, and all types of motor vehicles and covers different types of crashes. In this study, vehicle-related
+data are used to explore the effects of pre-crash events on the number of vehicles with or without injuries for curve-
+related crashes. Around 8% of the involved vehicles in these crashes are related to crashes on curves (7542 crashes
+out of 94718). Table 1 shows the frequency of crashes based on roadway alignment for cases with available injury
+levels (90269 crashes out of 94718). This table indicates the proportion of involved vehicles in curve-related crashes
+for vehicles without or with injuries.
+Table 1
+Frequency of crashes based on roadway alignment.
+Injury
+No
+Yes
+Total
+Roadway Alignment
+Row%
+Col%
+Cell%
+Row%
+Col%
+Cell%
+No.
+1,864
+81
+3
+2
+428
+19
+1
+0
+2,292
+Non-Trafficway
+51,97
+67
+87
+58
+25,604
+33
+84
+28
+77,574
+Straight
+1,758
+55
+3
+2
+1,436
+45
+5
+2
+3,194
+Curve Right
+1,502
+49
+3
+2
+1,581
+51
+5
+2
+3,083
+Curve Left
+563
+57
+1
+1
+429
+43
+1
+0
+992
+Curve-Unknown Direction
+1,998
+65
+3
+2
+1,085
+35
+4
+1
+3,083
+Not Reported
+35
+69
+0
+0
+16
+31
+0
+0
+51
+Total
+59,69
+66
+100
+66
+30,579
+34
+100
+34
+90,269
+Different pre-crash variables are considered predictors for the predicted variable, i.e., the number of vehicles with
+or without injury (0: vehicle without injury, 1: vehicle with injury) in the crashes that occurred in a curve. Table 2
+defines these variables after decoding. The selected variables in Table 2 present pre-crash events in addition to the
+driver behavior and some crash level data, such as the month of the crash. Since the crash rate may differ in different
+seasons, the month of the crash indicates the crashes in the winter. In addition, since more occupants in crashes may
+lead increased chance of vehicle injury, the effect of the number of occupants in each vehicle is also tested in this study.
+Some driver behavior-related factors may affect the number of vehicles with or without injury. These factors include
+driver errors (e.g., careless driving, aggressive driving, improper or erratic lane changing and overcorrecting), driving
+too fast, and avoidance maneuvers. Some factors represent environment and vehicle conditions. For example, the
+vehicle’s manufacturing year represents the vehicle’s condition assuming that newer and less damaged cars may lead
+to fewer injuries because of more advanced safety facilities. The driving environment condition is presented by traffic
+way description (one way or two ways and how the traffic ways are separated), speed limit, roadway surface condition,
+and the presence of traffic controls.
+The rest of the selected variables, such as the harmful events, the critical pre-crash events, the vehicle’s stability
+after the critical event, the location of the vehicle after the critical event, and the crash type, represent pre-crash events.
+As expected, all required data for the main variables are not reported for all curve-related crashes in the NHTSA report,
+and there are many not reported or unknown data for these variables. In addition, to focus on curve-related crashes,
+only the vehicles that negotiate a curve prior to realizing an impending critical event or just prior to impact are included.
+Therefore, after data preparation (removing incomplete information for the main variables and including the vehicles
+that negotiate a curve), the total number of curve-related crashes retained in this study equals 740.
+Page 3 of 14
+
+Table 2: Selected variables.
+Variable
+Description
+Coding
+Urban or rural
+The geographical area of the crash is essen-
+tially urban or rural
+1: urban
+2: rural
+Number of motor
+vehicles
+Number of motor vehicles involved in the
+crash
+0: 1
+1: >1
+Number of
+occupants
+The number of occupants in each vehicle
+0: 1
+1: >1
+First harmful
+event
+The first injury or damage producing event
+0: other events
+1: collision with motor vehi-
+cles in transport
+Vehicle’s Model
+Manufacturer’s model year of the vehicle
+0: <2010
+1: >=2010
+Initial contact
+point
+The area on the vehicle that produced the
+first instance of injury or damage
+0: other areas
+1: front
+Extent of
+damage
+The amount of damage sustained by the ve-
+hicle
+0: Not disabling damage
+1: Disabling damage
+Most harmful
+event
+The event that resulted in the most severe
+injury or the greatest damage
+0: other events
+1: collision with motor vehi-
+cles in transport
+Speeding-related
+The driver’s speed was related to the crash
+0: no
+1: yes
+Page 4 of 14
+
+Driver Error
+Factors related to the driver errors ex-
+pressed by the investigating officer
+0: no error
+1:
+error
+(e.g.,
+careless
+driving or aggressive driv-
+ing/road rage or operating
+the vehicle in an erratic,
+reckless, or negligent man-
+ner,
+improper
+or
+erratic
+lane changing or improper
+lane usage, or driving on
+the wrong side of two-way
+traffic way, etc.)
+Traffic way
+Trafficway description
+0: divided two-way and oth-
+ers
+1: not divided two-way
+Speed limit
+The posted speed limit in miles per hour
+0: <46
+1: >=46
+Roadway alignment
+The roadway alignment prior to the critical
+pre-crash event
+1: curve right
+2: curve left
+3: curve – unknown direc-
+tion
+Grade
+Roadway grade prior to the critical pre-
+crash event
+0: not level
+1: level
+Roadway surface con-
+dition
+Roadway surface condition prior to the crit-
+ical pre-crash event
+0: not dry
+1: dry
+Traffic control device
+The presence of traffic controls in the envi-
+ronment prior to the critical pre-crash event
+0: no
+1: yes
+Critical pre-crash event
+The critical event which made this crash
+imminent
+1:
+the vehicle itself (loss
+of control, traveling too fast,
+etc.)
+2: other vehicles (traveling
+in the opposite direction, en-
+croaching into the lane, etc.)
+3:
+others (pedestrian in
+the road, animal approach-
+ing road, etc.)
+Attempted
+avoidance
+maneuve r
+Movements/actions taken by the driver
+within the crash
+0: no action
+1: braking
+2: others.
+Page 5 of 14
+
+Pre-impact stability
+The stability of the vehicle after the critical
+event but before the impact.
+0:
+no tracking (skidding,
+loss-of-control, etc.)
+1: tracking
+Pre-impact location
+The location of the vehicle after the critical
+event but before the impact
+0: not departed roadway
+1: departed roadway
+Crash Type
+The type of crash
+0: others
+1:
+single driver involved
+(roadside departure,
+colli-
+sion with pedestrians, etc.)
+Different methods like multinomial logit models, general linear models, order prohibit models, linear regression
+((Clark and Cushing, 2004); (Levine, Kim and Nitz, 1995); (Abdel-Aty, 2003); (Yang, Zhibin, Pan and Liteng, 2011);
+(Fan, Kane and Haile, 2015)), Negative binomial (NB) models ((Abdel-Aty and Radwan, 2000); (Hadayeghi, Shalaby
+and Persaud, 2003); (Hadayeghi, Shalaby and Persaud, 2007); (Wei and Lovegrove, 2013); (Moeinaddini et al., 2014);
+(Moeinaddini et al., 2015)), Poisson models ((Movig et al., 2004)) and Zero-inflated Poisson and NB models ((Qin,
+Ivan and Ravishanker, 2004); (Shankar, Milton and Mannering, 1997)) have been used to analyze traffic fatalities and
+injuries related data. In addition to these methods, some studies have used decision tree approaches (e.g., ID3, C4.5,
+C5.0, C&R, CHAID) to find the major contributing factors to collision and the number of fatalities. For example,
+the Classification and Regression (C&R) Tree was used by (Tavakoli Kashani, Shariat-Mohaymany and Ranjbari,
+2011). One of the most common approaches for representing classifiers is Decision Trees (Maimon and Rokach, 2005).
+Researchers from different disciplines such as machine learning, statistics, pattern recognition, and data mining use
+decision trees to analyze data in a more comprehensive way (Maimon and Rokach, 2005).
+(Zhang and Fan, 2013) used data mining models using ID3 and C4.5 decision tree algorithms to evaluate the traffic
+collision data in Canada. (Chong, Abraham and Paprzycki, 2005) compared different machine learning paradigms,
+including neural networks trained using hybrid learning approaches, support vector machines, decision trees, and a
+concurrent hybrid model involving decision trees and neural networks to model the injury severity of traffic crashes.
+The results of their study show that for the non-incapacitating injury, the incapacitating injury, and the fatal injury
+classes, the hybrid approach performed better than a neural network, decision trees, and support vector machines.
+(da Cruz Figueira, Pitombo, Larocca et al., 2017) used the C&R algorithm as a useful tool for identifying potential
+sites of crashes with victims. (Chang and Wang, 2006) used C&R to find the relationships between crash severity with
+factors such as drivers’ and vehicles’ variables and the road and environment characteristics. The results of their study
+show that vehicle type is one of the most important factors that have an effect on the severity of the crash.
+The majority of traditional and parametric analysis techniques have different assumptions and pre-defined functions
+that describe the relationship between the selected and the predicted variables (Chang and Wang, 2006). If these
+assumptions are violated, the model power can be affected negatively (Griselda, Joaquín et al., 2012). Therefore,
+assumption-free models such as decision trees can be used to avoid this limitation (Griselda et al., 2012). (Kuhnert,
+Do and McClure, 2000) compared the results of different methods such as logistic regression, multivariate adaptive
+regression splines (MARS), and C&R in the analysis of data related to the injury in motor vehicle crashes. The findings
+of their study show the usefulness of non-parametric techniques such as C&R and MARS to provide more attractive
+and informative models (Griselda et al., 2012). (Pandya and Pandya, 2015) compared the results of ID3, C4.5, and
+C5.0 with each other. They found that among all these classifiers, C5.0 gives more efficient, accurate, and fast results
+with low memory usage (fewer rules compare to other techniques).
+Finding the most accurate prediction models can help planners and designers to develop better traffic safety control
+policies. In the current study, a variety of modeling techniques is envisaged as possible analysis methods, and the most
+appropriate model based on the accuracy rate is retained for further discussion of the results.
+The first step to finding the most appropriate model is applying random forest to identify the most influential
+variables among the selected variables. Then, the identified effective variables are used as selected variables to explore
+the effects of these selected variables on the predicted variable. Random forest is a common method for selecting the
+Page 6 of 14
+
+most effective variables in studies with a high number of predictors ((Jahangiri, Rakha and Dingus, 2016); (Kitali,
+Alluri, Sando, Haule, Kidando and Lentz, 2018); (Zhu, Li and Wang, 2018); (Aghaabbasi, Shekari, Shah, Olakunle,
+Armaghani and Moeinaddini, 2020); (Lu and Ma, 2020)). The random forest aggregates many binary decision trees.
+Cross-validation (10-fold cross-validation) which generally results in a less biased model than other methods like
+train and test split is applied to estimate the accuracy. Random and grid search for hyper-parameter optimization
+(Bergstra and Bengio, 2012) are used for hyper-parameter optimization. After applying random forests, the SHAP
+(SHapley Additive exPlanations) values (Lundberg and Lee, 2017) are used to select the most important variables. The
+SHAP explains the contribution of each observation and provides local interpretability but the traditional importance
+values explain each predictor’s effects that are based on the entire population. After estimating SHAP values, the not-
+important variables are excluded one by one. The accuracy rates for each step of exclusion are used to find the most
+effective variables.
+2.1. Models Description
+A couple of machine-learning algorithms were explored to identify the relationships between curve-related crashes
+and pre-crash events. To this end, identifying the significant features for training the models is an important step to
+ensure a good training process and better results.
+2.1.1. Feature selection
+Approximating the functional relationship between the input data and the output is one of the fundamental problems
+when applying machine learning methods. Selecting the significant feature for training the machine learning models
+is crucial to avoid overfitting and to induce high computational costs. Therefore, our approach utilized the power of
+random forest as a classifier and interpreted with Shapely values. Relying on SHAP values helps to perform feature
+selection based on ranking. This means that instead of using the embedded feature selection process of the random
+forest, we use the SHAP value to select the ones with the highest shapely values. This approach’s advantage is avoiding
+any bias in the native tree-based feature built by the random forest approach.
+2.1.2. C5.0 decision tree algorithm
+Decision trees are built using recursive partitioning. The algorithm starts by creating the root node, which in our
+case, is the actual data. Then, based on the most significant feature selected, the data is partitioned into groups. These
+groups are the distinct value of this feature, and this decision forms the first set that constitutes the tree branches. The
+algorithm divides the nodes until the criterion is reached (Algorithm 1). In practice, the C5.0 algorithm decides the
+split by using the concept of entropy for measuring purity. This means for a segment of data 푆, if the entropy is close
+to value 0 indicates that the data sample is homogenous, and the opposite if it is close to value 1 and it is defined as
+follows:
+Entropy(푆) =
+푚
+∑
+푖=1
+−푃푖 log2 푃푖;
+(1)
+where 푚 refers to the number of different class levels, and 푝푖 is the proportion of values falling into the class level
+푖. However, even after conducting this step, to understand the homogeneity of the data, the algorithm still needs to
+decide how to split the set. To solve this, the C5.0 uses the entropy to spot the variations of homogeneity resulting
+from the split. This measure is called Information Gain, which is defined using equation 2. It quantifies the gained
+information of an attribute 퐴, when selecting data set 푆 .
+InfoGain(푆, 퐴) = Entropy(푆) −
+∑
+푣∈푉 (퐴)
+|푆푣|
+|푆| Entropy(푆푣);
+(2)
+where 푉 (퐴) is the set of all possible attribute values for 퐴, and 푆푣 is the subset of 푆 for which 퐴 has value 푣.
+Hence, the creation of homogeneous groups after the split on specific feature is better when the information gain is
+high.
+Page 7 of 14
+
+Algorithm 1 C5.0 decision tree
+Require: Data 푇 = {(푥푖, 푦푗), 푖, 푗 ∈ {1, 2, ⋯ , 푛}}
+Require: Attributes 퐴 = {푎푙, 푙 ∈ {1, 2, ⋯ , 푝}}
+Require: InfoGain = {푔푘, 푘 ∈ {1, 2, ⋯ , 푑}}
+Create node 푁
+if 푆 are all the same class, 퐶 then
+label 푁 with class 퐶 → 푁(퐶)
+return 푁(퐶) as leaf node
+end if
+if 퐴 ≠ ∅ or 푎푖 = 푎푗 for all 푎푖, 푎푗 ∈ 퐴 then
+label 푁 with majority class 푀 in 푆 → 푁(푀) return 푁(푀) as leaf node
+end if
+select best attribute 푎푖 using InfoGain
+for every 푎푣
+푖 ∈ 푎푖 do
+label node 푁 with splitting criterion
+if 푆푣 ≠ ∅ then where 푆푣 is the set of data in 푆 equal to 푎푣
+푖
+label 푁 with majority class 푀 in 푆 → 푁(푀) return 푁(푀) as leaf node
+else return 푁 with splitting criterion (푆푣, 퐴{푎푖})
+end if
+end for
+2.1.3. Chi-squared Automatic Interaction Detection
+Chi-squared automatic interaction detection (CHAID) is one of the techniques based on a tree machine learning
+algorithm. Hence, the algorithm relies on the multiway split using Chi-square or F-test. The CHAID algorithm uses
+two approaches for separation reference depending on the variable. If the variable is categorical, Pearson’s Chi-square
+is adequate; otherwise, the likelihood ratio Chi-square statistic. The CHAID uses Pearson’s Chi-squared test of inde-
+pendence to test the existence of an association between two categorical variables (“true" or“false"). The main steps
+to calculate Chi-square for the split are as follows:
+1. Calculating the deviation for "true" and "false" in the node which constitutes the Chi-square computation.
+2. Getting the split by computing the sum of all the chi-square of "true" and "false" of each split node.
+2.1.4. Classification and Regression Tree node
+The Classification and Regression (C&R) Tree node algorithm is a classification algorithm that is based on a binary
+tree built by splitting nodes into two child nodes continually similarly to C5.0 method. The algorithm is designed in
+such a manner that it follows three major steps:
+1. Identifying each feature’s best split.
+2. Identifying the node’s best split.
+3. Based on the step 2 result, the node is split and repeats the process from step 1 till the stopping criterion is met.
+For performing the split, Gini’s impurity index criterion is used and it is defined as follows for a node 푡:
+Gini (푡) =
+∑
+푖,푗
+퐶(푖 ∣ 푗)푃 (푖 ∣ 푛)푃(푗 ∣ 푛);
+(3)
+where,
+• 퐶(푖 ∣ 푗) is the cost of classifying wrongly a class 푗 as a class 푖 and it is defined as follows:
+퐶(푖 ∣ 푗) =
+{
+1
+푖 ≠ 푗
+0
+푖 = 푗
+(4)
+• 푃 (푖 ∣ 푛) is the probability of 푖 falls into node 푛
+Page 8 of 14
+
+• 푃(푗 ∣ 푛) is the probability of 푗 falls into node 푛
+The splitting criterion is based on Gini’s impurity criterion, which follows a decrease of impurity using the following
+formula:
+ΔGini (푠, 푛) = Gini (푡)−푃푅Gini (푛푅)−푃퐿Gini (푛퐿);
+(5)
+where, ΔGini (푠, 푛) is the decrease of impurity at node 푛 with a split 푠. 푃푅 and 푃퐿 are, respectively, the probabilities of
+sending the case to the right or the left node 푛푅 or 푛퐿, and Gini (푛푅) and Gini (푛퐿) are respectively the Gini impurity
+index of the right and left child node.
+2.1.5. Bayesian network
+A Bayesian network is a compact graphical interpretation of the causal relationship between variables of a dataset.
+The structure is presented by a directed acyclic graph (DAG), and parameters are expressed as conditional probabilities.
+In order to learn the network, structure and conditional probabilities must be known. The structure is learned by DAG
+search algorithms and assigning prior probabilities. Then parameters are determined by the maximum likelihood
+estimation. Including the prior knowledge of the causal structure of DAG is a crucial step in learning parameters in
+this method.
+2.1.6. Logistic regression
+Logistic regression is a statistical classification algorithm that maps the results of a linear function onto the regres-
+sion function
+푃 (퐗) =
+푒훽0+훽퐗
+1 + 푒훽0+훽퐗 .
+(6)
+Based on the maximum likelihood method, the coefficients 훽0 and 훽 are estimated in the training phase. This algorithm
+is a suitable classifier when variables can be categorized into two or a few classes. In contrast to linear regression, the
+logistic function associates probabilities to each possible output class by producing an S-shape curve and a range of
+output between 0 and 1.
+2.1.7. Neural Network
+Our case study adopted neural network architecture based on five hidden layers using a multilayer perception
+model. The multilayer perceptron is the most straightforward feed-forward network. When the layers are increased, it
+can provide exciting performance in learning and precision. The units are arranged into a set of layers, and each layer
+contains a collection. The first layer is the input layer, populated by the value of input features. Then, the input is later
+connected to a very hidden layer in a fully connected fashion. The last layer is the output layer, which has one unit for
+each network output weight with a stopping rule on the error generated.
+2.1.8. QUEST algorithm
+Quick Unbiased Efficient Statistical Tree (QUEST) is a cost-effective classification method for building binary
+decision trees for categorical and quantitative predictors with a large number of variables. Instead of examining all
+possible splits, QUEST uses statistical analysis and a multi-way chi-square test to select the variable at each node.
+This leads to a significant reduction in the time complexity, compared to methods like R&C Tree, by avoiding ineffi-
+cient splits. Moreover, the split point at each node is selected based on a quadratic discriminant analysis of potential
+categories.
+2.1.9. Decision List
+Decision lists are a representation of Boolean functions that work as a collection of rule-based classifiers. Rules
+are learned sequentially and based on a greedy approach by identifying the rule that covers the maximum number of
+instances in the input space 푋. Then rules are appended to the decision tree one at a time, and the corresponding data
+is removed from the data set in each process. A new instance is classified by examining the rules in order, and if no
+rule is satisfied, the default rule is applied.
+Features are defined as Boolean functions 푓푖 that map the input space 푋 onto {0, 1}. For a given set of features
+ = {푓푖(푥)}, with 푥 ∈ 푋, and the training set  , learning algorithm returns a selection of features ′ ⊂ . Once the
+effective features are determined, for an arbitrary input 푥, the output of the decision tree is calculated according to a
+set of conditions to be satisfied.
+Page 9 of 14
+
+3. Results
+The overall accuracy for the applied random forest model with all predictors is 0.67. However, the overall accuracy
+for the applied random forest model with the 10 most important predictors based on SHAP values can reach 0.68.
+Therefore, these 10 predictors are selected as the most effective variables among the selected variables (the extent of
+the damage, critical pre-crash event, pre-impact location, the trafficway description, roadway surface condition, the
+month of the crash, the first harmful event, number of motor vehicles, attempted avoidance maneuver, and roadway
+grade). The identified effective variables are used as selected variables for a variety of possible modeling methods to
+find the most appropriate model based on the accuracy rate. The accuracy of the traditional logistic regression model
+is lower than non-parametric models like C5.0, CHAID, C&R Tree, and Bayesian network. In addition, potential high
+correlations between some of the selected crash-related variables in this study may lead to a multicollinearity concern.
+Therefore, it is better to use modeling techniques that can handle multicollinearity issues to be able to consider the
+effects of these variables. Since non-parametric models can handle multi-collinearity issues in crash-related data better
+than traditional and parametric models, based on the overall accuracy that is achieved for each model (refer Table 3),
+the C5.0 model with the highest accuracy score is used for modeling the most important predictors. To develop this
+C5.0 model, the minimum number of records per child branch number is considered to be 2 and the pruning severity
+is considered to be 75. To collapse weak subtrees, trees are pruned in local and global pruning stages. Cross-validate
+is used to estimate the accuracy of the model. This technique uses a set of models using subsets of the data to estimate
+the accuracy. C5.0 is an improved version of C4.5 that is an extension of ID3 algorithm ((Quinlan, 1993); (Witten,
+2011); (Kotsiantis, Zaharakis, Pintelas et al., 2007); (Quinlan, 1996)).
+Table 3
+The overall accuracy of the possible analysis methods.
+Applied model
+Overall accuracy (%)
+C5.0
+71.757
+CHAID
+70.135
+C&R Tree
+68.784
+Bayesian Network
+67.973
+Logistic Regression
+66.486
+Neural Network
+65.27
+Quest
+63.514
+Decision List
+63.108
+The applied C5.0 model is shown in Figure1. This figure shows the total percentage and the classification of the
+predicted variable for each node. The overall accuracy based on the results is more than 71%. Sixteen terminal nodes
+(the bottom nodes of the decision tree) have been shown in Figure 1 and it is clear that this model has 8 splitters, i.e. the
+extent of the damage, first harmful event, the month of the crash, critical pre-crash event, pre-impact location, roadway
+surface condition, the trafficway description, and roadway grade. The most important variable for data segmentation
+is the extent of the damage. The probability of having vehicles without injury in curve-related crashes is high in node
+1. Node 1 shows that not disabling damage results in a higher rate for vehicles without injury. In contrast, from node
+23 one can depict that disabling damage results in a higher rate for vehicles with injury. The findings show that all
+environmental and pre-crash events that lead to driving with extra caution are related to vehicles with a lower chance
+of having injuries in curve-related crashes. For example, for vehicles that have disabling damage (refer node 23), the
+model prediction is with injury if the month of the crash is not in the winter. The same prediction can be expected for
+the months in the winter if the surface is dry (refer node 27) and the pre-impact location is departed the roadway (refer
+node 29). However, the prediction for months in the winter can be without injury if the surface is not dry (refer node
+26) or the pre-impact location has not departed the roadway (refer node 28) for dry surfaces. The effects of driving with
+extra caution can also be noticed for vehicles that do not have collisions with motor vehicles in transport. Node 1 is
+divided into node 2 and node 20 which are related to the first harmful event. For vehicles that have no disabling damage
+(refer node 1) and do not have collisions with motor vehicles in transport (refer node 2), the model prediction is without
+injury if the critical pre-crash event is related to the vehicle itself (refer node 3), the surface is not dry (refer node 4),
+and the month of the crash is in the winter (refer node 8). The same prediction can be expected for the months that are
+not in the winter if the roadway is not a divided two-way road (refer node 7). The same prediction also can be expected
+Page 10 of 14
+
+for the critical pre-crash event that is related to the other vehicles (node 10) while the pre-impact location is departed
+the roadway (refer node 12) and for the other critical pre-crash events while the pre-impact location is not departed
+the roadway (refer node 14). The prediction is also without injury for departed the roadway in this case (refer node
+15) if the roadway is a divided two-way road (refer node 16) or a not level road (refer node 18) for a divided two-way
+road (refer node 17). For vehicles that have no disabling damage (refer node 1) and do not have collisions with motor
+vehicles in transport (refer node 2), the model prediction is with injury if the critical pre-crash event is related to the
+vehicle itself (refer node 3) and the surface is dry (refer node 9). The same prediction can be expected for the critical
+pre-crash event that is related to the other vehicles (node 10) while the pre-impact location is not departed the roadway
+(refer node 11). For vehicles that have no disabling damage (refer node 1) and have collisions with motor vehicles in
+transport (refer node 20), the model prediction is without injury if the pre-impact location is not departed the roadway
+(refer node 21) and with injury if the pre-impact location is departed the roadway (refer node 22). This finding shows
+that departing the roadway is a very important factor for collisions with motor vehicles in transport in curve-related
+crashes. The results confirm that out of all input selected variables, eight main variables play an important role in
+vehicles with or without injury in curve-related crashes. Table 4 shows the importance of these main predictors based
+on the proposed C5.0 algorithm. Higher importance scores mean a greater contribution of the variable in predicting
+the number of vehicles with or without injury. A breakdown of prediction accuracy is also estimated (refer Table 5).
+Table 4
+Importance of the predictors based on the proposed C5.0 algorithm.
+Nodes
+Importance
+Extent of damage
+0.3401
+Pre-impact location
+0.2303
+The first harmful event
+0.2056
+Month of crash
+0.1021
+Roadway surface condition
+0.0433
+Trafficway description
+0.0430
+Roadway grade
+0.0351
+Critical pre-crash event
+0.0006
+Table 5
+Coincidence matrix for predicted values .
+0
+1
+%
+0
+206
+140
+60
+1
+69
+325
+82
+4. Discussion and Conclusion
+To reduce the number of crash injuries and have better planning decisions and strategies, it is important to have
+deep knowledge about factors influencing crash injuries. The proposed C5.0 algorithm (with a higher overall accuracy
+rate compared to the other analysis methods) can help to identify the variables that have the most important impacts
+on the number of vehicles with or without injury in curve-related crashes. This study used the 2020 NHTSA data
+for different states in the USA to find the key variables that affect the number of vehicles with or without injury in
+curve-related crashes. The results show that the extent of the damage, critical pre-crash event, pre-impact location,
+the trafficway description, roadway surface condition, the month of the crash, the first harmful event, number of motor
+vehicles, attempted avoidance maneuver, and roadway grade affect the number of vehicles with or without injury the
+most. The C5.0 model shows that most of the important predictors are related to environmental and pre-crash events
+that lead to driving with extra caution. Analysis results also revealed that departing the roadway is a very important
+factor for collisions with motor vehicles in transport in curve-related crashes. This is in line with previous studies like
+(Wang et al., 2017) that identified traveling too fast on curves as one of the most important factors that contribute to
+Page 11 of 14
+
+Figure 1: The proposed C5.0 model.
+Page 12 of 14
+
+With or without injury
+Nodeo
+000:0
+1.000
+Extentof damage
+0.000
+1.000
+ Node 1 
+ Node 23 
+The first harmful event
+Month of the crash
+0.000
+1.000
+0.000
+1.000
+Node 2
+ Node 20
+Node 24
+ Node 25
+二
+-
+Critical pre-crash event
+Pre-impact location Roadway surface condition
+1.000
+2.000
+3.000
+000'0
+1.000
+000°0
+1.000
+Node 3
+Node10
+Node 13
+Node 21
+Node22
+Node 26
+Node 27
+一
+Roadway surface condition
+Pre-impact location
+Pre-impact location
+Pre-impact location
+0.000
+1.000
+0.000
+1.000
+0.000
+1.000
+0.000
+1.000
+Node 4
+Node 9
+Node 11
+Node 12
+Node 14
+Node 15
+Node 28
+Node 29
+Month of the crash
+Trafficway description
+0.000
+1.000
+0.000
+1.000
+ Node 5
+8apon
+ Node 16 
+Node17
+Trafficway description
+Roadway grade
+0.000
+1.0.00
+0.000
+1.000
+Nodeb
+Node 7
+ Noce 19crash fatalities. (Wang et al., 2017) considered the effects of driver behavior factors such as speeding in curve-related
+crashes. Still, this study did not consider factors such as critical events and pre-critical event factors in addition to the
+reaction or maneuvers of the driver during the crash.
+In (Wang et al., 2017), the icy and snowy road surface is another important factor that was associated with curve-
+related crashes, however, in our study, not dry surface was a significant factor for vehicles with injury for the months
+that are not in the winter. This is in line with the (Eisenberg and Warner, 2005) study that evaluated the impacts of
+snowy surfaces on traffic crash rates in the USA (1975-2000). They found that snow days are associated with fewer
+severe crashes, whereas more no severe crashes and property-damage crashes are reported on snow days. Therefore,
+although icy and snowy surfaces can be an important factor in the crash rate, they do not have a high association with
+severe crashes. Crash type is another important variable in similar research such as (da Cruz Figueira et al., 2017)
+and (Griselda et al., 2012). However, the proposed models in the current study show that crash type is not significant
+while considering critical pre-crash events. Based on the proposed final model, the extent of damage, the pre-impact
+location, the first harmful event, and the critical pre-crash event are among the significant pre-crash events that can
+affect the number of vehicles with or without injury in addition to the environmental factors like the month of the crash,
+roadway surface condition, the traffic way description, and roadway grade. There are limited studies about the impacts
+of these important pre-crash events and environmental factors on traffic injuries (Bärgman et al., 2017) and although
+curve related crashes are associated with a high proportion of severe crashes, there is no study about the effects of
+these important factors on the number of vehicles with or without injury in curve-related crashes. Applying non-
+parametric tree-based models like C5.0 has some advantages compared to traditional regression and other parametric
+models. (Chang and Wang, 2006) highlighted that C5.0 analysis does not require the specification of a functional form
+and also it can handle multi-collinearity problems, which often occur due to the high correlations between selected
+variables in traffic injury data (e.g., collision type and driver/vehicle action; weather condition and pavement condition).
+The proposed C5.0 model can be presented graphically, which is intuitively easy to interpret without complicated
+statistics. It also provides useful results by focusing on limited, yet most influential factors (Chang and Wang, 2006).
+However, non-parametric models have some disadvantages such as a lack of formal statistical inference procedures
+(Chang and Wang, 2006). These models also do not have a confidence interval for the risk factors (splitters) and
+predictions (Chang and Wang, 2006). The structure and accuracy can be changed significantly if different partitioning
+and sampling strategies (e.g., stratified random sampling) are applied for model testing. It is not recommended to
+have a generalization based on the results of nonparametric techniques. Therefore, the tree models are often applied to
+identify important variables, and other modeling techniques are needed to develop final models. Since sampling and
+different partitioning strategies are not applied to the proposed models in this study, this disadvantage is not a great
+concern for the current research.
+Acknowledgments
+This work was supported by the European Social Fund via IT Academy programme, and the Estonian Centre of
+Excellence in IT (EXCITE).
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+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised
+Learning
+Pranav Singh 1 Jacopo Cirrone 2
+Abstract
+Existing self-supervised techniques have extreme
+computational requirements and suffer a substan-
+tial drop in performance with a reduction in batch
+size or pretraining epochs. This paper presents
+Cross Architectural - Self Supervision (CASS),
+a novel self-supervised learning approach that
+leverages Transformer and CNN simultaneously.
+Compared to the existing state-of-the-art self-
+supervised learning approaches, we empirically
+show that CASS-trained CNNs and Transformers
+across four diverse datasets gained an average of
+3.8% with 1% labeled data, 5.9% with 10% la-
+beled data, and 10.13% with 100% labeled data
+while taking 69% less time. We also show that
+CASS is much more robust to changes in batch
+size and training epochs than existing state-of-the-
+art self-supervised learning approaches. We have
+opensourced our code at https://github.
+com/pranavsinghps1/CASS.
+1. Introduction
+Self-supervised learning has emerged as a powerful
+paradigm for learning representations that can be used
+for various downstream tasks like classification, object de-
+tection, and image segmentation. Pretraining with self-
+supervised techniques is label-free, allowing us to train
+even on unlabeled images. This is especially useful in fields
+with limited labeled data availability or if the cost and effort
+required to provide annotations are high. Medical Imaging
+is one field that can benefit from applying self-supervised
+techniques. Medical imaging is a field characterized by min-
+imal data availability. First, data labeling typically requires
+domain-specific knowledge. Therefore, the requirement of
+1Department of Computer Science, Tandon School of En-
+gineering, New York University, New York, NY 11202, USA
+2Center for Data Science, New York University, and Colton
+Center for Autoimmunity, NYU Grossman School of Medicine,
+New York, NY 10011, USA. Correspondence to: Pranav Singh
+<[email protected]>.
+Preliminary work. Under review. Copyright 2023 by the author(s).
+large-scale clinical supervision may be cost and time pro-
+hibitive. Second, due to patient privacy, disease prevalence,
+and other limitations, it is often difficult to release imag-
+ing datasets for secondary analysis, research, and diagnosis.
+Third, due to an incomplete understanding of diseases. This
+could be either because the disease is emerging or because
+no mechanism is in place to systematically collect data about
+the prevalence and incidence of the disease. An example
+of the former is COVID-19 when despite collecting chest
+X-ray data spanning decades, the samples lacked data for
+COVID-19 (Sriram et al., 2021). An example of the latter
+is autoimmune diseases. Statistically, autoimmune diseases
+affect 3% of the US population or 9.9 million US citizens.
+There are still major outstanding research questions for au-
+toimmune diseases regarding the presence of different cell
+types and their role in inflammation at the tissue level. The
+study of autoimmune diseases is critical because autoim-
+mune diseases affect a large part of society and because
+these conditions have been on the rise recently (Galeotti &
+Bayry, 2020; Lerner et al., 2015; Ehrenfeld et al., 2020).
+Other fields like cancer and MRI image analysis have bene-
+fited from the application of artificial intelligence (AI). But
+for autoimmune diseases, the application of AI is partic-
+ularly challenging due to minimal data availability, with
+the median dataset size for autoimmune diseases between
+99-540 samples (Tsakalidou et al., 2022; Stafford et al.,
+2020).
+To overcome the limited availability of annotations, we turn
+to self-supervised learning. Models extract representations
+that can be fine-tuned even with a small amount of labeled
+data for various downstream tasks (Sriram et al., 2021).
+As a result, this learning approach avoids the relatively ex-
+pensive and human-intensive task of data annotation. But
+self-supervised learning techniques suffer when limited data
+is available, especially in cases where the entire dataset size
+is smaller than the peak performing batch size for some
+of the leading self-supervised techniques. This calls for a
+reduction in the batch size; this again causes existing self-
+supervised techniques to drop performance; for example,
+state-of-the-art DINO (Caron et al., 2021) drops classifi-
+cation performance by 25% when trained with batch size
+8. Furthermore, existing self-supervised techniques are
+compute-intensive and trained using multiple GPU servers
+arXiv:2301.12025v1  [cs.CV]  27 Jan 2023
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+over multiple days. This makes them inaccessible to general
+practitioners.
+Existing approaches in the field of self-supervised learning
+rely purely on Convolutional Neural Networks (CNNs) or
+Transformers as the feature extraction backbone and learn
+feature representations by teaching the network to com-
+pare the extracted representations. Instead, we propose to
+combine a CNN and Transformer in a response-based con-
+trastive method. In CASS, the extracted representations of
+each input image are compared across two branches rep-
+resenting each architecture (see Figure 1). By transferring
+features sensitive to translation equivariance and locality
+from CNN to Transformer, our proposed approach - CASS,
+learns more predictive data representations in limited data
+scenarios where a Transformer-only model cannot find them.
+We studied this quantitatively and qualitatively in Section 5.
+Our contributions are as follows:
+• We introduce Cross Architectural - Self Supervision
+(CASS), a hybrid CNN-Transformer approach for
+learning improved data representations in a self-
+supervised setting in limited data availability problems
+in the medical image analysis domain 1
+• We propose the use of CASS for analysis of autoim-
+mune diseases such as dermatomyositis and demon-
+strate an improvement of 2.55% compared to the ex-
+isting state-of-the-art self-supervised approaches. To
+our knowledge, the autoimmune dataset contains 198
+images and is the smallest known dataset for self-
+supervised learning.
+• Since our focus is to study self-supervised techniques
+in the context of medical imaging. We evaluate CASS
+on three challenging medical image analysis problems
+(autoimmune disease cell classification, brain tumor
+classification, and skin lesion classification) on three
+public datasets (Dermofit Project Dataset (Fisher &
+Rees, 2017), brain tumor MRI Dataset (Cheng, 2017;
+Kang et al., 2021) and ISIC 2019 (Tschandl et al.,
+2018; Gutman et al., 2018; Combalia et al., 2019)) and
+find that CASS improves classification performance
+(F1 Score and Recall value) over the existing state of
+the art self-supervised techniques by an average of
+3.8% using 1% label fractions, 5.9 % with 10% label
+fractions and 10.13% with 100% label fractions.
+• Existing methods also suffer a severe drop in perfor-
+mance when trained for a reduced number of epochs
+or batch size ((Caron et al., 2021; Grill et al., 2020b;
+Chen et al., 2020a)). We show that CASS is robust to
+these changes in Sections 5.3.2 and 5.3.1.
+1We have opensourced our code at https://github.
+com/pranavsinghps1/CASS
+• New state-of-the-art self-supervised techniques often
+require significant computational requirements. This
+is a major hurdle as these methods can take around 20
+GPU days to train (Azizi et al., 2021b). This makes
+them inaccessible in limited computational resource
+settings. CASS, on average, takes 69% less time than
+the existing state-of-the-art methods. We further ex-
+pand on this result in Section 5.2.
+2. Background
+2.1. Neural Network Architectures for Image Analysis
+CNNs are a famous architecture of choice for many im-
+age analysis applications (Khan et al., 2020). CNNs learn
+more abstract visual concepts with a gradually increasing
+receptive field. They have two favorable inductive biases:
+(i) translation equivariance resulting in the ability to learn
+equally well with shifted object positions, and (ii) locality
+resulting in the ability to capture pixel-level closeness in the
+input data. CNNs have been used for many medical image
+analysis applications, such as disease diagnosis (Yadav &
+Jadhav, 2019) or semantic segmentation (Ronneberger et al.,
+2015). To address the requirement of additional context
+for a more holistic image understanding, the Vision Trans-
+former (ViT) architecture (Dosovitskiy et al., 2020) has been
+adapted to images from language-related tasks and recently
+gained popularity (Liu et al., 2021b; 2022a; Touvron et al.,
+2021). In a ViT, the input image is split into patches that
+are treated as tokens in a self-attention mechanism. Com-
+pared to CNNs, ViTs can capture additional image context
+but lack ingrained inductive biases of translation and loca-
+tion. As a result, ViTs typically outperform CNNs on larger
+datasets (d’Ascoli et al., 2021).
+2.1.1. CROSS-ARCHITECTURE TECHNQIUES
+Cross-architecture techniques aim to combine the features of
+CNNs and Transformers; they can be classified into two cat-
+egories (i) Hybrid cross architecture techniques and (ii) pure
+cross-architecture techniques. Hybrid cross-architecture
+techniques combine parts of CNNs and Transformers in
+some capacity, allowing architectures to learn unique repre-
+sentations. ConViT (d’Ascoli et al., 2021) combines CNNs
+and ViTs using gated positional self-attention (GPSA) to
+create a soft convolution similar to inductive bias and im-
+prove upon the capabilities of Transformers alone. More
+recently, the training regimes and inferences from ViTs
+have been used to design a new family of convolutional
+architectures - ConvNext (Liu et al., 2022b), outperforming
+benchmarks set by ViTs in classification tasks. (Li et al.,
+2021) further simplified the procedure to create an opti-
+mal CNN-Transformer using their self-supervised Neural
+Architecture Search (NAS) approach.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+On the other hand, pure cross-architecture techniques com-
+bine CNNs and Transformers without any changes to their
+architecture to help both of them learn better representations.
+(Gong et al., 2022) used CNN and Transformer pairs in a
+consistent teaching knowledge distillation format for audio
+classification and showed that cross-architecture distillation
+makes distilled models less prone to overfitting and also
+improves robustness. Compared with the CNN-attention
+hybrid models, cross-architecture knowledge distillation is
+more effective and does not require any model architecture
+change. Similarly, (Guo et al., 2022) also used a 3D-CNN
+and Transformer to learn strong representations and pro-
+posed a self-supervised learning module to predict an edit
+distance between two video sequences in the temporal order.
+Although their approach showed encouraging results on two
+datasets, their approach relies on both positive and nega-
+tive pairs. Furthermore, their proposed approach is batch
+statistic dependent.
+2.2. Self-Supervised Learning
+Most existing self-supervised techniques can be classified
+into contrastive and reconstruction-based techniques. Tra-
+ditionally, contrastive self-supervised techniques have been
+trained by reducing the distance between representations
+of different augmented views of the same image (‘positive
+pairs’) and increasing the distance between representations
+of augmented views from different images (‘negative pairs’)
+(He et al., 2020; Chen et al., 2020b; Caron et al., 2020b).
+But this is highly memory intensive as we need to track
+positive and negative pairs. Recently, Bootstrap Your Own
+Latent (BYOL) (Grill et al., 2020b) and DINO (Caron et al.,
+2021) have improved upon this approach by eliminating
+the memory banks. The premise of using negative pairs is
+to avoid collapse. Several strategies have been developed
+with BYOL using a momentum encoder, Simple Siamese
+(SimSiam) (Chen & He, 2021) a stop gradient, and DINO
+applying the counterbalancing effects of sharpening and cen-
+tering on avoiding collapse. Techniques relying only on the
+positive pairs are much more efficient than the ones using
+positive and negative pairs. Recently, there has been a surge
+in reconstruction-based self-supervised pretraining meth-
+ods with the introduction of MSN (Assran et al., 2022b),
+and MAE (He et al., 2021). These methods learn semantic
+knowledge of the image by masking a part of it and then
+predicting the masked portion.
+2.2.1. SELF-SUPERVISED LEARNING AND MEDICAL
+IMAGE ANALYSIS
+ImageNet is most commonly used for benchmarking and
+comparing self-supervised techniques. ImageNet is a bal-
+anced dataset that is not representative of real-world data,
+especially in the field of medical imaging, that has been
+characterized by class imbalance. Self-supervised methods
+that use batch-level statistics have been found to drop a
+significant amount of performance in image classification
+tasks when trained on ImageNet by artificially inducing
+class imbalance (Assran et al., 2022a). This prior of some
+self-supervised techniques like MSN (Assran et al., 2022b),
+SimCLR (Chen et al., 2020a), and VICreg (Bardes et al.,
+2021) limits their applicability on imbalanced datasets, es-
+pecially in the case of medical imaging.
+Existing self-supervised techniques typically require large
+batch sizes and datasets. When these conditions are not met,
+a marked reduction in performance is demonstrated (Caron
+et al., 2021; Chen et al., 2020a; Caron et al., 2020a; Grill
+et al., 2020b). Self-supervised learning approaches are prac-
+tical in big data medical applications (Ghesu et al., 2022; Az-
+izi et al., 2021a), such as analysis of dermatology and radiol-
+ogy imaging. In more limited data scenarios (3,662 images -
+25,333 images), Matsoukas et al. (2021) reported that ViTs
+outperform their CNN counterparts when self-supervised
+pre-training is followed by supervised fine-tuning. Trans-
+fer learning favors ViTs when applying standard training
+protocols and settings. Their study included running the
+DINO (Caron et al., 2021) self-supervised method over 300
+epochs with a batch size of 256. However, questions re-
+main about the accuracy and efficiency of using existing
+self-supervised techniques on datasets whose entire size
+is smaller than their peak performance batch size. Also,
+viewing this from the general practitioner’s perspective with
+limited computational power raises the question of how we
+can make practical self-supervised approaches more acces-
+sible. Adoption and faster development of self-supervised
+paradigms will only be possible when they become easy to
+plug and play with limited computational power.
+In this work, we explore these questions by designing CASS,
+a novel self-supervised approach developed with the core
+values of efficiency and effectiveness. In simple terms, we
+are combining CNN and Transformer in a response-based
+contrastive method by reducing similarity to combine the
+abilities of CNNs and Transformers. This approach was ini-
+tially designed for a 198-image dataset for muscle biopsies
+of inflammatory lesions from patients with dermatomyositis
+- an autoimmune disease. The benefits of this approach are
+illustrated by challenges in diagnosing autoimmune diseases
+due to their rarity, limited data availability, and heteroge-
+neous features. Consequently, misdiagnoses are common,
+and the resulting diagnostic delay plays a significant factor
+in their high mortality rate. Autoimmune diseases share
+commonalities with COVID-19 regarding clinical manifes-
+tations, immune responses, and pathogenic mechanisms.
+Moreover, some patients have developed autoimmune dis-
+eases after COVID-19 infection (Liu et al., 2020). Despite
+this increasing prevalence, the representation of autoim-
+mune diseases in medical imaging and deep learning is
+limited.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+3. Methodology
+We start by motivating our method before explaining it
+in detail (in Section 3.1). Self-supervised methods have
+been using different augmentations of the same image to
+create positive pairs. These were then passed through the
+same architectures but with a different set of parameters
+(Grill et al., 2020b). In (Caron et al., 2021) the authors
+introduced image cropping of different sizes to add local
+and global information. They also used different operators
+and techniques to avoid collapse, as described in Section 2.2.
+But there can be another way to create positive pairs -
+through architectural differences. (Raghu et al., 2021) in
+their study suggested that for the same input, Transformers
+and CNNs extract different representations. They conducted
+their study by analyzing the CKA (Centered Kernel Align-
+ment) for CNNs and Transformer using ResNet (He et al.,
+2016) and ViT (Vision Transformer) (Dosovitskiy et al.,
+2020) family of encoders, respectively. They found that
+Transformers have a more uniform representation across all
+layers as compared to CNNs. They also have self-attention,
+enabling global information aggregation from shallow lay-
+ers and skip connections that connect lower layers to higher
+layers, promising information transfer. Hence, lower and
+higher layers in Transformers show much more similarity
+than in CNNs. The receptive field of lower layers for Trans-
+formers is more extensive than in CNNs. While this recep-
+tive field gradually grows for CNNs, it becomes global for
+Transformers around the midway point. Transformers don’t
+attend locally in their earlier layers, while CNNs do. Using
+local information earlier is essential for solid performance.
+CNNs have a more centered receptive field as opposed to a
+more globally spread receptive field of Transformers. Hence,
+representations drawn from the same input will differ for
+Transformers and CNNs. Until now, self-supervised tech-
+niques have used only one kind of architecture at a time,
+either a CNN or Transformer. But differences in the repre-
+sentations learned with CNN and Transformers inspired us
+to create positive pairs by different architectures or feature
+extractors rather than using a different set of augmentations.
+This, by design, avoids collapse as the two architectures
+will never give the exact representation as output. By con-
+trasting their extracted features at the end, we hope to help
+the Transformer learn representations from CNN and vice
+versa. This should help both the architectures to learn better
+representations and learn from patterns that they would miss.
+We verify this by studying attention maps and feature maps
+from supervised and CASS-trained CNN and Transform-
+ers in Appendix C.4 and Section 5.3.3. We observed that
+CASS-trained CNN and Transformer were able to retain a
+lot more detail about the input image, which pure CNN and
+Transformers lacked.
+3.1. Description of CASS
+CASS’ goal is to extract and learn representations in a self-
+supervised way. To achieve this, an image is passed through
+a common set of augmentations. The augmented image is
+then simultaneously passed through a CNN and Transformer
+to create positive pairs. The output logits from the CNN
+and Transformer are then used to find cosine similarity loss
+(equation 1). This is the same loss function as used in BYOL
+(Grill et al., 2020a). Furthermore, the intuition of CASS is
+very similar to that of BYOL. In BYOL to avoid collapse
+to a trivial solution the target and the online arm are differ-
+ently parameterized and an additional predictor is used with
+the online arm. They compared this setup to that of GANs
+where joint of optimization of both arms to a common value
+was impossible due to differences in the arms. Analogously,
+In CASS instead of using an additional MLP on top of one
+of the arms and differently parameterizing them, we use
+two fundamentally different architectures. Since the two
+architectures give different output representations as men-
+tioned in (Raghu et al., 2021), the model doesn’t collapse.
+Additionally, to avoid collapse we introduced a condition
+where if the outputs from the CNN and Transformer are the
+same, artificial noise sampled from a Gaussian distribution
+is added to the model outputs and thereby making the loss
+non-zero. We also report results for CASS using a different
+set of CNNs and Transformers in Appendix B.6 and Section
+5, and not a single case of the model collapse was registered.
+loss = 2 − 2 × F(R) × F(T)
+(1)
+where, F(x) =
+N
+�
+i=1
+�
+x
+(max (∥x∥2) , ϵ)
+�
+We use the same parameters for the optimizer and learn-
+ing schedule for both architectures. We also use stochastic
+weigh averaging (SWA) (Izmailov et al., 2018) with Adam
+optimizer and a learning rate of 1e-3. For the learning rate,
+we use a cosine schedule with a maximum of 16 iterations
+and a minimum value of 1e-6. ResNets are typically trained
+with Stochastic Gradient Descent (SGD) and our use of
+the Adam optimizer is quite unconventional. Furthermore,
+unlike existing self-supervised techniques there is no param-
+eter sharing between the two architectures.
+We compare CASS against the state-of-the-art self-
+supervised technique DINO (DIstilation with NO labels).
+This choice was made based on two conditions (i) As already
+explained in Section 2.2.1, some self-supervised techniques
+use batch-level statistics that makes them less suitable for
+application on imbalanced datasets and imbalanced datasets
+are a feature of medical imaging. (ii) The self-supervised
+technique should be benchmarked for both CNNs and Trans-
+formers as both architectures have exciting properties and
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+apriori, it is difficult to predict which architecture will per-
+form better.
+In Figure 1, we show CASS on top, and DINO (Caron et al.,
+2021) at the bottom. Comparing the two, CASS does not use
+any extra mathematical treatment used in DINO to avoid col-
+lapse such as centering and applying the softmax function
+on the output of its student and teacher networks. We also
+provide an ablation study using a softmax and sigmoid layer
+for CASS in Appendix B. After training CASS and DINO
+for one cycle, DINO yields only one kind of trained architec-
+ture. In contrast, CASS provides two trained architectures
+(1 - CNN and 1 - Transformer). CASS-pre-trained architec-
+tures perform better than DINO-pre-trained architectures in
+most cases, as further elaborated in Section 5.
+Figure 1. (Top) In our proposed self-supervised architecture -
+CASS, R represents ResNet-50, a CNN and T in the other box
+represents the Transformer used (ViT); X is the input image, which
+becomes X’ after applying augmentations. Note that CASS applies
+only one set of augmentations to create X’. X’ is passed through
+both arms to compute loss, as in Equation 1. This differs from
+DINO, which passes different augmentation of the same image
+through networks with the same architecture but different param-
+eters. The output of the teacher network is centered on a mean
+computed over a batch. Another key difference is that in CASS, the
+loss is computed over logits; meanwhile, in DINO, it is computed
+over softmax output.
+4. Experimental Details
+4.1. Datasets
+We split the datasets into three splits - training, validation,
+and testing following the 70/10/20 split strategy unless spec-
+ified otherwise. We further expand upon our thought process
+for choosing datasets in Appendix C.4.5.
+• Autoimmune diseases biopsy slides (Singh & Cir-
+rone, 2022; Van Buren et al., 2022) consists of slides
+cut from muscle biopsies of dermatomyositis patients
+stained with different proteins and imaged to generate
+a dataset of 198 TIFF image set from 7 patients. The
+presence or absence of these cells helps to diagnose
+dermatomyositis. Multiple cell classes can be present
+per image; therefore this is a multi-label classification
+problem. Our task here was to classify cells based on
+their protein staining into TFH-1, TFH-217, TFH-Like,
+B cells, and others. We used F1 score as our metric for
+evaluation, as employed in previous works by (Singh
+& Cirrone, 2022; Van Buren et al., 2022). These RGB
+images have a consistent size of 352 by 469.
+• Dermofit dataset (Fisher & Rees, 2017) contains nor-
+mal RGB images captured through an SLR camera
+indoors with ring lightning. There are 1300 image sam-
+ples, classified into 10 classes: Actinic Keratosis (AK),
+Basal Cell Carcinoma (BCC), Melanocytic Nevus /
+Mole (ML), Squamous Cell Carcinoma (SCC), Sebor-
+rhoeic Keratosis (SK), Intraepithelial carcinoma (IEC),
+Pyogenic Granuloma (PYO), Haemangioma (VASC),
+Dermatofibroma (DF) and Melanoma (MEL). This
+dataset comprises images of different sizes and no two
+images are of the same size. They range from 205×205
+to 1020×1020 in size. Our pretext task is multi-class
+classification and we use the F1 score as our evaluation
+metric on this dataset.
+• Brain tumor MRI dataset (Cheng, 2017; Amin et al.,
+2022) 7022 images of human brain MRI that are classi-
+fied into four classes: glioma, meningioma, no tumor,
+and pituitary. This dataset combines Br35H: Brain tu-
+mor Detection 2020 dataset used in ”Retrieval of Brain
+tumors by Adaptive Spatial Pooling and Fisher Vector
+Representation” and Brain tumor classification curated
+by Navoneel Chakrabarty and Swati Kanchan. Out
+of these, the dataset curator created the training and
+testing splits. We followed their splits, 5,712 images
+for training and 1,310 for testing. Since this was a com-
+bination of multiple datasets, the size of images varies
+throughout the dataset from 512×512 to 219×234. The
+pretext of the task is multi-class classification, and we
+used the F1 score as the metric.
+• ISIC 2019 (Tschandl et al., 2018; Gutman et al., 2018;
+Combalia et al., 2019) consists of 25,331 images
+across eight different categories - melanoma (MEL),
+melanocytic nevus (NV), Basal cell carcinoma (BCC),
+actinic keratosis(AK), benign keratosis(BKL), der-
+matofibroma(DF), vascular lesion (VASC) and Squa-
+mous cell carcinoma(SCC). This dataset contains im-
+ages of size 600 × 450 and 1024 × 1024. The distri-
+bution of these labels is unbalanced across different
+
+R
+logits,
+Loss
+logits,Softmax
+Student
+(-p, log p,)
+Loss
+ema
+Softmax
+X.
+Teacher
+c
+SCross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Techniques
+Backbone
+Testing F1 score
+10%
+100%
+DINO
+ResNet-50
+0.8237±0.001
+0.84252±0.008
+CASS
+ResNet-50
+0.8158±0.0055
+0.8650±0.0001
+Supervised
+ResNet-50
+0.819±0.0216
+0.83895±0.007
+DINO
+ViT B/16
+0.8445±0.0008
+0.8639± 0.002
+CASS
+ViT B/16
+0.8717±0.005
+0.8894±0.005
+Supervised
+ViT B/16
+0.8356±0.007
+0.8420±0.009
+Table 1. Results for autoimmune biopsy slides dataset. In this table, we compare the F1 score on the test set. We observed that CASS
+outperformed the existing state-of-art self-supervised method using 100% labels for CNN as well as for Transformers. Although DINO
+outperforms CASS for CNN with 10% labeled fraction. Overall, CASS outperforms DINO by 2.2% for 100% labeled training for CNN
+and Transformer. For Transformers in 10% labeled training CASS’ performance was 2.7% better than DINO.
+Dataset
+DINO
+CASS
+Autoimmune
+1 H 13 M
+21 M
+Dermofit
+3 H 9 M
+1 H 11 M
+Brain MRI
+26 H 21 M
+7 H 11 M
+ISIC-2019
+109 H 21 M
+29 H 58 M
+Table 2. Self-supervised pretraining time comparison for 100
+epochs on a single RTX8000 GPU. In this table, H represents
+hour(s), and M represents minute(s).
+classes. For evaluation, we followed the metric fol-
+lowed in the official competition i.e balanced multi-
+class accuracy value, which is semantically equal to
+recall.
+4.2. Self-supervised learning
+We studied and compared results between DINO and CASS-
+pre-trained self-supervised CNNs and Transformers. For the
+same, we trained from ImageNet initialization (Matsoukas
+et al., 2021) for 100 epochs with a batch size of 16. We ran
+these experiments on an internal cluster with a single GPU
+unit (NVIDIA RTX8000) with 48 GB video RAM, 2 CPU
+cores, and 64 GB system RAM.
+For DINO, we used the hyperparameters and augmentations
+mentioned in the original implementation. For CASS, we
+describe the experimentation details in Appendix C.5.
+4.3. End-to-end fine-tuning
+In order to evaluate the utility of the learned representa-
+tions, we use the self-supervised pre-trained weights for
+the downstream classification tasks. While performing the
+downstream fine-tuning, we perform the entire model (E2E
+fine-tuning). The test set metrics were used as proxies for
+representation quality. We trained the entire model for a
+maximum of 50 epochs with an early stopping patience of
+5 epochs. For supervised fine-tuning, we used Adam opti-
+mizer with a cosine annealing learning rate starting at 3e-04.
+Since almost all medical datasets have some class imbalance
+we applied class distribution normalized Focal Loss (Lin
+et al., 2017) to navigate class imbalance.
+We fine-tune the models using different label fractions dur-
+ing E2E fine-tuning i.e 1%, 10%, and 100& label fractions.
+For example, if a model is trained with a 10% label fraction,
+then that model will have access only to 10% of the training
+dataset samples and their corresponding labels during the
+E2E fine-tuning after initializing weights using the CASS
+or DINO pretraining.
+5. Results and Discussion
+5.1. Compute and Time analysis Analysis
+We ran all the experiments on a single NVIDIA RTX8000
+GPU with 48GB video memory. In Table 2, we compare the
+cumulative training times for self-supervised training of a
+CNN and Transformer with DINO and CASS. We observed
+that CASS took an average of 69% less time compared
+to DINO. Another point to note is that CASS trained two
+architectures at the same time or in a single pass. While
+training a CNN and Transformer with DINO it would take
+two separate passes.
+5.2. Results on the four medical imaging datasets
+We did not perform 1% finetuning for the autoimmune dis-
+eases biopsy slides of 198 images because using 1% images
+would be too small a number to learn anything meaningful
+and the results would be highly randomized. Similarly, we
+also did not perform 1% fine-tuning for the dermofit dataset
+as the training set was too small to draw meaningful results
+with just 10 samples. We present the results on the four med-
+ical imaging datasets in Tables 1, 3, 4, and 5. From these
+tables, we observe that CASS improves upon the classifica-
+tion performance of existing state-of-the-art self-supervised
+method DINO by 3.8% with 1% labeled data, 5.9% with
+10% labeled data, and 10.13% with 100% labeled data.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Techniques
+Testing F1 score
+10%
+100%
+DINO (Resnet-50)
+0.3749±0.0011
+0.6775±0.0005
+CASS (Resnet-50)
+0.4367±0.0002
+0.7132±0.0003
+Supervised (Resnet-50)
+0.33±0.0001
+0.6341±0.0077
+DINO (ViT B/16)
+0.332± 0.0002
+0.4810±0.0012
+CASS (ViT B/16)
+0.3896±0.0013
+0.6667±0.0002
+Supervised (ViT B/16)
+0.299±0.002
+0.456±0.0077
+Table 3. This table contains the results for the dermofit dataset. We observe that CASS outperforms both supervised and existing
+state-of-the-art self-supervised methods for all label fractions. Parenthesis next to the techniques represents the architecture used, for
+example, DINO(ViT B/16) represents ViT B/16 trained with DINO. In this table, we compare the F1 score on the test set. We observed
+that CASS outperformed the existing state-of-art self-supervised method using all label fractions and for both the architectures.
+Techniques
+Backbone
+Testing F1 score
+1%
+10%
+100%
+DINO
+Resnet-50
+0.63405±0.09
+0.92325±0.02819
+0.9900±0.0058
+CASS
+Resnet-50
+0.40816±0.13
+0.8925±0.0254
+0.9909± 0.0032
+Supervised
+Resnet-50
+0.52±0.018
+0.9022±0.011
+0.9899± 0.003
+DINO
+ViT B/16
+0.3211±0.071
+0.7529±0.044
+0.8841± 0.0052
+CASS
+ViT B/16
+0.3345±0.11
+0.7833±0.0259
+0.9279± 0.0213
+Supervised
+ViT B/16
+0.3017 ± 0.077
+0.747±0.0245
+0.8719± 0.017
+Table 4. This table contains results on the brain tumor MRI classification dataset. While DINO outperformed CASS for 1% and 10%
+labeled training for CNN, CASS maintained its superiority for 100% labeled training, albeit by just 0.09%. Similarly, CASS outperformed
+DINO for all data regimes for Transformers, incrementally 1.34% in for 1%, 3.04% for 10%, and 4.38% for 100% labeled training. We
+observe that this margin is more significant than for biopsy images. Such results could be ascribed to the increase in dataset size and
+increasing learnable information.
+Techniques
+Backbone
+Testing Balanced multi-class accuracy
+1%
+10%
+100%
+DINO
+Resnet-50
+0.328±0.0016
+0.3797±0.0027
+0.493±3.9e-05
+CASS
+Resnet-50
+0.3617±0.0047
+0.41±0.0019
+0.543±2.85e-05
+Supervised
+Resnet-50
+0.2640±0.031
+0.3070±0.0121
+0.35±0.006
+DINO
+ViT B/16
+0.3676± 0.012
+0.3998±0.056
+0.5408±0.001
+CASS
+ViT B/16
+0.3973± 0.0465
+0.4395±0.0179
+0.5819±0.0015
+Supervised
+ViT B/16
+0.3074±0.0005
+0.3586±0.0314
+0.42±0.007
+Table 5. Results for the ISIC-2019 dataset.
+Comparable to the official metrics used in the challenge https://challenge.
+isic-archive.com/landing/2019/. The ISIC-2019 dataset is an incredibly challenging, not only because of the class im-
+balance issue but because it is made of partially processed and inconsistent images with hard-to-classify classes. We use balanced
+multi-class accuracy as our metric, which is semantically equal to recall value. We observed that CASS consistently outperforms DINO
+by approximately 4% for all label fractions with CNN and Transformer.
+5.3. Ablation Studies
+As mentioned in Section 2.2.1, existing self-supervised
+methods experience a drop in classification performance
+when trained for a reduced number of pretraining epochs
+and batch size. We performed ablation studies to study the
+effect of change in performance for CASS and DINO pre-
+trained ResNet-50 and ViTB/16 on the autoimmune dataset.
+Additional ablation studies have been provided in Appendix.
+5.3.1. CHANGE IN EPOCHS
+In this section, we compare the performance change in
+CASS and DINO pretrained and then E2E finetuned with
+100% labels over the autoimmune dataset. To study the
+robustness, we compare the mean-variance over CNN and
+Transformer trained with the two techniques. The recorded
+mean-variance in performance for ResNet-50 and ViTB-16
+trained with CASS and DINO with change in the number
+of pretraining epochs is 0.0001791 and 0.0002265, respec-
+tively. Based on these results, we observed that CASS-
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+(a)
+(b)
+Figure 2. In Figure a, we report the change in performance with
+respect to the change in the number of pretraining epochs for DINO
+and CASS for ResNet-50 and ViTB/16, respectively. In Figure b,
+we report the change in performance with respect to the change
+in the number of pretraining batch sizes for DINO and CASS for
+ResNet-50 and ViTB/16, respectively. These ablation studies were
+conducted on the autoimmune dataset, while keeping the other
+hyper-parameters the same during pretraining and downstream
+finetuning.
+trained models have less variance, i.e., they are more robust
+to change in the number of pretraining epochs.
+5.3.2. CHANGE IN BATCH SIZE
+Similar to Section 5.3.1, in this section, we study the change
+in performance concerning the batch size. As previously
+mentioned existing self-supervised techniques suffer a drop
+in performance when they are trained for small batch sizes;
+we studied the change in performance for batch sizes 8, 16,
+and 32 on the autoimmune dataset with CASS and DINO.
+We reported these results in Figure 2. We observe that the
+mean-variance in performance for ResNet-50 and ViTB-16
+trained with CASS and DINO with change in batch size for
+CASS and DINO is 5.8432e-5 and 0.00015003, respectively.
+Hence, CASS is much more robust to changes in pretraining
+batch size than DINO.
+5.3.3. ATTENTION MAPS
+To study the effect qualitatively we study the attention of a
+supervised and CASS-pre trained Transformer. From Figure
+3 we observe that the attention map of the CASS-pre-trained
+Transformer is a lot more connected than a supervised Trans-
+former due to the transfer of locality information from the
+CNN. We further expand on this Appendix C.4.
+Figure 3. This figure shows the attention maps over a single test
+sample image from the autoimmune dataset. The left image is
+the overall attention map over a single test sample for the super-
+vised Transformer, while the one on the right is for CASS trained
+Transformer.
+6. Conclusion
+Based on our experimentation on four diverse medical imag-
+ing datasets, we qualitatively concluded that CASS im-
+proves upon the classification performance of existing state-
+of-the-art self-supervised method DINO by 3.8% with 1%
+labeled data, 5.9% with 10% labeled data, and 10.13% with
+100% labeled data and trained in 69% less time. Further-
+more, we saw that CASS is robust to batch size changes and
+training epochs reduction. To conclude, for medical image
+analysis, CASS is computationally efficient, performs bet-
+ter, and overcomes some of the shortcomings of existing
+self-supervised techniques. This ease of accessibility and
+better performance will catalyze medical imaging research
+to help us improve healthcare solutions and propagate these
+advancements in state-of-the-art techniques to deep practical
+
+F1 score on test set vs Epodhs for ResNetso
+CASS
+DINO
+0.8766
+0.B7
+18
+0.865
+0.B6
+0.8534
+0.B5
+0.8521
+0.84252
+0.8335
+50
+140
+20D
+Number of Prebrainirg EpachaF1 score on test set vs Epochs for viTEl6
+CASS
+0.90
+0.9053
+DINO
+0.B9
+0.8894
+L score on test :
+0.BB
+0.876
+0.8765
+0.B7
+0.8639
+0.B6
+0.B5
+0.8391
+0.B4
+50
+240
+Number of Pretrainirg EpachsF1 score on test set vs Batch 5ize for ResNets0
+CASS
+DINO
+0.865
+0.8652
+0.B6
+18
+0.B5
+0.84715
+0.84252
+0.844
+0.B4
+0.B3
+0.8222
+8
+16
+32
+Betch SizeF1 score on test set vs Batch 5ize for viTB16
+0.B9
+0.8894
+0.89
+0.BB
+0.8844
+test set
+0.8711
+score ont
+0.B7
+0.8639
+0.B5
+0.8471
+CASS
+DINO
+8
+16
+32
+Betch Size1.0
+0
+0.8
+100
+0.6
+200
+0.4
+300
+0.2
+0
+100
+200
+300
+0.01.0
+0
+0.8
+100
+0.6
+200
+0.4
+300
+0.2
+0
+100
+200
+300
+0.0Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+learning in developing countries and practitioners with lim-
+ited resources to develop new solutions for underrepresented
+and emerging diseases.
+Acknowledgements
+We would like to thank Prof. Elena Sizikova (Moore Sloan
+Faculty Fellow, Center for Data Science (CDS), New York
+University (NYU)) for her valuable feedback and NYU HPC
+team for assisting us with our computational needs.
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+Immunological Methods, 505:113233, 2022. ISSN 0022-
+1759.
+doi: https://doi.org/10.1016/j.jim.2022.113233.
+URL
+https://www.sciencedirect.com/
+science/article/pii/S0022175922000205.
+Wightman, R. Pytorch image models. https://github.
+com/rwightman/pytorch-image-models,
+2019.
+Yadav, S. S. and Jadhav, S. M. Deep convolutional neural
+network based medical image classification for disease
+diagnosis. Journal of Big Data, 6(1):1–18, 2019.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Algorithm 1 CASS self-supervised Pretraining algorithm
+Input: Unlabeled same augmented images from the training set x′
+for epochs in range(num epochs) do
+for x in train loader: do
+R = cnn(x′) (taking logits output from CNN)
+T = vit(x′) (taking logits output from ViT)
+if R == T then
+Rnoise = X �→ N(10e−6, 10e−9)
+Tnoise = X �→ N(10e−10, 10e−15)
+R = R + Rnoise
+T = T + Tnoise
+Calculate loss using Equation 1
+else
+Calculate loss using Equation 1
+end if
+end for
+end for
+A. CASS Pretraining Algorithm
+The core self-supervised algorithm used to train CASS with a CNN (R) and a Transformer (T) is described in Algorithm 1.
+Here, num epochs represents the number of self-supervised epochs to run. CNN and Transformer represent our respective
+architecture; for example, CNN could be a ResNet50, and Transformer can be ViT Base/16. The loss used is described in
+Equation 1. Finally, after pretraining, we save the CNN and Transformer for downstream finetuning.
+B. Additional Ablation Studies
+B.1. Batch size
+We studied the effect of change in batch size on the autoimmune dataset in Section 5.3.2. Similarly, in this section, we study
+the effect of varying the batch size on the brain MRI classification dataset. In the standard implementation of CASS, we
+used a batch size of 16; here, we showed results for batch sizes 8 and 32. The largest batch size we could run was 34 on
+a single GPU of 48 GB video memory. Hence 32 was the biggest batch size we showed in our results. We present these
+results in Table 6. Similar to the results in Section 5.3.2, performance decreases as we reduce the batch size and increases
+slightly as we increase the batch size for both CNN and Transformer.
+Batch Size
+CNN F1 Score
+Transformer F1 Score
+8
+0.9895±0.0025
+0.9198±0.0109
+16
+0.9909± 0.0032
+0.9279± 0.0213
+32
+0.991±0.011
+0.9316±0.006
+Table 6. This table represents the results for different batch sizes on the brain MRI classification dataset. We maintain the downstream
+batch size constant in all three cases, following the standard experimental setup mentioned in Appendix C.5 and C.6. These results are on
+the test set after E2E fine-tuning with 100% labels.
+B.2. Change in pretraining epochs
+As standard, we pretrained CASS for 100 epochs in all cases. However, existing self-supervised techniques are plagued with
+a loss in performance with a decrease in the number of pretraining epochs. To study this effect for CASS, we reported results
+in Section 5.3.1. Additionally, in this section, we report results for training CASS for 300 epochs on the autoimmune and
+brain tumor MRI datasets. We reported these results in Table 7 and 8, respectively. We observed a slight gain in performance
+when we increased the epochs from 100 to 200 but minimal gain beyond that. We also studied the effect of longer pretraining
+on the brain tumor MRI classification dataset and presented these results in Table 8.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Epochs
+CNN F1 Score
+Transformer F1 Score
+50
+0.8521±0.0007
+0.8765± 0.0021
+100
+0.8650±0.0001
+0.8894±0.005
+200
+0.8766±0.001
+0.9053±0.008
+300
+0.8777±0.004
+0.9091±8.2e-5
+Table 7. Performance comparison over a varied number of epochs on the brain tumor MRI classification dataset, from 50 to 300 epochs,
+the downstream training procedure, and the CNN-Transformer combination is kept constant across all the four experiments, only the
+number of self-supervised pretraining epochs were changed.
+Epochs
+CNN F1 Score
+Transformer F1 Score
+50
+0.9795±0.0109
+0.9262±0.0181
+100
+0.9909± 0.0032
+0.9279± 0.0213
+200
+0.9864±0.008
+0.9476±0.0012
+300
+0.9920±0.001
+0.9484±0.017
+Table 8. Performance comparison over a varied number of epochs, from 50 to 300 epochs, the downstream training procedure, and the
+CNN-transformer combination is kept constant across all four experiments; only the number of self-supervised epochs has been changed.
+B.3. Augmentations
+Contrastive learning techniques are known to be highly dependent on augmentations. Recently, most self-supervised
+techniques have adopted BYOL (Grill et al., 2020b)-like a set of augmentations. DINO (Caron et al., 2021) uses the same
+set of augmentations as BYOL, along with adding local-global cropping. We use a reduced set of BYOL augmentations
+for CASS, along with a few changes. For instance, we do not use solarize and Gaussian blur. Instead, we use affine
+transformations and random perspectives. In this section, we study the effect of adding BYOL-like augmentations to CASS.
+We report these results in Table 9. We observed that CASS-trained CNN is robust to changes in augmentations. On the
+other hand, the Transformer drops performance with changes in augmentations. A possible solution to regain this loss in
+performance for Transformer with a change in augmentation is using Gaussian blur, which converges the results of CNN
+and the Transformer.
+Augmentation Set
+CNN F1 Score
+Transformer F1 Score
+CASS only
+0.8650±0.0001
+0.8894±0.005
+CASS + Solarize
+0.8551±0.0004
+0.81455±0.002
+CASS + Gaussian blur
+0.864±4.2e-05
+0.8604±0.0029
+CASS + Gaussian blur + Solarize
+0.8573±2.59e-05
+0.8513±0.0066
+Table 9. We report the F1 metric of CASS trained with a different set of augmentations for 100 epochs. While CASS-trained CNN
+fluctuates within a percent of its peak performance, CASS-trained Transformer drops performance with the addition of solarization and
+Gaussian blur. Interestingly, the two arms converged with the use of Gaussian blur.
+B.4. Optimization
+In CASS, we use Adam optimizer for both CNN and Transformer. This is a shift from using SGD or stochastic gradient
+descent for CNNs. In this Table 10, we report the performance of CASS-trained CNN and Transformer with the CNN using
+SGD and Adam optimizer. We observed that while the performance of CNN remained almost constant, the performance of
+the Transformer dropped by almost 6% with CNN using SGD.
+Optimiser for CNN
+CNN F1 Score
+Transformer F1 Score
+Adam
+0.8650±0.0001
+0.8894±0.005
+SGD
+0.8648±0.0005
+0.82355±0.0064
+Table 10. We report the F1 metric of CASS trained with a different set of optimizers for the CNN arm for 100 epochs. While there is no
+change in CNN’s performance, the Transformer’s performance drops around 6% with SGD.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+B.5. Using softmax and sigmoid layer in CASS
+As noted in Fig 1, CASS doesn’t use a softmax layer like DINO (Caron et al., 2021) before the computing loss. The output
+logits of the two networks have been used to combine the two architectures in a response-based knowledge distillation (Gou
+et al., 2021) manner instead of using soft labels from the softmax layer. In this section, we study the effect of using an
+additional softmax layer on CASS. Furthermore, we also study the effect of adding a sigmoid layer instead of a softmax
+layer and compare it with a CASS model that doesn’t use the sigmoid or the softmax layer. We present these results in Table
+11. We observed that not using sigmoid and softmax layers in CASS yields the best result for both CNN and Transformers.
+Techniques
+CNN F1 Score
+Transformer F1 Score
+Without Sigmoid or Softmax
+0.8650±0.0001
+0.8894±0.005
+With Sigmoid Layer
+0.8296±0.00024
+0.8322±0.004
+With Softmax Layer
+0.8188±0.0001
+0.8093±0.00011
+Table 11. We observe that performance reduces when we introduce the sigmoid or softmax layer.
+B.6. Change in architecture
+B.6.1. CHANGING TRANSFORMER AND KEEPING THE CNN SAME
+From Table 12 and 13, we observed that CASS-trained ViT Transformer with the same CNN consistently gained approxi-
+mately 4.7% over its supervised counterpart. Furthermore, from Table 13, we observed that although ViT L/16 performs
+better than ViT B/16 on ImageNet ( (Wightman, 2019)’s results), we observed that the trend is opposite on the autoimmune
+dataset. Hence, the supervised performance of architecture must be considered before pairing it with CASS.
+Transformer
+CNN F1 Score
+Transformer F1 Score
+ViT Base/16
+0.8650±0.001
+0.8894± 0.005
+ViT Large/16
+0.8481±0.001
+0.853±0.004
+Table 12. In this table, we show the performance of CASS for ViT large/16 with ResNet-50 and ViT base/16 with ResNet-50. We observed
+that CASS-trained Transformers, on average, performed 4.7% better than their supervised counterparts.
+Architecture
+Testing F1 Score
+ResnNet-50
+0.83895±0.007
+ViT Base/16
+0.8420±0.009
+ViT large/16
+0.80495±0.0077
+Table 13. Supervised performance of ViT family on the autoimmune dataset. We observed that as opposed to ImageNet performance, ViT
+large/16 performs worse than ViT Base/16 on the autoimmune dataset.
+We keep the CNN constant for this experiment and study the effect of changing the Transformer. For this experiment, we
+use ResNet as our choice of CNN and ViT base and large Transformers with 16 patches. Additionally, we also report
+performance for DeiT-B (Touvron et al., 2020) with ResNet-50. We report these results in Table 14. Similar to Table 12,
+we observe that by changing Transformer from ViT Base to Large while keeping the number of tokens the same at 16,
+performance drops. Additionally, for approximately the same size, out of DeiT base and ViT base Transformers, DeiT
+performs much better than ViT base.
+B.6.2. CHANGING CNN AND KEEPING THE TRANSFORMER SAME
+Table 15 and 16 we observed that similar to changing Transformer while keeping the CNN same, CASS-trained CNNs gained
+an average of 3% over their supervised counterparts. ResNet-200 (Wightman, 2019) doesn’t have ImageNet initialization
+hence using random initialization.
+For this experiment, we use the ResNet family of CNNs and ViT base/16 as our Transformer. We use ImageNet initialization
+for ResNet 18 and 50, while random initialization for ResNet-200 (As Timm’s library doesn’t have an ImageNet initialization).
+We present these results in Table 17. We observed that an increase in the performance of ResNet correlates to an increase in
+the performance of the Transformer, implying that there is information transfer between the two.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+CNN
+Transformer
+CNN F1 Score
+Transformer F1 Score
+ResNet-50
+(25.56M)
+DEiT Base/16 (86.86M)
+0.9902±0.0025
+0.9844±0.0048
+ViT Base/16 (86.86M)
+0.9909±0.0032
+0.9279± 0.0213
+ViT Large/16 (304.72M)
+0.98945±2.45e-5
+0.8896±0.0009
+Table 14. For the same number of Transformer parameters, DEiT-base with ResNet-50 performed much better than ResNet-50 with
+ViT-base. The difference in their CNN arm is 0.10%. On ImageNet DEiT-base has a top1% accuracy of 83.106 while ViT-base has an
+accuracy of 86.006. We use both Transformers with 16 patches. [ResNet-50 has an accuracy of 80.374]
+CNN
+Transformer
+100% Label Fraction
+CNN F1 score
+Transformer F1 score
+ResNet-18 (11.69M)
+ViT Base/16 (86.86M)
+0.8674±4.8e-5
+0.8773±5.29e-5
+ResNet-50 (25.56M)
+0.8680±0.001
+0.8894± 0.0005
+ResNet-200 (64.69M)
+0.8517±0.0009
+0.874±0.0006
+Table 15. F1 metric comparison between the two arms of CASS trained over 100 epochs, following the protocols and procedure listed in
+Appendix E. The numbers in parentheses show the parameters learned by the network. We use (Wightman, 2019) implementation of CNN
+and transformers, with ImageNet initialization except for ResNet-200.
+Architecture
+Testing F1 Score
+ResnNet-18
+0.8499±0.0004
+ResnNet-50
+0.83895±0.007
+ResnNet-200
+0.833±0.0005
+Table 16. Supervised performance of the ResNet CNN family on the autoimmune dataset.
+CNN
+Transformer
+100% Label Fraction
+CNN F1 score
+Transformer F1 score
+ResNet-18 (11.69M)
+ViT Base/16 (86.86M)
+0.9913±0.002
+0.9801±0.007
+ResNet-50 (25.56M)
+0.9909±0.0032
+0.9279± 0.0213
+ResNet-200 (64.69M)
+0.9898±0.005
+0.9276±0.017
+Table 17. F1 metric comparison between the two arms of CASS trained over 100 epochs, following the protocols and procedure listed
+in Appendix C.5 and C.6. The numbers in parentheses show the parameters learned by the network. We use (Wightman, 2019)
+implementation of CNN and transformers, with ImageNet initialization except for ResNet-200.
+B.6.3. USING CNN IN BOTH ARMS
+We have experimented using a CNN and a Transformer in CASS on the brain tumor MRI classification dataset. In this section,
+we present results for using two CNNs in CASS. We pair ResNet-50 with DenseNet-161. We observe that both CNNs fail to
+reach the benchmark set by ResNet-50 and ViT-B/16 combination. Although training the ResNet-50-DenseNet-161 pair
+takes 5 hours 24 minutes, less than the 7 hours 11 minutes taken by the ResNet-50-ViT-B/16 combination to be trained with
+CASS. We compare these results in Table 18.
+CNN
+Architecture in
+arm 2
+F1 Score of ResNet-50 arm
+F1 Score of arm 2
+ResNet-50
+ViT Base/16
+0.9909±0.0032
+0.9279± 0.0213
+DenseNet-161
+0.9743±8.8e-5
+0.98365±9.63e-5
+Table 18. We observed that for the ResNet-50-DenseNet-161 pair, we train 2 CNNs instead of 1 in our standard setup of CASS.
+Furthermore, none of these CNNs could match the performance of ResNet-50 trained with the ResNet-50-ViT base/16 combination.
+Hence, by adding a Transformer-CNN combination, we transfer information between the two architectures that would have been missed
+otherwise.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+B.6.4. USING TRANSFORMER IN BOTH ARMS
+Similar to the above section, we use a Transformer-Transformer combination instead of a CNN-Transformer combination.
+We use Swin-Transformer patch-4/window-12 (Liu et al., 2021a) alongside ViT-B/16 Transformer. We observe that
+the performance for ViT/B-16 improves by around 1.3% when we use Swin Transformer. However, this comes at a
+computational cost. The swin-ViT combination took 10 hours to train as opposed to 7 hours and 11 minutes taken by the
+ResNet-50-ViT-B/16 combination to be trained with CASS. Even with the increased time to train the Swin-ViT combination,
+it is still almost 50% less than DINO. We present these results in Table 19.
+Architecture in
+arm 1
+Transformer
+F1 Score of arm 1
+F1 Score of ViT-B/16 arm
+ResNet-50
+ViT Base/16
+0.9909±0.0032
+0.9279± 0.0213
+Swin Transformer
+0.9883±1.26e-5
+0.94±8.12e-5
+Table 19. We present the results for using Transformers in both arms and compare the results with the CNN-Transformer combination.
+B.7. Effect of Initialization
+Although the aim of self-supervised pretraining is to provide better initialization, we use ImageNet initialized CNN and
+Transformers for CASS and DINO pertaining as well as supervised training similar to (Matsoukas et al., 2021). We use
+Timm’s library for these initialization (Wightman, 2019). ImageNet initialization is preferred not because of feature reuse but
+because ImageNet weights allow for faster convergence through better weight scaling (Raghu et al., 2019). But sometimes
+pre-trained weights might be hard to find, so we study CASS’ performance with random and ImageNet initialization in this
+section. We observed that performance almost remained the same, with minor gains when the initialization was altered for
+the two networks. Table 20 presents the results of this experimentation.
+Initialisation
+CNN F1 Score
+Transformer F1 Score
+Random
+0.9907±0.009
+0.9116±0.027
+Imagenet
+0.9909±0.0032
+0.9279± 0.0213
+Table 20. We observe that the Transformer gains some performance with the random initialization, although performance has more
+variance when used with random initialization.
+Initialisation
+CNN F1 Score
+Transformer F1 Score
+Random
+0.8437±0.0047
+0.8815±0.048
+Imagenet
+0.8650±0.0001
+0.8894±0.005
+Table 21. We observe that the Transformer gains some performance with the random initialization, although performance has more
+variance when used with random initialization.
+C. Result Analysis
+C.1. Time complexity analysis
+In Section 5.1, we observed that CASS takes 69% less time than DINO. This reduction in time could be attributed to the
+following reasons:
+1. In DINO, augmentations are applied twice as opposed to just once in CASS. Furthermore, per application, CASS uses
+fewer augmentations than DINO.
+2. Since the architectures used are different, there is no scope for parameter sharing between them. A major chunk of
+time is saved by updating the two architectures after each epoch instead of re-initializing architectures with lagging
+parameters.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Figure 4. Sample image used from the test set of the autoimmune dataset.
+C.2. Qualitative analysis
+To qualitatively expand our study, in this section, we study the feature maps of CNN and attention maps of Transformers
+trained using CASS and supervised techniques. To reinstate, based on the study by (Raghu et al., 2021), since CNN and
+Transformer extract different kinds of features from the same input, combing the two of them would help us create positive
+pairs for self-supervised learning. In doing so, we would transfer between the two architectures, which is not innate. We
+have already seen that this yield better performance in most cases over four different datasets and with three different label
+fractions. In this section, we study this gain qualitatively with the help of feature maps and class attention maps. Also, we
+briefly discussed attention maps in Section 5.3.3, where we observed that CASS-trained Transformers have more local
+understanding of the image and hence a more connected attention map than purely-supervised Transformer.
+C.3. Feature maps
+In this section, we study the feature maps from the first five layers of the ResNet-50 model trained with CASS and
+supervision. We extracted feature maps after the Conv2d layer of ResNet-50. We present the extracted features in Figure 6.
+We observed that CASS-trained CNN could retain much more detail about the input image than supervised CNN.
+C.4. Class attention maps
+We have already studied the class attention maps over a single image in Section 5.3.3. This section will explore the average
+class attention maps for all four datasets. We studied the attention maps averaged over 30 random samples for autoimmune,
+dermofit, and brain MRI datasets. Since the ISIC 2019 dataset is highly unbalanced, we averaged the attention maps over
+100 samples so that each class may have an example in our sample. We maintained the same distribution as the test set,
+which has the same class distribution as the overall training set. We observed that CASS-trained Transformers were better
+able to map global and local connections due to Transformers’ ability to map global dependencies and by learning features
+sensitive to translation equivariance and locality from CNN. This helps the Transformer learn features and local patterns that
+it would have missed.
+C.4.1. AUTOIMMUNE DATASET
+We study the class attention maps averaged over 30 test samples for the autoimmune dataset in Figure 7. We observed
+that the CASS-trained Transformer has much more attention in the center than the supervised Transformer. This extra
+attention could be attributed to a Transformer on its own inability to map out due to the information transfer from CNN.
+Another feature to observe is that the attention map of the CASS-trained Transformer is much more connected than that of a
+supervised Transformer.
+
+0
+100
+200
+300
+0
+100
+200
+300
+400Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Figure 5. This figure shows the feature map extracted after the first Conv2d layer of ResNet-50 for CASS (on the left) and supervised
+CNN (on the right). The color bar shows the intensity of pixels retained. From the four circles, it is clear that CASS-trained CNN can
+retain more intricate details about the input image (Figure 4) more intensely so that they can be propagated through the architecture and
+help the model learn better representations as compared to the supervised CNN. We study the same feature map in detail for the first five
+layers after Conv2d in Figure 6.
+C.4.2. DERMOFIT DATASET
+We present the average attention maps for the dermofit dataset in Figure 8. We observed that the CASS-trained Transformer
+can pay much more attention to the center part of the image. Furthermore, the attention map of the CASS-trained Transformer
+is much more connected than the supervised Transformer. So, overall with CASS, the Transformer is not only able to map
+long-range dependencies which are innate to Transformers but is also able to make more local connections with the help of
+features sensitive to translation equivariance and locality from CNN.
+C.4.3. BRAIN TUMOR MRI CLASSIFICATION DATASET
+We present the average class attention maps results in Figure 9. We observed that a CASS-trained Transformer could better
+capture long and short-range dependencies than a supervised Transformer. Furthermore, we observed that a CASS-trained
+Transformer’s attention map is much more centered than a supervised Transformer’s. From Figure 13, we can observe that
+most MRI images are center localized, so having a more centered attention map is advantageous in this case.
+C.4.4. ISIC 2019 DATASET
+The ISIC-2019 dataset is one of the most challenging datasets out of the four datasets. ISIC 2019 consists of images from the
+HAM10000 and BCN 20000 datasets (Cassidy et al., 2022; Gessert et al., 2020). For the HAM1000 dataset, it isn’t easy to
+classify between 4 classes (melanoma and melanocytic nevus), (actinic keratosis, and benign keratosis). HAM10000 dataset
+contains images of size 600×450, centered and cropped around the lesion. Histogram corrections have been applied to only
+a few images. The BCN 20000 dataset contains images of size 1024×1024. This dataset is particularly challenging as many
+images are uncropped, and lesions are in difficult and uncommon locations. Hence, in this case, having more spread-out
+attention maps would be advantageous instead of a more centered one. From Figure 10, we observed that a CASS-trained
+Transformer has a lot more spread attention map than a supervised Transformer. Furthermore, a CASS-trained Transformer
+can also attend the corners far better than a supervised Transformer.
+From Figures 7, 8, 9 and 10, we observed that in most cases, the supervised Transformer had spread out attention, while the
+CASS trained Transformer has a more ”connected.” attention map. This is primarily because of local-level information
+transfer from CNN. Hence we could add some more image-level intuition, with the help of CNN, to the Transformer that it
+
+Conv2d
+0.15
+Conv2d
+0.15
+0.10
+0.10
+0.05
+0.05
+0.00
+-0.00Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Figure 6. At the top, we have features extracted from the top 5 layers of supervised ResNet-50, while at the bottom, we have features
+extracted from the top 5 layers of CASS-trained ResNet-50. We supplied both networks with the same input ( shown in Figure 4).
+Figure 7. To ensure the consistency of our study, we studied average attention maps over 30 sample images from the autoimmune dataset.
+The left image is the overall attention map averaged over 30 samples for the supervised Transformer, while the one on the right is for
+CASS pretrained Transformer (both after finetuning with 100% labels).
+would have rather missed on its own.
+C.4.5. CHOICE OF DATASETS
+We chose four medical imaging datasets with diverse sample sizes ranging from 198 to 25,336 and diverse modalities to
+study the performance of existing self-supervised techniques and CASS. Most of the existing self-supervised techniques have
+been studied on million image datasets, but medical imaging datasets, on average, are much smaller than a million images.
+Furthermore, they are usually imbalanced and some of the existing self-supervised techniques rely on batch statistics, which
+makes them learn skewed representations. We also include a dataset of emerging and underrepresented diseases with only a
+few hundred samples, the autoimmune dataset in our case (198 samples). To the best of our knowledge, no existing literature
+studies the effect of self-supervised learning on such a small dataset. Furthermore, we chose the dermofit dataset because
+all the images are taken using an SLR camera, and no two images are the same size. Image size in dermofit varies from
+205×205 to 1020×1020. MRI images constitute a large part of medical imaging; hence we included this dataset in our study.
+So, to incorporate them in our study, we had the Brain tumor MRI classification dataset. Furthermore, it is our study’s only
+black-and-white dataset; the other three datasets are RGB. The ISIC 2019 is a unique dataset as it contains multiple pairs
+
+Conv2d
+Conv2d
+Conv2d
+Conv2d
+Conv2d
+20
+-20
+_40
+60
+80
+-100Conv2d
+Conv2d
+Conv2d
+Conv2d
+Conv2d
+21
+V.
+-40
+-60
+80
+-1001.0
+0
+0.8
+100
+0.6
+200
+0.4
+300
+0.2
+0
+100
+200
+300
+0.01.0
+0
+0.8
+100
+0.6
+200
+0.4
+300
+0.2
+0
+100
+200
+300
+0.0Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+of hard-to-classify classes (Melanoma - melanocytic nevus and actinic keratosis - benign keratosis) and different image
+sizes - out of which only a few have been prepossessed. It is a highly imbalanced dataset containing samples with lesions in
+difficult and uncommon locations. To give an idea about the images used in our experiments, we provide sample images
+from the four datasets used in our experimentation in Figures 11, 12, 13 and 14.
+C.5. Self-supervised pretraining
+C.5.1. PROTOCOLS
+• Self-supervised learning was only done on the training data and not on the validation data. We used https:
+//github.com/PyTorchLightning/pytorch-lightning to set the pseudo-random number generators
+in PyTorch, NumPy, and (python.random).
+• We ran training over five seed values and reported mean results with variance in each table. We didn’t perform a seed
+value sweep to extract any more performance (Picard, 2021).
+• For DINO implementation, we use Phil Wang’s implementation:
+https://github.com/lucidrains/
+vit-pytorch.
+• For the implementation of CNNs and Transformers, we use Timm’s library (Wightman, 2019).
+• For all experiments, ImageNet (Deng et al., 2009) initialised CNN and Transformers were used.
+• After pertaining, an end-to-end finetuning of the pre-trained model was done using x% labeled data. Where x was
+either 1 or 10, or 100. When fine-tuned with x% labeled data, the pre-trained model was then fine-tuned only on x%
+data points with corresponding labels.
+C.5.2. AUGMENTATIONS
+• Resizing: Resize input images to 384×384 with bilinear interpolation.
+• Color jittering: change the brightness, contrast, saturation, and hue of an image or apply random perspective with
+a given probability. We set the degree of distortion to 0.2 (between 0 and 1) and use bilinear interpolation with an
+application probability of 0.3.
+• Color jittering or applying the random affine transformation of the image, keeping center invariant with degree 10, with
+an application probability of 0.3.
+Figure 8. Class attention maps averaged over 30 samples of the dermofit dataset for supervised Transformer (on the left), and CASS
+pretrained Transformer (on the right). Both after finetuning with 100% labels.
+
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+Figure 9. Class attention maps averaged over 30 samples of the brain tumor MRI classification dataset for supervised Transformer (on the
+left), and CASS pretrained Transformer (on the right). Both after finetuning with 100% labels.
+Figure 10. Class attention maps averaged over 100 samples from the ISIC-2019 dataset for the supervised Transformer (on the left) and
+CASS-trained Transformer (on the right). Both after finetuning with 100% labels.
+• Horizontal and Vertical flip. Each with an application probability of 0.3.
+• Channel normalization with a mean (0.485, 0.456, 0.406) and standard deviation (0.229, 0.224, 0.225).
+C.5.3. HYPER-PARAMETERS
+• Optimization: We use stochastic weighted averaging over Adam optimizer with learning rate (LR) set to 1e-3 for both
+CNN and vision transformer (ViT). This is a shift from SGD, which is usually used for CNNs.
+• Learning Rate: Cosine annealing learning rate is used with 16 iterations and a minimum learning rate of 1e-6. Unless
+mentioned otherwise, this setup was trained over 100 epochs. These were then used as initialization for the downstream
+supervised learning. The standard batch size is 16.
+C.6. Supervised training
+C.6.1. AUGMENTATIONS
+We use the same set of augmentations used in self-supervised pretraining.
+
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+Figure 11. Sample of autofluorescence slide images from the muscle biopsy of patients with dermatomyositis - a type of autoimmune
+disease.
+Figure 12. Sample images from the Dermofit dataset.
+Figure 13. Sample images of brain tumor MRI dataset, Each image corresponds to a prediction class in the data set glioma (Left),
+meningioma (Center), and No tumor (Right)
+C.6.2. HYPER-PARAMETERS
+• We use Adam optimizer with lr set to 3e-4 and a cosine annealing learning schedule.
+
+Cross-Architectural Positive Pairs improve the effectiveness of Self-Supervised Learning
+Figure 14. Sample images from the ISIC-2019 challenge dataset.
+• Since all medical datasets have class imbalance, we address it by using focal loss (Lin et al., 2017) as our choice of
+the loss function with the alpha value set to 1 and the gamma value to 2. In our case, it uses minimum-maximum
+normalized class distribution as class weights for focal loss.
+• We train for 50 epochs. We also use a five epoch patience on validation loss to check for early stopping. This
+downstream supervised learning setup is kept the same for CNN and Transformers.
+We repeat all the experiments with different seed values five times and then present the average results in all the tables.
+D. Miscellaneous
+D.1. Description of Metrics
+After performing downstream fine-tuning on the four datasets under consideration, we analyze the CASS, DINO, and
+Supervised approaches on specific metrics for each dataset. The choice of this metric is either from previous work or as
+defined by the dataset provider. For the Autoimmune dataset, Dermofit, and Brain MRI classification datasets based in
+prior works, we use the F1 score as our metric for comparing performance, which is defined as F1 = 2∗P recision∗Recall
+P recision+Recall =
+2∗T P
+2∗T P +F P +F N
+For the ISIC-2019 dataset, as mentioned by the competition organizers, we used the recall score as our comparison metric,
+which is defined as Recall =
+T P
+T P +F N
+For the above two equations, TP: True Positive, TN: True Negative, FP: False Positive, and FN: False Negative.
+D.2. Limitations
+Although CASS’ performance for larger and non-biological data can be hypothesized based on inferences, a complete
+study on large-sized natural datasets hasn’t been conducted. In this study, we focused extensively on studying the effects
+and performance of our proposed method for small dataset sizes and in the context of limited computational resources.
+Furthermore, all the datasets used in our experimentation are restricted to academic and research use only. Although CASS
+performs better than existing self-supervised and supervised techniques, it is impossible to determine at inference time
+(without ground-truth labels) whether to pick the CNN or the Transformers arm of CASS.
+D.3. Potential negative societal impact
+The autoimmune dataset is limited to a geographic institution. Hence the study is specific to a disease variant. Inferences
+drawn may or may not hold for other variants. Also, the results produced are dependent on a set of markers. Medical
+practitioners often require multiple tests before finalizing a diagnosis; medical history and existing health conditions also
+play an essential role. We haven’t incorporated the meta-data above in CASS. Finally, application on a broader scale -
+real-life scenarios should only be trusted after clearance from the concerned health and safety governing bodies.
+